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view texts/archimedesOldCVSRepository/archimedes/xml/galil_disco_069_en_1665.xml @ 27:8dce37600d38
New Special Instructions
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Wed, 30 Jul 2014 15:58:21 +0200 |
| parents | 22d6a63640c6 |
| children |
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<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" ><archimedes> <info> <author>Galilei, Galileo</author> <title>Discorsi</title> <date>1665</date> <place>London</place> <translator>Salusbury, Thomas</translator> <lang>en</lang> <cvs_file>galil_disco_069_en_1665.xml</cvs_file> <cvs_version></cvs_version> <locator>069.xml</locator> </info> <text> <front> <section> <pb xlink:href="069/01/001.jpg"/><p type="head"> <s>MATHEMATICAL <lb/>DISCOURSES <lb/>AND <lb/>DEMONSTRATIONS, <lb/>TOVCHING <lb/>Two <emph type="italics"/>NEW SCIENCES<emph.end type="italics"/>; pertaining to <lb/>THE <lb/>MECHANICKS <lb/>AND <lb/>LOCAL MOTION:</s></p><p type="head"> <s>BY <lb/><emph type="italics"/>GALILÆVS GALILÆVS LYNCEVS,<emph.end type="italics"/><lb/>Chiefe <emph type="italics"/>Phylo&longs;opher<emph.end type="italics"/> and <emph type="italics"/>Mathematitian<emph.end type="italics"/> to the mo&longs;t <lb/>Serene <emph type="italics"/>GRAND DVKE<emph.end type="italics"/> of <emph type="italics"/>TVSCANY.<emph.end type="italics"/><lb/>WITH <lb/><emph type="italics"/>AN APPENDIX OF THE<emph.end type="italics"/><lb/>Centre of Gravity <lb/>Of &longs;ome <emph type="italics"/>SOLIDS.<emph.end type="italics"/></s></p><p type="head"> <s>Engli&longs;hed from the Originall <emph type="italics"/>Latine<emph.end type="italics"/> and <emph type="italics"/>Italian, <lb/>By THOMAS SALUSBURY, E&longs;q<emph.end type="italics"/>;</s></p><p type="head"> <s><emph type="italics"/>LONDON,<emph.end type="italics"/><lb/>Printed by WILLIAM LEYBOURN, <emph type="italics"/>Anno Dom. <lb/> MDCLXV.<emph.end type="italics"/></s></p> <pb xlink:href="069/01/002.jpg" pagenum="1"/><p type="head"> <s>GALILEUS, <lb/>HIS <lb/>DIALOGUES <lb/>OF <lb/>MOTION.</s></p> </section> </front> <body> <chap> <pb xlink:href="069/01/003.jpg"/><p type="head"> <s>The Fir&longs;t Dialogue.</s></p><p type="head"> <s><emph type="italics"/>INTERLOCUTORS,<emph.end type="italics"/></s></p><p type="head"> <s>SALVIATUS, SAGREDUS, and SIMPLICIUS.</s></p><p type="main"> <s>SALVIATUS.</s></p><p type="main"> <s>The frequent re&longs;ort (Gentlemen) to <lb/><arrow.to.target n="marg984"></arrow.to.target><lb/>your Famous Ar&longs;enal of <emph type="italics"/>Venice,<emph.end type="italics"/> pre&longs;en­<lb/>teth, in my thinking, to your Speculative <lb/><arrow.to.target n="marg985"></arrow.to.target><lb/>Wits, a large field to Philo&longs;ophate in: <lb/>and more particularly, as to that part <lb/>which is called the <emph type="italics"/>Mechanicks:<emph.end type="italics"/> in re­<lb/>gard that there all kinds of Engines, and <lb/>Machines are continually put in u&longs;e, by a <lb/>huge number of Artificers of all &longs;orts; <lb/>among&longs;t whom, as well through the ob&longs;ervations of their Prede­<lb/>ce&longs;&longs;ors, as tho&longs;e, which through their own care they continually <lb/>are making, it's probable, that there are &longs;ome very learned, and <lb/>bravely di&longs;cours'd Men.</s></p><p type="margin"> <s><margin.target id="marg984"></margin.target><emph type="italics"/>A De&longs;cription of <lb/>the Ar&longs;enal of<emph.end type="italics"/><lb/>Venice.</s></p><p type="margin"> <s><margin.target id="marg985"></margin.target><emph type="italics"/>It is a large field <lb/>for Wits to Philo­<lb/>&longs;ophate in.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. Sir, you are not therein mi&longs;taken: and I my &longs;elf, out of <pb xlink:href="069/01/004.jpg" pagenum="2"/>a natural Curio&longs;itie, do frequentlie for my Recreation vi&longs;it that <lb/>place, and confer with the&longs;e per&longs;ons; which for a certain prehe­<lb/><arrow.to.target n="marg986"></arrow.to.target><lb/>minence that they have above the re&longs;t we call ^{*} <emph type="italics"/>Over&longs;eers<emph.end type="italics"/>: who&longs;e <lb/>di&longs;cour&longs;e hath oft helped me in the inve&longs;tigation of not only won­<lb/>derful, but ab&longs;truce, and incredible Effects: and indeed I have been <lb/>at a lo&longs;&longs;e &longs;ometimes, and de&longs;paired to penetrate how that could <lb/>po&longs;&longs;ibly come to pa&longs;&longs;e, which far from all expectation my &longs;en&longs;es <lb/>demon&longs;trated to be true; and yet that which not long &longs;ince that <lb/>good Old man told us, is a &longs;aying and propo&longs;ition, though com­<lb/><arrow.to.target n="marg987"></arrow.to.target><lb/>mon enough, yet in my opinion wholly vain, as are many others, <lb/>often in the mouths of unskilful per&longs;ons; introduced by them, as <lb/>I &longs;uppo&longs;e, to &longs;hew that they under&longs;tand how to &longs;peak &longs;omething <lb/>about that, of which neverthele&longs;&longs;e they are incapable.</s></p><p type="margin"> <s><margin.target id="marg986"></margin.target>* Proti.</s></p><p type="margin"> <s><margin.target id="marg987"></margin.target><emph type="italics"/>The Opinion of <lb/>Common Artificers <lb/>are often fal&longs;e.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>It may be Sir, you &longs;peak of that la&longs;t propo&longs;ition which <lb/>he affirmed, when we de&longs;ired to under&longs;tand, why they made <lb/><arrow.to.target n="marg988"></arrow.to.target><lb/>&longs;o much greater provi&longs;ion of &longs;upporters, and other provi&longs;ions, <lb/>and reinforcements about that Galea&longs;&longs;e, which was to be launcht <lb/>than is made about le&longs;&longs;er Ve&longs;&longs;els, and he an&longs;wered us, that they did <lb/>&longs;o to avoid the peril of breaking its Keel, through the mighty <lb/>weight of its va&longs;t bulk, an inconvenience to which le&longs;&longs;er &longs;hips are <lb/>not subject.</s></p><p type="margin"> <s><margin.target id="marg988"></margin.target><emph type="italics"/>Great Ships apter <lb/>than others to break <lb/>their Keels in <lb/>Launching, accor­<lb/>ding to &longs;ome.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>I do intend the &longs;ame, and chiefly that la&longs;t conclu&longs;ion, <lb/>which he added to his others, and which I alwaies e&longs;teemed a vain <lb/>conceit of the Vulgar, namely, That in the&longs;e and other Machines <lb/>we mu&longs;t not argue from the le&longs;&longs;e to the greater, becau&longs;e many <lb/>Mechanical Inventions take in little, which hold not in great. </s> <s>But <lb/>being that all the Rea&longs;ons of the Mechanicks, have their founda­<lb/>tions from Geometry; in which I &longs;ee not that greatne&longs;&longs;e and <lb/>&longs;malne&longs;&longs;e make Circles, Triangles, Cilinders, Cones, or any other <lb/>&longs;olid Figures &longs;ubject to different pa&longs;&longs;ions: when the great Ma­<lb/>chine is conformed in all its members to the proportions of the <lb/>le&longs;&longs;e that is u&longs;eful, and fit for exerci&longs;e to which it is de&longs;igned; I <lb/>cannot &longs;ee why it al&longs;o &longs;hould not be exempt from the unlucky, <lb/>&longs;ini&longs;ter, and de&longs;tructive accidents that may befall it.</s></p><p type="main"> <s>SALV The &longs;aying of the Vulgar is ab&longs;olutely vain, and &longs;o <lb/>fal&longs;e, that its contrary may be affirmed with equal truth, &longs;aying, <lb/><arrow.to.target n="marg989"></arrow.to.target><lb/>That many Machines may be made more perfect in great than lit­<lb/>tle: As for in&longs;tance, a Clock that &longs;hews and &longs;trikes the Houres, <lb/>may be made more exact in one certain &longs;ize, than in another le&longs;&longs;e. <lb/></s> <s>With better ground is that &longs;ame conclu&longs;ion u&longs;urped by other more <lb/>intelligent per&longs;ons, who refer the cau&longs;e of &longs;uch effects in the&longs;e <lb/>great Machines different from what is collected from the pure, and <lb/>ab&longs;tracted Demon&longs;trations of Geometry, to the imperfection of <lb/>the matter, which is &longs;ubject to many alterations, and defects. <lb/></s> <s>But here, I know not whether I may without contracting &longs;ome <pb xlink:href="069/01/005.jpg" pagenum="3"/>&longs;u&longs;pition of Arrogance &longs;ay, that thither al&longs;o doth the recour&longs;e to <lb/>the defects of the matter (able to blemi&longs;h the perfecte&longs;t Mathe­<lb/>matical Demon&longs;trations) &longs;uffice to excu&longs;e the di&longs;obedience of <lb/><arrow.to.target n="marg990"></arrow.to.target><lb/>Machines in concrete, to the &longs;ame ab&longs;tracted and Ideal: yet not­<lb/>with&longs;tanding I will &longs;peak it, affirming, That ab&longs;tracting all imper­<lb/>fections from the Matter, and &longs;uppo&longs;ing it mo&longs;t perfect, and unal­<lb/>terable, and from all accidental mutation exempt, yet neverthe­<lb/>le&longs;&longs;e its only being Material, cau&longs;eth, that the greater Machine, <lb/>made of the &longs;ame matter, and with the &longs;ame proportions, as the <lb/>le&longs;&longs;er; &longs;hall an&longs;wer in all other conditions to the le&longs;&longs;er in exact <lb/>Symetry, except in &longs;trength, and re&longs;i&longs;tance again&longs;t violent inva&longs;i­<lb/>ons: but the greater it is, &longs;o much in proportion &longs;hall it be wea­<lb/>ker. </s> <s>And becau&longs;e I &longs;uppo&longs;e the Matter to be unalterable, that is <lb/>alwaies the &longs;ame, it is manife&longs;t, that one may produce Demon&longs;tra­<lb/>tions of it, no le&longs;&longs;e &longs;imply and purely Mathematical, then of eter­<lb/>nal, and nece&longs;&longs;ary Affections: Therefore, <emph type="italics"/>Sagredus,<emph.end type="italics"/> Revoke the <lb/>opinion which you, and, it may be, all the re&longs;t hold, that have &longs;tu­<lb/>died the Mechanicks; that Machines, and Frames compo&longs;ed of the <lb/>&longs;ame Matter, with punctual ob&longs;ervation of the &longs;elf &longs;ame proporti­<lb/>on between their parts, ought to be equally, or to &longs;ay better, pro­<lb/>portionally di&longs;po&longs;ed to Re&longs;i&longs;t; and to yield to External injuries <lb/>and a&longs;&longs;aults: For if it may be Geometrically demon&longs;trated, that <lb/>the greater are alwaies in proportion le&longs;s able to re&longs;i&longs;t, than the <lb/>le&longs;&longs;e; &longs;o that in fine there is not only in all Machines & Fabricks <lb/>Arti&longs;icial, but Natural al&longs;o, a term nece&longs;&longs;arily a&longs;cribed, beyond <lb/>which neither Art, nor Nature may pa&longs;&longs;e; may pa&longs;&longs;e, I &longs;ay, al­<lb/>waies ob&longs;erving the &longs;ame proportions with the Identity of the <lb/>Matter.</s></p><p type="margin"> <s><margin.target id="marg989"></margin.target><emph type="italics"/>Many Machines <lb/>may be made more <lb/>exact in great than <lb/>in little.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg990"></margin.target><emph type="italics"/>Great Material <lb/>Machines, al­<lb/>though framed In <lb/>the &longs;ame proportion <lb/>as others of the <lb/>&longs;ame Matter that <lb/>are le&longs;&longs;er, are le&longs;&longs;e <lb/>&longs;trong and able to <lb/>re&longs;i&longs;t external Im­<lb/>petu&longs;s's than the <lb/>le&longs;&longs;er.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>I already feel my Brains to turn round, and my Mind, <lb/>(like a Cloud unwillingly opened by the Lightning,) I perceive <lb/>to be &longs;urprized with a tran&longs;cient, and unu&longs;ual Light, which from <lb/>affar off twinkleth, and &longs;uddenly a&longs;toni&longs;heth me; and with ab­<lb/>&longs;truce, &longs;trange, and indige&longs;ted imaginations. </s> <s>And from what hath <lb/>been &longs;poken, it &longs;eems to follow, that, it is a thing impo&longs;&longs;ible to <lb/>frame two Fabricks of the &longs;ame Matter, alike, and unequal, and <lb/>between them&longs;elves in proportion equally able to Re&longs;i&longs;t; and <lb/>were it to be done, yet it would be impo&longs;&longs;ible to find two only <lb/>Launces of the &longs;ame wood, alike between them&longs;elves in &longs;trength, <lb/><arrow.to.target n="marg991"></arrow.to.target><lb/>and toughne&longs;&longs;e, but unequal in bigne&longs;&longs;e.</s></p><p type="margin"> <s><margin.target id="marg991"></margin.target><emph type="italics"/>A Wooden Launce <lb/>fixed in a Wall at <lb/>Right-Angles, and <lb/>reduced to &longs;uch a <lb/>length and thick­<lb/>ne&longs;&longs;e as that it may <lb/>endure, but made a <lb/>hairs breadth big­<lb/>ger, breaketh with <lb/>its own weight, is <lb/>&longs;ingly one and no <lb/>more.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>So it is Sir; and the better to a&longs;&longs;ure you that we con­<lb/>cur in opinion, I &longs;ay, that if we take a Launce of wood of &longs;uch a <lb/>length and thickne&longs;&longs;e, that being fixed fa&longs;t <emph type="italics"/>(v. </s> <s>g.)<emph.end type="italics"/> in a Wall at <lb/>Right Angles, that is parallel to the Horizon, it is reduced to the <lb/>utmo&longs;t length, that it will hold at, &longs;o that lengthened never­<lb/>&longs;o-little more, it would break, being over-burthened with its own <pb xlink:href="069/01/006.jpg" pagenum="4"/>weight, there could not be another &longs;uch-a-one in the World: So <lb/>that if its length (for example) were Centuple to its thickne&longs;&longs;e, <lb/>there cannot be found another Launce of the &longs;ame Matter, that <lb/>being in length Centuple to its thickne&longs;&longs;e, &longs;hall be able to &longs;u&longs;tain <lb/>it &longs;elf preci&longs;ely, as that did, and no more: for all that are bigger <lb/>&longs;hall break, and the le&longs;&longs;er &longs;hall be able, be&longs;ides their own, to &longs;u&longs;tain <lb/>&longs;ome additional weight. </s> <s>And this that I &longs;ay of the <emph type="italics"/>State of bear­<lb/>ing it &longs;elf,<emph.end type="italics"/> I would have under&longs;tood to be &longs;poken of every other <lb/>Con&longs;titution, and thus if one Tran&longs;ome bear or &longs;u&longs;tain the force <lb/>often Tran&longs;omes equal to it, &longs;uch another Beam cannot bear the <lb/>weight of ten that are equal to it. </s> <s>Now be plea&longs;ed, Sir, and you <lb/><arrow.to.target n="marg992"></arrow.to.target><lb/><emph type="italics"/>Simplicius<emph.end type="italics"/> to ob&longs;erve, how true Conclu&longs;ions, though at the fir&longs;t <lb/>&longs;ight they &longs;eem improbable, yet never &longs;o little glanced at, do depo&longs;e <lb/>the Vailes which ob&longs;cure them, and make a voluntary &longs;hew of their <lb/>&longs;ecrets nakedly, and &longs;imply. </s> <s>Who &longs;ees not, that a Hor&longs;e falling <lb/><arrow.to.target n="marg993"></arrow.to.target><lb/>from a height of three or four yards, will break his bones, but a <lb/>Dog falling &longs;o many yards, or a Cat eight or ten, will receive no <lb/>hurt; nor likewi&longs;e a Gra&longs;hopper from a Tower, nor an Ant thrown <lb/>from the Orbe of the Moon? </s> <s>Little Children e&longs;cape all harm in <lb/>their falls, whereas per&longs;ons grown up break either their &longs;hins or <lb/>faces. </s> <s>And as le&longs;&longs;er Animals are in proportion more robu&longs;tious, <lb/>and &longs;trong than greater, &longs;o the le&longs;&longs;er Plants better &longs;upport them­<lb/>&longs;elves: and I already believe, that both of you think, that an Oake <lb/>two hundred foot high could not &longs;upport its branches &longs;pread like <lb/><arrow.to.target n="marg994"></arrow.to.target><lb/>one of an indifferent &longs;ize; and that Nature could not have made <lb/>an Hor&longs;e as big as twenty Hor&longs;es, nor a Giant ten times as tall as a <lb/>Man, unle&longs;&longs;e &longs;he did it either miraculou&longs;ly, or el&longs;e by much alte­<lb/>ring the proportion of the Members, and particularly of the Bones, <lb/>enlarging them very much above the Symetry of common Bones. <lb/></s> <s>To &longs;uppo&longs;e likewi&longs;e, that in Artificial Machines, the greater and <lb/>le&longs;&longs;er are with equal facility made, and pre&longs;erved, is a manife&longs;t Er­<lb/>rour: and thus for in&longs;tance, &longs;mall Spires, Pillars, and other &longs;olid <lb/>figures may be &longs;afely moved, laid along, and reared upright, with­<lb/>out danger of breaking them; but the very great upon every &longs;ini­<lb/>&longs;ter accident fall in pieces, and for no other rea&longs;on but their own <lb/>weight. </s> <s>And here it is nece&longs;&longs;ary that I relate an accident, worthy <lb/>of notice, as are all tho&longs;e events that occur unexpectedly, e&longs;pecial­<lb/>ly when the means u&longs;ed to prevent an inconvenience, proveth in <lb/><arrow.to.target n="marg995"></arrow.to.target><lb/>fine the mo&longs;t potent cau&longs;e of the di&longs;order. </s> <s>There was a very great <lb/>Pillar of Marble laid along, and two Rowlers were put under the <lb/>&longs;ame neer to the ends of it; it came into the mind of a certain In­<lb/>gineer &longs;ome time after, that it would be expedient, the better to <lb/>&longs;ecure it from breaking in the mid&longs;t through its own weight, to <lb/>put under it in that part yet another Rowler: the coun&longs;el &longs;eemed <lb/>generally very &longs;ea&longs;onable, but the &longs;ucce&longs;&longs;e demon&longs;trated it to be <pb xlink:href="069/01/007.jpg" pagenum="5"/>wholly contrary: for many moneths had not pa&longs;t, before the Pil­<lb/>lar crackt, and broke in the middle ju&longs;t upon the new Rowler.</s></p><p type="margin"> <s><margin.target id="marg992"></margin.target><emph type="italics"/>Truth upon a little <lb/>Courting, throweth <lb/>off her Vail, and <lb/>&longs;hews her Secrets <lb/>maked.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg993"></margin.target><emph type="italics"/>Great Animals <lb/>receive more harm <lb/>by a fall than le&longs;­<lb/>&longs;er.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg994"></margin.target><emph type="italics"/>Nature could not <lb/>have made of mea­<lb/>ner Hor&longs;es bigger, <lb/>and have retained <lb/>the &longs;ame &longs;trength, <lb/>but by altering <lb/>their Symetry.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg995"></margin.target><emph type="italics"/>A great Marble <lb/>Pillar broken by <lb/>its own weight, <lb/>and why.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. </s> <s>This was an accident truly &longs;trange, and indeed <emph type="italics"/>preter <lb/>&longs;pem,<emph.end type="italics"/> e&longs;pecially if it were derived from the addition of new &longs;up­<lb/>port in the middle.</s></p><p type="main"> <s>SALV. </s> <s>From that doubtle&longs;s it did proceed; and the known cau&longs;e <lb/>of the Effect removeth the wonder: for the two pieces of the Pillar <lb/>being taken from off the Rowlers, one of tho&longs;e bearers on which <lb/>one end of the Column had re&longs;ted, was by length of time rotten, and <lb/>&longs;unk away; and that in the mid&longs;t continuing &longs;ound, and &longs;trong, <lb/>occa&longs;ioned that half the Column lay hollow in the air without any <lb/>&longs;upport at the end; &longs;o that its own unweildy weight, made it do <lb/>that, which it would not have done, if it had re&longs;ted only upon the <lb/>two fir&longs;t Bearers, for as they had &longs;hrunk away it would have fol­<lb/>lowed. </s> <s>And here none can think that this would have faln out in <lb/>a little Column, though of the &longs;ame &longs;tone, and of a length an&longs;we­<lb/>rable to its thickne&longs;&longs;e, in the very &longs;ame proportion as the thick­<lb/>ne&longs;s, and length of the great Pillar.</s></p><p type="main"> <s>SAGR. </s> <s>I am now a&longs;&longs;ured of the effect, but do not yet compre­<lb/>hend the cau&longs;e, how in the augmentation of Matter, the Re&longs;i&longs;tance <lb/>and Strength ought not al&longs;o to multiply at the &longs;ame rate. </s> <s>And I <lb/>admire at it &longs;o much the more, in regard I &longs;ee, on the contrary, in <lb/>other ca&longs;es the &longs;trength of Re&longs;i&longs;tance again&longs;t Fraction to encrea&longs;e <lb/>much more than the enlargement of the matter encrea&longs;eth. </s> <s>For if <lb/>(for example) there be two Nailes fa&longs;tned in a Wall, the one twice <lb/>asthick as the other, that &longs;hall bear a weight not only double to this, <lb/>but triple, and quadruple.</s></p><p type="main"> <s>SALV. </s> <s>You may &longs;ay octuple, and not be wide of the truth: <lb/><arrow.to.target n="marg996"></arrow.to.target><lb/>nor is this effect contrary to the former, though in appearance it <lb/>&longs;eemeth &longs;o different.</s></p><p type="margin"> <s><margin.target id="marg996"></margin.target><emph type="italics"/>A Naile double <lb/>in thickne&longs;&longs;e to <lb/>another being fa&longs;t­<lb/>ned in a Wall, &longs;u­<lb/>&longs;tains a Weight <lb/>octuple to that of <lb/>the le&longs;&longs;er.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>Therefore <emph type="italics"/>Salviatus,<emph.end type="italics"/> explain unto us the&longs;e Riddles, and <lb/>level us the&longs;e Rocks, if you can do it: for indeed I gue&longs;&longs;e this mat­<lb/>ter of Re&longs;i&longs;tance to be a field repleni&longs;hed with rare, and u&longs;eful Con­<lb/>templations, and if you be content that this be the &longs;ubject of our <lb/>this-daies di&longs;cour&longs;e, it will be to me, and I believe to <emph type="italics"/>Simplicius,<emph.end type="italics"/><lb/>very acceptable.</s></p><p type="main"> <s>SALV. </s> <s>I cannot refu&longs;e to &longs;erve you, &longs;ince my Memory &longs;erveth <lb/><arrow.to.target n="marg997"></arrow.to.target><lb/>me, in minding me of that which I formerly learnt of our <emph type="italics"/>Accade­<lb/>mick,<emph.end type="italics"/> who hath made many Speculations on this &longs;ubject, and all <lb/>conformable (as his manner is) to Geometrical Demon&longs;tration: <lb/>in&longs;omuch that, not without rea&longs;on, this of his may be called a <emph type="italics"/>New <lb/>Science<emph.end type="italics"/>; for though &longs;ome of the Conclu&longs;ions have been ob&longs;erved <lb/><arrow.to.target n="marg998"></arrow.to.target><lb/>by others, and in the fir&longs;t place by <emph type="italics"/>Ari&longs;totle,<emph.end type="italics"/> yet neverthele&longs;&longs;e are <lb/>they not any of the mo&longs;t curious, or (which more importeth) <lb/>proved by nece&longs;&longs;ary Demon&longs;trations deduced from their primary, <pb xlink:href="069/01/008.jpg" pagenum="6"/>and indubitable fundamentals. </s> <s>And becau&longs;e, as I &longs;ay, I de&longs;ire de­<lb/>mon&longs;tratively to a&longs;&longs;ure you, and not with only probable di&longs;cour­<lb/>&longs;es to per&longs;wade you; pre&longs;uppo&longs;ing, that you have &longs;o much know­<lb/>ledge of the Mechanical Conclu&longs;ions, by others heretofore funda­<lb/>mentally handled, as &longs;ufficeth for our purpo&longs;e; it is requi&longs;ite, that <lb/>before we proceed any further, we con&longs;ider what effect that is which <lb/>opperates in the Fraction of a Beam of Wood, or other Solid, who&longs;e <lb/>parts are firmly connected; becau&longs;e this is the fir&longs;t <emph type="italics"/>Notion,<emph.end type="italics"/> where­<lb/>on the fir&longs;t and &longs;imple principle dependeth, which as familiarly <lb/>known, we may take for granted. </s> <s>For the clearer explanation <lb/>whereof; let us take the Cilinder, or Pri&longs;me, <emph type="italics"/>A. B.<emph.end type="italics"/> of Wood, or <lb/>other &longs;olid and coherent matter, fa&longs;tned above in <emph type="italics"/>A,<emph.end type="italics"/> and hanging <lb/>perpendicular; to which, at the other end <emph type="italics"/>B,<emph.end type="italics"/> let there hang the <lb/>Weight <emph type="italics"/>C<emph.end type="italics"/>: It is manife&longs;t, that how great &longs;oever the Tenacity and <lb/>coherence of the parts of the &longs;aid Solid to one another be, &longs;o it be <lb/>not infinite, it may be overcome by the <lb/>Force of the drawing Weight C: who&longs;e <lb/>Gravity I &longs;uppo&longs;e may be encrea&longs;ed as much <lb/><figure id="id.069.01.008.1.jpg" xlink:href="069/01/008/1.jpg"/><lb/>as we plea&longs;e; by the encrea&longs;e whereof the <lb/>&longs;aid Solid in fine &longs;hall break, like as if it had <lb/>been a Cord. </s> <s>And, as in a Cord, we under­<lb/>&longs;tand its re&longs;i&longs;tance to proceed from the mul­<lb/>titude of the &longs;trings or threads in the Hemp <lb/>that compo&longs;e it, &longs;o in Wood we &longs;ee its veins, <lb/>and grain di&longs;tended lengthwaies, that render <lb/>it far more re&longs;i&longs;ting again&longs;t Fraction, then any <lb/>Rope would be, of the &longs;ame thickne&longs;&longs;e: but <lb/>in a Cylinder of &longs;tone or Metal the Tenacity <lb/>of its parts, (which yet &longs;eemeth greater) de­<lb/>pendeth on another kind of Cement, <lb/>than of &longs;trings, or grains, and yet they al&longs;o <lb/>being drawn with equivalent force, break.</s></p><p type="margin"> <s><margin.target id="marg997"></margin.target><emph type="italics"/>By Accademick <lb/>here, as in his <lb/>Dialogues of the <lb/>Sy&longs;teme,<emph.end type="italics"/> Galile­<lb/>us <emph type="italics"/>meaneth him­<lb/>&longs;elf.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg998"></margin.target>Ari&longs;totle <emph type="italics"/>the fir&longs;t <lb/>Ob&longs;erver of Me­<lb/>chanical Conclu&longs;i­<lb/>ons, but they nei­<lb/>ther not the mo&longs;t <lb/>curious nor &longs;olidly <lb/>demon&longs;trated.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. </s> <s>If the thing &longs;ucceed as you &longs;ay, I under&longs;tand well <lb/>enough, that the thread or grain of the Wood which is as long as <lb/>the &longs;aid Wood may make it &longs;trong and able to Re&longs;i&longs;t a great vio­<lb/>lence done to it to break it: But a Cord compo&longs;ed of &longs;trings of <lb/>Hemp, no longer than two, or three foot a piece, how can it be &longs;o <lb/>&longs;trong when it is &longs;pun out, it may be, to a hundred times that <lb/>length? </s> <s>Now I would farther under&longs;tand your opinion concern­<lb/>ing the Connection of the parts of Metals, Stones, and other mat­<lb/>ters deprived of &longs;uch Ligatures, which neverthele&longs;&longs;e, if I be not <lb/>deceived, are yet more tenacious.</s></p><p type="main"> <s>SALV. </s> <s>We mu&longs;t be nece&longs;&longs;itated to digre&longs;&longs;e into new Specu­<lb/>lations, and not much to our purpo&longs;e, if we &longs;hould re&longs;olve tho&longs;e <lb/>difficulties you &longs;tart.</s></p><pb xlink:href="069/01/009.jpg" pagenum="7"/><p type="main"> <s>SAGR. </s> <s>But if Digre&longs;&longs;ions may lead us to the knowledge of <lb/>new Truths, what prejudice is it to us, that are not obliged to a <lb/>&longs;trict and conci&longs;e method, but that make our Congre&longs;&longs;ions only <lb/>for our diverti&longs;ement to digre&longs;&longs;e &longs;ometimes, le&longs;t we let &longs;lip tho&longs;e <lb/>Notions, which perhaps the offered occa&longs;ion being pa&longs;t, may never <lb/>meet with another opportunity of remembrance? </s> <s>Nay, who knows <lb/>not, that many times curio&longs;ity may thereby di&longs;cover hints of more <lb/>worth, than the primarily intended Conclu&longs;ions? </s> <s>Therefore I <lb/>entreat you to give &longs;atisfaction to <emph type="italics"/>Simplicius,<emph.end type="italics"/> and my &longs;elf al&longs;o, <lb/>no le&longs;&longs;e curious than he, and de&longs;irous to under&longs;tand what that <lb/>Cement is, that holdeth the parts of tho&longs;e Solids &longs;o tenaciou&longs;ly <lb/>conjoyned, which yet neverthele&longs;&longs;e in conclu&longs;ion are di&longs;&longs;oluble: <lb/>a knowledge which furthermore is nece&longs;&longs;ary for the under&longs;tanding <lb/>of the coherence of the parts of tho&longs;e very ligaments, whereof <lb/>&longs;ome Solids are compo&longs;ed.</s></p><p type="main"> <s>SALV. Well, &longs;ince it is your plea&longs;ure, I will herein &longs;erve you. <lb/><arrow.to.target n="marg999"></arrow.to.target><lb/>And the fir&longs;t difficulty is, how the threads of a Cord or Rope <lb/>an hundred foot long &longs;hould &longs;o clo&longs;ely connect together (none <lb/>of them exceeding two or three foot) that it requireth a great <lb/>violence to break them. </s> <s>But tell me, <emph type="italics"/>Simplicius,<emph.end type="italics"/> cannot you hold <lb/>one &longs;ingle &longs;tring of Hemp &longs;o fa&longs;t between your fingers by one <lb/><arrow.to.target n="marg1000"></arrow.to.target><lb/>end, that I pulling by the other end &longs;hould break it &longs;ooner than <lb/>get it from you? </s> <s>Que&longs;tionle&longs;&longs;e you might: when then, tho&longs;e <lb/>threads are not only at the end, but al&longs;o in every part of their <lb/>length, held fa&longs;t with much &longs;trength by him that gra&longs;peth them, is <lb/>it not apparent, that it is a much harder matter to pluck them <lb/>from him that holds them, then to break them? </s> <s>Now in the Cord, <lb/><arrow.to.target n="marg1001"></arrow.to.target><lb/>the &longs;ame act of twi&longs;ting, binds the threads mutually within one <lb/>another, in &longs;uch &longs;ort, that pulling the Cord with great force, the <lb/>threads of it break in&longs;under, but &longs;eparate and part not from one <lb/>another; as is plainly &longs;een by viewing the &longs;hort ends of the &longs;aid <lb/>threads in the broken place, that are not a &longs;pan long; as they <lb/>would be, if the divi&longs;ion of the Cord had been made by the &longs;ole <lb/>&longs;eperating of them in drawing the Cord, and not by breaking <lb/>them.</s></p><p type="margin"> <s><margin.target id="marg999"></margin.target><emph type="italics"/>What that Cement <lb/>is that Connecteth <lb/>the parts of Solids.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1000"></margin.target><emph type="italics"/>How a Rope or <lb/>Cord re&longs;i&longs;teth Fra­<lb/>ction.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1001"></margin.target><emph type="italics"/>In breaking a Rope <lb/>the parts are not <lb/>&longs;eparated, but bro­<lb/>kon.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>For confirmation of this, let me add, that the Cord is <lb/>&longs;ometimes &longs;een to break, not by pulling it length-waies, but by <lb/>over-twi&longs;ting it: an argument, in my judgment, concluding that <lb/>the threads are &longs;o enterchangeably compre&longs;t by one another, that <lb/>tho&longs;e compre&longs;&longs;ings permit not the compre&longs;&longs;ed to &longs;lip &longs;o very little, <lb/>as is requi&longs;ite to lengthen it out that it wind about the Cord, <lb/>which in the twining breaketh, and con&longs;equently in &longs;ome &longs;inall <lb/>mea&longs;ure &longs;wels in thickne&longs;&longs;e.</s></p><p type="main"> <s>SALV. </s> <s>You &longs;ay very well; but con&longs;ider by the way, how one <lb/>truth draweth on another. </s> <s>That thread, which griped between the <pb xlink:href="069/01/010.jpg" pagenum="8"/>fingers, did not yield to follow him that would have forceably <lb/>drawn it from between them, re&longs;i&longs;ted, becau&longs;e it was &longs;tayed by a <lb/>double compre&longs;&longs;ion, &longs;ince the upper finger pre&longs;t no le&longs;&longs;e again&longs;t <lb/>the nether, than it pre&longs;&longs;ed again&longs;t that. </s> <s>And there is no que&longs;tion, <lb/>that if of the&longs;e two pre&longs;&longs;ures, one alone might be retained, there <lb/>would remain half of that Re&longs;i&longs;tance, which depended conjunctive­<lb/>ly on them both: but becau&longs;e you cannot with removing, <emph type="italics"/>v.g.<emph.end type="italics"/> the <lb/>upper finger take away its pre&longs;&longs;ion, without taking away the other <lb/>part al&longs;o; it will be nece&longs;&longs;ary by &longs;ome new Artifice to retain one <lb/>of them, and to find a way that the &longs;ame thread may compre&longs;&longs;e it <lb/>&longs;elf again&longs;t the finger or other &longs;olid body upon which it is put; and <lb/>this is done by winding the &longs;ame thread about the Solid. </s> <s>For the <lb/>better under&longs;tanding whereof, I will briefly give it you in Figure; <lb/>and let <emph type="italics"/>A B<emph.end type="italics"/> and C<emph type="italics"/>D<emph.end type="italics"/> be two Cilinders, and between them let there <lb/>be di&longs;tended the thread <emph type="italics"/>E F,<emph.end type="italics"/> which for greater plainne&longs;&longs;e I will <lb/>repre&longs;ent to be a &longs;mall Cord: there is no doubt but that the two <lb/>Cylinders being pre&longs;&longs;ed hard one again&longs;t the other, the Cord <lb/><emph type="italics"/>E F<emph.end type="italics"/> pulled by the end <emph type="italics"/>F<emph.end type="italics"/> will Re&longs;i&longs;t no &longs;mal force before <lb/>it will &longs;lip from between the two Solids compre&longs;&longs;ing it: but if <lb/>we remove one of them, though the Cord <lb/><figure id="id.069.01.010.1.jpg" xlink:href="069/01/010/1.jpg"/><lb/>continue touching the other, yet &longs;hall it not <lb/>by &longs;uch contact be hindered from &longs;lipping <lb/>away. </s> <s>But if holding it fa&longs;t, though but <lb/>gently in the point A, towards the top of the <lb/>Cylinder, we wind, or belay it about the <lb/>&longs;ame &longs;pirally in A F L O T R, and pull it by <lb/>the end R: it is manife&longs;t, that it will begin <lb/>to pre&longs;&longs;e the Cylinder, and if the windings <lb/>and wreathes be many, it &longs;hall in its effectual <lb/>drawing alwaies pre&longs;&longs;e it &longs;o much the &longs;trai­<lb/>ter about the Cylinder: and by multiplying <lb/>the wreathes if you make the contact longer, <lb/>and con&longs;equently more invincible, the more <lb/>difficult &longs;till &longs;hall it be to withdraw the <lb/>Cord, and make it yield to the force that <lb/>pulls it. </s> <s>Now who &longs;eeth not, that the &longs;ame <lb/>Re&longs;i&longs;tance is in the threads, which with many thou&longs;and &longs;uch <lb/>twinings &longs;pin the thick Cord? </s> <s>Yea, the &longs;tre&longs;&longs;e of &longs;uch twi&longs;ting <lb/>bindeth with &longs;uch Tenacity, that a few Ru&longs;hes, and of no great <lb/>length, (&longs;o that the wreaths and windings are but few where­<lb/>with they entertwine) make very &longs;trong bands, called, as I take it, <lb/><arrow.to.target n="marg1002"></arrow.to.target><lb/>^{*} Thum-ropes.</s></p><p type="margin"> <s><margin.target id="marg1002"></margin.target>* Fu&longs;ta.</s></p><p type="main"> <s>SAGR. </s> <s>Your Di&longs;cour&longs;e hath removed the wonder out of my <lb/>mind at two effects, whereof I did not well under&longs;tand the rea­<lb/>&longs;on; One was to &longs;ee, how two, or at the mo&longs;t three twines of the <pb xlink:href="069/01/011.jpg" pagenum="9"/>Rope about the Axis of a Crane did not only hold it, that be­<lb/>ing drawn by the immen&longs;e force of the weight, which it held, it <lb/>&longs;lipt nor &longs;hrunk not; but that moreover turning the Crane about, <lb/>the &longs;aid Axis with the &longs;ole touch of the Rope which begirteth it, <lb/>did in the after-turnings, draw and rai&longs;e up va&longs;t &longs;tones, whil&longs;t the <lb/>&longs;trength of a little Boy &longs;ufficed to hold and &longs;tay the other end of <lb/>the &longs;ame Cord. </s> <s>The other is at a plain, but cunning, In&longs;trument found <lb/>out by a young Kin&longs;man of mine, by which with a Cord he could <lb/>let him&longs;elf down from a window without much gauling the palmes <lb/>of his hands, as to his great &longs;mart not long before he had done. </s> <s>For <lb/><arrow.to.target n="marg1003"></arrow.to.target><lb/>the better under&longs;tanding whereof, rake this Scheame: About &longs;uch <lb/>a Cylinder of Wood as A B, two Inches <lb/>thick, and &longs;ix or eight Inches long, he cut a <lb/>hollow notch &longs;pirally, for one turn and a <lb/><figure id="id.069.01.011.1.jpg" xlink:href="069/01/011/1.jpg"/><lb/>half and no more, and of widene&longs;&longs;e fit for <lb/>the Cord he would u&longs;e; which he made to <lb/>enter through the notch at the end A, and <lb/>to come out at the other B, incircling after­<lb/>wards the Cylinder in a barrel or &longs;ocket of <lb/>Wood, or rather Tin, but divided length­<lb/>waies, and made with Cla&longs;pes or Hinges to <lb/>open and &longs;hut at plea&longs;ure: and then gra&longs;p­<lb/>ing and holding the &longs;aid Barrel or Ca&longs;e with <lb/>both his hands, the rope being made fa&longs;t <lb/>above, he hung by his arms; and &longs;uch was <lb/>the compre&longs;&longs;ion of the Cord between the <lb/>moving Socket and the Cylinder, that at <lb/>plea&longs;ure griping his hands clo&longs;er he could <lb/>&longs;tay him&longs;elf without de&longs;cending, and &longs;lacking his hold a little, he <lb/>could let him&longs;elf down as he plea&longs;ed.</s></p><p type="margin"> <s><margin.target id="marg1003"></margin.target><emph type="italics"/>An Hand-Pully <lb/>or In&longs;trument in­<lb/>vented by an ama­<lb/>rous per&longs;on to let <lb/>him&longs;elf down from <lb/>any great height <lb/>with a Cord with­<lb/>out gauling his <lb/>hands.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>Aningenious invention verily, and for a full explanati­<lb/>on of its nature, me-thinks I di&longs;cover, as it were by a &longs;hadow, the <lb/>light of &longs;ome other additional di&longs;coveries: but I will not at this <lb/>time deviate any more from my purpo&longs;e upon this particular: and <lb/>the rather in regard you are de&longs;irous to hear my opinion of the <lb/>Re&longs;i&longs;tance of other Bodies again&longs;t Fraction, who&longs;e texture is not <lb/><arrow.to.target n="marg1004"></arrow.to.target><lb/>with threads, and fibrous &longs;trings, as is that of Ropes, and mo&longs;t <lb/>kinds of Wood: but the connection of their parts &longs;eem to de­<lb/>pend on other Cau&longs;es; which in my judgment may be reduced to <lb/>two heads; one is the much talked-of Repugnance that Nature <lb/>hath again&longs;t the admi&longs;&longs;ion of Vacuity: for another (this of Va­<lb/>cuity not &longs;ufficing) there mu&longs;t be introduced &longs;ome glue, vi&longs;cous <lb/>matter, or Cement, that tenaciou&longs;ly connecteth the Corpu&longs;cles of <lb/>which the &longs;aid Body is compacted.</s></p><p type="margin"> <s><margin.target id="marg1004"></margin.target><emph type="italics"/>Why &longs;uch Bodies <lb/>re&longs;i&longs;t Fraction that <lb/>are not connected <lb/>with Fibrous fila­<lb/>ments.<emph.end type="italics"/></s></p><p type="main"> <s>I will fir&longs;t &longs;peak of <emph type="italics"/>Vacuity,<emph.end type="italics"/> &longs;hewing by plain experiments, <pb xlink:href="069/01/012.jpg" pagenum="10"/><arrow.to.target n="marg1005"></arrow.to.target><lb/>what and how great its virtue is. </s> <s>And fir&longs;t of all the &longs;eeing at <lb/>plea&longs;ure two flat pieces of either Marble, Metal, or Gla&longs;&longs;e, exqui­<lb/>&longs;itely planed, &longs;lickt, and poli&longs;hed, that being laid upon one the <lb/>other, without any difficulty &longs;lide along upon each other, if drawn <lb/><arrow.to.target n="marg1006"></arrow.to.target><lb/>&longs;idewaies, (a certain argument that no glue connects them,) but <lb/>that going about to &longs;eperate them, keeping them equidi&longs;tant, <lb/>there is found &longs;uch repugnance, that the uppermo&longs;t will be lif­<lb/>ted up, and will draw the other after it, and keep it perperually <lb/>rai&longs;ed, though it be pretty thick, and heavy, evidently proveth to <lb/>us, how much Nature abhorreth to admit, though for a &longs;hort mo­<lb/>ment of time, the void &longs;pace, that would be between them, till <lb/>the concour&longs;e of the parts of the Circum-Ambient Air &longs;hould have <lb/>po&longs;&longs;e&longs;t, and repleated it. </s> <s>We &longs;ee likewi&longs;e, that if tho&longs;e two Plates <lb/>be not exactly poli&longs;hed, and con&longs;equently their contact not every <lb/>where exqui&longs;ite; in going about to &longs;eparate them gently, there will <lb/>be found no Renitence more than that of their meer weight, but in <lb/>the &longs;udden rai&longs;ing, the nether Stone will ri&longs;e, and in&longs;tantly fall <lb/>down again, following the upper only for that very &longs;mall time <lb/>which &longs;erveth for the expan&longs;ion of that little Air which interpo­<lb/>&longs;eth betwixt the Plates, that did not every where touch, and for <lb/>the ingre&longs;&longs;ion of the other circumfu&longs;ed. </s> <s>The like Re&longs;i&longs;tance, which <lb/>&longs;o &longs;en&longs;ibly di&longs;covers it &longs;elf betwixt the two Plates, cannot be <lb/>doubted to re&longs;ide al&longs;o between the parts of a Solid, and that it en­<lb/>tereth into their connection, at lea&longs;t in part, and as their Concomi­<lb/>tant Cau&longs;e.</s></p><p type="margin"> <s><margin.target id="marg1005"></margin.target><emph type="italics"/>The fir&longs;t Cau&longs;e of <lb/>the Cohorence of <lb/>Bodies is their Re­<lb/>pugnance to Vacu­<lb/>ity.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1006"></margin.target><emph type="italics"/>This is proved by <lb/>the Coherence of <lb/>two poli&longs;hed Mar­<lb/>bles.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. Hold, I pray you, and permit me to impart unto you a <lb/>particular Con&longs;ideration, ju&longs;t now come into my Mind, and this it <lb/>is; That &longs;eeing how the lower Plate followeth the upper, and is <lb/>by a &longs;peedy motion rai&longs;ed, we are thereby a&longs;certained that (con­<lb/>trary to the &longs;aying of many Philo&longs;ophers, and perchance of <emph type="italics"/>Ari­<lb/>&longs;totle<emph.end type="italics"/> him&longs;elf) the Motion in <emph type="italics"/>Vacuity<emph.end type="italics"/> would not be In&longs;tantaneous; <lb/>for &longs;hould in be &longs;uch, the propo&longs;ed Plates without the lea&longs;t repug­<lb/>nance would Seperate; &longs;ince the &longs;elf &longs;ame in&longs;tant of time would <lb/>&longs;uffice for their &longs;eparation, and for the concour&longs;e of the Ambient <lb/>Air to repleat that <emph type="italics"/>Vacuity,<emph.end type="italics"/> which might remain between them. <lb/></s> <s>By the Inferiour Plates following the Superiour therefore may be <lb/>gathered, that in the <emph type="italics"/>Vacuity<emph.end type="italics"/> the Motion would not be In&longs;tanta­<lb/><arrow.to.target n="marg1007"></arrow.to.target><lb/>neous. </s> <s>And al&longs;o it may be inferred, that even betwixt tho&longs;e Plates <lb/>there re&longs;teth &longs;ome <emph type="italics"/>Vacuity,<emph.end type="italics"/> at lea&longs;t for &longs;ome very &longs;hort time; that <lb/>is, for &longs;o long as the Ambient Air is moving whil&longs;t it concurreth to <lb/>replete the <emph type="italics"/>Vacuum:<emph.end type="italics"/> for if there did no <emph type="italics"/>Vacuity<emph.end type="italics"/> remain, there <lb/>would be no need either of the Concour&longs;e, or Motion of the Am­<lb/>bient We mu&longs;t therefore &longs;ay that <emph type="italics"/>Vacuity<emph.end type="italics"/> &longs;ometimes is admit­<lb/>ted, though by Violence or again&longs;t Nature, (albeit it is my opi­<lb/>nion, that nothing is contrary to Nature, but that which is im­<pb xlink:href="069/01/013.jpg" pagenum="11"/>po&longs;&longs;ible, which again never is.) But here &longs;tarts up another diffi­<lb/>culty, and it is, That though Experience a&longs;&longs;ures me of the truth of <lb/>the Conclu&longs;ion, yet my Judgment is not thorowly &longs;atisfied of the <lb/>Cau&longs;e, to which &longs;uch an effect may be a&longs;cribed. </s> <s>For as much as <lb/>the effect of the Seperation of the two Plates, is in time before the <lb/>Vacuity which &longs;hould &longs;ucceed by con&longs;equence upon the Separa­<lb/>tion. </s> <s>And becau&longs;e, in my opinion, the Cau&longs;e ought, if not in <lb/><arrow.to.target n="marg1008"></arrow.to.target><lb/>Time, at lea&longs;t in Nature, to precede the Effect: and that of a Po­<lb/>&longs;itive Effect, the Cau&longs;e ought al&longs;o to be Po&longs;itive; I cannot con­<lb/>ceive, how the Cau&longs;e of the Adhe&longs;ion of the two Plates, and of <lb/>their Repugnance to Separation, (Effects that are already in <lb/>Act) &longs;hould be a&longs;&longs;igned to Vacuity, which yet is not, but &longs;hould <lb/>follow. </s> <s>And of things that are not in being, there can be no Ope­<lb/><arrow.to.target n="marg1009"></arrow.to.target><lb/>ration; according to the infallible Maxime of Philo&longs;ophy.</s></p><p type="margin"> <s><margin.target id="marg1007"></margin.target><emph type="italics"/>Vacuity partly the <lb/>cau&longs;e of the Cohe­<lb/>rence between the <lb/>parts of Solids.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1008"></margin.target><emph type="italics"/>Of a Po&longs;itive Ef­<lb/>fect the Cau&longs;e is <lb/>Po&longs;itive.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1009"></margin.target><emph type="italics"/>Non-entity is at­<lb/>tended with Non­<lb/>operation.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. </s> <s>But &longs;ince you grant <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/> this Axiome, I do not <lb/>think you will deny another that is mo&longs;t excellent, and true; to <lb/><arrow.to.target n="marg1010"></arrow.to.target><lb/>wit, That Nature doth not attempt Impo&longs;&longs;ibilities: Upon which <lb/>Axiom I think the Solution of our doubt depends: becau&longs;e there­<lb/>fore a void &longs;pace is of it &longs;elf impo&longs;&longs;ible, Nature forbids the doing <lb/>that, in con&longs;equence of which Vacuity would nece&longs;&longs;arily &longs;ucceed; <lb/>and &longs;uch an act is the &longs;eparation of the two Plates.</s></p><p type="margin"> <s><margin.target id="marg1010"></margin.target><emph type="italics"/>Nature doth not <lb/>attempt Impo&longs;&longs;ibi­<lb/>lities.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. Now, (admitting this which <emph type="italics"/>Simplicius<emph.end type="italics"/> alledgeth is a <lb/>&longs;ufficient Solution of my Doubt) in per&longs;uance of the di&longs;cour&longs;e <lb/>with which I began, it &longs;eemeth to me, that this &longs;ame Repugnance <lb/>to Vacuity &longs;hould be a &longs;ufficient Cement in the parts of a Solid of <lb/>Stone, Metal, or what other &longs;ub&longs;tance is more firmly conjoyned, <lb/>and aver&longs;e to Divi&longs;ion. </s> <s>For if a &longs;ingle Effect, hath but one &longs;ole <lb/>Cau&longs;e, as I under&longs;tand, and think; or if many be a&longs;&longs;igned, they <lb/>are reducible to one alone: why &longs;hould not this of Vacuity, which <lb/>certainly is one, be &longs;ufficient to an&longs;wer all Re&longs;i&longs;tances?</s></p><p type="main"> <s>SALV. </s> <s>I will not at this time enter upon this conte&longs;t, whether <lb/>Vacuity, without other Cement, be in it &longs;elf alone &longs;ufficient to <lb/>keep together the &longs;eparable parts of firm Bodies; but yet this I <lb/>&longs;ay, that the Rea&longs;on of the Vacuity, which is of force, and con­<lb/>oluding in the two Plates, &longs;ufficeth not of it &longs;elf alone for the <lb/>firm connection of the parts of a &longs;olid Cylinder of Marble, or <lb/>Metal, the which forced with great violence, pulling them &longs;treight <lb/>out, in fine, divide and &longs;eparate. </s> <s>And in ca&longs;e I have found a way <lb/>to di&longs;tingui&longs;h this already-known Re&longs;i&longs;tance dependent on Va­<lb/>ouity, from all others what&longs;oever that may concur with it in <lb/>&longs;trengthening the Connection, and make you &longs;ee how that it alone <lb/>is not neer &longs;ufficient for &longs;uch an Effect, would not you grant that <lb/>it would be nece&longs;&longs;ary to introduce &longs;ome other? </s> <s>Help him out, <emph type="italics"/>Sim­<lb/>plicius,<emph.end type="italics"/> for he &longs;tands &longs;tudying what to an&longs;wer.</s></p><p type="main"> <s>SIMP. </s> <s>The Su&longs;pen&longs;ion of <emph type="italics"/>Sagredus<emph.end type="italics"/> mu&longs;t needs be upon ano­<pb xlink:href="069/01/014.jpg" pagenum="12"/>ther account, there being no place left for doubting of &longs;o clear, and <lb/>nece&longs;&longs;ary a Con&longs;equence.</s></p><p type="main"> <s>SAGR. </s> <s>You Divine <emph type="italics"/>Simplicius,<emph.end type="italics"/> I was thinking if a Million of <lb/>Gold <emph type="italics"/>per annum,<emph.end type="italics"/> coming from <emph type="italics"/>Spaine,<emph.end type="italics"/> not being &longs;ufficient to pay <lb/>the Army, whether it was nece&longs;&longs;ary to make any other provi&longs;ion <lb/>than of Money to pay the Souldiers. </s> <s>But proceed, <emph type="italics"/>Salviatus,<emph.end type="italics"/> and <lb/>&longs;uppo&longs;ing that I admit of your Con&longs;equence, &longs;hew us how to &longs;e­<lb/>parate the opperation of Vacuity from the other, that mea&longs;uring <lb/>it we may &longs;ee how it's in&longs;ufficient for the Effect of which we &longs;peak.</s></p><p type="main"> <s>SALV. </s> <s>Your Genius hath prompted you. </s> <s>Well, I will tell you <lb/>the way to part the Virtue of Vacuity from the re&longs;t, and then how <lb/>to mea&longs;ure it. </s> <s>And to &longs;ever it, we will take a continuate matter, <lb/><arrow.to.target n="marg1011"></arrow.to.target><lb/>who&longs;e parts are de&longs;titute of all other Re&longs;i&longs;tance to Separation, &longs;ave <lb/>only that of Vacuity, &longs;uch as Water at large hath been demon­<lb/>&longs;trated to be in a certain Tractate of our <emph type="italics"/>Accademick.<emph.end type="italics"/> So that <lb/>when ever a Cylinder of Water is &longs;o di&longs;po&longs;ed, that being drawn <lb/>we find a Re&longs;i&longs;tance again&longs;t the &longs;eparation of its parts, this mu&longs;t <lb/>be acknowledged to proceed from no other cau&longs;e, but from re­<lb/>pugnance to Vacuity. </s> <s>But to make &longs;uch an experiment, I have <lb/>imagined a device, which with the help of a &longs;mall Diagram, may <lb/>be better expre&longs;t than by my bare words. </s> <s>Let this Figure C A B D <lb/>be the Profile of a Cylinder of Metal, or of Gla&longs;s, which mu&longs;t <lb/>be made hollow within, but turned exactly round; into who&longs;e <lb/>Concave mu&longs;t enter a Cylinder of Wood, exqui&longs;itely fitted to <lb/>touch every where, who&longs;e Profile is noted by <lb/>E G H F, which Cylinder may be thru&longs;t up­<lb/><figure id="id.069.01.014.1.jpg" xlink:href="069/01/014/1.jpg"/><lb/>wards, and downwards: and this I would <lb/>have bored in the middle, &longs;o that there may <lb/>a rod of Iron pa&longs;s thorow, hooked in the end <lb/>K, and the other end I, &longs;hall grow thicker in <lb/>fa&longs;hion of a Cone, or Top; and let the <lb/>hole made for the &longs;ame thorow the Cylinder <lb/>of Wood be al&longs;o cut hollow in the upper <lb/>part, like a Conical Superficies, and exactly <lb/>fitted to receive the Conick end I, of the <lb/>Iron I K, as oft as it is drawn down by the <lb/>part K. </s> <s>Then I put the Cylinder of Wood <lb/>E H into the Concave Cylinder A D, and <lb/>would not have it come to touch the upper­<lb/>mo&longs;t Superficies of the &longs;aid hollow Cylinder, <lb/>but that it &longs;tay two or three fingers breadth <lb/>from it: and I would have that &longs;pace filled with Water; which <lb/>&longs;hould be put therein, holding the Ve&longs;&longs;el with the mouth C D up­<lb/>wards; and thereupon pre&longs;s down the Stopper E H, holding the <lb/>Conical part I &longs;omewhat di&longs;tant from the hollow that was made <pb xlink:href="069/01/015.jpg" pagenum="13"/>for it in the Wood, to leave way for the Air to go out, which in <lb/>thru&longs;ting down the Stopper will i&longs;&longs;ue out by the hole of the <lb/>Wood, which therefore &longs;hould be made a little wider than the <lb/>thickne&longs;s of the Hook of Iron I K. </s> <s>The Air being let out, and the <lb/>Iron pull'd back, which clo&longs;e &longs;toppeth the wood with its Conick <lb/>part I, then turn the ve&longs;&longs;el with its mouth downwards, and fa&longs;ten to <lb/>the hook K a Bucket that may receive into it &longs;and, or other weigh­<lb/>ty matter, and you may hang &longs;o much weight thereat, that at length <lb/>the Superiour &longs;urface of the Stopper E F will &longs;eparate and for&longs;ake <lb/>the inferiour part of the Water; to which nothing el&longs;e held it con­<lb/>nected but the Repugnance again&longs;t Vacuity: afterwards weighing <lb/>the Stopper with the Iron, the Bucket, and all that was in it, you <lb/>will have the quantity of the Force of the Vacuity. </s> <s>And if affixing <lb/>to a Cylinder of Marble, or Chri&longs;tal, as thick as the Cylinder of <lb/>Water, &longs;uch a weight, that together with the proper weight of the <lb/>Marble or Chri&longs;tal it &longs;elf, equalleth the gravity of all tho&longs;e fore­<lb/>named things, a Rupture follow thereupon; we may without <lb/>doubt affirm, that the only rea&longs;on of Vacuity holdeth the parts of <lb/>Marble and Chri&longs;tal conjoyned: but not &longs;ufficing; and &longs;eeing <lb/>that to break it there mu&longs;t be added four times as much weight, <lb/>it mu&longs;t be confe&longs;&longs;ed, that the Re&longs;i&longs;tance of Vacuity is one part of <lb/>&longs;ive, and that the other Re&longs;i&longs;tance is quadruple to that of Vacuity.</s></p><p type="margin"> <s><margin.target id="marg1011"></margin.target><emph type="italics"/>How to mea&longs;ure <lb/>the Virtue of Va­<lb/>cuity in Solids di­<lb/>&longs;tinct from other <lb/>convenient Cau&longs;es <lb/>of their Coherence. <lb/></s> <s>Water a Continu­<lb/>ate Matter, and <lb/>void of all other a­<lb/>ver&longs;ion to &longs;eparati­<lb/>on, &longs;ave that of Va­<lb/>cuity.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. </s> <s>It cannot be denied, but that the Invention is Ingen­<lb/>ous: but I hold it to be &longs;ubject to many difficulties, which makes <lb/>me que&longs;tion it; for who &longs;hall a&longs;&longs;ure us, that the Air cannot pene­<lb/>trate between the Gla&longs;s, and the Stopper, though it be clo&longs;e &longs;topt <lb/>with Flax, or other pliant matter? </s> <s>And al&longs;o it's a Que&longs;tion, whe­<lb/>ther Wax or Turpentine will &longs;erve to make the Cone I, &longs;top the <lb/>hole clo&longs;e: Again, Why may not the parts of the Water with­<lb/>draw and rarefie them&longs;elves? </s> <s>Why may not the Air, or Exhalati­<lb/>ons, or other more &longs;ubtil Sub&longs;tances penetrate through the Poro&longs;i­<lb/>ties of the Wood, or Gla&longs;s it &longs;elf?</s></p><p type="main"> <s>SALV. <emph type="italics"/>Simplicius<emph.end type="italics"/> is very nimble at rai&longs;ing doubts, and, in part, <lb/>helping us to re&longs;olve them, as to the Penetration of the Air through <lb/>the Wood, or between the Wood and Gla&longs;s. </s> <s>But I moreover <lb/>ob&longs;erve, that we may at the &longs;ame time &longs;ecure our &longs;elves, and with­<lb/>all acquire new Notions, if the fore-named doubts take place; for <lb/>if the Water be by Nature, howbeit with violence, capable of ex­<lb/>tention, as it falleth out in Air, you &longs;hall &longs;ee the Stopper to de­<lb/>&longs;cend: and if in the upper part of the Gla&longs;s we make a &longs;mall pro­<lb/>minent Bo&longs;s, as this V; in ca&longs;e any Air, or other more Tenuous or <lb/>Spirituous Matter &longs;hould penetrate thorow the Sub&longs;tance, or Poro&longs;i­<lb/>ty of the Gla&longs;s, or Wood, it would be &longs;een to reunite (the water <lb/>giving place) in the eminence V: which things not being percei­<lb/>ved, we re&longs;t a&longs;&longs;ured that the Experiment was made with due <pb xlink:href="069/01/016.jpg" pagenum="14"/>caution: and &longs;ee that the Water is not capable o&longs; exten&longs;ion, nor <lb/>the Gla&longs;s permeable by any matter, though never &longs;o &longs;ubtil.</s></p><p type="main"> <s>SAGR. </s> <s>And I, by means of the&longs;e Di&longs;cour&longs;es have found the <lb/>Cau&longs;e of an Effect, that hath for a long time puzled my mind <lb/><arrow.to.target n="marg1012"></arrow.to.target><lb/>with wonder, and kept it in Ignorance. </s> <s>I have heretofore ob­<lb/>&longs;erved a Ci&longs;tern, wherein, for the drawing thence of Water, there <lb/>was made a Pump, by &longs;ome one that thought, perhaps, (but in <lb/>vain) to be thereby able to draw, with le&longs;s labour, the &longs;ame, or <lb/>greater quantity of Water, than with the ordinary Buckets; and <lb/>this Pump had its Sucker and Value on high, &longs;o that the Water <lb/>was made to a&longs;cend by Attraction, and not by Impul&longs;e, as do the <lb/>Pumps that work below. </s> <s>This, whil&longs;t there is any Water in the <lb/>Ci&longs;tern to &longs;uch a determinate height, will draw it plentifully; but <lb/>when the Water ebbeth below a certain Mark, the Pump will <lb/>work no more. </s> <s>I conceited, the fir&longs;t time that I ob&longs;erved this ac­<lb/>cident, that the Engine ____ had been &longs;poyled, and looking for <lb/>the Workman, that he might amend it; he told me, that there was <lb/>no defect at all, other than what was in the Water, which being <lb/>fallen too low, permitted not it &longs;elf to be rai&longs;ed to &longs;uch a height; <lb/><arrow.to.target n="marg1013"></arrow.to.target><lb/>and farther &longs;aid, that neither Pump, or other Machine, that rai&longs;eth <lb/>the water by Attraction, was po&longs;&longs;ibly able to make it ri&longs;e a hair <lb/>more than eighteen Braces, and be the Pumps wide or narrow, this <lb/>is the utmo&longs;t limited mea&longs;ure of their height. </s> <s>And I have hitherto <lb/>been &longs;o dull of apprehen&longs;ion, that though I knew that a Rope, a <lb/>Stick, and a Rod of Iron might be &longs;o and &longs;o lengthened, that at <lb/>la&longs;t, holding it up on high in the Air, its own weight would break <lb/>it, yet I never remembred, that the &longs;ame would much more ea&longs;ily <lb/>happen in a Rope, or Thread of Water. </s> <s>And what other is that <lb/>which is attracted in the Pump than a Cylinder of Water, which <lb/>having its contraction above, prolonged more and more, in the end <lb/>arriveth to that term, beyond which being drawn, it breaketh by <lb/>its foregoing over-weight, ju&longs;t as if it was a Rope.</s></p><p type="margin"> <s><margin.target id="marg1012"></margin.target><emph type="italics"/>The Nature of the <lb/>attraction of Wa­<lb/>ter by Pumps.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1013"></margin.target><emph type="italics"/>Water rai&longs;ed or at­<lb/>tracted by a Pump <lb/>ri&longs;eth not above <lb/>eleven yards.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>It is even &longs;o as you &longs;ay; and becau&longs;e the &longs;aid height of <lb/>eighteen Braces is the prefixed term of the Elevation, to which any <lb/>quantity of Water, be it (that is to &longs;ay, be the Pump) broad, <lb/>narrow, or even, &longs;o narrow as to the thickne&longs;s of a &longs;traw, can &longs;u­<lb/>&longs;tain it &longs;elf; when ever we weigh the water contained in eighteen <lb/>Braces of Pipe, be it broad or narrow, we have the value of Re&longs;i­<lb/>&longs;tance of Vacuity in Cylinders of what&longs;oever &longs;olid matter, of the <lb/>thickne&longs;s of the propo&longs;ed Pipes. </s> <s>And &longs;ince I have &longs;aid &longs;o much, <lb/><arrow.to.target n="marg1014"></arrow.to.target><lb/>we will &longs;hew, that a man may ea&longs;ily find in all Metals, Stones, Tim­<lb/>bers, Gla&longs;&longs;es, <emph type="italics"/>&c.<emph.end type="italics"/> How far one may lengthen out Cylinders, <lb/>&longs;trings, or rods of any thickne&longs;s, beyond which, being oppre&longs;t with <lb/>their own weight, they can no longer hold, but break in pieces. <lb/></s> <s>Take for example a Bra&longs;s wyer of any certain thickne&longs;s, and length, <pb xlink:href="069/01/017.jpg" pagenum="15"/>and fixing one of its ends on high, add gradually more and more <lb/>weight to the other, till at la&longs;t it break, and let the greate&longs;t weight <lb/>that it can bear be <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> fifty pounds. </s> <s>It is manife&longs;t that fifty <lb/>pound of Bra&longs;s more than its own weight, which let us &longs;uppo&longs;e, <lb/>for example, to be one eighth of an Ounce, drawn out into a <lb/>Wyer of the like thickne&longs;s, would be the greate&longs;t length of the <lb/>Wyer that could bear it &longs;elf. </s> <s>Then mea&longs;ure how long the Wyer <lb/>was which brake, and let it be for in&longs;tance a y ard; and becau&longs;e it <lb/>weighed one eighth of an Ounce; and poi&longs;ed, or bore it &longs;elf, and <lb/>fifty pounds more; which are Four Thou&longs;and Eight Hundred <lb/>eighths of Ounces; we &longs;ay, that all Wyers of Bra&longs;s, whatever <lb/>thickne&longs;s they be of, can hold, at the length of Four Thou&longs;and <lb/>Eight Hundred and one yards, and no more: and &longs;o, a Bra&longs;s Wyer <lb/>being able to hold to the length of 4801 yards; the Re&longs;i&longs;tance it <lb/>findeth dependent on Vacuity, in re&longs;pect of the remainder, is as <lb/>much as is equivalent to the weight of a Rope of Water eighteen <lb/>Braces long, and of the &longs;ame thickne&longs;s with the &longs;aid Bra&longs;s Wyer: <lb/>and finding Bra&longs;s to be <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> nine times heavier than Water, in <lb/>any Wyer of Bra&longs;s, the Re&longs;i&longs;tance again&longs;t Fraction dependent on <lb/>the rea&longs;on of Vacuity, importeth as much as two Braces of the <lb/>&longs;ame Wyer weigheth. </s> <s>And thus arguing, and operating, we may <lb/>find the length of the Wyers, or Threads of all Solid Matters re­<lb/>duced to the utmo&longs;t length that they can &longs;ub&longs;i&longs;t of, and al&longs;o what <lb/>part Vacuity hath in their Re&longs;i&longs;tance.</s></p><p type="margin"> <s><margin.target id="marg1014"></margin.target><emph type="italics"/>To what length Cy­<lb/>linders or Ropes of <lb/>any Matter may <lb/>be prolonged, be­<lb/>yond which being <lb/>charged they break <lb/>by their own weight<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>It re&longs;teth now, that you declare to us wherein con&longs;i&longs;ts <lb/>the remainder of that Tenacity, that is, what that Glue or Reni­<lb/>tence is, which connecteth together the parts of a Solid, be&longs;ides <lb/>that which is derived from Vacuity; becau&longs;e I cannot imagine <lb/>what that Cement is, that cannot be burnt, or con&longs;umed in a ve­<lb/>ry hot Furnace in two, three, or four Moneths, nor ten, nor an hun­<lb/>dred; and yet Gold, Silver, and Gla&longs;s, &longs;tanding &longs;o long Liqui&longs;i­<lb/>ed, when it is taken out, its parts return, upon cooling, to reunite, <lb/>and conjoyn, as before. </s> <s>And again, becau&longs;e the &longs;ame difficulty <lb/>which I meet within the Connection of the parts of the Gla&longs;s, I <lb/>find al&longs;o in the parts of the Cement, that is, what thing that <lb/>&longs;hould be which maketh them cleave &longs;o clo&longs;s together.</s></p><p type="main"> <s>SALV. </s> <s>I told you but even now, that your Genius prompted <lb/>you: I am al&longs;o in the &longs;ame &longs;trait: and al&longs;o whereas I have in gene­<lb/>ral told you, how that Repugnance again&longs;t Vacuity is unque&longs;ti­<lb/>onably that which permits not, nnle&longs;s with great violence, the &longs;e­<lb/>paration of the two Plates, and moreover of the two great pieces of <lb/>the Pillar of Marble, or Bra&longs;s, I cannot &longs;ee why it &longs;hould not al&longs;o <lb/>take place, and be likewi&longs;e the Cau&longs;e of the Coherence of the le&longs;­<lb/>&longs;er parts, and even of the very lea&longs;t and la&longs;t, of the &longs;ame Matters: <lb/><arrow.to.target n="marg1015"></arrow.to.target><lb/>and being that of one &longs;ole Effect, there is but one only true, and <pb xlink:href="069/01/018.jpg" pagenum="16"/>mo&longs;t potent Cau&longs;e; if I can find no other Cement, why may I not <lb/>try whether this of Vacuity, which I have already found, may be <lb/>&longs;ufficient?</s></p><p type="margin"> <s><margin.target id="marg1015"></margin.target><emph type="italics"/>There is but one <lb/>&longs;ole Cau&longs;e of one <lb/>&longs;ole Effect.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. </s> <s>But when you have already demon&longs;trated the Re&longs;i­<lb/>&longs;tance of the great Vacuity in the &longs;eparation of the two great <lb/>parts of a Solid to be very &longs;mall in compari&longs;on of that which con­<lb/>necteth, and con&longs;olidates the little Particles, or Atomes, why will <lb/>you not &longs;till hold, for certain, that this is extreamly differing from <lb/>that?</s></p><p type="main"> <s>SALV. </s> <s>To this <emph type="italics"/>Sagredus<emph.end type="italics"/> an&longs;wereth, That every particular <lb/>Souldier is &longs;till paid with money collected by the general Impo&longs;i­<lb/>tions of Shillings and Pence, although a Million of Gold &longs;ufficeth <lb/>not to pay the whole Army. </s> <s>And who knows, but that other ex­<lb/>ceeding &longs;mall Vacuities may operate among&longs;t tho&longs;e &longs;mall Atomes, <lb/>(even like as that was of the &longs;elf-&longs;ame money) wherewith all <lb/>the parts are connected? </s> <s>I will tell you what I have &longs;ometimes <lb/>fancied: and I give it you, not as an unque&longs;tionable Truth, but as a <lb/>kind of Conjecture very undige&longs;ted, &longs;ubmitting it to exacter con­<lb/>&longs;iderations: Pick out of it what plea&longs;eth you, and judge of the re&longs;t <lb/><arrow.to.target n="marg1016"></arrow.to.target><lb/>as you think fit. </s> <s>Con&longs;idering &longs;ometimes how the Fire, penetra­<lb/>ting and in&longs;inuating between the &longs;mall Atomes of this or that Me­<lb/>tal, which were before &longs;o clo&longs;ely con&longs;olidated, in the end &longs;epa­<lb/>rates, and di&longs;unites them; and how, the Fire being gone, they re­<lb/>turn with the &longs;ame Tenacity as before to Con&longs;olidation, without <lb/>dimini&longs;hing in quantity, (at all in Gold, and very little in other <lb/>Metals,) though they continue a long time melted; I have thought <lb/>that that might happen, by rea&longs;on the extream &longs;mall parts of the <lb/>Fire, penetrating through the narrow pores of the Metal (through <lb/>which the lea&longs;t parts of Air, or of many other Fluids, could not <lb/>for their clo&longs;ene&longs;s perforate) by repleating the &longs;mall interpo&longs;ing <lb/>Vacuities might free the minute parts of the &longs;ame from the vio­<lb/>lence, wherewith the &longs;aid Vacuities attract them one to another, <lb/>prohibiting their &longs;eparation: and thus becoming able to move <lb/>freely, their Ma&longs;s might become fluid, and continue &longs;uch, as long <lb/>as the &longs;mall parts of the Fire &longs;hould abide betwixt them: and that <lb/>tho&longs;e departing, and leaving the former Vacuities, their wonted <lb/>attractions might return, and con&longs;equently the Cohe&longs;ion of the <lb/>parts. </s> <s>And, as to the Allegation made by <emph type="italics"/>Simplicius,<emph.end type="italics"/> it may, in <lb/>my opinion, be thus re&longs;olved; That although &longs;uch Vacuities &longs;hould <lb/>be very &longs;mall, and con&longs;equently each of them ea&longs;ie to be over­<lb/>come, yet neverthele&longs;s their innumerable multitude innumerably <lb/><arrow.to.target n="marg1017"></arrow.to.target><lb/>(if it be proper &longs;o to &longs;peak) multiplieth the Re&longs;i&longs;tances: and we <lb/>have an evident proof what, and how great is the Force that re&longs;ul­<lb/>teth from the conjunction of an immen&longs;e number of very weak <lb/>Moments, in &longs;eeing a Weight of many thou&longs;ands of pounds, held <pb xlink:href="069/01/019.jpg" pagenum="17"/>by mighty Cables, to yield, and &longs;uffer it &longs;elf at la&longs;t to be over­<lb/>come by the a&longs;&longs;ault of the innumerable Atomes of Water; which, <lb/>either carryed by the South-wind, or el&longs;e by being di&longs;tended into <lb/>very thin Mi&longs;ts that move to and fro in the Air, in&longs;inuate them­<lb/>&longs;elves between &longs;tring and &longs;tring of the Hemp of the harde&longs;t twi­<lb/>&longs;ted Cables; nor can the immen&longs;e force of the pendent Weight <lb/>prohibit their enterance; &longs;o that perforating the &longs;trict pa&longs;&longs;ages be­<lb/>tween the Pores, they &longs;well the Ropes, and by con&longs;equence &longs;hor­<lb/>ten them, whereupon that huge Ma&longs;s is forcibly rai&longs;ed.</s></p><p type="margin"> <s><margin.target id="marg1016"></margin.target><emph type="italics"/>Mo&longs;t &longs;mall Va­<lb/>cuities di&longs;&longs;emina­<lb/>ted and interpo&longs;ed <lb/>between the &longs;mall <lb/>Corpu&longs;cles of So­<lb/>lids the probable <lb/>cau&longs;e of the con&longs;i­<lb/>&longs;tence or connecti­<lb/>on of tho&longs;e Corpu&longs;­<lb/>cles to one another,<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1017"></margin.target><emph type="italics"/>Innumerable A­<lb/>tomes of Water in­<lb/>&longs;inuating into Ca­<lb/>bles draw and rai&longs;e <lb/>an immen&longs;e weight<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. There's no doubt but that &longs;o long as a Re&longs;i&longs;tance is not <lb/><arrow.to.target n="marg1018"></arrow.to.target><lb/>infinite, it may by a multitude of mo&longs;t minute Forces be over­<lb/>come; in&longs;omuch that a competent number even of Ants would <lb/>be able to carry to &longs;hore a whole &longs;hips lading of Corn: for Sen&longs;e <lb/>giveth us quotidian examples, that an Ant carrieth a &longs;ingle grain <lb/>with ea&longs;e; and its cleer, that in the Ship there are not infinite <lb/>grains, but that they are compri&longs;ed in a certain number; and if you <lb/>take another number four or &longs;ix times bigger than that, and take <lb/>al&longs;o another of Ants equal to it, and &longs;et them to work, they &longs;hall <lb/>carry the Corn, and the Ship al&longs;o. </s> <s>It is true indeed, that it will be <lb/>needful that the number be great, as al&longs;o in my judgment that of <lb/>the <emph type="italics"/>Vacuities,<emph.end type="italics"/> which hold together the &longs;inall parts of the <lb/>Mettal.</s></p><p type="margin"> <s><margin.target id="marg1018"></margin.target><emph type="italics"/>Any finite Re&longs;i­<lb/>&longs;tance is &longs;uperable <lb/>by any the lea&longs;t <lb/>Force, multiplied.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>But though they were required to be infinite, do you <lb/>think it impo&longs;&longs;ible?</s></p><p type="main"> <s>SAGR. </s> <s>Not if the Mettal were of an infinite ma&longs;&longs;e; other­<lb/>wi&longs;e ----</s></p><p type="main"> <s>SALV. </s> <s>Otherwi&longs;e what? </s> <s>Go to, feeing we are faln upon <lb/>Paradoxes, let us &longs;ee if we can any way demon&longs;trate, how that <lb/>in a continuate finite exten&longs;ion, it is not impo&longs;&longs;ible to finde infi­<lb/>nite <emph type="italics"/>Vacuities:<emph.end type="italics"/> and then, if we gain nothing el&longs;e, yet at lea&longs;t we <lb/><arrow.to.target n="marg1019"></arrow.to.target><lb/>&longs;hall finde a &longs;olution of that mo&longs;t admirable Problem propound­<lb/>ed by <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/> among&longs;t tho&longs;e which he him&longs;elf calleth admirable, <lb/>I mean among&longs;t his <emph type="italics"/>Mechanical Que&longs;tions<emph.end type="italics"/>; and the Solution may <lb/>haply be no le&longs;&longs;e plain and concluding, than that which he him&longs;elf <lb/>brings thereupon, and different al&longs;o from that which Learned <lb/><arrow.to.target n="marg1020"></arrow.to.target><lb/><emph type="italics"/>Mon&longs;ig. </s> <s>di Guevara<emph.end type="italics"/> very acutely di&longs;cu&longs;&longs;eth. </s> <s>But it is fir&longs;t requi&longs;ite <lb/>to declare a Propo&longs;ition not toucht by others, on which the &longs;olution <lb/>of the que&longs;tion dependeth, which afterwards, if I deceive not my <lb/>&longs;elf, will draw along with it other new and admirable Notions; for <lb/>under&longs;tanding whereof the more exactly, we will give it you in <lb/>a Scheme: We &longs;uppo&longs;e, therefore an equilateral, and equian­<lb/>gled Poligon of any number of Sides at plea&longs;ure, de&longs;cribed <lb/>about this Center G; and in this example let it be a Hexagon <lb/>A B C D E F; like to which, and concentrick with the &longs;ame <lb/>mu&longs;t be di&longs;tributed another le&longs;&longs;er, which we mark H I K L M N; <pb xlink:href="069/01/020.jpg" pagenum="18"/>and let one Side of the greater A B be prolonged indeterminately <lb/>towards S, and of the le&longs;&longs;e the corre&longs;pondent Side H I is to be <lb/>produced in like manner towards the &longs;ame part, repre&longs;enting the <lb/>Line H T, parallel to A S; and let another pa&longs;&longs;e by the Center <lb/>equidi&longs;tant from the former, namely G V. </s> <s>This done, we &longs;uppo&longs;e <lb/>the greater Poligon to turn about upon the Line A S, carrying <lb/>with it the other le&longs;&longs;er Poligon. </s> <s>It is manife&longs;t, that the point B, <lb/>the term of the Side A B, &longs;tanding &longs;till, whil&longs;t the Revolution <lb/>begins, the angle A ri&longs;eth, and the point C de&longs;cendeth, de&longs;cribing <lb/>the arch C <expan abbr="q;">que</expan> &longs;o that the Side B C is applyed to the line B Q, <lb/>equal to it &longs;elf: but in &longs;uch conver&longs;ion the angle I of the le&longs;&longs;er <lb/>Poligon ri&longs;eth above the Line I T. for that I B is oblique upon <lb/>A S: nor will the point I fall upon the parallel I T, before the <lb/>point C come to Q: and by that time I &longs;hall be de&longs;cended unto <lb/>O after it had de&longs;cribed the Arch I O, without the Line H T: and <lb/>at the &longs;ame time the Side I K &longs;hall have pa&longs;s'd to O P. </s> <s>But the Cen­<lb/>ter G &longs;hall have gone all this time out of the Line G V, on which it <lb/>&longs;hal not fall, until it &longs;hall fir&longs;t have de&longs;cribed the Arch G C. </s> <s>Having <lb/>made this fir&longs;t &longs;tep, the greater Poligon &longs;hall be tran&longs;po&longs;ed to re&longs;t <lb/>with the Side B C upon the Line B <expan abbr="q;">que</expan> the Side I K of the le&longs;&longs;er <lb/>upon the Line O P, having skipt all the Line I O without touching <lb/><figure id="id.069.01.020.1.jpg" xlink:href="069/01/020/1.jpg"/><lb/>it; and the Center G &longs;hall be removed to C, making its whole <lb/>cour&longs;e without the Parallel G V: And in fine all the Figure &longs;hall <lb/>be remitted into a Po&longs;ition like the fir&longs;t; &longs;o that the Revolution <lb/>being continued, and coming to the &longs;econd &longs;tep, the Side of the <lb/>greater Poligon D C &longs;hall remove to Q X; K L of the le&longs;&longs;er (ha­<lb/>ving fir&longs;t skipt the Arch P Y) &longs;hall fall upon Y Z, and the Center <lb/>proceeding evermore without G V &longs;hall fall on it in R, after the <lb/>great skip C R. </s> <s>And in the la&longs;t place, having fini&longs;hed an entire <lb/>Conver&longs;ion, the greater Poligon will have impre&longs;&longs;ed upon A S, &longs;ix <pb xlink:href="069/01/021.jpg" pagenum="19"/>Lines equal to its Perimeter without any interpo&longs;itions or skips: <lb/>the le&longs;&longs;er Poligon likewi&longs;e &longs;hall have traced &longs;ix Lines equal to its <lb/>Perimeter, but di&longs;continued by the interpo&longs;ition of five Arches, <lb/>under which are the Chords, parts of the parallel H T not toucht <lb/>by the Poligon: And la&longs;tly, the Center G never hath toucht the <lb/>Parallel G V except in &longs;ix points. </s> <s>From hence you may compre­<lb/>hend, how that the Space pa&longs;&longs;ed by the le&longs;&longs;er Poligon, is almo&longs;t <lb/>equal to that pa&longs;&longs;ed by the greater, that is the Line H T is almo&longs;t <lb/>equal to A S, then which it is le&longs;&longs;er only the quantity of one of <lb/>the&longs;e Arches, taking the Line H T, together with all its Arches. <lb/></s> <s>Now, this which I have declared and explained to you in the exam­<lb/>ple of the&longs;e Hexagons, I would have you under&longs;tand to hold true <lb/>in all other Poligons, of what number of Sides &longs;oever they be, &longs;o <lb/>that they be like Concentrick, and Conjoyned; and that at the <lb/>Conver&longs;ion of the greater, the other, how much &longs;oever le&longs;&longs;er, be <lb/>&longs;uppo&longs;ed to revolve therewith: that is, you mu&longs;t under&longs;tand, I &longs;ay, <lb/>that the Lines by them pa&longs;&longs;ed are very near equal, computing in­<lb/>to the Space pa&longs;t by the le&longs;&longs;er, the Intervals under the little Ar­<lb/>ches not toucht by any part of the Perimeter of the &longs;aid le&longs;&longs;er Po­<lb/>ligon. </s> <s>Let therefore the greater Poligon, of a thou&longs;and Sides, pa&longs;s <lb/>round, and mea&longs;ure out a continued Line equal to its Perimeter; <lb/>and in the &longs;ame time the le&longs;s pa&longs;&longs;eth a Line almo&longs;t as long, but <lb/>compounded of a thou&longs;and Particles equal to its thou&longs;and Sides, <lb/>but di&longs;continued with the interpo&longs;ition of a thou&longs;and void Spaces: <lb/>for &longs;uch may we call them, in relation to the thou&longs;and little Lines <lb/>toucht by the Sides of the Poligon. </s> <s>And what hath been &longs;poken <lb/>hitherto admits of no doubt or &longs;cruple. </s> <s>But tell me, in ca&longs;e that <lb/>about a Center, as &longs;uppo&longs;e the point A, (in the former Scheme) <lb/>we &longs;hould de&longs;cribe two Circles concentrick, and united together; <lb/>and that from the points C and B of their Semi-Diameters, there <lb/>be drawn the Tangents C E, and B F, and by the Center A the Pa­<lb/>rallel A D; &longs;uppo&longs;ing the greater Circle to be turned upon the <lb/>Line B F, (drawn equal to its Circumference, as likewi&longs;e the other <lb/>two C E, and A D;) when it hath compleated one Revolution, <lb/>what &longs;hall the le&longs;&longs;er Circle, and Center have done? </s> <s>The Center <lb/>&longs;hall doubtle&longs;s have run over, and touched the whole Line A D, <lb/>and the le&longs;s Circumference &longs;hall with its touches have mea&longs;ured <lb/>all C E, doing the &longs;ame as did the Poligons above; and different <lb/>only in this, that the Line H T was not touched in all its Parts by <lb/>the Perimeter of the le&longs;&longs;er Poligon, but there were as many parts <lb/>left untoucht with the interpo&longs;ition of &longs;alts, or skipped &longs;paces; as <lb/>were the&longs;e parts touched by the Sides: but here in the Circles, <lb/>the Circumference of the le&longs;&longs;er Circle, never &longs;eparates from the <lb/>Line C E, &longs;o as to leave any of its parts untou cht; nor is the parts <lb/>touching of the Circumference, le&longs;s than the part toucht of the <pb xlink:href="069/01/022.jpg" pagenum="20"/>Right-line. </s> <s>Now how is it po&longs;&longs;ible that the le&longs;&longs;er Circle &longs;hould <lb/>without skips run a Line &longs;o much bigger than its Circumfe­<lb/>rence?</s></p><p type="margin"> <s><margin.target id="marg1019"></margin.target>Ari&longs;totles <emph type="italics"/>admi­<lb/>rable Problem of <lb/>two Concentrick <lb/>Circles that turn <lb/>round, and its true <lb/>re&longs;olution.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1020"></margin.target>Mon&longs;ig. </s> <s>Gueva <lb/>ra <emph type="italics"/>honourably men­<lb/>tioned.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>I was con&longs;idering whether one might not &longs;ay, that like <lb/>as the Center of the Circle trailed alone upon A D toucht, it all <lb/>being yet but one &longs;ole Point; &longs;o likewi&longs;e might the Points of the <lb/>le&longs;&longs;er Circumference, drawn by the revolution of the greater, go <lb/>gliding along &longs;ome &longs;mall part of the Line C E.</s></p><p type="main"> <s><emph type="italics"/>S<emph.end type="italics"/>ALV. </s> <s>This cannot be, for two rea&longs;ons; fir&longs;t, becau&longs;e there is <lb/>no rea&longs;on why &longs;ome of the touches like to C &longs;hould go gliding <lb/>along &longs;ome part of the Line C E, more than others: and though <lb/>there &longs;hould; &longs;uch touches being (becau&longs;e they are points) in&longs;i­<lb/>nite, the glidings along upon C E would be infinite; and &longs;o being, <lb/>they would make an infinite Line, but the Line C E is finite. </s> <s>The <lb/>other rea&longs;on is, that the greater Circle, in its Revolution continu­<lb/>ally changing contact, the le&longs;&longs;er Circle mu&longs;t of nece&longs;&longs;ity do the <lb/>like; there being no other Point but B, by which a Right Line can <lb/>be drawn to the Center A, and pa&longs;&longs;ing through C; &longs;o that the <lb/>greater Circumference changing Contact, the le&longs;s doth change it <lb/>al&longs;o; nor doth any Point of the le&longs;s touch more than one Point of <lb/>its Right-Line C E: be&longs;ides, that al&longs;o in the conver&longs;ion of the Po­<lb/>ligons, no Point of the Perimeter of the le&longs;s falls on more than one <lb/>Point of the Line, which was by the &longs;aid Perimeter traced, as may <lb/>be ea&longs;ily under&longs;tood, con&longs;idering the Line I K is parallel to B C, <lb/>whereupon, till ju&longs;t that B C fall on B R, I K continueth elevated <lb/>above I P, and toucheth it not before B C is on the very Point of <lb/>uniting with B Q, and then all in the &longs;ame in&longs;tant I K uniteth <lb/>with O P, and afterwards immediately ri&longs;eth above it again.</s></p><p type="main"> <s>SAGR. </s> <s>The bu&longs;ine&longs;s is really very intricate, nor can I think on <lb/>any Solution of it, therefore do you explain it to us as far as you <lb/>judge needful.</s></p><p type="main"> <s>SALV. </s> <s>I &longs;hould, for the evincing hereof, have recour&longs;e to the <lb/>con&longs;ideration of the fore-de&longs;cribed Poligons, the effect of which is <lb/>intelligible and already comprehended, and would &longs;ay, that like as <lb/>in the Poligons of an hundred thou&longs;and Sides, the Line pa&longs;&longs;ed and <lb/>mea&longs;ured by the Perimeter of the greater, that is by its hundred <lb/>thou&longs;and Sides continually di&longs;tended, is not con&longs;iderably bigger <lb/>than that mea&longs;ured by the hundred thou&longs;and Sides of the le&longs;s, but <lb/>with the interpo&longs;ition of an hundred thou&longs;and void &longs;paces interve­<lb/>ning; fo I would &longs;ay in the Circles (which are Poligons of innu­<lb/>merable Sides) that the Line mea&longs;ured by the infinite Sides of the <lb/>great Circle, lying continued one with another, to be equalled in <lb/>length by the Line traced by the infinite Sides of the le&longs;s, but by <lb/>the&longs;e including the interpo&longs;ition of the like number of intervening <lb/>Spaces: and like as the Sides are not quantitative, but yet infinite <pb xlink:href="069/01/023.jpg" pagenum="21"/>in number, &longs;o the interpo&longs;ing Vacuitics are not quantitative, but <lb/>infinite in number; that is, tho&longs;e are infinite Points all filled, and <lb/>the&longs;e are infinite points, part filled, and part empty. </s> <s>And here I <lb/>would have you note, that re&longs;olving, and dividing a Line into quan­<lb/>titative parts, and con&longs;equently of a finite number, it is not po&longs;&longs;ible <lb/>to di&longs;po&longs;e them into a greater extention than that which they po&longs;­<lb/>&longs;e&longs;t whil&longs;t they were continued, and connected, without the inter­<lb/>po&longs;ition of a like number of void Spaces; but imagining it to be <lb/>re&longs;olved into parts not quantitative, namely, into its infinite indivi­<lb/>&longs;ibles, we may conceive it produced to immen&longs;ity without the in­<lb/>terpo&longs;ition of quantitative void &longs;paces, but yet of infinite indivi&longs;i­<lb/>ble Vacuities. </s> <s>And this which is &longs;poken of &longs;imple lines, &longs;hould al&longs;o <lb/>be under&longs;tood of Superficies, and Solid Bodies, con&longs;idering that they <lb/>are compo&longs;ed of infinite Atomes not non-quantitative; if we would <lb/>divide them into certain quantitative parts, there's no que&longs;tion, but <lb/>that we cannot di&longs;po&longs;e them into Spaces more ample than the Solid <lb/>before occupied, unle&longs;s with the interpo&longs;ition of a certain number <lb/>of quantitative void Spaces; void, I &longs;ay, at lea&longs;t of the matter of the <lb/>Solid: but if we &longs;hould propo&longs;e the highe&longs;t, and ultimate re&longs;olution <lb/>made into the fir&longs;t, non-quantitative, but infinite fir&longs;t compoun­<lb/>ding parts, we may be able to conceive &longs;uch compounding parts <lb/>extended unto an immen&longs;e Space without the interpo&longs;ition of <lb/>quantitative void Spaces; but only of infinite non-quantitative Va­<lb/>cuities: and in this manner a man may draw out, <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> a little Ball <lb/>of Gold into a very va&longs;t expan&longs;ion without admitting any quan­<lb/>titative void Spaces; yet neverthele&longs;s we may admit the Gold to <lb/>be compounded of infinite induci&longs;&longs;ible ones.</s></p><p type="main"> <s>SIMP. </s> <s>Me thinks that in this point you go the way of tho&longs;e di&longs;­<lb/>&longs;eminated Vacuities of a certain <emph type="italics"/>Ancient Philo&longs;opher<emph.end type="italics"/> ------</s></p><p type="main"> <s>SALV. </s> <s>But you add not: [<emph type="italics"/>who denied Divine Providence:)<emph.end type="italics"/><lb/>as on &longs;uch another occa&longs;ion, &longs;ufficiently be&longs;ides his purpo&longs;e, a cer­<lb/>tain Antagoni&longs;t of our <emph type="italics"/>Accademick<emph.end type="italics"/> did &longs;ubjoyn.</s></p><p type="main"> <s>SIMP. </s> <s>I &longs;ee very well, and not without indignation, the malice <lb/>of &longs;uch contradictors; but I &longs;hall forbear the&longs;e Cen&longs;ures, not only <lb/>upon the &longs;core of Good-Manners, but becau&longs;e I know how di&longs;a­<lb/>greeing &longs;uch Tenets are to the well-tempered, and well-di&longs;po&longs;ed <lb/>mind of a per&longs;on, &longs;o Religious and Pious, yea, Orthodox and Ho­<lb/>ly, as you, Sir. </s> <s>But returning to my purpo&longs;e; I find many &longs;cruples <lb/>to ari&longs;e in my mind about your la&longs;t Di&longs;cour&longs;e, which I know not <lb/>how to re&longs;olve. </s> <s>And this pre&longs;ents its &longs;elf for one, that if the Cir­<lb/>cumferences of two Circles are equall to the two Right Lines <lb/>C E, and B F, this taken continually, and that, with the interpo&longs;i­<lb/>tion of infinite void Points; how can A D, de&longs;cribed by the Center, <lb/>which is but one &longs;ole Point, be &longs;aid to be equal to the &longs;ame, it con­<lb/>taining infinite of them? </s> <s>Again, that &longs;ame compo&longs;ing the Line of <pb xlink:href="069/01/024.jpg" pagenum="22"/>Points, the divi&longs;ible of indivi&longs;ibles, the quantitative of non-quan­<lb/>titative, is a rock very hard, in my judgment, to pa&longs;s over: And <lb/>the very admitting of Vacuity, &longs;o thorowly confuted by <emph type="italics"/>Ari&longs;totle,<emph.end type="italics"/><lb/>no le&longs;s puzleth me than tho&longs;e difficulties them&longs;elves.</s></p><p type="main"> <s>SALV. </s> <s>There be, indeed, the&longs;e and other difficulties; but re­<lb/>member, that we are among&longs;t Infinites, and Indivi&longs;ibles: tho&longs;e in­<lb/>comprehen&longs;ible by our finite under&longs;tanding for their Grandure; <lb/>and the&longs;e for their minutene&longs;s: neverthele&longs;s we &longs;ee that Humane <lb/>Di&longs;cour&longs;e will not be beat off from ruminating upon them, in <lb/>which regard, I al&longs;o a&longs;&longs;uming &longs;ome liberty, will produce &longs;ome of <lb/>my conceits, if not nece&longs;&longs;arily concluding, yet for novelty &longs;ake, <lb/>which is ever the me&longs;&longs;enger of &longs;ome wonder: but perhaps the car­<lb/>rying you &longs;o far out of your way begun, may &longs;eem to you imper­<lb/>tinent, and con&longs;equently little plea&longs;ing.</s></p><p type="main"> <s>SAGR. </s> <s>Pray you let us enjoy the benefit, and priviledge, of free <lb/>&longs;peaking which is allowed to the living, and among&longs;t friends; e&longs;pe­<lb/>cially, in things arbitrary, and not nece&longs;&longs;ary; different from Di&longs;cour&longs;e <lb/>with dead Books, which &longs;tart us a thou&longs;and doubts, and re&longs;olve not <lb/>one of them. </s> <s>Make us therefore partakers of tho&longs;e Con&longs;iderations, <lb/>which the cour&longs;e of our Conferences &longs;ugge&longs;t unto you; for we <lb/>want no time, &longs;eeing we are di&longs;engaged from urgent bu&longs;ine&longs;&longs;es, to <lb/>continue and di&longs;cu&longs;&longs;e the other things mentioned; and particular­<lb/>ly, the doubts, hinted by <emph type="italics"/>Simplicius,<emph.end type="italics"/> mu&longs;t by no means e&longs;cape us.</s></p><p type="main"> <s>SAIV. </s> <s>It &longs;hall be &longs;o, &longs;ince it plea&longs;eth you: and beginning at <lb/>the fir&longs;t, which was, how it's po&longs;&longs;ible to imagine that a &longs;ingle Point <lb/>is equal to a Line; in regard I can do no more for the pre&longs;ent, I <lb/>will attempt to &longs;atisfie, or, at lea&longs;t, qualifie one improbability with <lb/>another like it, or greater; as &longs;ome times a Wonder is &longs;wallowed <lb/>up in a Miracle. </s> <s>And this &longs;hall be by &longs;hewing you two equal Su­<lb/>perficies, and at the &longs;ame time two Bodies, likewi&longs;e equal, and <lb/>placed upon tho&longs;e Superficies as their Ba&longs;es; and that go (both <lb/>the&longs;e and tho&longs;e) continually and equally dimini&longs;hing in the &longs;elf­<lb/><arrow.to.target n="marg1021"></arrow.to.target><lb/>&longs;ame time, and that in their remainders re&longs;t alwaies equal between <lb/>them&longs;elves, and (la&longs;tly) that, as well Super&longs;icies, as Solids, deter­<lb/>mine their perpetual precedent equalities, one of the Solids with <lb/>one of the Superficies in a very long Line; and the other Solid <lb/>with the other Superficies in a &longs;ingle Point: that is, the latter in <lb/>one Point alone, the other in infinite.</s></p><p type="margin"> <s><margin.target id="marg1021"></margin.target><emph type="italics"/>The equal Super­<lb/>ficies of two Solids <lb/>continually &longs;ub­<lb/>&longs;tracting from <lb/>them both equal <lb/>parts, are reduced, <lb/>the one into the <lb/>Circumference of a <lb/>Circle, and the o­<lb/>ther into a Point.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>S<emph.end type="italics"/>AGR. </s> <s>An admirable propo&longs;al, really, yet let us hear you ex­<lb/>plain and demon&longs;trate it.</s></p><p type="main"> <s>SALV. </s> <s>It is nece&longs;&longs;ary to give you it in Figure, becau&longs;e the proof <lb/>is purely Geometrical. </s> <s>Therefore &longs;uppo&longs;e the Semicircle A F B, <lb/>and its Center to be C, and about it de&longs;cribe the Rectangle <lb/>A D E B, and from the Center unto the Points D and E let there <lb/>be drawn the Lines C D, and C E; Then drawing the Semi-Dia­<pb xlink:href="069/01/025.jpg" pagenum="23"/>meter C F, perpendicular to one of the two Lines A B, or D E <lb/>and immoveable; we &longs;uppo&longs;e all this Figure to turn round about <lb/>that Perpendicular: It is manife&longs;t, that there will be de&longs;cribed by <lb/>the Parallelogram A D E B, a Cylinder; by the Semi-circle A F B, <lb/>an Hemi-Sphære; and by the Triangle C D E a Cone. </s> <s>This pre­<lb/>&longs;uppo&longs;ed, I would have you imagine the Hemi&longs;phære to be taken <lb/>away, leaving behind the Cone, and that which &longs;hall remain of <lb/>the Cylinder; which for the Figure, which it &longs;hall retain like to a <lb/>Di&longs;h, we will hereafter call a Di&longs;h: touching which, and the <lb/>Cone, we will &longs;ir&longs;t demon&longs;trate that they are equal; and next <lb/>a Plain being drawn parallel to the Circle, which is the foot or <lb/>Ba&longs;e of the Di&longs;h, who&longs;e Diameter is the Line D E, and its Center <lb/>F; we will demon&longs;trate, that &longs;hould the &longs;aid Plain pa&longs;s, <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> by <lb/>the Line G H, cutting the Di&longs;h in the points G I, and O N; and <lb/>the Cone in the points H and L; it would cut the part of the <lb/>Cone C H L, equal alwaies to the part of the Di&longs;h, who&longs;e Profile <lb/>is repre&longs;ented to us by the Triangles G A I, and B O N: and more­<lb/>over we will prove the Ba&longs;e al&longs;o of the &longs;ame Cone, (that is the <lb/>Circle, who&longs;e Diameter is H L) to be equal to that circular Su­<lb/>perficies, which is Ba&longs;e of the part of the Di&longs;h; which is, as we <lb/>may &longs;ay, a Rimme as broad as G I; (note here by the way what <lb/>Mathematical Definitions are: they be an impo&longs;ition of names, or, <lb/>we may &longs;ay, abreviations of &longs;peech, ordain'd and introduced to <lb/>prevent the trouble and pains, which you and I meet with, at pre­<lb/>&longs;ent, in that we have not agreed together to call <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> this Super­<lb/>ficies a circular Rimme, and that very &longs;harp Solid of the Di&longs;h a <lb/>round Razor:) now how&longs;oever you plea&longs;e to call them, it &longs;ufficeth <lb/>you to know, that the Plain produced to any di&longs;tance at plea&longs;ure, <lb/>&longs;o that it be parallel to the Ba&longs;e, <emph type="italics"/>viz.<emph.end type="italics"/> to the Circle who&longs;e Diame­<lb/>ter D E cuts alwaies the two Solids, namely, the part of the Cone <lb/>C H L, and the upper part of the Di&longs;h equal to one another: and <lb/>likewi&longs;e the two Superficies, Ba&longs;is of the &longs;aid Solids, <emph type="italics"/>viz.<emph.end type="italics"/> the &longs;aid <lb/>Rimme, and the Circle H L, equal al&longs;o to one another. </s> <s>Whence <lb/>followeth the forementioned Wonder; namely, that if we &longs;hould <lb/>&longs;uppo&longs;e the cutting-plain to be <lb/>&longs;ucce&longs;&longs;ively rai&longs;ed towards the <lb/><figure id="id.069.01.025.1.jpg" xlink:href="069/01/025/1.jpg"/><lb/>Line A B, the parts of the Solid <lb/>cut are alwaies equall, as al&longs;o the <lb/>Superficies, that are their Ba&longs;es, <lb/>are evermore equal; and, in <lb/>fine, rai&longs;ing the &longs;aid Plain higher <lb/>and higher, the two Solids (ever <lb/>equal) as al&longs;o their Ba&longs;es, (Su­<lb/>perficies ever equal) &longs;hall one couple of them terminate in a Cir­<lb/>cumference of a Circle, and the other couple in one &longs;ole point; <pb xlink:href="069/01/026.jpg" pagenum="24"/>for &longs;uch are the upper Verge or Rim of the Di&longs;h, and the Vertex <lb/>of the Cone. </s> <s>Now whil&longs;t that in the diminution of the two So­<lb/>lids, they till the very la&longs;t maintain their equality to one another, it <lb/>is, in my thoughts, proper to &longs;ay, that the highe&longs;t and ultimate terms <lb/>of &longs;uch Diminutions are equal, and not one infinitely bigger than <lb/>the other. </s> <s>It &longs;eemeth therefore, that the Circumference of an im­<lb/>men&longs;e Circle may be &longs;aid to be equal to one &longs;ingle point; and <lb/>this that befalls in Solids, holdeth likewi&longs;e in the Superficies their <lb/>Ba&longs;es; that they al&longs;o in the common Diminution con&longs;erving al­<lb/>waies equality, in fine, determine at the in&longs;tant of their ultimate <lb/>Diminution the one, (that is, that of the Di&longs;h) in their Circum­<lb/>ference of a Circle, the other (to wit, that of the Cone) in one <lb/>&longs;ole point. </s> <s>And why may not the&longs;e be called equal, if they be the <lb/>la&longs;t remainders, and foot&longs;teps left by equal Magnitudes? </s> <s>And note <lb/>again, that were &longs;uch Ve&longs;&longs;els capable of the immen&longs;e Cœle&longs;tial <lb/>Hemi&longs;pheres: both their upper Rims, and the points of the contai­<lb/>ned Cones (keeping evermore equally to one another) would fi­<lb/>nally determine, tho&longs;e, in Circumferences equal to tho&longs;e of the <lb/>greate&longs;t Circles of the Cœle&longs;tial Orbes, and the&longs;e in &longs;implo points. <lb/></s> <s>Whence, according to that which &longs;uch Speculations per&longs;wade us <lb/>to, all Circumferences of Circles, how unequal &longs;oever, may be <lb/>&longs;aid to be equal to one another, and each of them equal to one &longs;ole <lb/>point.</s></p><p type="main"> <s>SAGR. </s> <s>The Speculation is, in my e&longs;teem, &longs;o quaint and curi­<lb/>ous, that, for my part, though I could, yet would I not oppo&longs;e it, <lb/>for I take it for a piece of Sacriledge to deface &longs;o fine a Structure, <lb/>by &longs;purning at it with any pedantick contradiction; yet for our en­<lb/>tire &longs;atisfaction, give us the proof (which you &longs;ay is Geometrical) <lb/>of the equality alwaies retained between tho&longs;e Solids, and tho&longs;e <lb/>their Ba&longs;es, which I think mu&longs;t needs be very &longs;ubtil, the philo&longs;o­<lb/>phical Contemplation being &longs;o nice, which depends on the &longs;aid <lb/>Conclu&longs;ion.</s></p><p type="main"> <s>SALV. </s> <s>The Demon&longs;tration is but &longs;hort, and ea&longs;ie. </s> <s>Let us keep <lb/>to the former Figure, in which the Angle I P C being a Right An­<lb/>gle, the Square of the Semi-Diameter I C is equal to the two <lb/>Squares of the Sides I P, and P C. </s> <s>But the Semi-Diameter I C, is <lb/>equal to A C, and this to G P; and C P is equal to P H; therefore <lb/>the Square of the Line G P is equal to the two Squares of I P, and <lb/>P H, and the Quadruple to the Quadruples; that is, the Quadrate <lb/>of the Diameter G N is equal to the two Quadrates I O, and H L: <lb/>and becau&longs;e Circles are to each other, as the Squares of their Dia­<lb/>meters; the Circle who&longs;e Diameter is G N, &longs;hall be equall to the <lb/>two Circles who&longs;e Diameters are I O, and H L; and taking away <lb/>the Common Circle, who&longs;e Diameter is I O; the re&longs;idue of the <lb/>Circle G N &longs;hall be equal to the Circle, who&longs;e Diameter is H L. <pb xlink:href="069/01/027.jpg" pagenum="25"/>And this is as to the fir&longs;t part: Now as for the other part, we will, <lb/>for the pre&longs;ent, omit its Demon&longs;tration, as well becau&longs;e that if you <lb/>would &longs;ee it, you &longs;hall find it in the twelfth Propo&longs;ition of the Se­<lb/><arrow.to.target n="marg1022"></arrow.to.target><lb/>cond Book <emph type="italics"/>De centro Gravitatis Solidorum,<emph.end type="italics"/> publi&longs;hed by <emph type="italics"/>Signeur <lb/>Lucas Valerius,<emph.end type="italics"/> the new <emph type="italics"/>Archimedes<emph.end type="italics"/> of our Age; who upon ano­<lb/>ther occa&longs;ion hath made u&longs;e of it; as becau&longs;e in our ca&longs;e it &longs;uffi­<lb/>ceth to have &longs;een, how the Superficies, already explained, are ever­<lb/>more equal; and that alwaies dimini&longs;hing equally, they in the end <lb/>determine, one in a &longs;ingle point, and the other in the Circumfe­<lb/>rence of a Circle, be it never-&longs;omuch bigger, for in this lyeth our <lb/>Wonder.</s></p><p type="margin"> <s><margin.target id="marg1022"></margin.target>Lucas Valerius, <lb/><emph type="italics"/>the other<emph.end type="italics"/> Archi­<lb/>chimedes <emph type="italics"/>of our <lb/>Age, hath written <lb/>admirably,<emph.end type="italics"/> De <lb/>Centro Gravita­<lb/>tis Solidorum.</s></p><p type="main"> <s>SAGR. </s> <s>The Demon&longs;tration is as ingenious, as the reflection <lb/>grounded upon it is admirable. </s> <s>Now let us hear &longs;omewhat about <lb/>the other Doubt &longs;ugge&longs;ted by <emph type="italics"/>Simplicius,<emph.end type="italics"/> if you have any particu­<lb/>lars worth note to hint thereupon, but I &longs;hould incline to think it <lb/>impo&longs;&longs;ible to be, in regard it is a Controver&longs;ie that hath been &longs;o <lb/>canva&longs;&longs;ed.</s></p><p type="main"> <s>SALV. </s> <s>You &longs;hall have &longs;ome of my particular thoughts thereon; <lb/>fir&longs;t repeating what but even now I told you, namely, that Infini­<lb/>ty alone, as al&longs;o Indivi&longs;ibility, are things incompre hen&longs;ible to us: <lb/>now think how they will be conjoyned together: and yet if you <lb/>would compound the Line of indivi&longs;ible points, you mu&longs;t make <lb/>them infinite; and thus it will be requi&longs;ite to apprehend in the <lb/>&longs;ame in&longs;tant both Infinite, and Indivi&longs;ible. </s> <s>The things that ar &longs;e­<lb/>veral times have come into my mind, on this occa&longs;ion, are many; <lb/>part whereof, and the more con&longs;iderable, it may be, I cannot upon <lb/>&longs;uch a &longs;udden remember; but it may happen, that in the &longs;equal <lb/>of the Di&longs;cour&longs;e, coming to put que&longs;tions and doubts to you, and <lb/>particularly to <emph type="italics"/>Simplicius,<emph.end type="italics"/> they may, on the other &longs;ide, re-mind <lb/>me of that, which without &longs;uch excitement would have lain dor­<lb/>mant in my Fancy: and therefore, with my wonted freedom, per­<lb/>mit me that I produce any wild conjectures, for &longs;uch may we fitly <lb/>call them in compari&longs;on of &longs;upernatural Doctrines, the only true <lb/>and certain determiners of our Controver&longs;ies, and unerring guides <lb/>in our ob&longs;cure, and dubious paths, or rather Laberinths.</s></p><p type="main"> <s>Among&longs;t the fir&longs;t In&longs;tances that are wont to be produced <lb/><arrow.to.target n="marg1023"></arrow.to.target><lb/>again&longs;t tho&longs;e that compound <emph type="italics"/>Continuum<emph.end type="italics"/> of Indivi&longs;ibles, this is u&longs;u­<lb/>ally one; That an Indivi&longs;ible, added to another Indivi&longs;ible, produ­<lb/>ceth not a thing divi&longs;ible; for if that were &longs;o, it would follow, that <lb/>even the Indivi&longs;ibles were divi&longs;ible: for if two Indivi&longs;ibles, as for <lb/>example, two Points conjoyned, &longs;hould make a Quantity that <lb/>&longs;hould be a divi&longs;ible Line, much more &longs;uch &longs;hould one be that is <lb/>compounded of three, five, &longs;even, or others, that are odd num­<lb/>bers; the which Lines, being to be cut in two equal parts, render <lb/>divi&longs;ible that Indivi&longs;ible which was placed in the middle. </s> <s>In this <pb xlink:href="069/01/028.jpg" pagenum="26"/>and other Objections of this kind you may &longs;atisfie the propo&longs;er of <lb/>them, telling him, that neither two Indivi&longs;ibles, nor ten, nor an <lb/>hundred, no, nor a thou&longs;and can compound a Magnitude divi&longs;ible, <lb/>and quantitative, but being infinite they may.</s></p><p type="margin"> <s><margin.target id="marg1023"></margin.target>Continuum <emph type="italics"/>com­<lb/>pounded of Indivi­<lb/>&longs;ibles.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. </s> <s>Here already ri&longs;eth a doubt, which I think unre&longs;olvable; <lb/>and it is, that we being certain to find Lines one bigger than ano­<lb/>ther, although both contain infinite Points, we mu&longs;t of nece&longs;&longs;ity <lb/>confe&longs;s, that we have found in the &longs;ame Species a thing bigger than <lb/>infinite; becau&longs;e the Infinity of the Points of the greater Line, &longs;hall <lb/>exceed the Infinity of the Points of the le&longs;&longs;er. </s> <s>Now this a&longs;&longs;igning <lb/>of an Infinite bigger than an Infinite is, in my opinion, a conceit <lb/>that can never by any means be apprehended.</s></p><p type="main"> <s>SALV. </s> <s>The&longs;e are &longs;ome of tho&longs;e difficulties, which re&longs;ult from <lb/>the Di&longs;cour&longs;es that our finite Judgments make about Infinites, gi­<lb/>ving them tho&longs;e attributes which we give to things finite and ter­<lb/>minate; which I think is inconvenient; for I judge that the&longs;e <lb/>terms of Majority, Minority, and Equality &longs;ute not with Infinites, <lb/>of which we cannot &longs;ay that one is greater, or le&longs;s, or equal to ano­<lb/>ther: for proof of which there cometh to my mind a Di&longs;cour&longs;e, <lb/>which, the better to explain, I will propound by way of Interroga­<lb/>tories to <emph type="italics"/>Simplicius<emph.end type="italics"/> that &longs;tarted the que&longs;tion.</s></p><p type="main"> <s>I &longs;uppo&longs;e that you very well under&longs;tand which are Square Num­<lb/>bers, and which not Square.</s></p><p type="main"> <s>SIMP. </s> <s>I know very well, that the Square Number is that which <lb/>proceeds from the multiplication of another Number into it &longs;elf; <lb/>and &longs;o four, and nine, are Square Numbers, that ari&longs;ing from two, <lb/>and this from three multiplied into them&longs;elves.</s></p><p type="main"> <s>SALV. </s> <s>Very well; And you know al&longs;o, that as the Products are <lb/>called Squares: the Produ&longs;ors, that is, tho&longs;e that are multiplied, are <lb/>called Sides, or Roots; and the others, which proceed not from <lb/>Numbers multiplied into them&longs;elves, are not Squares. </s> <s>So that if I <lb/>&longs;hould &longs;ay, all Numbers comprehending the Square, and the not <lb/>Square Numbers, are more than the Square alone, I &longs;hould &longs;peak a <lb/>mo&longs;t unque&longs;tionable truth: Is it not &longs;o?</s></p><p type="main"> <s>SIMP. </s> <s>It cannot be denied.</s></p><p type="main"> <s>SALV. </s> <s>Farther que&longs;tioning, if I ask you how many are the <lb/>Numbers Square, you can an&longs;wer me truly, that they be as many, <lb/>as are their propper Roots; &longs;ince every Square hath its Root, and <lb/>every Root its Square, nor hath any Square more than one &longs;ole <lb/>Root, or any Root more than one &longs;ole Square.</s></p><p type="main"> <s>SIMP. True.<lb/><arrow.to.target n="marg1024"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1024"></margin.target><emph type="italics"/>An Infinite Num­<lb/>ber, as it contains <lb/>infinite Square <lb/>and Cupe Roots, &longs;o <lb/>it conta neth infi­<lb/>nite Square and <lb/>Cube Numbers.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>But if I &longs;hall demand how many Roots there be, you <lb/>cannot deny but that they be as many as all Numbers, &longs;ince there <lb/>is no Number that is not the Root of &longs;ome Square: And this be­<lb/>ing granted, it is requi&longs;ite to affirm, that Square Numbers are as <pb xlink:href="069/01/029.jpg" pagenum="27"/>many as their Roots, and Roots are all Numbers: and yet in the <lb/>beginning we &longs;aid, that all Numbers are far more than all Squares, <lb/>the greater part not being Squares: and yet neverthele&longs;s the num­<lb/>ber of the Squares goeth dimini&longs;hing alwaies with greater propor­<lb/>tion, by how much the greater number it ri&longs;eth to; for in an hun­<lb/>dred there are ten Squares, which is as much as to &longs;ay, the tenth <lb/>part are Squares: in ten thou&longs;and only the hundredth part are <lb/>Squares: in a Million only the thou&longs;andth, and yet in an Infinite <lb/>Number, if we are able to comprehend it, we may &longs;ay the Squares <lb/>are as many, as all Numbers put together.</s></p><p type="main"> <s>SAGR. </s> <s>What is to be re&longs;olved then on this occa&longs;ion?</s></p><p type="main"> <s>SALV. </s> <s>I &longs;ee no other deci&longs;ion that it may admit, but to &longs;ay, <lb/>that all Numbers are infinite, Squares are infinite, their Roots are <lb/>infinite; and that neither is the multitude of Squares le&longs;s than all <lb/>Numbers, nor this greater than that: and in conclu&longs;ion, that the <lb/>Attributes of Equality, Majority, and Minority, have no place <lb/>in Infinites, but only in terminate quantities. </s> <s>And therefore when <lb/><emph type="italics"/>Simplicius<emph.end type="italics"/> propoundeth to me many unequal <emph type="italics"/>L<emph.end type="italics"/>ines, and demand­<lb/>eth of me, how it can be, that in the greater there are no more <lb/>Points than in the le&longs;s: I an&longs;wer him, That there are neither more, <lb/>nor le&longs;s, nor ju&longs;t &longs;o many; but in each of them infinite. </s> <s>Or if I <lb/>had an&longs;wered him, that the Points in one, are as many as there are <lb/>Square Numbers; in another bigger, as many as all Numbers; in <lb/>a le&longs;s, as many as the Cubick Numbers, might not I have given &longs;a­<lb/>tisfaction, by a&longs;&longs;igning more to one, than to another, and yet to <lb/>every one infinite? </s> <s>And thus much as to the fir&longs;t difficulty.</s></p><p type="main"> <s>SAGR. Hold, I pray you, and give me leave to add unto what hath <lb/>been &longs;poken hitherto, a thought which I ju&longs;t now light on, and it <lb/>is this, that granting what hath been &longs;aid, me-thinks, that not on­<lb/>ly it's improper to &longs;ay, one Infinite is greater than another Infinite, <lb/>but al&longs;o, that it's greater than a Finite; for if an Infinite Number <lb/>were greater, <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> than a Million; it would thereupon follow, <lb/>that pa&longs;&longs;ing from the Million to others, and &longs;o to others continual­<lb/>ly greater, one &longs;hould pa&longs;s on towards Infinity; which is not &longs;o: but <lb/>on the contrary, to how much the greater Numbers we go, &longs;o <lb/>much the more we depart from Infinite Number; becau&longs;e in Num­<lb/>bers, the greater you take, &longs;o much the rarer and rarer alwaies are <lb/>Square Numbers contained in them; but in Infinite Number the <lb/>Squares can be no le&longs;s than all Numbers, as but ju&longs;t now was con­<lb/>cluded: therefore the going towards Numbers alwaies greater, and <lb/>greater, is a departing farther from Infinite Number.</s></p><p type="main"> <s>SALV. </s> <s>And &longs;o by your ingenious Di&longs;cour&longs;e we may conclude, <lb/>that the Attributes of Greater, Le&longs;&longs;er, or Equal, have no place, <lb/>not only among&longs;t Infinites; but al&longs;o betwixt Infinites, and Fi­<lb/>nites.</s></p><pb xlink:href="069/01/030.jpg" pagenum="28"/><p type="main"> <s>I pa&longs;s now to another Con&longs;ideration; and it is, that in regard <lb/>that the Line, and every continued quantity are divideable conti­<lb/>nually into divi&longs;ibles, I &longs;ee not how we can avoid granting that the <lb/>compo&longs;ition is of infinite Indivi&longs;ibles: becau&longs;e a divi&longs;ion and &longs;ub­<lb/>divi&longs;ion that may be pro&longs;ecuted perpetually &longs;uppo&longs;eth that the <lb/>parts are infinite; for otherwi&longs;e the &longs;ubdivi&longs;ion would be termina­<lb/>ble: and the parts being Infinite, it followeth of con&longs;equence <lb/>that they be non-quantitative; for infinite quantitative parts make <lb/>an infinite exten&longs;ion: and thus we have a <emph type="italics"/>Continuum<emph.end type="italics"/> compoun­<lb/>ded of infinite Indivi&longs;ibles.</s></p><p type="main"> <s>SIMP. </s> <s>But if we may continually pro&longs;ecute the divi&longs;ion in <lb/>quantitative parts, what need have we, for &longs;uch re&longs;pect, to intro­<lb/>duce the non-quantitative?</s></p><p type="main"> <s>SALV. </s> <s>The very po&longs;&longs;ibility of perpetually pro&longs;ecuting the di­<lb/>vi&longs;ion in quantitative parts induceth the nece&longs;&longs;ity of the compo&longs;iti­<lb/>on of infinite non-quantitative. </s> <s>Therefore, coming clo&longs;er to you, <lb/>I demand you to tell me re&longs;olutely, whether the quantitative parts <lb/>in <emph type="italics"/>Continuum<emph.end type="italics"/> be in your judgment finite or infinite?</s></p><p type="main"> <s>SIMP. </s> <s>I reply, that they are both Infinite, and Finite; Infinite <lb/>in Power, and Finite in Act. </s> <s>Infinite in Power, that is, before the <lb/>Divi&longs;ion; but Finite in Act, that is, after they are divided: for the <lb/>parts are not actually under&longs;tood to be in the whole, till it is di­<lb/>vided, or at lea&longs;t marked; otherwi&longs;e we &longs;ay that they are in <lb/>Power.</s></p><p type="main"> <s>SALV. </s> <s>So that a Line <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> twenty foot long, is not &longs;aid to <lb/>contain twenty Lines of one foot a piece, actually, but only after <lb/>it is divided into twenty equal parts: but is till then &longs;aid to contain <lb/>them only in power. </s> <s>Now be it as you plea&longs;e; and tell me whe­<lb/>ther, when the actual Divi&longs;ion of &longs;uch parts is made, that fir&longs;t <lb/>whole encrea&longs;eth or dimini&longs;heth, or el&longs;e continueth of the &longs;ame <lb/>bigne&longs;s?</s></p><p type="main"> <s>SIMP. </s> <s>It neither encrea&longs;eth, nor dimini&longs;heth.</s></p><p type="main"> <s>SALV. </s> <s>So I think al&longs;o. </s> <s>Therefore the quantitative parts in <emph type="italics"/>Con­<lb/>tinuum<emph.end type="italics"/> quantity, be they in Act, or be they in Power, make not its <lb/>quantity bigger or le&longs;&longs;er: but it is very plain that the&longs;e quantita­<lb/>tive parts, actually contained in their whole, if they be infinite, <lb/>make it an infinite Magnitude; therefore quantitative parts, <lb/>though infinite only in power, cannot be contained, but only in an <lb/>infinite Magnitude: therefore in a finite Magnitude infinite quan­<lb/>titative parts can be contained neither in Act, nor Power.</s></p><p type="main"> <s>SAGR. </s> <s>How then can it be true, that the <emph type="italics"/>Continuum<emph.end type="italics"/> may be <lb/>ince&longs;&longs;antly divided into parts &longs;till capable of new divi&longs;ions?</s></p><p type="main"> <s>SALV. </s> <s>It &longs;eems that that di&longs;tinction of Power, and Act, makes <lb/>that fea&longs;ible one way, which another way would be impo&longs;&longs;ible. <lb/></s> <s>But I will &longs;ee to adju&longs;t the&longs;e matters by making another account: <pb xlink:href="069/01/031.jpg" pagenum="29"/>And to the Que&longs;tion, which was put, Whether the quantitative <lb/>parts in a terminated <emph type="italics"/>Continuum<emph.end type="italics"/> be finite or infinite; I will an&longs;wer <lb/>directly contrary to that which <emph type="italics"/>Simplicius<emph.end type="italics"/> replied, namely, that <lb/>they be neither finite, nor infinite.</s></p><p type="main"> <s>SIMP. </s> <s>I &longs;hould never have found &longs;uch an an&longs;wer, not imagi­<lb/>ning that there was any mean term between finite and infinite; <lb/>&longs;o that the divi&longs;ion or di&longs;tinction which makes a thing to be either <lb/>Finite, or Infinite, is imperfect and deficient.</s></p><p type="main"> <s>SALV. </s> <s>In my opinion it is; and &longs;peaking of ^{*} Di&longs;crete Quan­</s></p><p type="main"> <s><arrow.to.target n="marg1025"></arrow.to.target><lb/>tities, me thinks that there is a third mean term between Finite and <lb/>Infinite, which is that which an&longs;wereth to every a&longs;&longs;igned Number: <lb/>So that being demanded in our pre&longs;ent ca&longs;e, Whether the quanti­<lb/>tative parts in <emph type="italics"/>Continuum<emph.end type="italics"/> be Finite, or Infinite, the mo&longs;t congru­<lb/>ous reply is to &longs;ay, that they are neither Finite, nor Infinite, but &longs;o <lb/>many, as that they <emph type="italics"/>An&longs;wer<emph.end type="italics"/> to any number a&longs;&longs;igned: the which to <lb/>do, it is nece&longs;&longs;ary that they be not comprehended in a limited <lb/>Number, for then they would not an&longs;wer to a greater: nor, again, <lb/>is it nece&longs;&longs;ary, that they be infinite, for no a&longs;&longs;igned Number is infi­<lb/>nite. </s> <s>And thus at the plea&longs;ure of the Demander, a Line being <lb/>propounded, we may be able to a&longs;&longs;ign in it an hundred quantita­<lb/>tive parts, or a thou&longs;and, or an hundred thou&longs;and, according to <lb/>the number which he be&longs;t likes; &longs;o that it be not divided into in­<lb/>finite. </s> <s>I grant therefore to the Philo&longs;ophers, that <emph type="italics"/>Continuum<emph.end type="italics"/> con­<lb/>taineth as many quantitative parts as they plea&longs;e, and grant them <lb/>that it containeth the &longs;ame either in Act, or in Power, which they <lb/>be&longs;t like: but this I add again, that in like manner, as in a Line of <lb/>ten yards, there are contained ten Lines of one yard a piece, and <lb/>thirty Lines of a foot a piece, and three hundred and &longs;ixty Lines <lb/>of an inch a piece, &longs;o it contains infinite Points; denominate them <lb/>in Act, or in Power, as you will: and I remit my &longs;elf in this matter <lb/>to your opinion and judgment, <emph type="italics"/>Simplicius.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1025"></margin.target><emph type="italics"/>Quantitative parts <lb/>in Di&longs;crete Quan­<lb/>tity are neither fi­<lb/>nite nor infinite, <lb/>but an&longs;werable to <lb/>every given Num­<lb/>ber.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. </s> <s>I cannot but commend your Di&longs;cour&longs;e: but am great­<lb/>ly afraid, that this parity of the Points, being contained in the like <lb/>manner as the quantitative parts, will not agree with ab&longs;olute ex­<lb/>actne&longs;s; nor &longs;hall it be &longs;o ea&longs;ie a matter for you to divide the gi­<lb/>ven Line into infinite Points, as for tho&longs;e Philo&longs;ophers to divide it <lb/>into ten yards, or thirty feet, nay, I hold it wholly impo&longs;&longs;ible to <lb/>effect &longs;uch a divi&longs;ion: &longs;o that this will be one of tho&longs;e Powers that <lb/>are never reduced to Act.</s></p><p type="main"> <s>SALV. </s> <s>The trouble, pains, and long time without which a <lb/>thing is not fea&longs;ible, render it not impo&longs;&longs;ible; for I think al&longs;o, that <lb/>you cannot &longs;o ea&longs;ily effect a divi&longs;ion to be made of a Line into a <lb/>thou&longs;and parts; and much le&longs;s being to divide it into 937, or &longs;ome <lb/>other great Prime Number. </s> <s>But if I di&longs;patch this, which you, it may <lb/>be, judge an impo&longs;&longs;ible divi&longs;ion, in as &longs;hort a time, as another <pb xlink:href="069/01/032.jpg" pagenum="30"/>would require to divide it into forty, you will be content more <lb/>willingly to admit of it in our future Di&longs;cour&longs;e?</s></p><p type="main"> <s>SIMP. </s> <s>I am plea&longs;ed with your way of arguing, as you now do <lb/>mix it with &longs;ome plea&longs;antne&longs;s: and to your que&longs;tion I reply, that <lb/>the facility would &longs;eem more than &longs;ufficient, if the re&longs;olving it into <lb/>Points were but as ea&longs;ie, as to divide it into a thou&longs;and parts.</s></p><p type="main"> <s>SALV. </s> <s>Here I will tell you a thing, which haply will make you <lb/>wonder in this matter of going about, or being able to re&longs;olve the <lb/>Line into its Infinites, keeping that order which others ob&longs;erve in <lb/>dividing it into forty, &longs;ixty, or an hundred parts; namely, by di­<lb/>viding it fir&longs;t into two, then into four: in which order he that <lb/>&longs;hould think to find its infinite Points would gro&longs;ly delude him&longs;elf; <lb/>for by that progre&longs;&longs;ion, though continued to eternity, he &longs;hould <lb/>never arrive to the divi&longs;ion of all its quantitative parts: yea, he is <lb/>in that way &longs;o far from being able to arrive at the intended term <lb/>of Indivi&longs;ibility, that he rather goeth farther from it; and whil&longs;t <lb/>he thinks by continuing the divi&longs;ion, and multiplying the multi­<lb/>tudes of the parts, to approach to Infinite, I am of opinion, that he <lb/>more and more removes from it: and my rea&longs;on is this; In the <lb/>Di&longs;cour&longs;e, we had even now, we concluded, that, in an infinite <lb/>Number, there was, of nece&longs;&longs;ity, as many Square, or Cube Num­<lb/>bers, as there were Numbers; &longs;ince that tho&longs;e and the&longs;e were as ma­<lb/>ny as their Roots, and Roots comprehend all Numbers: Next we <lb/>did &longs;ee, that the greater the Numbers were that were taken, the <lb/>&longs;eldomer are their Squares to be found in them, and &longs;eldomer yet <lb/>their Cubes: Therefore it is manife&longs;t, that the greater the Number <lb/>is to which you pa&longs;s, the farther you remove from Infinite Num­<lb/>ber: from whence it followeth, that turning backwards, (&longs;eeing <lb/>that &longs;uch a progre&longs;&longs;ion more removes us from the de&longs;ired term) if <lb/><arrow.to.target n="marg1026"></arrow.to.target><lb/>any number may be &longs;aid to be infinite it is the Unite: and, indeed, <lb/>there are in it tho&longs;e conditions, and nece&longs;&longs;ary qualities of the Infi­<lb/>nite Number, I mean, of containing in it as many Squares as Cubes, <lb/>and as Numbers.</s></p><p type="margin"> <s><margin.target id="marg1026"></margin.target><emph type="italics"/>The Unite of all <lb/>Numbers may <lb/>mo&longs;t properly be <lb/>&longs;aid to be Infinite.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. </s> <s>I do not apprehend very well, how this bu&longs;ine&longs;s &longs;hould <lb/>be under&longs;tood.</s></p><p type="main"> <s>SALV. </s> <s>The thing hath no difficulty at all in it, for the Unite <lb/>is a Square, a Cube, a Squared Square, and all other Powers; nor <lb/>is there any particular what&longs;oever e&longs;&longs;ential to the Square, or to the <lb/>Cube, which doth not agree with the Unite; as <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> one proper­<lb/>ty of two Square-numbers is to have between them a Number <lb/>mean-proportional; take any Square number for one of the terms, <lb/>and the Unite for the other, and you &longs;hall likewi&longs;e ever find be­<lb/>tween them a Number Mean-proportional. </s> <s>Let the two Square <lb/>Numbers be 9 and 4, you &longs;ee that between 9 and 1 the Mean­<lb/>proportional is 3, and between 4 and 1 the Mean-proportional <pb xlink:href="069/01/033.jpg" pagenum="31"/>is 2, and between the two Squares 9 and 4, 6 is the Mean. </s> <s>The <lb/>property of Cubes is to have nece&longs;&longs;arily between them two Num­<lb/>bers Mean-proportional. </s> <s>Suppo&longs;e 8, and 27, the Means between <lb/>them are 12 and 18; and between the Unite and 8 the Means <lb/>are 2 and 4; betwixt the Unite and 27 there are 3, and 9. We <lb/>therefore conclude, <emph type="italics"/>That there is no other Infinite Number but the <lb/>Vnite.<emph.end type="italics"/> And the&longs;e be &longs;ome of tho&longs;e Wonders, that &longs;urmount the <lb/>comprehen&longs;ion of our Imagination, and that advertize us how ex­<lb/>ceedingly they err, who di&longs;cour&longs;e about Infinites with tho&longs;e very <lb/>Attributes, that are u&longs;ed about Finites; the Natures of which have <lb/>no congruity with each other. </s> <s>In which affair I will not conceal <lb/>from you an admirable accident, that I met with &longs;ome time &longs;ince, <lb/>explaining the va&longs;t difference, yea, repugnance and contrariety of <lb/>Nature, that a terminate quantity would incur by changing or pa&longs;­<lb/>&longs;ing into Infinite. </s> <s>We a&longs;&longs;ign this Right Line A B, of any length at <lb/>plea&longs;ure, and any point in the &longs;ame, as C being taken, dividing it <lb/>into two unequal parts: I &longs;ay, that many couples Lines, (hold­<lb/>ing the &longs;ame proportion between them&longs;elves as have the parts <lb/>A C, and B C,) departing from the terms A and B to meet with <lb/>one another; the points of their Inter&longs;ection &longs;hall all fall in the <lb/>Circumference of one and the &longs;ame Circle: as for example, A L <lb/>and B L departing [or <emph type="italics"/>being drawn<emph.end type="italics"/>] from the Points A and B, and <lb/>having between them&longs;elves the &longs;ame proportion, as have the parts <lb/>A C and B C, and concurring in the point L: and the &longs;ame pro­<lb/>portion being between two others A K, and B K, concurring in K, <lb/>al&longs;o others as A I, and B I; A H, and B H; A G, and B G; A F, <lb/>and B F; A E, and B E: I &longs;ay, that the points of their Inter&longs;ecti­<lb/>on L, K, I, H, G, F, E, do all fall in the Circumference of one <lb/>and the &longs;ame Semi-circle: &longs;o that we &longs;hould imagine the point <lb/>C to mve conti­<lb/><figure id="id.069.01.033.1.jpg" xlink:href="069/01/033/1.jpg"/><lb/>nuallyafter &longs;uch <lb/>a &longs;ort, that the <lb/>Lines produced <lb/>from it to the fix­<lb/>ed terms A and <lb/>B retain alwaies <lb/>the &longs;ame propor­<lb/>tion that is be­<lb/>tween the fir&longs;t <lb/>parts A C and C B, that point C &longs;hall decribe the Circumference <lb/>of a Circle, as we &longs;hall &longs;hew you pre&longs;ently. </s> <s>And the Circle in &longs;uch <lb/>&longs;ort de&longs;cribed &longs;hall be alwaies greater and greater &longs;ucce&longs;&longs;ively, <lb/>according as the point C is taken nearer to the middle point <lb/>which is O; and the Circle &longs;hall be le&longs;&longs;er which &longs;hall be de&longs;cribed <lb/>from a point nearer to the extremity B, in&longs;omuch, that from the <pb xlink:href="069/01/034.jpg" pagenum="32"/>infinite Points which may be taken in the Line O B, there may be <lb/>de&longs;cribed Circles (moving them in &longs;uch &longs;ort as above is pre&longs;cri­<lb/>bed) of any Magnitude; le&longs;&longs;er than the Pupil of the eye of a <lb/>Flea, and bigger than the Equinoctial of the <emph type="italics"/>Primum Mobile.<emph.end type="italics"/><lb/>Now, if rai&longs;ing any of the Points comprehended betwixt the terms <lb/>O and B, from every one we may de&longs;cribe Circles, and va&longs;t ones <lb/>from the Points nearer to O; then if we rai&longs;e the Point O it &longs;elf, <lb/>and continue to move it in &longs;uch &longs;ort as afore&longs;aid, that is, that the <lb/>Lines drawn from it to the terms A and B keep the &longs;ame proporti­<lb/>on as have the fir&longs;t Lines A O, and O B, what Line &longs;hall be de&longs;cri­<lb/>bed? </s> <s>There would be de&longs;cribed the Circumference of a Circle, <lb/>but of a Circle bigger than the bigge&longs;t of all Circles, therefore of <lb/>a Circle that is infinite: but it doth al&longs;o de&longs;cribe a Right Line, and <lb/>perpendicular upon A B, erected from the Point O, and produced <lb/><emph type="italics"/>in infinitum<emph.end type="italics"/> without ever turning to reunite its la&longs;t term with the <lb/>fir&longs;t, as the others did; for the limited motion of the Point C, after <lb/>it had de&longs;igned the upper Semi-circle C H E, continued to de­<lb/>&longs;cribe the Lower E M C, reuniting its extream terms in the point <lb/>C: But the Point O being moved to de&longs;ign (as all the other Points <lb/>of the Line A B, for the Points taken in the other part O A <lb/>&longs;hall de&longs;ign their Circles, and tho&longs;e Points neare&longs;t to O the <lb/>greate&longs;t) its Circle; to make it the bigge&longs;t of all, and con&longs;e­<lb/>quently infinite, it can never return any more to its fir&longs;t term, and <lb/><arrow.to.target n="marg1027"></arrow.to.target><lb/>in a word de&longs;igneth an Infinite Right-Line for the Circumference <lb/>of its Infinite Circle. </s> <s>Con&longs;ider now, what difference there is be­<lb/>tween a finite Circle, and an infinite; &longs;eeing that this in &longs;uch man­<lb/>ner changeth its being that it wholly lo&longs;eth both its being, and <lb/>power of being; for we have already well comprehended, that <lb/>there cannot be a&longs;&longs;igned an infinite Circle; by which we may <lb/>con&longs;equently know that there can be no infinite Sphære, or other <lb/>Body, or figured Superficies. </s> <s>Now what &longs;hall we &longs;ay to this Meta­<lb/>morpho&longs;is in pa&longs;&longs;ing from Finite to Infinite? </s> <s>And why &longs;hould we <lb/>find greater repugnance, whil&longs;t &longs;eeking Infinity in Numbers, we <lb/><arrow.to.target n="marg1028"></arrow.to.target><lb/>come to conclude it to be in the Unite? </s> <s>And whil&longs;t that breaking <lb/>a Solid into many pieces, and pur&longs;uing to reduce it into very &longs;mall <lb/>powder, it were re&longs;olved into its infinite Atomes, admitting no far­<lb/>ther divi&longs;ion, why may we not &longs;ay that it is returned into one &longs;ole <lb/><emph type="italics"/>Continuum,<emph.end type="italics"/> but perhaps fluid, as the Water, or Quick&longs;ilver, or <lb/>other Metall melted? </s> <s>And do we not &longs;ee Stones liquified into <lb/>Gla&longs;s, and Gla&longs;s it &longs;elf with much Fire to become more fluid than <lb/>Water?</s></p><p type="margin"> <s><margin.target id="marg1027"></margin.target><emph type="italics"/>The difference be­<lb/>twixt a finite and <lb/>infinite Circle.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1028"></margin.target><emph type="italics"/>Vnity participates <lb/>of Infinity.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>Should we therefore think Fluids to be &longs;o called, be­<lb/>cau&longs;e they are re&longs;olved into their fir&longs;t, infinite, indivi&longs;ible com­<lb/>pounding parts?</s></p><p type="main"> <s>SALV. </s> <s>I know not how to find a better an&longs;wer to re&longs;olve cer­<pb xlink:href="069/01/035.jpg" pagenum="33"/>tain &longs;en&longs;ible appearances, among&longs;t which this is one: When I take <lb/>a hard Body, be it either Stone, or Metal, and with a Hammer, or <lb/>very fine File, endeavour to divide it, as much as is po&longs;&longs;ible, into <lb/>its mo&longs;t minute and impalpable powder; it is very clear, that its <lb/>lea&longs;t Atomes, albeit for their &longs;malne&longs;s they are imperceptible, one by <lb/>one, to our &longs;ight and touch; yet are they quantitative, figured, and <lb/>numerable: and it happens in them, that being accumulated to­<lb/>gether, they continue in heap; and being laid hollow, or with a <lb/>pit in the mid&longs;t, the hollowne&longs;s or pit remains, the parts heaped <lb/>about it not returning to fill it up; and being &longs;tirr'd, or &longs;haken, <lb/>they &longs;uddenly &longs;ettle &longs;o &longs;oon as their external mover leaves them, <lb/>And the like effects are &longs;een in all the Aggregates of &longs;mall Bodies, <lb/>bigger, and bigger, and of any kind of Figure, although Sphærical; <lb/>as we &longs;ee in heaps of Pea&longs;e, Wheat, Bird &longs;hot, and other matters. </s> <s>But <lb/>if we try to find the like accidents in Water, you will meet with <lb/>none of them; but, being rai&longs;ed, it in&longs;tantly returns to a level, if <lb/>it be not by a ve&longs;&longs;el, or &longs;ome other external &longs;tay upheld; being <lb/>made hollow, it pre&longs;ently diffu&longs;eth to fill up the Cavity; and be­<lb/>ing long moved, it continually undulates, and &longs;preads its waves very <lb/>far. </s> <s>From this, I think, we may very rationally infer, that the minute <lb/><arrow.to.target n="marg1029"></arrow.to.target><lb/>parts of Water, into which it &longs;eemeth to be re&longs;olved, (&longs;ince it hath <lb/>le&longs;s con&longs;i&longs;tence than any the fine&longs;t powder, yea, hath no con&longs;i­<lb/>&longs;tence at all) are va&longs;tly differing from Atomes quantitative and <lb/>divi&longs;ible; nor know I how to find any other difference therein <lb/>than that of being indivi&longs;ible. </s> <s>Methinks, al&longs;o, that its mo&longs;t exqui­<lb/>&longs;ite tran&longs;parency, affords us &longs;ufficient grounds to conjecture there­<lb/>of; for if we take the mo&longs;t diaphanous Chri&longs;tal that is, and begin <lb/>to break, and pound it to powder, when it is in powder it lo&longs;eth <lb/>its tran&longs;parency, and &longs;o much the more, the &longs;maller it is pounded; <lb/>but yet Water which is ground to the highe&longs;t degree, hath al&longs;o the <lb/>highe&longs;t degree of Diaphaneity Gold and Silver, reduced by <emph type="italics"/>Aqua­<lb/>fortis<emph.end type="italics"/> into a &longs;maller Powder than any File can make, yet they con­<lb/>tinue powder, and become not fluid; nor do they liquifie till the <lb/>Indivi&longs;ibles of the Fire, or of the Sun-beams di&longs;&longs;olve them, as, I be­<lb/>lieve, into their fir&longs;t and highe&longs;t infinite and indivi&longs;ible compoun­<lb/>ding parts.</s></p><p type="margin"> <s><margin.target id="marg1029"></margin.target><emph type="italics"/>Fluid Bodies are <lb/>&longs;uch, for that they <lb/>are re&longs;olved into <lb/>their fir&longs;t Indivi&longs;i­<lb/>ble Atomes.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>This which you have hinted of the Light I have many <lb/>times ob&longs;erved with admiration: I have &longs;een, I &longs;ay, a burning­<lb/>Gla&longs;s, of a foot Diameter, liquifie or melt lead in an in&longs;tant; <lb/>whence I came to be of opinion, that if the Gla&longs;&longs;es were very big, <lb/>and very polite, and of Parabolical Figure, they would no le&longs;s melt <lb/>every other Metal in a very &longs;hort time; &longs;eeing that that, not very <lb/>big, nor very clear, and of a Sphærical Concave, with &longs;uch force <lb/>melted Lead, and burnt every combu&longs;tible matter: effects, that <lb/>make the wonders, reported of the Burning-gla&longs;&longs;es of <emph type="italics"/>Archimedes,<emph.end type="italics"/><lb/>credible to me.<pb xlink:href="069/01/036.jpg" pagenum="34"/><arrow.to.target n="marg1030"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1030"></margin.target>Archimedes <emph type="italics"/>his <lb/>Burning — Gla&longs;&longs;es <lb/>admirable.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>Touching the Effects of the Gla&longs;&longs;es, invented by <emph type="italics"/>Ar­<lb/>chimedes,<emph.end type="italics"/> all the Miracles, that &longs;everal Writers record of them, <lb/>are to me rendred credible by the reading of <emph type="italics"/>Archimedes<emph.end type="italics"/> his own <lb/>Books, which I have with infinite amazement peru&longs;ed and &longs;tudied: <lb/>and if any doubts had been left me; that which la&longs;t of all Father </s></p><p type="main"> <s><arrow.to.target n="marg1031"></arrow.to.target><lb/><emph type="italics"/>Buonaventura Cavalieri<emph.end type="italics"/> hath publi&longs;hed, touching <emph type="italics"/>Lo Specehio <lb/>V&longs;torio,<emph.end type="italics"/> (or the Burning gla&longs;s) and which I have read with ad­<lb/>miration, is &longs;ufficient to re&longs;olve them all.</s></p><p type="margin"> <s><margin.target id="marg1031"></margin.target>Buonaventura <lb/>Cavalieri, <emph type="italics"/>the Je­<lb/>&longs;uate, a famous <lb/>Mathematician, <lb/>and his Book en­<lb/>titled,<emph.end type="italics"/> Lo Spec­<lb/>chio U&longs;torio.</s></p><p type="main"> <s>SAGR. </s> <s>I have al&longs;o &longs;een that Tract, and peru&longs;ed it with much <lb/>delight and wonder; and becau&longs;e I formerly had knowledge of <lb/>the Author, I was confirmed in the opinion which I had conceived <lb/>of him, that he was like to prove one of the principal Mathemati­<lb/>cians of our Age. </s> <s>But returning to the admirable effects of the <lb/>Sun-Beams in melting of Metals, are we to believe that &longs;uch, and <lb/>&longs;o violent an operation is without Motion, or el&longs;e that it is with <lb/>Motion, but extream &longs;wift?<lb/><arrow.to.target n="marg1032"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1032"></margin.target><emph type="italics"/>Burnings are per­<lb/>formed with a mo&longs;t <lb/>&longs;wift Motion.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>We &longs;ee other burnings, and meltings to be performed <lb/>with Motion, and with a mo&longs;t &longs;wift Motion. </s> <s>Ob&longs;erve the ope­<lb/>rations of Lightnings, of Powder in Mines, and in Petards, <lb/>and, in &longs;um, how by quickning the flame of Coles, mixt with <lb/>gro&longs;s and impure vapours, by Bellows, encrea&longs;eth its force in <lb/>the melting of Metals: &longs;o that I cannot &longs;ee how the Action of <lb/>Light, albeit mo&longs;t pure, can be without Motion, and that al&longs;o ve­<lb/>ry &longs;wift.</s></p><p type="main"> <s><emph type="italics"/>S<emph.end type="italics"/>AGR. </s> <s>But what and how great ought we to judge this Velo­<lb/>city of the Light? </s> <s>Is it haply <emph type="italics"/>In&longs;tantaneous,<emph.end type="italics"/> and done in a moment, <lb/>or, as the re&longs;t of Motions, performed in Time? </s> <s>May we not by <lb/>Experiment be a&longs;&longs;ured what it is?</s></p><p type="main"> <s>SIMP. </s> <s>Quotidian experience &longs;hews the expan&longs;ion of Light to <lb/>be <emph type="italics"/>In&longs;tantaneous<emph.end type="italics"/>; in that beholding a Cannon, let off at a great <lb/>di&longs;tance, the fla&longs;h of the fire, without interpo&longs;ition of time, is tran&longs;­<lb/>mitted to our eye, but &longs;o is not the Report to our ear untill a con­<lb/>&longs;iderable time after.</s></p><p type="main"> <s>SAGR. True, but, I pray you, what doth this obvious experi­<lb/>ment evince; but only this, that the Report is longer in arriving at <lb/>our Ear, than the Fla&longs;h at our Eye; but it a&longs;&longs;ures me not, that the <lb/>tran&longs;mi&longs;&longs;ion of the Light is therefore <emph type="italics"/>In&longs;tantaneous<emph.end type="italics"/> rather than in <lb/>Time, but only mo&longs;t &longs;wift. </s> <s>Nor doth &longs;uch an ob&longs;ervation con­<lb/>clude more than that other, of &longs;uch who &longs;ay, that as &longs;oon as the <lb/>Sun cometh to the Horizon, its Light arriveth at our eye: for who <lb/>&longs;hall a&longs;&longs;ure me, that its beams arrive not at the &longs;aid term, afore they <lb/>reach our &longs;ight?</s></p><p type="main"> <s>SALV. </s> <s>The inconcludency of the&longs;e, and other ob&longs;ervations of <lb/>the like Nature, made me once think of &longs;ome other way, whereby <lb/>we may without errour be a&longs;certained whether the illumination, <pb xlink:href="069/01/037.jpg" pagenum="35"/>that is, whether the expan&longs;ion of the Light were really <emph type="italics"/>In&longs;tantane­<lb/>ous<emph.end type="italics"/>; &longs;eeing that the very &longs;wift Motion of Sound, a&longs;&longs;ureth us, that <lb/>that of Light cannot but be extream &longs;wift. </s> <s>And the experiment I </s></p><p type="main"> <s><arrow.to.target n="marg1033"></arrow.to.target><lb/>hit upon, was this; I would have two per&longs;ons take each of them a <lb/>Light, which, by holding it in a Lanthorn, or other coverture, they <lb/>may cover, and di&longs;cover at plea&longs;ure by interpo&longs;ing their hand to the <lb/>fight of each other; and, that placing them&longs;elvs again&longs;t one another, <lb/>&longs;ome few paces di&longs;tance, they may practice the &longs;peedy di&longs;covery, <lb/>and occultation of their Lights from the &longs;ight of each other: So <lb/>that when one &longs;eeth the others Light, he immediatly di&longs;clo&longs;e his: <lb/>which corre&longs;pondence, after &longs;ome Re&longs;pon&longs;es mutually made, will <lb/>become &longs;o exactly In&longs;tantaneous, that, without &longs;en&longs;ible variation, <lb/>at the di&longs;covery of the one, the other &longs;hall at the &longs;ame time ap­<lb/>pear to the &longs;ight of him that di&longs;clos'd the fir&longs;t. </s> <s>Having adju&longs;ted <lb/>this practice at this &longs;mall di&longs;tance, let us place the two per&longs;ons with <lb/>two &longs;uch Lights at two or three miles di&longs;tance; and by night re­<lb/>newing the &longs;ame experiment; Let them inten&longs;ely ob&longs;erve if the <lb/>Re&longs;pon&longs;es of the di&longs;clo&longs;ures, and occultations do follow the &longs;ame <lb/>tenour which they did near hand: for if they keep the &longs;ame pro­<lb/>portion, it may be with certainty enough concluded, that the ex­<lb/>pan&longs;ion of Light is In&longs;tantaneous; but if it &longs;hould require time in <lb/>a di&longs;tance of three miles, which importeth &longs;ix for the going of <lb/>one, and return of the other, the &longs;tay would be &longs;ufficiently ob&longs;er­<lb/>vable. </s> <s>And if this Experiment be made at greater di&longs;tances, <lb/>namely, at eight or ten miles, we may make u&longs;e of the <emph type="italics"/>Tele&longs;cope,<emph.end type="italics"/><lb/>the Ob&longs;ervators accommodating each of them one at the places, <lb/>where by night the Lights are to be ob&longs;erved; which though not <lb/>very big, and &longs;o not vi&longs;ible, at that great di&longs;tance, to the eye at <lb/>large; (though ea&longs;ie to be di&longs;clo&longs;ed, and hid) by help of the <lb/><emph type="italics"/>Tele&longs;copes<emph.end type="italics"/> before admitted, and fixed they may commodiou&longs;ly be <lb/>di&longs;cerned.</s></p><p type="margin"> <s><margin.target id="marg1033"></margin.target><emph type="italics"/>The Velocity of <lb/>Light, how to find <lb/>by Experiment <lb/>whether it be In­<lb/>&longs;tantaneosu or not.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>The Invention &longs;eems to me no le&longs;s certain than ingenu­<lb/>ous; but tell us what upon experimenting it you concluded.</s></p><p type="main"> <s><emph type="italics"/>S<emph.end type="italics"/>ALV. Really, I have not tryed it, &longs;ave only at a &longs;mall di&longs;tance, <lb/>namely, le&longs;s than a Mile: whereby I could come to no certainty <lb/>whether the apparence of the oppo&longs;ite Light was truly In&longs;tantane­<lb/>ous; But if not In&longs;tantaneous, yet it was of exceeding great Velo­<lb/>city, and I may &longs;ay Momentary: and for the pre&longs;ent, I would re­<lb/>&longs;emble it to that Motion which we &longs;ee a fla&longs;h of Lightning make <lb/>in the Clouds ten or more Miles off: of which Light we di&longs;tin­<lb/>gui&longs;h the beginning, and, I may fay, the &longs;ource and ri&longs;e of it, in a <lb/>particular place in tho&longs;e Clouds; but yet its wide expan&longs;ion imme­<lb/>diatly &longs;ucceeds among&longs;t tho&longs;e adjacent: which to me &longs;eems an ar­<lb/>gument that it is &longs;ome &longs;mall time in doing; becau&longs;e had the illu­<lb/>mination been made all at once, and not by degrees, it feems to <pb xlink:href="069/01/038.jpg" pagenum="36"/>me that we could not have di&longs;tingui&longs;hed its original, or rather the <lb/>Center of its flake, and extream Dilatations. </s> <s>But into what Oceans <lb/>do we by degrees engage our &longs;elves? </s> <s>Among&longs;t <emph type="italics"/>Vacuities, Infinites, <lb/>Indivi&longs;ibles,<emph.end type="italics"/> and <emph type="italics"/>Instantaneous Motions<emph.end type="italics"/>; &longs;o that we &longs;hall not be <lb/>able by a thou&longs;and Di&longs;cour&longs;es to recover the Shore?</s></p><p type="main"> <s>SAGR. </s> <s>They are things, indeed, very di&longs;proportionate to our <lb/>under&longs;tanding. </s> <s>Behold Infinite, &longs;ought among&longs;t Numbers, &longs;eemeth <lb/>to determine in the Unite: From Indivi&longs;ibles ari&longs;eth things that <lb/>are continually divi&longs;ible: Vacuity &longs;eems only to re&longs;ide indivi&longs;ibly <lb/>mixt with Repletion: and, in brief, the&longs;e things &longs;o change the <lb/>nature of tho&longs;e under&longs;tood by us, that even the Circumference of <lb/>a Circle becometh an Infinite Right-Line; which, if I well re­<lb/>member, is that Propo&longs;ition which you, <emph type="italics"/>Salviatus,<emph.end type="italics"/> are to mani­<lb/>fe&longs;t by Geometrical Demon&longs;tration. </s> <s>Therefore, if you think fit, <lb/>it would be well, without any more digre&longs;&longs;ions, to make it out <lb/>to us.</s></p><p type="main"> <s>SALV. </s> <s>I am ready to &longs;erve you in demon&longs;trating the en&longs;uing <lb/>Problem for your fuller information.</s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s><emph type="italics"/>A Right-Line being given, divided, according to any <lb/>proportion, into unequal parts, to de&longs;cribe a Circle, to <lb/>the Circumference of which, at any point of the &longs;ame, <lb/>two Right-Lines being produced from the terms of <lb/>the given Right Line, they may retain the &longs;ame pro­<lb/>portion that the parts of the &longs;aid Line given have to <lb/>one another, &longs;o that tho&longs;e be Homologous which de­<lb/>part &longs;rom the &longs;ame terms.<emph.end type="italics"/></s></p><p type="main"> <s>Let the given Right-Line be AB, unequally divided ac­<lb/>cording to any proportion in the point C; it is required to <lb/>de&longs;cribe a Circle at any point of who&longs;e Circumference two <lb/>Right Lines, produced from the terms A and B, concurring, have <lb/>the &longs;ame proportion to each other, that A C, hath to B C, &longs;o that <lb/>tho&longs;e be Homologous which depart from the &longs;ame term. </s> <s>Upon <lb/>the Center C, at the di&longs;tance of the le&longs;&longs;er part C B, let a Circle be <lb/>&longs;uppo&longs;ed to be de&longs;cribed, to the Circumference of which from the <lb/>point A the Right-line A D is made a Tangent, and indetermi­<lb/>nately prolonged towards E: and let the Contact be in D, and <lb/>draw a Line from C to D, which &longs;hall be perpendicular to A E; <lb/>and let B E be perpendicular to B A, which produced, &longs;hall inter­<pb xlink:href="069/01/039.jpg" pagenum="37"/>&longs;ect A E, the Angle A being acute: Let the inter&longs;ection be in E, <lb/>from whence let fall a Perpendicular to A E, which produced, will <lb/>meet with A B infinitely prolonged in F. </s> <s>I &longs;ay, fir&longs;t, that the <lb/>Right-lines F E, and F C are equal: &longs;o that drawing the Line <lb/>E C, we &longs;hall, in the <lb/><figure id="id.069.01.039.1.jpg" xlink:href="069/01/039/1.jpg"/><lb/>two Triangles D E C, <lb/>B E C, have the two <lb/>Sides of the one, D E, <lb/>and C E, equal to the <lb/>two Sides of the other <lb/>B E, and E C; the <lb/>two Sides, D E, and <lb/>E B, being Tangents <lb/>to the Circle D B, <lb/>and the Ba&longs;es D C, <lb/>and C B, are likewi&longs;e <lb/>equal: wherefore the <lb/>two Angles D E C, <lb/>and B E C, &longs;hall be <lb/>equal. </s> <s>And becau&longs;e the Angle B C E wanteth of being a Right­<lb/>Angle, as much as the Angle B E C; and the Angle C E F, to <lb/>make it a Right-Angle, wants the Angle C E D, tho&longs;e Supple­<lb/>ments being equal, the Angles F C E, and F E C &longs;hall be equal, <lb/>and &longs;o con&longs;equently the Sides F E, and F C; wherefore making <lb/>the point F a Center, and at the di&longs;tance F E, de&longs;cribing a Circle, <lb/>it &longs;hall pa&longs;s by the point C. </s> <s>De&longs;cribe it, and let it be C E G. </s> <s>I &longs;ay, <lb/>that this is the Circle required, by any point of the Circumfe­<lb/>rence of which, any two Lines that &longs;hall inter&longs;ect, departing from <lb/>the terms A and B, &longs;hall be in proportion to each other, as are the <lb/>two parts A C, and B C, which be&longs;ore did concur in the point C. <lb/></s> <s>This is manife&longs;t in the two that concur or inter&longs;ect in the point E, <lb/>that is A E, and B E; the Angle E of the Triangle A E B being <lb/>divided in the mid&longs;t by C E; &longs;o that as A C is to C B, &longs;o is A E <lb/>to B E. </s> <s>The &longs;ame we prove in the two A G, and B G, determined <lb/>in the point G. </s> <s>Therefore being (by the Similitude of the Tri­<lb/>angles A F E, and E F B) that as A F is to E F, &longs;o is E F to F B; <lb/>that is, as A F is to F C, &longs;o is C F to F B: So by Divi&longs;ion; as A C <lb/>is to C F, (that is, to F G) &longs;o is C B to B F; and the whole A B <lb/>is to the whole B G, as the part C B to the part B F: and by Com­<lb/>po&longs;ition; as A G is to G B, &longs;o is C F to F B; that is, as E F to <lb/>F B, that is, as A E to E B, and A C to C B: Which was to be de­<lb/>mon&longs;trated. </s> <s>Again, let any other Point be taken in the Circum­<lb/>ference, as H; in which the two Lines A H and B H concur. </s> <s>I &longs;ay, in <lb/>like manner as before, that as A C is to C B, &longs;o is A H to B H. <lb/></s> <s>Continue H B untill it inter&longs;ect the Circumference in I, and draw <pb xlink:href="069/01/040.jpg" pagenum="38"/>a Line joyning I to F. </s> <s>And becau&longs;e it hath been proved already <lb/>that as A B is to B G, &longs;o is C B to B F, the Rectangle A B F &longs;hall be <lb/>equall to the Rectangle C B G, that is I B H: and therefore, as <lb/>A B is to B H, &longs;o is I B to B F, and the Angles at B are equal: <lb/>Therefore A H is to H B, as I F, that is E F, to F B, and as A E <lb/>to E B.</s></p><p type="main"> <s>I &longs;ay moreover, that it is impo&longs;&longs;ible, that the Lines, which have <lb/>this &longs;ame proportion, departing from the terms A and B, &longs;hould <lb/>meet in any point, either within or without the &longs;aid Circle: For­<lb/>a&longs;much as if it be po&longs;&longs;ible that two Lines &longs;hould concur in the <lb/>point L, placed without; let them be A L, and B L; and continue <lb/>L B to the Circumference in M, and conjoyn M to F. </s> <s>If therefore <lb/>A L is to B L, as A C to B C, that is, as M F to F B, we &longs;hall have <lb/>two Triangles A L B, and M F B, which about the two Angles <lb/>A L B and M F B have their Sides proportional, their upper Angles <lb/>in the point B equal, and the two remaining Angles F M B and <lb/>L A B le&longs;s than Right Angles (for that the Right-angle at the <lb/>point M hath for its Ba&longs;e the whole Diameter C G, and not the <lb/>&longs;ole part B F, and the other at the point A is acute by rea&longs;on the <lb/>Line A L Homologous to A C, is greater than B L Homologous to <lb/>B C) Therefore the Triangles A B L, and M B F are like: and <lb/>therefore as A B is to B L, &longs;o is M B to B F; Wherefore the <lb/>Rectangle A B F &longs;hall be equall to the Rectangle M B L. </s> <s>But the <lb/>Rectangle A B F hath been demon&longs;trated to be equal to that of <lb/>C B G: Therefore the Rectangle M B L is equal to the Rectangle <lb/>C B G, which is impo&longs;&longs;ible: Therefore the Concour&longs;e of the Lines <lb/>cannot fall without the Circle. </s> <s>And in like manner it may be de­<lb/>mon&longs;trated that it cannot fall within; Therefore all the Concour­<lb/>&longs;es fall in the Circumference it &longs;elf.</s></p><p type="main"> <s>But it is time that we return to give &longs;atisfaction to the Intreaty <lb/>of <emph type="italics"/>Simplicius,<emph.end type="italics"/> &longs;hewing him that the re&longs;olving the Line into its in­<lb/>finite Points is not only not impo&longs;&longs;ible, but that it hath in it no <lb/>more difficulty than to di&longs;tingui&longs;h its quantitative parts; pre&longs;up­<lb/>po&longs;ing one thing (notwith&longs;tanding) which I think, <emph type="italics"/>Simplicius,<emph.end type="italics"/><lb/>you will not deny me, and that is this; that you will not require me <lb/>to &longs;ever the Points one from another, and &longs;hew you them one by <lb/>one di&longs;tinctly upon this paper: for I my &longs;elfe &longs;hould be content, <lb/>if without enjoyning to pull the four or &longs;ix parts of a Line from <lb/>one another, you &longs;hould but &longs;hew me its divi&longs;ions marked, or at <lb/>mo&longs;t inclined to Angles, framing them into a Square, or a Hexa­<lb/>gon; therefore I per&longs;wade my &longs;elf, that for the pre&longs;ent you will <lb/>grant them then &longs;ufficiently, and actually di&longs;tingui&longs;hed.</s></p><p type="main"> <s>SIMP. </s> <s>I &longs;hall indeed.</s></p><p type="main"> <s>SALV. </s> <s>Now if the inclining of a Line to Angles, framing <lb/><arrow.to.target n="marg1034"></arrow.to.target><lb/>therewith &longs;ometimes a Square &longs;ometimes an Octagon, &longs;ometimes <pb xlink:href="069/01/041.jpg" pagenum="39"/>a Poligon of Forty, of an <emph type="italics"/>H<emph.end type="italics"/>undred, of a Thou&longs;and Angles be a <lb/>mutation &longs;ufficient to reduce into Act tho&longs;e four, eight, forty, <lb/>hundred, or thou&longs;and parts, which were, as you &longs;ay, Potentially <lb/>in the &longs;aid Line at fir&longs;t: if I make thereof a Poligon of infinite <lb/>Sides, namely, when I bend it into the Circumference of a Circle, <lb/>may not I, with the like leave, &longs;ay, that I have reduced tho&longs;e infi­<lb/>nite parts into Act, which you before, whil&longs;t it was &longs;traight, &longs;aid <lb/>were Potentially contained in it? </s> <s>Nor may &longs;uch a Re&longs;olution be <lb/>denied to be made into its Infinite Points, as well as that of its four <lb/>parts in forming thereof a Square, or into its thou&longs;and parts in <lb/>forming thereof a Mill-angular Figure; by rea&longs;on that there wants <lb/>not in it any of the Conditions found in the Poligon of a thou­<lb/>&longs;and, or of an hundred thou&longs;and Sides. </s> <s>This applied or layed to a <lb/>Right-Line covereth it, touching it with one of its Sides, that is, <lb/>with one of its hundred thou&longs;andth parts; the Circle, which is a <lb/>Poligon of infinite Sides, toucheth the &longs;aid Right-line with one of <lb/>its Sides, that is one &longs;ingle Point divers from all its Colaterals, and <lb/>therefore divided, and di&longs;tinct from them, no le&longs;s than a Side of <lb/>the Poligon from its Conterminals. </s> <s>And as the Poligon turned <lb/>round upon a Plane de&longs;cribes, with the con&longs;equent tacts of its Sides, <lb/>a Right-line equal to its Perimeter: &longs;o the Circle, rowled upon <lb/>&longs;uch a Plane, de&longs;cribes or &longs;tamps upon it, by its infinite &longs;ucce&longs;&longs;ive <lb/>Contacts, a Right-line, equall to its own Circumference. </s> <s>I know <lb/>not at pre&longs;ent, <emph type="italics"/>Simplicius,<emph.end type="italics"/> whether or no the Peripateticks, (to <lb/>whom I grant, as a thing mo&longs;t certain, that <emph type="italics"/>Continuum<emph.end type="italics"/> may be di­<lb/>vided into parts alwaies divi&longs;ible, &longs;o that continuing the divi&longs;ion <lb/>and &longs;ubdivi&longs;ion there can be no end thereof) will be content to <lb/>yield to me, that none of tho&longs;e divi&longs;ions are the ultimate, as in­<lb/>deed they be not, &longs;ince that there alwaies remains another; but <lb/>that only to be the la&longs;t, which re&longs;olves it into infinite Indivi&longs;ibles; <lb/>to which I yield we can never attain, dividing and &longs;ubdividing it <lb/>&longs;ucce&longs;&longs;ively into a greater, and greater multitude of parts: but <lb/>making u&longs;e of the way which I propound to di&longs;tingui&longs;h and re­<lb/>&longs;olve all the infinite parts at one only draught, (an Artifice which <lb/>ought not to be denied me) I could per&longs;wade my &longs;elf they <lb/>would &longs;atisfie them&longs;elves, and admit this compo&longs;ition of <emph type="italics"/>Continu-<emph.end type="italics"/><lb/><arrow.to.target n="marg1035"></arrow.to.target><lb/><emph type="italics"/>um<emph.end type="italics"/> to con&longs;i&longs;t of Atomes ab&longs;olutely indivi&longs;ible: And e&longs;pecially, <lb/>this one path being more current than any other to extricate us <lb/>out of very intricate Laberinths; &longs;uch as are, (be&longs;ides that alrea­<lb/>dy touched of the Coherence of the parts of Solids) the concei­<lb/>ving the bu&longs;ine&longs;s of Rarefaction and Conden&longs;ation, without <lb/>running into the inconvenience of being forced to admit forth of <lb/>void Spaces or Vacuities; and for this a Penetration of Bodies: in­<lb/>conveniences, which both, in my opinion, may ea&longs;ily be avoided, <lb/>by granting the fore&longs;aid Compo&longs;ition of Indivi&longs;ibles.</s></p><pb xlink:href="069/01/042.jpg" pagenum="40"/><p type="margin"> <s><margin.target id="marg1034"></margin.target><emph type="italics"/>How infinite points <lb/>are a&longs;&longs;igned in a <lb/>finite Line.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1035"></margin.target>Continuum <emph type="italics"/>com­<lb/>pounded of Indivi­<lb/>&longs;ibles.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. </s> <s>I know not what the Peripateticks would &longs;ay, in regard <lb/>that the Con&longs;iderations you have propo&longs;ed would be, for the mo&longs;t <lb/>part, new unto them, and as &longs;uch, it is requi&longs;ite that they be exa­<lb/>mined: and it may be, that they would find you an&longs;wers, and <lb/>powerful Solutions, to unty the&longs;e knots, which I, by rea&longs;on of the <lb/>want of time and ingenuity proportionate, cannot for the pre&longs;ent <lb/>re&longs;olve. </s> <s>Therefore, &longs;u&longs;pending this particular for this time, I <lb/>would gladly under&longs;tand how the introduction of the&longs;e Indivi&longs;i­<lb/>bles facilitateth the knowledge of Conden&longs;ation, and Rarefa­<lb/>ction, avoiding at the &longs;ame time a <emph type="italics"/>Vacuum,<emph.end type="italics"/> and the Penetration of <lb/>Bodies.</s></p><p type="main"> <s>SAGR. </s> <s>I al&longs;o much long to under&longs;tand the &longs;ame, it being to <lb/>my Capacity &longs;o ob&longs;cure: with this <emph type="italics"/>provi&longs;o,<emph.end type="italics"/> that I be not couzen­<lb/>ed of hearing (as <emph type="italics"/>Simplicius<emph.end type="italics"/> &longs;aid but even now) the Rea&longs;ons of <lb/><emph type="italics"/>Ari&longs;totle<emph.end type="italics"/> in confutation of a <emph type="italics"/>Vacuum,<emph.end type="italics"/> and con&longs;equently the Solu­<lb/>tions which you bring, as ought to be done, whil&longs;t that you ad­<lb/>mit what he denieth.</s></p><p type="main"> <s>SALV. </s> <s>I will do both the one and the other. </s> <s>And as to the fir&longs;t <lb/>it's nece&longs;&longs;ary, that like as in favour of Rarefaction, we make u&longs;e of <lb/>the Line de&longs;cribed by the le&longs;&longs;er Circle bigger than its own Cir­<lb/>cumference, whil&longs;t it was moved at the Revolution of the greater; <lb/>&longs;o, for the under&longs;tanding of Conden&longs;ation, we &longs;hall &longs;hew, how that, <lb/>at the conver&longs;ion made by the le&longs;&longs;er Circle, the greater de&longs;cribeth <lb/>a Right-line le&longs;s than its Circumference; for the clearer explicati­<lb/>on of which, let us &longs;et before us the con&longs;ideration of that which <lb/>befalls in the Poligons. </s> <s>In a de&longs;cription like to that other; &longs;up­<lb/>po&longs;e two Hexagons about the common Center L, which let be <lb/>A B C, and H I K, with the Parallel-lines H O M, and A B C, up­<lb/>on which they are to make their Revolutions; and the Angle I, of <lb/>the le&longs;&longs;er Poligon, re&longs;ting at a &longs;tay, turn the &longs;aid Poligon till &longs;uch <lb/>time as I K fall upon the Parallel, in which motion the point K <lb/>&longs;hall de&longs;cribe the Arch K M, and the Side K I, &longs;hall unite with the <lb/>part I M; while this is in doing, you mu&longs;t ob&longs;erve what the Side <lb/>C B of the greater Poligon will do. </s> <s>And becau&longs;e the Revolution <lb/>is made upon the Point I, the Line I B with its term B &longs;hall de­<lb/>&longs;cribe, turning backward the Arch B b, below the Parallel c A, &longs;o <lb/>that when the Side K I &longs;hall fall upon the Line M I, the Line B C <lb/>&longs;hall fall upon the Line b c, advancing forwards only &longs;o much as <lb/>is the Line B c, and retiring back the part &longs;ubtended by the Arch <lb/>B b, which falls upon the Line B A, and intending to continue af­<lb/>ter the &longs;ame manner the Revolution of the le&longs;&longs;er Poligon, this will <lb/>de&longs;cribe, and pa&longs;s upon its Parallel, a Line equal to its Perimeter; <lb/>but the greater &longs;hall pa&longs;s a Line le&longs;s than its Perimeter, the quan­<lb/>tity of &longs;o many of the lines <emph type="italics"/>B<emph.end type="italics"/> b as it hath Sides, wanting one; <lb/>and that &longs;ame line &longs;hall be very near equal to that de&longs;cribed by <pb xlink:href="069/01/043.jpg" pagenum="41"/>the le&longs;&longs;er Poligon, exceeding it only the quantity of b B. </s> <s>Here <lb/>then, without the lea&longs;t repugnance the cau&longs;e is &longs;een, why the grea­<lb/>ter Poligon pa&longs;&longs;eth or moveth not (being carried by the le&longs;s) <lb/>with its Sides a greater Line than that pa&longs;&longs;ed by the le&longs;s; that is, <lb/>becau&longs;e that one part of each of them falleth upon its next coter­<lb/>minal and precedent.</s></p><p type="main"> <s>But if we &longs;hould con&longs;ider the two Circles about the Center A, <lb/>re&longs;ting upon their Parallels, the le&longs;&longs;er touching his in the point B, <lb/>and the greater his in the <lb/><figure id="id.069.01.043.1.jpg" xlink:href="069/01/043/1.jpg"/><lb/>point C; here, in begin­<lb/>ning to make the Revolu­<lb/>tion of the le&longs;s, it &longs;hall not <lb/>occur as before, that the <lb/>point B re&longs;t for &longs;ome time <lb/>immoveable, &longs;o that the <lb/>Line B C giving back, <lb/>carry with it the point C, <lb/>as it befell in the Poligons, <lb/>which re&longs;ting fixed in the <lb/>point I till that the Side <lb/>K I falling upon the Line <lb/>I M, the Line I B carried <lb/>back B, the term of the <lb/>Side C B, as far as b, by <lb/>which means the Side B C <lb/>fell on b c, &longs;uper-po&longs;ing or <lb/>re&longs;ting the part B b upon <lb/>the Line B A, and advancing forwards only the part <emph type="italics"/>B<emph.end type="italics"/> c, equal to <lb/>I M, that is to one Side of the le&longs;&longs;er Poligon: by which &longs;uperpo&longs;i­<lb/>tions, which are the exce&longs;&longs;es of the greater Sides above the le&longs;s, the <lb/>advancements which remain equal to the Sides of the le&longs;&longs;er Poli­<lb/>gon come to compo&longs;e in the whole Revolution the Right-line <lb/>equal to that traced, and mea&longs;ured by the le&longs;&longs;er Poligon. </s> <s>But <lb/><arrow.to.target n="marg1036"></arrow.to.target><lb/>now, I &longs;ay, that if we would apply this &longs;ame di&longs;cour&longs;e to the ef­<lb/>fect of the Circles, it will be requi&longs;ite to confe&longs;s, that whereas the <lb/>Sides of what&longs;oever Poligon are comprehended by &longs;ome Number, <lb/>the Sides of the Circle are infinite; tho&longs;e are quantitative and di­<lb/>vi&longs;ible, the&longs;e non-quantitative and Indivi&longs;ible: the terms of the <lb/>Sides of a Poligon in the Revolution &longs;tand &longs;till for &longs;ome time, that <lb/>is, each &longs;uch part of the time of an entire Conver&longs;ion, as it is of <lb/>the whole Perimeter: in the Circles likewi&longs;e the &longs;tay o&longs; the terms <lb/><arrow.to.target n="marg1037"></arrow.to.target><lb/>of its infinite Sides are momentary, for a Moment is &longs;uch part of a <lb/>limited Time, as a Point is of a Line, which containeth infinite of <lb/>them; the regre&longs;&longs;ions made by the Sides of the greater Poligon, are <lb/>not of the whole Side, but only of its exce&longs;s above the Side of the <pb xlink:href="069/01/044.jpg" pagenum="42"/>le&longs;&longs;er, getting forwards as much &longs;pace as the &longs;aid le&longs;&longs;er Side: in <lb/>Circles, the Point, or Side C in the in&longs;tantaneous re&longs;t of B recedeth <lb/>as much as is its exce&longs;s above the Side B, advancing forward as <lb/>much as the quantity of the &longs;ame B: And in &longs;hort, the infinite <lb/>indivi&longs;ible Sides of the greater Circle with their infinite indivi&longs;ible <lb/>Regre&longs;&longs;ions, made in the infinite in&longs;tantaneous &longs;taies of the infi­<lb/>nite terms of the infinite Sides of the le&longs;&longs;er Circle, and with their <lb/>infinite Progre&longs;&longs;es, equal to the infinite Sides of the &longs;aid le&longs;&longs;er <lb/>Circle, they compo&longs;e and mea&longs;ure a Line equall to that de&longs;cribed <lb/>by the le&longs;&longs;er Circle, containing in it &longs;elf infinite &longs;uperpo&longs;itious <lb/>non-quantitative, which make a Con&longs;tipation and Conden&longs;ation <lb/>without any penctration of quantitative parts: which cannot be <lb/>contrived to be done in the Line divided into quantitative parts, <lb/>as is the Perimeter of any Poligon, which being di&longs;tended in a <lb/>Right-line at length, cannot be reduced to a le&longs;&longs;er length, unle&longs;s <lb/>the Sides fall upon and Penetrate one the other. </s> <s>This Con&longs;tipati­<lb/>on of parts non-quantitative, but infinite without Penetration of <lb/>quantitative parts, and the former Di&longs;traction above declared of <lb/><arrow.to.target n="marg1038"></arrow.to.target><lb/>infinite Indivi&longs;ibles by the interpo&longs;ition of indivi&longs;ible Vacui­<lb/>ties, I believe, is the mo&longs;t that can be &longs;aid for the Conden&longs;ation <lb/>and Rarefaction of Bodies, without being driven to introduce Pe­<lb/>netration of Bodies, or quantitative Void Spaces. </s> <s>If there be any <lb/>thing therein that plea&longs;eth you, make u&longs;e of it, if not, account it <lb/><arrow.to.target n="marg1039"></arrow.to.target><lb/>vain, and my di&longs;cour&longs;e al&longs;o; and &longs;eek out &longs;ome other explanation <lb/>that may better &longs;atisfie your Judgment. </s> <s>Only the&longs;e two words <lb/>by the way, let us remember that we are among&longs;t Infinites, and In­<lb/>divi&longs;ibles.</s></p><p type="margin"> <s><margin.target id="marg1036"></margin.target><emph type="italics"/>A Circle is a Poli­<lb/>gon of infinite in­<lb/>divi&longs;ible quantita­<lb/>tive Sides.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1037"></margin.target><emph type="italics"/>An In&longs;tant or Mo­<lb/>ment of quantita­<lb/>tive Time, is the <lb/>&longs;ame as a Point of <lb/>a quantitative <lb/>Line.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1038"></margin.target><emph type="italics"/>Rarefaction is the <lb/>di&longs;traction of infi­<lb/>nite Indivi&longs;ibles <lb/>by the interpo&longs;ition <lb/>of infinite indivi&longs;i­<lb/>ble Vaeuities.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1039"></margin.target><emph type="italics"/>Conden&longs;ation, ac­<lb/>cording to the ope­<lb/>ration of the Au­<lb/>thor, proceeds from <lb/>the Con&longs;tipation of <lb/>quantitative and <lb/>indivi&longs;ible parts.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>That the Conceit is ingenious, and to my eares wholly <lb/>new, and &longs;trange, I freely confe&longs;s, but whether or no Nature pro­<lb/>ceed in this order, I know not how to re&longs;olve; Truth is, that till <lb/>&longs;uch time as I hear &longs;omething that may better &longs;atisfie me, that I <lb/>may not &longs;tand &longs;ilent, I will adhere to this. </s> <s>But haply <emph type="italics"/>Simplicius<emph.end type="italics"/><lb/>may have &longs;omwhat, which I have not yet met with, to explicate <lb/>the explication, which is produced by Philo&longs;ophers in &longs;o ab&longs;truce <lb/>a matter; for, indeed, what I have hitherto read about Conden&longs;a­<lb/>tion, is to me &longs;o den&longs;e, and that of Rarefaction &longs;o &longs;ubtill, that <lb/>my weak &longs;ight neither penetrates the one, nor comprehends the <lb/>other.</s></p><p type="main"> <s>SIMP. </s> <s>I am full of confu&longs;ion, and find great Rubbs in the one <lb/>path, and in the other, and more particularly in this new one: for <lb/>according to this Rule, an Ounce of Gold might be rarefied and <lb/>drawn forth into a Ma&longs;s bigger than the whole Earth, and the <lb/>whole Earth conden&longs;ed and reduced into a le&longs;s Ma&longs;s than a Nut; <lb/>which I neither believe, nor think that you your &longs;elf do believe: <lb/>and the Con&longs;iderations and Demon&longs;trations by you hitherto de­<pb xlink:href="069/01/045.jpg" pagenum="43"/>livered, as they are things Mathematical, ab&longs;tract and &longs;eparate <lb/>from Sen&longs;ible Matter, I believe, that when they come to be apply­<lb/>ed to Matters Phy&longs;ical and Natural, they will not exactly comply <lb/>with the&longs;e Rules.</s></p><p type="main"> <s>SALV. </s> <s>It is not in my power, nor, as I believe, do you de&longs;ire, <lb/>that I &longs;hould make that vi&longs;ible which is invi&longs;ible; but as to &longs;uch <lb/>things as may be comprehended by our Sen&longs;es, in regard that you <lb/><arrow.to.target n="marg1040"></arrow.to.target><lb/>have in&longs;tanced in Gold, do we not &longs;ee an immen&longs;e exten&longs;ion to <lb/>be made of its parts? </s> <s>I know not whether you may have &longs;een the <lb/>Method that Wyer-drawers ob&longs;erve in di&longs;gro&longs;&longs;ing Gold Wyer: <lb/>which in reality is not Gold, &longs;ave only in the Superficies, for the <lb/>internal &longs;ub&longs;tance is Silver; and the way of di&longs;gro&longs;&longs;ing it is this. <lb/></s> <s>They take a Cylinder, or, if you will, Ingot of Silver, about half <lb/>a yard long, and about three or four Inches thick, and this they <lb/><arrow.to.target n="marg1041"></arrow.to.target><lb/>gild or over-lay with Leaves of beaten Gold, which, you know, <lb/>is &longs;o thin that the Wind will blow it to and again, and of the&longs;e <lb/>Leaves they lay on eight or ten, and no more. </s> <s>So &longs;oon as it is <lb/>gilded, they begin to draw it forth with extraordinary force, ma­<lb/>king it to pa&longs;s thorow the hole of the Drawing Iron, and then <lb/>reiterate this forceable di&longs;gro&longs;sment again and again thorow holes <lb/>&longs;ucce&longs;&longs;ively narrower, &longs;o that, after &longs;everal of the&longs;e di&longs;gro&longs;ments, <lb/>they bring it to the &longs;malne&longs;s of the hair of a womans head, if not <lb/>&longs;maller, and yet it &longs;till continueth gilded in its Superficies or out­<lb/>&longs;ide: Now I leave you to con&longs;ider to what a finene&longs;s and di&longs;ten&longs;i­<lb/>on the &longs;ub&longs;tance of the Gold is brought.</s></p><p type="margin"> <s><margin.target id="marg1040"></margin.target><emph type="italics"/>Gold in the gilding <lb/>of Silver is drawn <lb/>forth and di&longs;gro&longs;­<lb/>&longs;ed immen&longs;ly.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1041"></margin.target>* Or Thumb­<lb/>breadths.</s></p><p type="main"> <s>SIMP. </s> <s>I do not &longs;ee how it can be inferred from this Experi­<lb/>ment, that there may be a di&longs;gro&longs;ment of the matter of the Gold <lb/>&longs;ufficient to effect tho&longs;e wonders which you &longs;peak of: Fir&longs;t, For <lb/>that the fir&longs;t gilding was with ten Leaves of Gold, which make a <lb/>con&longs;iderable thickne&longs;s: Secondly, howbeit in the exten&longs;ion and <lb/>di&longs;gro&longs;ment that Silver encrea&longs;eth in length, it yet withall dimi­<lb/>ni&longs;heth &longs;o much in thickne&longs;s, that compen&longs;ating the one dimen&longs;i­<lb/>on with the other, the Superficies doth not &longs;o enlarge, as that for <lb/>overlaying the Silver with Gold, the &longs;aid Gold need to be reduced <lb/>to a greater thinne&longs;s than that of its fir&longs;t Leaves.</s></p><p type="main"> <s>SALV. </s> <s>You much deceive your &longs;elf, <emph type="italics"/>Simplicius,<emph.end type="italics"/> for the en­<lb/>crea&longs;e of the Superficies is Subduple to the exten&longs;ion in length, as <lb/>I could Geometrically demon&longs;trate to you.</s></p><p type="main"> <s>SAGR. </s> <s>I be&longs;eech you, both in the behalf of my &longs;elf, and of <lb/><emph type="italics"/>Simplicius,<emph.end type="italics"/> to favour us with that Demon&longs;tration, if &longs;o be you <lb/>think that we can comprehend it.</s></p><p type="main"> <s>SALV. </s> <s>I will &longs;ee whether I can, thus upon the &longs;udden, recall <lb/>it to mind. </s> <s>It is already manife&longs;t, that that &longs;ame fir&longs;t gro&longs;s Cylin­<lb/>der of Silver, and the Wyer di&longs;tended to &longs;o great a length are two <lb/>equal Cylinders, in regard that they are the &longs;ame Silver; &longs;o that <pb xlink:href="069/01/046.jpg" pagenum="44"/>if I &longs;hall &longs;hew you what proportion the Superficies of equall Cy­<lb/>linders have to one another, we &longs;hall obtain our de&longs;ire. </s> <s>I &longs;ay there­<lb/>fore, that</s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s><emph type="italics"/>The Superficies of Equal Cylinders, their Ba&longs;es being <lb/>&longs;ub&longs;tracted, are to one another in &longs;ubduple proportion <lb/>of their lengths.<emph.end type="italics"/></s></p><p type="main"> <s>Take two equall Cylinders, the heights of which let be A B, <lb/>and C D: and let the Line E be a Mean-proportional <lb/>between them. </s> <s>I &longs;ay, the Superficies of the Cylinder A B, <lb/>the Ba&longs;es &longs;ub&longs;tracted, hath the &longs;ame proportion to the Superficies <lb/>of the Cylinder C D, the Ba&longs;es in like manner &longs;ub&longs;tracted, as the <lb/>Line A B hath to the Line E, which is &longs;ubduple of the proportion <lb/>of A B to C D. </s> <s>Cut the part of the Cylinder A B in F, and let the <lb/>height A F be equal to C D: And becau&longs;e the Ba&longs;es of equal Cy­<lb/>linders an&longs;wer Reciprocally to their heights, the Circle, Ba&longs;e of <lb/>the Cylinder C D, to the Circle, Ba&longs;e of the <lb/><figure id="id.069.01.046.1.jpg" xlink:href="069/01/046/1.jpg"/><lb/>Cylinder A B, &longs;hall be as the height B A to <lb/>D C: And becau&longs;e Circles are to one ano­<lb/>ther as the Squares of their Diameters, the <lb/>&longs;aid Squares &longs;hall have the &longs;ame proportion, <lb/>that B A hath to C D: But as B A, is to <lb/>C D, &longs;o is the Square B A to the Square of <lb/>E: Therefore tho&longs;e four Squares are Pro­<lb/>portionals: And therefore their Sides &longs;hall <lb/>be Proportionals. </s> <s>And as the Line A B is to <lb/>E, &longs;o is the Diameter of the Circle C to the <lb/>Diameter of the Circle A: But as are the <lb/>Diameters, &longs;o are the Circumferences; and <lb/>as are the Circumferences, &longs;o likewi&longs;e are the Superficies of Cylin­<lb/>ders equal in Height. </s> <s>Therefore as the Line A B is to E, &longs;o is the <lb/>Superficies of the Cylinder C D to the Superficies of the Cylinder <lb/>A F. </s> <s>Becau&longs;e therefore the height A F to the height A B, is as the <lb/>Superficies A F to the Superficies A B: And as is the height A B <lb/>to the Line E, &longs;o is the Superficies C D to the Superficies A F: <lb/>Therefore by Perturbation of Proportion as the height A F is to <lb/>E, &longs;o is the Superficies C D to the Superficies A B: And, by Con­<lb/>ver&longs;ion, as the Superficies of the Cylinder A B is to the Superficies <lb/>of the Cylinder C D, &longs;o is the Line E to the Line A F; that is, to <lb/>the Line C D: or as A B to E: Which is in &longs;ubduple proportion <lb/>of A B to C D: Which is that which was to be proved.</s></p><pb xlink:href="069/01/047.jpg" pagenum="45"/><p type="main"> <s>Now if we apply this, that hath been demon&longs;trated, to our <lb/>purpo&longs;e; pre&longs;uppo&longs;ing that that &longs;ame Cylinder of Silver, that was <lb/>gilded whil&longs;t it was no more than half a yard long, and four or five <lb/>Inches thick, being di&longs;gro&longs;&longs;ed to the &longs;inene&longs;s of an hair, is prolon­<lb/>ged unto the exten&longs;ion of twenty thou&longs;and yards (for its length <lb/>would be much greater) we &longs;hall find its Superficies augmented <lb/>to two hundred times its former greatne&longs;s: and con&longs;equently, tho&longs;e <lb/>Leaves of Gold, which were laid on ten in number, being di&longs;ten­<lb/>ded on a Superficies two hundred times bigger, a&longs;&longs;ure us that the <lb/>Gold which covereth the Superficies of the &longs;o many yards of Wyer <lb/>is left of no greater thickne&longs;s than the twentieth part of a Leaf of <lb/>ordinary Beaten-Gold. </s> <s>Con&longs;ider, now, how great its thinne&longs;s is, and <lb/>whether it is po&longs;&longs;ible to imagine it done without an immen&longs;e di­<lb/>&longs;tention of its parts: and whether this &longs;eem to you an Experi­<lb/>ment, that tendeth likewi&longs;e towards a compo&longs;ition of infinite In­<lb/>divi&longs;ibles in Phy&longs;ical Matters: Howbeit there want not other more <lb/>&longs;trong and nece&longs;&longs;ary proofs of the &longs;ame.</s></p><p type="main"> <s>SAGR. </s> <s>The Demon&longs;tration &longs;eemeth to me &longs;o ingenuous, that <lb/>although it &longs;hould not be of force enough to prove that fir&longs;t intent <lb/>for which it was produced, (and yet, in my opinion, it plainly <lb/>makes it out) yet neverthele&longs;s that &longs;hort &longs;pace of time was well <lb/>&longs;pent which hath been employed in hearing of it.</s></p><p type="main"> <s>SALV. </s> <s>In regard I &longs;ee, that you are &longs;o well plea&longs;ed with the&longs;e <lb/>Geometrical Demon&longs;trations, which bring with them certain pro. <lb/></s> <s>fit, I will give you the fellow to this, which &longs;atisfieth to a very cu­<lb/>rious Que&longs;tion. </s> <s>In the former we have that which hapneth in <lb/>Cylinders that are equall, but of different heights or lengths: it <lb/>will be convenient, that you al&longs;o hear that which occurreth in Cy­<lb/>linders equal in Superficies, but unequal in heights; my meaning <lb/>alwaies is, in tho&longs;e Superficies only that encompa&longs;s them about, <lb/>that is, not comprehending the two Ba&longs;es &longs;uperiour and inferiour. <lb/></s> <s>I &longs;ay, therefore, that</s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s><emph type="italics"/>Upon Cylinders, the Superficies of which the Ba&longs;es be­<lb/>ing &longs;ub&longs;tracted are equal, have the &longs;ame proportion <lb/>to one another as their heights Reciprocally taken.<emph.end type="italics"/></s></p><p type="main"> <s>Let the Superficies of the two Cylinders A E and C F be <lb/>equall; but the height of this C D greater than the height <lb/>of the other A B. </s> <s>I &longs;ay, that the Cylinder A E hath the <lb/>&longs;ame proportion to the Cylinder C F, that the height C D hath <lb/>to A B. </s> <s>Becau&longs;e therefore the Superficies C F is equall to the <pb xlink:href="069/01/048.jpg" pagenum="46"/>&longs;uperficies A E, the Cylinder C F &longs;hall be le&longs;&longs;e than A E: For <lb/>if they were equal, its Superficies, by the la&longs;t Propo&longs;ition would <lb/>be greater than the Superficies A E, and <lb/><figure id="id.069.01.048.1.jpg" xlink:href="069/01/048/1.jpg"/><lb/>much the more, if the &longs;aid Cylinder C F <lb/>were greater than A E. </s> <s>Let the Cylinder <lb/>I D be &longs;uppo&longs;ed equal to A E: There­<lb/>fore, by the precedent Propo&longs;ition, the <lb/>Superficies of the Cylinder I D &longs;hall be <lb/>to the Superficies A E, as the height I F <lb/>to the Mean-proportional betwixt I F & <lb/>A B. </s> <s>But the Superficies A E being by <lb/>Suppo&longs;ition equal to C F and I D, ha­<lb/>ving the &longs;ame proportion to C F that the <lb/>height I F hath to C D: Therefore <lb/>C D is the Mean-Proportional between <lb/>I F and A B. Moreover, the Cylinder <lb/>I D being equal to the Cylinder A E, <lb/>they &longs;hall both have the &longs;ame proporti­<lb/>on to the Cylinder C F: But I D is to <lb/>C F, as the height I F is to C D: Therefore the Cylinder A E <lb/>&longs;hall have the &longs;ame proportion to the Cylinder C F, that the line <lb/>I F hath to C D; that is, that C D hath to A B: Which was to be <lb/>demon&longs;trated.<lb/><arrow.to.target n="marg1042"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1042"></margin.target><emph type="italics"/>Of Corn-&longs;acks <lb/>with a Board at <lb/>the Bottom, made <lb/>of the &longs;ame Stuffe, <lb/>but different in <lb/>height, which are <lb/>the more capa­<lb/>cious.<emph.end type="italics"/></s></p><p type="main"> <s>From hence is collected the Cau&longs;e of an Accident, which the <lb/>Vulgar do not hearken to without admiration; and it is, how it <lb/>is po&longs;&longs;ible that the &longs;ame piece of ^{*}Cloth, being longer one way than <lb/>another, if a Sack be made thereof to hold Corn, as the u&longs;ual <lb/>manner is, with a Board at the bottom, will hold more, making <lb/>u&longs;e of the le&longs;&longs;er breadth of the Cloth, for the height of the Sack, </s></p><p type="main"> <s><arrow.to.target n="marg1043"></arrow.to.target><lb/>and with the other encompa&longs;&longs;ing the Board at the bottom, than if <lb/>it be made up the other way: As if for Example, the Cloth were <lb/>one way &longs;ix foot, and the other way twelve, it will hold more, <lb/>when with the length of twelve one encompa&longs;&longs;eth the Board at the <lb/>bottom, the Sack being &longs;ix foot high, than if it encompa&longs;&longs;ed a <lb/>bottom of &longs;ix foot, having twelve for its height. </s> <s>Now, by what <lb/>hath been demon&longs;trated, there is added to the Knowledge in ge­<lb/>neral that it holds more that way than this, the Specifick, and <lb/>particular Knowledge of how much it holdeth more: which is, <lb/>That it will hold more in proportion as it is lower, and le&longs;&longs;er, as <lb/>it is higher. </s> <s>And thus in the mea&longs;ures afore taken, the Cloth be­<lb/>ing twice as long as broad, when it is &longs;ewed the length-ways it will <lb/>hold but half &longs;o much, as it will do the other way. </s> <s>And likewi&longs;e <lb/><arrow.to.target n="marg1044"></arrow.to.target><lb/>having a Mat to make a ^{*} Frale or Basket twenty five foot long, <lb/>and &longs;uppo&longs;e &longs;even broad; made up the long-way it will hold but <lb/>onely &longs;even of tho&longs;e mea&longs;ures, whereof the other way it will hold <lb/>five and twenty.</s></p><pb xlink:href="069/01/049.jpg" pagenum="47"/><p type="margin"> <s><margin.target id="marg1043"></margin.target>* Or Sacking.</s></p><p type="margin"> <s><margin.target id="marg1044"></margin.target>* Bugnola, any <lb/>Ve&longs;&longs;el made of <lb/>Rushes or Wick­<lb/>er.</s></p><p type="main"> <s><emph type="italics"/>S<emph.end type="italics"/>AGR. </s> <s>And thus to our particular content we continually di&longs;­<lb/>cover new Notions of great Curio&longs;ity, and not unaccompanyed <lb/>with Utility. </s> <s>But in the particular glanced at but even now, I <lb/>really believe, that among&longs;t &longs;uch as are altogether void of the <lb/>knowledge of Geometry, there would not be found one in twen­<lb/>ty, but at the fir&longs;t da&longs;h would not be mi&longs;taken, and wonder <lb/>that tho&longs;e Bodies that are contained within equal Superficies, <lb/>&longs;hould not likewi&longs;e be in every re&longs;pect equal; like as they run in­<lb/>to the &longs;ame errour, &longs;peaking of the Superficies, when for deter­<lb/>mining, as it frequently falls out, of the amplene&longs;&longs;e of &longs;everal <lb/>Cities, they think they have obtained their de&longs;ire &longs;o &longs;oon as they <lb/>know the &longs;pace of their Circuits, not knowing that one Circuit <lb/>may be equal to another, and yet the place conteined by this <lb/>much larger than the place of that: which befalleth not onely in <lb/>irregular Superficies, but in the regular; among&longs;t which tho&longs;e <lb/>of more Sides are alwayes more capacious than tho&longs;e of fewer; <lb/>&longs;o that in fine, the Circle, as being a Poligon of infinite Sides, is <lb/>more capacious than all other Poligons of equal Perimeter; of <lb/>which I remember, that I with particular delight &longs;aw the Demon­<lb/>&longs;tration on a time when I &longs;tudied the Sphere of <emph type="italics"/>Sacrobo&longs;co,<emph.end type="italics"/> with <lb/>a very learned Commentary upon the &longs;ame.</s></p><p type="main"> <s>SALV. </s> <s>It is mo&longs;t certain; and I having likewi&longs;e light upon <lb/>that very place, it gave me occa&longs;ion to inve&longs;tigate, how it may <lb/>with one &longs;ole Demon&longs;tration be concluded, that the Circle is <lb/>greater than all the re&longs;t of regular I&longs;operemitral Figures, and of <lb/>others, tho&longs;e of more Sides bigger than tho&longs;e of fewer.</s></p><p type="main"> <s>SAGR. </s> <s>And I that take great plea&longs;ure in certain &longs;elect and no­<lb/>wi&longs;e-trivial Demon&longs;trations, entreat you with all importunity to <lb/>make me a partaker therein.</s></p><p type="main"> <s>SALV. </s> <s>I &longs;hall di&longs;patch the &longs;ame in few words, demon&longs;trating <lb/>the following Theorem, namely;</s></p><pb xlink:href="069/01/050.jpg" pagenum="48"/><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s><emph type="italics"/>The Circle is a Mean-Proportional betwixt any two <lb/>Regular Homogeneal Poligons, one of which is cir­<lb/>cum&longs;cribed about it, and the other I&longs;operimetral to <lb/>it: Moreover, it being le&longs;&longs;e than all the circum&longs;cri­<lb/>bed, it is, on the contrary, bigger than all the I&longs;operi­<lb/>metral. </s> <s>And, again of the circum&longs;cribed, tho&longs;e that <lb/>have more angles are le&longs;&longs;er than tho&longs;e that have <lb/>fewer; and on the other &longs;ide of the I&longs;operimetral, <lb/>tho&longs;e of more angles are bigger.<emph.end type="italics"/></s></p><p type="main"> <s>Of the two like Poligons A and B, let A be circum&longs;cribed <lb/>about the Circle A, and let the other B, be I&longs;operime­<lb/>tral to the &longs;aid Circle: I &longs;ay, that the Circle is the Mean­<lb/>proportional betwixt them. </s> <s>For that (having drawn the Semidi­<lb/>ameter A C) the Circle being equal to that Right-angled Trian­<lb/>gle, of who&longs;e Sides including the Right angle, the one is equal <lb/><figure id="id.069.01.050.1.jpg" xlink:href="069/01/050/1.jpg"/><lb/>to the Semidiameter A C, and the other to the Circumference: <lb/>And likewi&longs;e the Poligon A being equal to the right angled Tri­<lb/>angle, that about the right angle hath one of its Sides equal to <lb/>the &longs;aid right line A C, and the other to the Perimeter of the &longs;aid <lb/>Poligon: It is manife&longs;t, that the circum&longs;cribed Poligon hath the <lb/>&longs;ame proportion to the Circle, that its Perimeter hath to the Cir­<lb/>cumference of the &longs;aid Circle; that is, to the Perimeter of the <lb/>Poligon B, which is &longs;uppo&longs;ed equal to the &longs;aid Circumference: <lb/>But the Poligon A hath a proportion to the Poligon B, double to <lb/>that of its Perimeter, to the Perimeter of B (they being like Fi­<lb/>gures:) Therefore the Circle A is the Mean-proportional be­<lb/>tween the two Poligons A and B. </s> <s>And the Poligon A being <lb/>bigger than the Circle A, it is manife&longs;t that the &longs;aid Circle <lb/>A is bigger than the Poligon B, its I&longs;operimetral, and con&longs;e­<lb/>quently the greate&longs;t of all Regular Poligons that are I&longs;operimetral <pb xlink:href="069/01/051.jpg" pagenum="49"/>to it. </s> <s>As to the other particular, that is to prove, that of the <lb/>Poligons circum&longs;cribed about the &longs;ame Circle, that of fewer <lb/>Sides is bigger than that of more Sides; but that, on the contrary, of <lb/>the I&longs;operimetral Poligons, that of more Sides is bigger than that <lb/>of fewer Sides, we will thus demon&longs;trate. </s> <s>In the Circle who&longs;e <lb/>Center is O, and Semidiameter O A, let there be a Tangent <lb/>A D, and in it let it be &longs;uppo&longs;ed, for example, that A D is the <lb/>half of the Side of the Pentagon circum&longs;cribed, and A C the half <lb/>of the Side of the Heptagon, and draw the right lines O G C, <lb/>and O F D; and on the Center O, at the di&longs;tance O C, draw the <lb/>Arch E C I: And becau&longs;e the Triangle D O C is greater than the <lb/>Sector E O C, and the Sector C O I greater than the Triangle <lb/>C O A; the Triangle D O C &longs;hall have greater proportion to <lb/>the Triangle C O A, than the Sector E O C, to the Secant C O I, <lb/>that is, than the Secant F O G to the Secant G O A. And, by <lb/>Compo&longs;ition, Permutation of Proportion, the Triangle D O A <lb/>&longs;hall have greater proportion to the Secant F O A, than the Tri­<lb/>angle C O A to the Secant G O A: And ten Triangles D O A <lb/>&longs;hall have greater proportion to ten Secants F O A, than four­<lb/>teen Triangles C O A to fourteen Sectors G O A: That is the <lb/>circum&longs;cribed Pentagon &longs;hall have greater proportion to the Cir­<lb/>cle, than hath the Heptagon: And therefore the Pentagon &longs;hall <lb/>be greater than the Heptagon. </s> <s>Let us now &longs;uppo&longs;e an Hep­<lb/>tagon and a Pentagon I&longs;operimetral to the &longs;ame Circle. </s> <s>I &longs;ay, that <lb/>the Heptagon is bigger than the Pentagon. </s> <s>For that the &longs;aid Cir­<lb/>cle being the Mean proportional between the Pentagon circum­<lb/>&longs;cribed and the Pentagon its I&longs;operimetral, and likewi&longs;e the Mean <lb/>between the Circum&longs;cribed and I&longs;operimetral Heptagon: It ha­<lb/>ving been proved that the Circum&longs;cribed Pentagon is greater then <lb/>the Circum&longs;cribed Heptagon, the &longs;aid Pentagon &longs;hall have greater <lb/>proportion to the Circle, than the Heptagon: that is, the Circle <lb/>&longs;hall have greater proportion to its I&longs;operimetral Pentagon, than <lb/>to its I&longs;operimetral Heptagon: Therefore the Pentagon is le&longs;&longs;er <lb/>than the I&longs;operimetral Heptagon. </s> <s>Which was to be demon­<lb/>&longs;trated</s></p><p type="main"> <s>SAGR. </s> <s>A mo&longs;t ingenious Demon&longs;tration, and very acute. </s> <s>But <lb/>whither are we run to ingulph our &longs;elves in Geometry, when as <lb/>we were about to con&longs;ider the Difficulties propo&longs;ed by <emph type="italics"/>Simpli­<lb/>cius,<emph.end type="italics"/> which indeed are very con&longs;iderable, and in particular, that <lb/>of Conden&longs;ation, is in my opinion, very ab&longs;truce.</s></p><p type="main"> <s>SALV. </s> <s>If Conden&longs;ation and Rarefaction are oppo&longs;ite Motions, <lb/>where there is &longs;een an immen&longs;e Rarefaction, one cannot deny an <lb/>extraordinary Conden&longs;ation: but immen&longs;e Rarefactions, and, <lb/>which encrea&longs;eth the wonder, almo&longs;t Momentary, we &longs;ee every <lb/>day: for what a boundle&longs;&longs;e Rarefaction is that of a little quan­<pb xlink:href="069/01/052.jpg" pagenum="50"/><arrow.to.target n="marg1045"></arrow.to.target><lb/>tity of Gunpowder re&longs;olved into a va&longs;t ma&longs;&longs;e of Fire? </s> <s>And what, <lb/>beyond this, the (I could almo&longs;t &longs;ay) indeterminate Expan&longs;ion <lb/>of its Light? </s> <s>And if that Fire and this Light &longs;hould reunite toge­<lb/>ther, which yet is no impo&longs;&longs;ibility, in regard, that at the fir&longs;t <lb/>they lay in that little room, what a Conden&longs;ation would this be? <lb/></s> <s>If you &longs;tudy for them, you will find hundreds of &longs;uch Rarefacti­<lb/>ons, which are much more readily ob&longs;erved, than Conden&longs;ati­<lb/>ons: for Den&longs;e matters are more tractable, and &longs;ubject to our <lb/>Sen&longs;es. </s> <s>For we can ea&longs;ily order Wood at plea&longs;ure, and we &longs;ee <lb/>it re&longs;olved into Fire, and into Light, but we do not in the &longs;ame <lb/>manner &longs;ee the Fire and the Light Conden&longs;e to the making of <lb/>Wood: We &longs;ee Fruits, Flowers, and many other &longs;olid matters <lb/>re&longs;olved in a great mea&longs;ure into Odors, but we do not after the <lb/>&longs;ame manner &longs;ee the odoriferous Atomes concurre to the con&longs;titu­<lb/>tion of the Oderate Solids; but where Sen&longs;ible Ob&longs;ervation is <lb/>wanting, we are to &longs;upply it with Rea&longs;on, which will &longs;uffice to <lb/>make us apprehen&longs;ive, no le&longs;&longs;e of the Motion to the Rarefaction <lb/>and re&longs;olution of Solids, than, to the Conden&longs;ation of rare and <lb/>mo&longs;t tenuous Sub&longs;tances. </s> <s>Moreover, we que&longs;tion how to effect <lb/>the Conden&longs;ation and Rarefaction of the Bodies which may be <lb/>rarefied and conden&longs;ed, &longs;tudying in what manner it may be done <lb/>without introducing of a <emph type="italics"/>Vacuum,<emph.end type="italics"/> and Penetration of Bodies; <lb/>which doth not hinder, but that in Nature there may be matters <lb/>which admit no &longs;uch accidents, and con&longs;equently do not allow <lb/>roome for tho&longs;e things which you phra&longs;e inconvenient and im­<lb/>po&longs;&longs;ible. </s> <s>And la&longs;tly, <emph type="italics"/>Simplicius,<emph.end type="italics"/> I have on the the &longs;core of &longs;atis­<lb/>fying you, and tho&longs;e Philo&longs;ophers that hold with you, taken <lb/>&longs;ome pains in con&longs;idering how Conden&longs;ation and Rarefaction <lb/>may be under&longs;tood to be performed without admitting Penetra­<lb/>tion of Bodies, and introducing the Void Spaces called Vacuities, <lb/>Effects which you deny and abhorre: for if you would but grant <lb/>them, I would no longer &longs;o re&longs;olutely contradict you. </s> <s>There­<lb/>fore either admit the&longs;e Inconveniences, or accept of my Spe­<lb/>culations, or el&longs;e finde out others more conducing to the <lb/>purpo&longs;e.</s></p><p type="margin"> <s><margin.target id="marg1045"></margin.target><emph type="italics"/>Rarefaction im­<lb/>min&longs;e is that of <lb/>a little Gunpow­<lb/>der into a va&longs;t <lb/>ma&longs;s of Fire.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>As to the denying of Penetration, I am wholly of opi­<lb/>nion with the Peripatetick Philo&longs;ophers; as to that of a <emph type="italics"/>Vacuum,<emph.end type="italics"/><lb/>I would &longs;ee the Demon&longs;tration of <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/> thorowly examined, <lb/>wherewith he oppo&longs;eth the &longs;ame, and what you, <emph type="italics"/>Salviatus,<emph.end type="italics"/> will <lb/>an&longs;wer to it. <emph type="italics"/>Simplicius<emph.end type="italics"/> &longs;hall do me the favour punctually to <lb/>recite the proof of the Philo&longs;opher; and you, <emph type="italics"/>Salviatus,<emph.end type="italics"/> to an­<lb/>&longs;wer it.</s></p><p type="main"> <s>SIMP. <emph type="italics"/>Ari&longs;totle,<emph.end type="italics"/> as neer as I can remember, breaks out again&longs;t <lb/>certain of the Ancients, who introduced Vacuity, as nece&longs;&longs;ary <lb/>to Motion, &longs;aying, that this without that could not be effected; <pb xlink:href="069/01/053.jpg" pagenum="51"/>to this <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/> making oppo&longs;ition, demon&longs;trateth, that on the <lb/>contrary, the effecting of Motion (as we &longs;ee) de&longs;troyeth the Po&longs;iti­<lb/>on of <emph type="italics"/>Vacuum<emph.end type="italics"/>; and his method therein is this. </s> <s>He maketh two <lb/>Suppo&longs;itions, one is touching Moveables different in Gravity <lb/>moved in the &longs;ame <emph type="italics"/>Medium:<emph.end type="italics"/> the other is concerning the &longs;ame <lb/>Moveable moved in &longs;everal <emph type="italics"/>Medium's.<emph.end type="italics"/> As to the fir&longs;t, he &longs;uppo­<lb/>&longs;eth that Moveables different in Gravity, move in the &longs;ame <lb/><emph type="italics"/>Medium<emph.end type="italics"/> with unequal Velocities, which bear to each other the <lb/>&longs;ame proportion as their Gravities: &longs;o that, for example, a Move­<lb/>able ten times heavier than another, moveth ten times more &longs;wift­<lb/>ly. </s> <s>In the other Po&longs;ition he a&longs;&longs;umes, that the Velocity of the <lb/>&longs;ame Moveable in different <emph type="italics"/>Medium's<emph.end type="italics"/> are in Reciprocal to that of <lb/>the thickne&longs;&longs;e or Den&longs;ity of the &longs;aid <emph type="italics"/>Medium's<emph.end type="italics"/>: &longs;o that &longs;uppo­<lb/>&longs;ing <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> that the Cra&longs;&longs;itude of the Water was ten times as great <lb/>as that of the Air, he will have the Velocity in the Air to be <lb/>ten times more than the Velocity in the Water. </s> <s>And from this &longs;e­<lb/>cond A&longs;&longs;umption he draweth his Demon&longs;tration in this manner. <lb/></s> <s>Becau&longs;e the tenuity of <emph type="italics"/>Vacuum<emph.end type="italics"/> infinitely &longs;urpa&longs;&longs;eth the corpu­<lb/>lence, though never &longs;o &longs;ubtil, of any whatever Replete <emph type="italics"/>Medi­<lb/>um,<emph.end type="italics"/> every Moveable that in the Replete <emph type="italics"/>Medium<emph.end type="italics"/> moveth a cer­<lb/>tain &longs;pace in a certain time, in a <emph type="italics"/>Vacuum<emph.end type="italics"/> would pa&longs;&longs;e the &longs;ame <lb/>in an in&longs;tant: But to make a Motion in an in&longs;tant is impo&longs;&longs;ible: <lb/>Therefore to introduce Vacuity in the accompt of Motion is im­<lb/>po&longs;&longs;ible.</s></p><p type="main"> <s>SALV. </s> <s>The Argument one may &longs;ee to be <emph type="italics"/>ad hominem,<emph.end type="italics"/> that is, <lb/><arrow.to.target n="marg1046"></arrow.to.target><lb/>again&longs;t tho&longs;e who would make a <emph type="italics"/>Vacuum<emph.end type="italics"/> nece&longs;&longs;ary to Motion; <lb/>but if I &longs;hall admit of the Argument as concludent, granting <lb/>withal, that in Vacuity there would be no Motion; yet the Po&longs;i­<lb/>tion of Vacuity taken ab&longs;olutely, and not in relation to Motion, <lb/>is not thereby overthrown. </s> <s>But to tell you what tho&longs;e Ancients, <lb/>peradventure, might an&longs;wer, that &longs;o we may the better di&longs;cover <lb/>how far the Demon&longs;tration of <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/> holds good, methinks that <lb/>one might oppo&longs;e his A&longs;&longs;umptions, denying them both. </s> <s>And as <lb/>to the fir&longs;t: I greatly doubt that <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/> never experimented <lb/>how true it is, that two &longs;tones, one ten times heavier than the o­<lb/>ther, let fall in the &longs;ame in&longs;tant from an height, <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> of an hun­<lb/>dred yards, were &longs;o different in their Velocity, that upon the <lb/>arrival of the greater to the ground, the other was found not to <lb/>have de&longs;cended &longs;o much as ten yards.</s></p><p type="margin"> <s><margin.target id="marg1046"></margin.target>Ari&longs;totle's <emph type="italics"/>Argu­<lb/>ment again&longs;t a<emph.end type="italics"/><lb/>Vacuum <emph type="italics"/>is<emph.end type="italics"/> ad <lb/>hominem.</s></p><p type="main"> <s>SIMP. Why, it may be &longs;een by his own words, that he confe&longs;­<lb/>&longs;eth he had made the Experiment, for he &longs;aith, [<emph type="italics"/>We &longs;ee the more <lb/>grave<emph.end type="italics"/>] now that <emph type="italics"/>Seeing<emph.end type="italics"/> implieth that he had tried the Experi­<lb/>ment.</s></p><p type="main"> <s>SAGR. </s> <s>But I, <emph type="italics"/>Simplicius,<emph.end type="italics"/> that have made proof thereof, do a&longs;­<lb/>&longs;ure you, that a Cannon bullet that weigheth one hundred, rwo <pb xlink:href="069/01/054.jpg" pagenum="52"/>hundred, and more pounds, will not one Palme anticipate the ar­<lb/>rival of a Musket-bullet to the ground, that weigheth but half <lb/>a pound, falling likewi&longs;e from an height of two hundred yards.</s></p><p type="main"> <s><emph type="italics"/>S<emph.end type="italics"/>ALV. </s> <s>But without any other Experiments, we may by &longs;hort <lb/>and nece&longs;&longs;ary Demon&longs;trations cleerly prove, that it is not true that <lb/>a Moveable more grave moveth more &longs;wiftly than another le&longs;&longs;e <lb/>grave, confining our meaning &longs;till to Moveables of the &longs;ame Mat­<lb/>ter; and, in &longs;hort, to tho&longs;e of which <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/> &longs;peaketh. </s> <s>For tell <lb/>me, <emph type="italics"/>Simplicius<emph.end type="italics"/> whether you admit, that to every cadent grave <lb/>Body there belongeth by nature one determinate Velocity; &longs;o <lb/>as that it cannot be encrea&longs;ed or dimini&longs;hed in it without u&longs;ing vi­<lb/>olence to it, or impo&longs;ing &longs;ome impediment upon it?</s></p><p type="main"> <s>SIMP. </s> <s>It cannot be doubted, but that the &longs;ame Moveable in <lb/>the &longs;ame <emph type="italics"/>Medium<emph.end type="italics"/> hath one e&longs;tabli&longs;hed and by-nature-determinate <lb/>Velocity, which cannot be increa&longs;ed, unle&longs;&longs;e with new <emph type="italics"/>Impetus<emph.end type="italics"/><lb/>conferred on it, or dimini&longs;hed, &longs;ave onely by &longs;ome impediment <lb/>that retards it.</s></p><p type="main"> <s>SALV. </s> <s>If therefore we had two Moveables, the natural Velo­<lb/>cities of which were unequal, it is manife&longs;t, that if we joyned the <lb/>&longs;lower with the &longs;wifter, this would be in part retarded by the <lb/>&longs;lower, and that in part accelerated by the other more &longs;wift. </s> <s>Do <lb/>not you concur with me in this opinion?</s></p><p type="main"> <s>SIMP. </s> <s>I think that it ought undoubtedly &longs;o to &longs;ucceed.</s></p><p type="main"> <s>SALV. </s> <s>But if this be &longs;o, and, it be likewi&longs;e true that a great <lb/>Stone moveth with (&longs;uppo&longs;e) eight degrees of Velocity, and a le&longs;­<lb/>&longs;er with fewer, then joyning them both together, the compound <lb/>of them will move with a Velocity le&longs;&longs;e than eight Degrees: But <lb/>the two Stones joyned together make one Stone greater than <lb/>that before, which moved with eight degrees of Velocity: There­<lb/>fore this greater Stone moveth le&longs;&longs;e &longs;wiftly than the le&longs;&longs;er, which <lb/>is contrary to your Suppo&longs;ition. </s> <s>You &longs;ee therefore, that from the <lb/>&longs;uppo&longs;ing that the more grave Moveable moveth more &longs;wiftly <lb/>than the le&longs;&longs;e grave, I prove unto you that the more grave mo­<lb/>veth le&longs;&longs;e &longs;wiftly.</s></p><p type="main"> <s>SIMP. </s> <s>I find my &longs;elf at a lo&longs;&longs;e, for the truth is, that the le&longs;­<lb/>&longs;er Stone being joyned to the greater, weight is added unto it, and <lb/>weight being added to it, I cannot &longs;ee why there &longs;hould not Ve­<lb/>locity be added to it, or at lea&longs;t why it &longs;hould be dimini&longs;hed <lb/>in it.</s></p><p type="main"> <s>SALV. </s> <s>Here you run into another errour, <emph type="italics"/>Simplicius,<emph.end type="italics"/> for it <lb/>is not true, that that &longs;ame le&longs;&longs;er Stone encrea&longs;eth the weight of <lb/>the greater.</s></p><p type="main"> <s>SIMP. </s> <s>Oh wonderful! this quite &longs;urpa&longs;&longs;eth my apprehen&longs;ion.</s></p><p type="main"> <s>SALV. </s> <s>Not at all, if you will but &longs;tay till I have di&longs;covered <lb/>to you the Equivokes, of which you are in doubt: Therefore <pb xlink:href="069/01/055.jpg" pagenum="53"/>you mu&longs;t know that it is nece&longs;&longs;ary to di&longs;tingui&longs;h betwixt grave <lb/>Bodies &longs;et on Moving, and the &longs;ame con&longs;tituted in Re&longs;t; a Stone <lb/>put into the Ballance not onely acquireth greater weight, by lay­<lb/>ing another Stone upon it, but al&longs;o the addition of, a Flake of <lb/>Hemp will make it weigh more by tho&longs;e &longs;ix or ten ounces that <lb/>the Hemp &longs;hall weigh; but if you &longs;hould freely let fall the Stone <lb/>tied to the Hemp from an high place, do you think that in the <lb/>Motion the Hemp weigheth down the Stone, &longs;o as to accelerate <lb/>its Motion; or el&longs;e do you believe that it will retard it, &longs;u&longs;tain­<lb/>ing it in part? </s> <s>We indeed feel our &longs;houlders laden, &longs;o long as we <lb/>will oppo&longs;e the Motion that the weight would make which lyeth <lb/>upon our backs; but if we &longs;hould de&longs;cend with the &longs;ame Velocity <lb/>wherewith that &longs;ame grave Body would naturally de&longs;cend, in what <lb/>manner will you that it pre&longs;&longs;e or bear upon us? </s> <s>Do not you &longs;ee <lb/>that this would be a wounding one with a Lance that runneth <lb/>before you, with as much or more &longs;peed than you pur&longs;ue him. <lb/></s> <s>You may conclude therefore that in the free and natural fall, the <lb/>le&longs;&longs;er Stone doth not bear upon the greater, and con&longs;equently doth <lb/>not encrea&longs;e their weight, as it doth in Re&longs;t.</s></p><p type="main"> <s>SIMP. </s> <s>But what if the greater was put upon the le&longs;&longs;er?</s></p><p type="main"> <s>SALV. </s> <s>It would encrea&longs;e their weight, in ca&longs;e its Motion were <lb/>more &longs;wift; but it hath been already concluded, that in ca&longs;e the <lb/>le&longs;&longs;er &longs;hould be more &longs;low it would in part retard the Velocity of <lb/>the greater, &longs;o that there Compound would move le&longs;&longs;e &longs;wiftly; <lb/>being greater than the other, which is contrary to your A&longs;&longs;umpti­<lb/>on: Let us conclude therefore, that great Moveables, and like­<lb/>wi&longs;e little, being of the &longs;ame Specifical Gravity, move with like <lb/>Velocity.</s></p><p type="main"> <s>SIMP. </s> <s>Your di&longs;cour&longs;e really is full of ingenuity, yet methinks <lb/>it is hard to conceive that a drop of Bird-&longs;hot, &longs;hould move as <lb/>&longs;wiftly as a Canon-bullet.</s></p><p type="main"> <s>SALV. </s> <s>You may &longs;ay a grain of Sand as fa&longs;t as a Mill-&longs;tone. <lb/></s> <s>I would not have you, <emph type="italics"/>Simplicius,<emph.end type="italics"/> to do as &longs;ome others are wont <lb/>to do, and diverting the di&longs;cour&longs;e from the principal de&longs;ign, fa­<lb/>&longs;ten upon &longs;ome one &longs;aying of mine that may want an hairs-breadth <lb/>of the truth, and under this hair hide a defect of another man as <lb/>big as the Cable of a Ship. <emph type="italics"/>Aristotle<emph.end type="italics"/> &longs;aith, a Ball of Iron of an <lb/>hundred pounds weight falling, from an height of an hundred yards, <lb/>commeth to the ground before that one of one pound is de&longs;cended <lb/>one &longs;ole yard: I &longs;ay, that they arrive at the earth both in the &longs;ame <lb/>time: You find, that the bigger anticipates the le&longs;&longs;er two Inches, <lb/>that is to &longs;ay, that when the great one falls to the ground, the o­<lb/>ther is di&longs;tant from it two inches: you go about to hide under <lb/>the&longs;e two inches the ninety nine yards of <emph type="italics"/>Ari&longs;totle,<emph.end type="italics"/> and &longs;peaking <lb/>onely to my &longs;mall errour, pa&longs;&longs;e over in &longs;ilence the other great one. <pb xlink:href="069/01/056.jpg" pagenum="54"/><emph type="italics"/>Ari&longs;totlee<emph.end type="italics"/> affirmeth, that Moveables of different Gravities in the <lb/>&longs;ame <emph type="italics"/>Medium<emph.end type="italics"/> move (as far as concerneth Gravity) with Veloci­<lb/>ties proportionate to their Weights; and exemplifieth it by <lb/>Moveables, wherein one may di&longs;cover the pure and ab&longs;olute effect <lb/>of Weight, omitting the other Con&longs;iderations, as well of Figures, <lb/>as of the minute Motions; which things receive great alteration <lb/>from the <emph type="italics"/>Medium,<emph.end type="italics"/> which altereth the &longs;imple effect of the &longs;ole <lb/>Gravity; wherefore we &longs;ee Gold, that is heavier than any other <lb/>matter, being reduced into a very thin Leaf, to go flying to and <lb/>again through the Air, the like do Stones beaten to very &longs;mall <lb/>Powder. </s> <s>But if you would maintain the Univer&longs;al Propo&longs;ition, it <lb/>is requi&longs;ite that you &longs;hew the proportion of the Velocities to be <lb/>ob&longs;erved in all grave Bodies, and that a Stone of twenty pounds <lb/>moveth ten times &longs;wifter than one of two: which, I tell you, is <lb/>fal&longs;e, and that falling from an height of fifty or an hundred yards, <lb/>they come to the ground in the &longs;ame in&longs;tant.</s></p><p type="main"> <s>SIMP. </s> <s>Perhaps in very great heights of Thou&longs;ands of yards <lb/>that would happen, which is not &longs;een to occur in the&longs;e le&longs;&longs;er <lb/>heights.</s></p><p type="main"> <s>SALV. </s> <s>If this was the Meaning of <emph type="italics"/>Ari&longs;totle,<emph.end type="italics"/> you have in­<lb/>volved him in another Errour, which will be found a Lie; for <lb/>there being no &longs;uch perpendicular altitudes found on the Earth, <lb/>its a clear ca&longs;e, that <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/> was not able to have made an Experi­<lb/>ment thereof; and yet would per&longs;wade us that he had, whil&longs;t he <lb/>&longs;aith, that the &longs;aid effect is <emph type="italics"/>&longs;een.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/> indeed makes no u&longs;e of this Principle, but of <lb/>that other, which I believe is not obnoxious to the&longs;e doubts.</s></p><p type="main"> <s>SALV. </s> <s>Why that al&longs;o is no le&longs;&longs;e fal&longs;e than this; and I admire <lb/>that you do not of your &longs;elf perceive the fallacy, and di&longs;cern, that <lb/>&longs;hould it be true, that the &longs;ame Moveable in <emph type="italics"/>Medium's<emph.end type="italics"/> of dif­<lb/>ferent Subtilty and Rarity, and, in a word, of different Ce&longs;&longs;ion, <lb/>&longs;uch, for example, as are Water and Air, move with a greater <lb/>Velocity in the Air than in the Water, according to the propor­<lb/>tion of the Airs Rarity to the Rarity of the Water, it would <lb/>follow that every Moveable that de&longs;cendeth in the Air would <lb/>de&longs;cend al&longs;o in the Water: Which is &longs;o fal&longs;e, that very many <lb/>Bodies de&longs;cend in the Air, that in the Water do not onely not <lb/>de&longs;cend, but al&longs;o ri&longs;e upwards.</s></p><p type="main"> <s>SIMP I do not under&longs;tand the nece&longs;&longs;ity of your Con&longs;equence: <lb/>and I will &longs;ay farther, that <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/> &longs;peaketh of tho&longs;e Grave­<lb/>bodies that de&longs;cend in the one <emph type="italics"/>Medium<emph.end type="italics"/> and in the other, and not <lb/>of tho&longs;e that de&longs;cend in the Air and a&longs;cend in the Water.</s></p><p type="main"> <s>SALV. </s> <s>You produce for the Phil&longs;opher &longs;uch Pleas as he, with­<lb/>out all doubt, would never alledge, for that they aggravate the <lb/>fir&longs;t mi&longs;take. </s> <s>Therefore tell me, if the Cra&longs;situde of the Water, <pb xlink:href="069/01/057.jpg" pagenum="55"/>or whatever it be that retardeth the Motion, hath any proporti­<lb/>on to the Cra&longs;&longs;itude of the Air that le&longs;&longs;e retards it; and if it have; <lb/>do you a&longs;&longs;ign it us, at plea&longs;ure.</s></p><p type="main"> <s>SIMP. </s> <s>It hath &longs;uch a proportion, and we will &longs;uppo&longs;e it to be <lb/>decuple; and that therefore the Velocity of a Grave Body, that <lb/>de&longs;cends in both the Elements, &longs;hall be ten times &longs;lower in the Wa­<lb/>ter than in the Air.</s></p><p type="main"> <s>SALV. </s> <s>I will take one of tho&longs;e Grave-Bodies that de&longs;cend in <lb/>the Air, but not in the Water; as for in&longs;tance, a Ball of Wood, <lb/>and de&longs;ire that you will a&longs;&longs;ign it what Velocity you plea&longs;e, whil&longs;t it <lb/>de&longs;cends through the Air.</s></p><p type="main"> <s>SIMP. </s> <s>Suppo&longs;e we, that it move with twenty degrees of Velo­<lb/>city.</s></p><p type="main"> <s>SALV. </s> <s>Very well: And it is manife&longs;t, that that Velocity to <lb/>&longs;ome other le&longs;&longs;er, may have the &longs;ame proportion, that the Cra&longs;&longs;i­<lb/>tude of the Water hath to that of the Air; and that this &longs;hall be <lb/>the Velocity of the two only degrees: &longs;o that exactly to an hair, <lb/>and in direct conformity to the A&longs;&longs;umption of <emph type="italics"/>Ari&longs;totle,<emph.end type="italics"/> it &longs;hould <lb/>be concluded, That the Ball of Wood, which in the Air, ten times <lb/>more yielding, moveth de&longs;cending with twenty degrees of Veloci­<lb/>ty, in the Water &longs;hould de&longs;cend with two, and not return from the <lb/>bottom to flote a-top, as it doth: unle&longs;s you will &longs;ay, that the <lb/>a&longs;cending of the Wood to the top is the &longs;ame in the Water, as its <lb/>&longs;inking to the bottom with two degrees of Velocity; which I do <lb/>not believe. </s> <s>But &longs;eeing that the Ball of Wood de&longs;cends not to the <lb/>bottom, I rather think that you will grant me, that &longs;ome other Ball, <lb/>of other matter different from Wood, might be found that de&longs;cends <lb/>in the Water with two degrees of Velocity.</s></p><p type="main"> <s>SIMP. </s> <s>Que&longs;tionle&longs;&longs;e there might; but it mu&longs;t be of a matter <lb/>con&longs;iderably more grave than Wood.</s></p><p type="main"> <s>SALV. </s> <s>This is that which I de&longs;ired to know. </s> <s>But this &longs;econd <lb/>Ball, which in the Water de&longs;cendeth with two degrees of Velocity, <lb/>with what Velocity will it de&longs;cend in the Air? </s> <s>It is requi&longs;ite (if <lb/>you will maintain <emph type="italics"/>Ari&longs;totles<emph.end type="italics"/> Rule) that you an&longs;wer that it will <lb/>move with twenty degrees: But you your &longs;elf have a&longs;&longs;igned twen­<lb/>ty degrees of Velocity to the Ball of Wood; Therefore this, and <lb/>the other that is much more grave, will move thorow the Air with <lb/>equall Velocity. </s> <s>Now how doth the Philo&longs;opher reconcile this <lb/>Conclu&longs;ion with that other of his, that the Moveables of different <lb/>Gravity, move in the &longs;ame <emph type="italics"/>Medium<emph.end type="italics"/> with different Velocities, and <lb/>&longs;o different as are their Gravities? </s> <s>But, without any deep &longs;tudies, <lb/>how comes it to pa&longs;s that you have not ob&longs;erved very frequent, <lb/>and very palpable Accidents, and not con&longs;idered two Bodies, that in <lb/>the Water will move one an hundred times more &longs;wiftly than the <lb/>other, but that again in the Air that &longs;wifter one will not out-go the <pb xlink:href="069/01/058.jpg" pagenum="56"/>other, one &longs;ole Cente&longs;m? </s> <s>As for example, an Egge of Marble will <lb/>de&longs;cend in the Water an hundred times fa&longs;ter than one of an Hen, <lb/>when as in the Air, at the height of twenty Yards it will not anti­<lb/>cipate it four Inches: and, in a word, &longs;uch a certain Grave Body <lb/>will &longs;ink to the bottom in three hours in ten fathom Water, that <lb/>in the Air will pa&longs;s the &longs;ame &longs;pace in one or two pul&longs;es, and &longs;uch <lb/>another (as for in&longs;tance a Ball of Lead) will pa&longs;s that number of <lb/>fathoms with ea&longs;e in le&longs;s than double the time. </s> <s>And here I &longs;ee <lb/>plainly, <emph type="italics"/>Simplicius,<emph.end type="italics"/> that you find, that herein there is no place left <lb/>for any di&longs;tinction, or reply. </s> <s>Conclude we therefore, that that <lb/>&longs;ame Argument concludeth nothing again&longs;t <emph type="italics"/>Vacuum<emph.end type="italics"/>; and if it <lb/>&longs;hould, it would only overthrow Spaces con&longs;iderably great, which <lb/>neither I, nor, as I take it, tho&longs;e <emph type="italics"/>Ancients<emph.end type="italics"/> did &longs;uppo&longs;e to be natu­<lb/>rally allowed, though, perhaps, with violence they may be effe­<lb/>cted, as, me thinks, one may collect from &longs;everal Experiments, which <lb/>it would be two tedious to go about at pre&longs;ent to produce.</s></p><p type="main"> <s>SAGR. </s> <s>Seeing that <emph type="italics"/>Simplicius<emph.end type="italics"/> is &longs;ilent, I will take leave to &longs;ay <lb/>&longs;omething. </s> <s>In regard you have with &longs;ufficient plainne&longs;&longs;e demon­<lb/>&longs;trated, that it is not true, That Moveables unequally grave move in <lb/>the &longs;ame <emph type="italics"/>Medium<emph.end type="italics"/> with Velocities proportionate to their Gravities, <lb/>but with equal: de&longs;iring to be under&longs;tood to &longs;peak of Bodies of the <lb/>&longs;ame Matter, or of the &longs;ame Specifick Gravity, but not (as I con­<lb/>ceive) of Gravities different <emph type="italics"/>in Spetie,<emph.end type="italics"/> (for I do not think that <lb/>you intend to prove unto us, that a Ball of Cork moveth with like <lb/>Velocity to one of Lead;) and having moreover very manife&longs;tly <lb/>demon&longs;trated, that it is not true, That the &longs;ame Moveable in <emph type="italics"/>Me­<lb/>diums<emph.end type="italics"/> of different Re&longs;i&longs;tances retain in their Velocities and Tardi­<lb/>ties the &longs;ame proportion as have their Re&longs;i&longs;tances: to me it would <lb/>be a very plea&longs;ing thing to hear, what tho&longs;e be which are ob&longs;erved <lb/>as well in the one ca&longs;e as in the other.</s></p><p type="main"> <s>SALV. </s> <s>The Que&longs;tions are ingenuous, and I have many times <lb/>thought of them: I will relate unto you the Contemplations made <lb/>upon them, and what at length I did from thence infer. </s> <s>After I <lb/>had a&longs;&longs;ured my &longs;elf that it was not true, That the &longs;ame Moveable <lb/>in <emph type="italics"/>Medium's<emph.end type="italics"/> of different Re&longs;i&longs;tance ob&longs;erveth in its Velocity the <lb/>proportion of the Ce&longs;&longs;ion of tho&longs;e <emph type="italics"/>Media<emph.end type="italics"/>; nor yet, again, That in <lb/>the &longs;ame <emph type="italics"/>Medium<emph.end type="italics"/> Moveables of different Gravity retain in their <lb/>Velocities the proportion of tho&longs;e Gravities (&longs;peaking alwaies of <lb/>Gravitles different <emph type="italics"/>in &longs;pecie<emph.end type="italics"/>) I began to put both the&longs;e Accidents <lb/>together, ob&longs;erving that which befell the Moveables different in <lb/>Gravity put into <emph type="italics"/>Mediums<emph.end type="italics"/> of different Re&longs;i&longs;tance, and I perceived <lb/>that the inequality of the Velocities were found to be alwaies <lb/>greater in the more re&longs;i&longs;ting <emph type="italics"/>Medium's,<emph.end type="italics"/> than in the more yielding; <lb/>and that with &longs;uch a diver&longs;ity, that of two Moveables that, de­<lb/>&longs;cending thorow the Air, differ very little in Velocity of Motion, <pb xlink:href="069/01/059.jpg" pagenum="57"/>one will, in the Water, move ten times fa&longs;ter than the other; <lb/>yea: that &longs;uch, as in the Air do &longs;wiftly de&longs;cend, in the Water not <lb/>only will not de&longs;cend, but will be wholly deprived of Motion, <lb/>and, which is yet more, will move upwards: for one &longs;hall &longs;ome­<lb/>times find &longs;ome kind of Wood, or &longs;ome knot, or root of the &longs;ame, <lb/>that in the Water will lye &longs;till, when as in the Air it will &longs;wiftly <lb/>de&longs;cend.</s></p><p type="main"> <s>SAGR. </s> <s>I have many times &longs;et my &longs;elf with an extream patience <lb/>to &longs;ee if I could reduce a Ball of Wax, (which of it &longs;elf doth not <lb/>go to the bottom) by adding to it grains of &longs;and, to &longs;uch a degree <lb/>of Gravity like to the Water, as to make it &longs;tand &longs;till in the <lb/>mid&longs;t of that Element; but I could never, by all the care I <lb/>u&longs;ed, &longs;ucceed in my attempt; &longs;o that I cannot tell, whether any <lb/>Solid matter may be found &longs;o naturally alike in Gravity to Wa­<lb/>ter, as that being put into any place of the &longs;ame, it can re&longs;t or lye <lb/>&longs;till.</s></p><p type="main"> <s>SALV. </s> <s>In this, as well as in a thou&longs;and other actions, many <lb/>Animals are more ingenuous than we. </s> <s>And, in this ca&longs;e, Fi&longs;hes <lb/><arrow.to.target n="marg1047"></arrow.to.target><lb/>would have been able to have given you &longs;ome light, being in this <lb/>affair &longs;o skilful, that at their plea&longs;ure they ^{*} equilibrate them&longs;elves, <lb/><arrow.to.target n="marg1048"></arrow.to.target><lb/>not only with one kind of Water, but with &longs;uch, as, either of their <lb/>own nature, or by means of &longs;ome &longs;upervenient muddine&longs;s, or for <lb/>their &longs;altne&longs;s (which maketh a great alteration) are very diffe­<lb/>rent; equilibrate them&longs;elves, I &longs;ay, &longs;o exactly, that without &longs;tir­<lb/>ring in the lea&longs;t they lye &longs;till in every place: and this, in my opi­<lb/>nion, they do, by making u&longs;e of the In&longs;trument given them by Na­<lb/>ture to that end, <emph type="italics"/>&longs;cilicet,<emph.end type="italics"/> of that Bladder which they have in their <lb/>Bodies, which by a very narrow neck an&longs;wereth to their mouth; <lb/>and by that they either, when they would &longs;tand &longs;till, &longs;end forth <lb/>part of the Air that is contained in the &longs;aid Bladders, or, &longs;wimming <lb/>to the top they draw in more, making them&longs;elves by that art one <lb/>while more, another while le&longs;s heavy than the Water, and at their <lb/>plea&longs;ures equilibrating them&longs;elves to the &longs;ame.</s></p><p type="margin"> <s><margin.target id="marg1047"></margin.target><emph type="italics"/>Fi&longs;hes equilibrate <lb/>them&longs;elves admi­<lb/>rably in the Water.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1048"></margin.target>* Or poi&longs;e.</s></p><p type="main"> <s>SAGR I deceived &longs;ome of my Friends with another device; <lb/>for I had made my boa&longs;t unto them, that I would reduce that Ball <lb/>of Wax to an exact <emph type="italics"/>equilibrium<emph.end type="italics"/> with the Water, and having put <lb/>&longs;ome &longs;alt Water in the bottom of the Ve&longs;&longs;el, and a-top of that &longs;ome <lb/>fre&longs;h, I &longs;hewed them the Ball, which in the mid&longs;t of the Water <lb/>&longs;tood &longs;till, and being thru&longs;t to the bottom, or to the top, &longs;taid nei­<lb/>ther in this nor that &longs;cituation, but returned to the mid&longs;t.</s></p><p type="main"> <s>SALV. </s> <s>This &longs;ame Experiment is not void of utility; for Phy&longs;i­<lb/><arrow.to.target n="marg1049"></arrow.to.target><lb/>cians, in particular, treating of &longs;undry qualities of Waters, and <lb/>among&longs;t other things, principally of the more or le&longs;s Gravity or <lb/>Levity of this or that: by &longs;uch a Ball, in &longs;uch manner poi&longs;ed and <lb/>adju&longs;ted that it may re&longs;t ambiguous, if I may &longs;o &longs;ay, between <pb xlink:href="069/01/060.jpg" pagenum="58"/>a&longs;cending and de&longs;cending in a Water, upon the lea&longs;t difference <lb/>of weight between two Waters, if that Ball &longs;hall de&longs;cend in the <lb/>one; in the other, that is more grave, it &longs;hall a&longs;cend. </s> <s>And the <lb/>Experiment is &longs;o exact, that the addition of but only two grains <lb/>of Salt, put into &longs;ix pounds of Water, &longs;hall make that Ball to <lb/>a&longs;cend from the bottom to the &longs;urface, which was but a little be­<lb/><arrow.to.target n="marg1050"></arrow.to.target><lb/>fore de&longs;cended thither. </s> <s>And moreover, I will tell you this in con­<lb/>firmation of the exactne&longs;s of this Experiment, and withall for a <lb/>clear proof of the Non-re&longs;i&longs;tance of Water to divi&longs;ion, that not <lb/>only the ingravitating it with the mixture of &longs;ome matter heavier <lb/>than it, maketh that &longs;o notable difference, but the warming or <lb/>cooling of it a little produceth the &longs;ame effect, and with &longs;o &longs;ubtil <lb/>an operation, that the infu&longs;ing four diops of other Water, a lit­<lb/>tle warmer, or a little colder, than the &longs;ix pounds, &longs;hall cau&longs;e the <lb/>Ball to ri&longs;e or &longs;ink in the &longs;ame; to &longs;ink in it upon the infu&longs;ion of <lb/>the warm, and to ri&longs;e at the infu&longs;ion of the cold. </s> <s>Now &longs;ee how <lb/>much tho&longs;e Philo&longs;ophers are deceived, who would introduce in <lb/>Water vi&longs;co&longs;ity, or other conjunction of parts which make it to <lb/>re&longs;i&longs;t Divi&longs;ion or Penetration.</s></p><p type="margin"> <s><margin.target id="marg1049"></margin.target><emph type="italics"/>A Ball of Wax <lb/>prepared to make <lb/>the Experiment of <lb/>the different Gra­<lb/>vities of Waters.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1050"></margin.target><emph type="italics"/>Water bath no <lb/>Re&longs;i&longs;tance to Di­<lb/>vi&longs;ion.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>I have &longs;een many Convincing Di&longs;cour&longs;es touching <lb/><arrow.to.target n="marg1051"></arrow.to.target><lb/>this Argument in a ^{*} Treati&longs;e of our <emph type="italics"/>Accademick<emph.end type="italics"/>; yet never the le&longs;s <lb/>there is re&longs;ting in me a &longs;trong &longs;cruple, which I know not how to <lb/>remove: For if nothing of Tenacity, or Coherence re&longs;ides among&longs;t <lb/>the parts of Water, how can it bear it &longs;elf up in rea&longs;onable big <lb/>and high Tumours; in particular, upon the leaves of Cole-worts <lb/>without di&longs;per&longs;ing or levelling?</s></p><p type="margin"> <s><margin.target id="marg1051"></margin.target>* The Tract cited <lb/>in this place is <lb/>that which we <lb/>di&longs;po&longs;e fir&longs;t in <lb/>Order, in the <lb/>fir&longs;t part of this <lb/>Tome,</s></p><p type="main"> <s>SALV. </s> <s>Although it be true, that he who is Ma&longs;ter of a true <lb/>Conclu&longs;ion, may re&longs;olve all Objections that can be brought again&longs;t <lb/>it, yet will not I arrogate to my &longs;elf the power &longs;o to do; nor <lb/>ought my in&longs;ufficiency becloud the &longs;plendour of Truth. </s> <s>Fir&longs;t, <lb/>therefore, I confe&longs;s that I know not how it cometh to pa&longs;s, that <lb/>tho&longs;e Globes of Water &longs;u&longs;tain them&longs;elves at &longs;uch an height and <lb/>bigne&longs;s, albeit I certainly know that it doth not proceed from any <lb/><arrow.to.target n="marg1052"></arrow.to.target><lb/>internal Tenacity that is between its parts; &longs;o that it remaineth <lb/>neœ&longs;&longs;ary, that the Cau&longs;e of that Effect do re&longs;ide without. </s> <s>That it <lb/>is not Internal, be&longs;ides tho&longs;e Experiments already &longs;hewn you, I can <lb/>prove by another mo&longs;t convincing one. </s> <s>If the parts of that Wa­<lb/>ter, which con&longs;erveth it &longs;elf in a Globe or Tumour whil&longs;t it is en­<lb/>compa&longs;&longs;ed by the Air, had an internal Cau&longs;e for &longs;o doing, they <lb/>would much better &longs;u&longs;tain them&longs;elves being environed by a <emph type="italics"/>Medi­<lb/>um,<emph.end type="italics"/> in which they had le&longs;s propen&longs;ion to de&longs;cend, than they have <lb/>in the Ambient Air: But every Fluid Body more grave than the <lb/>Air would be &longs;uch a <emph type="italics"/>Medium<emph.end type="italics"/>; as, for in&longs;tance, Wine: And there­<lb/>fore, infu&longs;ing Wine about that Globe of Water, it might rai&longs;e it <lb/>&longs;elf on every &longs;ide, and yet the parts of the Water, conglutinated <pb xlink:href="069/01/061.jpg" pagenum="59"/>by the internal Vi&longs;co&longs;ity, never di&longs;&longs;olve: But it doth not happen <lb/>&longs;o; nay, no &longs;ooner doth the circumfu&longs;ed liquor approach thereto, <lb/>but, without &longs;taying till it ri&longs;e much about it, the little globes of <lb/>Water will di&longs;&longs;olve and become flat, re&longs;ting under the Wine, if it <lb/>was red. </s> <s>The Cau&longs;e therefore of this Effect is External, and per­<lb/>haps in the Ambient Air: and, indeed, one may ob&longs;erve a great <lb/>di&longs;&longs;ention between the Air and Water; which I have ob&longs;erved <lb/>in another Experiment; and this it is: If I fill a ^{*} Ball of Chri&longs;tal, <lb/><arrow.to.target n="marg1053"></arrow.to.target><lb/>that hath a mouth as narrow as the hollow of a &longs;traw, with water, <lb/>and when it is thus full, turn it with its mouth downwards, yet will <lb/>not the Water, although very heavy, and prone to de&longs;cend tho­<lb/>row the Air, nor the Air, as much di&longs;po&longs;ed on the other hand, as <lb/>being very light, to a&longs;cend thorow the Waters, yet will they not <lb/>(I &longs;ay) agree that that &longs;hould de&longs;cend, i&longs;&longs;uing out at the mouth, <lb/>and this a&longs;cend, entering in at the &longs;ame: but they both continue <lb/>aver&longs;e and contumacious. </s> <s>Again, on the contrary, if I pre&longs;ent to <lb/>that mouth a ve&longs;&longs;el of red Wine, which is almo&longs;t in&longs;en&longs;ibly le&longs;s <lb/>grave than Water, we &longs;hall &longs;ee it in an in&longs;tant gently to a&longs;cend by <lb/>red &longs;treams thorow the Water, and the Water with like Tardity to <lb/>de&longs;cend through the Wine, without ever mixing with each other, <lb/>till that in the end, the Ball will be full of Wine, and the Water <lb/>Will all &longs;ink unto the bottom of the Ve&longs;&longs;el underneath. </s> <s>Now <lb/>what are we to &longs;ay, or what are we to infer, but a di&longs;agreement <lb/>between the Water and Air, occult to me, but perhaps -----</s></p><p type="margin"> <s><margin.target id="marg1052"></margin.target><emph type="italics"/>Water formed into <lb/>great drops upon <lb/>the Leaves of Col­<lb/>worts, how they <lb/>con&longs;i&longs;t.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1053"></margin.target>* Or bottle.</s></p><p type="main"> <s>SIMP. </s> <s>I can &longs;carce refrain my laughter to &longs;ee the great Anti­<lb/>pathy that <emph type="italics"/>Salviatus<emph.end type="italics"/> hath to Antipathy, &longs;o that he will not &longs;o much <lb/>as name it, and yet it is &longs;o accommodate to re&longs;olve the doubt.</s></p><p type="main"> <s>SALV. </s> <s>Now let this, for the &longs;ake of <emph type="italics"/>Simplicius<emph.end type="italics"/> be the &longs;oluti­<lb/>on of our &longs;cruple; and leaving the Digre&longs;&longs;ion, let us return to our <lb/>purpo&longs;e. </s> <s>Seeing that the difference of Velocity in Moveables of <lb/>divers Gravities is found to be more and more, as the <emph type="italics"/>Mediums<emph.end type="italics"/> are <lb/>more and more Re&longs;i&longs;ting: And withall, that in a <emph type="italics"/>Medium<emph.end type="italics"/> of <lb/>Quick&longs;ilver, Gold doth not only go to the bottom more &longs;wiftly <lb/>than Lead, but it alone de&longs;cends in it, and all other Metals and <lb/>Stones move upwards therein, and flote thereon; whereas between <lb/>Balls of Gold, Lead, Bra&longs;s, Porphiry, or other grave matters, the in­<lb/>equality of motion in the Air &longs;hall be almo&longs;t wholly in&longs;en&longs;ible, for <lb/>it is certain, that a Ball of Gold in the end of the de&longs;cent of an <lb/><arrow.to.target n="marg1054"></arrow.to.target><lb/>hundred yards &longs;hall not out-&longs;trip one of Bra&longs;s four Inches: &longs;eeing <lb/>this, I &longs;ay, I have thought, that if we wholly took away the <lb/>Re&longs;i&longs;tance of the <emph type="italics"/>Medium,<emph.end type="italics"/> all Matters would de&longs;cend with equall <lb/>Velocity.</s></p><p type="margin"> <s><margin.target id="marg1054"></margin.target><emph type="italics"/>Re&longs;i&longs;tance of the<emph.end type="italics"/><lb/>Medium <emph type="italics"/>remo­<lb/>ved, all Matters, <lb/>though of different <lb/>Gravities would <lb/>move with like <lb/>Velocity.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. </s> <s>This is a bold &longs;peech, <emph type="italics"/>Salviatus,<emph.end type="italics"/> I &longs;hall never believe <lb/>that in <emph type="italics"/>Vacuity<emph.end type="italics"/> it &longs;elf, if &longs;o be one &longs;hould allow Motion in it, a lock <lb/>of Wooll would move as &longs;wiftly as a piece of Lead.</s></p><pb xlink:href="069/01/062.jpg" pagenum="60"/><p type="main"> <s>SALV. </s> <s>Fair and &longs;oftly, <emph type="italics"/>Simplicius,<emph.end type="italics"/> your &longs;cruple is not &longs;o ab­<lb/>&longs;truce, nor I &longs;o incautelous, that you &longs;hould need to think that I <lb/>was not advi&longs;ed of it, and that con&longs;equently I have not found a re­<lb/>ply to it. </s> <s>Therefore, for my explanation, and your information, <lb/>hearken to what I &longs;hall &longs;ay. </s> <s>We are upon the examination of <lb/>what would befall Moveables exceeding different in weight in a <lb/><emph type="italics"/>Medium,<emph.end type="italics"/> in ca&longs;e it &longs;hould have no Re&longs;i&longs;tance, &longs;o that all the diffe­<lb/>rence of Velocity that is found between the &longs;aid Moveables ought <lb/>to be referred to the &longs;ole inequality of Weight. </s> <s>And becau&longs;e on­<lb/>ly a Space altogether void of Air, and of every other, though te­<lb/>nuous and yielding Body, would be apt &longs;en&longs;ibly to &longs;hew us what <lb/>we &longs;eek, &longs;ince we want &longs;uch a Space, let us &longs;ucce&longs;&longs;ively ob&longs;erve that <lb/>which happeneth in the more &longs;ubtill and le&longs;&longs;e re&longs;i&longs;ting <emph type="italics"/>Mediums,<emph.end type="italics"/><lb/>in compari&longs;on of that which we &longs;ee to happen in others le&longs;&longs;e &longs;ubtill <lb/>and more re&longs;i&longs;ting: for if we &longs;hould really find the Moveables <lb/>different in Gravity to differ le&longs;&longs;e and le&longs;&longs;e in Velocity, according <lb/>as the <emph type="italics"/>Mediums<emph.end type="italics"/> are found more and more yielding; and that, <lb/>finally, although extreamly unequal in weight, in a <emph type="italics"/>Medium<emph.end type="italics"/> more <lb/>tenuous than any other, though not void, the difference of Velo­<lb/>city di&longs;covers it &longs;elf to be very &longs;mall, and almo&longs;t unob&longs;ervable, I <lb/>conceive that we may, and that upon very probable conjecture, <lb/>believe, that in a <emph type="italics"/>Vacuum<emph.end type="italics"/> their Velocities would be exactly equal. <lb/></s> <s>Therefore let us con&longs;ider that which hapneth in the Air; wherein <lb/>to have a Figure of an uniform Superficies, and very light Matter, <lb/>I will that we take a blown Bladder, in which the included Air <lb/>will weigh little or nothing in a <emph type="italics"/>Medium<emph.end type="italics"/> of the Air it &longs;elf, becau&longs;e <lb/>it can make but very &longs;mall Compre&longs;&longs;ion therein, &longs;o that the Gravi­<lb/>ty is only that little of the &longs;aid film, which would not be the thou­<lb/>&longs;andth part of the weight of a lump of Lead of the bigne&longs;s of <lb/>the &longs;aid Bladder when blown. </s> <s>The&longs;e, <emph type="italics"/>Simplicius,<emph.end type="italics"/> being let fall <lb/>from the height of four or &longs;ix yards, how great a &longs;pace, do you <lb/>judge, that the Lead would anticipate the Bladder in its de&longs;cent? <lb/></s> <s>A&longs;&longs;ure your &longs;elf that would not move thrice, no nor twice as fa&longs;t, <lb/>although even now you would have had it to have been a thou­<lb/>&longs;and times more &longs;wift.</s></p><p type="main"> <s>SIMP. </s> <s>It is po&longs;&longs;ible that at the beginning of the Motion, that <lb/>is, in the fir&longs;t five or &longs;ix yards this might happen that you &longs;ay; but <lb/>in the progre&longs;&longs;e, and in a long continuation I believe, that the Lead <lb/>would leave it behind, not only &longs;ix, but al&longs;o eight and ten parts of <lb/>twelve.</s></p><p type="main"> <s>SALV. </s> <s>And I al&longs;o believe the &longs;ame: and make no que&longs;tion, <lb/>but that in very great di&longs;tances the Lead will have pa&longs;&longs;ed an hun­<lb/>dred miles of <emph type="italics"/>way,<emph.end type="italics"/> ere the Bladder will have pa&longs;&longs;ed &longs;o much as one. <lb/></s> <s>But this, <emph type="italics"/>Simplicius,<emph.end type="italics"/> which you propound, as an effect contrary to <lb/>my A&longs;&longs;ertion, is that which mo&longs;t e&longs;pecially confirmeth it. </s> <s>It is (I <pb xlink:href="069/01/063.jpg" pagenum="61"/>once more tell you) my intent to declare, That the difference of <lb/>Gravity is in no wi&longs;e the cau&longs;e of the divers velocities of Movea­<lb/>bles of different Gravity, but that the &longs;ame dependeth on exteri­<lb/>our accidents, & in particular, on the Re&longs;i&longs;tance of the <emph type="italics"/>Medium,<emph.end type="italics"/> &longs;o <lb/>that, this being removed, all Moveables move with the &longs;ame de­<lb/>grees of Velocity. </s> <s>And this I chiefly deduce from that which but <lb/>now you your &longs;elf did admit, and which is very true, namely, that <lb/>of two Moveables, very different in weight, the Velocities more and <lb/>more differ, according as the ^{*} Spaces are greater and greater that <lb/><arrow.to.target n="marg1055"></arrow.to.target><lb/>they pa&longs;&longs;e: an Effect which would not follow, if it did depend on <lb/>the different Gravities: for they being alwaies the &longs;ame, the pro­<lb/>portion betwixt the Spaces would likewi&longs;e alwaies continue the <lb/>&longs;ame, which proportion we &longs;ee &longs;till &longs;ucce&longs;&longs;ively to encrea&longs;e in the <lb/>continuance of the Motion; for that the heavie&longs;t Moveable in the <lb/>de&longs;cent of one yard will not anticipate the lighte&longs;t the tenth part <lb/>of that Space or Way, but in the fall of twelve yards will out-go <lb/>it a third part, in that of an hundred will out&longs;trip it 90/100.</s></p><p type="margin"> <s><margin.target id="marg1055"></margin.target>* Or Waies.</s></p><p type="main"> <s>SIMP. </s> <s>Very well: But following you &longs;tep by &longs;tep, if the dif­<lb/>ference of weight in Moveables of different Gravities cannot <lb/>cau&longs;e the difference of proportion in their Velocities, for that the <lb/>Gravities do not alter; neither then can the <emph type="italics"/>Medium,<emph.end type="italics"/> which is <lb/>&longs;uppo&longs;ed alwaies to continue the &longs;ame, cau&longs;e any alteration in the <lb/>proportion of the Velocities.</s></p><p type="main"> <s>SALV. </s> <s>You wittily bring an in&longs;tance again&longs;t my Po&longs;ition, that <lb/><arrow.to.target n="marg1056"></arrow.to.target><lb/>it is very nece&longs;&longs;ary to remove. </s> <s>I &longs;ay therefore, that a Grave Body <lb/>hath, by Nature, an intrin&longs;ick Principle of moving towards the <lb/>Common Center of heavy things, that is to that of our Terre&longs;trial <lb/>Globe, with a Motion continually accelerated, and accelerated <lb/>alwaies equally, <emph type="italics"/>&longs;cilicet,<emph.end type="italics"/> that in equal times there are made equal <lb/>^{*} additions of new Moments, and degrees of Velocities: and this <lb/>ought to be under&longs;tood to hold true at all times when all acciden­<lb/><arrow.to.target n="marg1057"></arrow.to.target><lb/>tal and external impediments are removed; among&longs;t which there <lb/>is one that we cannot obviate, that is the Impediment of the <emph type="italics"/>Me­<lb/>dium,<emph.end type="italics"/> which is Repleat, when as it &longs;hould be opened and latterally <lb/>moved by the falling Moveable, to which tran&longs;ver&longs;e Motion the <lb/><emph type="italics"/>Medium,<emph.end type="italics"/> though fluid, yielding and tranquile, oppo&longs;eth it &longs;elf <lb/>with a Re&longs;i&longs;tance one while le&longs;&longs;er, and another while greater and <lb/>greater, according as it is more &longs;lowly or ha&longs;tily to open to give <lb/>pa&longs;&longs;age to the Moveable, which, becau&longs;e, as I have &longs;aid, it goeth <lb/>of its own nature continually accelerating, it cometh of con&longs;e­<lb/>quence to encounter continually greater Re&longs;i&longs;tance in the <emph type="italics"/>Medi­<lb/>um,<emph.end type="italics"/> and therefore Retardment, and diminution in the acqui&longs;t of <lb/>new degrees of Velocity; &longs;o that in the end, the Velocity arriveth <lb/>to that &longs;wiftne&longs;&longs;e, and the Re&longs;i&longs;tance of the <emph type="italics"/>Medium,<emph.end type="italics"/> to that <lb/>&longs;trength, that ballancing each other, they take away all further <pb xlink:href="069/01/064.jpg" pagenum="62"/>Acceleration, and reduce the Moveable to an Equable and Uni­<lb/>form Motion, in which it afterwards continually abides. </s> <s>There is <lb/>therefore in the <emph type="italics"/>Medium<emph.end type="italics"/> augmentation of Re&longs;i&longs;tance, not becau&longs;e <lb/>it changeth its E&longs;&longs;ence, but becau&longs;e the Velocity altereth where­<lb/>with it ought to open, and laterally move, to give pa&longs;&longs;age to the <lb/>falling Body, which goeth continually accelerating. </s> <s>Now the <lb/>ob&longs;erving, that the Re&longs;i&longs;tance of the Air to the &longs;mall Moment or <lb/><emph type="italics"/>Impetus<emph.end type="italics"/> of the Bladder is very great, and to the great weight of <lb/>the Lead is very &longs;mall, makes me hold for certain, that if one &longs;hould <lb/>wholly remove it, by adding to the Bladder great a&longs;&longs;i&longs;tance, and <lb/>but very little to the Lead, their Velocities would equalize each <lb/>other. </s> <s>Taking this Principle therefore for granted, That in the <lb/><emph type="italics"/>Medium<emph.end type="italics"/> wherein, either by rea&longs;on of Vacuity, or otherwi&longs;e, there <lb/>were no Re&longs;i&longs;tance that might abate the Velocity of the Motion, <lb/>&longs;o that of all Moveables the Velocities were alike, we might con­<lb/><arrow.to.target n="marg1058"></arrow.to.target><lb/>gruou&longs;ly enough a&longs;&longs;ign the proportions of the Velocities of like <lb/>and unlike Moveables, in the &longs;ame and in different, Replear, and <lb/>therefore Re&longs;i&longs;ting <emph type="italics"/>Medium's.<emph.end type="italics"/> And this we might effect by &longs;tudy­<lb/>ing how much the Gravity of the <emph type="italics"/>Medium<emph.end type="italics"/> abateth from the Gra­<lb/>vity of the Moveable, which Gravity is the In&longs;trument wherewith <lb/>the Moveable makes its Way, repelling the parts of the <emph type="italics"/>Medium<emph.end type="italics"/><lb/>on each Side: an operation that doth not occur in void <emph type="italics"/>Mediums<emph.end type="italics"/>; <lb/>and therefore there is no difference to be expected from the di­<lb/>ver&longs;e Gravity: and becau&longs;e it is manife&longs;t, that the <emph type="italics"/>Medium<emph.end type="italics"/> abateth <lb/>from the Gravity of the Body by it contained, as much as is the <lb/>weight of &longs;uch another ma&longs;s of its own Matter, if the Velocities of <lb/>the Moveables that in a non-re&longs;i&longs;ting <emph type="italics"/>Medium<emph.end type="italics"/> would be (as hath <lb/>been &longs;uppo&longs;ed) equal, &longs;hould dimini&longs;h in that proportion, we <lb/>&longs;hould have what we de&longs;ired. </s> <s>As for example; &longs;uppo&longs;ing that <lb/>Lead be ten thou&longs;and times more grave than Air, but Ebony a <lb/>thou&longs;and times only; of the Velocities of the&longs;e two Matters, which <lb/>ab&longs;olutely taken, that is, all Re&longs;i&longs;tance being removed, would be <lb/>equal, the Air &longs;ub&longs;tracts from the ten thou&longs;and degrees of the <lb/>Lead one, and from the thou&longs;and degrees of the Ebony likewi&longs;e <lb/>abateth one, or, if you will, of its ten thou&longs;and, ten. </s> <s>If there­<lb/>fore the Lead and the Ebony &longs;hall de&longs;cend thorow the Air from <lb/>any height, which, the retardment of the Air removed, they would <lb/>have pa&longs;&longs;ed in the &longs;ame time, the Air will abate from the ten <lb/>thou&longs;and degrees of the Leads Velocity one, but from the ten <lb/>thou&longs;and degrees of Ebony's Velocity it will abate ten: which is <lb/>as much as to &longs;ay, that dividing that Altitude, from which tho&longs;e <lb/>Moveables departed into ten thou&longs;and parts, the Lead will arrive <lb/>at the Earth, the Ebony being left behind, ten, nay, nine of tho&longs;e <lb/>&longs;ame ten thou&longs;and parts. </s> <s>And what el&longs;e is this, but that a Ball of <lb/>Lead, falling from a Tower two hundred yards high, to find how <pb xlink:href="069/01/065.jpg" pagenum="63"/>much it will anticipate one of Ebony of le&longs;&longs;e than four Inches? <lb/></s> <s>The Ebony weigheth a thou&longs;and times more than the Air, but that <lb/>Bladder &longs;o blown, weigheth only four times &longs;o much; the Air <lb/>therefore from the intrin&longs;ick and natural Velocity of the Ebony <lb/>&longs;ubducteth one degree of a thou&longs;and, but from that, which al&longs;o in <lb/>the Bladder would ab&longs;olutely have been the &longs;ame, the Air &longs;ub­<lb/>ducts one part of four: &longs;o that by that time the Ball of Ebony <lb/>falling from the Tower, &longs;hall come to the ground, the Bladder <lb/>&longs;hall have pa&longs;&longs;ed but three quarters of that height. </s> <s>Lead is twelve <lb/>times heavier than Water, but Ivory only twice as heavy; the <lb/>Water therefore, from their ab&longs;olute Velocities which would be <lb/>equal, &longs;hall abate in the Lead the twelfth part, but in the Ivory <lb/>the half: when therefore, in the Water, the Lead &longs;hall have de­<lb/>&longs;cended eleven fathom, the Ivory &longs;hall have de&longs;cended &longs;ix. </s> <s>And, <lb/>arguing by this Rule, I believe, that we &longs;hall find the Experiment <lb/>much more exactly agree with this &longs;ame Computation, than with <lb/>that of <emph type="italics"/>Ari&longs;totle.<emph.end type="italics"/> By the like method we might find the Veloci­<lb/>ties of the &longs;ame Moveable in different fluid <emph type="italics"/>Mediums,<emph.end type="italics"/> not compa­<lb/>ring the different Re&longs;i&longs;tances of the <emph type="italics"/>Mediums,<emph.end type="italics"/> but con&longs;idering the <lb/>exce&longs;&longs;es of the Gravity of the Moveable over and above the Gra­<lb/>vities of the <emph type="italics"/>Mediums: v. </s> <s>gr.<emph.end type="italics"/> ^{*} Tin is a thou&longs;and times heavier than <lb/><arrow.to.target n="marg1059"></arrow.to.target><lb/>Air, and ten times heavier than Water; therefore dividing the ab­<lb/>&longs;olute Velocity of the Tin into a thou&longs;and degrees, it &longs;hall move <lb/>in the Air, (which deducteth from it the thou&longs;andth part,) with nine <lb/>hundred ninety nine, but in the Water with nine hundred only; <lb/>being that the Water abateth the tenth part of its Gravity, and <lb/>the Air the thou&longs;andth part. </s> <s>Take a Solid &longs;omewhat heavier than <lb/>Water, as for in&longs;tance, the Wood called Oake, a Ball of which <lb/>weighing, as we will &longs;uppo&longs;e, a thou&longs;and drams, a like quantity <lb/>of Water will weigh nine hundred and fifty, but &longs;o much Air will <lb/>weigh but two drams,: it is manife&longs;t, that &longs;uppo&longs;ing that its ab&longs;o­<lb/>lute Velocity were of a thou&longs;and degrees, in Air there would re­<lb/>main nine hundred ninety eight, but in the Water only fifty; be­<lb/>cau&longs;e that the Water of the thou&longs;and degrees of Gravity taketh <lb/>away nine hundred and fifty, and leaves fifty only; that Solid there­<lb/>fore would move well-near twenty times as fa&longs;t in the Air as Wa­<lb/>ter; like as the exce&longs;&longs;e of its Gravity above that of the Water is <lb/>the twentieth part of its own. </s> <s>And here I de&longs;ire that we may con­<lb/>&longs;ider, that no matters, having a power to move downwards in the <lb/>Water, but &longs;uch as are more grave in Species than it; and con&longs;e­<lb/>quently many hundreds of times, more grave than the Air, in <lb/>&longs;eeking what the proportions of their Velocities are in the Air and <lb/>Water, we may, without any con&longs;iderable errour, make account <lb/>that the Air doth not deduct any thing of moment from the ab&longs;o­<lb/>lute Gravity, and con&longs;equently, from the ab&longs;olute Velocity of &longs;uch <pb xlink:href="069/01/066.jpg" pagenum="66"/>matters: &longs;o that having ea&longs;ily found the exce&longs;&longs;e of their Gravi­<lb/>ty above the Gravity of the Water, we may &longs;ay that their Velo­<lb/>city in the Air, to their Velocity in the Water hath the &longs;ame propor­<lb/>tion, that their total Gravity hath to the exce&longs;&longs;e of this above <lb/>the Gravity of the Water. </s> <s>For example, a Ball of Ivory weigh­<lb/>eth twenty ounces, a like quantity of Water weigheth &longs;eventeen <lb/>ounces: therefore the Velocity of the Ivory in Air, to its Velocity <lb/>in Water is very neer as twenty to three.</s></p><p type="margin"> <s><margin.target id="marg1056"></margin.target><emph type="italics"/>The Velocity of <lb/>Grave Bodies de­<lb/>&longs;cending Natural­<lb/>ly to the Center do <lb/>go continually en­<lb/>crea&longs;ing till that <lb/>by the encrea&longs;e of <lb/>the Re&longs;i&longs;tance of <lb/>the<emph.end type="italics"/> Medium <emph type="italics"/>it <lb/>becometh uniform.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1057"></margin.target>* Or aqui&longs;ts.</s></p><p type="margin"> <s><margin.target id="marg1058"></margin.target><emph type="italics"/>To find the Pro­<lb/>portions of the Ve­<lb/>locities of different <lb/>Moveables in the <lb/>&longs;ame, and in diffe­<lb/>rent<emph.end type="italics"/> Mediums.</s></p><p type="margin"> <s><margin.target id="marg1059"></margin.target>* Or Pewter.</s></p><p type="main"> <s>SAGR. </s> <s>I have made a great acqui&longs;t in a bu&longs;ine&longs;&longs;e of it &longs;elf cu­<lb/>rious, and in which, but without any benefit, I have many times <lb/>wearied my-thoughts: nor would there any thing be wanting for <lb/>the putting the&longs;e Speculations in practice, &longs;ave onely the way <lb/>how one &longs;hould come to know of what Gravity the Air, is in com­<lb/>pari&longs;on to the Water, and con&longs;equently to other heavy matters.</s></p><p type="main"> <s>SIMP. </s> <s>But in ca&longs;e one &longs;hould finde, that the Air in&longs;tead of <lb/>Gravity had Levity, what ought one to &longs;ay of the foregoing di&longs;­<lb/>cour&longs;es, otherwi&longs;e very ingenuous?</s></p><p type="main"> <s>SALV. </s> <s>It would be nece&longs;&longs;ary to confe&longs;&longs;e that they were truly <lb/>Aerial, Light, and Vain. </s> <s>But will you que&longs;tion whether the Air <lb/>be heavy, having the expre&longs;&longs;e <emph type="italics"/>Text<emph.end type="italics"/> of <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/> that affirmeth it, <lb/>&longs;aying, That all the Elements have Gravity, even the Air it &longs;elf; <lb/><arrow.to.target n="marg1060"></arrow.to.target><lb/>a &longs;igne of which (&longs;ubjoyns he) we have in that a ^{*} Bladder blown, <lb/>weigheth heavier than un&longs;well'd.</s></p><p type="margin"> <s><margin.target id="marg1060"></margin.target>* Or <emph type="italics"/>Boracho<emph.end type="italics"/>; a <lb/>bottle made of a <lb/>Goat skin, u&longs;ed <lb/>to hold wine and <lb/>other Liquids.</s></p><p type="main"> <s>SIMP. </s> <s>That a <emph type="italics"/>Boracho,<emph.end type="italics"/> or Bladder blown, weigheth more, <lb/>might proceed, as I could &longs;uppo&longs;e, not from the Gravity that is <lb/>in the Air, but in the many gro&longs;&longs;e Vapours intermixed with it in <lb/>the&longs;e our lower Regions; by means whereof I might &longs;ay, that the <lb/>Gravity of the Bladder, or <emph type="italics"/>Boracho<emph.end type="italics"/> encrea&longs;eth.</s></p><p type="main"> <s>SALV. </s> <s>I would not have you &longs;ay it, and much le&longs;&longs;e that you <lb/>&longs;hould make <emph type="italics"/>Aristotle<emph.end type="italics"/> &longs;peak it, for he treating of the Elements, <lb/>and de&longs;iring to per&longs;wade me that the Element of Air is grave, <lb/>making me to &longs;ee it by an Experement: if in comming to the proof <lb/>he &longs;hould &longs;ay: Take a Bladder, and fill it with gro&longs;&longs;e Vapours; <lb/>and ob&longs;erve that its weight will encrea&longs;e; I would tell him that <lb/>it would weigh yet more if one &longs;hould fill it with bran; but would <lb/>afterwards adde; that tho&longs;e Experiments prove, that bran, and <lb/>gro&longs;&longs;e Vapours are grave: but as to the Element of Air, I &longs;hould <lb/>be left in the &longs;ame doubt as before. </s> <s>The Experiment of <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/><lb/>therefore is good, and the Propo&longs;ition true. </s> <s>But I will not &longs;ay &longs;o <lb/>much, for a certain other rea&longs;on taken expre&longs;ly out of a Philo&longs;o­<lb/>pher who&longs;e name I do not remember, but am &longs;ure that I have read <lb/>it, who argueth the Air to be more grave than light, becau&longs;e it <lb/>more ea&longs;ily carrieth grave Bodies downwards, than the light up­<lb/>wards.</s></p><p type="main"> <s>SAGR. </s> <s>Good i-faith. </s> <s>By this rea&longs;on then, the Air &longs;hall be <pb xlink:href="069/01/067.jpg" pagenum="65"/>much heavier than the Water, &longs;ince, that all Bodies are carried <lb/>more ea&longs;ily downwards thorow the Air than thorow the Water, <lb/>and all light Bodies more ea&longs;ily upwards in this than in that: nay, <lb/>infinite matters a&longs;cend in the Water, that in the Air de&longs;cend. <lb/></s> <s>But be the Gravity of the Bladder, <emph type="italics"/>Simplicius,<emph.end type="italics"/> either by rea&longs;on of <lb/>the gro&longs;&longs;e Vapours, or pure Air, this nothing concerns our pur­<lb/>po&longs;e, for we &longs;eek that which happeneth to Moveables that move <lb/>in this our Vaporous Region. </s> <s>Therefore, returning to that which <lb/>more concerneth me, I would for a full and ab&longs;olute informati­<lb/>on in the pre&longs;ent bu&longs;ine&longs;&longs;e, not onely be a&longs;&longs;ured that the Air is <lb/>grave, as I hold for certain, but I would, if it be po&longs;&longs;ible, know <lb/>what its Gravity is. </s> <s>Therefore, <emph type="italics"/>Salviatus,<emph.end type="italics"/> if you have wherewith <lb/>to &longs;atisfie me in this al&longs;o, I entreat you to favour me with the <lb/>&longs;ame.</s></p><p type="main"> <s>SALV. </s> <s>That there re&longs;ideth in the Air po&longs;itive Gravity, and <lb/><arrow.to.target n="marg1061"></arrow.to.target><lb/>not, as &longs;ome have thought, Levity, which haply is in no Mat­<lb/>ter to be found, the Experiment of the Blown-Bladder, alledged <lb/>by <emph type="italics"/>Ari&longs;totle,<emph.end type="italics"/> affordeth us a &longs;ufficiently-convincing Argument; for <lb/>if the quality of ab&longs;olute and po&longs;itive Levity were in the Air, <lb/>then the Air being multiplied and compre&longs;&longs;ed, the Levity would <lb/>encrea&longs;e, and con&longs;equently the propen&longs;ion of going upwards: <lb/>but Experience &longs;hews the contrary. </s> <s>As to the other demand, that <lb/><arrow.to.target n="marg1062"></arrow.to.target><lb/>is, of the Method how to inve&longs;tigate its Gravity, I have tried to <lb/>do it in this manner: I have taken a pretty bigge Gla&longs;&longs;e ^{*} Bottle, <lb/><arrow.to.target n="marg1063"></arrow.to.target><lb/>with its neck bended, and a Finger-&longs;tall of Leather fa&longs;t about <lb/>it, having in the top of the &longs;aid Finger-&longs;tall in&longs;erted and fa­<lb/>&longs;tened a Valve of Leather, by which with a Siringe I have made <lb/>pa&longs;&longs;e into the Bottle by force a great quantity of Air, of which, <lb/>becau&longs;e it admits of great Conden&longs;ation, it may take in two or <lb/>three other Bottles-ful over and above that which is naturally con­<lb/>tained therein. </s> <s>Then I have in an exact Ballance very preci&longs;ely <lb/>weighed that Bottle with the Air compre&longs;&longs;ed within it, adju&longs;ting <lb/>the weight with &longs;mall Sands. </s> <s>Afterwards, the Valve being opened, <lb/>and the Air let out, that was violently conteined in the Ve&longs;&longs;el, I <lb/>have put it again into the Scales, and finding it notably aleviated, <lb/>I have by degrees taken &longs;o much Sand from the other Scale, keep­<lb/>ing it by it &longs;elf, that the Ballance hath at la&longs;t &longs;tood <emph type="italics"/>in Equilibrio<emph.end type="italics"/><lb/>with the remaining counter-poi&longs;e, that is with the Bottle. </s> <s>And <lb/>here there is no que&longs;tion, but that the weight of the re&longs;erved Sand <lb/>is that of the Air that was forceably driven into the Bottle, and <lb/>which is at la&longs;t gone out thence. </s> <s>But this Experiment hitherto a&longs;­<lb/>&longs;ureth me of no more but this, that the Air violently deteined in <lb/>the Ve&longs;&longs;el, weigheth as much as the re&longs;erved Sand, but how much <lb/>the Air re&longs;olutely and determinately weigheth in re&longs;pect of the <lb/>Water, or other grave matter, I do not as yet know, nor can <pb xlink:href="069/01/068.jpg" pagenum="66"/>I tell, unle&longs;&longs;e I mea&longs;ure the quantity of the Air compre&longs;&longs;ed: and <lb/>for the di&longs;covering of this a Rule is nece&longs;&longs;ary, which I have <lb/>found may be performed two manner of wayes, one of which <lb/>is to take &longs;uch another Bottle or Flask as the former, and in like <lb/>manner bended, with a Finger-&longs;tall of Leather, the end of which <lb/>may clo&longs;ely imbrace the Volve of the other, and let it be very <lb/>fa&longs;t tied about it. </s> <s>It's requi&longs;ite, that this &longs;econd Bottle be bored in <lb/>the bottom, &longs;o that as by that hole we may thru&longs;t in a Wier, <lb/>wherewith we may, at plea&longs;ure, open the &longs;aid Volve, to let out <lb/>the &longs;uperfluous Air of the other Ve&longs;&longs;el, after it hath been weighed: <lb/>but this &longs;econd Bottle ought to be full of Water. </s> <s>All being pre­<lb/>pared in the manner afore&longs;aid, and with the Wier opening the <lb/>Volve, the Air i&longs;&longs;uing out with impetuo&longs;ity, and pa&longs;&longs;ing into the <lb/>Ve&longs;&longs;el of Water, &longs;hall drive it out by the hole at the Bottom: <lb/>and it is manife&longs;t, that the quantity of Water which &longs;hall be <lb/>thru&longs;t out, is equal to the Ma&longs;&longs;e and quantity of Air that &longs;hall <lb/>have i&longs;&longs;ued from th'other Ve&longs;&longs;el: that Water therefore being <lb/>kept, and returning to weigh the Ve&longs;&longs;el lightned of the Air com­<lb/>pre&longs;&longs;ed (which I &longs;uppo&longs;e to have been weighed likewi&longs;e fir&longs;t with <lb/>the &longs;aid forced Air) and the &longs;uperfluous &longs;and being laid by, as I <lb/>directed before; it is manife&longs;t, that this is the ju&longs;t weight of &longs;o <lb/>much Air in ma&longs;&longs;e, as is the ma&longs;&longs;e of the expul&longs;ed and re&longs;erved <lb/>Water; which we are to weigh, and &longs;ee how many times its <lb/>weight &longs;hall contain the weight of the re&longs;erved &longs;and: and we may <lb/>without errour affirme, that the Water is &longs;o many times heavier <lb/>than Air; which &longs;hall not be ten times, as it &longs;eemeth <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/><lb/>held, but very neer four hundred, as the &longs;aid Experiment &longs;heweth.</s></p><p type="margin"> <s><margin.target id="marg1061"></margin.target><emph type="italics"/>The Air hath Po­<lb/>&longs;itive Gravity.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1062"></margin.target><emph type="italics"/>How that Gravity <lb/>may be computed.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1063"></margin.target>* <emph type="italics"/>Un Fia&longs;co,<emph.end type="italics"/> tho&longs;e <lb/>long-neckt gla&longs;&longs;e <lb/>bottles in which <lb/>we have our <lb/><emph type="italics"/>Florence<emph.end type="italics"/> Wine <lb/>brought to us.</s></p><p type="main"> <s>The other way is more expeditious, and it may be done with <lb/>one Ve&longs;&longs;el onely, that is with the fir&longs;t accomodated after the man­<lb/>ner before directed, into which I will not that any other Air be <lb/>put, more than that which naturally is found therein; but I will, <lb/>that we inject Water without &longs;uffering any Air to come out, <lb/>which being forced to yield to the &longs;upervenient Water mu&longs;t of <lb/>nece&longs;&longs;ity be compre&longs;&longs;ed: having gotten in, therefore, as much <lb/>Water as is po&longs;&longs;ible, (but yet without great violence one cannot get <lb/>in three quarters of what the Bottle will hold) put it into the <lb/>Scales, and very carefully weigh it: which done, holding the <lb/>Ve&longs;&longs;el with the neck upwards, open the Volve, letting out the <lb/>Air, of which there will preci&longs;ely i&longs;&longs;ue forth &longs;o much as there is <lb/>Water in the Bottle. </s> <s>The Air being gone out, put the Ve&longs;&longs;el again <lb/>into the Scales, which by the departure of the Air will be found <lb/>lightened, and abating from the oppo&longs;ite Scale the &longs;uperfluous <lb/>weight, it &longs;hall give us the weight of as much Air as there is <lb/>Water in the Bottle.</s></p><p type="main"> <s>SIMP. </s> <s>The Contrivances you found out cannot but be con­<pb xlink:href="069/01/069.jpg" pagenum="67"/>fe&longs;&longs;ed to be witty and very ingenuous, but whil&longs;t, me thinks, they <lb/>fully &longs;atisfie my under&longs;tanding, they another way occa&longs;ion in <lb/>me much Confu&longs;ion, for it being undoubtedly true that the Ele­<lb/>ments in their proper Region are neither heavy nor light, I can­<lb/>not comprehend, how and which way that portion of Air, which <lb/>&longs;eemeth to have weighed <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> four drams of &longs;and, &longs;hould af­<lb/>terwards have that &longs;ame Gravity in the Air, in which the &longs;and is <lb/>contained that weigheth again&longs;t it: and therefore me thinks that <lb/>the Experiment ought not to be practiced in the Element of Air, <lb/>but in a <emph type="italics"/>Medium<emph.end type="italics"/> in which the Air it &longs;elf might exerci&longs;e its quality <lb/>of Gravitation, if it really be owner thereof.</s></p><p type="main"> <s>SALV. </s> <s>Certainly the Objection of <emph type="italics"/>Simplicius<emph.end type="italics"/> is very acute, <lb/>and therefore its nece&longs;&longs;ary, either that it be unan&longs;werable, or that <lb/>the Solution be no le&longs;&longs;e acute. </s> <s>That that Air, which compre&longs;­<lb/>&longs;ed, appeared to weigh as much as that &longs;and, left at liberty in its <lb/>Element is no longer to weigh any thing as the Sand doth, is a thing <lb/>manife&longs;t: and therefore for making of &longs;uch an Experiment, its <lb/>requi&longs;ite to choo&longs;e a place and <emph type="italics"/>Medium<emph.end type="italics"/> wherein the Air as well as <lb/>the Sand might weigh: for, as hath &longs;everal times been &longs;aid, the <lb/><emph type="italics"/>Medium<emph.end type="italics"/> &longs;ub&longs;tracts from the Weight of every Matter that is im­<lb/>merged therein, &longs;o much, as &longs;uch another quantity of the &longs;aid <lb/><emph type="italics"/>Medium,<emph.end type="italics"/> as is that of the ma&longs;&longs;e immer&longs;ed, weigheth: &longs;o that <lb/>the Air depriveth the Air of all its Gravity. </s> <s>The operation, there­<lb/><arrow.to.target n="marg1064"></arrow.to.target><lb/>fore, to the end it were made exactly, ought to be tried in a <emph type="italics"/>Va­<lb/>cuum,<emph.end type="italics"/> wherein every grave Body would exerci&longs;e its Moment <lb/>without any diminution. </s> <s>In ca&longs;e therefore, <emph type="italics"/>Simplicius,<emph.end type="italics"/> that we <lb/>&longs;hould weigh a portion of Air in a <emph type="italics"/>Vacuum,<emph.end type="italics"/> would you then be <lb/>convinced and a&longs;&longs;ured of the bu&longs;ine&longs;&longs;e?</s></p><p type="margin"> <s><margin.target id="marg1064"></margin.target><emph type="italics"/>The Air compre&longs;­<lb/>&longs;ed and violently <lb/>pent up, weigheth in <lb/>a<emph.end type="italics"/> Vacuum; <emph type="italics"/>and <lb/>how its weight is to <lb/>be e&longs;timated.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. </s> <s>Verily I &longs;hould: but this is to defire, or enjoyn that <lb/>which is impo&longs;&longs;ible.</s></p><p type="main"> <s>SALV. </s> <s>And therefore the obligation mu&longs;t needs be great that <lb/>you owe to me, when ever I &longs;hall for your &longs;ake effect an impo&longs;&longs;ibi­<lb/>lity: but I will not &longs;ell you that which I have already given you: <lb/>for we, in the foregoing Experiment, weigh the Air in a <emph type="italics"/>Vacuum,<emph.end type="italics"/><lb/>and not in the Air, or in any other Replete <emph type="italics"/>Medium.<emph.end type="italics"/> That from <lb/>the Ma&longs;s, <emph type="italics"/>Simplicius,<emph.end type="italics"/> that in the fluid <emph type="italics"/>Medium<emph.end type="italics"/> is immerged certain <lb/>Gravity is &longs;ub&longs;tracted by the &longs;aid <emph type="italics"/>Medium,<emph.end type="italics"/> this commeth to pa&longs;s <lb/>by rea&longs;on that it re&longs;i&longs;teth its being opened, driven back, and in a <lb/>word commoved; a &longs;ign of which is its pronene&longs;s to return in&longs;tant­<lb/>ly to fill the Space up again, that the immer&longs;ed ma&longs;s occupied in it, <lb/>as &longs;oon as ever it departeth thence; for if it &longs;uffered not by that <lb/>immer&longs;ion, it would not operate again&longs;t the &longs;ame. </s> <s>Now tell me, <lb/>when you have in the Air the Bottle before filled with the &longs;ame Air <lb/>naturally contained therein, what divi&longs;ion, repul&longs;e, or, in &longs;hort, <lb/>what mutation doth the external ambient Air receive from the &longs;e­<pb xlink:href="069/01/070.jpg" pagenum="68"/>cond Air that was newly infu&longs;ed with force into the Ve&longs;&longs;el? </s> <s>Doth <lb/>it enlarge the Bottle, whereupon the Ambient ought the more to <lb/>retire it &longs;elf to make room for it? </s> <s>Certainly no: And therefore <lb/>we may &longs;ay, that the &longs;econd Air is not immer&longs;ed in the Ambient, <lb/>not occupying any Space therein; but is as if it was in a <emph type="italics"/>Vacuum,<emph.end type="italics"/><lb/>nay more, is really con&longs;tituted in it, and is placed in Vacuities that <lb/>were not repleted by the former un-conden&longs;ed Air. </s> <s>And, really, I <lb/>know not how to di&longs;cern any difference between the two Con&longs;ti <lb/>tutions of Inclo&longs;ed and <emph type="italics"/>Ambient,<emph.end type="italics"/> whil&longs;t in this the <emph type="italics"/>Ambient<emph.end type="italics"/> doth <lb/>no-ways pre&longs;s the Inclo&longs;ed, and in that the Inclo&longs;ed doth not re­<lb/>repul&longs;e the <emph type="italics"/>Ambient<emph.end type="italics"/>: and &longs;uch is the placing of any matter in a <lb/><emph type="italics"/>Vacuum,<emph.end type="italics"/> and the &longs;econd Air compre&longs;sed in the Flask. </s> <s>The weight <lb/>therefore that is found in that &longs;ame conden&longs;ed Air, is the &longs;ame that <lb/>it would have, were it freely di&longs;tended in a <emph type="italics"/>Vacuum.<emph.end type="italics"/> Tis true in­<lb/>deed, that the weight of the Sand that weigheth again&longs;t it, as ha­<lb/>ving been in the open Air, would in a <emph type="italics"/>Vacuum<emph.end type="italics"/> have been a little <lb/>more than ju&longs;t &longs;o heavy; and therefore it is nece&longs;&longs;ary to &longs;ay, that <lb/>the weighed Air is in reality &longs;omewhat le&longs;&longs;e heavy than the Sand <lb/>that counterpoi&longs;eth it, that is, &longs;o much, by how much the like <lb/>quantity of Air would weigh in a <emph type="italics"/>Vacuum.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. </s> <s>I had thought that there was &longs;omething to have been <lb/>wi&longs;hed for in the Experiments before produced; but now I am <lb/>thorowly &longs;atisfied.<lb/><arrow.to.target n="marg1065"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1065"></margin.target><emph type="italics"/>The difference, <lb/>though very great, <lb/>of the Gravity of <lb/>Moveables hath <lb/>no part in differer­<lb/>cing their Veloci­<lb/>ties.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>The things by me hitherto alledged, and in particular, <lb/>this, That the difference of Gravity, although exceeding great, <lb/>hath no part in diver&longs;ifying the Velocities of Moveables, &longs;o that, <lb/>notwith&longs;tanding any thing depending on that, they would all <lb/>move with equal Celerity, is &longs;o new, and at the fir&longs;t apprehen&longs;i­<lb/>on &longs;o remote from probability, that, were there not a way to de­<lb/>lucidate it, and make it as clear as the Sun, it would be better <lb/>to pa&longs;&longs;e it over in &longs;ilence, than to divulge it: therefore &longs;eeing <lb/>that I have let it e&longs;cape from me, its fit that I omit neither Expe­<lb/>riment nor Rea&longs;on that may corroborate it.</s></p><p type="main"> <s>SAGR. </s> <s>Not onely this, but many other al&longs;o of your A&longs;&longs;erti­<lb/>ons are &longs;o remote from the Opinions and Doctrines commonly <lb/>received, that &longs;ending them abroad, you would &longs;tir up a great <lb/>number of Antagoni&longs;ts: in regard, that the innate Di&longs;po&longs;ition of <lb/>Men doth not &longs;ee with good eyes, when others in their Studies <lb/>di&longs;cover Truths or Fallacies, that were not di&longs;covered by them­<lb/>&longs;elves: and with the title of Innovators of Doctrines, little plea­<lb/>&longs;ing to the ears of many, they &longs;tudy to cut tho&longs;e knots which <lb/>they cannot untie, and with &longs;ub-terranean Mines to blow up <lb/>tho&longs;e Structures, which have been with the ordinary Tools by <lb/>patient Architects erected: but with us here, who are far from <lb/>any &longs;uch thoughts, your Experiments and Arguments are <pb xlink:href="069/01/071.jpg" pagenum="69"/>&longs;ufficient to give full &longs;atisfaction: yet neverthele&longs;&longs;e, if &longs;o be you <lb/>have other more palpable Experiments, and more convincing <lb/>Rea&longs;ons we would very gladly hear them.</s></p><p type="main"> <s>SALV. </s> <s>The Experiment made with two Moveables, as different <lb/>in weight as may be, by letting them de&longs;cend from a place on <lb/>high, thereby to &longs;ee whether their Velocity be equal, meets with <lb/>&longs;ome difficulty: for if the height &longs;hall be great, the <emph type="italics"/>Medium,<emph.end type="italics"/><lb/>which is to be opened and laterally repelled by the <emph type="italics"/>Impetus<emph.end type="italics"/> of the <lb/>cadent Body, &longs;hall be of much greater prejudice to the &longs;mall Mo­<lb/>ment of the light Moveable, than to the violence of the heavy <lb/>one; whereupon in a long way the light one will be left behind: <lb/>and in a little altitude it might be doubted whether there were <lb/>really any difference, or if there were, whether it would be <lb/>&longs;en&longs;ible. </s> <s>Therefore I have oft been thinking to reiterate the de­<lb/>&longs;cent &longs;o many times from &longs;mall heights, and to accumulate toge­<lb/>ther &longs;o many of tho&longs;e minute differences of time, as might inter­<lb/>cede between the arrival or fall of the heavy Body to the ground, <lb/>and the arrival of the light one, which &longs;o conjoyned, would make <lb/>a time not onely ob&longs;ervable, but ob&longs;ervable with much facility <lb/>Moreover, that I might help my &longs;elf with Motions as &longs;low as po&longs;­<lb/>&longs;ible may be, in which the Re&longs;i&longs;tance of the <emph type="italics"/>Medium<emph.end type="italics"/> operates <lb/>le&longs;&longs;e in altering the effect that dependeth on &longs;imple Gravity, I <lb/>have had thoughts to cau&longs;e the Moveable to de&longs;cend upon a de­<lb/>clining Plane, not much rai&longs;ed above the Plane of the Horizon; <lb/>for upon this, no le&longs;&longs;e than in perpendicularity, we may di&longs;cover <lb/>that which is done by Grave Bodies different in weight: and pro­<lb/>ceeding farther, I have de&longs;ired to free my &longs;elf from any what&longs;o­<lb/>ever impediment, that might ari&longs;e from the Contact of the &longs;aid <lb/>Moveables upon the &longs;aid declining Plane: and la&longs;tly, I have ta­<lb/>ken two Balls, one of Lead, and one of Cork, that above an hun­<lb/>dred times more grave than this, and have fa&longs;tened them to two <lb/>&longs;mall threads, each equally four or five yards long, tyed on <lb/>high: and having removed a&longs;wel the one as the other Ball from <lb/>the &longs;tate of Perpendicularity, I have let them both go in the &longs;ame <lb/>Moment, and they de&longs;cending by the Circumferences of Circles <lb/>de&longs;cribed by the equal Strings their Semidiameters, and having <lb/>pa&longs;&longs;ed beyond the Perpendicular, they afterwards by the &longs;ame <lb/>way returned back, and reiterating the&longs;e Vibrations, and re­<lb/>turns of them&longs;elves neer an hundred times, they have &longs;hewn ve­<lb/>ry &longs;en&longs;ibly, that the grave <emph type="italics"/>Pendulum<emph.end type="italics"/> moveth &longs;o exactly under the <lb/>time of the light one, that it doth not in an hundred, no nor in a <lb/>thou&longs;and Vibrations, anticipate the time of one &longs;mall moment, <lb/>but that they keep an equal pa&longs;&longs;e in their Recur&longs;ions. </s> <s>They al&longs;o <lb/>&longs;hew the Operation of the <emph type="italics"/>Medium,<emph.end type="italics"/> which conferring &longs;ome im­<lb/>pediment on the Motion, doth much more dimini&longs;h the Vibrati­<pb xlink:href="069/01/072.jpg" pagenum="70"/>ons of the Cork, than that of the Lead: not that it maketh them <lb/>more or le&longs;&longs;e frequent, nay, when the Arches pa&longs;&longs;ed by the Cork <lb/>were not of above five or &longs;ix degrees, and tho&longs;e of the Lead fif­<lb/>ty, they did pa&longs;s them under the &longs;ame times.</s></p><p type="main"> <s>SIMP. </s> <s>If this be &longs;o, how is it then that the Velocity of the <lb/>Lead is not greater than that of the Cork? </s> <s>that pa&longs;&longs;ing a jour­<lb/>ney of &longs;ixty degrees, in the time that this pa&longs;seth hardly &longs;ix?</s></p><p type="main"> <s><emph type="italics"/>S<emph.end type="italics"/>ALV. </s> <s>But what would you &longs;ay, <emph type="italics"/>Simplicius,<emph.end type="italics"/> in ca&longs;e they <lb/>&longs;hould both di&longs;patch their Recur&longs;ions in the &longs;ame time, when the <lb/>Cork being removed thirty degrees from the Perpendicular, <lb/>&longs;hould pa&longs;s an arch of &longs;ixty, and the Lead removed from the <lb/>&longs;ame middle point onely two degrees, &longs;hould run an arch of four? <lb/></s> <s>would not then the Cork be &longs;o much more &longs;wift than the Lead? <lb/></s> <s>and yet Experience &longs;hews that &longs;o it happeneth: therefore ob&longs;erve, <lb/>The <emph type="italics"/>Pendulum<emph.end type="italics"/> of Lead being carried <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> fifty degrees from the <lb/>Perpendicular, and thence let go, &longs;wingeth, and pa&longs;&longs;ing beyond <lb/>the Perpendicular, neer fifty more degrees, de&longs;cribeth an arch <lb/>of well neer an hundred degrees; and returning of its &longs;elf back <lb/>again, it de&longs;cribeth another arch, not much le&longs;&longs;e than the former, <lb/>and continuing its Vibrations, after a great number of them, it <lb/>finally returneth to Re&longs;t: Each of tho&longs;e Vibrations are made un­<lb/>der equal times a&longs;wel tho&longs;e of ninety degrees, as tho&longs;e of fifty, <lb/>twenty, ten, or four; &longs;o that by con&longs;equence, the Velocity of the <lb/>Moveable doth &longs;ucce&longs;&longs;ively langui&longs;h and abate, in regard, that <lb/>under equal times it doth &longs;ucce&longs;&longs;ively pa&longs;&longs;e arches continually <lb/>le&longs;&longs;er and le&longs;&longs;er. </s> <s>The like, yea the &longs;elf &longs;ame effect is performed <lb/>by the Cork, hanging by a &longs;tring of the like length, &longs;ave that <lb/>in a le&longs;&longs;e number of Vibracions it returneth to Re&longs;t, as being le&longs;s <lb/>apt, by means of its Levity, to overcome the ob&longs;tacle of the Air: <lb/>and yet neverthele&longs;s all the Vibrations, both great and &longs;mall, are <lb/>made under times equal to one another, and equal al&longs;o to the <lb/>times of the times of the Vibrations of the Lead. </s> <s>Whereupon it <lb/>is true, that if whil&longs;t the Lead pa&longs;&longs;eth an arch of fifty degrees, <lb/>the Cork pa&longs;seth one but of ten, the Cork is then more &longs;low <lb/>than the Lead: but it will al&longs;o happen on the other &longs;ide, that the <lb/>Cork pa&longs;seth the arch of fifty degrees, when the Lead pa&longs;seth <lb/>but that of ten or &longs;ix; and &longs;o in &longs;everal times the Lead &longs;hall be <lb/>&longs;wifter onewhile, and the Cork another while: but if the &longs;ame <lb/>Moveables &longs;hall al&longs;o under the &longs;ame equal times, pa&longs;s arches that <lb/>are equal, one may then very &longs;afely &longs;ay, that their Velocities are <lb/>equal.</s></p><p type="main"> <s>SIMP. </s> <s>This di&longs;cour&longs;e &longs;eems to me concluding, and not con­<lb/>cluding, and I finde in my thoughts &longs;uch a Confu&longs;ion, ari&longs;ing <lb/>from the one-while &longs;wift, another-while &longs;low, another-while ex­<lb/>treme &longs;low motion of both the one and other Moveable; as that <pb xlink:href="069/01/073.jpg" pagenum="71"/>it permits me not to di&longs;cern clearly, whether it be true, That their <lb/>Velocities are alwaies equal.</s></p><p type="main"> <s>SAGR. </s> <s>Give me leave, I pray you, <emph type="italics"/>Salviatus,<emph.end type="italics"/> to interpo&longs;e two <lb/>words. </s> <s>And tell me, <emph type="italics"/>Simplicius,<emph.end type="italics"/> whether you admit, that it may be <lb/>&longs;aid with ab&longs;olute verity that the Velocities of the Cork and of <lb/>the Lead are equal, in ca&longs;e, that both of them departing at the <lb/>&longs;ame moment from Re&longs;t, and moving by the &longs;ame declivities, they <lb/>&longs;hould alwaies pa&longs;&longs;e equal Spaces in equal times?</s></p><p type="main"> <s>SIMP. </s> <s>This admits of no doubt, nor can it be contradicted.</s></p><p type="main"> <s>SAGR. </s> <s>It hapneth now in the Pendulums that each of them <lb/>pa&longs;&longs;eth now &longs;ixty degrees, now fifty, now thirty, now ten, now <lb/>eight, four, and two; and when each of them pa&longs;&longs;eth the Arch of <lb/>&longs;ixty degrees they pa&longs;&longs;e it in the &longs;ame time; in the Arch of fifty the <lb/>&longs;ame time is &longs;pent by both the one and the other Moveable; &longs;o in <lb/>the Arch of thirty, of ten, and of the re&longs;t: and therefore it is con­<lb/>cluded, that the Velocity of the Lead in the Arch of &longs;ixty degrees, <lb/>is equal to the Velocity of the Cork in the &longs;ame Arch of &longs;ixty de­<lb/>grees: and that the Velocities in the Arch of fifty, are likewi&longs;e <lb/>equal to one the other, and &longs;o in the re&longs;t. </s> <s>But it is not &longs;aid, that the <lb/>Velocity that is exerci&longs;ed in the Arch of &longs;ixty is equal to the Ve­<lb/>locity that is exerci&longs;ed in the Arch of fifty, nor this to that of the <lb/>Arch of thirty. </s> <s>But the Velocities are alwaies le&longs;&longs;er, in the le&longs;&longs;er <lb/>Arches. </s> <s>And this is collected from our &longs;en&longs;ibly &longs;eeing the &longs;ame <lb/>Moveable con&longs;ume as much time in pa&longs;&longs;ing the great Arch of &longs;ixty <lb/>degrees, as in pa&longs;&longs;ing the le&longs;&longs;er of fifty, or the lea&longs;t of ten: and, in a <lb/>word, in their being all pa&longs;&longs;ed alwaies under equal times. </s> <s>It is true <lb/>therefore, that both the Lead and the Cork &longs;ucce&longs;&longs;ively retard the <lb/>Motion, according to the Diminution of the Arches, but yet do <lb/>not alter their harmony in keeping the equality of Velocity in all <lb/>the &longs;ame Arches by them pa&longs;&longs;ed. </s> <s>I de&longs;ired to &longs;ay thus much, more <lb/>to try whether I have rightly apprehended the Conceit of <emph type="italics"/>Salvia­<lb/>tus,<emph.end type="italics"/> than out of any nece&longs;&longs;ity that I thought <emph type="italics"/>Simplicius<emph.end type="italics"/> to &longs;tand in <lb/>of a more plain Explanation than that of <emph type="italics"/>Salviatus,<emph.end type="italics"/> which is, as <lb/>in all other things, extreamly clear, and &longs;uch, that, it being fre­<lb/>quent with him to re&longs;olve Que&longs;tions, in appearance not only ob­<lb/>&longs;cure, but repugnant to Nature, and to the Truth, with Rea&longs;ons, <lb/>or Ob&longs;ervations, or Experiments very trite and familiar to every <lb/>one, it hath (as I have under&longs;tood from divers) given occa&longs;ion to <lb/>one of the mo&longs;t e&longs;teemed Profe&longs;&longs;ors of our Age to put the le&longs;&longs;e <lb/>e&longs;teem upon his Novelties, holding them to have as much of Sor­<lb/>didne&longs;&longs;e, for that they depend on over low and popular Funda­<lb/>mentals: as if the mo&longs;t admirable and mo&longs;t-to-be-prized Proper­<lb/>ty of the Demon&longs;trative Sciences, were not to &longs;pring and ari&longs;e <lb/>from Principles known, under&longs;tood, and granted by every one. <lb/></s> <s>But let us, for all that, continue to banquet our &longs;elves with this diet <pb xlink:href="069/01/074.jpg" pagenum="72"/>that is &longs;o light of dige&longs;tion; and &longs;uppo&longs;ing that <emph type="italics"/>Simplicius<emph.end type="italics"/> is fully <lb/>&longs;atisfied in under&longs;tanding and admitting, That the intern Gravity <lb/>of different Moveables hath no &longs;hare in differencing their Veloci­<lb/>ties, &longs;o that all of them, for ought that dependeth on that, would <lb/>move with the &longs;ame Velocities; tell us, <emph type="italics"/>Salviatus,<emph.end type="italics"/> in what you <lb/>place the &longs;en&longs;ible and apparent inequalities of Motion; and an­<lb/>&longs;wer to that In&longs;tance that <emph type="italics"/>Simplicius<emph.end type="italics"/> produceth, and which I like­<lb/>wi&longs;e confirm, I mean, of &longs;eeing a Cannon Bullet move more &longs;wift­<lb/>ly than a drop of Bird-&longs;hot, for the difference of Velocity &longs;hall be <lb/>but &longs;mall, in re&longs;pect of that which I object again&longs;t you of Movea­<lb/>bles of the &longs;ame matter, of which &longs;ome of the greater will de&longs;cend <lb/>in a <emph type="italics"/>Medium,<emph.end type="italics"/> in le&longs;&longs;e than one beat of the Pul&longs;e, that &longs;pace, that <lb/>others which are le&longs;&longs;er will not pa&longs;&longs;e in an hour, nor in four, nor in <lb/>twenty; &longs;uch are pebbles and minute gravel-&longs;tones, e&longs;pecially, <lb/>that &longs;mall &longs;and which muddieth the Water; in which <emph type="italics"/>Medium<emph.end type="italics"/><lb/>they will not de&longs;cend in many hours &longs;o much as two fathoms, <lb/>which Stones, and tho&longs;e of no great bigne&longs;&longs;e, do pa&longs;&longs;e in one beat <lb/>of the Pul&longs;e.</s></p><p type="main"> <s>SALV. </s> <s>That which the <emph type="italics"/>Medium<emph.end type="italics"/> operates, in retarding Movea­<lb/>bles, the more according as they are compared to one another, le&longs;s <lb/>grave <emph type="italics"/>in &longs;pecie,<emph.end type="italics"/> hath been already declared, &longs;hewing that it pro­<lb/>ceeds from the &longs;ub&longs;traction of weight. </s> <s>But how one and the &longs;ame <lb/><emph type="italics"/>Medium<emph.end type="italics"/> can with &longs;o great difference dimini&longs;h the Velocity in <lb/>Moveables that differ only in Magnitude, although they are of <lb/>the &longs;ame Matter, and of the &longs;ame Figure, requireth for its expli­<lb/>cation a more &longs;ubtil di&longs;cour&longs;e, than that which &longs;ufficeth for under­<lb/>&longs;tanding how the more dilated Figure of the Moveable, or the <lb/>Motion of the <emph type="italics"/>Medium<emph.end type="italics"/> that is made contrary to the Moveable, re­</s></p><p type="main"> <s><arrow.to.target n="marg1066"></arrow.to.target><lb/>tardeth the Velocity of the &longs;aid Moveable. </s> <s>I reduce the cau&longs;e of <lb/>the &longs;aid Problem to the Scabro&longs;ity, and Poro&longs;ity, that is common­<lb/>ly, and, for the mo&longs;t part, nece&longs;&longs;arily found in the Superficies of <lb/>Solid Bodies, the which Scabro&longs;ities, in their Motion, go repul&longs;ing <lb/>and commoving the Air, or other Ambient <emph type="italics"/>Medium<emph.end type="italics"/>: of which we <lb/>have an evident te&longs;timony, in that we hear the Bodies, though made <lb/>as round as is po&longs;&longs;ible for them to be, to hum whil&longs;t they pa&longs;&longs;e ve­<lb/>ry &longs;wiftly thorow the Air; and they are not only heard to hum, but <lb/>to whir and whi&longs;tle, if there be but in them &longs;ome more than ordi­<lb/>nary cavity or prominency. </s> <s>We &longs;ee al&longs;o, that in turning round <lb/>every rotund Solid maketh a little wind: And what need more? <lb/></s> <s>Do we not hear a notable whirring, and in a very &longs;harp Accent, <lb/>made by a Top, while it turneth round on the ground with great <lb/>Celerity? </s> <s>The &longs;hrilne&longs;s of which whizzing groweth flatter accor­<lb/>ding as the Velocity of the <emph type="italics"/>Vertigo<emph.end type="italics"/> doth by degrees more and <lb/>more &longs;lacken: a nece&longs;&longs;ary Argument likewi&longs;e of the commotion <lb/>and percu&longs;&longs;ion of the Air by tho&longs;e (though very &longs;mall) Scabro&longs;i­<pb xlink:href="069/01/075.jpg" pagenum="73"/>ties of their Superficies. </s> <s>It is not to be doubted, but that the&longs;e in the <lb/>de&longs;cent of Moveables, grating upon, and repul&longs;ing the fluid Am­<lb/>bient, procure retardment in the Velocity, and &longs;o much the greater, <lb/>by how much the Superficies &longs;hall be greater, as is that of le&longs;&longs;er <lb/>Solids compared to bigger.</s></p><p type="margin"> <s><margin.target id="marg1066"></margin.target><emph type="italics"/>The greater or le&longs;s <lb/>Scabro&longs;ity and Po­<lb/>ro&longs;ity of the Super­<lb/>ficies of Movea­<lb/>bles, a probable <lb/>cau&longs;e of their grea­<lb/>ter or le&longs;&longs;er Retar­<lb/>dation.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. Stay, I pray you, for here I begin to be at a lo&longs;&longs;e: for <lb/>though I under&longs;tand and admit, that the Confrication of the <emph type="italics"/>Medi­<lb/>um<emph.end type="italics"/> with the Superficies of the Moveable retardeth the Motion, <lb/>and that it more retardeth it where <emph type="italics"/>(ceteris paribus)<emph.end type="italics"/> the Superficies <lb/>is greater, yet do I not comprehend upon what ground you call the <lb/>Superficies of le&longs;&longs;er Solids greater: & farthermore if, as you affirm, the <lb/>greater Superficies ought to cau&longs;e greater retardment, the greater <lb/>Solids ought to be the &longs;lower, which is not &longs;o: but this Objection <lb/>may ea&longs;ily be removed, by &longs;aying, that although the greater hath <lb/>a greater Superficies, it hath al&longs;o a greater Gravity, upon which <lb/>the impediment of the greater Superficies hath not &longs;o much more <lb/>prevalent influence, than the impediment of the le&longs;&longs;er Superficies <lb/>hath upon the le&longs;&longs;er Gravity, as that the Velocity of the greater <lb/>Solid &longs;hould become the le&longs;&longs;er. </s> <s>And therefore I &longs;ee no rea&longs;on why <lb/>one &longs;hould alter the equality of the Velocities, whil&longs;t, that looking <lb/>how much the Moving Gravity dimini&longs;heth, the faculty of the Re­<lb/>tarding Superficies doth dimini&longs;h at the &longs;ame rate.</s></p><p type="main"> <s>SALV. </s> <s>I will re&longs;olve all that which you object in one word. <lb/></s> <s>Therefore, <emph type="italics"/>Simplicius,<emph.end type="italics"/> you will without controver&longs;ie admit, that <lb/>when, of two equal Moveables of the &longs;ame Matter, and alike in Fi­<lb/>gure (which undoubtedly would move with equal &longs;wiftne&longs;&longs;e) as <lb/>well the Gravity, as the Superficies of one of them dimini&longs;heth, <lb/>(yet &longs;till retaining the &longs;imilitude of Figure) the Velocity like­<lb/>wi&longs;e, for the &longs;ame rea&longs;on, would not be dimini&longs;hed in that which <lb/>was le&longs;&longs;ened.</s></p><p type="main"> <s>SIMP. Really, I think, that it ought &longs;o to follow as you &longs;ay, <lb/>granting the pre&longs;ent Doctrine with a <emph type="italics"/>&longs;alvo<emph.end type="italics"/> &longs;till to our Doctrine, <lb/>which teacheth, that the greater or le&longs;&longs;er Gravity hath no operati­<lb/>on in accelerating or retarding Motion.</s></p><p type="main"> <s>SALV. </s> <s>And this I confirm; and grant you likewi&longs;e your Po­<lb/>&longs;ition, from whence, in my opinion, may be inferred, That in ca&longs;e <lb/>the Gravity dimini&longs;heth more than the Superficies, there may be <lb/>introduced in the Moveable, in that manner dimini&longs;hed, &longs;ome re­<lb/>tardment of Motion, and that greater and greater, by how much in <lb/>proportion, the diminution of the Weight was greater than the di­<lb/>minution of the Superficies</s></p><p type="main"> <s>SIMP. </s> <s>I make not the lea&longs;t que&longs;tion of it.<lb/><arrow.to.target n="marg1067"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1067"></margin.target><emph type="italics"/>Solids cannot be <lb/>dimini&longs;hed at the <lb/>&longs;ame rate in Super­<lb/>ficies as in Weight, <lb/>retaining the &longs;imi­<lb/>litude of the Fi­<lb/>gures.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>Now know, <emph type="italics"/>Simplicius,<emph.end type="italics"/> that in Solids one cannot di­<lb/>mini&longs;h the Superficies &longs;o much as the Weight keeping the &longs;imili­<lb/>tude of Figure. </s> <s>For it being manife&longs;t, that in dimini&longs;hing of grave <pb xlink:href="069/01/076.jpg" pagenum="74"/>Solids, the Weight le&longs;&longs;eneth as much as the Bulk, when ever the <lb/>Bulk happens to be dimini&longs;hed more than the Superficies, (care <lb/>being had to retain the &longs;imilitude of Figure) the Gravity likewi&longs;e <lb/>would come to be more dimini&longs;hed than the Superficies. </s> <s>But <emph type="italics"/>Geo­<lb/>metry<emph.end type="italics"/> teacheth us, that there is much greater proportion between <lb/>the Bulk and the Bulk in like Solids, than between their Superfi­<lb/>cies. </s> <s>Which for your better under&longs;tanding, I &longs;hall explain in &longs;ome <lb/>particular ca&longs;e. </s> <s>Therefore fancy to your &longs;elf, for example, a Dye, <lb/>one of the Sides of which is <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> two Inches long, &longs;o that one of <lb/>its Surfaces &longs;hall be four Square Inches, and all &longs;ix, that is, all its <lb/>Superficies twenty four Square Inches. </s> <s>Then &longs;uppo&longs;e the &longs;ame <lb/>Dye at three &longs;awings cut into eight &longs;mall Dice, the Side of every <lb/>one of which will be one Inch, and one of its Surfaces an Inch <lb/>Square, and its whole Superficies &longs;ix Square Inches, of which the <lb/>whole Dye contained twenty four in its Superficial content. </s> <s>Now, <lb/>you &longs;ee, that the Superficial content of the little Dye is the fourth <lb/>part of the Superficial content of the great one, (for &longs;ix is the <lb/>fourth part of twenty four) but the Solid content of the &longs;aid Dye <lb/>is only the eighth part: therefore the Bulk, and con&longs;equently the <lb/>Weight, doth much more dimini&longs;h than the Superficies. </s> <s>And if <lb/>you &longs;ubdivide the little Dye into eight others, we &longs;hall have for <lb/>the whole Superficial content of one of the&longs;e, one and an half <lb/>Square Inches, which is the &longs;ixteenth part of the Superficies of the <lb/>fir&longs;t Dye; but its Bulk, or Ma&longs;s, is only the &longs;ixty fourth part of that. <lb/></s> <s>You &longs;ee therefore, how that in only the&longs;e two divi&longs;ions the Bulks <lb/>decrea&longs;e four times fa&longs;ter than their Superficies: and if we &longs;hould <lb/>pro&longs;ecute the Subdivi&longs;ion, untill that we had reduced the fir&longs;t So­<lb/>lid into a &longs;mall powder, we &longs;hould find the Gravity of the minute <lb/>Atomes to be le&longs;&longs;ened an hundred and an hundred times more <lb/>than their Superficies. </s> <s>And this which I have exemplified in <lb/>Cubes, hapneth in all like Solids, the Bulks of which are in Se&longs;­<lb/>quialter proportion of their Superficies. </s> <s>You &longs;ee, therefore, in how <lb/>much greater proportion the Impediment of the Contact of the <lb/>Superficies of the Moveable with the <emph type="italics"/>Medium<emph.end type="italics"/> encrea&longs;eth in &longs;mall <lb/>Moveables, than in greater: and if we &longs;hould add, that the Sca­<lb/>bro&longs;ities in the very &longs;mall Superficies of the minute Atomes are <lb/>not happily le&longs;&longs;er than tho&longs;e of the Superficies of greater Solids, <lb/>that are diligently poli&longs;hed, ob&longs;erve how fluid, and void of all Re­<lb/>&longs;i&longs;tance being opened, the <emph type="italics"/>Medium<emph.end type="italics"/> is required to be, when it is to <lb/>give pa&longs;&longs;age to &longs;o feeble a Virtue. </s> <s>And therefore take notice, <emph type="italics"/>Sim­<lb/>plicius,<emph.end type="italics"/> that I did not equivocate, when even now I &longs;aid, That the <lb/>Superficies of le&longs;&longs;er Solids is greater, in compari&longs;on of that of <lb/>bigger.</s></p><p type="main"> <s>SIMP. </s> <s>I am wholly &longs;atisfied: and I verily believe, that if I were <lb/>to begin my Studies again, I &longs;hould follow the Coun&longs;el of <emph type="italics"/>Plato,<emph.end type="italics"/><pb xlink:href="069/01/077.jpg" pagenum="75"/>and enter my &longs;elf fir&longs;t in the Mathematicks, which I &longs;ee to proceed <lb/>very &longs;crupulou&longs;ly, and refu&longs;e to admit any thing for certain, &longs;ave <lb/>that which they nece&longs;&longs;arily demon&longs;trate.</s></p><p type="main"> <s>SAGR. </s> <s>I have taken great delight in this Di&longs;cour&longs;e; but, be­<lb/>fore we pa&longs;&longs;e any further, I would be glad to be &longs;atisfied in one <lb/>particular, which newly came into my thoughts, when but ju&longs;t <lb/>now you &longs;aid, that Like-Solids are in Se&longs;quialter proportion to <lb/>their Superficies for I have &longs;een, and under&longs;tood the Propo&longs;ition </s></p><p type="main"> <s><arrow.to.target n="marg1068"></arrow.to.target><lb/>with its Demon&longs;tration, in which it is proved, That the Superficies <lb/>of Like-Solids are in duplicate proportion of their Sides; and ano­<lb/>ther that proveth the &longs;ame Solids to be in triple proportion of the <lb/>&longs;ame Sides; but the proportion of Solids to their Superficies, I do <lb/>not remember that I ever &longs;o much as heard it mentioned.</s></p><p type="margin"> <s><margin.target id="marg1068"></margin.target><emph type="italics"/>Solids are to each <lb/>other in Se&longs;quial­<lb/>ter proportion to <lb/>their Superficies.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>You your &longs;elf have an&longs;wered and declared the doubt. <lb/></s> <s>For that which is triple of a thing of which another is double, doth <lb/>it not come to be Se&longs;quialter of this double? </s> <s>Yes doubtle&longs;&longs;e. </s> <s>Now, <lb/>if Superficies are in double proportion of the Lines, of which the <lb/>Solids are in triple proportion, may not we &longs;ay, That the Solids are <lb/>in Se&longs;quialter proportion of their Superficies?</s></p><p type="main"> <s>SAGR. </s> <s>I under&longs;tand you very well. </s> <s>And although other par­<lb/>ticulars, pertaining to the matter of which we have treated, do re­<lb/>main for me to ask, yet if we &longs;hould thus run from one Digre&longs;&longs;ion <lb/>to another, it will be late before we &longs;hould come to the Que&longs;tions <lb/>principally intended, which concern the diver&longs;ities of the Acci­<lb/>dents of the Re&longs;i&longs;tances of Solids again&longs;t Fraction; and therefore, <lb/>if you &longs;o plea&longs;e, we may return to the fir&longs;t Theme, which we pro­<lb/>po&longs;ed in the beginning.</s></p><p type="main"> <s>SALV. </s> <s>You &longs;ay very well; but the &longs;o many, and &longs;o different <lb/>things that have been examined, have &longs;toln &longs;o much of our time, <lb/>that there is but little of it left in this day to &longs;pend in our other <lb/>principal Argument, which is full of Geometrical Demon&longs;trati­<lb/>ons that are to be con&longs;idered with attention: &longs;o that I &longs;hould think <lb/>it were better to adjourn our meeting till to morrow, as well for <lb/>this which I have told you, as al&longs;o becau&longs;e I might bring with me <lb/>&longs;ome Papers, on which I have, in order, &longs;et down the Theorems and <lb/>Problems, in which are propo&longs;ed and demon&longs;trated the different <lb/>Pa&longs;&longs;ions of this Subject, which, it may be, would not otherwi&longs;e <lb/>with requi&longs;ite Method come into my mind.</s></p><p type="main"> <s>SAGR. </s> <s>I very gladly comply with your advice, and &longs;o much the <lb/>more willingly, in regard that, for a Conclu&longs;ion of this daies Con­<lb/>ference, I &longs;hall have time to hear you re&longs;olve &longs;ome doubts that I <lb/>find in my mind concerning the Point la&longs;t handled. </s> <s>Of which one <lb/>is, Whether we are to hold, that the Impediment of the <emph type="italics"/>Medium<emph.end type="italics"/><lb/>may be &longs;ufficient to a&longs;&longs;ign bounds to the Acceleration of Bodies of <lb/>very grave Matter, that are of great Bulk, and of a Spherical Figure: <pb xlink:href="069/01/078.jpg" pagenum="76"/>and I in&longs;tance in the Spherical Figure, that I might take that which <lb/>is contained under the lea&longs;t Superficies, and therefore le&longs;&longs;e &longs;ubject <lb/>to Retardment. </s> <s>Another &longs;hall be, touching the Vibrations of Pen­<lb/>dulums, and this hath many heads: One &longs;hall be, Whether all, <lb/>both Great, Mean, and Little, are made really and preci&longs;ely under <lb/>equal Times: And another, What is the proportion of the Times <lb/>of Moveables, &longs;u&longs;pended at unequal &longs;trings, of the Times of their <lb/>Vibrations I mean.</s></p><p type="main"> <s>SALV. </s> <s>The Que&longs;tions are ingenious, and, like as it is incident <lb/>to all Truths, I &longs;uppo&longs;e, that, which ever of them we handle, it will <lb/>draw after it &longs;o many other Truths, and curious Con&longs;equences, <lb/>that I cannot tell whether the remainder of this day may &longs;uffice <lb/>for the di&longs;cu&longs;&longs;ing of them all.</s></p><p type="main"> <s>SAGR. </s> <s>If they &longs;hall be but as delightful as the precedent, it <lb/>would be more grateful for me to employ as many daies, not to &longs;ay, <lb/>hours, as it is unto night, and I believe that <emph type="italics"/>Simplicius<emph.end type="italics"/> will not be <lb/>cloy'd with &longs;uch Argumentations as the&longs;e.</s></p><p type="main"> <s>SIMP. </s> <s>No certainly: and e&longs;pecially, when the Que&longs;tions trea­<lb/>ted of are Phy&longs;ical, touching which we read not the Opinions or <lb/>Di&longs;cour&longs;es of other Philo&longs;ophers.<lb/><arrow.to.target n="marg1069"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1069"></margin.target><emph type="italics"/>Any Body, of any <lb/>Figure, Greatne&longs;s, <lb/>and Gravity, is <lb/>checked by the Re­<lb/>nitence of the<emph.end type="italics"/> Me­<lb/>dium, <emph type="italics"/>though ne­<lb/>ver &longs;o tenuous, in <lb/>&longs;uch &longs;ort, that the <lb/>Motion continuing, <lb/>it is reduced to <lb/>equability.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>I come therefore to the fir&longs;t, affirming without any <lb/>hæ&longs;itation, that there is not a Sphere &longs;o big, nor of Matter &longs;o grave, <lb/>but that the Renitence of the <emph type="italics"/>Medium,<emph.end type="italics"/> though very tenuous, checks <lb/>its Acceleration, and in the continuation of the Motion reduceth <lb/>it to Equability, of which we may draw a very clear Argument <lb/>from Experience it &longs;elf. </s> <s>For if any falling Moveable were able in <lb/>its continuation of Motion to attain any degree of Velocity, no <lb/>Velocity that &longs;hould be conferred upon it, could be &longs;o great but <lb/>that it would depo&longs;e it, and free it &longs;elf of it by help of the Impe­<lb/>diment of the <emph type="italics"/>Medium.<emph.end type="italics"/> And thus, a Cannon-bullet, that had de <lb/>&longs;cended through the Air, <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> four yards, and had, for example, <lb/>acquired ten degrees of Velocity, and that with the&longs;e &longs;hould enter <lb/>into the Water, in ca&longs;e the Impediment of the Water were not <lb/>able to prohibit &longs;uch a certain <emph type="italics"/>Impetus<emph.end type="italics"/> in the Ball, it would en­<lb/>crea&longs;e it, or at lea&longs;t would continue it unto the bottom; which is <lb/>not ob&longs;erved to en&longs;ue: nay, the Water, although it were but a few <lb/>fathoms in depth, would impede and debilitate it in &longs;uch a man­<lb/>ner, that it will make but a &longs;mall impre&longs;&longs;ion in the bottom of the <lb/>River or Lake. </s> <s>It is therefore manife&longs;t, that that Velocity, of <lb/>which the Water had ability to deprive it in a very &longs;hort way, <lb/>would never be permitted to be acquired by it, though in a depth <lb/>of a thou&longs;and Fathoms. </s> <s>And why &longs;hould it be permitted to gain <lb/>it in a thou&longs;and, to be taken from it again in four? </s> <s>What need we <lb/>more? </s> <s>Do we not &longs;ee the immen&longs;e <emph type="italics"/>Impetus<emph.end type="italics"/> of the Ball, &longs;hot from <lb/>the Cannon it &longs;elf, to be in &longs;uch a manner flatted by the interpo­<pb xlink:href="069/01/079.jpg" pagenum="77"/>&longs;ition of a few Fathom of Water, that without any harm to the <lb/>Ship, it but very hardly reacheth to make a dent in it? </s> <s>The Air al­<lb/>&longs;o, though very yielding, doth neverthele&longs;&longs;e repre&longs;&longs;e the Velocity <lb/>of the falling Moveable, although it be very heavy, as we may by <lb/>&longs;uch like Experiments collect; for if from the top of a very high <lb/>Tower we &longs;hould di&longs;charge a Mu&longs;quet downwards, this will make <lb/>a le&longs;&longs;er impre&longs;&longs;ion on the ground, than if we &longs;hould di&longs;charge the <lb/>Mu&longs;quet at the height of four or &longs;ix yards above the Plane: an <lb/>evident &longs;ign, that the <emph type="italics"/>Impetus,<emph.end type="italics"/> wherewith the Bullet i&longs;&longs;ueth from <lb/>the Gun, di&longs;charged on the top of the Tower, doth gradually di­<lb/>mini&longs;h in de&longs;cending thorow the Air: therefore the de&longs;cending <lb/>from any what&longs;oever great height will not &longs;uffice to make it ac­<lb/>quire that <emph type="italics"/>Impetus,<emph.end type="italics"/> of which the Re&longs;i&longs;tance of the Air deprived <lb/>it, when it had in any manner been conferred upon it. </s> <s>The batte­<lb/>ry likewi&longs;e that the force of a Bullet, &longs;hot from a Culverin, &longs;hall <lb/>make in a Wall at the di&longs;tance of twenty Paces, would not, I be­<lb/>lieve, be &longs;o great, if the Bullet was &longs;hot perpendicularly from any <lb/>immen&longs;e Altitude. </s> <s>I believe, therefore, that there is a Bound or <lb/>term belonging to the Acceleration of every Natural Moveable <lb/>that departs from Re&longs;t, and that the Impediment of the <emph type="italics"/>Medium<emph.end type="italics"/> in <lb/>the end reduceth it to ^{*} Equality, in which it afterwards alwaies </s></p><p type="main"> <s><arrow.to.target n="marg1070"></arrow.to.target><lb/>continueth.</s></p><p type="margin"> <s><margin.target id="marg1070"></margin.target>* Or Equability.</s></p><p type="main"> <s>SAGR. </s> <s>The Experiments are really, in my opinion, much to <lb/>the purpo&longs;e: nor doth any thing remain, unle&longs;&longs;e the Adver&longs;ary <lb/>&longs;hould fortifie him&longs;elf, by denying, that they will hold true in great <lb/>and ponderous Ma&longs;&longs;es, and that a Cannon-bullet coming from the <lb/>Concave of the Moon, or from the upper Region of the Air, <lb/>would make a greater percu&longs;&longs;ion than coming from the Cannon.</s></p><p type="main"> <s>SALV. </s> <s>There is no que&longs;tion, but that many things may be <lb/>objected, and that they may not be all &longs;alved by Experiments; ne­<lb/>verthele&longs;&longs;e in this contradiction, me thinks, there is &longs;omething that <lb/>may fall under con&longs;ideration; <emph type="italics"/>&longs;cilicet,<emph.end type="italics"/> that it is very probable, <lb/><arrow.to.target n="marg1071"></arrow.to.target><lb/>that the Grave Body, falling from an Altitude, acquireth &longs;o much <lb/><emph type="italics"/>Impetus,<emph.end type="italics"/> at its arrival to the ground, as would &longs;uffice to return it <lb/>to that height, as is plainly &longs;een in a <emph type="italics"/>Pendulum<emph.end type="italics"/> rea&longs;onable weighty, <lb/>that being removed fifty or &longs;ixty degrees from the Perpendicular, <lb/>gaineth that Velocity and Virtue which exactly &longs;ufficeth to force it <lb/>to the like Recur&longs;ion, that little abated, which is taken from it by <lb/>the Impediment of the Air. </s> <s>To con&longs;titute, therefore, the Cannon­<lb/>bullet in &longs;uch an Altitude as may &longs;uffice for the acqui&longs;t of an <emph type="italics"/>Impe­<lb/>tus,<emph.end type="italics"/> as great as that which the Fire giveth it in its i&longs;&longs;uing from the <lb/>Piece, it would &longs;uffice to &longs;hoot it upwards perpendicularly with <lb/>the &longs;aid Cannon, and then ob&longs;erving, whether in its fall it maketh <lb/>an impre&longs;&longs;ion equal to that of the percu&longs;&longs;ion made near at hand in <lb/>its i&longs;&longs;uing forth; but, indeed, I believe, that it would not be any <pb xlink:href="069/01/080.jpg" pagenum="78"/>whit near &longs;o forcible. </s> <s>And therefore I hold that the Velocity, <lb/>which the Bullet hath near to its going out of the Piece, would <lb/>be one of tho&longs;e that the Impediment of the Air would never &longs;uffer <lb/>it to acquire, whil&longs;t it &longs;hould with a natural Motion de&longs;cend, leaving <lb/>the &longs;tate of Re&longs;t, from any great height. </s> <s>I come now to the other <lb/>Que&longs;tions belonging to <emph type="italics"/>Pendulums,<emph.end type="italics"/> matters which to many would <lb/>&longs;eem very frivolous, and more e&longs;pecially to tho&longs;e Philo&longs;ophers that <lb/>are continually bu&longs;ied in the more profound Que&longs;tions of Natural <lb/>Philo&longs;ophy: yet, notwith&longs;tanding, will not I contemn them, being <lb/>encouraged by the Example of <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/> him&longs;elf, in whom I admire <lb/>this above all things; that he hath not, as one may &longs;ay, omitted any <lb/>matter that any waies merited con&longs;ideration, which he hath not <lb/>&longs;poken of: and now upon the Que&longs;tions you propounded, I think <lb/>I can tell you a certain conceit of mine upon &longs;ome Problems con­<lb/>cerning Mu&longs;ick, a noble Subject, of which &longs;o many famous men, <lb/>and <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/> him&longs;elf, have written; and touching it, he con&longs;ide­<lb/>reth many curious Problems: &longs;o that if I likewi&longs;e &longs;hall from &longs;o fa­<lb/>miliar and &longs;en&longs;ible Experiments, draw Rea&longs;ons of admirable acci­<lb/>dents on the Argument of Sounds, I may hope that my di&longs;cour&longs;es <lb/>will be accepted by you.</s></p><p type="margin"> <s><margin.target id="marg1071"></margin.target><emph type="italics"/>A Grave Body, <lb/>falling from an <lb/>Altitude, acqui­<lb/>reth &longs;o much<emph.end type="italics"/> Im­<lb/>petus <emph type="italics"/>at its arri­<lb/>val to the ground, <lb/>as in all probabili­<lb/>ty, would &longs;uffice to <lb/>recarry it to the <lb/>&longs;ame height from <lb/>whence it fell.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR Not only accepted, but by me, in particular, mo&longs;t pa&longs;­<lb/>&longs;ionately de&longs;ired, in regard that I taking a great delight in all Mu­<lb/>&longs;ical In&longs;truments, and being rea&longs;onably well in&longs;tructed concerning <lb/>Con&longs;onances, have alwaies been ignorant and perplexed with <lb/>endeavouring to know, whence it cometh that one &longs;hould more <lb/>plea&longs;e and delight me than another; and that &longs;ome not only pro­<lb/>cure me no delight, but highly di&longs;plea&longs;e me: the trite Ptoblem al­<lb/>&longs;o of the two Chords &longs;et to an Uni&longs;on, one of which moveth and <lb/>actually &longs;oundeth at the touching of the other, I al&longs;o am unre&longs;ol­<lb/>ved in: nor am I very clearly informed concerning the Forms of <lb/>Con&longs;onances, and other particularities.</s></p><p type="main"> <s><emph type="italics"/>S<emph.end type="italics"/>ALV. </s> <s>We will &longs;ee, if from the&longs;e our <emph type="italics"/>Peudulums<emph.end type="italics"/> one may ga­<lb/>ther any &longs;atisfaction in all the&longs;e Doubts. </s> <s>And as to the fir&longs;t Que­<lb/>&longs;tion, that is, Whether the &longs;ame <emph type="italics"/>Pendulum<emph.end type="italics"/> doth really and punctu­<lb/>ally perform all its Vibrations, great, le&longs;&longs;er, and lea&longs;t, under Times <lb/>preci&longs;ely equal; I refer my &longs;elf to that which I have heretofore <lb/>learnt from our <emph type="italics"/>Academian,<emph.end type="italics"/> who plainly demon&longs;trateth, that the <lb/><arrow.to.target n="marg1072"></arrow.to.target><lb/>Moveable that &longs;hould de&longs;cend along the Chords, that are Subten­<lb/>&longs;es to any Arch, would nece&longs;&longs;arily pa&longs;&longs;e them all in equal Times, <lb/>as well the Subten&longs;e under an hundred and eighty degrees, (that <lb/>is, the whole Diameter) as the Subten&longs;es of an hundred, &longs;ixty, ten, <lb/>two, or half a degree, or of four minutes: &longs;till &longs;uppo&longs;ing that they <lb/>all determine in the lowe&longs;t Point touching the Horizontal Plane. <lb/></s> <s>Next as to the de&longs;cendents by the Arches of the &longs;ame Chords eli­<lb/>vated above the Horizon, and that are not greater than a Qua­<pb xlink:href="069/01/081.jpg" pagenum="79"/>drant, that is, than ninety degrees, Experience likewi&longs;e &longs;hews, that <lb/><arrow.to.target n="marg1073"></arrow.to.target><lb/>they pa&longs;&longs;e all in Times equal, but yet &longs;horter than the Times of <lb/>the pa&longs;&longs;ages by the Chords: an effect which hath &longs;o much of won­<lb/>der in it, by how much at the fir&longs;t apprehen&longs;ion one would think <lb/>the contrary ought to follow: For the terms of the beginning, <lb/>and the end of the Motion being common, and the Right-Line be­<lb/>ing the &longs;horte&longs;t, that can be comprehended between the &longs;aid <lb/>Terms, it &longs;eemeth rea&longs;onable, that the Motion made by it &longs;hould <lb/>be fini&longs;hed in the &longs;horte&longs;t Time, which yet is not &longs;o: but the &longs;hor­<lb/>te&longs;t Time, and con&longs;equently, the &longs;wifte&longs;t Motion, is that made by <lb/>the Arch of which the &longs;aid Right-Line is Chord. </s> <s>In the next <lb/><arrow.to.target n="marg1074"></arrow.to.target><lb/>place, as to the Times of the Vibrations of Moveables, &longs;u&longs;pended <lb/>by &longs;trings of different lengths, tho&longs;e Times are in Subduple pro­<lb/>portion to the lengths of the &longs;trings, or, if you will, the lengths <lb/>are in duplicate proportion to the Times, that is, are as the Squares <lb/>of the Times: &longs;o that if, for example, the Time of a Vibration <lb/>of one <emph type="italics"/>Pendulum<emph.end type="italics"/> is double to the Time of a Vibration of another, <lb/>it followeth, that the length of the &longs;tring of that is quadruple to <lb/>the length of the &longs;tring of this. </s> <s>And in the Time of one Vibration <lb/>of that, another &longs;hall then make three Vibrations, when the &longs;tring <lb/>of that &longs;hall be nine times as long as the other. </s> <s>From whence doth <lb/>follow, that the length of the &longs;trings have to each other the &longs;ame <lb/>proportion, that the Squares of the Numbers of the Vibrations that <lb/>are made in the &longs;ame Times have.</s></p><p type="margin"> <s><margin.target id="marg1072"></margin.target><emph type="italics"/>Moveables de&longs;cen­<lb/>ding along the <lb/>Chords, that are <lb/>Subten&longs;es to any <lb/>Arch of a Circle, <lb/>pa&longs;&longs;e as well the <lb/>greater as the le&longs;­<lb/>&longs;er Chords in equal <lb/>Times.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1073"></margin.target><emph type="italics"/>Moveables and<emph.end type="italics"/><lb/>Pendula <emph type="italics"/>de&longs;cend­<lb/>ing along the Ar­<lb/>ches of the &longs;ame <lb/>Chords, elivated as <lb/>far as 90 deg. </s> <s>pa&longs;s <lb/>the &longs;aid Arches in <lb/>Times equal, but <lb/>that are &longs;horter <lb/>than the tran&longs;iti­<lb/>ons along the <lb/>Chords.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1074"></margin.target><emph type="italics"/>The Times of the <lb/>Vibrations of Mo­<lb/>vables, hanging at <lb/>alonger or &longs;horter <lb/>thread, are to one <lb/>another in propor­<lb/>tion &longs;ubduple the <lb/>lengths of the <lb/>&longs;trings, at which <lb/>they hang.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. Then, if I have rightly under&longs;tood you, I may ea&longs;ily <lb/><arrow.to.target n="marg1075"></arrow.to.target><lb/>know the length of a &longs;tring, hanging at any never-&longs;o-great height, <lb/>although the &longs;ublime term of the &longs;u&longs;pen&longs;ion were invi&longs;ible to me, <lb/>and I only &longs;aw the other lower extream. </s> <s>For if I &longs;hall fa&longs;ten a <lb/>weight of &longs;ufficient Gravity to the &longs;aid &longs;tring here below, and &longs;et <lb/>it on vibrating to and again, and a friend telling &longs;ome of its Recur­<lb/>&longs;ions, and I at the &longs;ame time tell the Recur&longs;ions of another Movea­<lb/>ble, &longs;u&longs;pended at a &longs;tring that is preci&longs;ely a yard long, by the <lb/>Numbers of the Vibrations of the&longs;e <emph type="italics"/>Pendula,<emph.end type="italics"/> made in the &longs;ame <lb/>Time, I will find the length of the &longs;tring. </s> <s>As for example, &longs;uppo&longs;e <lb/>that in the time that my friend hath counted twenty Recur&longs;ions of <lb/>the long &longs;tring, I had told two hundred and forty of my &longs;tring, <lb/>that is one yard long: &longs;quaring the two numbers twenty and two <lb/>hundred and forty, which are 400, and 57600, I will &longs;ay, that the <lb/>long &longs;tring containeth 57600 of tho&longs;e Mea&longs;ures, of which my <lb/>&longs;tring containeth 400. and becau&longs;e the &longs;tring is one &longs;ole yard, I will <lb/>divide 57600 by 400, and the quotient will be 144, and I will af­<lb/>firm that &longs;tring to be 144 yards long.</s></p><p type="margin"> <s><margin.target id="marg1075"></margin.target><emph type="italics"/>To find the Length <lb/>of any Rope, or <lb/>&longs;tring, at which a <lb/>Moveable hang­<lb/>eth, by the frequen­<lb/>cy of its Vibrations<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>Nor will you be mi&longs;taken one Inch; and e&longs;pecially, if <lb/>you take a great Number of Vibrations.</s></p><p type="main"> <s>SAGR. </s> <s>You give me frequent occa&longs;ion to admire the Riches, <pb xlink:href="069/01/082.jpg" pagenum="80"/>and withal the extraordinary bounty of Nature, whil'&longs;t by things <lb/>&longs;o common, and, I might in a certain &longs;ence &longs;ay, vile, you go col­<lb/>lecting of Notions very curious, new, and oftentimes, remote <lb/>from all imagination. </s> <s>I have an hundred times con&longs;idered the Vi­<lb/>brations, in particular, of the Lamps in &longs;ome Churches, hanging <lb/>by very long ropes, when they have been unawares &longs;tirred by <lb/>any one: but the mo&longs;t that I inferred from that &longs;ame Ob&longs;ervati­<lb/>on, was the improbability of the Opinion of tho&longs;e who hold, <lb/>that &longs;uch-like Motions are maintained and continued by the <emph type="italics"/>Medi­<lb/>um,<emph.end type="italics"/> that is by the Air: for it &longs;hould &longs;eem to me, that the Air had <lb/>a great judgment, and withal but little bu&longs;ine&longs;&longs;e to &longs;pend &longs;o ma­<lb/>ny hours time in vibrating an hanging Weight with &longs;o much Regu­<lb/>larity: but that I &longs;hould have learnt, that that &longs;ame Moveable, <lb/>&longs;u&longs;pended at a &longs;tring of an hundred yards long, being removed <lb/>from Perpendicularity one while ninety degrees, and another <lb/>while one degree onely, or half a degree, &longs;hould &longs;pend as much time <lb/>in pa&longs;&longs;ing this little, as in pa&longs;&longs;ing that great Arch, certainly would <lb/>never have come into my head, for I &longs;till think, that it bordereth <lb/>upon Impo&longs;sibility. </s> <s>Now I am in expectation to hear that the&longs;e <lb/>petty Notions will a&longs;sign me &longs;uch Rea&longs;ons of tho&longs;e Mu&longs;ical Pro­<lb/>blems, as may, in part at lea&longs;t, give me &longs;atisfaction.<lb/><arrow.to.target n="marg1076"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1076"></margin.target><emph type="italics"/>Every<emph.end type="italics"/> Pendulum <lb/><emph type="italics"/>hath the Time of <lb/>its Vibration &longs;o li­<lb/>mited; that it is <lb/>not po&longs;&longs;ible to make <lb/>it move under any <lb/>other Period.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>Above all things, you are to know, that every <emph type="italics"/>Pendu­<lb/>lum<emph.end type="italics"/> hath the Time of its Vibrations &longs;o limited, and prefixed, that <lb/>it is impo&longs;&longs;ible to make it move under any other Period, than that <lb/>onely one, which is natural unto it. </s> <s>Let any one take the &longs;tring in <lb/>hand, to which the Weight is fa&longs;tened, and trie all the wayes <lb/>he can to encrea&longs;e or decrea&longs;e the frequency of its Vibrations, <lb/>and he &longs;hall finde it labour in vain: but we may, on the contrary, <lb/>on a <emph type="italics"/>Pendulum,<emph.end type="italics"/> though grave and at re&longs;t, by onely blowing up­<lb/>on it, conferre a Motion, and a Motion con&longs;iderably great, by <lb/>reiterating the bla&longs;ts, but under the Time that is properly be­<lb/>longing to its Vibrations: for if at the fir&longs;t bla&longs;t we &longs;hould have re­<lb/>moved it from Perpendicularity half an Inch, adding a &longs;econd, <lb/>after that it being returned towards us, is ready to begin the &longs;e­<lb/>cond Vibration, we &longs;hould conferre new Motion on it, and &longs;o <lb/>&longs;ucce&longs;&longs;ively with other bla&longs;ts, but given in Time, and not when <lb/>the <emph type="italics"/>Pendulum<emph.end type="italics"/> is comming towards us (for &longs;o we &longs;hould impede; <lb/>and not help the Motion) and &longs;o continuing with many Impul­<lb/>&longs;es, we &longs;hould confer upon it &longs;uch an <emph type="italics"/>Impetus,<emph.end type="italics"/> that a greater <lb/>force by much than that of a bla&longs;t of our breath, will be required <lb/>to &longs;tay it.</s></p><p type="main"> <s>SAGR. </s> <s>I have, from my childhood, ob&longs;erved, that one man a­<lb/>lone, by means of the&longs;e Impul&longs;es, given in Time, hath been able <lb/>to towl a very great Bell, and when it was to cea&longs;e, I have &longs;een <lb/>four or &longs;ix men more lay hold on the Bell-rope, and they have all <pb xlink:href="069/01/083.jpg" pagenum="81"/>been rai&longs;ed from the ground: &longs;o many together being unable to <lb/>arre&longs;t that <emph type="italics"/>Impetus,<emph.end type="italics"/> which one alone, with regular Pulls, had con­<lb/>ferred upon the Bell.</s></p><p type="main"> <s>SALV. </s> <s>An example, that declareth my meaning with no le&longs;&longs;e </s></p><p type="main"> <s><arrow.to.target n="marg1077"></arrow.to.target><lb/>propriety than this that I have premi&longs;ed, doth &longs;ute to render the <lb/>rea&longs;on of the admirable Problem of the Chord of the Lute or Viol, <lb/>which moveth, and maketh not onely that really to &longs;ound, which <lb/>is tuned to the Uni&longs;on, but that al&longs;o which is &longs;et to an Eighth <lb/>and a Fifth. </s> <s>The Chord being toucht, its Vibrations begin, and <lb/>continue all the Time that its Sound is heard to endure: the&longs;e <lb/>Vibrations make the Air neer adjacent to vibrate and tremble, <lb/>who&longs;e tremblings and quaverings di&longs;tend them&longs;elves a great way, <lb/>and &longs;trike upon all the Chords of the In&longs;trument, and al&longs;o of o­<lb/><arrow.to.target n="marg1078"></arrow.to.target><lb/>thers neer unto it: the Chord that is &longs;et to an Uni&longs;on, with that <lb/>which is toucht, being di&longs;po&longs;ed to make its Vibrations ^{*} in the <lb/>&longs;ame Time, beginneth at the fir&longs;t impul&longs;e to move a little, and <lb/><arrow.to.target n="marg1079"></arrow.to.target><lb/>a &longs;econd, a third, a twentieth, and many more, overtaking it, all <lb/>in ju&longs;t and Periodick Times, it receiveth at la&longs;t, the &longs;ame Tre­<lb/>mulation, with that fir&longs;t touched, and one may clearly &longs;ee it go, <lb/>dilating its Vibrations exactly according to the Pace of its Mo­<lb/>ver. </s> <s>This Undulation that di&longs;tendeth it &longs;elf thorow the Air, mo­<lb/>veth, and makes to vibrate, not onely the Chords, but likewi&longs;e <lb/>any other Body di&longs;po&longs;ed to trembling, and to vibrate in the very <lb/>Time of the trembling Chord: &longs;o that if we fix in the Sides of <lb/>the In&longs;trument &longs;everal &longs;mall pieces of Bri&longs;tles, or of other flexible <lb/>matters, you &longs;hall &longs;ee upon the &longs;ounding of the Viol, now one, <lb/>now another of tho&longs;e Corpu&longs;cles tremble, according as that <lb/>Chord is toucht, who&longs;e Vibrations return in the &longs;ame Time: the <lb/>others will not move at the &longs;triking of this Chord, nor will that <lb/>Bri&longs;tle tremble at the &longs;triking of another Chord. </s> <s>If with the Bow <lb/>one &longs;martly &longs;trike the Ba&longs;e-Chord of a Viol, and &longs;et a drinking <lb/>Gla&longs;&longs;e, thin and &longs;mooth, neer unto it, if the Tone of the Chord <lb/>be an Uni&longs;on to the Tone of the Gla&longs;&longs;e, the Gla&longs;&longs;e &longs;hall dance, <lb/>and &longs;en&longs;ibly re-&longs;ound. </s> <s>Again, the ample dilating of the Tremor <lb/>or Undulation of the <emph type="italics"/>Medium<emph.end type="italics"/> about the Body re&longs;ounding, is ap­<lb/>parently &longs;een in making the Gla&longs;&longs;e to &longs;ound, by putting a little <lb/>Water in it, and then chafing the brim or edge of it with the tip <lb/>of the finger: for the included Water is ob&longs;erved to undulate in <lb/>a mo&longs;t regular order: and the &longs;ame effect will be yet more clearly <lb/>&longs;een, by &longs;etting the foot of the Gla&longs;&longs;e in the bottom of a rea&longs;o­<lb/>nable large Ve&longs;&longs;el, in which there is Water as high almo&longs;t as to <lb/>the brim of the Gla&longs;&longs;e, for making it to &longs;ound, as before, with <lb/>the Confrication of the finger, we &longs;hall &longs;ee the trembling of the <lb/>Water to diffu&longs;e it &longs;elf mo&longs;t regularly, and with great Velocity, <lb/>to a great di&longs;tance round about the Gla&longs;&longs;e; and it hath many <pb xlink:href="069/01/084.jpg" pagenum="82"/>times been my fortune, in making a rea&longs;onable big Gla&longs;&longs;e, almo&longs;t <lb/>full of Water, to &longs;ound as afore&longs;aid, to &longs;ee the Waves in the <lb/>Water, at fir&longs;t formed with an exact equality; and it hapning <lb/>&longs;ometimes, that the Tone of the Gla&longs;&longs;e ri&longs;eth an Eighth higher, at <lb/>the &longs;ame in&longs;tant, I have &longs;een every one of the &longs;aid Waves to divide <lb/>them&longs;elves in two: an accident that very clearly proveth the <lb/>forme of the Octave to be the double.</s></p><p type="margin"> <s><margin.target id="marg1077"></margin.target><emph type="italics"/>The Chord of a <lb/>Mu&longs;ical In&longs;tru­<lb/>ment touched, mo­<lb/>veth, and maketh <lb/>the Chords &longs;et to an <lb/>Uni&longs;on, Fifth and <lb/>Eighth, with it to <lb/>&longs;ound; and why.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1078"></margin.target><emph type="italics"/>Sundry Problems <lb/>touching Mu&longs;ical <lb/>Proportions, and <lb/>their Solutions.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1079"></margin.target>* Or under.</s></p><p type="main"> <s>SAGR. </s> <s>The &longs;ame hath al&longs;o befaln me more than once, to my <lb/>delight, and al&longs;o benefit: for I &longs;tood a long time perplexed a­<lb/>bout the&longs;e Forms of Con&longs;onants, not conceiving, that the Rea­<lb/>&longs;on, commonly given thereof by the Authours that have hither­<lb/>to written learnedly of Mu&longs;ick, were &longs;ufficiently convincing, <lb/>they tell us, that the Diapa&longs;on, that is the Eighth, is contained <lb/>by the double, the Diapente, which we call the Fifth, by the <lb/>Se&longs;quialter: for a Chord being di&longs;tended on the ^{*} Monochord, <lb/><arrow.to.target n="marg1080"></arrow.to.target><lb/>&longs;triking it all; and afterwards &longs;triking but the half of it, by pla­<lb/>cing a Bridge in the middle, one heareth an Eighth; and if the <lb/>Bridge be placed at a third of the whole Chord, touching the <lb/>whole, and then the two thirds, it &longs;oundeth a Fifth; whereupon <lb/>they infer, that the Eighth is contained between two and one, and <lb/>the Fifth between three and two. </s> <s>This Rea&longs;on, I &longs;ay, &longs;eemed to <lb/>me not nece&longs;&longs;arily concluding for the a&longs;&longs;igning ju&longs;tly the double <lb/>and the Se&longs;quialter, for the natural Forms of the Diapa&longs;on and <lb/>the Diapente. </s> <s>And that which moved me &longs;o to think, was this. <lb/></s> <s>There are three ways, by which we may &longs;harpen the Tone of a <lb/>Chord: one is, by making it &longs;horter, the other is by di&longs;tending; <lb/>or making it more ten&longs;e; and the third is by making it thinner. </s> <s>If, <lb/>retaining the &longs;ame Tention and thickne&longs;&longs;e, we would hear an <lb/>Eighth, it is nece&longs;&longs;ary to &longs;horten it to one half, which is done by <lb/>&longs;triking it all, and then half. </s> <s>But if, retaining the &longs;ame length <lb/>and thickne&longs;&longs;e, we would have it ri&longs;e to an Eighth, by &longs;crewing <lb/>it higher, it will not &longs;uffice to &longs;tretch it double as much, but we <lb/>&longs;hall need the quadruple, &longs;o that, if before it was &longs;tretched by a <lb/>Weight of one pound, it will be needful to fa&longs;ten four pound <lb/>to it to &longs;harpen it to an Eighth. </s> <s>And la&longs;tly, if, keeping the &longs;ame <lb/>length and Tention, we would have a Chord, that by being &longs;mal­<lb/>ler, rendereth an Eighth, it will be nece&longs;&longs;ary, that it retain onely <lb/>a fourth part of the thickne&longs;&longs;e of the other more Grave. </s> <s>And this <lb/>which I &longs;peak of the Eighth, that is, that its form taken from the <lb/>Tention, or from the thickne&longs;&longs;e of the Chord, is in duplicate <lb/>proportion to that which it receiveth from the length, is to be <lb/>under&longs;toood of all other Mu&longs;ical Intervals: for that which the <lb/>length giveth us in a Se&longs;quialter proportion, <emph type="italics"/>i. </s> <s>e.<emph.end type="italics"/> by &longs;triking it all, <lb/>and then the two thirds, if you would have it proceed from the <lb/>Tention, or from the di&longs;gro&longs;&longs;ing, you mu&longs;t double the Se&longs;qui­<pb xlink:href="069/01/085.jpg" pagenum="83"/>alter proportion, taking the double Se&longs;quiquartan: and if the <lb/>Grave Chord were &longs;tretched by four pound weight, fa&longs;ten to the <lb/>Acute not &longs;ix, but nine: and, as to the thickne&longs;&longs;e, make the Grave <lb/>Chord thicker than the Acute, according to the proportion of <lb/>nine to four, to have the Fifth. </s> <s>The&longs;e being mo&longs;t exact Experi­<lb/>ments, I thought, that I &longs;aw no rea&longs;on, why the&longs;e Sage Philo&longs;o­<lb/>phers &longs;hould e&longs;tabli&longs;h the form of the Eighth to be rather the dou­<lb/>ble, than quadruple; and the Form of the Fifth to be rather the <lb/>Se&longs;quialter, than the double Se&longs;quiquartan. </s> <s>But becau&longs;e the <lb/>numbring of the Vibrations of a Chord, which in giving a &longs;ound, <lb/>are extreme frequent, is altogether impo&longs;&longs;ible, I &longs;hould always <lb/>have been in doubt, whether or no it were true, that the more <lb/>Acute Chord of the Eighth, made in the &longs;ame time, double the <lb/>number of the Vibrations of the more Grave, if the Waves, <lb/>which may be continued as long as you plea&longs;e, by making the <lb/>Gla&longs;s to &longs;ound and vibrate, had not &longs;en&longs;ibly &longs;hewn me, that in <lb/>the &longs;elf &longs;ame moment that (&longs;ometimes) the Sound is heard to ri&longs;e <lb/>to an Eighth, there are &longs;een to ari&longs;e other Waves more minute, <lb/>which with infinite &longs;moothne&longs;s cut in the middle each of tho&longs;e <lb/>fir&longs;t.</s></p><p type="margin"> <s><margin.target id="marg1080"></margin.target>* An In&longs;trument <lb/>of but one &longs;tring; <lb/>called by <emph type="italics"/>Mar­<lb/>&longs;ennus la Tromper­<lb/>te Marine.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>An excellent Ob&longs;ervation for di&longs;tingui&longs;hing one by <lb/>one the Undulations ari&longs;ing from the Tremulation of the re­<lb/>&longs;ounding Body: which are tho&longs;e that diffu&longs;ing them&longs;elves tho­<lb/>row the Air, make the titillation upon the Drum of our Ear, that <lb/>in our Soul becommeth a Sound: But whereas beholding and ob­<lb/>&longs;erving them in the Water, endure no longer than the confrica­<lb/>tion of the finger la&longs;teth, and al&longs;o in that time they are not per­<lb/>manent, but are continually made and di&longs;&longs;olved, would it not <lb/>be an ingenious undertaking, if one could make, with much <lb/>exqui&longs;itene&longs;&longs;e, &longs;uch, as would continue a long time; I mean <lb/>Moneths and Years, &longs;o as to give a man opportunity mea&longs;ure, <lb/>and with ea&longs;e to number them?</s></p><p type="main"> <s>SAGR. </s> <s>I a&longs;&longs;ure you I &longs;hould highly value &longs;uch an Invention.</s></p><p type="main"> <s>SALV. </s> <s>The di&longs;covery was accidental, and the Ob&longs;ervation <lb/>and applicative improvement of it onely were mine, and I hold <lb/>it to be a Circum&longs;tance of noble Contemplation, althongh a bu&longs;i­<lb/>ne&longs;&longs;e in its &longs;elf &longs;ufficiently homely. </s> <s>Scraping a Bra&longs;&longs;e Plate with <lb/>an Iron Chizzel to fetch out &longs;ome Spots, in moving the Chizzel to <lb/>and again upon it pretty quick, I heard it (once or twice among&longs;t <lb/>many gratings) to Sibilate and &longs;end forth a whi&longs;tling noi&longs;e, very <lb/>&longs;hrill and audible: and looking upon the Plate, I &longs;aw a long <lb/>row of &longs;mall &longs;treaks, parallel to one another, and di&longs;tant from <lb/>one another by mo&longs;t equal Intervals: returning to my &longs;craping <lb/>again, I perceived by &longs;everal trials, that in tho&longs;e &longs;crapings, and <lb/>tho&longs;e onely that whi&longs;tled, the Chizzel left the &longs;treaks upon the <pb xlink:href="069/01/086.jpg" pagenum="84"/>Plate: but when the Scraping pa&longs;&longs;ed without any Sibilation, <lb/>there was not &longs;o much as the lea&longs;t &longs;ign of any &longs;uch &longs;treaks. </s> <s>Re­<lb/>peating the Experiment &longs;everal times afterwards, &longs;craping now <lb/>with greater, now with le&longs;&longs;e velocity, the Sibilation hapned to <lb/>be of a Tone &longs;ometimes acuter, &longs;ometimes graver; and I ob&longs;erved <lb/>the marks made in the more acute &longs;ounds to be clo&longs;er together, <lb/>and tho&longs;e of the more grave farther a&longs;under: and &longs;ometimes al&longs;o, <lb/>according as the &longs;elf &longs;ame &longs;crape was made towards the end, with <lb/>greater velocity than at the beginning, the &longs;ound was heard to <lb/>grow &longs;harper, and the &longs;treaks were ob&longs;erved to &longs;tand thicker, <lb/>but ever with extream neatne&longs;&longs;e, and marked with exact equidi­<lb/>&longs;tance: and farther-more, in the Sibilating &longs;crapes; I felt the <lb/>Chizzel to &longs;hake or tremulate in my hand, and a certain chilne&longs;&longs;e <lb/>to run along my arm; and in &longs;hort, I &longs;aw the &longs;ame effected upon <lb/>the Toole, which we u&longs;e to ob&longs;erve in whi&longs;pering, and after­<lb/>wards &longs;peaking aloud, for &longs;ending forth the breath without <lb/>forming a &longs;ound, we do not perceive any moving in the throat <lb/>and mouth, in compari&longs;on of that which we di&longs;cern to be in the <lb/>Wind-pipe and Throat of every one, in &longs;ending forth the voice; <lb/>and e&longs;pecially in grave and loud Tones. </s> <s>I have likewi&longs;e &longs;ome­<lb/>times among&longs;t the Chords of the Viols, ob&longs;erved two that were <lb/>Uni&longs;ons to the Sibilations made by &longs;craping after the manner I <lb/>told you, and that were mo&longs;t different in Tone, from which two <lb/>they preci&longs;ely were di&longs;tant a perfect Fifth, and then mea&longs;uring <lb/>the intervals of the &longs;treaks of both the Scrapes, I &longs;aw the di­<lb/>&longs;tance that conteined forty five &longs;paces of the one, conteined <lb/>thirty of the other: which, indeed, is the Form attributed to the <lb/>Diapente. </s> <s>But here, before I proceed any farther, I will tell you, <lb/>that of the three manners of rendring a Sound Acute, that which <lb/>you refer to the &longs;lenderne&longs;&longs;e or finene&longs;&longs;e of the Chord, may <lb/>with more truth be a&longs;cribed to the Weight. </s> <s>For the alteration ta­<lb/>ken from the thickne&longs;&longs;e, an&longs;wereth, when the Chords are of the <lb/>&longs;ame matter; and &longs;o a Gut-&longs;tring to make an Eighth, ought to be <lb/>four times thicker than the other Gut-&longs;tring; and one of Wier four <lb/>times thicker than another of Wier. </s> <s>But if I would make an Eighth <lb/>with one of Wier to one of Gut-&longs;tring, I am not to make it four <lb/>times thicker, but four times graver, &longs;o that, as to thickne&longs;&longs;e, <lb/>this of Wier &longs;hall not be four times thicker, but quadruple in <lb/>Gravity, for &longs;ome times it &longs;hall be more &longs;mall than its re&longs;pon­<lb/>dent to the Acuter Eighth, that is of Gut-&longs;tring. </s> <s>Hence it com­<lb/>meth to pa&longs;&longs;e that, &longs;tringing an In&longs;trument with Chords of Gold, <lb/>and another with Chords of Bra&longs;&longs;e, if they &longs;hall be of the &longs;ame <lb/>length, thickne&longs;&longs;e, and Tention, Gold being almo&longs;t twice as <lb/>heavy, the Strings &longs;hall prove about a Fifth more Grave. </s> <s>And <lb/>here it is to be noted, that the Gravity of the Moveable more re­<pb xlink:href="069/01/087.jpg" pagenum="85"/>&longs;i&longs;teth the Velocity, than the thickne&longs;&longs;e doth; contrary to what <lb/>others at the fir&longs;t would think: for indeed, in appearance, its more <lb/>rea&longs;onable, that the Velocity &longs;hould be retarded by the Re&longs;i&longs;tance <lb/>of the <emph type="italics"/>Medium<emph.end type="italics"/> again&longs;t Opening in a Moveable thick and light, <lb/>than in one grave and &longs;lender: and yet in this ca&longs;e it happeneth <lb/>quite contrary. </s> <s>But pur&longs;uing our fir&longs;t Intent, I &longs;ay, That the <lb/>ncere&longs;t and immediate rea&longs;ons of the Forms of Mu&longs;ical Intervals, <lb/>is neither the length of the Chord, nor the Tention, nor the <lb/>thickne&longs;&longs;e, but the proportion of the numbers of the Vibrations, <lb/>and Percu&longs;&longs;ions of the Undulations of the Air that beat upon the <lb/>Drum of our Ear, which it &longs;elf al&longs;o doth tremulate under the <lb/>&longs;ame mea&longs;ures of Time. </s> <s>Having e&longs;tabli&longs;hed this Point, we may, <lb/>perhaps, a&longs;&longs;ign a very apt rea&longs;on, whence it commeth, that of <lb/>tho&longs;e Sounds that are different in Tone, &longs;ome Couples are re­<lb/>ceived with great delight by our Sence, others with le&longs;s, and <lb/>others occa&longs;ion in us a very great di&longs;turbance; which is to &longs;eek a <lb/>rea&longs;on of the Con&longs;onances more or le&longs;&longs;e perfect, and of Di&longs;lo­<lb/>nances. </s> <s>The mole&longs;tation and har&longs;hne&longs;&longs;e of the&longs;e proceeds, as I <lb/>believe, from the di&longs;cordant Pul&longs;ations of two different Tones, <lb/>which di&longs;proportionally &longs;trike the Drum of our Ear: and the <lb/>Di&longs;&longs;onances &longs;hall be extreme har&longs;h, in ca&longs;e the Times of the Vi­<lb/>brations were incommen&longs;urable. </s> <s>For one of which take that, <lb/>when of two Chords &longs;et to an Uni&longs;on, one is &longs;ounded, and &longs;uch <lb/>a part of another, as is the Side of the Square of its Diameter; <lb/>a Di&longs;&longs;onance like to the ^{*} Tritone, or Semi-diapente. </s> <s>Con&longs;onan­<lb/><arrow.to.target n="marg1081"></arrow.to.target><lb/>ces, and with plea&longs;ure received, &longs;hall tho&longs;e Couples of Sounds <lb/>be, that &longs;hall &longs;trike in &longs;ome order upon the Drum; which order <lb/>requireth, fir&longs;t, that the Pul&longs;ations made in the &longs;ame Time be <lb/>commen&longs;urable in number, to the end, the Cartillage of the Drum, <lb/>may not &longs;tand in the perpetual Torment of a double inflection of <lb/>allowing and obeying the ever di&longs;agreeing Percu&longs;&longs;ions. </s> <s>Therefore <lb/>the fir&longs;t and mo&longs;t grateful Con&longs;onance &longs;hall be the Eighth, being, <lb/>that for every &longs;troke, that the Grave-&longs;tring or Chord giveth upon <lb/>the Drum, the Acute giveth, two; &longs;o that both beat together <lb/>in every &longs;econd Vibration of the Acute Chord; and &longs;o of the <lb/>whole number of &longs;trokes, the one half accord to &longs;trike together, <lb/>but the &longs;trokes of the Chords that are Uni&longs;ons, alwayes joyn <lb/>both together, and therefore they are, as if they were of the <lb/>&longs;ame Chord, nor make they a Con&longs;onance. </s> <s>The Fifth delighteth <lb/>likewi&longs;e, in regard, that for every two &longs;troaks of the Grave <lb/>Chord, the Acute giveth three: from whence it followeth, that <lb/>numbering the Vibrations of the Acute Chord, the third part of <lb/>that number will agree to beat together; that is, two Solitary ones <lb/>interpo&longs;e between every couple of Con&longs;onances; and in the Di­<lb/>ate&longs;&longs;eron there interpo&longs;e three. </s> <s>In the &longs;econd, that is in the <emph type="italics"/>Se&longs;-<emph.end type="italics"/><pb xlink:href="069/01/088.jpg" pagenum="86"/><emph type="italics"/>quioctave<emph.end type="italics"/> Tone for every nine Pul&longs;ations, one onely &longs;trikes in Con­<lb/>&longs;ort with the other of the Graver Chord; all the re&longs;t are Di&longs;cords, <lb/>and received upon the Drum with regret, and are judged Di&longs;&longs;o­<lb/>nances by the Ear.</s></p><p type="margin"> <s><margin.target id="marg1081"></margin.target>* Or a fal&longs;e Fifth.</s></p><p type="main"> <s>SIMP. </s> <s>I could wi&longs;h this Di&longs;cour&longs;e were a little explained.</s></p><p type="main"> <s>SALV. </s> <s>Suppo&longs;e this line A B the Space, and dilating of a Vi­<lb/>bration of the Grave Chord; and the line C D that of the Acute <lb/>Chord, which with the other giveth the Eighth: and let A B be <lb/>divided in the mid&longs;t in E. </s> <s>It is manife&longs;t, that the Chords begin­<lb/>ing to move at the terms A and C, by that time the Acute Vibra­<lb/>tion &longs;hall be come to the term D, the other <lb/><figure id="id.069.01.088.1.jpg" xlink:href="069/01/088/1.jpg"/><lb/>&longs;hall be di&longs;tended onely to the half E, which <lb/>not being the bound or term of the Motion, <lb/>it &longs;trikes not: but yet a &longs;troak is made in D. <lb/></s> <s>The Vibrations afterwards returning from D <lb/>to C, the other pa&longs;&longs;eth from E to B, where­<lb/>upon the two Percu&longs;&longs;ions of B and C &longs;trike <lb/>both together upon the Drum: and &longs;o con­<lb/>tinuing to reiterate the like &longs;ub&longs;equent Vi­<lb/>brations; one &longs;hall &longs;ee, that the union of the <lb/>Percu&longs;&longs;ions of the Vibrations C D with tho&longs;e of A B, happen al­<lb/>ternately every other time: but the Pullations of the terms A B <lb/>are alwayes accompanied with one of C D, and that alwayes the <lb/>&longs;ame: which is manife&longs;t, for &longs;uppo&longs;ing that A and C &longs;trike to­<lb/>gether; in the time that A is pa&longs;&longs;ing to B, C goeth to D, and <lb/>returneth back to C: &longs;o that the &longs;troaks at B and C are al&longs;o <lb/>together. </s> <s>But now let the two Vibrations A B and C D be tho&longs;e <lb/>that produce the Diapente, the times of which are in proportion <lb/>Se&longs;quialter, and divide A B of the Grave Chord, in three equal <lb/>parts in E and O; And &longs;uppo&longs;e the Vibrations to begin at the <lb/>&longs;ame moment from the terms A and C: It is manife&longs;t, that at the <lb/>&longs;troke that &longs;hall be made in D, the Vibration of A B &longs;hall have <lb/>got no farther than O, the Drum therefore receiveth the Pul&longs;a­<lb/>tion D onely: again in the return from D to C, the other Vibra­<lb/>tion pa&longs;&longs;eth from O to B, and returneth to O, making the Pul­<lb/>&longs;ation in B, which likewi&longs;e is &longs;olitary, and in Counter-time, (an <lb/>accident to be con&longs;idered:) for we having &longs;uppo&longs;ed the fir&longs;t <lb/>Pul&longs;ations to be made at the &longs;ame moment in the terms A and C, <lb/>the &longs;econd, which was onely by the term D, was made as long after <lb/>as the time of the tran&longs;ition C D, that is A O, imports; but <lb/>that which followeth, made in B, is di&longs;tant from the other one­<lb/>ly &longs;o much as is the time O B, which is the half: afterwards con­<lb/>tinuing the Recur&longs;ion from O to A, whil&longs;t the other goeth from <lb/>C to D, the two Pul&longs;ations come to be made both at once in A <lb/>and D. </s> <s>There afterwards follow other Periods like to the&longs;e, that <pb xlink:href="069/01/089.jpg" pagenum="87"/>is, with the interpo&longs;ition of two &longs;ingle and &longs;olitary Pul&longs;ations of <lb/>the Acute Chord, and one of the Grave Chord, likewi&longs;e &longs;olita­<lb/>ry, is interpo&longs;ed between the two &longs;olitary &longs;trokes of the Acute. </s> <s>So <lb/>that if we did but &longs;uppo&longs;e the Time divided into Moments, that is, <lb/>into &longs;mall equal Particles: &longs;uppo&longs;ing that in the two fir&longs;t moments, <lb/>I pa&longs;&longs;ed from the Concordant Pul&longs;ations made in A and C to O <lb/>and D, and that in D, I make a Percu&longs;&longs;ion: and that in the third <lb/>and fourth moment I return from D to C, &longs;triking in C, and <lb/>that from O, I pa&longs;t to B, and returned to O, &longs;triking in B; and <lb/>that la&longs;tly in the fifth and &longs;ixth moment from O and C, I pa&longs;t to <lb/>A and D &longs;triking in both: we &longs;hall have the Pul&longs;ations di&longs;tributed <lb/>with &longs;uch order upon the Drum, that &longs;uppo&longs;ing the Pul&longs;ations of <lb/>the two Chords in the &longs;ame in&longs;tant, it &longs;hall two moments after <lb/>receive a &longs;olitary Percu&longs;&longs;ion, in the third moment anothor, &longs;oli­<lb/>tary likewi&longs;e, in the fourth another &longs;ingle one, and two moments <lb/>after, that is, in the &longs;ixth, two together; and here ends the <lb/>Period, and, if I may &longs;o &longs;ay, Anomaly; which Period is oft-times <lb/>afterwards replicated.</s></p><p type="main"> <s>SAGR. </s> <s>I can hold no longer, but mu&longs;t needs expre&longs;&longs;e the con­<lb/>tent I take in hearing rea&longs;ons &longs;o appo&longs;itely a&longs;&longs;igned of effects that <lb/>have &longs;o long time held me in darkne&longs;&longs;e and blindne&longs;&longs;e. </s> <s>Now I <lb/>know why the Uni&longs;on differeth not at all from a &longs;ingle Tone: I <lb/>&longs;ee why the Eighth is the principal Con&longs;onance, but withal &longs;o <lb/>like to an Uni&longs;on, that, as an Uni&longs;on, it is taken and cojoyned <lb/>with others: it re&longs;embleth an Uni&longs;on, for that whereas the Pul­<lb/>&longs;ations of Chords &longs;et to an Uni&longs;on, keep time in &longs;triking, the&longs;e <lb/>of the Grave Chord in an Eighth alwayes keep time with tho&longs;e <lb/>of the Acute, and of the&longs;e one interpo&longs;eth alone, and in equal <lb/>di&longs;tances, and as, one may &longs;ay, without any variety, whereupon <lb/>that Con&longs;onance is over &longs;weet. </s> <s>But the Fifth, with tho&longs;e its <lb/>Counter-times, and with the interpo&longs;ures of two &longs;olitary Pul&longs;a­<lb/>tions of the Acute Chord, and one of the Grave Chord, <lb/>between the Couples of Di&longs;cordant Pul&longs;ations, and tho&longs;e <lb/>three &longs;olitary ones, with an interval of time, as great as the half of <lb/>that which interpo&longs;eth between each Couple, and the &longs;olitary <lb/>Percu&longs;&longs;ions of the Acute Chord, maketh &longs;uch a Titillation and <lb/>Tickling upon the Cartillage of the Drum of the Ear, that al­<lb/>laying the Dulcity with a mixture of Acrimony, it &longs;eemeth at <lb/>one and the &longs;ame time to ki&longs;&longs;e and bite.</s></p><p type="main"> <s>SALV. </s> <s>It is convenient, in regard I &longs;ee, that you take &longs;uch de­<lb/>light in the&longs;e Novelties, that I &longs;hew you the way whereby the Eye <lb/>al&longs;o, and not the Ear alone, may recreate it &longs;elf in beholding <lb/>the &longs;ame &longs;ports that the Ear feeleth. </s> <s>Su&longs;pend Balls of Lead or o­<lb/>ther heavy matter on three &longs;trings of different lengths, but in <lb/>&longs;uch proportion, that while the longer maketh two Vibrations, <pb xlink:href="069/01/090.jpg" pagenum="88"/>the &longs;horter may make four, and the middle one three; which <lb/>will happen, when the longe&longs;t containeth &longs;ixteen feet, or other <lb/>mea&longs;ures, of which the middle one containeth nine, and the <lb/>&longs;horte&longs;t four: and removing them all together from Perpendi­<lb/>cularity, and then letting them go, you &longs;hall &longs;ee a plea&longs;ing In­<lb/>termixtion of the &longs;aid <emph type="italics"/>Pendulums<emph.end type="italics"/> with various encounters, but <lb/>&longs;uch, that, at every fourth Vibration of the longe&longs;t, all the three <lb/>will concurre in one and the &longs;ame term together, and then again <lb/>will depart from it, reiterating anew the &longs;ame Period: the which <lb/>commixture of Vibrations, is the &longs;ame, that being made by the <lb/>Chords, pre&longs;ents to the Ear an Eighth, with a Fifth in the mid&longs;t. <lb/></s> <s>And if you qualifie the length of other &longs;trings in the like di&longs;po­<lb/>&longs;ure, &longs;o that their Vibrations an&longs;wer to tho&longs;e of other Mu&longs;ical, <lb/>but Con&longs;onant Intervals, you &longs;hall &longs;ee other and other Inter­<lb/>weavings, and alwaies &longs;uch, that in determinate times, and after <lb/>determinate numbers of Vibrations, all the &longs;trings (be they three, <lb/>or be they four) will agree to joyn in the &longs;ame moment, in the <lb/>term of their Recur&longs;ions, and from thence to begin &longs;uch another <lb/>Period: but if the Vibrations of two or more &longs;trings are either <lb/>Incommen&longs;urable, &longs;o, that they never return harmoniou&longs;ly to ter­<lb/>minate determinate numbers of Vibrations, or though they be <lb/>not Incommen&longs;urable, yet if they return not till after a long time, <lb/>and after a great number of Vibrations, then the &longs;ight is con­<lb/>founded in the di&longs;orderly order of irregular Intermixtures, and <lb/>the Ear with wearine&longs;&longs;e and regret receiveth the intemperate Im­<lb/>pul&longs;es of the Airs Tremulations, that without Order or Rule, <lb/>&longs;ucce&longs;&longs;ively beat upon its Drum.</s></p><p type="main"> <s>But whither, my Ma&longs;ters, have we been tran&longs;ported for &longs;o <lb/>many hours by various Problems, and unlook't for Di&longs;cour&longs;es? <lb/></s> <s>We have made it Night, and yet we have handled few or none of <lb/>the points propounded; nay we have &longs;o lo&longs;t our way, that I <lb/>&longs;car&longs;e remember our fir&longs;t entrance, and that &longs;mall Introduction, <lb/>which we laid down, as the Hypothe&longs;is and beginning of the fu­<lb/>ture Demon&longs;trations.</s></p><p type="main"> <s>SAGR It will be convenient, therefore, that we break up our <lb/>Conference for this time, giving our Minds leave to compo&longs;e <lb/>them&longs;elves in the Nights Repo&longs;e, that we may to Morrow (if <lb/>you plea&longs;e &longs;o far to favour us) rea&longs;&longs;ume the Di&longs;cour&longs;es de&longs;ired, <lb/>and chiefly intended.</s></p><p type="main"> <s>SALV. </s> <s>I &longs;hall not fail to be here to Morrow at the u&longs;ual <lb/>hour, to &longs;erve and enjoy you.</s></p><p type="head"> <s><emph type="italics"/>The End of the Fir&longs;t Dialogue.<emph.end type="italics"/></s></p></chap><chap><pb xlink:href="069/01/091.jpg" pagenum="89"/><p type="head"> <s>GALILEUS, <lb/>HIS <lb/>DIALOGUES <lb/>OF <lb/>MOTION.</s></p><p type="head"> <s>The Second Dialogue.</s></p><p type="head"> <s><emph type="italics"/>INTERLOCUTORS,<emph.end type="italics"/></s></p><p type="head"> <s>SALVIATUS, SAGREDUS, and SIMPLICIUS.</s></p><p type="main"> <s>SAGREDUS.</s></p><p type="main"> <s><emph type="italics"/>Simplicius,<emph.end type="italics"/> and I, &longs;taid expecting your com­<lb/>ing, and we have been trying to recall to <lb/>memory our la&longs;t Con&longs;ideration, which, as <lb/>the Principle and Suppo&longs;ition, on which <lb/>you ground the Conclu&longs;ions that you in­<lb/>tended to Demon&longs;trate to us, was that <lb/>Re&longs;i&longs;tance, that all Bodies have to <emph type="italics"/>Fracti­<lb/>on,<emph.end type="italics"/> depending on that Cement, that con­<lb/>nects and glutinates the parts, &longs;o, as that <lb/>they do not &longs;eparate and divide without a powerful attraction: <lb/>and our enquiry hath been, what might be the Cau&longs;e of that <lb/>Coherence, which in &longs;ome Solids is very vigorous; propounding <lb/>that of <emph type="italics"/>Vacuum<emph.end type="italics"/> for the principal, which afterwards occa&longs;ioned &longs;o <lb/>many Digre&longs;&longs;ions as held us the whole day, and far from the <pb xlink:href="069/01/092.jpg" pagenum="90"/>matter at fir&longs;t propo&longs;ed, which was the Contemplation of the Re­<lb/>&longs;i&longs;tances of Solids to Fraction.</s></p><p type="main"> <s>SALV. </s> <s>I remember all that hath been &longs;aid, and returning to <lb/>our begun di&longs;cour&longs;e; What ever this Re&longs;i&longs;tance of Solids to brea­<lb/>king by a violent attraction, is &longs;uppo&longs;ed to be, it is &longs;ufficient, that it <lb/>is to be found in them: which, though it be very great again&longs;t the <lb/>&longs;trength of one that draweth them &longs;treight out, it is ob&longs;erved to be <lb/>le&longs;&longs;e in forcing them tran&longs;ver&longs;ely, or &longs;idewaies: and thus we &longs;ee, <lb/>for example, a rod of Steel, or Gla&longs;&longs;e to &longs;u&longs;tain the length-waies a <lb/>weight of a thou&longs;and pounds, which, fa&longs;tned at Right-Angles in­<lb/>to a Wall, will break if you hang upon it but only fifty. </s> <s>And of <lb/>this &longs;econd Re&longs;i&longs;tance we are to &longs;peak, enquiring, according to <lb/>what proportions it is found in Pri&longs;mes, and Cylinders of like and <lb/>unlike figure, length, and thickne&longs;s, and, withal, of the &longs;ame mat­<lb/>ter. </s> <s>In which Speculation, I take for a known Principle, that which <lb/>in the Mechanicks is demon&longs;trated among&longs;t the Pa&longs;&longs;ions of the <lb/>Vectis, which we call the Leaver: namely, That in that u&longs;e of the <lb/>Leaver, the Force is to the Re&longs;i&longs;tance in Reciprocal proportion, <lb/>as the Di&longs;tances from the Fulciment to the &longs;aid Force and the Re­<lb/>&longs;i&longs;tance.</s></p><p type="main"> <s>SIMP. </s> <s>This <emph type="italics"/>Ari&longs;totle,<emph.end type="italics"/> in his Mechanicks, demon&longs;trated before <lb/>any other man.</s></p><p type="main"> <s>SALV. </s> <s>I am content to grant him the precedency in time, but <lb/>for the firmne&longs;&longs;e o&longs; Demon&longs;tration, I think, that <emph type="italics"/>Archimedes<emph.end type="italics"/><lb/>ought to be preferred far before him, on one &longs;ole Propo&longs;ition of <lb/>whom, by him demon&longs;trated in his Book, <emph type="italics"/>De Equiponderantium,<emph.end type="italics"/><lb/>depend the Rea&longs;ons, not only of the Leaver, but of the greater <lb/>part of the other Mechanick In&longs;truments.</s></p><p type="main"> <s>SAGR. </s> <s>But &longs;ince that this Principle is the foundation of all <lb/>that which you intend to demon&longs;trate to us, it would be very re­<lb/>qui&longs;ite, that you produce us the proof of this &longs;ame Suppo&longs;ition, <lb/>if it be not too long a work, giving us a full and perfect informati­<lb/>on thereof.</s></p><p type="main"> <s>SALV. </s> <s>Though I am to do this, yet it will be better, that I lead <lb/>you into the field of all our future Speculations, by an enterance <lb/>&longs;omewhat different from that of <emph type="italics"/>Archimedes<emph.end type="italics"/>; and that, &longs;uppo­<lb/>&longs;ing no more, but only that equal Weights, put into a Ballance of <lb/>equal Arms, make an <emph type="italics"/>Equilibrium,<emph.end type="italics"/> (a Principle likewi&longs;e &longs;uppo&longs;ed <lb/>by <emph type="italics"/>Archimedes<emph.end type="italics"/> him&longs;elf.) I come, in the next place, to demon­<lb/>&longs;trate to you, that not only it is as true as the other, That unequal <lb/>Weights make an <emph type="italics"/>Equilibrium<emph.end type="italics"/> in a Stiliard of Armes unequal, ac­<lb/>cording to the proportion of tho&longs;e Weights Reciprocally &longs;u&longs;pen­<lb/>ded, but that it is one and the &longs;ame thing to place equal Weights <lb/>at equal di&longs;tances, as to place unequal Weights at di&longs;tances that <lb/>are in Reciprocal Proportion to the Weights. </s> <s>Now for a plain <pb xlink:href="069/01/093.jpg" pagenum="91"/>Demon&longs;tration of what I &longs;ay, de&longs;cribe a Solid Pri&longs;m or Cylinder <lb/>A B, [<emph type="italics"/>as in<emph.end type="italics"/> Figure 1. <emph type="italics"/>at the end of this Dialogue,<emph.end type="italics"/>] &longs;u&longs;pended by <lb/>its ends at the Line H I, and &longs;u&longs;tained by two Cords, H A, and I B. <lb/></s> <s>It is manife&longs;t, that if I &longs;u&longs;pend the whole by the Cord C, placed <lb/>in the middle of the Beam or Ballance H I, the Pri&longs;m A B will be <lb/>equilibrated, one half of its weight, being on one &longs;ide, and the other <lb/>half on the other &longs;ide of the Point of Su&longs;pen&longs;ion C by the Princi­<lb/>ple that we pre&longs;uppo&longs;ed. </s> <s>Now let the Pri&longs;m be divided into un­<lb/>equal parts by the Line D, and let the part D A be grea­<lb/>ter, and D B le&longs;&longs;er; and to the end, that &longs;uch divi&longs;ion being made, <lb/>the Parts of the Pri&longs;m may re&longs;t in the &longs;ame &longs;cituation and con&longs;ti­<lb/>tution, in re&longs;pect of the Line H I, let us help it with a Cord E D, <lb/>which, being fa&longs;tened in the Point E, &longs;u&longs;taineth the parts A D, and <lb/>D B: It is not to be doubted, but that there being no local muta­<lb/>tion in the Pri&longs;m, in re&longs;pect of the Ballance H I, it &longs;hall remain in <lb/>the &longs;ame &longs;tate of Equilibration. </s> <s>But it will re&longs;t in the &longs;ame Con­<lb/>&longs;titution likewi&longs;e, if the Part of the Pri&longs;m, that is now &longs;u&longs;pended at <lb/>the two extreams, or ends with Cords A H and D E, be hanged at <lb/>one &longs;ole Cord G L, placed in the mid&longs;t: and likewi&longs;e the other <lb/>part D B, will not change &longs;tate, if &longs;u&longs;pended by the middle, and <lb/>&longs;u&longs;tained by the Cord F M. </s> <s>So that the Cords H A, E D, and I B <lb/>being untied, and only the two Cords G L, and F M being left, the <lb/><emph type="italics"/>Equilibrium<emph.end type="italics"/> will &longs;till remain, the Su&longs;pen&longs;ion being &longs;till made at <lb/>the Point C. Now, here let us confider, that we have two Grave <lb/>Bodies A D, and D B, hanging at the terms G and F of a Beam <lb/>G F, in which the <emph type="italics"/>Equilibrium<emph.end type="italics"/> is made at the Point C: in &longs;uch <lb/>manner, that the di&longs;tance of the &longs;u&longs;pen&longs;ion of the Weight A D <lb/>from the Point C, is the Line C G, and the other part C F, is the <lb/>di&longs;tance at which the other Weight D B hangeth. </s> <s>It remaineth, <lb/>therefore, only to be demon&longs;trated, that tho&longs;e Di&longs;tances have the <lb/>&longs;ame proportion to one another, as the Weights them&longs;elves have, <lb/>but reciprocally taken: that is, that the di&longs;tance G C is to the di­<lb/>&longs;tance C F, as the Pri&longs;m D B to the Pri&longs;m D A, which we prove <lb/>thus. </s> <s>The Line G E being the half of E H, and E F the half of <lb/>E I, all G F &longs;hall be equall to all H I, and therefore equal to C I: <lb/>and taking away the common part C F, the remainder G C &longs;hall <lb/>be equal to the remainder F I, that is, to F E: and C E taken in <lb/>common, the two Lines G E and C F &longs;hall be equal: and, there­<lb/>fore, as G E, is to E F, &longs;o is F C, to C G: but as G C is to E F, &longs;o is <lb/>the double to the double; that is H E to E I; that is, the Pri&longs;m <lb/>A D to the Pri&longs;m D B. </s> <s>Therefore by Equality of proportion, <lb/>and by Conver&longs;ion, as the di&longs;tance G C is to the di&longs;tance C F, &longs;o <lb/>is the Weight B D to the Weight D A: which is that that I was to <lb/>demon&longs;trate. </s> <s>If you under&longs;tand this, I believe that you will not <lb/>&longs;cruple to admit, that the two Pri&longs;mes A D, and D B make an <pb xlink:href="069/01/094.jpg" pagenum="92"/><emph type="italics"/>Equilibrium<emph.end type="italics"/> in th Point C, for the half of the whole Solid A B is <lb/>on the right hand of the Su&longs;pen&longs;ion C, and the other half on the <lb/>left; and that in this manner there are repre&longs;ented two equal <lb/>Weights, di&longs;po&longs;ed and di&longs;tended at two equal di&longs;tances. </s> <s>Again, <lb/>that the two Pri&longs;mes A D, and D B, being reduced into two Dice, <lb/>or two Balls, or into any two other Figures, (provided that they <lb/>keep the &longs;ame Su&longs;pen&longs;ions G and F) do continue to make their <lb/><emph type="italics"/>Equilibrium<emph.end type="italics"/> in the Point C, I believe none can deny, for that it is <lb/>mo&longs;t manife&longs;t, that Figures change not weight, where the &longs;ame <lb/>quantity of matter is retained. </s> <s>From which we may gather the <lb/>general Conclu&longs;ion, That two Weights, whatever they be, make <lb/>an <emph type="italics"/>Equilibrium<emph.end type="italics"/> at Di&longs;tances reciprocally an&longs;wering to their Gra­<lb/>vities. </s> <s>This Principle, therefore, being e&longs;tabli&longs;hed, before we pa&longs;s <lb/>any farther, I am to propo&longs;e to Con&longs;ideration, how the&longs;e Forces, <lb/>Re&longs;i&longs;tances, Moments, Figures, may be con&longs;idered in Ab&longs;tract, <lb/>and &longs;eparate from Matter, as al&longs;o in Concrete and conjoyned <lb/>with Matter; and in this manner tho&longs;e Accidents that agree with <lb/>Figures, con&longs;idered as Immaterial, &longs;hall receive certain Modifica­<lb/>tions, when we &longs;hall come to add Matter to them, and con&longs;equent­<lb/>ly Gravity. </s> <s>As for example, if we take a Leaver, as for in&longs;tance <lb/>B A [<emph type="italics"/>as in<emph.end type="italics"/> Fig. </s> <s>2.] which, re&longs;ting upon the Fulciment E, we ap­<lb/>ply to rai&longs;e the heavy Stone D: It is manife&longs;t by the Principle de­<lb/>mon&longs;trated, that the Force placed at the end B, &longs;hall &longs;uffice to <lb/>equal the Re&longs;i&longs;tance of the Weight D, if &longs;o be, that its Moment <lb/>have the &longs;ame proportion to the Moment of the &longs;aid D, that the <lb/>Di&longs;tance A C hath to the Di&longs;tance C B: and this is true, if we <lb/>confider no other Moments than tho&longs;e of the &longs;imple Force in B, <lb/>and of the Re&longs;i&longs;tance in D, as if the &longs;aid Leaver were immaterial, <lb/>and void of Gravity. </s> <s>But if we bring to account the Gravity al&longs;o <lb/>of the In&longs;trument or Leaver it &longs;elf, which hapneth &longs;ometimes to be <lb/>of Wood, and &longs;ometimes of Iron; it is manife&longs;t, that the weight <lb/>of the Leaver, being added to the Force in B, it will alter the pro­<lb/>portion, which it will be requi&longs;ite to deliver in other terms. </s> <s>And <lb/>therefore before we pa&longs;&longs;e any farther, it is nece&longs;&longs;ary, that we di­<lb/>&longs;tingui&longs;h between the&longs;e two waies of Con&longs;ideration, calling that a <lb/>taking it ab&longs;olutely, when we &longs;uppo&longs;e the In&longs;trument to be taken <lb/>in Ab&longs;tract, that is, disjunct from the Gravity of its own Matter; <lb/>but conjoyning the Matter, as al&longs;o the Gravity, with &longs;imple and <lb/>ab&longs;olute Figures, we will phra&longs;e the Figures conjoyn'd with the <lb/>Matter, Moment, or Force compounded.</s></p><p type="main"> <s>SAGR I mu&longs;t of nece&longs;&longs;ity break the Re&longs;olution I had taken, <lb/>not to give occa&longs;ion of digre&longs;&longs;ing, for I &longs;hould not be able to &longs;et <lb/>my &longs;elf to hear what remaines with attention, if a certain &longs;cruple <lb/>were not removed that cometh into my head; and it is this, That <lb/>I gue&longs;&longs;e you make compari&longs;on between the Force placed in B, and <pb xlink:href="069/01/095.jpg" pagenum="93"/>the total Gravity of the Stone D, of which Gravity me thinks, that <lb/>one, and that, very probably, the greater part, re&longs;teth upon the <lb/>Plane of the Horizon: &longs;o that----</s></p><p type="main"> <s>SALV. </s> <s>I have rightly apprehended you, &longs;o that you need &longs;ay <lb/>no more, but only take notice, that I named not the total Gravity <lb/>of the Stone, but &longs;pake of the Moment that it hath, and exerci&longs;eth <lb/>at the Point A, the extream term of the Leaver B A, which is ever <lb/>le&longs;s than the entire weight of the Stone; and is variable according <lb/>to the Figure of the Stone, and according as it hapneth to be more <lb/>or le&longs;&longs;e elevated.</s></p><p type="main"> <s>SAGR. </s> <s>I am &longs;atisfied in that particular, but I have one thing <lb/>more to de&longs;ire, namely, that for my perfect information, you would <lb/>demon&longs;trate to me the way, if there be one, how I may find what <lb/>part of the total weight that is, which cometh to be born by the <lb/>&longs;ubjacent Plane, and what that which gravitates upon the Leaver <lb/>at the extream A.</s></p><p type="main"> <s><emph type="italics"/>S<emph.end type="italics"/>ALV. </s> <s>Becau&longs;e I can give you &longs;atisfaction in few words, I will <lb/>not fail to &longs;erve you: therefore, de&longs;cribing a &longs;light Figure thereof, <lb/>be plea&longs;ed to &longs;uppo&longs;e, that the Weight, who&longs;e Center of Gravity is <lb/>A, [<emph type="italics"/>as in<emph.end type="italics"/> Fig. </s> <s>3.] re&longs;teth upon the Horizon with the term B, and <lb/>at the other end is born up by the Leaver C G, on the Fulciment <lb/>N, by a Power placed in G: and that from the Center A, and term <lb/>C, Perpendiculars be let fall to the Horizon, A O, and C F. </s> <s>I &longs;ay, <lb/>That the Moment of the whole Weight &longs;hall have to the Moment <lb/>of the whole Power in G, a proportion compounded of the Di­<lb/>&longs;tance G N to the Di&longs;tance N C, and of F B to B O. Now, as the <lb/>Line F B is to B O, &longs;o let N C be to X. </s> <s>And the whole Weight A <lb/>being born by the two Powers placeed in B and C, the Power B is <lb/>to C, as the di&longs;tance F O to O B: and by Compo&longs;ition, the <lb/>two Powers B and C together, that is, the total Moment of <lb/>the whole Weight A, is to the Power in C, as the Line F B is <lb/>to the Line B O; that is, as N C to X: But the Moment of <lb/>the Power in C is to the Moment of the Power in G, as the Di­<lb/>&longs;tance G N is to N C: Therefore, by Perturbation of proportion, <lb/>the whole Weight A is to the Moment of the Power in G, as G N <lb/>to X: But the proportion of G N to X is compounded of the pro­<lb/>portion G N to N C, and of that of N C to X; that is, of F B to <lb/>B O: Therefore the Weight A is to the Power that bears it up in <lb/>G, in a proportion compounded of G N to N C, and of that of <lb/>F B to B O: which is that that was to be demon&longs;trated. </s> <s>Now re­<lb/>turning to our fir&longs;t intended Argument, all things hitherto decla­<lb/>red being under&longs;tood, it will not be hard to know the rea&longs;on, <lb/>whence it cometh to pa&longs;&longs;e that</s></p><pb xlink:href="069/01/096.jpg" pagenum="94"/><p type="head"> <s>PROPOSITION I.</s></p><p type="main"> <s><emph type="italics"/>A Solid Pri&longs;m or Cylinder of Gla&longs;&longs;e, Steel, Wood, or <lb/>other Frangible Matter, that being &longs;u&longs;pended length­<lb/>waies, will &longs;u&longs;tain a very great Weight hanged <lb/>Thereat, will, Sidewaies, (as we &longs;aid even now) be <lb/>broken in pieces by a far le&longs;&longs;er Weight, according as <lb/>its length &longs;hall exceed its thickne&longs;s.<emph.end type="italics"/></s></p><p type="main"> <s>Wherefore let us de&longs;cribe the Solid Pri&longs;m A B C D, <lb/>fixed into a Wall by the Part A B, and in the <lb/>other extream &longs;uppo&longs;e the Force of the Weight E; <lb/>(alwaies under&longs;tanding the Wall to be erect to the Horizon, <lb/>and the Pri&longs;m or Cylinder fa&longs;tened in the Wall at Right-An­<lb/>gles) it is manife&longs;t, that being to break, it will be broken in the place <lb/>B, where the Mortace in the Wall &longs;erveth for Fulciment, and B C <lb/>for the part of the Leaver in which lieth the force, and the thick­<lb/>ne&longs;&longs;e of the Solid B A is the other part of the Leaver, in which <lb/>lieth the Re&longs;i&longs;tance, which con&longs;i&longs;teth in the unfa&longs;tening, or divi­<lb/>ding, that is to be made of the part of the Solid B D, that is with­<lb/>out the Wall from that which is within: and by what hath been <lb/>declared, the Moment <lb/><figure id="id.069.01.096.1.jpg" xlink:href="069/01/096/1.jpg"/><lb/>of the Force placed in <lb/>C, is to the Moment of <lb/>the Re&longs;i&longs;tance that lieth <lb/>in the thickne&longs;&longs;e of the <lb/>Pri&longs;m, that is, in the <lb/>Connection of the Ba&longs;e <lb/>B A, with the parts con­<lb/>tiguous to it, as the <lb/>length C B is to the half <lb/>of B A: And therefore <lb/>the ab&longs;olute Re&longs;i&longs;tance <lb/>again&longs;t Fraction that is <lb/>in the Pri&longs;m B D, <lb/>(which ab&longs;olute Re&longs;i­<lb/>&longs;tance is that which is <lb/>made by drawing it <lb/>downwards, for at that <lb/>time the motion of the Mover is the &longs;ame with that of the Body <lb/>Moved) again&longs;t the fracture to be made by help of the Leaver <pb xlink:href="069/01/097.jpg" pagenum="95"/>B C, is as the Length B C to the half of A B in the Pri&longs;m, which <lb/>in the Cylinder is the Semidiameter of its Ba&longs;e. </s> <s>And this is our fir&longs;t <lb/>Propo&longs;ition. </s> <s>And ob&longs;erve, that what I have &longs;aid ought to be un­<lb/>der&longs;tood, when the Confideration of the proper Weight of the So­<lb/>lid B D is removed: which Solid I have taken as weighing nothing. <lb/></s> <s>But in ca&longs;e we would bring its Gravity to account, conjoyning it <lb/>with the Weight E, we ought to add to the Weight E the half of <lb/>the Weight of the Solid B D: &longs;o that the Weight B D being <lb/><emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> two pounds, and the Weight of E ten pounds, we are to <lb/>take the Weight E, as if it were eleven pounds.</s></p><p type="main"> <s>SIMP. </s> <s>And why not as if it were twelve?</s></p><p type="main"> <s>SALV. </s> <s>The Weight E, <emph type="italics"/>Simplicius,<emph.end type="italics"/> hanging at the term C, <lb/>gravitates in re&longs;pect of B C, with all its Moment of ten pounds, <lb/>whereas if only B D were pendent, it would weigh with its whole <lb/>Moment of two pounds; but, as you &longs;ee, that Solid is di&longs;tributed <lb/>thorow all the length B C, uniformly, &longs;o that its parts near to the <lb/>extream B, gravitate le&longs;&longs;e than the more remote: &longs;o that, in a word, <lb/>compen&longs;ating tho&longs;e with the&longs;e, the weight of the whole Pri&longs;m is <lb/>brought to operate under the Center of its Gravity, which an&longs;we­<lb/>reth to the middle of the Leaver B C: But a Weight hanging at <lb/>the end C, hath a Moment double to that which it would have <lb/>hanging at the middle: And therefore the half of the Weight of <lb/>the Pri&longs;m ought to be added to the Weight E, when we would u&longs;e <lb/>the Moment of both, as placed in the Term C.</s></p><p type="main"> <s>SIMP. </s> <s>I apprehend you very well, and, if I deceive not my &longs;elf, <lb/>me thinks, that the Power of both the Weights B D and E, &longs;o placed, <lb/>would have the &longs;ame Moment, as if the whole Weight of B D, and <lb/>the double of E were hanged in the mid&longs;t of the Leaver B C.</s></p><p type="main"> <s>SALV. </s> <s>It is exactly &longs;o, and you are to bear it in mind. </s> <s>Here we <lb/>may immediatly under&longs;tand</s></p><p type="head"> <s>PROPOSITION II.</s></p><p type="main"> <s><emph type="italics"/>How, and with what proportion, a Ruler, or Pri&longs;m, <lb/>more broad than thick, re&longs;i&longs;teth Fraction, better if it <lb/>be forced according to its breadth, than according to <lb/>its thickne&longs;&longs;e.<emph.end type="italics"/></s></p><p type="main"> <s>For under&longs;tanding of which, let a Pri&longs;m be &longs;uppo&longs;ed A D: <lb/>[<emph type="italics"/>as in<emph.end type="italics"/> Fig. </s> <s>4.] who&longs;e breadth is A C, and its thickne&longs;s much <lb/>le&longs;&longs;er C B: It is demanded, why we would attempt to break <lb/>it edge-waies, as in the fir&longs;t Figure it will re&longs;i&longs;t the great Weight <lb/>T, but placed flat-waies, as in the &longs;econd Figure, it will not re&longs;i&longs;t <pb xlink:href="069/01/098.jpg" pagenum="96"/>X, le&longs;&longs;er than T: Which is manife&longs;ted, &longs;ince we under&longs;tand the <lb/>Fulciment, one while under the Line B C, and another while under <lb/>C A, and the Di&longs;tances of the Forces to be alike in both Ca&longs;es, to <lb/>wit, the length <emph type="italics"/>B<emph.end type="italics"/> D. </s> <s>But in the fir&longs;t Ca&longs;e, the Di&longs;tance of the Re­<lb/>&longs;i&longs;tance from the Fulciment, which is the half of the Line C A, is <lb/>greater than the Di&longs;tance in the other Ca&longs;e, which is the half of B <lb/>C: Therefore the Force of the Weight T, mu&longs;t of nece&longs;&longs;ity be grea­<lb/>ter than X, as much as the half of the breadth C A is greater than <lb/>half the thichne&longs;&longs;e B C, the fir&longs;t &longs;erving for the Counter-Leaver of <lb/>C A, and the &longs;econd of C B to overcome the &longs;ame Re&longs;i&longs;tance, that <lb/>is the quantity of the <emph type="italics"/>Fibres,<emph.end type="italics"/> or &longs;trings of the whole Ba&longs;e A B. <lb/></s> <s>Conclude we therefore, that the &longs;aid Pri&longs;m or Ruler, which is <lb/>broader than it is thick, re&longs;i&longs;teth, bresking more the edge-waies <lb/>than the flat-waies, according to the Proportion of the breadth to <lb/>the thickne&longs;s.</s></p><p type="main"> <s>It is requi&longs;ite that we begin in the next place</s></p><p type="head"> <s>PROPOSITION III.</s></p><p type="main"> <s><emph type="italics"/>To find according to what proportion the encrea&longs;e of the <lb/>Moment of the proper Gravity is made in a Pri&longs;m <lb/>or Cylinder, in relation to the proper Re&longs;i&longs;tance <lb/>again&longs;t Fraction, whil&longs;t that being parallel to the <lb/>Horizon, it is made longer and longer: Which Mo­<lb/>ment I find to encrea&longs;e &longs;ucce&longs;sively in duplicate Pro­<lb/>portion to that of the prolongation.<emph.end type="italics"/></s></p><p type="main"> <s>For demon&longs;tration whereof, de&longs;cribe the Pri&longs;m or Cylin­<lb/>der A D, firmly fa&longs;tned in the Wall at the end A, and let <lb/>it be equidi&longs;tant from the Horizon, and let the &longs;ame be <lb/>under&longs;tood to be prolonged as far as E, adding thereto the part <lb/><emph type="italics"/>B<emph.end type="italics"/> E. </s> <s>It is manife&longs;t, that the prolongation of the Leaver A <emph type="italics"/>B<emph.end type="italics"/><lb/>to C encrea&longs;eth, by it &longs;elf alone, that is taken ab&longs;olutely, the <lb/>Moment of the Force pre&longs;&longs;ing again&longs;t the Re&longs;i&longs;tance of the <lb/>Separation and Rupture to be made in A, according to the pro­<lb/>portion of C A to <emph type="italics"/>B<emph.end type="italics"/> A: but, moreover, the Weight of the Solid <lb/>affixed <emph type="italics"/>B<emph.end type="italics"/> E, encrea&longs;eth the Moment of the pre&longs;&longs;ing Gravity of <lb/>the Weight of the Solid A <emph type="italics"/>B,<emph.end type="italics"/> according to the Proportion of <lb/>the Pri&longs;m A E to the Pri&longs;m A <emph type="italics"/>B<emph.end type="italics"/>; which proportion is the &longs;ame <lb/>as that of the length A C, to the length A <emph type="italics"/>B<emph.end type="italics"/>: Therefore it is clear <pb xlink:href="069/01/099.jpg" pagenum="97"/>that the two augmentations of the Lengths and of the Gravities <lb/>being put together, the Moment compounded of both is in double <lb/><figure id="id.069.01.099.1.jpg" xlink:href="069/01/099/1.jpg"/><lb/>proportion to ei­<lb/>ther of them. </s> <s>We <lb/>conclude there­<lb/>fore, That the Mo­<lb/>ments of the For­<lb/>ces of Pri&longs;mes and <lb/>Cylinders of equal <lb/>thickne&longs;&longs;e, but of <lb/>unequal length, are <lb/>to one another in <lb/>duplicate proporti­<lb/>on to that of their <lb/>Lengths; that is, <lb/>are as the Squares of <lb/>their Lengths.</s></p><p type="main"> <s>We will &longs;hew, in <lb/>the &longs;econd place, <lb/>according to what proportion the Re&longs;i&longs;tance of Fraction in Pri&longs;mes <lb/>and Cylinders encrea&longs;eth, when they continue of the &longs;ame length, <lb/>and encrea&longs;e in thickne&longs;s. </s> <s>And here I &longs;ay, that</s></p><p type="head"> <s>PROPOSITION IV.</s></p><p type="main"> <s><emph type="italics"/>In Pri&longs;mes and Cylinders of equal length, but unequal <lb/>thickne&longs;s, the Re&longs;i&longs;tance again&longs;t Fraction encrea&longs;eth <lb/>in a proportion iriple to the Diameters of their <lb/>Thickne&longs;&longs;es, that is, of their Ba&longs;es.<emph.end type="italics"/></s></p><p type="main"> <s>Let the two Cylinders be the&longs;e A and <emph type="italics"/>B, [as in<emph.end type="italics"/> Fig. </s> <s>5.] <lb/>who&longs;e equal lengths are D G, and F H, the unequal <emph type="italics"/>B<emph.end type="italics"/>a&longs;es <lb/>the Circles, who&longs;e Diameters are C D, and E F. </s> <s>I &longs;ay, <lb/>that the Re&longs;i&longs;tance of the Cylinder <emph type="italics"/>B<emph.end type="italics"/> is to the Re&longs;i&longs;tance of the <lb/>Cylinder A again&longs;t Fraction, in a proportion triple to that which <lb/>the Diameter F E hath to the Diameter D C. </s> <s>For if we con&longs;ider <lb/>the ab&longs;olute and &longs;imple Re&longs;i&longs;tance that re&longs;ides in the <emph type="italics"/>B<emph.end type="italics"/>a&longs;es, that <lb/>is, in the Circles E F, and D C to breaking, offering them vio­<lb/>lence by pulling them end-waies, without all doubt, the Re&longs;i&longs;tance <lb/>of the Cylinder <emph type="italics"/>B,<emph.end type="italics"/> is &longs;o much greater than that of the Cylinder A, <lb/>by how much the Circle E F is greater than C D; for the Fibres, <lb/>Filaments, or tenacious parts, which hold together the Parts of the <lb/>Solid, are &longs;o many the more. <emph type="italics"/>B<emph.end type="italics"/>ut if we con&longs;ider, that in offering <pb xlink:href="069/01/100.jpg" pagenum="98"/>them violence tran&longs;ver&longs;ly we make u&longs;e of two Leavers; of which <lb/>the Parts or Di&longs;tances, at which the Forces are applied are the Lines <lb/>D G, and F H, the Fulciments are in the Points D and F; but the <lb/>other Parts or Di&longs;tances, at which the Re&longs;i&longs;tances are placed, are <lb/>the Semidiameters of the Circles D C and E F, becau&longs;e the Fila­<lb/>ments di&longs;per&longs;ed thorow the whole Superficies of the Circles are as <lb/>if they were all reduced into the Centers: con&longs;idering, I &longs;ay, tho&longs;e <lb/>Leavers, we would be under&longs;tood to intend, that the Re&longs;i&longs;tance in <lb/>the Center of the Ba&longs;e E F again&longs;t the Force of H, is &longs;o much grea­<lb/>ter than the Re&longs;i&longs;tance of the Ba&longs;e C D, again&longs;t the Force placed <lb/>in G, (and the Forces in G and H are of equal Leavers D G, and <lb/>F H) as the Semidiameter F E is greater than the Semidiameter <lb/>D C, the Re&longs;i&longs;tance again&longs;t Fraction, therefore, in the Cylinder <lb/>B, encrea&longs;eth above the Re&longs;i&longs;tance of the Cylinder A, according <lb/>to both the proportions of the Circles E F and D C, and of their <lb/>Semidiameters, or, if you will, Diameters: <emph type="italics"/>B<emph.end type="italics"/>ut the proportion of <lb/>the Circles is double of that of the Diameters; Therefore the pro­<lb/>portion of the Re&longs;i&longs;tances, which is compounded of them, is in <lb/>triplicate proportion of the &longs;aid Diameters: Which is that which <lb/>I was to prove. <emph type="italics"/>B<emph.end type="italics"/>ut becau&longs;e al&longs;o the Cubes are in triplicate pro­<lb/>portion to their Sides, we may likewi&longs;e conclude, <emph type="italics"/>That the Re&longs;i­<lb/>&longs;tances of Cylinders of equal Length, are to one another as the Cubes <lb/>of their Diameters.<emph.end type="italics"/></s></p><p type="main"> <s>From that which we have Demon&longs;trated we may likewi&longs;e con­<lb/>clude, that</s></p><p type="head"> <s>COROLARY.</s></p><p type="main"> <s><emph type="italics"/>The Re&longs;i&longs;tances of Pri&longs;ms, and Cylinders of equal length are in <lb/>Se&longs;quialter proportion to that of the &longs;aid Cylinders.<emph.end type="italics"/></s></p><p type="main"> <s>The which is manife&longs;t, becau&longs;e the Pri&longs;ms and Cylinders, <lb/>equal in height, are to one another, in the &longs;ame proportion as <lb/>their <emph type="italics"/>B<emph.end type="italics"/>a&longs;es; that is, the double of the Sides or Diameters of the <lb/>&longs;aid <emph type="italics"/>B<emph.end type="italics"/>a&longs;es: <emph type="italics"/>B<emph.end type="italics"/>ut the Re&longs;i&longs;tances (as hath been demon&longs;trated) are <lb/>in triplicate proportion to the &longs;aid Sides or Diameters: Therefore <lb/>the proportion of the Re&longs;i&longs;tances is Se&longs;quialter to the proportion <lb/>of the &longs;aid Solids, and, con&longs;equently, to the Weights of the &longs;aid <lb/>Solids.</s></p><p type="main"> <s>SIMP. </s> <s>It is convenient, that, before we proceed any farther, I <lb/>be re&longs;olved of a certain Doubt, and this it is, That I have not hi­<lb/>therto heard propo&longs;ed to Con&longs;ideration another certain kind of <lb/>Re&longs;i&longs;tance, that, in my opinion, is &longs;ucce&longs;&longs;ively dimini&longs;hed in So­<lb/>lids, according as they are more and more prolonged, and not on­<lb/>ly in u&longs;ing them &longs;idelongs, but al&longs;o leng thwaies, in the &longs;elf &longs;ame <pb xlink:href="069/01/101.jpg" pagenum="99"/>manner ju&longs;t as we &longs;ee a very long Cord to be much le&longs;&longs;e apt to <lb/>&longs;u&longs;tain a great weight, than if it were &longs;hort: &longs;o that I believe, that <lb/>a Ruler of Wood or Iron will &longs;u&longs;tain a much greater weight, if it <lb/>&longs;hall be &longs;hort, than if it &longs;hall be very long; under&longs;tanding it al­<lb/>waies to be u&longs;ed lengthwaies, and not tran&longs;ver&longs;ly; and al&longs;o <lb/>its own weight being accounted for, which in the longer is <lb/>greater.</s></p><p type="main"> <s>SALV. </s> <s>I fear, <emph type="italics"/>Simplicius,<emph.end type="italics"/> that in this Point you, with many <lb/>others, are deceived, if &longs;o be, that I have rightly apprehended your <lb/>meaning, &longs;o that you would &longs;ay, that a Cord <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> forty yards <lb/>long cannot &longs;u&longs;tain &longs;o much, as if u&longs;e were made but of one or two <lb/>yards of the &longs;ame Rope.</s></p><p type="main"> <s>SIMP. </s> <s>That is it, which I would have &longs;aid, and as yet it &longs;eemeth <lb/>a very probable Propo&longs;ition.</s></p><p type="main"> <s>SALV. </s> <s>But I hold it not only improbable, but fal&longs;e: and think <lb/>that I can very ea&longs;ily reclaim you from your Errour. </s> <s>Therefore <lb/>let us &longs;uppo&longs;e this Rope A B, [<emph type="italics"/>as in<emph.end type="italics"/> Fig. </s> <s>6.] fa&longs;tned on high by <lb/>the end A, and by the other end let there hang the Weight C, <lb/>by the force of which, the &longs;aid Rope is to be broken. </s> <s>Do you <lb/>a&longs;&longs;ign me the particular place, <emph type="italics"/>Simplicius,<emph.end type="italics"/> where the Rupture is <lb/>to happen.</s></p><p type="main"> <s>SIMP. </s> <s>Let it be in the place D.</s></p><p type="main"> <s>SALV. </s> <s>I ask what is the cau&longs;e why it &longs;hould break in D.</s></p><p type="main"> <s>SIMP. </s> <s>The rea&longs;on thereof is, becau&longs;e the Rope was not &longs;trong <lb/>enough in that part, to &longs;u&longs;tain <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> an hundred pounds of weight, <lb/>for &longs;o much is the Rope D B with the Stone C.</s></p><p type="main"> <s>SALV. </s> <s>Therefore when ever &longs;uch a Rope &longs;hall come to be vio­<lb/>lently &longs;tretched by tho&longs;e hundred pounds of weight, it &longs;hall break <lb/>in that place.</s></p><p type="main"> <s>SIMP So I think.</s></p><p type="main"> <s>SALV. </s> <s>But tell me now; if one did hang the &longs;ame Weight, not <lb/>at the end of the Rope <emph type="italics"/>B,<emph.end type="italics"/> but near to the point D, as for in&longs;tance, <lb/>in E, or el&longs;e did tye the Rope not at the height A, but very near, <lb/>and almo&longs;t at the Point D it &longs;elf, as in F, tell me, I &longs;ay, whether <lb/>the Point D would feel the &longs;ame weight of an hundred pounds.</s></p><p type="main"> <s>SIMP. </s> <s>It would &longs;o, &longs;till joyning the piece of Rope E <emph type="italics"/>B<emph.end type="italics"/> to the <lb/>Stone C.</s></p><p type="main"> <s>SALV. </s> <s>If then the Rope in the Point D commeth to be drawn <lb/>by the &longs;aid hundred pounds of weight, it will break by your con­<lb/>ce&longs;&longs;ion. </s> <s>And yet F E, is a &longs;mall piece of the length A <emph type="italics"/>B<emph.end type="italics"/>: why do <lb/>you &longs;ay then, that the long Rope is weaker than the &longs;hort one? <lb/><emph type="italics"/>B<emph.end type="italics"/>e content, therefore, to &longs;uffer your &longs;elf to be reclaimed from an <lb/>Errour, in which you have had many Companions, and tho&longs;e in <lb/>other things very knowing. </s> <s>And let us go on: and having demon­<lb/>&longs;trated, that Pri&longs;ms and Cylinders encrea&longs;e their Moments above <pb xlink:href="069/01/102.jpg" pagenum="100"/>their Re&longs;i&longs;tances, according to the Squares of their Lengths (alwaies <lb/>provided, that they retain the &longs;ame thickne&longs;&longs;e) and that likewi&longs;e, <lb/>the&longs;e that are equally long, but different in thickne&longs;&longs;e, encrea&longs;e <lb/>their Re&longs;i&longs;tances according to the proportion of the Cubes of the <lb/>Sides or Diameters of their Ba&longs;es, we may enquire what befal­<lb/>leth to tho&longs;e Solids, being different in length and thickne&longs;s, in which <lb/>I ob&longs;erve, that</s></p><p type="head"> <s>PROPOSITION V.</s></p><p type="main"> <s><emph type="italics"/>Pri&longs;ms and Cylinders, of different length and thickne&longs;s, <lb/>have their Re&longs;i&longs;tances again&longs;t Fraction, in a propor­<lb/>tion compounded of the proportion of the Cubes of the <lb/>Diameters of their Ba&longs;es, and of the proportion of <lb/>their lengths reciprocally taken.<emph.end type="italics"/></s></p><p type="main"> <s>Let the&longs;e two A B C, and D E F, [<emph type="italics"/>as in<emph.end type="italics"/> Fig. </s> <s>7.] be &longs;uch Cy­<lb/>linders. </s> <s>I &longs;ay, the Re&longs;i&longs;tance of the Cylinder A C &longs;hall be to <lb/>the Re&longs;i&longs;tance of the Cylinder D F, in a proportion com­<lb/>pounded of the proportion of the Cube of the Diameter A B, to <lb/>the Gube of the Diameter D E, and of the proportion of the <lb/>Length E F to the Length B C. </s> <s>Suppo&longs;e E G equal to B C, and to <lb/>the Lines A B, and D E, let C H be a third proportional, and I, <lb/>a fourth; and as E F is to B C, &longs;o let I be to S. </s> <s>And becau&longs;e the <lb/>Re&longs;i&longs;tance of the Cylinder A C is to the Re&longs;i&longs;tance of the Cylin­<lb/>der D G, as the Cube A B to the Cube D E; that is, as the Line <lb/>A B to the Line I: and the Re&longs;i&longs;tance of the Cylinder G D is to <lb/>the Re&longs;i&longs;tance of the Cylinder D F, as the Length F E is to the <lb/>Length E G; that is, as the Line I is to S: Therefore by Equali­<lb/>ty of proportion, as the Re&longs;i&longs;tance of the Cylinder A C is to the <lb/>Re&longs;i&longs;tance of the Cylinder D F, &longs;o is the Line A B to S: But the <lb/>Line A B is to S, in a proportion compounded of A B to I, and of <lb/>I to S: Therefore the Re&longs;i&longs;tance of the Cylinder A C is to the Re­<lb/>&longs;i&longs;tance of the Cylinder D F, in a proportion compounded of A B <lb/>to I, that is, as the Cube of A B to the Cube of D E, and of the <lb/>proportion of the Line I to S; that is, of the Length E F to the <lb/>Length B C: Which was to be demon&longs;trated.</s></p><p type="main"> <s>After the Propo&longs;ition la&longs;t demon&longs;trated, we will con&longs;ider what <lb/>hapneth between like Cylinders and Pri&longs;ms, of which we will de­<lb/>mon&longs;trate, how that</s></p><pb xlink:href="069/01/103.jpg" pagenum="101"/><p type="head"> <s>PROPOSITION VI.</s></p><p type="main"> <s><emph type="italics"/>Of like Cylinders and Pri&longs;ms the Moments compoun­<lb/>ded, that is to &longs;ay, re&longs;ulting from their Gravities, <lb/>and from their Lengths, which are, as it were, Lea­<lb/>vers, have to one another a proportion Se&longs;quialter to <lb/>that which is between the Re&longs;i&longs;tances of their &longs;ame <lb/>Ba&longs;es.<emph.end type="italics"/></s></p><p type="main"> <s>For demon&longs;tration of which let us de&longs;cribe the two like Cy­<lb/>linders A B, and C D, [<emph type="italics"/>as in<emph.end type="italics"/> Fig. </s> <s>8.] I &longs;ay, that the Mo­<lb/>ment of the Cylinder A B, to overcome the Re&longs;i&longs;tance of its <lb/>Ba&longs;e B, hath to the Moment of C D, to overcome the Re&longs;i&longs;tance <lb/>of its Ba&longs;e C, a proportion Se&longs;quialter to that which the &longs;ame Re­<lb/>&longs;i&longs;tance of the Ba&longs;e B, hath to the Re&longs;i&longs;tance of the Ba&longs;e D: <lb/>And becau&longs;e the Moments of the Solids A B, and C D, to over­<lb/>come the Re&longs;i&longs;tances of their Ba&longs;es B and D, are compounded of <lb/>their Gravities, and of the Forces of their Leavers, and the Force <lb/>of the Leaver A B is equal to the Force of the Leaver C D, and <lb/>that becau&longs;e the length A B hath the &longs;ame proportion to the Semi­<lb/>diameter of the Ba&longs;e B, (by the &longs;imilitude of the Cylinders) that <lb/>the Length C D hath to the Semidiameter of the Ba&longs;e D; it re­<lb/>maineth, that the total Moment of the Cylinder A B, be to the <lb/>total Moment of C D, as the &longs;ole Gravity of the Cylinder A B is <lb/>to the &longs;ole Gravity of the Cylinder C D; that is, as the &longs;aid Cy­<lb/>linder A B is to the &longs;aid C D: But the&longs;e are in triplicate propor­<lb/>tion to the Diameters of their Ba&longs;es <emph type="italics"/>B<emph.end type="italics"/> and D; and the Re&longs;i&longs;tances <lb/>of the &longs;ame <emph type="italics"/>B<emph.end type="italics"/>a&longs;es, being to one another as the &longs;aid <emph type="italics"/>B<emph.end type="italics"/>a&longs;es, they are <lb/>con&longs;equently in duplicate proportion to their &longs;ame <emph type="italics"/>B<emph.end type="italics"/>a&longs;es: There­<lb/>fore the Moments of Cylinders are in Se&longs;quialter proportion to <lb/>the Re&longs;i&longs;tances of their <emph type="italics"/>B<emph.end type="italics"/>a&longs;es.</s></p><p type="main"> <s>SIMP. </s> <s>This Propo&longs;ition, indeed, is not only new, but unexpe­<lb/>cted to me, and at fir&longs;t &longs;ight, very remote from the judgment that <lb/>I &longs;hould have conjecturally pa&longs;t upon it: for in regard, that the&longs;e <lb/>Figures are in all other re&longs;pects alike, I &longs;hould have thought that <lb/>their Moments likewi&longs;e &longs;hould have retained the &longs;ame proportion <lb/>towards their proper Re&longs;i&longs;tances.</s></p><p type="main"> <s>SAGR. </s> <s>This is the Demon&longs;tration of that Propo&longs;ition, that in <lb/>the beginning of our Di&longs;cour&longs;es, I &longs;aid, I thought------I had &longs;ome <lb/>glimps of.</s></p><p type="main"> <s>SALV. </s> <s>That which now befalleth, <emph type="italics"/>Simplicius,<emph.end type="italics"/> hapned for &longs;ome <pb xlink:href="069/01/104.jpg" pagenum="102"/>time to my &longs;elf, believing, that the Re&longs;i&longs;tances of like Solids were <lb/>alike, till that a certain, and that no very fixed or accurate Ob&longs;er­<lb/>vation &longs;eemed to repre&longs;ent unto me, that Solids do not contain <lb/>an equal tenure in their Toughne&longs;s, but that the bigger are le&longs;&longs;e <lb/>apt to &longs;uffer violent accidents, as lu&longs;ty men are more damnified by <lb/>their falls than little children; and, as in the begining we &longs;aid, we <lb/>&longs;ee a great <emph type="italics"/>B<emph.end type="italics"/>eam or Column break to pieces falling from the &longs;ame <lb/>height, and not a &longs;mall Pri&longs;in or little Cylinder of Marble. </s> <s>This <lb/>&longs;ame Ob&longs;ervation gave me the hint for finding of that which I am <lb/>now about to demon&longs;trate; a Quality truly admirable, for that <lb/>among&longs;t the infinite Solid-like Figures, there are not &longs;o much <lb/>as two, who&longs;e Moments retain the &longs;ame proportion towards their <lb/>proper Re&longs;i&longs;tances.</s></p><p type="main"> <s>SIMP. </s> <s>Now you put me in mind of &longs;omething in&longs;erted by <emph type="italics"/>Ari­<lb/>&longs;totle<emph.end type="italics"/> among&longs;t his Mechanical Que&longs;tions, where he would give a <lb/>Rea&longs;on, whence it is, that <emph type="italics"/>B<emph.end type="italics"/>eams the longer they are, they are by &longs;o <lb/>much the more weak, and bend more and more, although the &longs;hort <lb/>ones be the &longs;lendere&longs;t, and the long ones thicke&longs;t: and, if I well re­<lb/>member, he reduceth the Rea&longs;on to the &longs;imple Leaver.</s></p><p type="main"> <s>SALV. </s> <s>It is very true, and becau&longs;e the Solution &longs;eemeth not <lb/>wholly to remove the cau&longs;e of doubting <emph type="italics"/>Mon&longs;ignore di Guevara,<emph.end type="italics"/><lb/>who, the truth is, with his mo&longs;t learned <emph type="italics"/>Commentaries<emph.end type="italics"/> hath highly <lb/>enobled and illu&longs;trated that Work, enlargeth him&longs;elf with other <lb/>accute Speculations for the obviating all difficulties, yet him&longs;elf <lb/>al&longs;o remaining perplexed in this point, whether, the lengths and <lb/>thickne&longs;&longs;es of &longs;uch Solid Figures, encrea&longs;ing with the &longs;elf &longs;ame <lb/>proportion, they ought to retain the &longs;ame tenure in their Tough­<lb/>ne&longs;&longs;es and Re&longs;i&longs;tances again&longs;t their breaking, and likewi&longs;e again&longs;t <lb/>their bending. </s> <s>After I had long con&longs;idered thereon, I have, in <lb/>this manner found, that which I am about to tell you. </s> <s>And fir&longs;t <lb/>I will demon&longs;trate that</s></p><pb xlink:href="069/01/105.jpg" pagenum="103"/><p type="head"> <s>PROPOSITION VII.</s></p><p type="main"> <s><emph type="italics"/>Of like and heavy Pri&longs;ms or Cylinders there is one only, <lb/>and no more, that is reduced (being charged with its <lb/>own weight) to the ultimate &longs;tate between breaking <lb/>and holding it &longs;elf together: &longs;othat every greater, as <lb/>being unable to re&longs;i&longs;t its own weight, will break, <lb/>and every le&longs;&longs;er re&longs;i&longs;teth &longs;ome Force that is employed <lb/>again&longs;t it to break, it.<emph.end type="italics"/></s></p><p type="main"> <s>Let the heavy Pri&longs;m be A B [<emph type="italics"/>as in<emph.end type="italics"/> Fig 9.] reduced to the <lb/>utmo&longs;t length of its Con&longs;i&longs;tance, &longs;o that being lengthned <lb/>never &longs;o little more it will break: I &longs;ay, that this is the only <lb/>one among&longs;t all tho&longs;e that are like unto it, (which yet are infinite) <lb/>that is capable of being reduced to that dubious and tickli&longs;h &longs;tate; <lb/>&longs;o that every greater being oppre&longs;&longs;ed with its own weight will <lb/>break, and every le&longs;&longs;er not, nay, will be able to re&longs;i&longs;t &longs;ome additi­<lb/>on of a new violence, over and above that of its own weight. <lb/></s> <s>Fir&longs;t, take the Pri&longs;m C E, like to, and greater than A B. </s> <s>I &longs;ay, that <lb/>this cannot con&longs;i&longs;t, but will break, being overcome by its own <lb/>Gravity. </s> <s>Suppo&longs;e the part C D as long as A B. </s> <s>And becau&longs;e the <lb/>Re&longs;i&longs;tance C D is to that of A B, as the Cube of the thickne&longs;&longs;e of <lb/>C D to the Cube of the thickne&longs;s of A B; that is, as the Pri&longs;m <lb/>C E to the Pri&longs;m A B (being alike:) Therefore the Weight of <lb/>C E is the greate&longs;t that can be &longs;u&longs;tained at the length of the Pri&longs;m <lb/>C D: But the Length C E is greater: Therefore the Pri&longs;m C E <lb/>will break. </s> <s>But let F G be le&longs;&longs;et: it &longs;hall be demon&longs;trated like­<lb/>wi&longs;e (&longs;uppo&longs;ing F H equal to B A) that the Re&longs;i&longs;tance of F G is <lb/>to that of A B, as the Pri&longs;m F G is to the Pri&longs;m A B, in ca&longs;e that the <lb/>Di&longs;tance A B, that is F H, were equal to F G, but it is greater: <lb/>Therefore the Moment of the Pri&longs;m F G, placed in G, doth not <lb/>&longs;uffice to break the Pri&longs;m F G.</s></p><p type="main"> <s>SAGR. </s> <s>A mo&longs;t manife&longs;t and brief Demon&longs;tration, inferring the <lb/>truth and nece&longs;&longs;ity of a Propo&longs;ition that at fir&longs;t &longs;ight &longs;eemeth far <lb/>from probability. </s> <s>It would be requi&longs;ite, therefore, to alter much <lb/>the proportion betwixt the Length and Thickne&longs;&longs;e of the greater <lb/>Pri&longs;m by making it thicker or &longs;horter, to the end it might be re­<lb/>duced to that nice &longs;tate of indifferency between holding and brea­<lb/>king; and the Inve&longs;tigation of that &longs;ame State, as I think, would <lb/>be no le&longs;&longs;e ingenuous.</s></p><p type="main"> <s>SALV. Nay, rather more, as it is al&longs;o more laborious: and I am <pb xlink:href="069/01/106.jpg" pagenum="104"/>&longs;ure I have &longs;pent no &longs;mall time to find it; and I will now impart it <lb/>to you: Therefore</s></p><p type="head"> <s>PROP. VIII. PROBL. I.</s></p><p type="main"> <s><emph type="italics"/>A Cylinder or Pri&longs;m of the utmo&longs;t length not to be bro­<lb/>ken by its own weight, and al&longs;o a greaver length, be­<lb/>ing given, to find the thickne&longs;&longs;e of another Cylinder <lb/>or Pri&longs;m that under-given length is the only one, and <lb/>bigge&longs;t, that can re&longs;i&longs;t its own weight.<emph.end type="italics"/></s></p><p type="main"> <s>Let the Cylinder B C [<emph type="italics"/>as in<emph.end type="italics"/> Fig. </s> <s>10.] be the bigge&longs;t that <lb/>can re&longs;i&longs;t its own weight, and let D E be a Length greater <lb/>than A C; it is required to find the Thickne&longs;&longs;e of the Cylin­<lb/>der, that under the Length D E is the greate&longs;t re&longs;i&longs;ting its own <lb/>weight. </s> <s>Let I be a third proportional to the Lengths D E, and <lb/>A C; and as D E is to I, &longs;o let the Diameter F D be to the Dia­<lb/>meter B A: and make the Cylinder F E. </s> <s>I &longs;ay, that this is the big­<lb/>ge&longs;t, and only one among&longs;t all that are like to it that re&longs;i&longs;teth its <lb/>own weight. </s> <s>To the Lines D C and I let M be a third propor­<lb/>tional, and O a fourth. </s> <s>And &longs;uppo&longs;e F G equal to A C. </s> <s>And be­<lb/>cau&longs;e the Diameter F D is to the Diameter A B, as the Line D E <lb/>to I, and O is a fourth proportional to D E and I, the Cube of <lb/>F D &longs;hall be to the Cube of B A as D E is to O: But as the Cube of <lb/>F D is to the Cube of B A, &longs;o is the Re&longs;i&longs;tance of the Cylinder <lb/>D G to the Re&longs;i&longs;tance of the Cylinder B C: Therefore the Re&longs;i­<lb/>&longs;tance of the Cylinder D G is to that of the Cylinder B C, as the <lb/>Line D F is to O. </s> <s>And becau&longs;e the Moment of the Cylinder B C <lb/>is equal to its Re&longs;i&longs;tance, if we &longs;hew that the Moment of the Cylin­<lb/>der F E is to the Moment of the Cylinder B C, as the Re&longs;i&longs;tance <lb/>D F to the Re&longs;i&longs;tance B A; that is, as the Cube of F D to the Cube <lb/>of B A; that is, as the Line D E to O, we &longs;hall have our intent: <lb/>that is, that the Moment of the Cylinder F E is equal to the Re&longs;i­<lb/>&longs;tance placed in F D. </s> <s>The Moment of the Cylinder F E is to the <lb/>Moment of the Cylinder D G, as the Square of D E is to the <lb/>Square of A C; that is, as the Line D E to I: But the Moment of <lb/>the Cylinder D G is to the Moment of the Cylinder B C, as the <lb/>Square D F to the Square B A; that is, as the Square of D E to the <lb/>Square of I; that is, as the Square of I to the Square of M; that <lb/>is, as I to O: Therefore, by Equality of proportion, as the Mo­<lb/>ment of the Cylinder F E is to the Moment of the Cylinder B C, <lb/>&longs;o is the Line D E to O; that is, the Cube D F to the Cube <lb/>B A; that is, the Re&longs;i&longs;tance of the Ba&longs;e D F to the Re&longs;i&longs;tance <pb xlink:href="069/01/107.jpg" pagenum="105"/>of the Ba&longs;e B A: Which is that that was &longs;ought.</s></p><p type="main"> <s>SAGR This, <emph type="italics"/>Salviatus,<emph.end type="italics"/> is a long Demon&longs;tration, and very hard <lb/>to be born in mind at the fir&longs;t hearing, therefore I could wi&longs;h, that <lb/>you would plea&longs;e to repeat it.</s></p><p type="main"> <s>SALV. </s> <s>I will do what you &longs;hall command; but haply it would <lb/>be better to produce one more conci&longs;e and &longs;hort: but then it will <lb/>be requi&longs;ite to de&longs;cribe a Figure &longs;omewhat different.</s></p><p type="main"> <s>SAGR. </s> <s>The favour will then be the greater: and be&longs;tow upon <lb/>me the draught of that already explained, that I may at my lea&longs;ure <lb/>con&longs;ider it again.</s></p><p type="main"> <s>SALV. </s> <s>I will not fail to &longs;erve you. </s> <s>Now, &longs;uppo&longs;e a Cylinder A, <lb/><arrow.to.target n="marg1082"></arrow.to.target><lb/>[<emph type="italics"/>as in<emph.end type="italics"/> Fig. </s> <s>11.] the Diameter of who&longs;e Ba&longs;e let be the Line D C, <lb/>and let this A be the greate&longs;t that can &longs;u&longs;tain it &longs;elf and not break, <lb/>than which we will find a bigger, which likewi&longs;e &longs;hall be the big­<lb/>ge&longs;t al&longs;o, and the only one that &longs;u&longs;taineth it &longs;elf. </s> <s>Let us de&longs;ire one <lb/>like to the &longs;aid A, and as long as the a&longs;&longs;igned Line, and let this be <lb/><emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> E, the Diameter of who&longs;e Ba&longs;e let be K L; and to the two <lb/>Lines D C, and K L let M N be a third proportional; which let be <lb/>the Diameter of the Ba&longs;e of the Cylinder X, in length equal to E. <lb/></s> <s>I &longs;ay, that this X is that which we &longs;eek. </s> <s>And becau&longs;e the Re&longs;i­<lb/>&longs;tance D C is to the Re&longs;i&longs;tance K L, as the Square D C to the <lb/><emph type="italics"/>S<emph.end type="italics"/>quare K L; that is, as the Square K L to the Square M N; that <lb/>is, as the Cylinder E to the Cylinder X; that is, as the Moment E <lb/>to the Moment X: But the Re&longs;i&longs;tance K L is to M N, as the Cube <lb/>of K L is to the Cube of M N; that is, as the Cube B C to the <lb/>Cube K L; that is, as the Cylinder A to the Cylinder E; that is, <lb/>as the Moment A to the Moment E: Therefore, by Perturbation <lb/>of proportion, as the Re&longs;i&longs;tance D C is to M N, &longs;o is the Moment <lb/>A to the Moment X: Therefore the Pri&longs;m X, is in the &longs;ame Con&longs;ti­<lb/>tution of Moment and Re&longs;i&longs;tance as the Pri&longs;m A.</s></p><p type="margin"> <s><margin.target id="marg1082"></margin.target><emph type="italics"/>The la&longs;t Problem <lb/>performed another <lb/>way.<emph.end type="italics"/></s></p><p type="main"> <s>But let us make the Problem more general, and let the Propo­<lb/>&longs;ition be this:</s></p><p type="main"> <s><emph type="italics"/>The Cylinder<emph.end type="italics"/> A C <emph type="italics"/>being given, and its Moment to-<emph.end type="italics"/><lb/><arrow.to.target n="marg1083"></arrow.to.target><lb/><emph type="italics"/>wards its Re&longs;i&longs;tance being &longs;uppo&longs;ed at plea&longs;ure, and <lb/>any Length<emph.end type="italics"/> D E <emph type="italics"/>being a&longs;signed, to find the Thick­<lb/>ne&longs;&longs;e af the Cylinder who&longs;e Length is<emph.end type="italics"/> D E, <emph type="italics"/>and who&longs;e <lb/>Moment towards its Re&longs;i&longs;tance retaineth the &longs;ame <lb/>proportion, that the Moment of the Cylinder<emph.end type="italics"/> A C <lb/><emph type="italics"/>doth to its Re&longs;i&longs;tance.<emph.end type="italics"/></s></p><pb xlink:href="069/01/108.jpg" pagenum="106"/><p type="margin"> <s><margin.target id="marg1083"></margin.target><emph type="italics"/>The la&longs;t Propo&longs;i­<lb/>tion made more ge­<lb/>neral.<emph.end type="italics"/></s></p><p type="main"> <s>Rea&longs;&longs;uming the above &longs;aid Figure and almo&longs;t the &longs;ame Me­<lb/>thod, we will &longs;ay: Becau&longs;e the Moment of the Cylinder <lb/>F E hath the &longs;ame proportion to the Moment of the part <lb/>D G, that the Square E D hath to the Square F G; that is that <lb/>the Line D E hath to I: and becau&longs;e the Moment of the Cylinder <lb/>F G is to the Moment of the Cylinder A C, as the Square F D to <lb/>the Square A B; that is, as the Square D E to the Square I; that <lb/>is, as the Square I to the Square M; that is, as the Line I to O: <lb/>Therefore, <emph type="italics"/>ex æquali,<emph.end type="italics"/> the Moment of the Cylinder F E hath the <lb/>&longs;ame proportion to the Moment of the Cylinder A C, that the <lb/>Line D E hath to the Line O; that is, that the Cube D E hath <lb/>to the Cube of I; that is, that the Cube of F D hath to the <lb/>Cube of A B; that is, that the Re&longs;i&longs;tance of the Ba&longs;e F D hath to <lb/>the Re&longs;i&longs;tance of the Ba&longs;e A B: Which was to be performed.</s></p><p type="main"> <s>Now, let it be ob&longs;erved, that from the things hitherto demon&longs;tra­<lb/>ted, we may plainly gather, how Impo&longs;&longs;ible it is, not only for Art, but <lb/><arrow.to.target n="marg1084"></arrow.to.target><lb/>for Nature her &longs;elf to encrea&longs;e her Machines to an immen&longs;e Va&longs;t­<lb/>ne&longs;&longs;e: &longs;o that it would be impo&longs;&longs;ible by Art to build extraordina­<lb/>ry va&longs;t Ships, Palaces, or Temples, who&longs;e ^{*} Oars, Sail-yards, Beams, <lb/>Iron Bolts, and, in a word, their other parts &longs;hould con&longs;i&longs;t or hold <lb/>together: neither again could Nature make Trees of unmea&longs;ura­<lb/><arrow.to.target n="marg1085"></arrow.to.target><lb/>ble greatne&longs;&longs;e, for that their Arms or Bows being oppre&longs;&longs;ed with <lb/>their own weight would at la&longs;t break: and likewi&longs;e it would be <lb/>impo&longs;&longs;ible for her to make &longs;tructures of Bones for men, Hor&longs;es, or <lb/>other Animals, that might &longs;ub&longs;i&longs;t, and proportionatly perform <lb/>their Offices, when tho&longs;e Animals &longs;hould be augmented to im­<lb/>men&longs;e heights, unle&longs;&longs;e &longs;he &longs;hould take Matter much more hard and <lb/>Refi&longs;ting than that which &longs;he commonly u&longs;eth, or el&longs;e &longs;hould de­<lb/>form tho&longs;e Bones by augmenting them beyond their due Symetry, <lb/>and making the Figure or &longs;hape of the Animal to become mon­<lb/>&longs;trou&longs;ly big: Which haply was hinted by my mo&longs;t Witty Poet, <lb/>where de&longs;cribing an huge Giant, he &longs;aith,</s></p><p type="margin"> <s><margin.target id="marg1084"></margin.target>* Oares are u&longs;ed <lb/>in the Ships or <lb/>Gallies of the <lb/>Mediterrane, up­<lb/>on which our <lb/>Author was a <lb/>Coa&longs;ter.</s></p><p type="margin"> <s><margin.target id="marg1085"></margin.target><emph type="italics"/>Bones of Animals <lb/>magnified beyond <lb/>their ratural &longs;ize, <lb/>would not &longs;ub&longs;i&longs;t, if <lb/>it be required to <lb/>retain the &longs;ame <lb/>proportion of thick­<lb/>ne&longs;s and hardne&longs;s <lb/>in them that is in <lb/>tho&longs;e of Natural <lb/>Animals.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Non &longs;i puo compartir quanto &longs;ia lungo,<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Si &longs;mi&longs;uratamente è tutto gro&longs;&longs;o.<emph.end type="italics"/><lb/><arrow.to.target n="marg1086"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1086"></margin.target><emph type="italics"/>Example of the <lb/>Bone of an Animal <lb/>enlarged to thrice <lb/>the Natural pro­<lb/>portion, how much <lb/>thicker it ought to <lb/>be to perform its <lb/>office.<emph.end type="italics"/></s></p><p type="main"> <s>And for a &longs;hort example of this that I &longs;ay, [<emph type="italics"/>as in<emph.end type="italics"/> Fig. </s> <s>12.] I <lb/>have heretofore drawn the Figure of a Bone only trebled in <lb/>Length, and augmented in Thickne&longs;&longs;e in &longs;uch proportion, as that <lb/>it may in its great Animal perform the office proportionate to that <lb/>of the le&longs;&longs;er Bone in a &longs;maller Animal, and the Figures are the&longs;e: <lb/>whereby you &longs;ee what a di&longs;proportionate Figure that of the aug­<lb/>mented Bone becometh. </s> <s>Whence it is manife&longs;t, that he that would <lb/>in an huge Giant keep the proportions that the Members have in <pb xlink:href="069/01/109.jpg" pagenum="107"/>an ordinary Man, mu&longs;t either find Matter much more hard and re­<lb/>&longs;i&longs;ting to make Bone of, or el&longs;e mu&longs;t admit that its Strength is in <lb/>proportion much more weak than in Men of middle Stature: other­<lb/>wi&longs;e, encrea&longs;ing the Giant to an immea&longs;urable height he would be <lb/>oppre&longs;&longs;ed, and fall under his own weight. </s> <s>Whereas on the con­<lb/>trary, in dimini&longs;hing of Bodies we do not &longs;ee the Strength and <lb/>Forces to dimini&longs;h in the &longs;ame proportion, nay, in the le&longs;&longs;er the <lb/>Robu&longs;tiou&longs;ne&longs;&longs;e encrea&longs;eth with a great proportion. </s> <s>So that I <lb/>believe, that a little Dog could carry on his back two or three Dogs <lb/>equal to him&longs;elf, but I do not think that an Hor&longs;e could carry &longs;o <lb/>much as one &longs;ingle Hor&longs;e of his own &longs;ize.</s></p><p type="main"> <s>SIMP. </s> <s>But if it be &longs;o, I have great rea&longs;on to doubt the Im­<lb/>men&longs;e bulks that we &longs;ee in Fi&longs;hes, for (if I rightly under&longs;tand <lb/>you) a Whale &longs;hall be as big as ten Elephants, and yet they &longs;u­<lb/>&longs;tain them&longs;elves.</s></p><p type="main"> <s>SALV. </s> <s>Your doubt, <emph type="italics"/>Simplicius,<emph.end type="italics"/> prompts me with another Con­<lb/>dition which I perceived not before, which is al&longs;o able to make <lb/>Giants and other very big Animals to con&longs;i&longs;t, and act them&longs;elves <lb/>no le&longs;&longs;e than &longs;maller, and this will happen when not only Strength <lb/>is added to the Bones and other Parts, who&longs;e office it is to &longs;u&longs;tain <lb/>their own and the &longs;upervenient weight; but the &longs;tructure of the <lb/>Bones being left with the &longs;ame proportions, the &longs;ame Fabricks <lb/>would ju&longs;t in the &longs;ame manner, yea, with much more ea&longs;e, con­<lb/>&longs;i&longs;t, when the Gravity of the matter of tho&longs;e Bones, or that of <lb/>the Fle&longs;h, or whatever el&longs;e is to re&longs;t it &longs;elf upon the Bones is dimini­<lb/>&longs;hed in that proportion: and of this &longs;econd Artifice, Nature hath <lb/>made u&longs;e in the framing of Fi&longs;hes, making their Bones, and Pulps, <lb/>not only very light, but without any Gravity.</s></p><p type="main"> <s>SIMP. </s> <s>I &longs;ee very well, <emph type="italics"/>Salviatus,<emph.end type="italics"/> whither your Di&longs;cour&longs;e ten­<lb/>deth: you will &longs;ay, that becau&longs;e the Element of Water is the Ha­<lb/>bitation of Fi&longs;hes, which by its Corpulence, or, as others will, by <lb/>its Gravity dimini&longs;heth the weight of Bodies demerged in it, for <lb/>that rea&longs;on the Matter of Fi&longs;hes, not weighing any thing, may be <lb/>&longs;u&longs;tained without &longs;urcharging their Bones: but this doth not &longs;uf­<lb/>fice, for although the re&longs;t of the &longs;ub&longs;tance of the Fi&longs;h weigh not, <lb/>yet without doubt the matter of their Bones hath its weight: <lb/>and who will &longs;ay, that the Rib of a Whale that is as big as a <lb/>Beam doth not weigh very much, and in Water &longs;inketh to the Bot­<lb/>tom? </s> <s>The&longs;e therefore &longs;hould not be able to &longs;ub&longs;i&longs;t in &longs;o va&longs;t a <lb/>Bulk.</s></p><p type="main"> <s>SALV. </s> <s>You argue very cunningly; and for an an&longs;wer to your <lb/>Doubt, tell me, whether you have ob&longs;erved Fi&longs;hes to &longs;tand im­<lb/>moveable under water at their plea&longs;ures, and not to de&longs;cend to­<lb/>wards the Bottom, or rai&longs;e them&longs;elves towards the top without <lb/>making &longs;ome motion with their Fins?</s></p><pb xlink:href="069/01/110.jpg" pagenum="108"/><p type="main"> <s>SIMP. </s> <s>This is a very manife&longs;t Ob&longs;ervation.</s></p><p type="main"> <s><arrow.to.target n="marg1087"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1087"></margin.target><emph type="italics"/>The Cau&longs;e why <lb/>Fi&longs;hes do equili­<lb/>brate them&longs;elves <lb/>in the Water.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>This power therefore that the Fi&longs;hes have to &longs;tay them­<lb/>&longs;elves, as if they were immoveable in the mid&longs;t of the Water, is a <lb/>mo&longs;t infallible argument, that the Compofition of their Corporeal <lb/>Ma&longs;&longs;e equalleth the Specifick Gravity of the Water, &longs;o that if <lb/>there be found in them &longs;ome parts that are more grave than the <lb/>Water, it is nece&longs;&longs;arily requi&longs;ite that they have others &longs;o much <lb/>le&longs;&longs;e grave, &longs;o that the <emph type="italics"/>Equilibrium<emph.end type="italics"/> may be ballanced. </s> <s>If therefore <lb/>the Bones be more grave, it is nece&longs;&longs;ary that the Pulps, or other <lb/>Matters that are in them, be more light; and the&longs;e will with their <lb/>lightne&longs;&longs;e counterpoi&longs;e and compen&longs;ate the weight of the Bones. <lb/></s> <s>So that in Aquatick Animals the quite contrary hapneth to that <lb/>which befals the Terre&longs;trial, namely, that in the latter it is the of­<lb/>fice of the Bones to &longs;u&longs;tain their own weight, and the weight of <lb/>the Fle&longs;h; and in the former, the <emph type="italics"/>Fle&longs;h [if one may &longs;o call it]<emph.end type="italics"/></s></p><p type="main"> <s><arrow.to.target n="marg1088"></arrow.to.target><lb/>beareth up its own weight, and that of the Bones. </s> <s>And therefore <lb/>cea&longs;e to wonder how there may be mo&longs;t va&longs;t Animals in the Wa­<lb/>ter, but not on the Earth, that is, in the Air.</s></p><p type="margin"> <s><margin.target id="marg1088"></margin.target><emph type="italics"/>Aquatick Animals <lb/>greater than the <lb/>Terre&longs;trial, and for <lb/>what Rea&longs;on.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. </s> <s>I am &longs;atisfied, and moreover ob&longs;erve, that the&longs;e which <lb/>we call Terre&longs;trial Animals, ought with more rea&longs;on to be called <lb/>Aerial; becau&longs;e in the Air they really live, and by the Air they are <lb/>environ'd, and of the Air they breath.</s></p><p type="main"> <s>SAGR. </s> <s>The Di&longs;cour&longs;e of <emph type="italics"/>Simplicius<emph.end type="italics"/> plea&longs;eth me, as al&longs;o his <lb/>Doubt and its Solution. </s> <s>And farthermore I comprehend very ea­<lb/>&longs;ily, that one of the&longs;e huge Fi&longs;hes being haul'd on &longs;hore, could not <lb/>perchance be able to &longs;u&longs;tain it &longs;elf for any time; but that the Con­<lb/>nections of the Bones being relaxed, its Ma&longs;&longs;e would be cru&longs;h'd un­<lb/>der its own weight.</s></p><p type="main"> <s>SALV. </s> <s>For the pre&longs;ent, I encline to the &longs;ame Opinion: nor am <lb/>I far from thinking that the &longs;ame would happen to that huge Ship, <lb/>which floating in the Sea is not di&longs;&longs;olved by its weight, and the bur­<lb/>den of its Lading and Artilery, but on dry ground, and environed <lb/>with Air, it perhaps would fall in pieces. </s> <s>But let us pur&longs;ue our bu­<lb/>&longs;ine&longs;&longs;e, and demon&longs;trate, that</s></p><pb xlink:href="069/01/111.jpg" pagenum="109"/><p type="head"> <s>PROP. IX. PROBL. II.</s></p><p type="main"> <s><emph type="italics"/>A Pri&longs;me or Cylinder with its weight, and the great­<lb/>e&longs;t Weight &longs;u&longs;tained by it being given, to find the <lb/>greate&longs;t Length, beyond which being prolonged. </s> <s>it <lb/>would break under its own Weight.<emph.end type="italics"/></s></p><p type="main"> <s>Let there be given the Pri&longs;me A C (<emph type="italics"/>as in<emph.end type="italics"/> Fig. </s> <s>13.) with its <lb/>weight, and likewi&longs;e let the Weight D be given, the great­<lb/>e&longs;t that can be &longs;u&longs;tained by the extreme C: it is required to <lb/>finde the greate&longs;t Length unto which the &longs;aid Pri&longs;me may be pro­<lb/>longed, without breaking. </s> <s>As the weight of the Pri&longs;me A C is to <lb/>the Compound of the weights A C, with the double of the <lb/>Weight D, &longs;o let the length C A be to C A H: between which <lb/>let A G be a Mean-Proportional. </s> <s>I &longs;ay that A G is the Length <lb/>&longs;ought. </s> <s>For the depre&longs;&longs;ing Moment of the Weight D in C, is <lb/>equal to the Moment of the double weight D, if it be placed in <lb/>the middle of A C, where is al&longs;o the Center of the Moment of <lb/>the Pri&longs;me A C: The Moment, therefore, of the Re&longs;i&longs;tance of <lb/>the Pri&longs;me A C, which re&longs;ides in A, is equivalent to the gravi­<lb/>tation of the double of the Weight D with the weight A C, but <lb/>hanged in the mid&longs;t of A C. </s> <s>And becau&longs;e it hath been made, <lb/>that as the Moment of the &longs;aid Weights &longs;o &longs;ituated, that is, of <lb/>the double of D, with A C, is to the Moment of A C, &longs;o is H A <lb/>to A C, between which A G is a Mean Proportional: There­<lb/>fore the Moment of D doubled with the Moment of A C, is to <lb/>the Moment A C, as the Square G A to the Square A C: But the <lb/>pre&longs;&longs;ing Moment of the Pri&longs;me G A, is to the Moment of A C, <lb/>as the Square G A to the Square A C: Therefore the Length <lb/>A G is the greate&longs;t that was &longs;ought, namely, that unto which the <lb/>Pri&longs;me A G being prolonged, it would &longs;u&longs;tain it &longs;elf, but beyond <lb/>it would break.</s></p><p type="main"> <s>Hitherto we have con&longs;idered the Moments and Re&longs;i&longs;tances of <lb/>&longs;olid Pri&longs;mes and Cylinders, one end of which is &longs;uppo&longs;ed im­<lb/>moveable, and to the other onely the Force of a pre&longs;&longs;ing weight <lb/>is applyed, con&longs;idering it by it &longs;elf alone, or joyned with the <lb/>Gravity of the &longs;ame Solid, or el&longs;e the &longs;ole Gravity of the &longs;aid <lb/>Solid. </s> <s>Now I de&longs;ire that we may &longs;peak &longs;omething of tho&longs;e &longs;ame <lb/>Pri&longs;mes or Cylinders, in ca&longs;e they were &longs;u&longs;tained at both ends, or <lb/>did re&longs;t upon one &longs;ole point taken between the ends. </s> <s>And fir&longs;t, <lb/>I &longs;ay that,</s></p><pb xlink:href="069/01/112.jpg" pagenum="110"/><p type="head"> <s>PROPOSITION X.</s></p><p type="main"> <s><emph type="italics"/>The Cylinder that being charged with its own Weight <lb/>&longs;hall be reduced to its greate&longs;t Length, beyond which <lb/>it would not &longs;u&longs;tain it &longs;elf, whether it be born up in <lb/>the middle by one &longs;ole Fulciment, or el&longs;e by two at <lb/>the ends, may be double in length to that which <lb/>&longs;hould be fa&longs;tned in the Wall, that is &longs;u&longs;tained at but <lb/>one end.<emph.end type="italics"/></s></p><p type="main"> <s>Which of it &longs;elt is very obvious; for if we &longs;hall &longs;up­<lb/>po&longs;e of the Cylinder which I de&longs;cribe A B C, its <lb/>half A B to be the utmo&longs;t Length that is able to be <lb/>&longs;u&longs;tained, being fa&longs;tened at the end B, it &longs;hall be &longs;u&longs;tained in the <lb/>&longs;ame manner, if being laid upon the Fulciment G, it &longs;hall be <lb/>counterpoi&longs;ed by its other half B C. </s> <s>And likewi&longs;e, if of the Cy­<lb/>linder D E F, the Length &longs;hall be &longs;uch that onely one half of it <lb/>can be &longs;u&longs;tained, being fa&longs;tened at the end D, and con&longs;equent­<lb/>ly the other E F, fixed at the end F; it is manife&longs;t, that placing <lb/>the Fulciments H and I under the ends D and F, every Moment <lb/>of Force or of Weight that is added in E, will there make the <lb/>Fracture.</s></p><p type="main"> <s>That which requireth a more &longs;ubtil Speculation is, when &longs;ub­<lb/>&longs;tracting from the proper Gravity of &longs;uch Solids, it were pro­<lb/>pounded to us</s></p><p type="head"> <s>PROP. XI. PROBL. III.</s></p><p type="main"> <s><emph type="italics"/>To find whether that Force or weight, that being ap­<lb/>plied to the middle of a Cylinder &longs;u&longs;tained at the <lb/>ends, would &longs;uffice to break it, could do the &longs;ame, <lb/>applied in any other place, neerer to one end than to <lb/>the other.<emph.end type="italics"/></s></p><p type="main"> <s>As for Example, whether we de&longs;iring to break a Staffe <lb/>and took it with the ends in our hands, and &longs;etting our <lb/>knee, to the mid&longs;t of it, the &longs;ame Force that &longs;hould &longs;uf­<lb/>fice to break it in that manner, would al&longs;o &longs;uffice in ca&longs;e the knee <pb xlink:href="069/01/113.jpg" pagenum="111"/>were &longs;et, not in the mid&longs;t, but neerer to one of the ends.</s></p><p type="main"> <s>SAGR. </s> <s>I think the Problem is toucht upon by <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/> in his <lb/><emph type="italics"/>Mechanical Que&longs;tions.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>The Que&longs;tion of <emph type="italics"/>Aristotle<emph.end type="italics"/> is not preci&longs;ely the &longs;ame, <lb/>for he &longs;eeks no more, but to render a rea&longs;on why le&longs;&longs;e labour is <lb/>required to break the Staffe, holding the hands at the ends of it, <lb/>that is, far di&longs;tant from the Knee, than if we held them neerer: <lb/>and he giveth a general Rea&longs;on of the &longs;ame, reducing the cau&longs;e <lb/>of it to the Leavers, which are longer when the Arms are ex­<lb/>tended, gra&longs;ping the ends. </s> <s>Our Que&longs;tion addeth &longs;omething <lb/>more, &longs;eeking whether, &longs;etting the Knee in the mid&longs;t, or in ano­<lb/>ther place, but alwayes keeping the hands at the ends, the &longs;ame <lb/>Force &longs;erveth in all &longs;ituations.</s></p><p type="main"> <s>SAGR. </s> <s>At fir&longs;t apprehen&longs;ion it &longs;hould &longs;eem that it doth, for <lb/>that the two Leavers retain in a certain fa&longs;hion the &longs;ame Moment, <lb/>&longs;eeing that as the one is &longs;hortned, the other is lengthened.</s></p><p type="main"> <s>SALV. </s> <s>Now you &longs;ee, how ea&longs;ie it is to make Equivocations, <lb/>and with what caution and circum&longs;pection we are to walk, lea&longs;t <lb/>we run into them. </s> <s>This that you &longs;ay, and which indeed at the <lb/>fir&longs;t &longs;ight carrieth with it &longs;o much of probability, is in the &longs;trict­<lb/>ne&longs;&longs;e of it &longs;o fal&longs;e, that whether the Knee, which is the Fulci­<lb/>ment of the two Leavers, be placed or not placed in the mid&longs;t, <lb/>it maketh &longs;uch alteration, that of that Force which would &longs;uffice <lb/>to make the Fracture in the mid&longs;t, it being to be made in &longs;ome <lb/>other place, it will not &longs;uffice to apply four times &longs;o much, nor <lb/>ten, nor an hundred, no nor a thou&longs;and. </s> <s>Upon this we will <lb/>make &longs;ome general Con&longs;ideration, and then we will come to the <lb/>Specifick Determination of the Propo&longs;ition, according to which, <lb/>the Forces for making of Fractures gradually vary more in one <lb/>point than in another.</s></p><p type="main"> <s>Let us fir&longs;t de&longs;igne this Truncheon A B to be broken in the <lb/>mid&longs;t upon the Fulciment C, and neer unto that let us de&longs;igne <lb/>it again, but under the Characters D E, to be broken on the <lb/>Fulciment F, remote from the middle. </s> <s>Fir&longs;t it is manife&longs;t, that <lb/>the Di&longs;tances A C and C B being equal, the Force &longs;hall be &longs;ha­<lb/>red equally in the ends B and A. Again, according as the Di­<lb/>&longs;tance D F groweth le&longs;&longs;e than the Di&longs;tance A C, the Moment <lb/>of the Force placed in D groweth le&longs;&longs;e than the Moment in A, <lb/>that is placed at the Di&longs;tance C A, and le&longs;&longs;eneth according to <lb/>the proportion of the Line D F to A C; and con&longs;equently, it is <lb/>requi&longs;ite to encrea&longs;e it to equalize or exceed the Re&longs;i&longs;tance of F: <lb/>But the Di&longs;tance D F may dimini&longs;h <emph type="italics"/>in infinitum,<emph.end type="italics"/> in relation to <lb/>the Di&longs;tance A C: Therefore it is requi&longs;ite, that it be po&longs;&longs;ible for <lb/>the Force to be applyed in D, to encrea&longs;e <emph type="italics"/>in infinitum,<emph.end type="italics"/> that it <lb/>may countervail the Re&longs;i&longs;tance in F. But, on the contrary, ac­<pb xlink:href="069/01/114.jpg" pagenum="112"/>cording as the Di&longs;tance F E encrea&longs;eth above C B, it is requi&longs;ite <lb/>to dimini&longs;h the Force in E, that it may compen&longs;ate the Re&longs;i­<lb/>&longs;tance in F: But the Di&longs;tance F E in relation to C B, cannot en­<lb/>crea&longs;e <emph type="italics"/>in infinitum,<emph.end type="italics"/> by drawing the Fulciment F towards the end <lb/>D, no nor yet to the double: Therefore, the Force in E, that it <lb/>may compen&longs;ate the Re&longs;i&longs;tance in F, &longs;hall be alwayes more than <lb/>half of the Force in B. </s> <s>We may comprehend, therefore, the ne­<lb/>ce&longs;&longs;ity of augmenting the Moments of the Collected Forces in E <lb/>and D infinitely to equalize or exceed the Re&longs;i&longs;tance placed in F, <lb/>according as the Fulciment F &longs;hall approach neerer and neerer <lb/>to the end D.</s></p><p type="main"> <s><emph type="italics"/>S<emph.end type="italics"/>AGR. </s> <s>What will <emph type="italics"/>Simplicius<emph.end type="italics"/> &longs;ay to this? </s> <s>Mu&longs;t he not con­<lb/>fe&longs;&longs;e the Virtue of Geometry to be a more powerful in&longs;trument <lb/>than all others, to &longs;harpen the Wit, and di&longs;po&longs;e it to di&longs;cour&longs;e <lb/>and &longs;peculate well? </s> <s>and that <emph type="italics"/>Plato<emph.end type="italics"/> had great rea&longs;on to de&longs;ire that <lb/>his Scholars &longs;hould be well grounded in the Mathematicks? </s> <s>I <lb/>have very well under&longs;tood the nature of the Leaver, and how <lb/>that its Length encrea&longs;ing or decrea&longs;ing, the Moment of the <lb/>Force and of the Re&longs;i&longs;tance augmented or dimini&longs;hed, and yet in <lb/>the determination of the pre&longs;ent Problem I deceived my &longs;elf, and <lb/>that not a little, but infinitely much.</s></p><p type="main"> <s>SIMP. </s> <s>The truth is, I begin to &longs;ee that Logick, although it <lb/>be a mo&longs;t appo&longs;ite In&longs;trument to regulate our Di&longs;cour&longs;e, doth <lb/>not attain, as to the prompting of the Mind with Invention, <lb/>unto the acutene&longs;&longs;e of Geometry.</s></p><p type="main"> <s>SAGR. </s> <s>In my conceit, Logick giveth us to under&longs;tand, whe­<lb/>ther the Di&longs;courfes and Demon&longs;trations already made and found <lb/>are concluding, but that it teacheth us how to finde concluding <lb/>Di&longs;cour&longs;es and Demon&longs;trations; the truth is, I do not believe: <lb/>But it will be better, that <emph type="italics"/>Salviatus<emph.end type="italics"/> &longs;hew us according to what pro­<lb/>portion the Moments of the Forces do go increa&longs;ing, to overcome <lb/>the Re&longs;i&longs;tance of the &longs;ame Piece of Wood, according to the &longs;e­<lb/>veral places of the Fracture.</s></p><p type="main"> <s>SALV. </s> <s>The proportion that you &longs;eek, proceedeth after &longs;uch <lb/>a manner, that</s></p><pb xlink:href="069/01/115.jpg" pagenum="113"/><p type="head"> <s>PROPOSITION XII.</s></p><p type="main"> <s><emph type="italics"/>If in the length of a Cylinder we &longs;hall marke two places, <lb/>upon which we would make the Fracture of the &longs;aid <lb/>Cylinder, the Re&longs;i&longs;tances of tho&longs;e two places have <lb/>the &longs;ame proportion to each other, as have the Re­<lb/>ctangles made by the Di&longs;tances of tho&longs;e places <lb/>reciprocally taken.<emph.end type="italics"/></s></p><p type="main"> <s>Let the two Forces (<emph type="italics"/>as in<emph.end type="italics"/> Fig. </s> <s>16.) be A and B the lea&longs;t, to <lb/>break in C, and E and F likewi&longs;e the lea&longs;t, to break in D. <lb/></s> <s>I &longs;ay the Forces A and B have the &longs;ame proportion to the <lb/>Forces E and F, that the Rectangle A D B hath to the Rectan­<lb/>gle A C B. </s> <s>For the Forces A and B, have to the Forces E and F, a <lb/>proportion compounded of the Forces A and B, to the Force <lb/>B, of B to F, and of F to E and E: But as the Forces A and <lb/>B are to the Force B, &longs;o is the Length B A to A C; and as the <lb/>Force B is to F, &longs;o is the Line D B to B C; and as the Force F is <lb/>to F and E, &longs;o is the Line D A to A B: Therefore the Forces A <lb/>and B have to the Forces E and F a proportion compounded of <lb/>the&longs;e three, namely, of B A to A C, of D B to B C, and of D A <lb/>A B. </s> <s>But of the two proportions D A to A B, and A B to A C, <lb/>is compounded the proportion of D A to A C: Therefore the <lb/>Forces A and B have to the Forces E and F, the proportion com­<lb/>pounded of this D A to A C, and of the other D B to D C. <lb/></s> <s>But the Rectangle A D B hath to the Rectangle A C B, a pro­<lb/>portion compounded of the &longs;ame D A to A C, and of D B to <lb/>B C: Therefore the Forces A and B are to the Forces E and F, <lb/>as the Rectangle A D B is to the Rectangle A C B; which is as <lb/>much as to &longs;ay, the Re&longs;i&longs;tance again&longs;t Fraction in C, hath the <lb/>&longs;ame proportion to the Re&longs;i&longs;tance again&longs;t Fraction in D, that <lb/>the Rectangle A D B hath to the Rectangle A C B: Which was <lb/>to be demon&longs;trated.</s></p><p type="main"> <s>In con&longs;equence of this Theorem we may re&longs;olve a Problem of <lb/>great Curio&longs;ity; and it is this:</s></p><pb xlink:href="069/01/116.jpg" pagenum="114"/><p type="head"> <s>PROP. XIII. PROBL. IV.</s></p><p type="main"> <s><emph type="italics"/>There being given the greate&longs;t Weight that can be &longs;up­<lb/>ported at the middle of a Cylinder or Pri&longs;me, where <lb/>the Re&longs;i&longs;tance is leafl; and there being given a <lb/>Weight greater than that, to find in the &longs;aid Cylin­<lb/>der, the point at which the given greater Weight may <lb/>be &longs;upporited as the greate&longs;t Weight.<emph.end type="italics"/></s></p><p type="main"> <s>Let the given weight greater than the greate&longs;t weight that <lb/>can be &longs;upported at the middle of the Cylinder A B, have <lb/>unto the &longs;aid greate&longs;t weight, the proportion of the line E <lb/>to F: it is required to find the point in the Cylinder at which the <lb/>&longs;aid given weight commeth to be &longs;upported as the bigge&longs;t. </s> <s>Be­<lb/>tween E and F let G be a Mean-Proportional; and as E is to G, <lb/>&longs;o let A D be to S, S &longs;hall be le&longs;&longs;er than A D. </s> <s>Let A D be the <lb/>Diameter of the Semicircle A H D: in which &longs;uppo&longs;e A H equal <lb/>to S; and joyn together H and D, and take D R equal to it. <lb/></s> <s>I &longs;ay that R is the point &longs;ought, at which the given weight, <lb/>greater than the greate&longs;t that can be &longs;upported at the middle of the <lb/>Cylinder D, would become as the greate&longs;t weight. </s> <s>On the length <lb/><emph type="italics"/>B<emph.end type="italics"/>A de&longs;cribe the Semicircle A N B, and rai&longs;e the Perpendicular <lb/>RN, and conjoyn N and <lb/>D: And becau&longs;e the two <lb/><figure id="id.069.01.116.1.jpg" xlink:href="069/01/116/1.jpg"/><lb/>Squares N R and R D are <lb/>equal to the Square N D; <lb/>that is, to the Square A D; <lb/>that is, to the two A H and <lb/>and H D; and H D is equal <lb/>to the Square D R: There­<lb/>fore the Square N R, that <lb/>is, the Rectangle A R B <lb/>&longs;hall be equal to the Square A H; that is, to the Square S: But <lb/>the Square S is to the Square A D, as F to E; that is, as the <lb/>greate&longs;t &longs;upportable Weight at D to the given greater Weight: <lb/>Therefore this greater &longs;hall be &longs;upported at R, as the greate&longs;t <lb/>that can be there &longs;u&longs;tained. </s> <s>Which is that that we &longs;ought.</s></p><p type="main"> <s>SAGR. </s> <s>I under&longs;tand you very well, and am con&longs;idering that <lb/>the Pri&longs;me A B having alwayes more &longs;trength and re&longs;i&longs;tance a­<lb/>gain&longs;t Pre&longs;&longs;ion in the parts that more and more recede from the <lb/>middle, whether in very great and heavy Beams one may take <pb xlink:href="069/01/117.jpg" pagenum="115"/>away a pretty big part towards the end with a notable alleviation <lb/>of the weight; which in Beams of great Rooms would be commo­<lb/>dious, and of no &longs;mall pro&longs;it. </s> <s>And it would be pretty, to find what <lb/>Figure that Solid ought to have, that it might have equal Re&longs;i­<lb/>&longs;tance in all its parts; &longs;o as that it were not with more ea&longs;e to be <lb/>broken by a weight that &longs;hould pre&longs;&longs;e it in the mid&longs;t, than in any <lb/>other place.</s></p><p type="main"> <s>SALV. </s> <s>I was ju&longs;t about to tell you a thing very notable and <lb/>plea&longs;ant to this purpo&longs;e. </s> <s>I will a&longs;&longs;ume a brief Scheme for the bet­<lb/>ter explanation of my meaning. </s> <s>This Figure D B is a Pri&longs;m, who&longs;e <lb/>Re&longs;i&longs;tance again&longs;t Fraction in the term A D by a Force pre&longs;&longs;ing <lb/>at the term B, is le&longs;&longs;e than the Re&longs;i&longs;tance that would be found in <lb/>the place C I, by how much the length C B is le&longs;&longs;er than B A; as <lb/>hath already been demon­<lb/>&longs;trated. </s> <s>Now &longs;uppo&longs;e the <lb/><figure id="id.069.01.117.1.jpg" xlink:href="069/01/117/1.jpg"/><lb/>&longs;aid Pri&longs;me to be &longs;awed <lb/>Diagonally according to the <lb/>Line FB, &longs;o that the oppo­<lb/>&longs;ite Surfaces may be two <lb/>Triangles, one of which to­<lb/>wards us is F A B. </s> <s>This So­<lb/>lid obtains a quality contrary to the Pri&longs;me, to wit, that it le&longs;&longs;e re­<lb/>&longs;i&longs;teth Fraction by the Force placed in B at the term C than at A, <lb/>by as much the Length C <emph type="italics"/>B<emph.end type="italics"/> is le&longs;&longs;e than <emph type="italics"/>B<emph.end type="italics"/> A; Which we will ea <lb/>&longs;ily prove: For imagining the Section C N O parallel to the other <lb/>A F D, the Line <emph type="italics"/>F<emph.end type="italics"/> A &longs;hall be to C N in the Triangle F A <emph type="italics"/>B<emph.end type="italics"/> in the <lb/>&longs;ame proportion, as the Line A <emph type="italics"/>B<emph.end type="italics"/> is to <emph type="italics"/>B<emph.end type="italics"/> C: and therefore if we <lb/>&longs;uppo&longs;e the Fulciment of the two Leavers to be in the Points A <lb/>and C, who&longs;e Di&longs;tances are <emph type="italics"/>B<emph.end type="italics"/> A, A F, <emph type="italics"/>B<emph.end type="italics"/> C, and C N, the&longs;e, I &longs;ay, <lb/>&longs;hall be like: and therefore that Moment which the <emph type="italics"/>F<emph.end type="italics"/>orce placed <lb/>at <emph type="italics"/>B<emph.end type="italics"/> hath at the Di&longs;tance <emph type="italics"/>B<emph.end type="italics"/> A above the Re&longs;i&longs;tance placed at the <lb/>Di&longs;tance A <emph type="italics"/>F<emph.end type="italics"/>, the &longs;aid <emph type="italics"/>F<emph.end type="italics"/>orce at <emph type="italics"/>B<emph.end type="italics"/> &longs;hall have at the Di&longs;tance <emph type="italics"/>B<emph.end type="italics"/>C <lb/>above the &longs;ame Re&longs;i&longs;tance, were it placed at the Di&longs;tance C N: <lb/><emph type="italics"/>B<emph.end type="italics"/>ut the Re&longs;i&longs;tance to be overcome at the <emph type="italics"/>F<emph.end type="italics"/>ulciment C, being pla­<lb/>ced at the Di&longs;tance C N, from the <emph type="italics"/>F<emph.end type="italics"/>orce in <emph type="italics"/>B<emph.end type="italics"/> is le&longs;&longs;er than the <lb/>Re&longs;i&longs;tance in A &longs;o much as the Rectangle C O is le&longs;&longs;e than the <lb/>Rectangle A D; that is, &longs;o much as the Line C N is le&longs;s than A <emph type="italics"/>F<emph.end type="italics"/>; <lb/>that is, C <emph type="italics"/>B<emph.end type="italics"/> than B A: Therefore the Re&longs;i&longs;tance of the part O C B <lb/>again&longs;t <emph type="italics"/>F<emph.end type="italics"/>raction in C is &longs;o much le&longs;s than the Re&longs;i&longs;tance of the <lb/>whole D A O again&longs;t <emph type="italics"/>F<emph.end type="italics"/>racture in O, as the Length C B is le&longs;s than <lb/>A B. </s> <s>We have therefore from the Beam or Pri&longs;me D B, taken <lb/>away a part, that is half, cutting it Diagonally, and left the Wedge <lb/>or triangular Pri&longs;m <emph type="italics"/>F<emph.end type="italics"/> B A; and they are two Solids of contrary <lb/>Qualities, namely, that more re&longs;i&longs;ts the more it is &longs;hortned, and this <lb/>in &longs;hortning lo&longs;eth its toughne&longs;s as fa&longs;t. </s> <s>Now this being granted, <pb xlink:href="069/01/118.jpg" pagenum="116"/>it &longs;eemeth very rea&longs;onable, nay, nece&longs;&longs;ary, that one may give it <lb/>a cut, by which taking away that which is &longs;uperfluous, there remai­<lb/>neth a Solid of &longs;uch a <emph type="italics"/>F<emph.end type="italics"/>igure, as in all its parts hath equal Re&longs;i­<lb/>&longs;tance.</s></p><p type="main"> <s>SIMP. </s> <s>It mu&longs;t needs be &longs;o; for where there is a tran&longs;ition from <lb/>the greater to the le&longs;&longs;er, one meeteth al&longs;o with the equal.</s></p><p type="main"> <s>SAGR. </s> <s>But the bu&longs;ine&longs;&longs;e is to find how we are to guide the <lb/>Saw for making of this Section.</s></p><p type="main"> <s>SIMP. </s> <s>This &longs;eemeth to me as if it were a very ea&longs;ie bu&longs;ine&longs;&longs;e; <lb/>for if in &longs;awing the Pri&longs;m diagonally, taking away half of it, the <lb/>Figure that remains retaineth a contrary quality to that of the <lb/>whole Pri&longs;m, &longs;o as that in all places wherein this acquireth &longs;trength, <lb/>that as fa&longs;t lo&longs;eth it, me thinks, that keeping the middle way, that <lb/>is, taking only the half of that half, which is the fourth part of the <lb/>whole, the remaining Figure will not gain or lo&longs;e &longs;trength in any <lb/>of all tho&longs;e places wherein the lo&longs;&longs;e and the gain of the other two <lb/>Figures were alwaies equal.</s></p><p type="main"> <s>SALV. </s> <s>You have not hit the mark, <emph type="italics"/>Simplicius<emph.end type="italics"/>; and as I &longs;hall <lb/>&longs;hew you, it will appear in reality, that that which may be cut off <lb/>from the Pri&longs;m, and taken away without weakening it is not its <lb/>fourth part, but the third. </s> <s>Now it remaineth (which is that that <lb/>was hinted by <emph type="italics"/>Sagredus<emph.end type="italics"/>)</s></p><p type="head"> <s>PROP. XIV. PROBL. V.</s></p><p type="main"> <s><emph type="italics"/>To find according to what Line the Section is to be <lb/>made; Which I will prove to be a Parabolical <lb/>Line.<emph.end type="italics"/></s></p><p type="main"> <s>But fir&longs;t it is nece&longs;&longs;ary to demon&longs;trate a certain Lemma, which <lb/>is this:</s></p><p type="head"> <s>LEMMA I.</s></p><p type="main"> <s><emph type="italics"/>If there &longs;hall be two Ballances or Leavers divided by their Fulci­<lb/>ments in &longs;uch &longs;ort that the two Distances where at the Forces <lb/>are to be placed, have to each other double the proportion of <lb/>the Di&longs;tances at which the Re&longs;i&longs;tances &longs;ball be, which Re&longs;i­<lb/>&longs;tances are to each other as their Di&longs;tances, the &longs;u&longs;taining <lb/>Powers &longs;hall be equal.<emph.end type="italics"/></s></p><p type="main"> <s>Let A B and C D be two Leavers divided upon their Fulciments <lb/>E and F, in &longs;uch &longs;ort that the Di&longs;tance E B hath to F D a pro­<lb/>portion double to that which the Di&longs;tance E A hath to F C. </s> <s>I &longs;ay, <pb xlink:href="069/01/119.jpg" pagenum="117"/>the Powers that in BD &longs;hall &longs;u&longs;tain the Re&longs;i&longs;tances A and C &longs;hall <lb/>be equal to each other. </s> <s>Let E G be &longs;uppo&longs;ed a Mean-Proporti­<lb/>onal between E B and F D; therefore as B E is to E G, &longs;o &longs;hall <lb/>G E be to F D, and A E to C <emph type="italics"/>F<emph.end type="italics"/>; and &longs;o is &longs;uppo&longs;ed the Re&longs;i&longs;tance <lb/>of A to the Re&longs;i&longs;tance of C. </s> <s>And becau&longs;e that as E G is to <emph type="italics"/>F<emph.end type="italics"/> D, <lb/>&longs;o is A E to C <emph type="italics"/>F<emph.end type="italics"/>; by Permutation as G E is to E A, &longs;o &longs;hall D <emph type="italics"/>F<emph.end type="italics"/><lb/>be to <emph type="italics"/>F<emph.end type="italics"/> C: And therefore (in <lb/>regard that the two Leavers <lb/><figure id="id.069.01.119.1.jpg" xlink:href="069/01/119/1.jpg"/><lb/>D C and G A are divided pro­<lb/>portionally in the Points <emph type="italics"/>F<emph.end type="italics"/> and <lb/>E) in ca&longs;e the Power that being <lb/>placed at D compen&longs;ates the <lb/>Re&longs;i&longs;tance of C were at G, it <lb/>would countervail the &longs;ame Re&longs;i&longs;tance of C placed in A: But by <lb/>what hath been granted, the Re&longs;i&longs;tance of A hath the &longs;ame propor­<lb/>tion to the Re&longs;i&longs;tance of C, that AE hath to C <emph type="italics"/>F<emph.end type="italics"/>; that is, B E <lb/>hath to E G: Therefore the Power G, or if you will D, placed at <lb/>B will &longs;u&longs;tain the Re&longs;i&longs;tance placed at A: Which was to be de­<lb/>mon&longs;trated.</s></p><p type="main"> <s>This being under&longs;tood: in the Surface <emph type="italics"/>F<emph.end type="italics"/> B of the Pri&longs;me D B, <lb/>let the Parabolical Line <emph type="italics"/>F<emph.end type="italics"/> N B be drawn, who&longs;e Vertex is B, ac­<lb/>cording to which let the &longs;aid Pri&longs;me be &longs;uppo&longs;ed to be &longs;awed, the <lb/>Solid compri&longs;ed between the Ba&longs;e A D, the Rectangular Plane <lb/>A G, the Bight Line B G, and the Superficies D G B <emph type="italics"/>F<emph.end type="italics"/> being le&longs;t <lb/>incurvated according to the Curvity of the Parabolical Line <lb/><emph type="italics"/>F<emph.end type="italics"/> N B. </s> <s>I &longs;ay, that <lb/>that Solid is through­<lb/><figure id="id.069.01.119.2.jpg" xlink:href="069/01/119/2.jpg"/><lb/>out of equal Re&longs;i­<lb/>&longs;tance. </s> <s>Let it be cut <lb/>by the Plane C O pa­<lb/>rallel to A D; and <lb/>imagine two Leavers <lb/>divided and &longs;uppor­<lb/>ted upon the Fulciments A and C; and let the Di&longs;tances of one <lb/>be B A and A F, and of the other B C, and C N. </s> <s>And becau&longs;e in <lb/>the Parabola <emph type="italics"/>F B<emph.end type="italics"/> A, A <emph type="italics"/>B<emph.end type="italics"/> is to <emph type="italics"/>B<emph.end type="italics"/> C, as the Square of <emph type="italics"/>F<emph.end type="italics"/> A to the <lb/>Square of C N, it is manife&longs;t, that the Di&longs;tance <emph type="italics"/>B<emph.end type="italics"/> A of one Leaver, <lb/>hath to the Di&longs;tance <emph type="italics"/>B<emph.end type="italics"/> C of the other a proportion double to that <lb/>which the other Di&longs;tance A <emph type="italics"/>F<emph.end type="italics"/> hath to the other C N, And be­<lb/>cau&longs;e the Re&longs;i&longs;tance that is to be equal by help of the Leaver <lb/><emph type="italics"/>B<emph.end type="italics"/> A hath the &longs;ame proportion to the Re&longs;i&longs;tance that is to be <lb/>equal by help of the Leaver <emph type="italics"/>B<emph.end type="italics"/> C, that the Rectangle D A hath to <lb/>the Rectangle O C; which is the &longs;ame that the Line A <emph type="italics"/>F<emph.end type="italics"/> hath to <lb/>N C, which are the other two Di&longs;tances of the Leavers; it is ma­<lb/>nife&longs;t by the fore going Lemma, that the &longs;ame Force that being <pb xlink:href="069/01/120.jpg" pagenum="118"/>applyed to the Line <emph type="italics"/>B<emph.end type="italics"/> G will equal the Re&longs;i&longs;tance D A, will like­<lb/>wi&longs;e equal the Re&longs;i&longs;tance C O. </s> <s>And the &longs;ame may be demon&longs;tra­<lb/>ted, if one cut the Solid in any other place: therefore that Parabo­<lb/>lical Solid is throughout of equal Re&longs;i&longs;tance. </s> <s>In the next place, <lb/>that cutting the Pri&longs;me according to the Parabolical Line F N B, <lb/>the third part of it is taken away, appeareth, For that the Semi­<lb/>Parabola F N <emph type="italics"/>B<emph.end type="italics"/> A and the Rectangle F <emph type="italics"/>B<emph.end type="italics"/> are Ba&longs;es of two Solids <lb/>contained between two parallel Planes, that is, between the Rect­<lb/>angles F B and D G, whereby they retain the &longs;ame Proportion, as <lb/>tho&longs;e their Ba&longs;es: But the Rectangle F <emph type="italics"/>B<emph.end type="italics"/> is Se&longs;quialter to the Se­<lb/>miparabola F N <emph type="italics"/>B<emph.end type="italics"/> A: Therefore cutting the Pri&longs;ine according to <lb/>the Parabolick Line, we take away the third part of it. </s> <s>Hence we <lb/>&longs;ee, that <emph type="italics"/>B<emph.end type="italics"/>eams may be made with the diminution of their Weight <lb/>more than thirty three in the hundred, without dimini&longs;hing their <lb/>Strength in the lea&longs;t; which in great Ships, in particular, for bea­<lb/>ring the Decks may be of no &longs;mall benefit; for that in &longs;uch kind <lb/>of Fabricks Lightne&longs;&longs;e is of infinite importance.</s></p><p type="main"> <s>SAGR. </s> <s>The Commodities are &longs;o many, that it would be tedi­<lb/>ous, if not impo&longs;&longs;ible, to mention them all. <emph type="italics"/>B<emph.end type="italics"/>ut I, laying a&longs;ide <lb/>the&longs;e, would more gladly under&longs;tand that the alleviation is made <lb/>according to the a&longs;&longs;igned proportions. </s> <s>That the Section, according <lb/>to the Diagonal Line, cuts away half of the weight I very well <lb/>know: but that the other Section according to the Parabolical Line <lb/>takes away the third part of the Pri&longs;me I can believe upon the <lb/>word of <emph type="italics"/>Salviatus,<emph.end type="italics"/> who evermore &longs;peaks the truth, but in this <lb/>Ca&longs;e Science would better plea&longs;e me than Faith.</s></p><p type="main"> <s>SALV. </s> <s>I &longs;ee then that you would have the Demon&longs;tration, <lb/>whether or no it be true, that the exce&longs;&longs;e of the Pri&longs;me over and <lb/>above this, which for this time we will call a Parabolical Solid, is <lb/>the third part of the whole Pri&longs;me. </s> <s>I am certain that I have for­<lb/>merly demo&longs;trated it; I will try now whether I can put the <lb/>Demon&longs;tration together again: to which purpo&longs;e I do remember <lb/>that I made u&longs;e of a Certain Lemma of <emph type="italics"/>Archimedes,<emph.end type="italics"/> in&longs;erted by <lb/>him in his <emph type="italics"/>B<emph.end type="italics"/>ook <emph type="italics"/>de Spiralibus,<emph.end type="italics"/> and it is this:</s></p><p type="head"> <s>LEMMA II.</s></p><p type="main"> <s><emph type="italics"/>If any number of Lines at plea&longs;ure &longs;hall exceed one another equal­<lb/>ly, and the exce&longs;&longs;es be equal to the lea&longs;t of them, and there be as <lb/>many more, each of them equal to the greate&longs;t; the Squares of all <lb/>the&longs;e &longs;hall be le&longs;&longs;e than the triple of the Squares of tho&longs;e that <lb/>exceed one another: but they &longs;hall be more than triple to tho&longs;e <lb/>others that remain, the Square of the greate&longs;t being &longs;ub­<lb/>&longs;tracted.<emph.end type="italics"/></s></p><pb xlink:href="069/01/121.jpg" pagenum="119"/><p type="main"> <s>This being granted: Let the Parabolick Line A <emph type="italics"/>B<emph.end type="italics"/> be in&longs;cribed <lb/>in this Rectangle A C <emph type="italics"/>B<emph.end type="italics"/> P: we are to prove the Mixt Triangle <lb/><emph type="italics"/>B<emph.end type="italics"/> A P, who&longs;e &longs;ides are <emph type="italics"/>B<emph.end type="italics"/> P and P A, and <emph type="italics"/>B<emph.end type="italics"/>a&longs;e the Parabolical Line <lb/><emph type="italics"/>B<emph.end type="italics"/> A, to be the third part of the whole Rectangle C P. </s> <s>For if it be <lb/>not &longs;o, it will be either more than the third part, or le&longs;&longs;e. </s> <s>Let it be <lb/>&longs;uppo&longs;ed that it may be <lb/>le&longs;&longs;e, and to that which is <lb/><figure id="id.069.01.121.1.jpg" xlink:href="069/01/121/1.jpg"/><lb/>wanting &longs;uppo&longs;e the Space <lb/>X to be equal. </s> <s>Then di­<lb/>viding the Rectangle con­<lb/>tinually into equal parts <lb/>with Lines parallel to the <lb/>Sides <emph type="italics"/>B<emph.end type="italics"/> P and C A, we <lb/>&longs;hall in the end arrive at <lb/>&longs;uch parts, as that one of them &longs;hall be le&longs;&longs;e than the Space X. <lb/></s> <s>Now let one of them be the Rectangle O <emph type="italics"/>B,<emph.end type="italics"/> and by the Points <lb/>where the other Parallels inter&longs;ect the Parabolick Line, let the Pa­<lb/>rallels to A P pa&longs;&longs;e: and here I will &longs;uppo&longs;e a Figure to be cir­<lb/>cum&longs;cribed about our Mixt-Triangle, compo&longs;ed of Rectangles, <lb/>which are <emph type="italics"/>B<emph.end type="italics"/> O, I N, H M, F L, E K, G A; which Figure &longs;hall al&longs;o <lb/>yet be le&longs;s than the third part of the Rectangle C P, in regard that <lb/>the exce&longs;&longs;e of that Figure over and above the Mixed Triangle is <lb/>much le&longs;&longs;e than the Rectangle <emph type="italics"/>B<emph.end type="italics"/> O, which yet again is le&longs;&longs;e than <lb/>the Space X.</s></p><p type="main"> <s>SAGR. Softly, I pray you, for I do not &longs;ee how the exce&longs;&longs;e of <lb/>this circum&longs;cribed Figure above the Mixt Triangle is con&longs;iderably <lb/>le&longs;&longs;er than the Rectangle <emph type="italics"/>B<emph.end type="italics"/> O.</s></p><p type="main"> <s>SALV. </s> <s>Is not the Rectangle <emph type="italics"/>B<emph.end type="italics"/> O equal to all the&longs;e &longs;mall Rect­<lb/>angles by which our Parabolical Line pa&longs;&longs;eth; I mean the&longs;e, <emph type="italics"/>B<emph.end type="italics"/> I, <lb/>I H, H F, F E, E G, and G A, of which one part only lyeth with­<lb/>out the Mixt Triangle? </s> <s>And the Rectangle <emph type="italics"/>B<emph.end type="italics"/> O, is it not al&longs;o &longs;up­<lb/>po&longs;ed to be le&longs;&longs;e than the Space X? </s> <s>Therefore if the Triangle to­<lb/>gether with X did, as the Adver&longs;ary &longs;uppo&longs;eth, equalize the third <lb/>part of the Rectangle C P the circum&longs;cribed Figure that adjoyns <lb/>to the Triangle &longs;o much le&longs;&longs;e than the Space X, will remain even <lb/>yet le&longs;&longs;e than the third part of the &longs;aid Rectangle C P. <emph type="italics"/>B<emph.end type="italics"/>ut this <lb/>cannot be, for it is more than a third part, therefore it is not true <lb/>that our Mixt Triangle is le&longs;&longs;e than one third of the Rectangle.</s></p><p type="main"> <s>SAGR. </s> <s>I under&longs;tand the Solution of my Doubt. <emph type="italics"/>B<emph.end type="italics"/>ut it is <lb/>requi&longs;ite now to prove unto us, that the Circum&longs;cribed Figure is <lb/>more than a third part of the Rectangle C P; which, I believe, will <lb/>be harder to do.</s></p><p type="main"> <s>SALV. </s> <s>Not at all. </s> <s>For in the Parabola the Square of the Line <lb/><arrow.to.target n="marg1089"></arrow.to.target><lb/>D E hath the &longs;ame proportion to the Square of Z G, that the Line <pb xlink:href="069/01/122.jpg" pagenum="120"/>D A hath to A Z; which is the &longs;ame that the Rectangle K E hath to <lb/>the Rectangle A G, their heights A K and K L being equal. </s> <s>There­<lb/>fore the proportion that the Square E D hath to the Square Z G; <lb/>that is, the Square L A hath to the Square A K, the Rectangle K E <lb/>hath likewi&longs;e to the Rectangle K Z. </s> <s>And in the &longs;elf-&longs;ame manner <lb/>we might prove that the other Rectangles L F, M H, N I, O B are <lb/>to one another as the Squares of the Lines M A, N A, O A, P A. <lb/></s> <s>Con&longs;ider we in the next place, how the Circum&longs;cribed Figure is <lb/>compounded of certain Spaces that are to one another as the <lb/>Squares of the Lines that exceed with Exce&longs;&longs;es equal to the lea&longs;t, <lb/>and how the Rectangle C P is compounded of &longs;o many other Spa­<lb/>ces each of them equal to the Greate&longs;t, which are all the Rectan­<lb/>gles equal to O B. Therefore, by the Lemma of <emph type="italics"/>Archimedes,<emph.end type="italics"/> the <lb/>Circum&longs;cribed Figure is more than the third part of the Rectangle <lb/>C P: But it was al&longs;o le&longs;&longs;e, which is impo&longs;&longs;ible: Therefore the <lb/>Mixt-Triangle is not le&longs;&longs;e than one third of the Rectangle C P. <lb/></s> <s>I &longs;ay likewi&longs;e, that it is not more: For if it be more than one <lb/>third of the Rectangle C P, &longs;uppo&longs;e the Space X equal to the ex­<lb/>ce&longs;&longs;e of the Triangle above the third part of the &longs;aid Rectangle <lb/>C P, and the divi&longs;ion and &longs;ubdivi&longs;ion of the Rectangle into Rect­<lb/>angolets, but alwaies equal, being made, we &longs;hall meet with &longs;uch as <lb/>that one of them is le&longs;&longs;er than the Space X; which let be done: <lb/>and let the Rectangle <emph type="italics"/>B<emph.end type="italics"/> O be le&longs;&longs;er than X; and, having de&longs;cribed <lb/>the Figure as before, we &longs;hall have in&longs;cribed in the Mixt-Triangle <lb/>a Figure compounded of the Rectangles V O, T N, S M, N L, Q K, <lb/>which yet &longs;hall not be le&longs;s <lb/><figure id="id.069.01.122.1.jpg" xlink:href="069/01/122/1.jpg"/><lb/>than the third part of the <lb/>great Rectangle C P, for <lb/>the Mixt Triangle doth <lb/>much le&longs;&longs;e exceed the In­<lb/>&longs;cribed Figure than it doth <lb/>exceed the third part of <lb/>the Rectangle C P; Be­<lb/>cau&longs;e the exce&longs;&longs;e of the <lb/>Triangle above the third part of the Rectangle C P is equal to <lb/>the Space X which is greater than the Rectangle <emph type="italics"/>B<emph.end type="italics"/> O, and this al­<lb/>&longs;o is con&longs;iderably greater than the exce&longs;&longs;e of the Triangle above <lb/>the In&longs;cribed Figure: For to the Rectangle <emph type="italics"/>B<emph.end type="italics"/> O, all the Rectan­<lb/>golets A G, G E, E <emph type="italics"/>F,<emph.end type="italics"/> F H, H I, I <emph type="italics"/>B<emph.end type="italics"/> are equal, of which the Ex­<lb/>ce&longs;&longs;es of the Triangle above the In&longs;cribed <emph type="italics"/>F<emph.end type="italics"/>igure are le&longs;&longs;e than <lb/>half: And therefore the Triangle exceeding the third part of the <lb/>Rectangle C P, by much more (exceeding it by the Space X) <lb/>than it exceedeth its in&longs;cribed <emph type="italics"/>F<emph.end type="italics"/>igure, that &longs;ame <emph type="italics"/>F<emph.end type="italics"/>igure &longs;hall al&longs;o <lb/>be greater than the third part of the Rectangle C P: <emph type="italics"/>B<emph.end type="italics"/>ut it is le&longs;&longs;er, <lb/>by the Lemma pre&longs;uppo&longs;ed: <emph type="italics"/>F<emph.end type="italics"/>or that the Rectangle C P, as being <pb xlink:href="069/01/123.jpg" pagenum="127"/>the Aggregate of all the bigge&longs;t Rectangles, hath the &longs;ame pro­<lb/>portion to the Rectangles compounding the In&longs;cribed <emph type="italics"/>F<emph.end type="italics"/>igure, that <lb/>the Aggregate of of all the Squares of the Lines equal to the big­<lb/>ge&longs;t, hath to the Squares of the Lines that exceed equally, &longs;ub&longs;tra­<lb/>cting the Square of the bigge&longs;t: And therefore (as it hapneth in <lb/>Squares) the whole Aggregate of the bigge&longs;t (that is the Rectan­<lb/>gle C P) is more than triple the Aggregate of the exceeding <lb/>ones, the bigge&longs;t deducted, that compound the In&longs;cribed <emph type="italics"/>F<emph.end type="italics"/>i­<lb/>gure. </s> <s>Therefore the Mixt-Triangle is neither greater nor le&longs;&longs;er <lb/>than the third part of the Rectangle C P: It is therefore equal.</s></p><p type="margin"> <s><margin.target id="marg1089"></margin.target><emph type="italics"/>The Quadrature of <lb/>the Parabola &longs;hewn <lb/>by one &longs;ingle De­<lb/>mon&longs;tration.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>A pretty and ingenuous Demon&longs;tration: and &longs;o much <lb/>the more, in that it giveth us the Quadrature of the Parabola, &longs;hew­<lb/>ing it to be <emph type="italics"/>Se&longs;quitertial<emph.end type="italics"/> of the Triangle in&longs;cribed in the &longs;ame; <lb/>proving that which <emph type="italics"/>Archimedes<emph.end type="italics"/> demon&longs;trateth by two very diffe­<lb/>rent, but both very admirable, methods of a great number of Pro­<lb/>po&longs;itions. </s> <s>As hath likewi&longs;e been demon&longs;trated lately by <emph type="italics"/>Lucas <lb/>Valerius,<emph.end type="italics"/> another &longs;econd <emph type="italics"/>Archimedes<emph.end type="italics"/> of our Age, which Demon­<lb/>&longs;tration is &longs;et down in the Book that he writ of the Center of the <lb/>Gravity of Solids.</s></p><p type="main"> <s><emph type="italics"/>S<emph.end type="italics"/>ALV. </s> <s>A Treati&longs;e which verily is not to come behind any one <lb/>that hath been written by the mo&longs;t <emph type="italics"/>F<emph.end type="italics"/>amous Geometricians of the <lb/>pre&longs;ent and all pa&longs;t Ages: which when it was read by our <emph type="italics"/>Acade­<lb/>mick,<emph.end type="italics"/> it made him de&longs;i&longs;t from pro&longs;ecuting his Di&longs;coveries that he <lb/>was then proceeding to write on the &longs;ame Subject: in regard he <lb/>&longs;aw the whole bu&longs;ine&longs;s &longs;o happily found and demon&longs;trated by the <lb/>&longs;aid <emph type="italics"/>Valerius.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>I was informed of all the&longs;e things by our <emph type="italics"/>Academick<emph.end type="italics"/>; <lb/>and have be&longs;ought him withall that he would one day let me &longs;ee <lb/>his Demon&longs;trations that he had &longs;ound at the time when he met <lb/>with the <emph type="italics"/>B<emph.end type="italics"/>ook of <emph type="italics"/>Valerius:<emph.end type="italics"/> but I never was &longs;o happy as to &longs;ee them.</s></p><p type="main"> <s>SALV. </s> <s>I have a Copy of them, and will impart them to you, <lb/>for you will be much plea&longs;ed to &longs;ee the variety of Methods, which <lb/>the&longs;e two Authors take to inve&longs;tigate the &longs;ame Conclu&longs;ions, and <lb/>their Demon&longs;trations: wherein al&longs;o &longs;ome of the Conclu&longs;ions have <lb/>different Explanations, howbeit in effect equally true.</s></p><p type="main"> <s>SAGR. </s> <s>I &longs;hall be very glad to &longs;ee them, therefore when you re­<lb/>turn to our wonted Conferences you may do me the favour to <lb/>bring them with you. <emph type="italics"/>B<emph.end type="italics"/>ut in the mean time, this &longs;ame of the Re­<lb/>fi&longs;tance of the Solid taken from the Pri&longs;me by a Parabolick Secti­<lb/>on, being an Operation no le&longs;&longs;e ingenuous than beneficial in many <lb/>Mechanical Works, it would be good that Artificers had &longs;ome ea­<lb/>&longs;ie and expedite Rule how they may draw the &longs;aid Parabolick <lb/>Line upon the Plane of the Pri&longs;me.</s></p><p type="main"> <s>SALV. </s> <s>There are &longs;everal waies to draw tho&longs;e Lines, but two <lb/><arrow.to.target n="marg1090"></arrow.to.target><lb/>that are more expedite than all the re&longs;t, I will de&longs;cribe unto you. <pb xlink:href="069/01/124.jpg" pagenum="122"/>One of which is truly admirable, &longs;ince that thereby, in le&longs;&longs;e time <lb/>than another can with Compa&longs;&longs;es &longs;lightly draw upon a paper <lb/>four or &longs;ix Circles of different &longs;izes, I can de&longs;ign thirty or forty <lb/>Parabolick Lines no le&longs;&longs;e exact, &longs;mall, and &longs;mooth than the Cir­<lb/>cumferences of tho&longs;e Circles. </s> <s>I have a <emph type="italics"/>B<emph.end type="italics"/>all of <emph type="italics"/>B<emph.end type="italics"/>ra&longs;&longs;e exqui&longs;itely <lb/>round, no bigger than a Nut, this thrown upon a Steel Mirrour <lb/>held, not erect to the Horizon, but &longs;omewhat inclined, &longs;o that the <lb/><emph type="italics"/>B<emph.end type="italics"/>all in its motion may run along pre&longs;&longs;ing lightly upon it, leaveth <lb/>a Parabolical Line finely and &longs;moothly de&longs;cribed, and wider or <lb/>narrower according as the Projection &longs;hall be more or le&longs;s elevated. <lb/></s> <s>Whereby al&longs;o we have a clear and &longs;en&longs;ible Experiment that the <lb/>Motion of Projects is made by Parabolick Lines: an Effect ob&longs;er­<lb/>ved by none before our <emph type="italics"/>Academick,<emph.end type="italics"/> who al&longs;o layeth down the <lb/>Demon&longs;tration of it in his <emph type="italics"/>B<emph.end type="italics"/>ook of Motion, which we will joynt­<lb/>ly peru&longs;e at our next meeting. </s> <s>Now the <emph type="italics"/>B<emph.end type="italics"/>all, that it may de&longs;cribe <lb/>by its motion tho&longs;e Parabola's, mu&longs;t be rouled a little in the hands <lb/>that it may be warmed, and &longs;omewhat moy&longs;tned, for by this <lb/>means it will leave its track more apparent upon the Mirrour. </s> <s>The <lb/>other way to draw the Line that we de&longs;ire upon the Pri&longs;me is after <lb/>this manner. </s> <s>Let two Nailes be fa&longs;tned on high in a Wall, at an <lb/>equal di&longs;tance from the Horizon, and remote from one another <lb/>twice the breadth of the Rectangle upon which we would trace the <lb/>Semiparabola, and to the&longs;e two Nails tye a &longs;mall thread of &longs;uch a <lb/>length that its doubling may reach as far as the length of the <lb/>Pri&longs;me; this &longs;tring will hang in a Parabolick <emph type="italics"/>F<emph.end type="italics"/>igure: &longs;o that tra­<lb/>cing out upon the Wall the way that the &longs;aid String maketh on it, <lb/>we &longs;hall have a whole Parabola de&longs;cribed: which a Perpendicular <lb/>that hangeth in the mid&longs;t between the&longs;e two Nailes will divide <lb/>into two equal parts. </s> <s>And for the transferring or &longs;etting off of <lb/>that Line afterwards upon the oppo&longs;ite Surfaces of the Pri&longs;me it is <lb/>not difficult at all, &longs;o that every indifferent Arti&longs;t will know how <lb/>to do it. </s> <s>The &longs;ame Line might be drawn upon the &longs;aid Sur­<lb/>face of the Pri&longs;me by help of the Geometrical Lines delineated up­<lb/>on the <emph type="italics"/>Compa&longs;&longs;e<emph.end type="italics"/> of our <emph type="italics"/>Friend,<emph.end type="italics"/> without any more ado.</s></p><p type="margin"> <s><margin.target id="marg1090"></margin.target><emph type="italics"/>Several waies to <lb/>de&longs;cribe a Para­<lb/>bola.<emph.end type="italics"/></s></p><p type="main"> <s>We have hitherto demon&longs;trated &longs;o many Conclu&longs;ions touching <lb/>the Contemplation of the&longs;e Re&longs;i&longs;tances of Solids again&longs;t Fraction <lb/>by having fir&longs;t opened the way unto the Science with &longs;uppo&longs;ing the <lb/>direct Re&longs;i&longs;tance for known, that we may in pur&longs;uance of them <lb/>proceed forwards to the finding of other, and other Conclu&longs;ions, <lb/>with their Demon&longs;trations of tho&longs;e which in Nature are infinite. <lb/></s> <s>Only at pre&longs;ent, for a final conclu&longs;ion of this daies Conferences, <lb/>I will add the Speculation of the Re&longs;i&longs;tances of the Hollow Solids <lb/>which Art, and chiefly Nature, u&longs;eth in an hundred Operations, <lb/>when without encrea&longs;ing the weight &longs;he greatly augmenteth the <lb/>&longs;trength: as is &longs;een in the Bones of Birds, and in many Canes that <pb xlink:href="069/01/125.jpg" pagenum="123"/>are light and of great Re&longs;i&longs;tance again&longs;t bending and breaking. <lb/></s> <s>For if a Wheat Straw that &longs;upports an Ear that is heavier than the <lb/>whole Stalk, were made of the &longs;ame quantity of matter but were <lb/>ma&longs;&longs;ie or &longs;olid, it would be much le&longs;&longs;e repugnant to Fraction or <lb/>Flection. </s> <s>And with the &longs;ame Rea&longs;on Art hath ob&longs;erved, and Ex­<lb/>perience confirmed, that an hollow Cane, or a Trunk of Wood <lb/>or Metal, is much more firm and tough than if being of the &longs;ame <lb/>weight and length it were &longs;olid, which con&longs;equently would be <lb/>more flender, and therefore Art hath contrived to make Lances hol­<lb/>low within when they are de&longs;ired to be &longs;trong and light. </s> <s>We will <lb/>&longs;hew therefore, that</s></p><p type="head"> <s>PROPOSITION XV.</s></p><p type="main"> <s><emph type="italics"/>The Re&longs;i&longs;tances of two Cylinders, equall, and equally <lb/>long, one of which is Hollow, and the other Ma&longs;sie, <lb/>have to each other the &longs;ame proportion, as their Dia­<lb/>meters.<emph.end type="italics"/></s></p><p type="main"> <s>Let the Cane or Hollow Cylinder be A E, [<emph type="italics"/>as in<emph.end type="italics"/> Fig. </s> <s>17.] <lb/>and the Cylinder I N Ma&longs;&longs;ie, and equall in weight and length. <lb/></s> <s>I &longs;ay, the Re&longs;i&longs;tance of the Cane A E hath the &longs;ame propor­<lb/>tion to the Re&longs;i&longs;tance of the &longs;olid Cylinder, as the Diameter <lb/>A B hath to the Diameter I L. </s> <s>Which is very manife&longs;t; For the <lb/>Cane and the Cylinder I N being equal, and of equal lengths, the <lb/>Circle I L that is Ba&longs;e of the Cylinder &longs;hall be equal to the Ring <lb/>A B that is Ba&longs;e of the Cane A E, (I call the Superficies that re­<lb/>maineth when a le&longs;&longs;er Circle is taken out of a greater that is Con­<lb/>centrick with it a Ring:) and therefore their Ab&longs;olute Re&longs;i&longs;tan­<lb/>ces &longs;hall be equal: but becau&longs;e in breaking cro&longs;&longs;e-waies we make <lb/>u&longs;e in the Cylinder I N of the length L N for a Leaver, and of the <lb/>point L for a Fulciment, and of the Semidiameter or Diameter L I <lb/>for a Counter-Leaver; and in the Cane the part of the Leaver, <lb/>that is the Line B E is equal to L N; but the Counter-Leaver at <lb/>the Fulciment B is the Diameter or Semidiameter A B: It is mani­<lb/>fe&longs;t therefore that the Re&longs;i&longs;tance of the Cane exceedeth that of <lb/>the Solid Cylinder as much as the Diameter A B exceeds the Dia­<lb/>meter I L; Which is that that we &longs;ought. </s> <s>Toughne&longs;s therefore is ac­<lb/>quired in the hollow Cane above the Toughne&longs;s of the &longs;olid Cylin­<lb/>der according to the proportion of the Diameters: provided al­<lb/>waies that they be both of the &longs;ame matter, weight, and length.</s></p><p type="main"> <s>It would be well, that in con&longs;equence of this we try to inve&longs;tigate <lb/>that which hapneth in other Ca&longs;es indifferently between all Canes <lb/>and &longs;olid Cylinders of equal length, although unequal in quantity <lb/>of weight, and more or le&longs;s evacuated. </s> <s>And fir&longs;t we will demon­<lb/>&longs;trate, that</s></p><pb xlink:href="069/01/126.jpg"/><figure id="id.069.01.126.1.jpg" xlink:href="069/01/126/1.jpg"/><p type="caption"> <s><emph type="italics"/>Place this at the end of the &longs;econd Dialogue pag:<emph.end type="italics"/> 124,</s></p><pb xlink:href="069/01/127.jpg" pagenum="124"/><p type="head"> <s>PROP. XVI. PROBL. VI.</s></p><p type="main"> <s><emph type="italics"/>A Trunk or Hollow Cane being given, a Solid Cylinder <lb/>may be found equal to it.<emph.end type="italics"/></s></p><p type="main"> <s>This Operation is very ea&longs;ie. </s> <s>For let the Line A B, be the Dia­<lb/>meter of the Cane, and C D the Diameter of the Hollow or <lb/>Cavity. </s> <s>Let the Line A E be &longs;et off upon the greater Circle <lb/>equal to the Diameter C D, and conjoyn E B. </s> <s>And becau&longs;e in <lb/><figure id="id.069.01.127.1.jpg" xlink:href="069/01/127/1.jpg"/><lb/>the Semicircle A E B the Angle E is Right­<lb/>Angle, the Circle who&longs;e Diameter is A B <lb/>&longs;hall be equall to the two Circles of the Di­<lb/>ameters A E and E B: But A E is the Dia­<lb/>meter of the Hollow of the Cane: Therefore <lb/>the Circle who&longs;e Diameter is E B, &longs;hall be <lb/>equal to the Ring A C B D: And therefore <lb/>the &longs;olid Cylinder, the Circle of who&longs;e Ba&longs;e <lb/>hath the Diameter E B &longs;hall be equal to the <lb/>Cane, they being of the &longs;ame length. </s> <s>This demon&longs;trated, we may <lb/>pre&longs;ently be able</s></p><p type="head"> <s>PROP. XVII. PROBL. VII.</s></p><p type="main"> <s><emph type="italics"/>To find what proportion is betwixt the Re&longs;i&longs;tances of <lb/>any what&longs;oever Cane and Cylinder, their lengths be­<lb/>ing equal.<emph.end type="italics"/></s></p><p type="main"> <s>LET the Cane A B E, and the Cylinder R S M, be of equal <lb/>length: it is required to find what proportion the Re&longs;i&longs;tances <lb/>have to each other. </s> <s>By the precedent let the Cylinder I L N <lb/>be found equal to the Cane, and of the &longs;ame length; and to the <lb/>Lines I L and R S (Diameters of the Ba&longs;es of the Cylinders I N and <lb/><figure id="id.069.01.127.2.jpg" xlink:href="069/01/127/2.jpg"/><lb/>R M) let the Line V be a fourth <lb/>Proportional. </s> <s>I &longs;ay, the Re&longs;i&longs;tance <lb/>of the Cane A E is to the Re&longs;i­<lb/>&longs;tance of the Cylinder R M, as the <lb/>Line A B is to V. </s> <s>For the Cane <lb/>A E being equal to, and of the <lb/>&longs;ame length with the Cylinder <lb/>I N, the Re&longs;i&longs;tance of the Cane <lb/>&longs;hall be to the Re&longs;i&longs;tance of the <lb/>Cylinder, as the Line A B is to I L: <lb/>But the Re&longs;i&longs;tance of the Cylinder I N is to the Re&longs;i&longs;tance of the <lb/>Cylinder R M, as the Cube I L is to the Cube R S; that is, as the <lb/>Line I L to V: Therefore, <emph type="italics"/>ex æquali,<emph.end type="italics"/> the Re&longs;i&longs;tance of the Cane <lb/>A E hath the &longs;ame proportion to the Re&longs;i&longs;tance of the Cylinder <lb/>R M, that the Line A B hath to V: Which is that that was &longs;ought.</s></p><p type="head"> <s><emph type="italics"/>The End of the Second Dialogue.<emph.end type="italics"/></s></p></chap><chap><pb xlink:href="069/01/128.jpg" pagenum="125"/><p type="head"> <s>GALILEUS, <lb/>HIS <lb/>DIALOGUES <lb/>OF <lb/>MOTION.</s></p><p type="head"> <s>The Third Dialogue.</s></p><p type="head"> <s><emph type="italics"/>INTERLOCUTORS,<emph.end type="italics"/></s></p><p type="head"> <s>SALVIATUS, SAGREDUS, and SIMPLICIUS.</s></p><p type="head"> <s>OF LOCAL MOTION.</s></p><p type="main"> <s><emph type="italics"/>We promote a very new Science, but of a very <lb/>old Subject. </s> <s>There is nothing in Nature more <lb/>antient than<emph.end type="italics"/> MOTION, <emph type="italics"/>of which <lb/>many and great Volumns have been written <lb/>by Philo&longs;ophers: But yet there are &longs;undry <lb/>Symptomes and Properties in it worthy of <lb/>our Notice, which I find not to have been hi­<lb/>therto ob&longs;erved, much le&longs;&longs;e demon&longs;trated by <lb/>any. </s> <s>Some &longs;light particulars have been no­<lb/>ted: as that the Natural Motion of Grave Bodies continually accelle-<emph.end type="italics"/><pb xlink:href="069/01/129.jpg" pagenum="126"/><emph type="italics"/>rateth, as they de&longs;cend towards their Center: but it hath not been as yet <lb/>declared in what proportion that Acceleration is made. </s> <s>For no man, <lb/>that I know, hath ever demon&longs;trated, That there is the &longs;ame proportion <lb/>between the Spaces, thorow which a thing moveth in equal Times, as <lb/>there is between the Odde Numbers which follow in order after a Vnite. <lb/></s> <s>It hath been ob&longs;erved that Projects [or things thrown or darted with vi­<lb/>olence] make a Line that is &longs;omewhat curved; but that this line is a Pa­<lb/>rabola, none have hinted: Yet the&longs;e, and &longs;undry other things, no <lb/>le&longs;&longs;e worthy of our knowledg, will I here demon&longs;trate: And which <lb/>is more, I will open a way to a mo&longs;t ample and excellent Science, <lb/>of which the&longs;e our Labours &longs;hall be the Elements: into which more <lb/>&longs;ubtil and piercing Wits than mine will be better able to dive.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>We divide this Treati&longs;e into three parts. </s> <s>In the fir&longs;t part we con&longs;ider <lb/>&longs;uch things as re&longs;pect Equable or Vniforme Motion. </s> <s>In the &longs;econd we <lb/>write of Motion naturally accelerate. </s> <s>In the third we treat of Violent <lb/>Motion, or<emph.end type="italics"/> De Projectis.</s></p><p type="head"> <s>OF EQVABLE MOTION.</s></p><p type="main"> <s><emph type="italics"/>Concerning Equable or Vniform Motion we have need of onely one <lb/>Definition, which I thus deliver.<emph.end type="italics"/></s></p><p type="head"> <s>DEFINITION.</s></p><p type="main"> <s>By an Equable or Uniform Motion, I under&longs;tand that by which a <lb/>Moveable in all equal Times pa&longs;&longs;eth thorow equal Spaces.</s></p><p type="head"> <s>ADVERTISEMENT.</s></p><p type="main"> <s><emph type="italics"/>I thought good to add to the old Definition (which &longs;imply termeth <lb/>that an Equable Motion, whereby equal Spaces are pa&longs;t in equal <lb/>Times) this Particle<emph.end type="italics"/> All, <emph type="italics"/>that is, any what&longs;oever Times that are equal: <lb/>for it may happen, that a Moveable may pa&longs;&longs;e thorow equal Spaces in cer­<lb/>tain equal Times, though the Spaces be not equal which it hath gone in <lb/>le&longs;&longs;er, though equal parts of the &longs;ame Time. </s> <s>From this our Definition <lb/>follow the&longs;e four Axiomes:<emph.end type="italics"/> &longs;cilicet,</s></p><p type="head"> <s>AXIOMEL</s></p><p type="main"> <s>In the &longs;ame Equable Motion that Space is greater which is pa&longs;&longs;ed <lb/>in a longer Time, and that le&longs;&longs;er which is pa&longs;t in a &longs;horter.</s></p><pb xlink:href="069/01/130.jpg" pagenum="127"/><p type="head"> <s>AXIOME II.</s></p><p type="main"> <s>In the &longs;ame Equable Motion, the greater the Space is that hath <lb/>been gone thorow, the longer was the Time in which the Move­<lb/>able was going it.</s></p><p type="head"> <s>AXIOME III.</s></p><p type="main"> <s>The Space which a greater Velocity pa&longs;&longs;eth in any Time, is great­<lb/>er than the Space which a le&longs;&longs;er Velocity pa&longs;&longs;eth in the &longs;ame <lb/>Time.</s></p><p type="head"> <s>AXIOME IV.</s></p><p type="main"> <s>The Velocity which pa&longs;&longs;eth a greater Space, is greater than the <lb/>Velocity which pa&longs;&longs;eth a le&longs;&longs;er Space in the &longs;ame Time.</s></p><p type="head"> <s>THEOR. I. PROP. I.</s></p><p type="main"> <s>If a Moveable moving with an Equable Motion, <lb/>and with the &longs;ame Velocity pa&longs;&longs;e two &longs;everal <lb/>Spaces, the Times of the Motion &longs;hall be to <lb/>one another as the &longs;aid Spaces.</s></p><p type="main"> <s><emph type="italics"/>Let the Moveable by an Equable Motion with the &longs;ame Velocity paß <lb/>the two Spaces A B and B C: and let D E be the Time of the Moti­<lb/>on thorow A B; and let the Time of the Motion thorow B C be E F <lb/>I &longs;ay that the Time D E to the Time E F, is as the Space A B to the <lb/>Space B C. </s> <s>Protract the Spaces and Times on both &longs;ides, towards <lb/>G H and I K, and in A G take any number of Spaces equal to A B,<emph.end type="italics"/><lb/><figure id="id.069.01.130.1.jpg" xlink:href="069/01/130/1.jpg"/><lb/><emph type="italics"/>and in D I the like number of Times equal to D E. Again, in C H take <lb/>any number of Spaces equal to B C, and in F K take the &longs;ame number <lb/>of Times equal to the Time E F. </s> <s>This done, the Space B G will con­<lb/>tain ju&longs;t as many Spaces equal to B A, as the Time E I containeth <lb/>Times equal to E D, equimultiplied according to what ever Rate; And <lb/>likewi&longs;e the Space B H will contain as many Spaces equal to B C, as<emph.end type="italics"/><pb xlink:href="069/01/131.jpg" pagenum="128"/><emph type="italics"/>the Time K E containeth Times equal to F E, at what ever rate equi­<lb/>multiplied. </s> <s>And fora&longs;much as D E is the Time of the Motion thorow <lb/>A B, the whole Time E I, &longs;hall be the Time of the whole Space of the <lb/>Motion thorow B G, by rea&longs;on that the Motion is Equable, and that the <lb/>number of the Times in E I equal to D E, is the &longs;ame with the number <lb/>of Spaces in B G, equal to B A: For the &longs;ame rea&longs;on E K is the Time <lb/>of the Motion thorow H B. </s> <s>Now in regard the Motion is Equable, if the <lb/>Space G B were equal to H B, the Time I E would be equal to E K: <lb/>and if G B be greater than B H, I E &longs;hall likewi&longs;e be greater than E K: <lb/>and if le&longs;&longs;er, le&longs;&longs;er. </s> <s>They are therefore four Magnitudes; A B the fir&longs;t, <lb/>B C the &longs;econd, D E the third, and E F the Fourth; and the fir&longs;t <lb/>and third, to wit, the Space A B, and Time D E, there were taken the <lb/>Time I E, and the Space G B equimultiple, according to any multi­<lb/>plication; and it hath been demon&longs;trated that the&longs;e do at once either <lb/>equal, or fall &longs;hort of, or el&longs;e exceed the Time E K, and Space B H, <lb/>which are equimultiple of the &longs;econd and fourth: Therefore the fir&longs;t <lb/>bath to the &longs;econd, to wit the Space A B to the Space B C, the &longs;ame <lb/>proportion that the third hath to the fourth, to wit, the Time D E to <lb/>the Time E F. </s> <s>Which was to be demon&longs;trated.<emph.end type="italics"/></s></p><p type="head"> <s>THEOR. II. PROP. II.</s></p><p type="main"> <s>If a Moveable in equal Times pa&longs;&longs;e thorow two <lb/>Spaces, the &longs;aid Spaces will be to each other, <lb/>as the Velocities. </s> <s>And if the Spaces are to each <lb/>other as the Velocities, the Times will be <lb/>equal.</s></p><p type="main"> <s><emph type="italics"/>Let us &longs;uppo&longs;e A B and B C in the former Figure, to be two <lb/>Spaces pa&longs;t, by the Moveable in equal times; the Space A B with <lb/>the Velocity D E, and the Space B C with the Velocity E F. </s> <s>I <lb/>&longs;ay, that the Space A B is to the Space B C, as the Velocity D E is to <lb/>the Velocity E F: and thus I prove it. </s> <s>Take as before, of the Spaces <lb/>and Velocities equi-multiples, accordieg to any what ever Rate, &longs;ci­<lb/>licet G B and I E, of A B and D E, and likewi&longs;e H B and K E, of <lb/>B C and E F: It may be concluded as above, that G B and I E are <lb/>both at once either equal to, or fall &longs;hort of, or el&longs;e exceed the equi-mul­<lb/>tiples of D H and E K. </s> <s>Therefore the Propo&longs;ition is proved.<emph.end type="italics"/></s></p><pb xlink:href="069/01/132.jpg" pagenum="129"/><p type="head"> <s>THEOR. III. PROP. III.</s></p><p type="main"> <s>The Times in which the &longs;ame Space is pa&longs;t tho­<lb/>row by unequal Velocities, have the &longs;ame pro­<lb/>portion to each other as their Velocities contra­<lb/>rily taken.</s></p><p type="main"> <s><emph type="italics"/>Let the two unequal Velocities be A the greater, and B the le&longs;&longs;e: <lb/>and according to both the&longs;e let a Motion be made thorow the &longs;ame <lb/>Space C D. </s> <s>I &longs;ay the Time in which the Velocity A pa&longs;&longs;eth the <lb/>Space C D, &longs;hall be to the Time in which the Velocity B pa&longs;&longs;eth the <lb/>&longs;aid Space, as the Velocity B to the Velocity A. </s> <s>As A is to B, &longs;o let <lb/>C D be to C E: Then, by the <lb/>former Propo&longs;ition, the Time in<emph.end type="italics"/><lb/><figure id="id.069.01.132.1.jpg" xlink:href="069/01/132/1.jpg"/><lb/><emph type="italics"/>which the Velocity A pa&longs;&longs;eth <lb/>C D, &longs;hall be the &longs;ame with <lb/>the Time in which B pa&longs;&longs;eth <lb/>C E. </s> <s>But the Time in which <lb/>the Velocity B pa&longs;&longs;eth C E, is <lb/>to the Time in which it pa&longs;&longs;eth C D, as C E is to C D: Therefore <lb/>the Time in which the Velocity A pa&longs;&longs;eth C D, is to the Time in which <lb/>the Velocity B pa&longs;&longs;eth the &longs;ame C D, as C E is to C D; that is, the Ve­<lb/>locity B is to the Velocity A: Which was to be proved.<emph.end type="italics"/></s></p><p type="head"> <s>THEOR. IV. PROP. IV.</s></p><p type="main"> <s>If two Moveables move with an Equable Mo­<lb/>tion, but with unequal Velocities, the Spaces <lb/>which they pa&longs;&longs;e in unequal Times, are to each <lb/>other in a proportion compounded of the pro­<lb/>portion of the Velocities, and of the propor­<lb/>tion of the Times.</s></p><p type="main"> <s><emph type="italics"/>Let the two Moveables moving with an Equable Motion, be E and <lb/>F: And let the proportion of the Velocity of the Moveable E be <lb/>to the Velocity of the Moveable F, as A is to B: And let the Time <lb/>in which E is moved, be unto the Time in which F is moved, as C is <lb/>to D. </s> <s>I &longs;ay the Space pa&longs;&longs;ed by E, with the Velocity A in the Time C, is to <lb/>the Space pa&longs;&longs;ed by F, with the Velocity B in the Time D, in a proportion <lb/>compounded of the proportion of the Velocity A to the Velocity B, and of<emph.end type="italics"/><pb xlink:href="069/01/133.jpg" pagenum="130"/><emph type="italics"/>the proportion of the Time C to the Time D. </s> <s>Let the Space pa&longs;&longs;ed by the <lb/>Moveable E, with the Velocity A in the Time C, be G: And as the <lb/>Velocity A is to the Velocity B, <lb/><figure id="id.069.01.133.1.jpg" xlink:href="069/01/133/1.jpg"/><lb/>&longs;o let G be to I: And as the <lb/>Time C is to the Time D, &longs;o <lb/>let I be to L: It is manife&longs;t, <lb/>that I is the Space pa&longs;&longs;ed by F <lb/>in the &longs;ame Time in which E <lb/>pa&longs;&longs;eth thorow G; &longs;eeing that <lb/>the Spaces G and I are as the <lb/>Velocities A and B; and &longs;eeing that as the Time C is to the Time D, &longs;o <lb/>is I unto L; and &longs;ince that I is the Space pa&longs;&longs;ed by the Moveable F in the <lb/>Time C: Therefore L &longs;hall be the Space that F pa&longs;&longs;eth in the Time D, <lb/>with the Velocity B: But the proportion of G to L, is compounded of the <lb/>proportions of G to I, and of I to L; that is, of the proportions of the <lb/>Velocity A to the Velocity B, and of the Time C to the Time D: <lb/>Therefore the Propo&longs;ition is demon&longs;trated.<emph.end type="italics"/></s></p><p type="head"> <s>THEOR. V. PROP. V.</s></p><p type="main"> <s>If two Moveables move with an Equable Motion, <lb/>but with unequal Velocities, and if the Spaces <lb/>pa&longs;&longs;ed be al&longs;o unequal, the Times &longs;hall be to <lb/>each other in a proportion compounded of the <lb/>proportion of the Spaces, and of the proporti­<lb/>on of the Velocities contrarily taken.</s></p><p type="main"> <s><emph type="italics"/>Let A and B be the two Moveables, and let the Velocity of A be to <lb/>the Velocity of B, as V to T, and let the Spaces pa&longs;&longs;ed, be as S to <lb/>R. </s> <s>I &longs;ay the proportion of the Time in which A is moved to the <lb/>Time in which B is moved, &longs;hall be compounded of the proportions of the <lb/>Velocity T to the Velocity V, and of the Space S to the Space R. </s> <s>Let C be <lb/>the Time of the Motion A;<emph.end type="italics"/><lb/><figure id="id.069.01.133.2.jpg" xlink:href="069/01/133/2.jpg"/><lb/><emph type="italics"/>and as the Velocity T is to <lb/>the Velocity V, &longs;o let the <lb/>Time C be to the Time E: <lb/>And for a&longs;much as C is the <lb/>Time in which A with <lb/>the Velocity V pa&longs;&longs;eth the <lb/>Space S; and that the <lb/>Time C is to the Time E, as the Velocity T of the Moveable B is to the <lb/>Velocity V, E &longs;hall be the Time in which the Moveable B would pa&longs;&longs;e<emph.end type="italics"/><pb xlink:href="069/01/134.jpg" pagenum="131"/><emph type="italics"/>the &longs;ame Space S. </s> <s>Again as the Space S is to the Space R, &longs;o let the <lb/>Time E be to the Time G: Therefore G is the Time in which B would <lb/>pa&longs;&longs;e the Space R. </s> <s>And becau&longs;e the proportion of C to G is compounded <lb/>of the proportions of C to E, and of E to G; And &longs;ince the proportion <lb/>of C to E is the &longs;ame with that of the Velocities of the Moveables A and <lb/>B contrarily taken; that is, with that of T and V; And the proportion <lb/>of E to G is the &longs;ame with the proportion of the Spaces S and R: There­<lb/>fore the Propo&longs;ition is demon&longs;trated.<emph.end type="italics"/></s></p><p type="head"> <s>THEOR. VI. PROP. VI.</s></p><p type="main"> <s>If two Moveables move with an Equable Motion, <lb/>the proportion of their Velocities &longs;hall be com­<lb/>pounded of the proportion of the Spaces pa&longs;­<lb/>&longs;ed, and of the proportion of the Times con­<lb/>trarily taken.</s></p><p type="main"> <s><emph type="italics"/>Let A and B be the two Moveables moving with an Equable <lb/>Motion; and let the Spaces by them pa&longs;&longs;ed, be as V to T; and <lb/>let the Times be as S to R. </s> <s>I &longs;ay that the proportion of the Ve­<lb/>locity of the Moveable A, to that of the Velocity of B, &longs;hall be <lb/>compounded of the proportions of the Space V to the Space T, and <lb/>of the Time R to the Time S. </s> <s>Let C be the Velocity with which the <lb/>Moveable A pa&longs;&longs;eth the Space V in the Time S: And let the Velocity C <lb/>be to the Velo-<emph.end type="italics"/><lb/><figure id="id.069.01.134.1.jpg" xlink:href="069/01/134/1.jpg"/><lb/><emph type="italics"/>city E, as the <lb/>Space V is to <lb/>the Space T; <lb/>And E &longs;hall <lb/>be the Veloci­<lb/>ty with which <lb/>the Moveable <lb/>B pa&longs;&longs;eth the Space T in the Time S: Again, let the Velocity E be to the <lb/>other Velocity G, as the Time R is to the Time S; And G &longs;hall be the <lb/>Velocity with which the Moveable B pa&longs;&longs;eth the Space T in the Time R. <lb/></s> <s>We have therefore the Velocity C, wherewith the Moveable A pa&longs;&longs;eth <lb/>the Space V in the Time S; and the Velocity G, wherewith the Move­<lb/>able B pa&longs;&longs;eth the Space T in the Time R: And the proportion of C to <lb/>G is compounded of the proportions of C to E and of E to G: But the <lb/>proportion of C to E, is &longs;uppo&longs;ed the &longs;ame with that of the Space V to <lb/>the Space T; and the proportion of E to G, is the &longs;ame with that of R <lb/>to S: Therefore the Propo&longs;ition is manifest.<emph.end type="italics"/><pb xlink:href="069/01/135.jpg" pagenum="132"/><arrow.to.target n="marg1091"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1091"></margin.target>* That is the A­<lb/>cademick, <emph type="italics"/>i. </s> <s>e. <lb/></s> <s>Galileus.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>This that we have read, is what our ^{*} <emph type="italics"/>Author<emph.end type="italics"/> hath written <lb/>of the Equable Motion. </s> <s>We will pa&longs;s therefore to a more &longs;ubtil and <lb/>new Contemplation touching the Motion Naturally Accelerate: <lb/>and behold here the Title and Introduction.</s></p><p type="head"> <s>OF MOTION <lb/>NATVRALLY ACCELERATE.</s></p><p type="main"> <s><emph type="italics"/>In the former Book we have con&longs;idered the Accidents which ac­<lb/>company Equable Motion; we are now to treat of another kind of <lb/>Motion which we call Accelerate. </s> <s>And fir&longs;t it will be expedient to <lb/>find out and explain a Definition be&longs;t agreeing to that which Nature <lb/>makes u&longs;e of. </s> <s>For though it be not nconvenient to feign a Motion at plea­<lb/>&longs;ure, and then to con&longs;ider the Accidents that attend it (as tho&longs;e have <lb/>done, who having framed in their imagination Helixes and Conchoi­<lb/>des, which are Lines ari&longs;ing from certain Motions, although not u&longs;ed <lb/>by Nature, and upon that Suppo&longs;ition have laudably demon&longs;trated the <lb/>Symptomes thereof) yet in regard that Nature maketh u&longs;e of a certain <lb/>kind of Acceleration in the de&longs;cent of Grave Bodies, we are re&longs;olved to <lb/>&longs;earch out and contemplate the pa&longs;&longs;ions thereof, and &longs;ee whether the <lb/>Definition that we are about to produce of this our Accelerate Motion, <lb/>doth aptly and congruou&longs;ly &longs;ute with the E&longs;&longs;ence of Motion Naturally <lb/>Accelerate. </s> <s>After many long and laborious Studies we have found out <lb/>a Definition which &longs;eemeth to expre&longs;&longs;e the true nature of this Accelerate <lb/>Motion, in regard that all the Natural Experiments that fall under <lb/>the Ob&longs;ervation of our Sen&longs;es, do agree with tho&longs;e its properties that <lb/>we intend anon to demon&longs;trate. </s> <s>In this Di&longs;qui&longs;ition we have been a&longs;&longs;i­<lb/>&longs;ted, and as it were led by the hand by that ob&longs;ervation of the u&longs;ual <lb/>Method and common procedure of Nature her &longs;elf in her other Operati­<lb/>ons, wherein &longs;he con&longs;tantly makes u&longs;e of the Fir&longs;t, Simple&longs;t, and Ea­<lb/>&longs;ie&longs;t Means that are: for I believe that no man can think that Swim­<lb/>ming or flying can be performed in a more &longs;imple or ea&longs;ie way, than that <lb/>which Fi&longs;hes and Birds do u&longs;e out of a Natural In&longs;tinct. </s> <s>Why there­<lb/>fore &longs;hall not I be per&longs;waded, that, when I &longs;ee a Stone to acquire conti­<lb/>nually new additions of Velocity in its de&longs;cending from its Re&longs;t out of &longs;ome <lb/>high place, this encrea&longs;e made in the &longs;imple&longs;t ea&longs;ie&longs;t and mo&longs;t obvious <lb/>manner that we can imagine? </s> <s>Now if we &longs;eriou&longs;ly examine all the ways <lb/>that can be devi&longs;ed, we &longs;hall find no encrea&longs;es, no acqui&longs;itions <lb/>le&longs;&longs;e intricate or more intelligible than that which ever encrea&longs;eth or <lb/>makes its additions after the &longs;ame manner. </s> <s>This appeareth by the great<emph.end type="italics"/><lb/>Affinity <emph type="italics"/>that is between Time and Motion. </s> <s>For as the Equability or <lb/>Vniformity of Motion is defined and expre&longs;&longs;ed by the Equability of the<emph.end type="italics"/><pb xlink:href="069/01/136.jpg" pagenum="133"/><emph type="italics"/>Times and Spaces, (for we call that Motion or Lation Equable, by which <lb/>equal Spaces are pa&longs;t in equal Times) &longs;o by the &longs;ame Equability of the <lb/>parts of Time, we may perceive, that the encrea&longs;e of Celerity in the Natu­<lb/>ral Motion of Grave Bodies, is made after a Simple and plain manner; <lb/>conceiving in our Mind that their Motion is continually accelerated uni­<lb/>formly and at the &longs;ame Rate, whil&longs;t equal additions of Celerity are <lb/>conferred upon them in all equal Times. </s> <s>So that taking any equal par­<lb/>ticles of Time beginning from the fir&longs;t In&longs;tant in which the Moveable <lb/>departeth from Re&longs;t, and entereth upon its De&longs;cent, the Degree of <lb/>Velocity acquired in the fir&longs;t and &longs;econd Particles of Time, is double the <lb/>degree of Velocity that the Moveable acquired in the fir&longs;t Particle: and <lb/>the degree of Velocity that it acquireth in three Particles, is triple, and <lb/>that in four quadruple to the &longs;ame Degree of the fir&longs;t Time: As, for <lb/>our better under&longs;tanding, if a Moveable &longs;hould continue its Motion <lb/>according to the degree or moment of Velocity acquired in the fir&longs;t Parti­<lb/>cle of Time, and &longs;hould extend its cour&longs;e equably with that &longs;ame De­<lb/>gree; this Motion would be twice as &longs;low as that which it would obtain <lb/>according to the degree of Velocity acquired in two Particles of Time: <lb/>So that it will not be improper if we under&longs;tand the Intention of the Ve­<lb/>locity, to proceed according to the Exten&longs;ion of the Time. </s> <s>From whence <lb/>we may frame this Definition of the Motion of which we are about to <lb/>treat.<emph.end type="italics"/></s></p><p type="head"> <s>DEFINITION.</s></p><p type="main"> <s>Motion Accelerate in an Equable or Vniform <lb/>Proportion, I call that which departing from <lb/>Re&longs;t, &longs;uperaddeth equal moments of Velocity <lb/>in equal Times.</s></p><p type="main"> <s>SAGR. </s> <s>Though it were Irrational for me to oppo&longs;e this or any <lb/>other Definition a&longs;&longs;igned by any what&longs;oever Author, they being all <lb/>Arbitrary, yet I may very well, without any offence, que&longs;tion whe­<lb/>ther this Definition, which is under&longs;tood and admitted in Ab&longs;tract, <lb/>doth &longs;ute, agree, and hold true in that &longs;ort of Accelerate Motion, <lb/>which Grave Bodies de&longs;cending naturally do exerci&longs;e. </s> <s>And becau&longs;e <lb/>the Authour &longs;eemeth to promi&longs;e us, that the Natural Motion of <lb/>Grave Bodies is &longs;uch as he hath defined it, I could wi&longs;h that &longs;ome <lb/>Scruples were removed that trouble my mind; that &longs;o I might apply <lb/>my &longs;elf afterwards with greater attention to the Proportions and <lb/>Demon&longs;trations which are expected.</s></p><p type="main"> <s>SALV. </s> <s>I like well, that you and <emph type="italics"/>Simplicius<emph.end type="italics"/> do propound <lb/>Doubts as they come in the way: which I do imagine will be the <pb xlink:href="069/01/137.jpg" pagenum="134"/>&longs;ame that I my &longs;elf did meet with when I fir&longs;t read this Treati&longs;e, <lb/>and that, either were re&longs;olved by conferring with the Author, or <lb/>removed by my own con&longs;idering of them.</s></p><p type="main"> <s>SAGR. </s> <s>Whil&longs;t I am fancying to my &longs;elf a Grave De&longs;cending <lb/>Moveable to depart from Re&longs;t, that is from the privation of all <lb/>Velocity, and to enter into Motion, and in that to go encrea­<lb/>&longs;ing, according to the proportion after which the Time encrea&longs;eth <lb/>from the fir&longs;t in&longs;tant of the Motion; and to have <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> in eight <lb/>Pul&longs;ations, acquired eight degrees of Velocity, of which in the <lb/>fourth Pul&longs;ation it had gained four, in the &longs;econd two, in the <lb/>fir&longs;t one, Time being &longs;ubdivi&longs;ible <emph type="italics"/>in infinitum,<emph.end type="italics"/> it followeth, that <lb/>the Antecedent Velocity alwayes dimini&longs;hing at that Rate, there <lb/>will bt no degree of Velocity &longs;o &longs;mall, or, if you will, of Tardity <lb/>&longs;o great, in which the &longs;aid Moveable is not found to be con&longs;ti­<lb/>tuted, after its departure from infinite Tardity, that is, from <lb/>Re&longs;t. </s> <s>So that if that degree of Velocity which it had at four Pul­<lb/>&longs;ations of Time, was &longs;uch, that maintaining it Equable, it would <lb/>have run two Miles in an hour, and with the degree of Velocity <lb/>that it had in the &longs;econd Pul&longs;ation, it would have gone one mile <lb/>an hour, it mu&longs;t be granted, that in the In&longs;tants of Time neeter <lb/>and neerer to its fir&longs;t In&longs;tant of moving from Re&longs;t, it is &longs;o &longs;low, <lb/>as that (continuing to move with that Tardity) it would not have <lb/>pa&longs;&longs;ed a Mile in an hour, nor in a day, nor in a year, nor in a <lb/>thou&longs;and; nay, nor have gone one &longs;ole foot in a greater time: <lb/>An accident to which me thinks the Imagination but very unea­<lb/>&longs;ily accords, &longs;eeing that Sen&longs;e &longs;heweth us, that a Grave Falling <lb/>Body commeth down &longs;uddenly, and with great Velocity.</s></p><p type="main"> <s>SALV. </s> <s>This is one of tho&longs;e Doubts that al&longs;o fell in my way <lb/>upon my fir&longs;t thinking on this affair, but not long after I remo­<lb/>ved it: and that removal was the effect of the &longs;elf &longs;ame Expe­<lb/>riment which at pre&longs;ent &longs;tarts it to you. </s> <s>You &longs;ay, that in your <lb/>opinion, Experience &longs;heweth that the Moveable hath no &longs;ooner <lb/>departed from Re&longs;t, but it entereth into a very notable Velocity: <lb/>and I &longs;ay, that this very Experiment proves it to us, that the fir&longs;t <lb/>Impetus's of the Cadent Body, although it be very heavy, are <lb/>mo&longs;t &longs;lack and &longs;low. </s> <s>Lay a Grave Body upon &longs;ome yielding mat­<lb/>ter, and let it continue upon it till it hath pre&longs;&longs;ed into it as far as <lb/>it can with its &longs;imple Gravity; it is manife&longs;t, that rai&longs;ing it a yard <lb/>or two, and then letting it fall upon the &longs;ame matter, it &longs;hall <lb/>with its percu&longs;&longs;ion make a new pre&longs;&longs;ure, and greater than that <lb/>made at fir&longs;t by its meer weight: and the effect &longs;hall be cau&longs;ed <lb/>by the falling Moveable conjoyned with the Velocity acquired in <lb/>the Fall: which impre&longs;&longs;ion &longs;hall be greater and greater, accord­<lb/>ing as the Percu&longs;&longs;ion &longs;hall come from a greater height; that is, <lb/>according as the Velocity of the Percutient &longs;hall be greater. </s> <s>We <pb xlink:href="069/01/138.jpg" pagenum="135"/>may therefore without mi&longs;take conjecture the quantity of the Ve­<lb/>locity of a falling heavy Body; by the quality and quantity of <lb/>the Percu&longs;&longs;ion. </s> <s>But tell me Sirs, that Beetle which being let fall <lb/>upon a Stake from an height of four yards, driveth it into the <lb/>ground, <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> four inches, comming from an height of two yards, <lb/>&longs;hall drive it much le&longs;&longs;e, and le&longs;&longs;e from an height of one, and <lb/>le&longs;&longs;e from a foot; and la&longs;tly lifting it up an inch, what will it do <lb/>more than if without any blow it were laid upon it? </s> <s>Certainly <lb/>but very little, and the operation would be wholly impercep­<lb/>tible, if it were rai&longs;ed the thickne&longs;&longs;e of a leaf. </s> <s>And becau&longs;e the <lb/>effect of the Percu&longs;&longs;ion is regulated by the Velocity of the Percu­<lb/>tient, who will que&longs;tion but that the Motion is very &longs;low, and <lb/>the Velocity extreme &longs;mall, where its operation is impercep­<lb/>tible? </s> <s>See now of what power Truth is, &longs;ince the &longs;ame Experi­<lb/>ment that &longs;eemed at the fir&longs;t blu&longs;h to hold forth one thing, be­<lb/>ing better con&longs;idered, a&longs;certains us of the contrary. </s> <s>But without <lb/>having recour&longs;e to that Experiment (which without doubt is mo&longs;t <lb/>per&longs;wa&longs;ive) me-thinks that it is not hard to penetrate &longs;uch a <lb/>Truth as this by meer Di&longs;cour&longs;e. </s> <s>We have an heavy &longs;tone &longs;u­<lb/>&longs;tained in the Air at Re&longs;t: let it be di&longs;engaged from its uphol­<lb/>der, and &longs;et at liberty; and, as being more grave than the Air, it <lb/>goeth de&longs;cending downwards, and that not with a Motion Equa­<lb/>ble, but &longs;low in the beginning, and continually afterwards ac­<lb/>celerate: and &longs;eeing that the Velocity is Augmentable and Di­<lb/>mini&longs;hable <emph type="italics"/>in infinitum,<emph.end type="italics"/> what Rea&longs;on &longs;hall per&longs;wade me, that that <lb/>Moveable departing from an infinite Tardity (for &longs;uch is Re&longs;t) <lb/>entereth immediately into ten degrees of Velocity, rather than in <lb/>one of four, or in this more than in one of two, of one, of half <lb/>one, or of the hundredth part of one; and to be &longs;hort, in all <lb/>the infinite le&longs;&longs;er? </s> <s>Pray you hear me. </s> <s>I do not think that you <lb/>would &longs;cruple to grant me, that the acqui&longs;t of the Degrees of Ve­<lb/>locity of the falling Stone may be made with the &longs;ame Order as <lb/>is the Diminution and lo&longs;&longs;e of the &longs;ame degrees, when with an <lb/>impellent Virtue it is driven upwards to the &longs;ame height: But if <lb/>that be &longs;o, I do not &longs;ee how it can be &longs;uppo&longs;ed that in the diminu­<lb/>tion of the Velocity of the a&longs;cendent Stone, &longs;pending it all, it <lb/>can come to the &longs;tate of Re&longs;t before it hath pa&longs;&longs;ed thorow all the <lb/>degrees of Tardity.</s></p><p type="main"> <s>SIMP. </s> <s>But if the greater and greater degrees of Tardity are <lb/>infinite, it &longs;hall never &longs;pend them all; &longs;o that the a&longs;cendent <lb/>Grave will never attain to Re&longs;t, but will move <emph type="italics"/>ad infinitum,<emph.end type="italics"/> &longs;till <lb/>retarding: a thing which we &longs;ee not to happen.</s></p><p type="main"> <s>SALV. </s> <s>This would happen, <emph type="italics"/>Simplicius,<emph.end type="italics"/> in ca&longs;e the Moveable <lb/>&longs;hould &longs;tay for &longs;ome time in each degree: but it pa&longs;&longs;eth thorow <lb/>them, without &longs;taying longer than an in&longs;tant in any of them. <pb xlink:href="069/01/139.jpg" pagenum="136"/>And becau&longs;e in every quantitative Time, though never &longs;o &longs;mall, <lb/>there are infinite In&longs;tants, therefore they are &longs;ufficient to an&longs;wer <lb/>to the infinite degrees of Velocity dimini&longs;hed. </s> <s>And that the <lb/>a&longs;cendent Grave Body per&longs;i&longs;ts not for any quantitative Time in <lb/>one and the &longs;ame degree of Velocity, may thus be made out: <lb/>Becau&longs;e, a certain quantitative Time being a&longs;&longs;igned it in the fir&longs;t <lb/>in&longs;tant of that Time, and likewi&longs;e in the la&longs;t, the Moveable <lb/>&longs;hould be found to have one and the &longs;ame degree of Velocity, it <lb/>might by this &longs;econd degree be likewi&longs;e driven upwards &longs;uch an­<lb/>other Space, like as from the fir&longs;t it was tran&longs;ported to the &longs;e­<lb/>cond; and by the &longs;ame rea&longs;on it would pa&longs;&longs;e from the &longs;econd to <lb/>the third, and, in &longs;hort, would continue its Motion Uniform <emph type="italics"/>ad <lb/>infinitum.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>From this Di&longs;cour&longs;e, as I conceive, one might derive a <lb/>very appo&longs;ite Rea&longs;on of the Que&longs;tion controverted among&longs;t Philo. <lb/></s> <s>&longs;ophers, Touching what &longs;hould be the Cau&longs;e of the acceleration <lb/>of the Natural Motion of Grave Moveables. </s> <s>For when I confider <lb/>in the Grave Body driven upwards, its continual Diminution of <lb/>that Virtue impre&longs;&longs;ed upon it by the Projicient, which &longs;o long as <lb/>it was &longs;uperiour to that other contrary one of Gravity, forced it <lb/>upwards, this and that being come to an <emph type="italics"/>Equilibrium,<emph.end type="italics"/> the Move­<lb/>able cea&longs;eth to ri&longs;e any higher, and pa&longs;&longs;eth thorow the &longs;tate of <lb/>Re&longs;t, in which the <emph type="italics"/>Impetus<emph.end type="italics"/> impre&longs;&longs;ed is not annihilated, but one­<lb/>ly that exce&longs;&longs;e is &longs;pent, which it before had above the Gravity of <lb/>the Moveable, whereby prevailing over the &longs;ame, it did drive <lb/>it upwards. </s> <s>And the Diminution of this forrein <emph type="italics"/>Impetus<emph.end type="italics"/> continu­<lb/>ing, and con&longs;equently the advantage beginning to be on the part <lb/>of the Gravity, the De&longs;cent al&longs;o beginneth but &longs;low, in regard <lb/>of the oppo&longs;ition of the Virtue impre&longs;&longs;ed, a con&longs;iderable part of <lb/>which &longs;till remaineth in the Moveable: but becau&longs;e it doth go <lb/>continually dimini&longs;hing, and is &longs;till with a greater and greater <lb/>proportion overcome by the Gravity, hence ari&longs;eth the continual <lb/>Acceleration of the Motion.</s></p><p type="main"> <s>SIMP. </s> <s>The conceit is witty, but more &longs;ubtil than &longs;olid: for in <lb/>ca&longs;e it were concludent, it &longs;alveth onely tho&longs;e Natural Motions <lb/>to which a Violent Motion preceded, in which part of the extern <lb/>Virtue &longs;till remains in force: but where there is no &longs;uch remaining <lb/>impul&longs;e, as where the Moveable departeth from a long Quie&longs;­<lb/>cence, the &longs;trength of your whole Di&longs;cour&longs;e vani&longs;heth.</s></p><p type="main"> <s>SAGR. </s> <s>I believe that you are in an Errour, and that this Di­<lb/>&longs;tinction of Ca&longs;es which you make, is needle&longs;&longs;e, or, to &longs;ay bet­<lb/>ter, <emph type="italics"/>Null.<emph.end type="italics"/> Therefore tell me, whether may there be impre&longs;&longs;ed <lb/>on the Project by the Projicient &longs;ometimes much, and &longs;ometimes <lb/>little Vertue; &longs;o as that it may be &longs;tricken upwards an hundred <lb/>yards, and al&longs;o twenty, or four, or one?</s></p><pb xlink:href="069/01/140.jpg" pagenum="137"/><p type="main"> <s>SIMP. </s> <s>No doubt but there may.</s></p><p type="main"> <s>SAGR. </s> <s>And no le&longs;&longs;e po&longs;&longs;ible is it, that the &longs;aid Virtue impre&longs;&longs;ed <lb/>&longs;hall &longs;o little &longs;eperate the Re&longs;i&longs;tance of the Gravity, as not to <lb/>rai&longs;e the Project above an inch: and finally the Virtue of the <lb/>Projicient may be onely &longs;o much, as ju&longs;t to equalize and com­<lb/>pen&longs;ate the Re&longs;i&longs;tance of the Gravity, &longs;o as that the Moveable <lb/>is not driven upwards, but onely &longs;u&longs;tained. </s> <s>So that when you <lb/>hold a Stone in your hand, what el&longs;e do you, but impre&longs;&longs;e on it <lb/>&longs;o much Virtue impelling upwards, as is the faculty of its Gra­<lb/>vity drawing downwards? </s> <s>And this your Virtue, do you not <lb/>continue to keep it impre&longs;&longs;ed on the Stone all the time that you <lb/>hold it in your hand? </s> <s>What &longs;ay you, is it dimini&longs;hed by your <lb/>long holding it? </s> <s>And this &longs;u&longs;tention which impedeth the Stones <lb/>de&longs;cent, what doth it import, whether it be made by your hand, <lb/>or by a Table, or by a Rope, that &longs;u&longs;pends it? </s> <s>Doubtle&longs;&longs;e no <lb/>thing at all. </s> <s>Conclude with your &longs;elf therefore, <emph type="italics"/>Simplicius,<emph.end type="italics"/> that <lb/>the precedence of a long, a &longs;hort, or a Momentary Re&longs;t to the <lb/>Fall of the Stone, makes no alteration at all, &longs;o that the Stone <lb/>&longs;hould not alwaies depart affected with &longs;o much Virtue contrary <lb/>to Gravity, as did exactly &longs;uffice to have kept it in Re&longs;t.</s></p><p type="main"> <s>SALV. </s> <s>I do not think it a &longs;ea&longs;onable time at pre&longs;ent to enter <lb/>upon the Di&longs;qui&longs;ition of the Cau&longs;e of the Acceleration of Natu­<lb/>ral Motion: touching which &longs;undry Philo&longs;ophers have produced <lb/>&longs;undry opinions: &longs;ome reducing it to the approximation unto <lb/>the Center others to the le&longs;&longs;e parts of the <emph type="italics"/>Medium<emph.end type="italics"/> &longs;ucce&longs;&longs;ively re­<lb/>maining to be perforated; others to a certain Extru&longs;ion of the <lb/>Ambient <emph type="italics"/>Medium,<emph.end type="italics"/> which in reuniting upon the back of the <lb/>Moveable, goeth driving and continually thru&longs;ting it; which <lb/>Fancies, and others of the like nature, it would be nece&longs;&longs;ary to <lb/>examine, and with &longs;mall benefit to an&longs;wer. </s> <s>It &longs;erveth our Au­<lb/>thours turn at the pre&longs;ent, that we under&longs;tand that he will de­<lb/>clare and demon&longs;trate to us &longs;ome Pa&longs;&longs;ions of an Accelerate Mo­<lb/>tion (be the Cau&longs;e of its Acceleration what it will) &longs;o as that the <lb/>Moments of its Velocity do go encrea&longs;ing, after its departure from <lb/>Re&longs;t with that mo&longs;t &longs;imple proportion wherewith the Continua­<lb/>tion of the Time doth encrea&longs;e: which is as much as to &longs;ay, that <lb/>in equal Times there are made equal additaments of Velocity. <lb/></s> <s>And if it &longs;hall be found, that the Accidents that &longs;hall hereafter <lb/>be demon&longs;trated, do hold true in the Motion of Naturally De­<lb/>&longs;cendent and Accelerate Grave Moveables, we may account, <lb/>that the a&longs;&longs;umed Definition taketh in that Motion of Grave Bo­<lb/>dies, and that it is true, that their Acceleration doth encrea&longs;e ac­<lb/>cording as the Time and Duration of the Motion encrea&longs;eth.</s></p><p type="main"> <s>SAGR. </s> <s>By what as yet is &longs;et before my Intellectuals, it appears <lb/>to me that one might with (haply) more plainne&longs;&longs;e define, and yet <pb xlink:href="069/01/141.jpg" pagenum="138"/>never alter the Conceit; &longs;aying that, A Motion uniformly accele­<lb/>rate is that in which the Velocity goeth encrea&longs;ing according as <lb/>the Space encrea&longs;eth that is pa&longs;&longs;ed thorow: So that, for example, <lb/>the degree of Velocity acquired by the Moveable in a de&longs;cent of <lb/>four yards &longs;hould be double to that that it would have after it had <lb/>de&longs;cended a Space of two, and this double to that acquired in the <lb/>Space of the fir&longs;t Yard. </s> <s>For I do not think that it can be doubted, <lb/>but that that Grave Moveable which falleth from an height of &longs;ix <lb/>yards hath, and percu&longs;&longs;eth with an <emph type="italics"/>Impetus<emph.end type="italics"/> double to that which <lb/>it had when it had de&longs;cended three yards, and triple to that which <lb/>it had at two, and &longs;extuple to that had in the Space of one.</s></p><p type="main"> <s>SALV. </s> <s>I comfort my &longs;elf in that I have had &longs;uch a Companion <lb/>in my Errour: and I will tell you farther, that your Di&longs;cour&longs;e hath <lb/>&longs;o much of likelihood and probability in it, that our Author him&longs;elf <lb/>did not deny unto me, when I propo&longs;ed it to him, that he likewi&longs;e <lb/>had been for &longs;ome time in the &longs;ame mi&longs;take. <emph type="italics"/>B<emph.end type="italics"/>ut that which I af­<lb/>terwards extreamly wondred at, was to &longs;ee in four plain words, <lb/>di&longs;covered, not only the falfity, but impo&longs;&longs;ibility of two Propo&longs;i­<lb/>tions that carry with them &longs;o much of &longs;eeming truth, that having <lb/>propounded them to many, I never met with any one but did freely <lb/>admit them to be &longs;o.</s></p><p type="main"> <s>SIMP. </s> <s>Certainly I &longs;hould be of the number, and that the De­<lb/>&longs;cendent Grave Moveable <emph type="italics"/>vires acquir at eundo,<emph.end type="italics"/> encrea&longs;ing its Ve­<lb/>locity at the rate of the Space, and that the Moment of the &longs;ame <lb/>Percutient is double, coming from a double height, &longs;eem to me Pro­<lb/>po&longs;itions to be granted without any hæ&longs;itation or controver&longs;ie.</s></p><p type="main"> <s>SALV. </s> <s>And yet they are as fal&longs;e and impo&longs;&longs;ible, as that Moti­<lb/>on is made in an in&longs;tant. </s> <s>And hear a clear proof of the &longs;ame. </s> <s>In <lb/>ca&longs;e the Velocities have the &longs;ame proportion as the Spaces pa&longs;&longs;ed, <lb/>or to be pa&longs;&longs;ed, tho&longs;e Spaces &longs;hall be pa&longs;&longs;ed in equal Times: if <lb/>therefore the Velocities with which the falling Moveable pa&longs;&longs;eth <lb/>the Space of four yards, were double to the Velocities with which it <lb/>pa&longs;&longs;eth the two fir&longs;t yards (like as the Space is double to the Space) <lb/>then the Times of tho&longs;e Tran&longs;itions are equal: but the &longs;ame Move­<lb/>able's pa&longs;&longs;ing the four yards, and the two in one and the &longs;ame Time, <lb/>hath place only in In&longs;tantaneous Motion. <emph type="italics"/>B<emph.end type="italics"/>ut we &longs;ee, that the <lb/>falling grave <emph type="italics"/>B<emph.end type="italics"/>ody maketh its Motion in Time, and pa&longs;&longs;eth the two <lb/>yards in a le&longs;&longs;er than it doth the four. </s> <s>Therefore it is fal&longs;e that its <lb/>Velocity encrea&longs;eth as its Space. </s> <s>The other Propo&longs;ition is demon­<lb/>&longs;trated to be fal&longs;e with the &longs;ame per&longs;picuity. </s> <s>For that which per­<lb/>cu&longs;&longs;eth being the &longs;ame, the difference and Moment of the Percu&longs;&longs;ton <lb/>cannot be determined but by the difference of Velocity; If there­<lb/>fore the percutient, coming from a double height, make a Percu&longs;&longs;i­<lb/>on with a double Moment, it is nece&longs;&longs;ary that it &longs;trike with a dou­<lb/>ble Velocity: <emph type="italics"/>B<emph.end type="italics"/>ut the double Velocity pa&longs;&longs;eth the double Space in <pb xlink:href="069/01/142.jpg" pagenum="139"/>the &longs;ame Time; and we &longs;ee the Time of the De&longs;cent from the grea­<lb/>ter altitude to be longer.</s></p><p type="main"> <s>SAGR. </s> <s>This is too great an Evidence, too great a Facility <lb/>wherewith you manife&longs;t ab&longs;truce Conclu&longs;ions: this extream ea&longs;i­<lb/>ne&longs;s rendreth them of le&longs;&longs;e value than they were whil&longs;t they lay hid <lb/>under contrary appearances. </s> <s>I believe that the Generality of men <lb/>little pre&longs;&longs;e tho&longs;e Notions which are ea&longs;ily obtained, in compari­<lb/>&longs;on of tho&longs;e about which men make &longs;o long and inexplicable alter­<lb/>cations.</s></p><p type="main"> <s>SALV. </s> <s>To tho&longs;e which with great brevity and clarity &longs;hew the <lb/>fallacies of Propo&longs;itions that have been commonly received for <lb/>true by the generality of people, it would be a very tolerable in­<lb/>jury to return them only &longs;lighting in&longs;tead of thanks: but there is <lb/>much di&longs;plea&longs;ure and mole&longs;tation in another certain affection <lb/>&longs;ometimes found in &longs;ome men, that pretending in the &longs;ame Studies <lb/>at lea&longs;t Parity with any whom&longs;oever, do &longs;ee that they have let <lb/>pa&longs;s &longs;uch and &longs;uch for true Conclu&longs;ions, which afterwards by <lb/>another, with a &longs;hort and ea&longs;ie di&longs;qui&longs;ition, have been detected and <lb/>convicted for fal&longs;e. </s> <s>I will not call that affection Envy, that is ac­<lb/>cu&longs;tomed to convert in time to hatred and de&longs;pite again&longs;t the di&longs;­<lb/>coverers of &longs;uch Fallacies, but I will call it an itch, and a de&longs;ire to <lb/>be able rather to maintain their inveterate Errours, than to per­<lb/>mit the reception of new-di&longs;covered Truths. </s> <s>Which humour &longs;ome­<lb/>times induceth them to write in contradiction of tho&longs;e truths <lb/>which are but too perfectly known unto them&longs;elves only to keep <lb/>the Reputation of others low in the opinion of the numerous and <lb/>ill-informed Vulgar. </s> <s>Of &longs;uch fal&longs;e Conclu&longs;ions received for true, <lb/>and very ea&longs;ie to be confuted, I have heard no &longs;mall number from <lb/>our <emph type="italics"/>Academick,<emph.end type="italics"/> of &longs;ome of which I have kept account.</s></p><p type="main"> <s>SAGR. </s> <s>And you mu&longs;t not deprive us of them; but in due time <lb/>impart them to us, when a particular Meeting &longs;hall be appointed <lb/>for them. </s> <s>For the pre&longs;ent, continuing the di&longs;cour&longs;e we are about, <lb/>I think that by this time we have e&longs;tabli&longs;hed the Definition of Mo­<lb/>tion uniformly Accelerate, treated of in the en&longs;uing di&longs;cour&longs;es, <lb/>and it is this;</s></p><p type="main"> <s><emph type="italics"/>A Motion Equable, or Vniformly Accelerate, we call that which <lb/>departing from Re&longs;t &longs;uperadds equal Moments of Velocity in <lb/>equal Times.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>That Definition being confirmed, the Author asketh <lb/>and &longs;uppo&longs;eth but one only Principle to be true, namely:</s></p><pb xlink:href="069/01/143.jpg" pagenum="140"/><p type="head"> <s>SVPPOSITION.</s></p><p type="main"> <s><emph type="italics"/>I &longs;uppo&longs;e that the degrees of Velocity acquired by the <lb/>&longs;ame Moveable upon Planes of different inclinations <lb/>are equal then, when the Elevations of the &longs;aid <lb/>Planes are equal.<emph.end type="italics"/></s></p><p type="main"> <s>By the Elevation of an inclined Plane he meaneth the Per­<lb/>pendicular, which from the higher term of the &longs;aid Plane <lb/>falleth upon the Horizontal Line produced along by the <lb/>lower term of the &longs;aid Plane inclined: as for better under&longs;tanding; <lb/>the Line A B being parallel to the Horizon, upon which let the two <lb/><figure id="id.069.01.143.1.jpg" xlink:href="069/01/143/1.jpg"/><lb/>Planes C A, and C D be inclined: <lb/>the Perpendicular C B falling up­<lb/>on the Horizontal Line B A the <lb/>Author calleth the Elevation <lb/>of the Planes C A and C D; <lb/>and &longs;uppo&longs;eth that the degrees of <lb/>Velocity of the &longs;ame Moveable <lb/>de&longs;cending along the inclined Planes C A and C D, acqui­<lb/>red in the Terms A and D are equal, for that their Elevation is <lb/>the &longs;ame C B. </s> <s>And &longs;o great al&longs;o ought the degree of Velocity be <lb/>under&longs;tood to be which the &longs;ame Moveable falling from the Point <lb/>C would acquire in the term B.</s></p><p type="main"> <s>SAGR. </s> <s>The truth is, this Suppo&longs;ition hath in it &longs;o much of pro­<lb/>bability, that it de&longs;erveth to be granted without di&longs;pute, alwaies <lb/>pre&longs;uppo&longs;ing that all accidental and extern Impediments are re­<lb/>moved, and that the Planes be very Solid and Ter&longs;e, and the Move­<lb/>able in Figure mo&longs;t perfectly Rotund, &longs;o that neither the Plane, <lb/>nor the Moveable have any unevenne&longs;s. </s> <s>All Contra&longs;ts and Im­<lb/>pediments, I &longs;ay, being removed, the light of Nature dictates to <lb/>me without any difficulty, that a Ball heavy and perfectly round <lb/>de&longs;cending by the Lines C A, C D, and C B would come to the <lb/>terms A D, and B with equal <emph type="italics"/>Impetus's.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>You argue very probably; but over and above the pro­<lb/>bability, I will by an Experiment &longs;o increa&longs;e the likelihood, as that <lb/>it wants but little of being equal to a very nece&longs;&longs;ary Demon&longs;trati­<lb/>on. </s> <s>Imagine this leafe of Paper to be a Wall erect at Right-angles <lb/>to the Horizon, and at a Nail, fa&longs;tned in the &longs;ame, hang a Ball or <lb/>Plummet of Lead, weighing an ounce or two, &longs;u&longs;pended by the <lb/>&longs;mall thread A B, two or three yards long, perpendicular to the <lb/>Horizon: and on the Wall draw an Horizontal Line D C, cutting <pb xlink:href="069/01/144.jpg" pagenum="141"/>the Perpendicular A B at Right angles, which A B mu&longs;t hang two <lb/>Inches, or thereabouts, from the Wall: Then transferring the <lb/>&longs;tring A B with the Ball into C, let go the &longs;aid Ball; which you will <lb/><figure id="id.069.01.144.1.jpg" xlink:href="069/01/144/1.jpg"/><lb/>&longs;ee fir&longs;t to de&longs;cend <lb/>de&longs;cribing C B D, and <lb/>to pa&longs;s &longs;o far beyond <lb/>the Term B, that run­<lb/>ning along the Arch <lb/>B D it will ri&longs;e almo&longs;t <lb/>as high as the de&longs;igned <lb/>Parallel C D, wanting <lb/>but a very &longs;mall mat­<lb/>ter of reaching to it, <lb/>the preci&longs;e arrival thi­<lb/>ther being denied it by <lb/>the Impediment of the Air, and of the Thread. </s> <s>From which we <lb/>may truly conclude, that the <emph type="italics"/>Impetus<emph.end type="italics"/> acquired in the point B by <lb/>the Ball in its de&longs;cent along the Arch C B, was &longs;o much as &longs;ufficed <lb/>to carry it upwards along &longs;uch another Arch B D unto the &longs;ame <lb/>height: having made, and often reiterated this Experiment, let <lb/>us drive into the Wall, along which the Perpendicular A B pa&longs;&longs;eth, <lb/>another Nail, as in E or in F, which is to &longs;tand out five or &longs;ix In­<lb/>ches; and this to the end that the thread A B, returning as before <lb/>to carry back the Ball C along the Arch C B, when it is come to <lb/>B, the Thread &longs;topping at the Nail E may be con&longs;trained to move <lb/>along the Circumference B G, de&longs;cribed about the Center E: by <lb/>which we &longs;hall &longs;ee what that &longs;ame <emph type="italics"/>Impetus<emph.end type="italics"/> is able to do, which be­<lb/>fore, being conceived in the &longs;ame term B, carried the &longs;ame Move­<lb/>able along the Arch B D unto the height of the Horizontal Line <lb/>C D. Now, Sirs, you &longs;hall with delight &longs;ee the Ball carried unto <lb/>the Horizontal Line in the Point G; and the &longs;ame will happen if <lb/>the &longs;top be placed lower, as in F, where the Ball would de&longs;cribe <lb/>the Arch B I, evermore terminating its a&longs;cent exactly in the Line <lb/>C D: and in ca&longs;e the Check were &longs;o low that the overplus of the <lb/>thread beneath it cannot reach to the height of C D, (which would <lb/>happen if it were nearer to the point B than to the inter&longs;ection of <lb/>A B with the Horizontal Line C D) then the thread would <lb/>whirle and twine about the Nail. </s> <s>This experiment leaveth no <lb/>place for our doubting of the truth of the Suppo&longs;ition: for the <lb/>two Arches C B and D B being equall, and &longs;cituate alike, the <lb/>acqui&longs;t of Moment made along the De&longs;cent in the Arch C B, is <lb/>the &longs;ame with that made along the De&longs;cent in the Arch D B. </s> <s>But <lb/>the Moment acquired in <emph type="italics"/>B,<emph.end type="italics"/> along the Arch C <emph type="italics"/>B,<emph.end type="italics"/> is able to carry the <lb/>&longs;ame Moveable upwards along the Arch <emph type="italics"/>B<emph.end type="italics"/> D: Therefore the Mo­<lb/>ment acquired in the De&longs;cent D <emph type="italics"/>B<emph.end type="italics"/> is equall to that which driveth <pb xlink:href="069/01/145.jpg" pagenum="142"/>the &longs;ame Moveable along the &longs;ame Arch from <emph type="italics"/>B<emph.end type="italics"/> to D: So that ge­<lb/>nerally every Moment acquired along the De&longs;cent of an Arch is <lb/>equall to that which hath power to make the &longs;ame Moveable re­<lb/>a&longs;cend along the &longs;ame Arch: <emph type="italics"/>B<emph.end type="italics"/>ut all the Moments that make the <lb/>Moveable a&longs;cend along all the Arches <emph type="italics"/>B<emph.end type="italics"/> D, <emph type="italics"/>B<emph.end type="italics"/> G, <emph type="italics"/>B<emph.end type="italics"/> I are equal, <lb/>&longs;ince they are made by one and the &longs;ame Moment acquired along <lb/>the De&longs;cent C <emph type="italics"/>B,<emph.end type="italics"/> as Experience &longs;hews: Therefore all the Moments <lb/>that are acquired by the De&longs;cents along the Arches D <emph type="italics"/>B,<emph.end type="italics"/> G <emph type="italics"/>B,<emph.end type="italics"/> and <lb/>I <emph type="italics"/>B<emph.end type="italics"/> are equal.</s></p><p type="main"> <s>SAGR. </s> <s>Your Di&longs;cour&longs;e is in my Judgment very Rational, and <lb/>the Experiment &longs;o appo&longs;ite and pertinent to verifie the <emph type="italics"/>Po&longs;tulatum,<emph.end type="italics"/><lb/>that it very well de&longs;erveth to be admitted as if it were Demon­<lb/>&longs;trated.</s></p><p type="main"> <s>SALV. </s> <s>I will not con&longs;ent, <emph type="italics"/>Sagredus,<emph.end type="italics"/> that we take more to our <lb/>&longs;elves than we ought; and the rather for that we are chiefly to <lb/>make u&longs;e of this A&longs;&longs;umption in Motions made upon &longs;treight and <lb/>not curved Superficies; in which the Acceleration proceedeth with <lb/>degrees very different from tho&longs;e wherewith we &longs;uppo&longs;e it to pro­<lb/>ceed in &longs;treight Planes. </s> <s>In&longs;omuch, that although the Experiment <lb/>alledged &longs;hews us, that the de&longs;cent along the Arch C <emph type="italics"/>B<emph.end type="italics"/> conferreth <lb/>on the Moveable &longs;uch a Moment, as that it is able to re-carry it <lb/>to the &longs;ame height along any other Arch <emph type="italics"/>B<emph.end type="italics"/> C, <emph type="italics"/>B<emph.end type="italics"/> G, and <emph type="italics"/>B<emph.end type="italics"/> I, yet <lb/>we cannot with the like evidence &longs;hew, that the &longs;ame would hap­<lb/>pen in ca&longs;e a mo&longs;t exact <emph type="italics"/>B<emph.end type="italics"/>all were to de&longs;cend by &longs;treight Planes in­<lb/>clined according to the inclinations of the Chords of the&longs;e &longs;ame <lb/>Arches: yea, it is credible, that Angles being formed by the &longs;aid <lb/>Right Planes in the term <emph type="italics"/>B,<emph.end type="italics"/> the <emph type="italics"/>B<emph.end type="italics"/>all de&longs;cended along the Declivi­<lb/>ty according to the Chord C <emph type="italics"/>B,<emph.end type="italics"/> finding a &longs;top in the Planes a&longs;cend­<lb/>ing according to the Chords <emph type="italics"/>B<emph.end type="italics"/> D, <emph type="italics"/>B<emph.end type="italics"/> G, and <emph type="italics"/>B<emph.end type="italics"/> I, in ju&longs;tling again&longs;t <lb/>them, would lo&longs;e of its <emph type="italics"/>Impetus,<emph.end type="italics"/> and could not be able in ri&longs;ing to <lb/>attain the height of the Line C D. <emph type="italics"/>B<emph.end type="italics"/>ut the Ob&longs;tacle being remo­<lb/>ved, which prejudiceth the Experiment, I do believe, that the un­<lb/>der&longs;tanding may conceive, that the <emph type="italics"/>Impetus<emph.end type="italics"/> (which in effect de­<lb/>riveth vigour from the quantity of the De&longs;cent) would be able to <lb/>remount the Moveable to the &longs;ame height. </s> <s>Let us therefore take <lb/>this at pre&longs;ent for a <emph type="italics"/>Po&longs;tulatum<emph.end type="italics"/> or Petition, the ab&longs;olute truth of <lb/>which will come to be e&longs;tabli&longs;hed hereafter by &longs;eeing other Con­<lb/>clu&longs;ions rai&longs;ed upon this Hypothe&longs;is to an&longs;wer, and exactly jump <lb/>with the Experiment. </s> <s>The Author having &longs;uppo&longs;ed this only Prin­<lb/>ciple, he pa&longs;&longs;eth to the Propo&longs;itions, demon&longs;tratively proving them; <lb/>of which the fir&longs;t is this;</s></p><pb xlink:href="069/01/146.jpg" pagenum="143"/><p type="head"> <s>THEOR. I. PROP. I.</s></p><p type="main"> <s>The time in which a Space is pa&longs;&longs;ed by a Movea­<lb/>ble with a Motion Vniformly Accelerate, out of <lb/>Re&longs;t, is equal to the Time in which the &longs;ame <lb/>Space would be pa&longs;t by the &longs;ame Moveable <lb/>with an Equable Motion, the degree of who&longs;e <lb/>Velocity is &longs;ubduple to the greate&longs;t and ulti <lb/>mate degree of the Velocity of the former Vni­<lb/>formly Accelerate Motion.</s></p><p type="main"> <s><emph type="italics"/>Let us by the exten&longs;ion A B repre&longs;ent the Time, in which the <lb/>Space<emph.end type="italics"/> C D <emph type="italics"/>is pa&longs;&longs;ed by a Moveable with a Motion Vniformly <lb/>Accelerate, out of Re&longs;t in C: and let the greate&longs;t and la&longs;t de-<emph.end type="italics"/><lb/><figure id="id.069.01.146.1.jpg" xlink:href="069/01/146/1.jpg"/><lb/><emph type="italics"/>gree of Velocity acquired in the In&longs;tants of the Time<emph.end type="italics"/><lb/>A B <emph type="italics"/>be repre&longs;ented by<emph.end type="italics"/> E B; <emph type="italics"/>and con&longs;titute at plea­<lb/>&longs;ure upon<emph.end type="italics"/> A B <emph type="italics"/>any number of parts, and thorow the <lb/>points of divi&longs;ion draw as many Lines, continued <lb/>out unto the Line<emph.end type="italics"/> A E, <emph type="italics"/>and equidi&longs;tant to<emph.end type="italics"/> B E, <lb/><emph type="italics"/>which will repre&longs;ent the encrea&longs;e of the degrees of <lb/>Velocity after the fir&longs;t In&longs;tant A. </s> <s>Then divide<emph.end type="italics"/> B E <lb/><emph type="italics"/>into two equall parts in<emph.end type="italics"/> F, <emph type="italics"/>and draw<emph.end type="italics"/> F G <emph type="italics"/>and<emph.end type="italics"/> A G <lb/><emph type="italics"/>parallel to B A and<emph.end type="italics"/> B F<emph type="italics"/>: The Parallelogram<emph.end type="italics"/> A G <lb/>F B <emph type="italics"/>&longs;hall be equall to the Triangle<emph.end type="italics"/> A E B, <emph type="italics"/>its Side<emph.end type="italics"/><lb/>G F <emph type="italics"/>dividing<emph.end type="italics"/> A E <emph type="italics"/>into two equall parts in I: For <lb/>if the Parallels of the Triangle<emph.end type="italics"/> A E <emph type="italics"/>B be continued <lb/>out unto<emph.end type="italics"/> I G F, <emph type="italics"/>we &longs;hall have the Aggregate of all <lb/>the Parallels contained in the Quadrilatural Figure <lb/>equal to the Aggregate of all the Parallels compre­<lb/>hended in the Triangle<emph.end type="italics"/> A E <emph type="italics"/>B; For tho&longs;e in the Triangle<emph.end type="italics"/> I E F <emph type="italics"/>are equal <lb/>to tho&longs;e contained in the Triangle<emph.end type="italics"/> G I A, <emph type="italics"/>and tho&longs;e that are in the<emph.end type="italics"/> Tra­<lb/>pezium <emph type="italics"/>are in common. </s> <s>Now &longs;ince all and &longs;ingular the In&longs;tants of Time <lb/>do an&longs;wer to all and &longs;ingular the Points of the Line A B; and &longs;ince the <lb/>Parallels contained in the Triangle<emph.end type="italics"/> A E <emph type="italics"/>B do repre&longs;ent the degrees of Ac­<lb/>celeration or encrea&longs;ing Velocity, and the Parallels contained in the Pa­<lb/>rallelogram do likewi&longs;e repre&longs;ent as many degrees of Equable Motion or <lb/>unencrea&longs;ing Velocity: It appeareth, that as many Moments of Velocity <lb/>pa&longs;&longs;ed in the Accelerate Motion according to the encrea&longs;ing Parallels of the <lb/>Triangle A E B, as in the Equable Motion according to the Parallels of <lb/>the Parallelogram G B: Becau&longs;e what is wanting in the fir&longs;t half of the<emph.end type="italics"/><pb xlink:href="069/01/147.jpg" pagenum="144"/><emph type="italics"/>Accelerate Motion of the Velocity of the Equable Motion (which defi­<lb/>cient Moments are repre&longs;ented by the Parallels of the Triangle A<emph.end type="italics"/> G I) <lb/><emph type="italics"/>is made up by the moments repre&longs;ented by the Parallels of the Triangle<emph.end type="italics"/><lb/>I E F. <emph type="italics"/>It is manife&longs;t, therefore, that tho&longs;e Spaces are equal which are <lb/>in the &longs;ame Time by two Moveables, one whereof is moved with a Mo­<lb/>tion uniformly Accelerated from Re&longs;t, the other with a Motion Equable <lb/>according to the Moment &longs;ubduple of that of the greate&longs;t Velocity of the <lb/>Accelerated Motion: Which was to be demon&longs;trated.<emph.end type="italics"/></s></p><p type="head"> <s>THEOR. II. PROP. II.</s></p><p type="main"> <s>If a <emph type="italics"/>M<emph.end type="italics"/>oveable de&longs;cend out of Re&longs;t with a <emph type="italics"/>M<emph.end type="italics"/>oti­<lb/>on uniformly Accelerate, the Spaces which it <lb/>pa&longs;&longs;eth in any what&longs;oever Times are to each <lb/>other in a proportion Duplicate of the &longs;ame <lb/>Times; that is, they are as the Squares of <lb/>them.</s></p><p type="main"> <s><emph type="italics"/>Let<emph.end type="italics"/> A B <emph type="italics"/>repre&longs;ent a length of Time beginning at the fir&longs;t In&longs;tant A; <lb/>and let<emph.end type="italics"/> A D <emph type="italics"/>and<emph.end type="italics"/> A E <emph type="italics"/>repre&longs;ent any two parts of the &longs;aid Time; <lb/>and let<emph.end type="italics"/> H I <emph type="italics"/>be a Line in which the Moveable out of H, (as the fir&longs;t <lb/>beginning of the Motion) de&longs;cendeth uniformly accelerating; and let the<emph.end type="italics"/><lb/><figure id="id.069.01.147.1.jpg" xlink:href="069/01/147/1.jpg"/><lb/><emph type="italics"/>Space<emph.end type="italics"/> H L <emph type="italics"/>be pa&longs;&longs;ed in the fir&longs;t Time<emph.end type="italics"/> A D; <emph type="italics"/>and let<emph.end type="italics"/> H M <lb/><emph type="italics"/>be the Space that it &longs;hall de&longs;cend in the Time<emph.end type="italics"/> A E. <emph type="italics"/>I &longs;ay, <lb/>the Space<emph.end type="italics"/> M H <emph type="italics"/>is to the Space<emph.end type="italics"/> H L <emph type="italics"/>in duplicate propor­<lb/>tion of that which the Time<emph.end type="italics"/> E A <emph type="italics"/>hath to the Time<emph.end type="italics"/> A D<emph type="italics"/>: <lb/>Or, if you will, that the Spaces<emph.end type="italics"/> M H <emph type="italics"/>and<emph.end type="italics"/> H L <emph type="italics"/>are to one <lb/>another in the &longs;ame proportion as the Squares<emph.end type="italics"/> E A <emph type="italics"/>and<emph.end type="italics"/><lb/>A D. <emph type="italics"/>Draw the Line<emph.end type="italics"/> A C <emph type="italics"/>at any Angle with<emph.end type="italics"/> A B, <emph type="italics"/>and <lb/>from the points D and E draw the Parallels<emph.end type="italics"/> D O <emph type="italics"/>and<emph.end type="italics"/><lb/>P E<emph type="italics"/>: of which<emph.end type="italics"/> D O <emph type="italics"/>will repre&longs;ent the greate&longs;t degree <lb/>of Velocity acquired in the In&longs;tant D of the Time<emph.end type="italics"/> A D; <lb/><emph type="italics"/>and<emph.end type="italics"/> P <emph type="italics"/>the greate&longs;t degree of Velocity acquired in the In­<lb/>&longs;tant E of the Time<emph.end type="italics"/> A E. <emph type="italics"/>And becau&longs;e we have de­<lb/>mon&longs;trated in the la&longs;t Propo&longs;ition concerning Spaces, that <lb/>tho&longs;e are equal to one another, of which two Moveables <lb/>have pa&longs;t in the &longs;ame Time, the one by a Moveable out <lb/>of Re&longs;t with a Motion uniformly Accelerate, and the <lb/>other by the &longs;ame Moveable with an Equable Motion, <lb/>who&longs;e Velocity is &longs;ubduple to the greate&longs;t acquired by the <lb/>Accelerate Motion: Therefore<emph.end type="italics"/> M H <emph type="italics"/>and<emph.end type="italics"/> H L <emph type="italics"/>are the Spaces that two <lb/>Lquable Motions, who&longs;e Velocities &longs;hould be as the half of<emph.end type="italics"/> P E, <emph type="italics"/>and<emph.end type="italics"/><pb xlink:href="069/01/148.jpg" pagenum="145"/><emph type="italics"/>half of<emph.end type="italics"/> O D, <emph type="italics"/>would pa&longs;&longs;e in the Times<emph.end type="italics"/> E A <emph type="italics"/>and<emph.end type="italics"/> D A. <emph type="italics"/>If it be proved <lb/>therefore that the&longs;e Spaces<emph.end type="italics"/> M H <emph type="italics"/>and<emph.end type="italics"/> L H <emph type="italics"/>are in duplicate proportion to <lb/>the Times<emph.end type="italics"/> E A <emph type="italics"/>and<emph.end type="italics"/> D A; <emph type="italics"/>We &longs;hall have done that which was intended. <lb/></s> <s>But in the fourth Propo&longs;ition of the Fir&longs;t Book we have demon&longs;trated: <lb/>That the Spaces pa&longs;t by two Moveables with an Equable Motion are <lb/>to each other in a proportion compounded of the proportion of the Velo­<lb/>cities and of the proportion of the Times: But in this ca&longs;e the propor­<lb/>tion of the Velocities and the proportion of the Times is the &longs;ame<emph.end type="italics"/> (<emph type="italics"/>for <lb/>as the half of<emph.end type="italics"/> P E <emph type="italics"/>is to the half of<emph.end type="italics"/> O D, <emph type="italics"/>or the whole<emph.end type="italics"/> P E <emph type="italics"/>to the whole<emph.end type="italics"/><lb/>O D, <emph type="italics"/>&longs;o is<emph.end type="italics"/> A E <emph type="italics"/>to<emph.end type="italics"/> A D<emph type="italics"/>: Therefore the proportion of the Spaces pa&longs;­<lb/>&longs;ed is double to the proportion of the Times. </s> <s>Which was to be demon­<lb/>&longs;trated.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Hence likewi&longs;e it is manife&longs;t, that the proportion of the &longs;ame Spaces <lb/>is double to the proportions of the greate&longs;t degrees of Velocity: that is, <lb/>of the Lines<emph.end type="italics"/> P E <emph type="italics"/>and<emph.end type="italics"/> O D<emph type="italics"/>: becau&longs;e<emph.end type="italics"/> P E <emph type="italics"/>is to<emph.end type="italics"/> O D, <emph type="italics"/>as<emph.end type="italics"/> E A <emph type="italics"/>to<emph.end type="italics"/> D A.</s></p><p type="head"> <s>COROLARY I.</s></p><p type="main"> <s><emph type="italics"/>Hence it is manife&longs;t, that if there were many equal Times taken in or­<lb/>der from the fir&longs;t In&longs;tant or beginniug of the Motion, as &longs;uppo&longs;e<emph.end type="italics"/><lb/>A D, D E, E F, F G, <emph type="italics"/>in which the Spaces<emph.end type="italics"/> H L, L M, M N, N I <lb/><emph type="italics"/>are pa&longs;&longs;ed, tho&longs;e Spaces &longs;hall be to one another as the odd numbers <lb/>from an Vnite:<emph.end type="italics"/> &longs;cilicet, <emph type="italics"/>as 1, 3, 5, 7. For this is the Rate or pro­<lb/>portion of the exce&longs;&longs;es of the Squares of Lines that equally exceed <lb/>one another, and the exce&longs;&longs;e of which is equal to the least of them, <lb/>or, if you will, of Squares that follow one another, beginning<emph.end type="italics"/> ab <lb/>Unitate. <emph type="italics"/>Whil&longs;t therefore the degree of Velocity is encrea&longs;ed ac­<lb/>cording to the &longs;imple Series of Numbers in equal Times, the Spaces <lb/>pa&longs;t in tho&longs;e Times make their encrea&longs;e according to the Series of <lb/>odd Numbers from an Vnite.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>Be plea&longs;ed to &longs;tay your Reading, whil&longs;t I do paraphra&longs;e <lb/>touching a certain Conjecture that came into my mind <lb/>but even now; for the explanation of which, unto your under­<lb/>&longs;tanding and my own, I will de&longs;cribe a &longs;hort Scheme: in which I <lb/>fan&longs;ie by the Line A I the continuation of the Time after the fir&longs;t <lb/>In&longs;tant, applying the Right Line A F unto A according to any <lb/>Angle: and joyning together the Terms I F, I divide the Time A I <lb/>in half at C, and then draw C B parallel to I F. <emph type="italics"/>A<emph.end type="italics"/>nd then con&longs;ide­<lb/>ring B C, as the greate&longs;t degree of Velocity which beginning from <lb/>Re&longs;t in the fir&longs;t In&longs;tant of the Time <emph type="italics"/>A<emph.end type="italics"/> goeth augmenting accord­<lb/>ing to the encrea&longs;e of the Parallels to B C, drawn in the Triangle <lb/><emph type="italics"/>A<emph.end type="italics"/> B C, (which is all one as to encrea&longs;e according to the encrea&longs;e <lb/>of the Time) I admit without di&longs;pute, upon what hath been &longs;aid <lb/>already, That the Space pa&longs;t by the falling Moveable with the <pb xlink:href="069/01/149.jpg" pagenum="146"/>Velocity encrea&longs;ed in the manner afore&longs;aid would be equal to the <lb/>Space that the &longs;aid Moveable would pa&longs;&longs;e, in ca&longs;e it were in the <lb/>&longs;ame Time <emph type="italics"/>A<emph.end type="italics"/> C, moved with an Uniform Motion, who&longs;e degree of <lb/>Velocity &longs;hould be equal to E C, the half of B C. </s> <s>I now proceed <lb/>farther, and imagine the Moveable; having de&longs;cended with an <lb/><emph type="italics"/>A<emph.end type="italics"/>ccelerate Motion, to have in the In&longs;tant <lb/>C the degree of Velocity B C: It is ma­<lb/><figure id="id.069.01.149.1.jpg" xlink:href="069/01/149/1.jpg"/><lb/>nife&longs;t, that if it did continue to move <lb/>with the &longs;ame degree of Velocity B C, <lb/>without farther <emph type="italics"/>A<emph.end type="italics"/>cceleration, it would <lb/>pa&longs;&longs;e in the following Time C I, a Space <lb/>double to that which it pa&longs;&longs;ed in the equal <lb/>Time <emph type="italics"/>A<emph.end type="italics"/> C, with the degree of Uniform <lb/>Velocity E C, the half of the Degree B C. <lb/></s> <s>But becau&longs;e the Moveable de&longs;cendeth <lb/>with a Velocity encrea&longs;ed alwaies Uni­<lb/>formly in all equal Times; it will add to <lb/>the degree C B in the following Time <lb/>C I, tho&longs;e Tame Moments of Velocity <lb/>that encrea&longs;e according to the Parallels of <lb/>the Triangle B F G, equal to the Triangle <lb/><emph type="italics"/>A<emph.end type="italics"/> B C. </s> <s>So that adding to the degree of <lb/>Velocity G I, the half of the degree F G, the greate&longs;t of tho&longs;e ac­<lb/>quired in the <emph type="italics"/>A<emph.end type="italics"/>ccelerate Motion, and regulated by the Parallels of <lb/>the Triangle B F G, we &longs;hall have the degree of Velocity I N, with <lb/>which, with an Uniform Motion, it would have moved in the <lb/>Time C I: Which degree I N, being triple the degree E C, pro­<lb/>veth that the Space pa&longs;&longs;ed in the &longs;econd Time C I ought to be tri­<lb/>ple to that of the fir&longs;t Time C <emph type="italics"/>A. A<emph.end type="italics"/>nd if we &longs;hould &longs;uppo&longs;e to be <lb/>added to <emph type="italics"/>A<emph.end type="italics"/> I another equal part of Time I O, and the Triangle to <lb/>be enlarged unto <emph type="italics"/>A<emph.end type="italics"/> P O; it is manife&longs;t, that if the Motion &longs;hould <lb/>continue for all the Time I O with the degree of Velocity I F, <lb/>acquired in the <emph type="italics"/>A<emph.end type="italics"/>ccelerate Motion in the Time <emph type="italics"/>A<emph.end type="italics"/> I, that degree <lb/>I F being Quadruple to E C, the Space pa&longs;&longs;ed would be Quadruple <lb/>to that pa&longs;&longs;ed in the equal fir&longs;t Time <emph type="italics"/>A<emph.end type="italics"/> C: But continuing the <lb/>encrea&longs;e of the Uniform <emph type="italics"/>A<emph.end type="italics"/>cceleration in the Triangle F P Q like <lb/>to that of the Triangle <emph type="italics"/>A<emph.end type="italics"/> B C, which being reduced to equable <lb/>Motion addeth the degree equal to E C, Q R being added, equal <lb/>to E C, we &longs;hall have the whole Equable Velocity exerci&longs;ed in the <lb/>Time I O, quintuple to the Equable Velocity of the fir&longs;t Time <emph type="italics"/>A<emph.end type="italics"/> C, <lb/>and therefore the Space pa&longs;&longs;ed quintuple to that pa&longs;t in the fir&longs;t <lb/>Time <emph type="italics"/>A<emph.end type="italics"/> C. </s> <s>We &longs;ee therefore, even by this familiar computation, <lb/>That the Spaces pa&longs;&longs;ed in equal Times by a Moveable which <lb/>departing from Re&longs;t goeth acquiring Velocity, according to the <lb/>encrea&longs;e of the Time, are to one another as the odd Numbers <emph type="italics"/>ab<emph.end type="italics"/><pb xlink:href="069/01/150.jpg" pagenum="147"/><emph type="italics"/>unitate 1, 3, 5: A<emph.end type="italics"/>nd that the Spaces pa&longs;&longs;ed being conjunctly taken, <lb/>that pa&longs;&longs;ed in the double Time is quadruple to that pa&longs;&longs;ed in the <lb/>&longs;ubduple, that pa&longs;&longs;ed in the triple Time is nonuple; and, in a word, <lb/>that the Spaces pa&longs;&longs;ed are in duplicate proportion to their Times; <lb/>that is, as the Squares of the &longs;aid Times.</s></p><p type="main"> <s>SIMP. </s> <s>I mu&longs;t confe&longs;&longs;e that I have taken more plea&longs;ure in this <lb/>plain and clear di&longs;cour&longs;e of <emph type="italics"/>Sagredus,<emph.end type="italics"/> than in the to-me-more <lb/>ob&longs;cure Demon&longs;tration of the <emph type="italics"/>A<emph.end type="italics"/>uthor: &longs;o that I am very well <lb/>&longs;atisfied, that the bu&longs;ine&longs;&longs;e is to &longs;ucceed as hath been &longs;aid, the <lb/>Definition of Uniformly <emph type="italics"/>A<emph.end type="italics"/>ccelerate Motion being &longs;uppo&longs;ed, and <lb/>granted. </s> <s>But whether this be the <emph type="italics"/>A<emph.end type="italics"/>cceleration of which Nature <lb/>maketh u&longs;e in the Motion of its de&longs;cending Grave Bodies, I yet <lb/>make a que&longs;tion: and therefore for information of me, and of <lb/>others like unto me, me thinks it would be &longs;ea&longs;onable in this place <lb/>to produce &longs;ome Experiment among&longs;t tho&longs;e which were &longs;aid to be <lb/>many, which in &longs;undry Ca&longs;es agree with the Conclu&longs;ions demon­<lb/>&longs;trated.</s></p><p type="main"> <s>SALV. You, like a true <emph type="italics"/>A<emph.end type="italics"/>rti&longs;t, make a very rea&longs;onable demand, <lb/>and &longs;o it is u&longs;ual and convenient to do in Sciences that apply <lb/>Mathematical Demon&longs;trations to Phy&longs;ical Conclu&longs;ions, as we &longs;ee <lb/>in the Profe&longs;&longs;ors of Per&longs;pection, <emph type="italics"/>A<emph.end type="italics"/>&longs;tronomy, Mechanicks, Mu&longs;ick, <lb/>and others, who with Sen&longs;ible Experiments confirm tho&longs;e their <lb/>Principles that are as the foundations of all the following Structure: <lb/>and therefore I de&longs;ire that it may not be thought &longs;uperfluous, that <lb/>we di&longs;cour&longs;e with &longs;ome prolixity upon this fir&longs;t and grand funda­<lb/>mental on which we lay the weight of the Immen&longs;e Machine of <lb/>infinite Conclu&longs;ions, of which we have but a very &longs;mall part &longs;et <lb/>down in this Book by our <emph type="italics"/>A<emph.end type="italics"/>uthor, who hath done enough to open <lb/>the way and door that hath been hitherto &longs;hut unto all Specula­<lb/>tive Wits. </s> <s>Touching Experiments, therefore, the <emph type="italics"/>A<emph.end type="italics"/>uthor hath <lb/>not omitted to make &longs;everal; and to a&longs;&longs;ure us, that the <emph type="italics"/>A<emph.end type="italics"/>ccelerati­<lb/>on of natural-de&longs;cending Graves hapneth in the afore&longs;aid propor­<lb/>tion, I have many times in his company &longs;et my &longs;elf to make a triall <lb/>thereof in the following Method.</s></p><p type="main"> <s>In a pri&longs;me or Piece of Wood, about twelve yards long, and <lb/>half a yard broad one way, and three Inches the other, we made, <lb/>upon the narrow Side or edge a Groove of little more than an Inch <lb/>wide; we &longs;hot it with the Grooving Plane very &longs;traight, and to <lb/>make it very &longs;mooth and &longs;leek, we glued upon it a piece of Vellum, <lb/>poli&longs;hed and &longs;moothed as exactly as can be po&longs;&longs;ible: and in it we <lb/>have let a brazen Ball, very hard, round, and &longs;mooth, de&longs;cend. <lb/></s> <s>Having placed the &longs;aid Pri&longs;me Pendent, rai&longs;ing one of its ends <lb/>above the Horizontal Plane a yard or two at plea&longs;ure, we have let <lb/>the Ball (as I &longs;aid) de&longs;cend along the Grove, ob&longs;erving, in the <lb/>manner that I &longs;hall tell you pre&longs;ently, the Time which it &longs;pent in <pb xlink:href="069/01/151.jpg" pagenum="148"/>runing it all; repeating the &longs;ame ob&longs;ervation again and again to <lb/>a&longs;&longs;ure our &longs;elves of the Time, in which we never found any diffe­<lb/>rence, no not &longs;o much as the tenth part of one beat of the Pul&longs;e. <lb/></s> <s>Having done, and preci&longs;ely ordered this bu&longs;ine&longs;&longs;e, we made the <lb/>&longs;ame Ball to de&longs;cend only the fourth part of the length of that <lb/>Grove: and having mea&longs;ured the time of its de&longs;cent, we alwaies <lb/>found it to be punctually half the other. </s> <s>And then making trial of <lb/>other parts, examining one while the Time of the whole Length <lb/>with the Time of half the Length, or with that of 2/3, or of 3/4, or, in <lb/>brief, with any whatever other Divi&longs;ion, by Experiments repeated <lb/>near an hundred Times, we alwaies found the Spaces to be to one <lb/>another as the Squares of the Times. </s> <s>And this in all Inclinations <lb/>of the Plane, that is, of the Grove in which the Ball was made to <lb/>de&longs;cend. </s> <s>In which we ob&longs;erved moreover, that the Times of the <lb/>De&longs;cents along &longs;undry Inclinations did retain the &longs;ame proportion <lb/>to one another, exactly, which anon you will &longs;ee a&longs;&longs;igned to them, <lb/>and demon&longs;trated by the Author. </s> <s>And as to the mea&longs;uring of the <lb/>Time; we had a good big Bucket full of Water hanged on high, <lb/>which by a very &longs;mall hole, pierced in the bottom, &longs;pirted, or, as <lb/>we &longs;ay, &longs;pin'd forth a &longs;mall thread of Water, which we received <lb/>with a &longs;mall cup all the while that the Ball was de&longs;cending in the <lb/>Grove, and in its parts; and then weighing from time to time the <lb/>&longs;mall parcels of Water, in that manner gathered, in an exact pair <lb/>of &longs;cales, the differences and proportions of their Weights gave <lb/>ju&longs;tly the differences and proportions of the Times; and this with <lb/>&longs;uch exactne&longs;&longs;e, that, as I &longs;aid before, the trials being many <lb/>and many times repeated, they never differed any con&longs;iderable <lb/>matter.</s></p><p type="main"> <s>SIMP. </s> <s>I &longs;hould have received great &longs;atisfaction by being pre&longs;ent <lb/>at tho&longs;e Experiments: but being confident of your diligence in <lb/>making them, and veracity in relating them, I content my &longs;elf, and <lb/>admit them for true and certain.</s></p><p type="main"> <s>SALV. </s> <s>We may, then, rea&longs;&longs;ume our Reading, and go on.</s></p><p type="head"> <s>COROLLARY II.</s></p><p type="main"> <s>It is collected in the &longs;econd place, that if any two Spaces are ta­<lb/>ken from the beginning of the Motion, pa&longs;&longs;ed in any Times, <lb/>tho&longs;e Times &longs;hall be unto each other as one of them is to a <lb/>Space that is the Mean proportional between them.</s></p><p type="main"> <s><emph type="italics"/>For taking two Spaces S T, and S V from the beginning of the Mo­<lb/>tion S, to which S X is a Mean-proportional, the Time of the de&longs;cent <lb/>along S T, &longs;hall be to the Time of the de&longs;cent along S V, as S T to S X; <lb/>or, if you will, the Time along S V &longs;hall be to the Time along S T,<emph.end type="italics"/><pb xlink:href="069/01/152.jpg" pagenum="149"/><figure id="id.069.01.152.1.jpg" xlink:href="069/01/152/1.jpg"/><lb/><emph type="italics"/>as VS is to SX. </s> <s>For it is demon&longs;trated, that Spaces <lb/>pa&longs;&longs;ed are in duplicate proportion to the Times, or, (which <lb/>is the &longs;ame) are as the Squares of the Times: But the pro­<lb/>portion of the Space VS to the Space ST is double to the <lb/>proportion of V S to SX, or is the &longs;ame that V S, and S X <lb/>&longs;quared have to one another: Therefore, the proportion of <lb/>the Times of the Motion by V S, and ST, is as the Spaces or <lb/>Lines V S to S X.<emph.end type="italics"/></s></p><p type="head"> <s>SCHOLIUM.</s></p><p type="main"> <s><emph type="italics"/>That which is demon&longs;trated in Motions that are made Perpendicu­<lb/>larly, may be under&longs;tood al&longs;o to hold true in the Motions made along <lb/>Planes of any whatever Inclination; for it is &longs;uppo&longs;ed, that in them <lb/>the degree of Acceleration encrea&longs;eth in the &longs;ame proportion; that <lb/>is, according to the encrea&longs;e of the Time; or, if you will, according <lb/>to the &longs;imple and primary Series of Numbers.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>Here I de&longs;ire <emph type="italics"/>Sagredus,<emph.end type="italics"/> that I al&longs;o may be allowed, al­<lb/>beit perhaps with too much tediou&longs;ne&longs;&longs;e in the opinion of <emph type="italics"/>Simplici­<lb/>us,<emph.end type="italics"/> to defer for a little time the pre&longs;ent Reading, untill I may have <lb/>explained what from that which hath been already &longs;aid and de­<lb/>mon&longs;trated, and al&longs;o from the knowledge of certain Mechanical <lb/>Conclu&longs;ions heretofore learnt of our <emph type="italics"/>Academick,<emph.end type="italics"/> I now remember <lb/>to adjoyn for the greater confirmation of the truth of the Princi­<lb/>ple, which hath been examined by us even now with probable <lb/>Rea&longs;ons and Experiments: and, which is of more importance, for <lb/>the Geometrical proof of it, let me fir&longs;t demon&longs;trate one &longs;ole Ele­<lb/>mental <emph type="italics"/>Lemma<emph.end type="italics"/> in the Contemplation of <emph type="italics"/>Impetus's.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>If our advantage &longs;hall be &longs;uch as you promi&longs;eus, there <lb/>is no time that I would not mo&longs;t willingly &longs;pend in di&longs;cour&longs;ing <lb/>about the confirmation and thorow e&longs;tabli&longs;hing the&longs;e Sciences of <lb/>Motion: and as to my own particular, I not only grant you liber­<lb/>ty to &longs;atisfie your &longs;elf in this particular, but moreover entreat you <lb/>to gratifie, as &longs;oon as you can, the Curio&longs;ity which you have begot <lb/>in me touching the &longs;ame: and I believe that <emph type="italics"/>Simplicius<emph.end type="italics"/> al&longs;o is of the <lb/>&longs;ame mind.</s></p><p type="main"> <s>SIMP. </s> <s>I cannot deny what you &longs;ay.</s></p><p type="main"> <s><emph type="italics"/>S<emph.end type="italics"/>ALV. </s> <s>Seeing then that I have your permi&longs;&longs;ion, I will in the <lb/>fir&longs;t place con&longs;ider, as an Effect well known, That</s></p><pb xlink:href="069/01/153.jpg" pagenum="150"/><p type="head"> <s>LEMMA.</s></p><p type="main"> <s><emph type="italics"/>That the Moments or Velocities of the &longs;ame Moveable are different <lb/>upon different Inclinations of Planes, and the greate&longs;t is by the <lb/>Line elevated perpendicularly above the Horizon, and by the <lb/>others inclined, the &longs;aid Velocity dimini&longs;heth according as they <lb/>more and more depart from Perpendicularity, that is, as they in­<lb/>cline more obliquely: &longs;o that the Impetus, Talent, Energy, or, we <lb/>may &longs;ay, Moment of de&longs;cending is dimini&longs;hed in the Moveable by <lb/>the &longs;ubjected Plane, upon which the &longs;aid Moveable lyeth and <lb/>de&longs;cendeth.<emph.end type="italics"/></s></p><p type="main"> <s>And the better to expre&longs;s my &longs;elf, let the Line A B be perpen­<lb/>dicularly erected upon the Horizon A C: then &longs;uppo&longs;e the <lb/>&longs;ame to be declined in &longs;undry Inclinations towards the Horizon, as <lb/>in A D, A E, A F, <emph type="italics"/>&c.<emph.end type="italics"/> I &longs;ay, that the greate&longs;t and total <emph type="italics"/>Impetus<emph.end type="italics"/><lb/>of the Grave Body in de&longs;cending is along the Perpendicular B A, <lb/>and le&longs;s than that along D A, <lb/><figure id="id.069.01.153.1.jpg" xlink:href="069/01/153/1.jpg"/><lb/>and yet le&longs;s along E A; and <lb/>&longs;ucce&longs;&longs;ively dimini&longs;hing along <lb/>the more inclined F <emph type="italics"/>A,<emph.end type="italics"/> and fi­<lb/>nally is wholly extinct in the <lb/>Horizontal C <emph type="italics"/>A,<emph.end type="italics"/> where the <lb/>Moveable is indifferent either <lb/>to Motion or Re&longs;t, and hath not <lb/>of it &longs;elf any Inclination to <lb/>move one way or other, nor yet <lb/>any Re&longs;i&longs;tance to its being mo­<lb/>ved: for as it is impo&longs;&longs;i­<lb/>ble that a Grave Body, or a <lb/>Compound thereof &longs;hould move naturally upwards, receding from <lb/>the Common Center, towards which all Grave Matters con&longs;pire <lb/>to go, &longs;o it is impo&longs;&longs;ible that it do &longs;pontaneou&longs;ly move, unle&longs;s <lb/>with that Motion its particular Center of Gravity do acquire Proxi­<lb/>mity to the &longs;aid Common Center: &longs;o that upon the Horizontal <lb/>which here is under&longs;tood to be a Superficies equidi&longs;tant from the <lb/>&longs;aid Center, and therefore altogether void of Inclination, the <emph type="italics"/>Im­<lb/>petus<emph.end type="italics"/> or Moment of that &longs;ame Moveable &longs;hall be nothing at all. <lb/></s> <s>Having under&longs;tood this mutation of <emph type="italics"/>Impetus,<emph.end type="italics"/> I am to explain that <lb/>which, in an old Treati&longs;e of the Mechanicks, written heretofore <lb/>in <emph type="italics"/>Padona<emph.end type="italics"/> by our <emph type="italics"/>Academick,<emph.end type="italics"/> only for the u&longs;e of his Scholars, was <lb/>diffu&longs;ely and demon&longs;tratively proved, upon the occa&longs;ion of con­<lb/>&longs;idering the Original and Nature of the admirable In&longs;trument cal­<lb/>led the Screw, and it is, With what proportion that mutation of <pb xlink:href="069/01/154.jpg" pagenum="151"/><emph type="italics"/>Impetus<emph.end type="italics"/> is made along &longs;everal Inclinations or Declivities of <lb/>Planes.</s></p><p type="main"> <s>As, for example, in the inclined Plane A F, drawing its Eleva­<lb/>tion above the Horizontal, that is, the Line F C, along the which <lb/>the <emph type="italics"/>Impetus<emph.end type="italics"/> of a Grave Body, and the Moment of De&longs;cent is the <lb/>greate&longs;t; it is &longs;ought what proportion this Moment hath to the <lb/>Moment of the &longs;ame Moveable along the Declivity F A: Which <lb/>Proportion, I &longs;ay, is Reciprocal to the &longs;aid Lengths. </s> <s>And this is <lb/>the <emph type="italics"/>Lemma<emph.end type="italics"/> that was to go before the Theorem, which I hope to be <lb/>able anon to Demon&longs;trate. </s> <s>Hence it is manife&longs;t, That the <emph type="italics"/>Impetus<emph.end type="italics"/><lb/>of De&longs;cent of a Grave Body is as much as the Re&longs;i&longs;tance or lea&longs;t <lb/>force that &longs;ufficeth to arre&longs;t and &longs;tay it. </s> <s>For this Force or Re&longs;i­<lb/>&longs;tance, and its mea&longs;ure, I will make u&longs;e of the Gravity of another <lb/>Moveable. </s> <s>Let us now upon the Plane F A put the Moveable G <lb/>tyed to a thread which &longs;liding over F hath fa&longs;tned at its other end <lb/>the Weight H: and let us con&longs;ider that the Space of the De&longs;cent <lb/>or A&longs;cent of the Weight H along the Perpendicular, is alwaies <lb/>equal to the whole A&longs;cent or De&longs;cent of the other Moveable G <lb/>along the ^{*} Declivity A F, but yet not to the A&longs;cent or De&longs;cent </s></p><p type="main"> <s><arrow.to.target n="marg1092"></arrow.to.target><lb/>along the Perpendicular, in which only the &longs;aid Moveable G (like <lb/>as every other Moveable) exerci&longs;eth its Re&longs;i&longs;tance. </s> <s>Which is <lb/>manife&longs;t: for con&longs;idering in the Triangle AFC the Motion of <lb/>the Moveable G, as for example, upwards from A to F, to be com­<lb/>po&longs;ed of the tran&longs;ver&longs;e Horizontal Line A C, and of the Perpendi­<lb/>cular C F: <emph type="italics"/>A<emph.end type="italics"/>nd in regard, that as to the Horizontal Plane along <lb/>which the Moveable, as hath been &longs;aid, hath no Re&longs;i&longs;tance to mo­<lb/>ving (it not making by that Motion any lo&longs;s, nor yet acqui&longs;t in <lb/>regard of its particular di&longs;tance from the Common Center of Grave <lb/>Matters, which in the Horizon continueth &longs;till the &longs;ame) it remai­<lb/>neth that the Re&longs;i&longs;tance be only in re&longs;pect of the <emph type="italics"/>A<emph.end type="italics"/>&longs;cent that it is to <lb/>make along the Perpendicular C F. </s> <s>Whil&longs;t therefore the Grave <lb/>Moveable G, moving from <emph type="italics"/>A<emph.end type="italics"/> to F, hath only the Perpendicular <lb/>Space C F to re&longs;i&longs;t in its <emph type="italics"/>A<emph.end type="italics"/>&longs;cent, and whil&longs;t the other Grave Move­<lb/>able H de&longs;cendeth along the Perpendicular of nece&longs;&longs;ity as far as <lb/>the whole Space F <emph type="italics"/>A,<emph.end type="italics"/> and that the &longs;aid proportion of <emph type="italics"/>A<emph.end type="italics"/>&longs;cent and <lb/>De&longs;cent maintains it &longs;elf alwaies the &longs;ame, be the Motion of the <lb/>&longs;aid Moveables little or much (by rea&longs;on they are tyed toge­<lb/>ther) we may confidently affirm, that in ca&longs;e there were an <emph type="italics"/>Equi­<lb/>librium,<emph.end type="italics"/> that is Re&longs;t, to en&longs;ue betwixt the &longs;aid Moveables, the Mo­<lb/>ments, the Velocities, or their Propen&longs;ions to Motion, that is the <lb/>Spaces which they would pa&longs;s in the &longs;ame Time &longs;hould an&longs;wer re­<lb/>ciprocally to their Gravities, according to that which is demon&longs;tra­<lb/>ted in all ca&longs;es of Mechanick Motions: &longs;o that it &longs;hall &longs;uffice to <lb/>impede the de&longs;cent of G, if H be but &longs;o much le&longs;s grave than it, as <lb/>in proportion the Space C F is le&longs;&longs;er than the Space F <emph type="italics"/>A.<emph.end type="italics"/> Therefore <pb xlink:href="069/01/155.jpg" pagenum="152"/>&longs;uppo&longs;e that the Moveable G is to the Moveable H, as F <emph type="italics"/>A<emph.end type="italics"/> is to <lb/>F C; and then the <emph type="italics"/>Equilibrium<emph.end type="italics"/> &longs;hall follow, that is, the Moveables <lb/>H and G &longs;hall have equal Moments, and the Motion of the &longs;aid <lb/>Moveables &longs;hall cea&longs;e. <emph type="italics"/>A<emph.end type="italics"/>nd becau&longs;e we &longs;ee that the <emph type="italics"/>Impetus,<emph.end type="italics"/><lb/>Energy, Moment, or Propen&longs;ion of a Moveable to Motion is the <lb/>&longs;ame as is the Force or &longs;malle&longs;t Re&longs;i&longs;tance that &longs;ufficeth to &longs;top it; <lb/>and becau&longs;e it hath been concluded, that the Grave Body H is &longs;uf. <lb/></s> <s>ficient to arre&longs;t the Motion of <lb/><figure id="id.069.01.155.1.jpg" xlink:href="069/01/155/1.jpg"/><lb/>the Grave Body G: Therefore <lb/>the le&longs;&longs;er Weight H, which in <lb/>the Perpendicular F C imploy­<lb/>eth its total Moment, &longs;hall be <lb/>the preci&longs;e mea&longs;ure of the par­<lb/>tial Moment that the greater <lb/>Weight G exerci&longs;eth along the <lb/>inclined Plane F <emph type="italics"/>A<emph.end type="italics"/>: But the <lb/>mea&longs;ure of the total Moment of <lb/>the &longs;aid Grave Body G, is the <lb/>&longs;elf &longs;ame, (&longs;ince that to impede <lb/>the Perpendicular De&longs;cent of a <lb/>Grave Body there is required the oppo&longs;ition of &longs;uch another Grave <lb/>Body, which likewi&longs;e is at liberty to move Perpendicularly:) <lb/>Therefore the partial <emph type="italics"/>Impetus<emph.end type="italics"/> or Moment of G along the inclined <lb/>Plane F A &longs;hall be to the grand and total <emph type="italics"/>Impetus<emph.end type="italics"/> of the &longs;ame G <lb/>along the Perpendicular F C, as the Weight H to the Weight G: <lb/>that is, by Con&longs;truction, as the &longs;aid Perpendicular F C, the Eleva­<lb/>tion of the inclined Plane, is to the &longs;ame inclined Plane F A: <lb/>Which is that that by the <emph type="italics"/>Lemma<emph.end type="italics"/> was propo&longs;ed to be demon­<lb/>&longs;trated, and which by our Author, as we &longs;hall &longs;ee, is &longs;uppo&longs;ed as <lb/>known in the &longs;econd part of the Sixth Propo&longs;ition of the pre&longs;ent <lb/>Treati&longs;e.</s></p><p type="margin"> <s><margin.target id="marg1092"></margin.target>* Or inclined <lb/>Plane.</s></p><p type="main"> <s>SAGR. </s> <s>From this that you have already concluded I conceive <lb/>one may ea&longs;ily deduce, arguing <emph type="italics"/>ex æquali<emph.end type="italics"/> by perturbed Proportion, <lb/>that the Moments of the &longs;ame Moveable, along Planes variou&longs;ly <lb/>inclined (as F A and F I) that have the &longs;ame Elevation, are to each <lb/>other in Reciprocal proportion to the &longs;ame Planes.</s></p><p type="main"> <s>SALV. <emph type="italics"/>A<emph.end type="italics"/> mo&longs;t certain Conclu&longs;ion. </s> <s>This being agreed on, we <lb/>will pa&longs;s in the next place to demon&longs;trate the <emph type="italics"/>Theoreme,<emph.end type="italics"/> namely, <lb/>that</s></p><pb xlink:href="069/01/156.jpg" pagenum="153"/><p type="head"> <s>THEOREM.</s></p><p type="main"> <s><emph type="italics"/>The degrees of Velocity of a Moveable de&longs;cending with a Natural <lb/>Motion from the &longs;ame height along Planes in any manner inclined <lb/>at the arrival to the Horizon are alwaies equal, Impediments be­<lb/>ing removed.<emph.end type="italics"/></s></p><p type="main"> <s>Here we are in the fir&longs;t place to adverti&longs;e you, that it having <lb/>been proved, that in any Inclination of the Plane the Move­<lb/>able from its rece&longs;&longs;ion from Quie&longs;&longs;ence goeth encrea&longs;ing its Ve­<lb/>locity, or quantity of its <emph type="italics"/>Impetus,<emph.end type="italics"/> with the proportion of the <lb/>Time (according to the Definition which the Author giveth of <lb/>Motion naturally Accelerate) whereupon, as he hath by the pre­<lb/>cedent Propo&longs;ition demon&longs;trated, the Spaces pa&longs;&longs;ed are in dupli­<lb/>cate proportion to the Times, and, con&longs;equently, to the degrees <lb/>of Velocity: look what the <emph type="italics"/>Impetus's<emph.end type="italics"/> were in that which was fir&longs;t <lb/>moved, &longs;uch proportionally &longs;hall be the degrees of Velocity gai­<lb/>ned in the &longs;ame Time; &longs;eeing that both the&longs;e and tho&longs;e encrea&longs;e <lb/>with the &longs;ame proportion in the &longs;ame Time.</s></p><p type="main"> <s>Now let the inclined Plane be A B, its elevation above the Ho <lb/>rizon the Perpendicular A C, and the Horizontal Plane C B: and <lb/>becau&longs;e, as was even now concluded, the <emph type="italics"/>Impetus<emph.end type="italics"/> of a Moveable <lb/>along the Perpendicular A C is to the <emph type="italics"/>Impetus<emph.end type="italics"/> of the &longs;ame along <lb/>the inclined Plane A B, as A B is to A C, let there be taken in the <lb/>inclined Plane A B, A D a third proportional to A B and A C: <lb/>The <emph type="italics"/>Impetus,<emph.end type="italics"/> therefore, along A C is to the <emph type="italics"/>Impetus<emph.end type="italics"/> along A B, <lb/>that is along A D, as A C is to <lb/><figure id="id.069.01.156.1.jpg" xlink:href="069/01/156/1.jpg"/><lb/>A D: And therefore the Move­<lb/>able in the &longs;ame Time that it <lb/>would pa&longs;s the Perpendicular <lb/>Space AC, &longs;hall likewi&longs;e pa&longs;s the <lb/>Space A D, in the inclined Plane <lb/>A B, (the Moments being as <lb/>the Spaces:) And the degree of Velocity in C &longs;hall have the &longs;ame <lb/>proportion to the degree of Velocity in D, as A C hath to A D: <lb/>But the degree of Velocity in B is to the &longs;ame degree in D, as the <lb/>Time along A B is to the Time along AD, by the definition of <lb/>Accelerate Motion; And the Time along AB is to the Time along <lb/>A D, as the &longs;ame A C, the Mean Proportional between B A and <lb/>A D, is to A D, by the la&longs;t Corollary of the &longs;econd Propo&longs;ition: <lb/>Therefore the degrees of Velocity in B and in C have to the de­<lb/>gree in D, the &longs;ame Proportion as A C hath to A D; and therefore <lb/>are equal: Which is the <emph type="italics"/>Theorem<emph.end type="italics"/> intended to be demon&longs;trated.</s></p><p type="main"> <s>By this we may more concludingly prove the en&longs;uing third <pb xlink:href="069/01/157.jpg" pagenum="154"/>Propo&longs;ition of the Author, in which he makes u&longs;e of this Princi­<lb/>ple; and it is, That the Time along the inclined Plane, hath to the <lb/>Time along the Perpendicular, the &longs;ame proportion as the &longs;aid In­<lb/>clined Plane and Perpendicular. </s> <s>For if we put the ca&longs;e that BA <lb/>be the Time along A B, the Time along A D &longs;hall be the Mean <lb/>between them, that is A C, by the &longs;econd Corollary of the &longs;econd <lb/>Propo&longs;ition: But if C A be the Time along A D, it &longs;hall likewi&longs;e <lb/>be the Time along <emph type="italics"/>A<emph.end type="italics"/> C, by rea&longs;on that <emph type="italics"/>A<emph.end type="italics"/> D and <emph type="italics"/>A<emph.end type="italics"/> C are pa&longs;t in <lb/>equal Times: And therefore in ca&longs;e B <emph type="italics"/>A<emph.end type="italics"/> be the Time along A B, <lb/><emph type="italics"/>A<emph.end type="italics"/> C &longs;hall be the Time along <emph type="italics"/>A<emph.end type="italics"/> C: Therefore, as <emph type="italics"/>A<emph.end type="italics"/> B is to A C, &longs;o <lb/>is the Time along <emph type="italics"/>A<emph.end type="italics"/> B to the Time along <emph type="italics"/>A<emph.end type="italics"/> C.</s></p><p type="main"> <s>By the &longs;ame di&longs;cour&longs;e one &longs;hall prove, that the Time along <emph type="italics"/>A<emph.end type="italics"/> C <lb/>is to the Time along the inclined Plane <emph type="italics"/>A<emph.end type="italics"/> E, as <emph type="italics"/>A<emph.end type="italics"/> C is to <emph type="italics"/>A<emph.end type="italics"/> E: <lb/>Therefore, <emph type="italics"/>ex æquali,<emph.end type="italics"/> the Time along the inclined Plane <emph type="italics"/>A B<emph.end type="italics"/> is, <lb/>Directly, to the Time along the inclined Plane <emph type="italics"/>A<emph.end type="italics"/> E as <emph type="italics"/>A B<emph.end type="italics"/> to <lb/><emph type="italics"/>A E, &c.<emph.end type="italics"/></s></p><p type="main"> <s>One might al&longs;o by the &longs;ame application of the <emph type="italics"/>Theorem,<emph.end type="italics"/> as <emph type="italics"/>Sa­<lb/>gredus<emph.end type="italics"/> &longs;hall very evidently &longs;ee anon, immediately demon&longs;trate the <lb/>&longs;ixth Propo&longs;ition of the <emph type="italics"/>A<emph.end type="italics"/>uthor: <emph type="italics"/>B<emph.end type="italics"/>ut let this Digre&longs;&longs;ion &longs;uffice <lb/>for the pre&longs;ent, which he perhaps thinketh too tedious, though in­<lb/>deed it is of &longs;ome importance in the&longs;e matters of Motion.</s></p><p type="main"> <s>SAGR. </s> <s>You may &longs;ay extreamly delightful, and mo&longs;t nece&longs;&longs;ary <lb/>to the perfect under&longs;tanding of that Principle.</s></p><p type="main"> <s>SALV. </s> <s>I will go on, then, in my Reading of the Text.</s></p><p type="head"> <s>THEOR. III. PROP. III.</s></p><p type="main"> <s>If a Moveable departing from Re&longs;t do move along <lb/>an Inclined Plane, and al&longs;o along the Perpendi­<lb/>cular who&longs;e heights are the &longs;ame, the Times of <lb/>their Motions &longs;hall be to one another as the <lb/>Lengths of the &longs;aid Plane and Perpendicular.</s></p><p type="main"> <s><emph type="italics"/>Let the inclined Plane be A C, and the Perpendicular A B, <lb/>who&longs;e heights are the &longs;ame above the Horizon C B, to wit, <lb/>the &longs;elf &longs;ame Line B A. </s> <s>I &longs;ay, that the Time of the De&longs;cent <lb/>of the &longs;ame Moveable upon the Plane A C, hath the &longs;ame Proporti­<lb/>on to the Time of the De&longs;cent along the Perpendicular A B, as the <lb/>Length of the Plane A C hath to the Length of the &longs;aid Perpendi­<lb/>cular. </s> <s>For let any number of Lines D G, E I, F L, be drawn, Paral­<lb/>lel to the Horizon C B: It is manife&longs;t from the A&longs;&longs;umption fore­<lb/>going, that the degrees of Velocity of the Moveable, departing from <lb/>A the beginning of Motion, acquired in the Points G and D are<emph.end type="italics"/><pb xlink:href="069/01/158.jpg" pagenum="155"/><emph type="italics"/>equal, their exce&longs;&longs;e or elevation above the Horizon being equal; <lb/>and &longs;o the degrees in the Points I and E; as al&longs;o the degrees in L <lb/>and F. </s> <s>And if not only the&longs;e Parallels, but many more were &longs;up­<lb/>po&longs;ed to be drawn from all the points imagined to be in the Line <lb/>A B, untill they meet the Line A C, the Mo-<emph.end type="italics"/><lb/><figure id="id.069.01.158.1.jpg" xlink:href="069/01/158/1.jpg"/><lb/><emph type="italics"/>ments, or degrees of the Velocities along the <lb/>extreams [or ends] of every one of tho&longs;e <lb/>Parallels, &longs;hall be alwaies equal to one ano­<lb/>ther: Therefore the two Spaces A C and A B <lb/>are pa&longs;t with the &longs;ame degree of Velocity: <lb/>But it hath been demon&longs;trated, that if two <lb/>Spaces be pa&longs;&longs;ed by a Moveable with one <lb/>and the &longs;ame degree of Velocity, the Times <lb/>of the Motions have the &longs;ame proportion as <lb/>tho&longs;e Spaces: Therefore the Time of the Motion along A C is to the <lb/>Time along A B, as the Length of the Plane A C to the length of the <lb/>Perdendicular A B. </s> <s>Which was to be demon&longs;trated.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>It &longs;eemeth to me, that the &longs;ame might very clearly and <lb/>conci&longs;ely be concluded, it having fir&longs;t been proved that the &longs;um of <lb/>the Accelerate Motion of the Tran&longs;itions along A C and A B, is <lb/>as much as the Equable Motion, who&longs;e degree of Velocity is &longs;ub­<lb/>duple to the greate&longs;t degree C B: Therefore the two Spaces AC <lb/>and A B being pa&longs;&longs;ed with the &longs;ame Equable Motion, it hath been <lb/>&longs;hewn, by the Fir&longs;t Propo&longs;ition of the fir&longs;t, that the Times of the <lb/>Tran&longs;itions &longs;hall be as the &longs;aid Spaces.</s></p><p type="head"> <s>COROLLARY.</s></p><p type="main"> <s>Hence is collected, that the Times of the De&longs;cents along Planes <lb/>of different Inclination, but of the &longs;ame Elevation, are to <lb/>one another according to their Lengths.</s></p><p type="main"> <s><emph type="italics"/>For if we &longs;uppo&longs;e another Plane A M, coming from A, and ter­<lb/>minated by the &longs;ame Horizontal C B; it &longs;hall in like manner be <lb/>demon&longs;trated, that the Time of the De&longs;cent along A M, is to the <lb/>Time along A B, as the Line A M to A B: But as the Time A B is <lb/>to the Time along A C, &longs;o is the Line A B to A C: Therefore,<emph.end type="italics"/> ex <lb/>æquali, <emph type="italics"/>as A M is to A C, &longs;o is the Time along A M to the Time <lb/>along A C.<emph.end type="italics"/></s></p><pb xlink:href="069/01/159.jpg" pagenum="156"/><p type="head"> <s>THEOR. IV. PROP. IV.</s></p><p type="main"> <s>The Times of the Motions along equal Planes, <lb/>but unequally inclined, are to each other in <lb/>&longs;ubduple proportion of the Elevations of tho&longs;e <lb/>Planes Reciprocally taken.</s></p><p type="main"> <s><emph type="italics"/>Let there proceed from the term B two equal Planes, but une­<lb/>qually inclined, B A and B C, and let A E and C D be Hori­<lb/>zontal Lines, drawn as far as the Perpendicular B D: Let the <lb/>Elevation of the Plane B A be B E; and let the Elevation of the <lb/>Plane B C be B D: And let B I be a Mean Proportional between the <lb/>Elevations D B and B E: It is manife&longs;t<emph.end type="italics"/><lb/><figure id="id.069.01.159.1.jpg" xlink:href="069/01/159/1.jpg"/><lb/><emph type="italics"/>that the proportion of D B to B I, is &longs;ub­<lb/>duple the proportion of D B to B E. </s> <s>Now <lb/>I &longs;ay, that the proportion of the Times <lb/>of the De&longs;cents or Motions along the <lb/>Planes B A and B C, are the &longs;ame with <lb/>the proportion of D B to B I Reciprocal­<lb/>ly taken: So that to the Time B A the <lb/>Elevation of the other Plane B C, that is <lb/>B D be Homologal; and to the Time along <lb/>B C, B I be Homologal: Therefore it is <lb/>to be demon&longs;trated, That the Time along B A is to the Time along <lb/>B C, as D B is to B I. </s> <s>Let I S be drawn equidi&longs;tant from D C. </s> <s>And <lb/>becau&longs;e it hath been demon&longs;trated that the Time of the De&longs;cent <lb/>along B A, is to the Time of the De&longs;cent along the Perpendicular <lb/>B E, as the &longs;aid B A is to B E; and the Time along B E is to the <lb/>Time along B D, as B E is to B I; and the Time along B D is to the <lb/>Time along B C, as B D to B C, or as B I to B S: Therefore,<emph.end type="italics"/> ex æqua­<lb/>li, <emph type="italics"/>the Time along B A &longs;hall be to the Time along B C as B A to B S, <lb/>or as C B to BS: But C B is to B S, as D B to B I: Therefore the <lb/>Propo&longs;ition is manife&longs;t:<emph.end type="italics"/></s></p><pb xlink:href="069/01/160.jpg" pagenum="157"/><p type="head"> <s>THEOR. V. PROP. V.</s></p><p type="main"> <s>The proportion of the Times of the De&longs;cents <lb/>along Planes that have different Inclinations <lb/>and Lengths, and the Elivations unequal, is <lb/>compounded of the proportion of the Lengths <lb/>of tho&longs;e Planes, and of the &longs;ubduple proporti­<lb/>on of their Elevations Reciprocally taken.</s></p><p type="main"> <s><emph type="italics"/>Let A B and A C be Planes inclined after different manners, <lb/>who&longs;e Lengths are unequal, as al&longs;o their Elevations. </s> <s>I &longs;ay, <lb/>the proportion of the Time of the De&longs;cent along A C to the <lb/>Time along A B, is compounded of the proportion of the &longs;aid A C <lb/>to A B, and of the &longs;ubduple proportion of their Elevation Recipro­<lb/>cally taken. </s> <s>For let the Perpendicular A D be drawn, with which <lb/>let the Horizontal Lines B G and C D inter&longs;ect, and let A L be a <lb/>Mean-proportional between C A and A E; and from the point L let <lb/>a Parallel be drawn to the Horizon inter&longs;ecting<emph.end type="italics"/><lb/><figure id="id.069.01.160.1.jpg" xlink:href="069/01/160/1.jpg"/><lb/><emph type="italics"/>the Plane A C in F; and A F &longs;hall be a Mean <lb/>proportional between C A and A E. </s> <s>And becau&longs;e <lb/>the Time along A C is to the Time along A E, as <lb/>the Line F A to A E; and the Time along A E is <lb/>to the Time along A B, as the &longs;aid A E to the &longs;aid <lb/>A B: It is manife&longs;t that the Time along A C is to <lb/>the Time along A B, as A F to A B. </s> <s>It remaineth, <lb/>therefore, to be demon&longs;trated, that the proportion <lb/>of A F to A B is compounded of the proportion of <lb/>C A to A B, and of the proportion of G A to A L; <lb/>which is the &longs;ubduple proportion of the Elevati­<lb/>ons D A and A G Reciprocally taken. </s> <s>But that is manife&longs;t, C A <lb/>being put between F A and A B: For the proportion of F A to A C <lb/>is the &longs;ame as that of L A to A D, or of G A to A L; which is &longs;ub­<lb/>duple of the proportion of the Elevations G A and A D; and the <lb/>proportion of C A to A B is the proportion of the Lengths: Therefore <lb/>the Propo&longs;ition is manife&longs;t.<emph.end type="italics"/></s></p><pb xlink:href="069/01/161.jpg" pagenum="158"/><p type="head"> <s>THEOR. VI. PROP. VI.</s></p><p type="main"> <s>If from the highe&longs;t or lowe&longs;t part of a Circle, <lb/>erect upon the Horizon, certain Planes be <lb/>drawn inclined towards the Circumference, <lb/>the Times of the De&longs;cents along the &longs;ame <lb/>&longs;hall be equal.</s></p><p type="main"> <s><emph type="italics"/>Let the Circle be erect upon the Horizon G H, who&longs;e Diameter <lb/>recited upon the lowe&longs;t point, that is upon the contact with the <lb/>Horizon, let be F A, and from the highe&longs;t point A let certain <lb/>Planes A B and A C incline towards the Circumference: I &longs;ay that the <lb/>Times of the De&longs;cents along the &longs;ame are equal. </s> <s>Let B D and C E be <lb/>two Perpendiculars let fall unto the Diameter; and let A I be a Mean­<lb/>Proportional between the Altitudes<emph.end type="italics"/><lb/><figure id="id.069.01.161.1.jpg" xlink:href="069/01/161/1.jpg"/><lb/><emph type="italics"/>of the Planes E A and A D. </s> <s>And <lb/>becau&longs;e the Rectangles F A E and <lb/>F A D are equal to the Squares of <lb/>A C and A B; And al&longs;o becau&longs;e <lb/>that as the Rectangle F A E, is to <lb/>the Rectangle F A D, &longs;o is E A to <lb/>A D. </s> <s>Therefore as the Square of <lb/>C A is to the Square of B A, <lb/>&longs;o is the Line E A to the Line <lb/>A D. </s> <s>But as the Line E A is to <lb/>D A, &longs;o is the Square of I A to the Square of A D: Therefore <lb/>the Squares of the Lines C A and A B are to each other as the Squares <lb/>of the Lines I A and A D: And therefore as the Line C A is to A B, <lb/>&longs;o is I A to A D: But in the precedent Propo&longs;ition it hath been demon­<lb/>&longs;trated that the proportion of the Time of the De&longs;cent along A C to the <lb/>Time of the De&longs;cent by A B, is compounded of the proportions of C A <lb/>to A B, and of D A to A I, which is the &longs;ame with the proportion of <lb/>B A to A C: Therefore the proportion of the Time of the De&longs;cent along <lb/>A C, to the Time of the De&longs;cent along A B, is compounded of the pro­<lb/>portions of C A to A B, and of B A to A C: Therefore the proporti­<lb/>on of tho&longs;e Times is a proportion of equality: Therefore the Propo&longs;ition <lb/>is evident.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>The &longs;ame is another way demon&longs;trated from the Mechanicks: Name­<lb/>ly that in the en&longs;uing Figure the Moveable pa&longs;&longs;eth in equal Times along <lb/>C A and D A. </s> <s>For let B A be equal to the &longs;aid D A, ond let fall the <lb/>Perpendiculars B E and D F: It is manife&longs;t by the Elements of the<emph.end type="italics"/><pb xlink:href="069/01/162.jpg" pagenum="159"/><emph type="italics"/>Mechanicks: That the Moment of the Weight elevated upon the Plane <lb/>according to the Line A B C, is <lb/>to its total Moment, as B E to B A;<emph.end type="italics"/><lb/><figure id="id.069.01.162.1.jpg" xlink:href="069/01/162/1.jpg"/><lb/><emph type="italics"/>And that the Moment of the &longs;ame <lb/>Weight upon the Elevation A D, <lb/>is to its total Moment, as D F to <lb/>D A or B A: Therefore the Mo­<lb/>ment of the &longs;aid Weight upon the <lb/>Plane inclined according to D A, <lb/>is to the Moment upon the Plane <lb/>inclined according to A B C, as <lb/>the Line D F to the Line B E: <lb/>Therefore the Spaces which the <lb/>&longs;aid Weight &longs;hall pa&longs;&longs;e in equal <lb/>Times along the Inclined Planes C A and D A, &longs;hall be to each other as <lb/>the Line B E to D F; by the &longs;econd Propo&longs;ition of the Fir&longs;t Book: <lb/>But as B E is to D F, &longs;o A C is demon&longs;trated to be to D A: <lb/>Therefore the &longs;ame Moveable will in equal Times pa&longs;&longs;e the Lines <lb/>C A and D A.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>And that C A is to D A as B E is to D F, is thus demon&longs;trated.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Draw a Line from C to D; and by D and B draw the Lines <lb/>D G L, (cutting C A in the point I) and B H, Parallels to A F: <lb/>And the Angle A D I &longs;hall be equal to the Angle D C A, for that <lb/>the parts L A and A D of the Circumference &longs;ubtending them, are <lb/>equal, and the Angle D A C common to them both: Therefore of <lb/>the equiangled Triangles C A D and D A I, the &longs;ides about the <lb/>equal Angles &longs;hall be proportional: And as C A is to A D, &longs;o is <lb/>D A to A I, that is B A to A I, or H A to A G; that is, B E to <lb/>D F: Which was to be proved.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Or el&longs;e the &longs;ame &longs;hall be demon&longs;trated more &longs;peedily thus.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Vnto the Horizon A B, let a Circle be erect, who&longs;e Diameter is <lb/>perpendicular to the Horizon: and <lb/>from the highe&longs;t Term D let a Plane<emph.end type="italics"/><lb/><figure id="id.069.01.162.2.jpg" xlink:href="069/01/162/2.jpg"/><lb/><emph type="italics"/>at plea&longs;ure D F, be inclined to the <lb/>Circumference. </s> <s>I &longs;ay that the De­<lb/>&longs;cent along the Plane D F, and the <lb/>Fall along the Diameter B C, will <lb/>be pa&longs;&longs;ed by the &longs;ame Moveable in <lb/>equal Times. </s> <s>For let F G be drawn <lb/>parallel to the Horizon A B, which <lb/>&longs;hall be perpendicular to the Diameter <lb/>D C, and let a Line conjoyn F and <lb/>C: and becau&longs;e the Time of the Fall <lb/>along D C, is to the Time of the Fall along D G, as the Mean <lb/>Proportional between C D and D G, is to the &longs;aid D G; and the<emph.end type="italics"/><pb xlink:href="069/01/163.jpg" pagenum="160"/><emph type="italics"/>Mean between C D and D G being D F, (for that the Angle D F C <lb/>in the Semicircle, is a Right Angle, and F G perpendicular to D C:) <lb/>Therefore the Time of the Fall along D C is to the Time of the Fall <lb/>along D G, as the Line F D to D G: But it hath been demon&longs;trated <lb/>that the Time of the De&longs;cent along D F, is to the Time of the Fall <lb/>along D G, as the &longs;ame Line D F is to D G: The Times, therefore, <lb/>of the De&longs;cent along D F and Fall along D C, are to the Time of the <lb/>Fall along the &longs;aid D G in the &longs;ame proportion: Therefore they are <lb/>equal. </s> <s>It will likewi&longs;e be demon&longs;trated, if from the lowe&longs;t Term C, <lb/>one &longs;hould rai&longs;e the Chord C E, and draw E H parallel to the Hori­<lb/>zon, and conjoyn E and D, that the Time of the De&longs;cent along E C <lb/>equals the Time of the Fall along the Diameter D C.<emph.end type="italics"/></s></p><p type="head"> <s>COROLLARY I.</s></p><p type="main"> <s>Hence is collected that the Times of the De&longs;cents along all the <lb/>Chords drawn from the Terms C or D are equal to one <lb/>another.</s></p><p type="head"> <s>COROLLARY II.</s></p><p type="main"> <s>It is al&longs;o collected that if the Perpendicular and inclined Plane <lb/>de&longs;cend from the &longs;ame point along which the De&longs;cents are <lb/>made in equal Times, they are in a Semicircle who&longs;e Dia­<lb/>meter is the &longs;aid Perpendicular.</s></p><p type="head"> <s>COROLLARY III.</s></p><p type="main"> <s>Hence it is collected that the Times of the Motions along inclined <lb/>Planes, are then equal, where the Elevations of equal parts of <lb/>tho&longs;e Planes &longs;hall be to one another as their Longitudes.</s></p><p type="main"> <s><emph type="italics"/>For it hath been &longs;hewn that the Times C A and D A in the la&longs;t Fi­<lb/>gure &longs;ave one are equal, the Elevation of the part A B being equal <lb/>to A D, that is, that B E &longs;hall be to the Elevation D F, as C A <lb/>to D A.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>Pray you Sir be plea&longs;ed to &longs;tay your Reading of what <lb/>followeth until that I have &longs;atisfied my &longs;elf in a Contemplation <lb/>that ju&longs;t now cometh into my mind, which if it be not a delu&longs;i­<lb/>on, is not far from being a plea&longs;ing diverti&longs;ement: as are all &longs;uch <lb/>that proceed from Nature or nece&longs;&longs;ity.</s></p><p type="main"> <s>It is manife&longs;t, that if from a point a&longs;&longs;igned in an Horizontal <lb/>Plane, one &longs;hall produce along the &longs;ame Plane infinite right Lines <lb/>every way, upon each of which a point is under&longs;tood to move with <lb/>an Equable Motion, all beginning to move in the &longs;ame in&longs;tant <pb xlink:href="069/01/164.jpg" pagenum="161"/>of Time from the a&longs;&longs;igned point, and the Velocities of them all <lb/>being equal, there &longs;hall con&longs;equently be de&longs;cribed by tho&longs;e move­<lb/>able points Circumferences of Circles alwayes bigger and bigger, <lb/>all concentrick about the fir&longs;t point a&longs;&longs;igned: ju&longs;t in the &longs;ame <lb/>manner as we &longs;ee it done in the Undulations of &longs;tanding Water, <lb/>when a &longs;tone is dropt into it; the percu&longs;&longs;ion of which &longs;erveth to <lb/>give the beginning to the Motion on every &longs;ide, and remaineth <lb/>as the Center of all the Circles that happen to be de&longs;igned &longs;ucce&longs;­<lb/>&longs;ively bigger and bigger by the &longs;aid Undulations. </s> <s>But if we ima­<lb/>gine a Plane erect unto the Horizon, and a point be noted in the <lb/>&longs;ame on high, from which infinite Lines are drawn inclined, ac­<lb/>cording to all inclinations, along which we fancy grave Movea­<lb/>bles to de&longs;cend, each with a Motion naturally Accelerate <lb/>with tho&longs;e Velocities that agree with the &longs;everal Inclinations; <lb/>&longs;uppo&longs;ing that tho&longs;e de&longs;cending Moveables were continually vi&longs;i­<lb/>ble, in what kind of Lines &longs;hould we &longs;ee them continually di&longs;po&longs;ed? <lb/></s> <s>Hence my wonder ari&longs;eth, &longs;ince that the precedent Demon&longs;trati­<lb/>ons a&longs;&longs;ure me, that they &longs;hall all be alwayes &longs;een in one and the <lb/>&longs;ame Circumference of Circles &longs;ucce&longs;&longs;ively encrea&longs;ing, according <lb/>as the Moveables in de&longs;cending go more and more &longs;ucce&longs;&longs;ively re­<lb/>ceding from the highe&longs;t point in which their Fall began: And the <lb/>better to declare my &longs;elf, let the chiefe&longs;t point A be marked, from <lb/>which Lines de&longs;cend according to any Inclinations A F, A H, and <lb/>the Perpendicular A B, in which taking the points C and D, de­<lb/>&longs;cribe Circles about them that pa&longs;s by <lb/><figure id="id.069.01.164.1.jpg" xlink:href="069/01/164/1.jpg"/><lb/>the point A, inter&longs;ecting the inclined <lb/>Lines in the points F, H, B, and E, G, <lb/>I. </s> <s>It is manife&longs;t, by the fore-going <lb/>Demon&longs;trations, that Moveables de­<lb/>&longs;cendent along tho&longs;e Lines departing <lb/>at the &longs;ame Time from the term A, <lb/>one &longs;hall be in E, the other &longs;hall be in <lb/>G, and the other in I; and &longs;o con­<lb/>tinuing to de&longs;cend they &longs;hall arrive <lb/>in the &longs;ame moment of Time at F, H, <lb/>and B: and the&longs;e and infinite others continuing to move along the <lb/>infinite differing Inclinations, they &longs;hall alwayes &longs;ucce&longs;&longs;ively arrive <lb/>at the &longs;elf-&longs;ame Circumferences made bigger & bigger <emph type="italics"/>in infinitum.<emph.end type="italics"/><lb/>From the two Species, therefore, of Motion of which Nature makes <lb/>u&longs;e, ari&longs;eth, with admirable harmonious variety, the generation of in­<lb/>&longs;inite Circles. </s> <s>She placeth the one as in her Seat, and original be­<lb/>ginning, in the Center of infinite concentrick Circles; the other <lb/>is con&longs;tituted in the &longs;ublime or highe&longs;t Contact of infinite Circum­<lb/>ferences of Circles, all excentrick to one another: Tho&longs;e proceed <lb/>from Motions all equal and Equable; The&longs;e from Motions all al­<pb xlink:href="069/01/165.jpg" pagenum="162"/>wayes Inequable to them&longs;elves, and all unequal to one another, <lb/>that de&longs;cend along the infinite different Inclinations. </s> <s>But we fur­<lb/>ther adde, that if from the two points a&longs;&longs;igned for the Emanations, <lb/>we &longs;hall &longs;uppo&longs;e Lines to proceed, not onely along two Superfi­<lb/>cies Horizontal and Upright [or erect] but along all every ways <lb/>like as from tho&longs;e, beginning at one &longs;ole point, we pa&longs;&longs;ed to the <lb/>production of Circles from the lea&longs;t to the greate&longs;t, &longs;o beginning <lb/>from one &longs;ole point we &longs;hall &longs;ucce&longs;&longs;ively produce in&longs;inite Spheres, <lb/>or we may &longs;ay one Sphere, that &longs;hall <emph type="italics"/>gradatim<emph.end type="italics"/> increa&longs;e to infinite <lb/>bigne&longs;&longs;es: And this in two fa&longs;hions; that is, either with placing <lb/>the original in the Center, or el&longs;e in the Circumference of tho&longs;e <lb/>Spheres.</s></p><p type="main"> <s>SALV. </s> <s>The Contemplation is really ingenuous, and adequate <lb/>to the Wit of <emph type="italics"/>Sagredus.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. </s> <s>Though I am at lea&longs;t capable of the Speculation, ac­<lb/>cording to the two manners of the production of Circles and <lb/>Spheres, with the two different Natural Motions, howbeit I do <lb/>not perfectly under&longs;tand the production depending on the Acce­<lb/>lerate Motion and its Demon&longs;tration, yet notwith&longs;tanding that <lb/>licence of a&longs;&longs;igning for the place of that Emanation as well the <lb/>lowe&longs;t Center, as the highe&longs;t Spherical Superficies, maketh me to <lb/>think that its po&longs;&longs;ible that &longs;ome great Mi&longs;tery may be contained <lb/>in the&longs;e true and admirable Conclu&longs;ions: &longs;ome Mi&longs;tery I &longs;ay <lb/>touching the Creation of the Univer&longs;e, which is held to be of <lb/>Spherical form, and concerning the Re&longs;idence of the Fir&longs;t <lb/>Cau&longs;e.</s></p><p type="main"> <s>SALV. </s> <s>I am not unwilling to think the &longs;ame: but &longs;uch pro­<lb/>found Speculations are to be expected from Sharper Wits than <lb/>ours. </s> <s>And it &longs;hould &longs;uffice us, that if we be but tho&longs;e le&longs;&longs;e noble <lb/>Workmen that di&longs;cover and draw forth of the Quarry the <lb/>Marbles, in which the Indu&longs;trious Statuaries afterwards make <lb/>wonderful Images appear, that lay hid under rude and mi&longs;haped <lb/>Cru&longs;ts. </s> <s>Now, if you plea&longs;e, we will go on.</s></p><p type="head"> <s>THEOR. VII. PROP. VII.</s></p><p type="main"> <s>If the Elevations of two Planes &longs;hall have a pro­<lb/>portion double to that of their Lengths, the <lb/>Motions in them from Re&longs;t &longs;hall be fini&longs;hed in <lb/>equal Times.</s></p><p type="main"> <s><emph type="italics"/>Let A E and A B be two unequal Planes, and unequally inclined, <lb/>and let their Elevations be F A and D A, and let F A have the <lb/>&longs;ame proportion to D A, as A E hath to A B. </s> <s>I &longs;ay that the Times <lb/>of the Motions along the Planes A E and A B, out of Re&longs;t in A are<emph.end type="italics"/><pb xlink:href="069/01/166.jpg" pagenum="163"/><emph type="italics"/>equal. </s> <s>Draw Horizontal Parallels to the Line of Elevation E F and <lb/>B D, which cutteth A E in G. </s> <s>And be-<emph.end type="italics"/><lb/><figure id="id.069.01.166.1.jpg" xlink:href="069/01/166/1.jpg"/><lb/><emph type="italics"/>cau&longs;e the proportion of F A to A D, is <lb/>double the proportion of E A to A B; and <lb/>as F A to A D, &longs;o is E A to A G: There­<lb/>fore the proportion of E A to A G, is dou­<lb/>ble the proportion of E A to A B: There­<lb/>fore A B is a Mean-Proportional between <lb/>E A and A G: And becau&longs;e the Time of the <lb/>De&longs;cent along A B, is to the Time of the De­<lb/>&longs;cent along A G, as A B to A G; and the <lb/>Time of the De&longs;cent along AG, is to the Time of the De&longs;cent along A E, as <lb/>A G is to the Mean-proportional between A G and A E, which is A B: <lb/>Therefore<emph.end type="italics"/> ex equali, <emph type="italics"/>the Time along A B is to the Time along A E, as A B <lb/>unto it &longs;elf: Therefore the Times are equal: Which was to be demon&longs;trated.<emph.end type="italics"/></s></p><p type="head"> <s>THEOR. VIII. PROP. VIII.</s></p><p type="main"> <s>In Planes cut by the &longs;ame Circle, erect to the <lb/>Horizon, in tho&longs;e which meet with the end of <lb/>the erect Diameter, whether upper or lower, <lb/>the Times of the Motions are equal to the <lb/>Time of the Fall along the Diameter: and in <lb/>tho&longs;e which fall &longs;hort of the Diameter, the <lb/>Times are &longs;horter; and in tho&longs;e which inter­<lb/>&longs;ect the Diameter, they are longer.</s></p><p type="main"> <s><emph type="italics"/>Let A B be the Perpendicular Diameter of the Circle erect to the <lb/>Horizon. </s> <s>That the Times of the Motions along the Planes pro­<lb/>duced out of the Terms A and B unto the Circumference are equal, <lb/>hath already been demon&longs;trated: That the Time of the De&longs;cent along <lb/>the Plane D F, not reaching to the<emph.end type="italics"/><lb/><figure id="id.069.01.166.2.jpg" xlink:href="069/01/166/2.jpg"/><lb/><emph type="italics"/>Diameter is &longs;borter, is demon&longs;trated <lb/>by drawing the Plane D B, which <lb/>&longs;hall be both longer and le&longs;&longs;e decli­<lb/>ning than D F. </s> <s>Therefore the Time <lb/>along D F is &longs;horter than the Time <lb/>along D B, that is, along A B. </s> <s>And <lb/>that the Time of the De&longs;cent along <lb/>the Plane that inter&longs;ecteth the Dia­<lb/>meter, as C O is longer, doth in the <lb/>&longs;ame manner appear, for that it is <lb/>longer and le&longs;&longs;e declining than C B: Therefore the Propo&longs;ition is de­<lb/>mon&longs;trated.<emph.end type="italics"/></s></p><pb xlink:href="069/01/167.jpg" pagenum="164"/><p type="head"> <s>THEOR. IX. PROP. IX.</s></p><p type="main"> <s>If two Planes be inclined at plea&longs;ure from a point <lb/>in a Line parallel to the Horizon, and be inter­<lb/>&longs;ected by a Line which may make Angles Al­<lb/>ternately equal to the Angles contained be­<lb/>tween the &longs;aid Planes and Horizontal Parallel, <lb/>the Motion along the parts cut off by the &longs;aid <lb/>Line, &longs;hall be performed in equal Times.</s></p><p type="main"> <s><emph type="italics"/>From off the point C of the Horizontal Line X, let any two Planes <lb/>be inclined at plea&longs;ure C D and C E, and in any point of the <lb/>Line C D make the Angle C D F equal to the Angle X C E: <lb/>and let the Line D F cut the Plane C E in F, in &longs;uch a manner that <lb/>the Angles C D F and C F D may be equal to the Angles X C E, L C D <lb/>Alternately taken. </s> <s>I &longs;ay, that<emph.end type="italics"/><lb/><figure id="id.069.01.167.1.jpg" xlink:href="069/01/167/1.jpg"/><lb/><emph type="italics"/>the Times of the De&longs;cents along <lb/>C D and C F are equal. </s> <s>And <lb/>that (the Angle C D F being <lb/>&longs;uppo&longs;ed equal to the Angle <lb/>X C E) the Angle C F D is <lb/>equal to the Angle D C L, is <lb/>manife&longs;t. </s> <s>For the Common An­<lb/>gle D C F being taken from the <lb/>three Angles of the Triangle <lb/>C D F equal to two Right An­<lb/>gles, to which are equal all the Angles made with to the Line L X <lb/>at the point C, there remains in the Triangle two Angles C D F and <lb/>C F D, equal to the two Angles X C E and L C D: But it was &longs;up­<lb/>po&longs;ed that C D F is equal to the Angle X C E: Therefore the remaining <lb/>Angle C F D is equal to the remaining angle D C L. </s> <s>Let the Plane <lb/>C E be &longs;uppo&longs;ed equal to the Plane C D, and from the points D and <lb/>E rai&longs;e the Perpendiculars D A and E B, unto the Horizontal Paral­<lb/>lel X L; and from C unto D F let fall the Perpendicular C G. </s> <s>And <lb/>becau&longs;e the Angle C D G is equal to the Angle E C B; and becau&longs;e <lb/>D G C and C B E are Right Angles; The Triangles C D G and <lb/>C B E &longs;hall be equiangled: And as D C is to C G, &longs;o let C E be <lb/>to E B: But D C is equal to C E: Therefore C G &longs;hall be equal to <lb/>E B. </s> <s>And inregard that of the Triangles D A C and C G F, the An­<lb/>gles C and A are equal to the Angles F and G: Therefore as C D is to <lb/>D A, &longs;o &longs;hall F C be to C G; and Alternately, as D C is to C F, &longs;o<emph.end type="italics"/><pb xlink:href="069/01/168.jpg" pagenum="165"/><emph type="italics"/>is D A to C G, or B E. </s> <s>The proportion therefore of the Elevations <lb/>of the Planes equal to C D and C E, is the &longs;ame with the proportion <lb/>of the Longitudes D C and C E: Therefore, by the fir&longs;t Corollary of <lb/>the precedent Sixth Propo&longs;ition, the Times of the Dc&longs;cent along the <lb/>&longs;ame &longs;hall be equal: Which mas to be proved.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Take the &longs;ame another way: Draw F S perpendicular to the <lb/>Horizontal Parallel A S. </s> <s>Becau&longs;e the Triangle C S F is like to <lb/>the Triangle D G C, it &longs;hall be, that as S F is to F C, &longs;o is G C <lb/>to C D. </s> <s>And becau&longs;e the Triangle C F G is like to the Triangle <lb/>D C A, it &longs;hall be, that as F C is to C G, &longs;o is C D to D A: <lb/>Therefore,<emph.end type="italics"/> ex æquali, <emph type="italics"/>as <lb/>S F is to C G, &longs;o is C G to <lb/>D A: Thorefore C G is a<emph.end type="italics"/><lb/><figure id="id.069.01.168.1.jpg" xlink:href="069/01/168/1.jpg"/><lb/><emph type="italics"/>Mean-proportional between <lb/>S F and D A: And as DA <lb/>is to S F, &longs;o is the Square <lb/>D A unto the Square C G <lb/>Again, the Triangle A C D <lb/>being like to the Triangle <lb/>C G F, it &longs;hall be, that as <lb/>D A is to D C, &longs;o is G C <lb/>to C F: and, Alternately, <lb/>as D A is to G C, &longs;o is D C to C F; and as the Square of D A <lb/>is to the Square of C G, &longs;o is the Square of D C to the Square of <lb/>C F. </s> <s>But it hath been proved that the Square D A is to the <lb/>Square C G as the Line D A is to the Line F S: Therefore, as the <lb/>Square D C is to the Square C F, &longs;o is the Line D E to F S: There­<lb/>fore, by the &longs;eventh fore-going, in regard that the Elevations D A <lb/>and F S, of the Planes C D, and C F are in double proportion to <lb/>their Planes; the Times of the Motions along the &longs;ame &longs;hall be <lb/>equal.<emph.end type="italics"/></s></p><p type="head"> <s>THEOR. X. PROP. X.</s></p><p type="main"> <s>The Times of the Motions along &longs;everal Inclina­<lb/>tions of Planes who&longs;e Elevations are equal, <lb/>are unto one another as the Lengths of tho&longs;e <lb/>Planes, whether the Motions be made from <lb/>Re&longs;t, or there hath proceeded a Motion from <lb/>the &longs;ame height.</s></p><p type="main"> <s><emph type="italics"/>Let the Motions be made along A B C, and along A B D, until <lb/>they come to the Horizon D C, in &longs;uch &longs;ort as that the Motion <lb/>along A B precedeth the Motions along B D and B C. </s> <s>I &longs;ay, <lb/>that the Time of the Motion along B D, is to the Time along B C, as<emph.end type="italics"/><pb xlink:href="069/01/169.jpg" pagenum="166"/><emph type="italics"/>the Length B D is to B C. </s> <s>Let A F be drawn parallel to the Ho­<lb/>rizon, to which continue out D B, meeting it in F; and let F E be <lb/>a Mean-proportional between D F and F B; and draw E O parallel <lb/>to D C, and A O &longs;hall be a Mean-proportional between C A and <lb/>A B: But if we &longs;uppo&longs;e the Time <lb/>along A B, to be as A B, the Time a-<emph.end type="italics"/><lb/><figure id="id.069.01.169.1.jpg" xlink:href="069/01/169/1.jpg"/><lb/><emph type="italics"/>long F B &longs;hall be as F B. </s> <s>And the <lb/>Time along all A C, &longs;hall be as the <lb/>Mean-proportional A O; and along <lb/>all F D &longs;hall be F E: Wherefore the <lb/>Time along the remainder B C &longs;hall <lb/>be B O; and along the remainder <lb/>B D &longs;hall be B E. </s> <s>But as B E is to <lb/>B O, &longs;o is B D to B C: Therefore <lb/>the Times along B D and B C, after the De&longs;cent along A B and <lb/>F B, or which is the &longs;ame, along the Common part A B, &longs;hall be to <lb/>one another as the Lengths B D and B C: But that the Time along <lb/>B D, is to the Time along B C, out of Re&longs;t in B, as the Length <lb/>B D to B C, hath already been demon&longs;trated. </s> <s>Therefore the Times <lb/>of the Motions along different Planes who&longs;e Elevations are equal, are <lb/>to one another as the Lengths of the &longs;aid Planes, whether the Motion <lb/>be made along the &longs;ame out of Re&longs;t, or whether another Motion of <lb/>the &longs;ame Altitude do precede tho&longs;e Motions: Which was to be de­<lb/>mon&longs;trated.<emph.end type="italics"/></s></p><p type="head"> <s>THEOR. XI. PROP. XI.</s></p><p type="main"> <s>If a Plane, along which a Motion is made out of <lb/>Re&longs;t, be divided at plea&longs;ure, the Time of <lb/>the Motion along the fir&longs;t part, is to the Time <lb/>of the Motion along the &longs;econd, as the &longs;aid <lb/>fir&longs;t part is to the exce&longs;&longs;e whereby the &longs;ame <lb/>part &longs;hall be exceeded by the Mean-Propor­<lb/>tional between the whole Plane and the &longs;ame <lb/>fir&longs;t part.</s></p><p type="main"> <s><emph type="italics"/>Let the Motion be along the whole Plane A B, ex quiete in A, <lb/>which let be divided at plea&longs;ure in C; and let A F be a Mean <lb/>proportional between the whole B A and the fir&longs;t part A C; <lb/>C F &longs;hall be the exce&longs;&longs;e of the Mean proportional F A above the part <lb/>A C. </s> <s>I &longs;ay the Time of the Motion along A C is to the Time of the <lb/>following Motion along C B, as A C to C F. </s> <s>Which is manife&longs;t;<emph.end type="italics"/><pb xlink:href="069/01/170.jpg" pagenum="167"/><emph type="italics"/>For the Time along A C is to the Time along all <lb/>A B, as A C to the Mean-proportional A F: There-<emph.end type="italics"/><lb/><figure id="id.069.01.170.1.jpg" xlink:href="069/01/170/1.jpg"/><lb/><emph type="italics"/>fore, by Divi&longs;ion, the Time along A C, &longs;hall be to <lb/>the Time along the remainder C B as A C to C F: <lb/>If therefore the Time along A C be &longs;uppo&longs;ed to be <lb/>the &longs;aid A C, the Time along C B &longs;hall be C F: <lb/>Which was the Propo&longs;ition.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>But if the Motion be not made along the continu­<lb/>ate Plane A C B, but by the inflected Plane A C D <lb/>until it come to the Horizon B D, to which from F a Parallel is <lb/>drawn F E. </s> <s>It &longs;hall in like manner be<emph.end type="italics"/><lb/><figure id="id.069.01.170.2.jpg" xlink:href="069/01/170/2.jpg"/><lb/><emph type="italics"/>demon&longs;trated, that the Time along <lb/>A C is to the Time along the reflected <lb/>Plane C D, as A C is to C E. </s> <s>For <lb/>the Time along A C is to the Time a­<lb/>long C B, as A C is to C F: But the <lb/>Time along C B, after A C hath been <lb/>demon&longs;trated to be to the Time along <lb/>C D, after the &longs;aid De&longs;oent along <lb/>A C, as C B is to C D; that is, as <lb/>C F to C E: Therefore,<emph.end type="italics"/> ex æquali, <emph type="italics"/>the Time along A C &longs;hall be to <lb/>the Time along C D, as the Line A C to C E.<emph.end type="italics"/></s></p><p type="head"> <s>THEOR. XII. PROP. XII.</s></p><p type="main"> <s>If the Perpendicular and Plane Inclined at plea­<lb/>&longs;ure, be cut between the &longs;ame Horizontal <lb/>Lines, and Mean-Proportionals between <lb/>them and the parts of them contained betwixt <lb/>the common Section and upper Horizontal <lb/>Line be given; the Time of the Motion a­<lb/>long the Perpendicular &longs;hall have the &longs;ame <lb/>proportion to the Time of the Motion along <lb/>the upper part of the Perpendicular, and af­<lb/>terwards along the lower part of the inter&longs;e­<lb/>cted Plane, as the Length of the whole Per­<lb/>pendicular hath to the Line compounded of <lb/>the Mean-Proportional given upon the Per­<lb/>pendicular, and of the exce&longs;&longs;e by which the <lb/>whole Plane exceeds its Mean-Proporttonal.</s></p><pb xlink:href="069/01/171.jpg" pagenum="168"/><p type="main"> <s><emph type="italics"/>Let the Horizontal Lines be A F the upper, and C D the low­<lb/>er; between which let the Perpendicular A C, and inclined <lb/>Plane D F, be cut in B; and let A R be a Mean-Proportional <lb/>between the whole Perpendicular C A, and the upper part A B; and <lb/>let F S be a Mean-proportional between the whole Inclined Plane D F, <lb/>and the upper part B F. </s> <s>I &longs;ay, that the Time of the Fall along the <lb/>whole Perpendicular A C hath the &longs;ame proportion to the Time along <lb/>its upper part A B, with the lower of the Plane, that is, with B D, <lb/>as A C hath to the Mean-proporti­<lb/>onal of the Perpendicular, that is<emph.end type="italics"/><lb/><figure id="id.069.01.171.1.jpg" xlink:href="069/01/171/1.jpg"/><lb/><emph type="italics"/>A R, with S D, which is the ex­<lb/>ce&longs;&longs;e of the whole Plane D F above <lb/>its Mean-proportional F S. </s> <s>Let a <lb/>Line be drawn from R to S, which <lb/>&longs;hall be parallel to the two Horizon­<lb/>tal Lines. </s> <s>And becau&longs;e the Time of <lb/>the Fall along all A C, is to the <lb/>Time along the part A B, as C A is <lb/>to the Mean proportional A R, if we &longs;uppo&longs;e A C to be the Time of <lb/>the Fall along A C, A R &longs;hall be the Time of the Fall along A B, <lb/>and R C that along the remainder B C. </s> <s>For if the Time along A C <lb/>be &longs;uppo&longs;ed, as was done, to be A C it &longs;elf the Time along F D &longs;hall <lb/>be F D; and in like manner D S may be concluded to be the Time a­<lb/>long B D, after F B, or after A B. </s> <s>The Time therefore along the <lb/>whole A C, is A R, with R C; And the Time along the inflected <lb/>Plane A B D, &longs;hall be A R, with S D: Which was to be proved.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>The &longs;ame happeneth, if in&longs;tead of the Perpendicular, another <lb/>Plane were taken, as &longs;uppo&longs;e N O; and the Demonstration is the <lb/>&longs;ame.<emph.end type="italics"/></s></p><p type="head"> <s>PROBL I. PROP. XIII.</s></p><p type="main"> <s>A Perpendicular being given, to Inflect a Plane <lb/>unto it, along which, when it hath the &longs;ame <lb/>Elevation with the &longs;aid Perpendicular, it may <lb/>make a Motion after its Fall along the Per­<lb/>pendicular in the &longs;ame Time, as along the <lb/>&longs;ame Perpendicular <emph type="italics"/>ex quiete.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Let the Perpendicular given be A B, to which extended to C, <lb/>let the part B C be equal; and draw the Horizontal Lines <lb/>C E and A G. </s> <s>It is required from B to inflect a Plane reach­<lb/>ing to the Horizon C E, along which a Motion, after the Fall out<emph.end type="italics"/><pb xlink:href="069/01/172.jpg" pagenum="169"/><emph type="italics"/>of A, &longs;hall be made in the &longs;ame Time, as along A B from Re&longs;t in A. </s> <s>Let <lb/>C D be equal to C B, and drawing B D, let B E be applied equal to both <lb/>B D and D C. </s> <s>I &longs;ay B E is the Plane required. </s> <s>Continue out E B to <lb/>meet the Horizontal Line A G in G;<emph.end type="italics"/><lb/><figure id="id.069.01.172.1.jpg" xlink:href="069/01/172/1.jpg"/><lb/><emph type="italics"/>and let G F be a Mean-Proportional be­<lb/>tween the &longs;aid E G and G B. </s> <s>E F &longs;hall <lb/>be to F B, as E G is to G F; and the <lb/>Square E F &longs;hall be to the Square F B, as <lb/>the Square E G is to the Square G F; <lb/>that is as the Line E G to G B: But <lb/>E G is double to G B: Therefore the <lb/>Square of E F is double to the Square of F B: But al&longs;o the Square of <lb/>D B is double to the Square of B C: Therefore, as the Line E F is to <lb/>F B, &longs;o is D B to B C: And by Compo&longs;ition and Permutation, as E B is <lb/>to the two D B and B C, &longs;o is B F to B C: But B E is equal to the two <lb/>D B and B C: Therefore B F is equal to the &longs;aid B C, or B A. </s> <s>If there­<lb/>fore A B be under&longs;tood to be the Time of the Fall along A B, G B &longs;hall <lb/>be the Time along G B, and G F the Time along the whole G E: There­<lb/>fore B F &longs;hall be the Time along the remainder B E, after the Fall from <lb/>G, or from A, which was the Propo&longs;ition.<emph.end type="italics"/></s></p><p type="head"> <s>PROBL. II. PROP. XIV.</s></p><p type="main"> <s>A <emph type="italics"/>P<emph.end type="italics"/>erpendicular and a <emph type="italics"/>P<emph.end type="italics"/>lane inclined to it being <lb/>given, to find a part in the upper <emph type="italics"/>P<emph.end type="italics"/>erpendicu­<lb/>lar which &longs;hall be pa&longs;t <emph type="italics"/>ex quiete<emph.end type="italics"/> in a Time <lb/>equal to that in which the inclined <emph type="italics"/>P<emph.end type="italics"/>lane is <lb/>pa&longs;t after the Fall along the part found in the <lb/>Perpendicular.</s></p><p type="main"> <s><emph type="italics"/>Let the Perpendicular be D B, and the Plane inclined to it A C. </s> <s>It is <lb/>required in the Perpendicular A D to find a part which &longs;hall be <lb/>pa&longs;t<emph.end type="italics"/> ex quiete <emph type="italics"/>in a Time equal to that in which the Plane A C is <lb/>pa&longs;t after the Fall along the &longs;aid part. </s> <s>Draw the Horizontal Line C B; <lb/>and as B A more twice A C is to A C, &longs;o let E A be to A R; And from <lb/>R let fall the Perpendicular R X unto D B. </s> <s>I &longs;ay X is the point requi­<lb/>red. </s> <s>And becau&longs;e as B A more twice A C is to A C, &longs;o is C A to A E, <lb/>by Divi&longs;ion it &longs;hall be that as B A more A C is to A C, &longs;o is C E to E A: <lb/>And becau&longs;e as B A is to A C, &longs;o is E A to A R, by Compo&longs;ition it &longs;hall <lb/>be that as B A more A C is to A C, &longs;o is E R to R A: But as B A more <lb/>A C is to A C, &longs;o is C E to E A: Therefore, as C E is to E A, &longs;o is E R, <lb/>to R A, and both the Antecedents to both the Con&longs;equents, that is, C R<emph.end type="italics"/><pb xlink:href="069/01/173.jpg" pagenum="170"/><emph type="italics"/>to R E: Therefore C R, R E, and R A are Proportionals. </s> <s>Farther­<lb/>more, becau&longs;e as B A is to A C, &longs;o E A is &longs;uppo&longs;ed to be to A R, and,<emph.end type="italics"/><lb/><figure id="id.069.01.173.1.jpg" xlink:href="069/01/173/1.jpg"/><lb/><emph type="italics"/>in regard of the likene&longs;&longs;e of the Triangles, <lb/>as B A is to A C, &longs;o is X A to A R: There­<lb/>fore, as E A is to A R, &longs;o is X A to A R: <lb/>Therefore E A and X A are equal. </s> <s>Now if <lb/>we under&longs;tand the Time along R A to be as <lb/>R A, the Time along R C &longs;hall be R E, the <lb/>Mean-Proportional between C R and R A: <lb/>And A E &longs;hall be the Time along A C after <lb/>R A or after X A: But the Time along X A <lb/>is X A, &longs;o long as R A is the Time along R <lb/>A: But it hath been proved that X A and <lb/>A E are equal: Therefore the Propo&longs;ition is proved.<emph.end type="italics"/></s></p><p type="head"> <s>PROBL. III. <emph type="italics"/>P<emph.end type="italics"/>RO<emph type="italics"/>P.<emph.end type="italics"/> XV.</s></p><p type="main"> <s>A <emph type="italics"/>P<emph.end type="italics"/>erpendicular and a <emph type="italics"/>P<emph.end type="italics"/>lane inflected to it being <lb/>given, to find a part in the <emph type="italics"/>P<emph.end type="italics"/>erpendicular ex­<lb/>tended downwards which &longs;hall be pa&longs;&longs;ed in the <lb/>&longs;ame. </s> <s>Time as the inflected <emph type="italics"/>P<emph.end type="italics"/>lane after the Fall <lb/>along the given Perpendicular.</s></p><p type="main"> <s><emph type="italics"/>Let the Perpendicular be A B, and the Plane In&longs;lected to it B C. </s> <s>It <lb/>is required in the Perpendicular extended downwards to find a <lb/>part which from the Fall out of A &longs;hall be pa&longs;t in the &longs;ame Time as <lb/>B C is pa&longs;&longs;ed from the &longs;ame Fall out of A. </s> <s>Draw the Horizontal Line <lb/>A D, with which let C B meet extended to D; and let D E be a Mean­<lb/>proportional between C D and D B;<emph.end type="italics"/><lb/><figure id="id.069.01.173.2.jpg" xlink:href="069/01/173/2.jpg"/><lb/><emph type="italics"/>and let B F be equal to B E; and let <lb/>A G be a third Proportional to B A and <lb/>A F. </s> <s>I &longs;ay, B G is the Space that after <lb/>the Fall A B &longs;hall be pa&longs;t in the &longs;ame <lb/>Time, as the Plane B C &longs;hall be pa&longs;t af­<lb/>ter the &longs;ame Fall. </s> <s>For if we &longs;uppo&longs;e <lb/>the Time along A B to be as A B, the <lb/>Time along D B &longs;hall be as D B: And <lb/>becau&longs;e D E is the Mean-proportional <lb/>between B D and D C, the &longs;ame D E <lb/>&longs;hall be the Time along the whole D C, and B E the Time along the Part <lb/>or Remainder B C<emph.end type="italics"/> ex quiete, <emph type="italics"/>in D, or<emph.end type="italics"/> ^{*} ex ca&longs;u <emph type="italics"/>A B: And it may in<emph.end type="italics"/><lb/><arrow.to.target n="marg1093"></arrow.to.target><lb/><emph type="italics"/>like manner be proved, that B F is the Time along B G, after the &longs;ame <lb/>Fall: But B F is equal to B E: Which was the Propo&longs;ition to be proved.<emph.end type="italics"/></s></p><pb xlink:href="069/01/174.jpg" pagenum="171"/><p type="margin"> <s><margin.target id="marg1093"></margin.target>* From or after <lb/>the Fall A B.</s></p><p type="head"> <s>THEOR. XIII. <emph type="italics"/>P<emph.end type="italics"/>RO<emph type="italics"/>P.<emph.end type="italics"/> XVI.</s></p><p type="main"> <s>If the parts of an inclined <emph type="italics"/>P<emph.end type="italics"/>lane and <emph type="italics"/>P<emph.end type="italics"/>erpendicu­<lb/>lar, the Times of who&longs;e Motions <emph type="italics"/>ex quiete<emph.end type="italics"/> are <lb/>equal, be joyned together at the &longs;ame point, a <lb/>Moveable coming out of any &longs;ublimer Height <lb/>&longs;hall &longs;ooner pa&longs;&longs;e the &longs;aid part of the inclined <lb/><emph type="italics"/>P<emph.end type="italics"/>lane, than that part of the Perpendicular.</s></p><p type="main"> <s><emph type="italics"/>Let the Perpendicular be E B, and the Inclined Plane C E, joyned <lb/>at the &longs;ame Point E, the Times of who&longs;e Motions from off Re&longs;t in <lb/>E are equal, and in the Perpendicular continued out, let a &longs;ublime <lb/>point A be taken at plea&longs;ure, out of which the Moveables may be let <lb/>fall. </s> <s>I &longs;ay, that the Inclined Plane E C &longs;hall be pa&longs;&longs;ed in a le&longs;&longs;e Time <lb/>than the Perpendicular E B, after the Fall A E. </s> <s>Draw a Line from C <lb/>to B, and having drawn the Horizontal Line A D continue out C E till <lb/>it meet the &longs;ame in D; and let D F be a Mean-Proportional between <lb/>C D and D E; and let A G be a<emph.end type="italics"/><lb/><figure id="id.069.01.174.1.jpg" xlink:href="069/01/174/1.jpg"/><lb/><emph type="italics"/>Mean-Proportional between B A and <lb/>A E; and draw F G and D G. </s> <s>And <lb/>becau&longs;e the Time of the Motion along <lb/>E C and E B out of Re&longs;t in E are <lb/>equal, the Angle C &longs;hall be a Right <lb/>Angle, by the &longs;econd Corollary of the <lb/>Sixth Propo&longs;ition; and A is a Right <lb/>Angle, and the Vertical Angles <lb/>at E are equal: Therefore the Tri­<lb/>angles A E D and C E B are equian­<lb/>gled, and the Sides about equal An­<lb/>gles are Proportionals: Therefore as <lb/>B E is to E C, &longs;o is D E to E A. <lb/></s> <s>Therefore the Rectangle B E A is <lb/>equal to the Rectangle C E D: And <lb/>becau&longs;e the Rectangle C D E ex­<lb/>ceedeth the Rectangle C E D, by the Square E D, and the Rectangle <lb/>B A E doth exceed the Rectangle B E A, by the Square E A: The <lb/>exce&longs;&longs;e of the Rectangle C D E above the Rectangle B A E, that is of <lb/>the Square F D above the Square A G &longs;hall be the &longs;ame as the exce&longs;&longs;e <lb/>of the Square D E above the Square A E; which exce&longs;s is the <lb/>Square D A: Therefore the Square F D is equal to the two Squares <lb/>G A and A D, to which the Square G D is al&longs;o equal: Therefore the<emph.end type="italics"/><pb xlink:href="069/01/175.jpg" pagenum="172"/><emph type="italics"/>Line D F is equal to D G, and the Angle D G F is equal to the An­<lb/>gle D F G, and the Angle E G F is le&longs;&longs;c than the Angle E F G, and <lb/>the oppo&longs;ite Side E F le&longs;&longs;e than the Side E G. </s> <s>Now if we &longs;uppo&longs;e <lb/>the Time of the Fall along A E to be as A E, the Time by D E &longs;hall <lb/>be as D E; and A G being a Mean-Proportional between B A and A E, <lb/>A G &longs;hall be the Time along the whole A B, and the part E G &longs;hall be <lb/>the Time along the Part E B<emph.end type="italics"/> ex quiete <emph type="italics"/>in A. </s> <s>And it may in like man­<lb/>ner be proved that E F is the Time along E C after the De&longs;cent D E, or <lb/>after the Fall A E: But E F is proved to be le&longs;&longs;er than E G: Therefore <lb/>the Propo&longs;ition is proved.<emph.end type="italics"/></s></p><p type="head"> <s>COROLLARY.</s></p><p type="main"> <s>By this and the precedent it appears, that the Space that is pa&longs;­<lb/>&longs;ed along the Perpendicular after the Fall from above in the <lb/>&longs;ame Time in which the Inclined Plane is pa&longs;t, is le&longs;&longs;e than <lb/>that which is pa&longs;t in the &longs;ame Time as in the Inclined, no fall <lb/>from above preceding, yet greater than the &longs;aid Inclined <lb/>Plane.</s></p><p type="main"> <s><emph type="italics"/>For it having been proved, but now, that of the Moveables coming <lb/>from the &longs;ublime Term A the Time of the Conver&longs;ion along E C is <lb/>&longs;horter than the Time of the Progre&longs;&longs;ion along E B; It is manife&longs;t that <lb/>the Space that is pa&longs;t along E B in a Time equal to the Time along E C <lb/>is le&longs;s than the whole Space E B. </s> <s>And that the &longs;ame Space along the <lb/>Perpendicular is greater than E C is mani­<lb/>fe&longs;ted by rea&longs;&longs;uming the Figure of the pre-<emph.end type="italics"/><lb/><figure id="id.069.01.175.1.jpg" xlink:href="069/01/175/1.jpg"/><lb/><emph type="italics"/>cedent Propo&longs;ition, in which the part of the <lb/>Perpendicular B G hath been demon&longs;trated <lb/>to be pa&longs;&longs;ed in the &longs;ame Time as B C after <lb/>the Fall A B: But that B G is greater than <lb/>B C is thus collected. </s> <s>Becau&longs;e B E and F B <lb/>are equal, and B A le&longs;&longs;er than B D, F B, <lb/>hath greater proportion to B A, than E B <lb/>hath to B D: And, by Compo&longs;ition, F A <lb/>hath greater proportion to A B, than E D <lb/>to D B: But as F A is to A B, &longs;o is G F <lb/>to F B, (for A F is the Mean-Proportional <lb/>between B A and A G:) And in like man­<lb/>ner, as E D is to B D, &longs;o is C E to E B: Therefore G B hath greater <lb/>proportion to B F, than C B hath to B E: Therefore G B is greater <lb/>than B C.<emph.end type="italics"/></s></p><pb xlink:href="069/01/176.jpg" pagenum="173"/><p type="head"> <s>PROBL. IV. <emph type="italics"/>P<emph.end type="italics"/>RO<emph type="italics"/>P.<emph.end type="italics"/> XVII.</s></p><p type="main"> <s>A <emph type="italics"/>P<emph.end type="italics"/>erpendicular and <emph type="italics"/>P<emph.end type="italics"/>lane Inflected to it being <lb/>given, to a&longs;&longs;ign a part in the given <emph type="italics"/>P<emph.end type="italics"/>lane, in <lb/>which after the Fall along the <emph type="italics"/>P<emph.end type="italics"/>erpendicular <lb/>the Motion may be made in a Time equal to <lb/>that in which the Moveable <emph type="italics"/>ex quiete<emph.end type="italics"/> pa&longs;&longs;eth <lb/>the <emph type="italics"/>P<emph.end type="italics"/>erpendicular given.</s></p><p type="main"> <s><emph type="italics"/>Let the Perpendicular be A B, and a Plane Inflected to it B E: It is <lb/>required in B E to a&longs;&longs;ign a Space along which the Moveable af­<lb/>ter the Fall along A B may move in a Time equal to that in which <lb/>the &longs;aid Perpendicular A B is pa&longs;&longs;ed<emph.end type="italics"/> ex quiete. <emph type="italics"/>Let the Line A D be <lb/>parallel to the Horizon, with which let the Plane prolonged meet in D; <lb/>and &longs;uppo&longs;e F B equal to B A; and as B D <lb/>is to D F, &longs;o let F D be to D E. </s> <s>I &longs;ay, that<emph.end type="italics"/><lb/><figure id="id.069.01.176.1.jpg" xlink:href="069/01/176/1.jpg"/><lb/><emph type="italics"/>the Time along B E after the Fall along A B <lb/>equalleth the Time along A B, out of Re&longs;t <lb/>in A. </s> <s>For if we &longs;uppo&longs;e A B to be the Time <lb/>along A B, D B &longs;hall be the Time along <lb/>D B. </s> <s>And becau&longs;e, as B D is to D F, &longs;o is <lb/>F D to D E, D F &longs;hall be the Time along <lb/>the whole Plane D E, and B F along the <lb/>part B E out of D: But the Time along <lb/>B E after D B, is the &longs;ame as after A B: Therefore the Time along B E <lb/>after A B &longs;hall be B F, that is, equal to the Time<emph.end type="italics"/> ex quiete <emph type="italics"/>in A: <lb/>Which was the Propo&longs;ition.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>P<emph.end type="italics"/>ROBL. V. PROP. XVIII.</s></p><p type="main"> <s>Any Space in the Perpendicular being given from <lb/>the a&longs;&longs;igned beginning of Motion that is <lb/>pa&longs;&longs;ed in a Time given, and any other le&longs;&longs;er <lb/>Time being al&longs;o given, to find another Space in <lb/>the &longs;aid Perpendicular that may be pa&longs;&longs;ed in <lb/>the given le&longs;&longs;er Time.</s></p><pb xlink:href="069/01/177.jpg" pagenum="174"/><p type="main"> <s><emph type="italics"/>Let the Perpendicular be A D, in which let the Space a&longs;&longs;igned be <lb/>A B, who&longs;e Time from the beginning A let be A B: and let the <lb/>Horizon be C B E, and let a Time be given le&longs;s than A B, to <lb/>which let B C be noted equal in the Horizon: It is required in the <lb/>&longs;aid Perpendicular to find a Space equal to the &longs;ame A B that &longs;hall be <lb/>pa&longs;&longs;ed in the Time B C. </s> <s>Draw a Line from A to<emph.end type="italics"/><lb/><figure id="id.069.01.177.1.jpg" xlink:href="069/01/177/1.jpg"/><lb/><emph type="italics"/>C. </s> <s>And becau&longs;e B C is le&longs;&longs;e than B A, the Angle <lb/>B A C &longs;hall be le&longs;&longs;e than the Angle B C A. </s> <s>Let <lb/>C A E be made equal to it, and the Line A E meet <lb/>with the Horizon in the Point E, to which &longs;up­<lb/>po&longs;e E D a Perpendicular, cutting the Perpendi­<lb/>cular in D, and let D F be cut equal to B A. </s> <s>I <lb/>&longs;ay, that the &longs;aid F D is a part of the Perpendi­<lb/>cular along which the Lation from the beginning <lb/>of Motion in A, the Time B C given will be &longs;pent. <lb/></s> <s>For if in the Right-angled Triangle A E D, a <lb/>Perpendicular to the oppo&longs;ite Side A D, be drawn <lb/>E B, A E &longs;hall be a Mean-Proportional betwixt <lb/>D A and A B, and B E a Mean-Proportional betwixt D B and B A, <lb/>or betwixt F A and A B (for F A is equal to D B.) And in regard <lb/>A B hath been &longs;uppo&longs;ed to be the Time along A B, A E, or E C &longs;hall be <lb/>the Time along the whole A D, and E B the Time along A F: There­<lb/>fore the part B C &longs;hall be the Time along the part F D: Which was <lb/>intended.<emph.end type="italics"/></s></p><p type="head"> <s>PROBL. VI. PROP. XIX.</s></p><p type="main"> <s>Any Space in the Perpendicular pa&longs;&longs;ed from the <lb/>beginning of the Motion being given, and the <lb/>Time of the Fall being a&longs;&longs;igned, to find the <lb/>Time in which another Space. </s> <s>equal to the gi­<lb/>ven one, and taken in any part of the &longs;aid Per­<lb/>pendicular, &longs;hall be afterwards pa&longs;t by the <lb/>&longs;ame Moveable.</s></p><p type="main"> <s><emph type="italics"/>In the Perpendicular A B let A C be any Space taken from the be­<lb/>ginning of the Motion in A, to which let D B be another equal Space <lb/>taken any where at plea&longs;ure, and let the Time of the Motion along <lb/>A C be given, and let it be A C. </s> <s>It is required to &longs;ind the Time of the<emph.end type="italics"/><pb xlink:href="069/01/178.jpg" pagenum="175"/><emph type="italics"/>Motion along D B after the Fall from A. </s> <s>About the whole A B de­<lb/>&longs;cribe a Semicircle A E B, and from<emph.end type="italics"/><lb/><figure id="id.069.01.178.1.jpg" xlink:href="069/01/178/1.jpg"/><lb/><emph type="italics"/>C let fall C E, a Perpendicular to A <lb/>B, and draw a Line from A to E; <lb/>which &longs;hall be greater than E C. <lb/></s> <s>Let E F be out equall to E C: I &longs;ay, <lb/>that the remainder F A is the Time <lb/>of the Motion along D B. </s> <s>For be­<lb/>cau&longs;e A E is a Mean-proportional be­<lb/>twixt B A and and A C, and A C <lb/>is the Time of the Fall along A C; <lb/>A E &longs;hall be the Time along the <lb/>Whole A B. </s> <s>And becau&longs;e C E is a <lb/>Mean-proportional betwixt D A and <lb/>A C, (for D A is equal to B C) <lb/>C E, that is E F &longs;hall be the Time <lb/>along A D: Therefore the Remainder A F &longs;hall be the Time along the <lb/>Remainder B B: Which is the Propo&longs;ition.<emph.end type="italics"/></s></p><p type="head"> <s>COROLLARY.</s></p><p type="main"> <s>Hence is gathered, that if the Time of any Space <emph type="italics"/>ex quiete<emph.end type="italics"/> be <lb/>as the &longs;aid Spaec, the Time thereof after another Space is ad­<lb/>ded &longs;hall be the exce&longs;&longs;e of the Mean-proportional betwixt <lb/>the Addition and Space taken together, and the &longs;aid Space <lb/>above the Mean-proportional betwixt the fir&longs;t Space and the <lb/>Addition.</s></p><p type="main"> <s><emph type="italics"/>As for example, it being &longs;uppo&longs;ed that the Time along<emph.end type="italics"/><lb/><figure id="id.069.01.178.2.jpg" xlink:href="069/01/178/2.jpg"/><lb/><emph type="italics"/>A B, out of Re&longs;t in A, be A B; A S being another Space <lb/>added, The Time along A B after S A &longs;hall be the exce&longs;&longs;e of <lb/>the Mean-proportional betwixt S B and B A above the <lb/>Mean-proportional betwixt B A and A S.<emph.end type="italics"/></s></p><p type="head"> <s>PROBL VII. PROP. XX.</s></p><p type="main"> <s>Any Space and a part therein after the begining <lb/>of the Motion being given, to find another <lb/>part towards the end that &longs;hall be pa&longs;t in the <lb/>&longs;ame Time as the fir&longs;t part given.</s></p><p type="main"> <s><emph type="italics"/>Let the Space be C B, and let the part in it given after the begin­<lb/>ing of the Motion in C be C D. </s> <s>It is required to find another <lb/>part towards the end B, which &longs;hall be pa&longs;t in the &longs;ame Time as<emph.end type="italics"/><pb xlink:href="069/01/179.jpg" pagenum="176"/><emph type="italics"/>the given part C D. </s> <s>Take a Mean-proportional betwixt B C and C D, <lb/>to which &longs;uppo&longs;e B A equal; and let C E be a third proportional be-<emph.end type="italics"/><lb/><figure id="id.069.01.179.1.jpg" xlink:href="069/01/179/1.jpg"/><lb/><emph type="italics"/>tween B C and C A. </s> <s>I &longs;ay, that E B is the Space that after <lb/>the Fall out of C &longs;hall be past in the &longs;ame Time as the &longs;aid <lb/>C D is pa&longs;&longs;ed. </s> <s>For if we &longs;uppo&longs;e the Time along C B <lb/>to be as C B; B A (that is the Mean-proportional betwixt <lb/>B C and C D) &longs;hall be the Time along C D. </s> <s>And becau&longs;e <lb/>C A is the Mean proportional betwixt B C and C E, C A <lb/>&longs;hall be the Time along C E: But the whole B C is the <lb/>Time along the Whole C B: Therefore the part B A &longs;hall be <lb/>the Time along the part E B, after the Fall out of C: But <lb/>the &longs;aid B A was the Time along C D: Therefore C D and <lb/>E B &longs;hall be pa&longs;t in equal Times out of Re&longs;t in C: Which <lb/>was to be done.<emph.end type="italics"/></s></p><p type="head"> <s>THEOR. XIV. PROP. XXI.</s></p><p type="main"> <s>If along the Perpendicular a Fall be made <emph type="italics"/>ex quie­<lb/>te,<emph.end type="italics"/> in which from the begining of the Motion <lb/>a part is taken at plea&longs;ure, pa&longs;&longs;ed in any Time, <lb/>after which an Inflex Motion followeth along <lb/>any Plane however Inclined, the Space which <lb/>along that Plane is pa&longs;&longs;ed in a Time equal to <lb/>the Time of the Fall already made along the <lb/>Perpendicular &longs;hall be to the Space then pa&longs;­<lb/>&longs;ed along the Perpendicular more than double, <lb/>and le&longs;&longs;e than triple.</s></p><p type="main"> <s><emph type="italics"/>From the Horizon A E let fall a Perpendicular A B, along which <lb/>from the begining A let a Fall be made, of which let a part A C <lb/>be taken at plea&longs;ure; then out of C let any Plane G be inclined at <lb/>plea&longs;ure: along which after the Fall along A C let the Motion be con­<lb/>tinued. </s> <s>I &longs;ay, the Space pa&longs;&longs;ed by that Motion along C G in a Time <lb/>equall to the Time of the Fall along A C, is more than double, and le&longs;s <lb/>than triple that &longs;ame Space A C. </s> <s>For &longs;uppo&longs;e C F equal to A C, and <lb/>extending out the Plane G C as far as the Horizon in E, and as C E <lb/>is to E F, &longs;o let F E be to E G. </s> <s>If therefore we &longs;uppo&longs;e the Time of<emph.end type="italics"/><pb xlink:href="069/01/180.jpg" pagenum="177"/><emph type="italics"/>the Fall along A C to be as the Line A C; C E &longs;hall be the Time along <lb/>E C, and C F or C A the Time of the Motion along C G. </s> <s>Therefore <lb/>it is to be proved that the<emph.end type="italics"/><lb/><figure id="id.069.01.180.1.jpg" xlink:href="069/01/180/1.jpg"/><lb/><emph type="italics"/>Space C G is more than <lb/>double, and le&longs;&longs;e than <lb/>triple the &longs;aid C A. </s> <s>For <lb/>in regard that as C E is <lb/>to E F, &longs;o is F E to E G; <lb/>therefore al&longs;o &longs;o is C F to <lb/>F G. </s> <s>But E C is le&longs;&longs;e <lb/>than E F: Therefore C F <lb/>&longs;hall be le&longs;&longs;e than F G, and <lb/>G C more than double to <lb/>F C or A C. </s> <s>And moreover, in regard that F E is le&longs;&longs;e than double to <lb/>E C, (for E C is greater than C A or C F) G F &longs;hall al&longs;o be le&longs;&longs;e <lb/>than double to F C, and G C le&longs;&longs;e than triple to C F or C A: Which <lb/>was to be demon&longs;trated.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>And the &longs;ame may be more generally propounded: for that which <lb/>hapneth in the Perpendicular and Inclined Plane, holdeth true al&longs;o if <lb/>after the Motion a Plane &longs;omewhat inclined it be inflected along a more <lb/>inclining Plane, as is &longs;een in the other Figure: And the Demon&longs;tration <lb/>is the &longs;ame.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>P<emph.end type="italics"/>ROBL. VIII. <emph type="italics"/>P<emph.end type="italics"/>RO<emph type="italics"/>P.<emph.end type="italics"/> XXII.</s></p><p type="main"> <s>Two unequall Times being given, and a Space <lb/>that is pa&longs;t <emph type="italics"/>ex quiete<emph.end type="italics"/> along the Perpendicular <lb/>in the &longs;horte&longs;t of tho&longs;e given Times, to inflect <lb/>a Plane from the highe&longs;t point of the Perpen­<lb/>dicular unto the Horizon, along which the <lb/>Moveable may de&longs;cend in a Time equal to the <lb/>longe&longs;t of tho&longs;e Times given.</s></p><p type="main"> <s><emph type="italics"/>Let the unsqual Times be A the greater, and B the le&longs;&longs;er; and let <lb/>the Space that is pa&longs;t<emph.end type="italics"/> ex quiete <emph type="italics"/>along the Perpendicular in the <lb/>Time B, be C D. </s> <s>It is required from the Term C to inflect<emph.end type="italics"/> [or <lb/><figure id="id.069.01.180.2.jpg" xlink:href="069/01/180/2.jpg"/><lb/>bend] <emph type="italics"/>a Plane untill it reach the Horizon that may be pa&longs;&longs;ed in the<emph.end type="italics"/><pb xlink:href="069/01/181.jpg" pagenum="178"/><emph type="italics"/>Time A. </s> <s>As B is to A, &longs;o let C D be to another Line, to which let C X <lb/>be equal that de&longs;cendeth from C unto the Horizon: It is manife&longs;t that <lb/>the Plane C X is that along which the Moveable de&longs;cendeth in the Gi­<lb/>ven Time A. </s> <s>For it hath been demon&longs;trated, that the Time along the <lb/>inclined Plane hath the &longs;ame proportion to the Time along its ^{*} Eleva-<emph.end type="italics"/><lb/><arrow.to.target n="marg1094"></arrow.to.target><lb/><emph type="italics"/>tion, as the Length of the Plane hath to the Length of its Elevation,. <lb/>The Time, therefore, along C X is to the Time along C D, as C X is to <lb/>C D, that is, as the Time A is to the Time B: But the Time B is that <lb/>in which the Perpendicular is pa&longs;t<emph.end type="italics"/> ex quiete: <emph type="italics"/>Therefore the Time A is <lb/>that in which the Plane C X is pa&longs;&longs;ed.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1094"></margin.target>* Or Perpendi­<lb/>cular.</s></p><p type="head"> <s><emph type="italics"/>P<emph.end type="italics"/>ROBL. IX. PROP. XXIII.</s></p><p type="main"> <s>A Space pa&longs;t <emph type="italics"/>ex quiete<emph.end type="italics"/> along the Perpendicular in <lb/>any Time being given, to inflect a Plane from <lb/>the lowe&longs;t term of that Space, along which, <lb/>after the Fall along the Perpendicular, a Space <lb/>equal to any Space given may be pa&longs;&longs;ed in the <lb/>&longs;ame Time: which neverthele&longs;&longs;e is more than <lb/>double, and le&longs;&longs;e than triple the Space pa&longs;&longs;ed <lb/>along the Perpendicular.</s></p><p type="main"> <s><emph type="italics"/>Along the Perpendicular A S, in the Time A C, let the Space <lb/>A C be pa&longs;t<emph.end type="italics"/> ex quiete <emph type="italics"/>in A; to which let I R be more than <lb/>double, and le&longs;&longs;e than triple. </s> <s>It is required from the Terme C <lb/>to inflect a Plane, along which a Moveable after the Fall along A C <lb/>may in the &longs;ame Time A C pa&longs;&longs;e a Space equal to the &longs;aid I R. </s> <s>Let <lb/>R N, and N M be equal to A C: And look what proportion the part <lb/>I M hath to M N, the &longs;ame &longs;hall the Line A C have to another, equal <lb/>to which draw C E from C to<emph.end type="italics"/><lb/><figure id="id.069.01.181.1.jpg" xlink:href="069/01/181/1.jpg"/><lb/><emph type="italics"/>the Horizon A E, which con­<lb/>tinue out towards O, and take <lb/>C F, F G, and G O, equal to <lb/>the &longs;aid R N, N M, and M I. <lb/></s> <s>I &longs;ay, that the Time along the <lb/>inflected Plane C O, after the <lb/>Fall A G, is equal to the Time <lb/>A C out of Re&longs;t in A. </s> <s>For in <lb/>regard that as O G is to G F, <lb/>&longs;o is F C to C E by Compo&longs;ition it &longs;hall be that as O F is to F G or F C, <lb/>&longs;o is F E to E C; and as one of the Antecedents is to one of the Con­<lb/>&longs;equents, &longs;o are all to all; that is, the whole O E is to E F as F E to <lb/>E C: Therefore O E, E F, and E C are Continual Proportionals:<emph.end type="italics"/><pb xlink:href="069/01/182.jpg" pagenum="179"/><emph type="italics"/>And &longs;ince it was &longs;uppo&longs;ed that the Time along A C is as A C, C E &longs;hall <lb/>be the Time along E C; and E F the Time along the whole E O; and <lb/>the part C F that along the part C O: But C F is equal to the &longs;aid C A: <lb/>Therefore that is done which was required: For the Time C A is the <lb/>Time of the Fall along A C<emph.end type="italics"/> ex quiete <emph type="italics"/>in A; and C F (which is equal <lb/>to C A) is the Time along C O, after the De&longs;cent along E C, or after <lb/>the Fall along A C: Which was the Propo&longs;ition.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>And here it is to be noted, that the &longs;ame may happen if the preceding <lb/>Motion be not made along the Perpendicular, but along an Inclined Plane: <lb/>As in the following Figure, in which let the preceding Lation be made <lb/>along the inclined Plane A S beneath the Horizon A E: And the Demon­<lb/>&longs;tration is the very &longs;ame.<emph.end type="italics"/></s></p><p type="head"> <s>SCHOLIUM.</s></p><p type="main"> <s>If one ob&longs;erve well, it &longs;hall be manife&longs;t, that the le&longs;&longs;e the given <lb/>Line I R wanteth of being triple to the &longs;aid A C, the nearer <lb/>&longs;hall the Inflected Plane, along which the &longs;econd Motion is <lb/>to be made, which &longs;uppo&longs;e to be C O, come to the Perpen­<lb/>dicular, along which in a Time equal to A C a Space &longs;hall <lb/>be pa&longs;&longs;ed triple to A C.</s></p><p type="main"> <s><emph type="italics"/>For in ca&longs;e I R were very near the triple of A C, I M &longs;hould be well­<lb/>near equal to M N: And if, as I M is to M N by Con&longs;truction, &longs;o <lb/>A C is to C E, then it is evident that the &longs;aid C E will be found but <lb/>little bigger than C A, and, which followeth of con&longs;equence, the point E <lb/>&longs;hall be found very near the point A, and C O to containe a very acute<emph.end type="italics"/><lb/><figure id="id.069.01.182.1.jpg" xlink:href="069/01/182/1.jpg"/><lb/><emph type="italics"/>Angle with C S, and <lb/>almo&longs;t to concur both in <lb/>one Line. </s> <s>And on the <lb/>contrary, if the &longs;aid I R <lb/>were but a very little <lb/>more than double the <lb/>&longs;aid A C, I M &longs;hould <lb/>be a very &longs;hort Line. <lb/></s> <s>Hence it may happen <lb/>al&longs;o that A C may come <lb/>to be very &longs;hort in re&longs;pect of C E which &longs;hall be very long, and &longs;hall ap­<lb/>proach very near the Horizontal Parallel drawn from C. </s> <s>And from <lb/>hence we may collect, that if in the pre&longs;ent Figure after the De&longs;cent along <lb/>the inclined Plane A C, a Reflexion be made along the Horizontal Line, <lb/>as<emph.end type="italics"/> v. </s> <s>gr. <emph type="italics"/>C T, the Space along which the Moveable afterwards moved <lb/>in a Time equal to the Time of the De&longs;cent along A C would be exactly <lb/>double to the Space A C. </s> <s>And it appears that the like Di&longs;cour&longs;e may be <lb/>here applied: For it is apparent by what hath been &longs;aid, that &longs;ince O E<emph.end type="italics"/><pb xlink:href="069/01/183.jpg" pagenum="180"/><emph type="italics"/>is to E F, as F E is to E C, that F C determineth the Time along C O: <lb/>And if a part of the Horizontal Line T C double to C A be divided in <lb/>two equal parts in V, the exten&longs;ion towards X &longs;hall be prolonged<emph.end type="italics"/> in in­<lb/>finitum, <emph type="italics"/>whil&longs;t it &longs;eeks to meet with the prolonged Line A E: And the <lb/>proportion of the Infinite Line T X to the Infinite Line V X, &longs;hall be <lb/>no other than the proportion of the Infinite Line V X to the Infinite <lb/>Line X C.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>We may conclude the &longs;elf-&longs;ame thing another way by rea&longs;&longs;uming the <lb/>&longs;ame Rea&longs;oning that we u&longs;ed in the Demon&longs;tration of the fir&longs;t Propo&longs;i­<lb/>tion. </s> <s>For re&longs;uming the Triangle A B C, repre&longs;enting to us by its Pa­<lb/>rallels to the Ba&longs;e B C the Degrees of Velocity continually encrea&longs;ed ac­<lb/>cording to the encrea&longs;es of the Time; from which, &longs;ince they are infi­<lb/>nite, like as the Points are infinite in the Line A C, and the In&longs;tants <lb/>in any Time, &longs;hall re&longs;ult the Superficies of that &longs;ame Triangle, if we <lb/>under&longs;tand the Motion to continue for &longs;uch another Time, but no far­<lb/>ther with an Accelerate, but with an Equable Motion, according to the <lb/>greate&longs;t degree of Velocity acquired, which degree is repre&longs;ented <lb/>by the Line B C. </s> <s>Of &longs;uch degrees &longs;hall be made up an Aggregate like to <lb/>a Parallelogram A D B C, which is the double of<emph.end type="italics"/><lb/><figure id="id.069.01.183.1.jpg" xlink:href="069/01/183/1.jpg"/><lb/><emph type="italics"/>the Triangle A B C. </s> <s>Wherefore the Space which <lb/>with degrees like to tho&longs;e &longs;hall be pa&longs;&longs;ed in the &longs;ame <lb/>Time, &longs;hall be double to the Space pa&longs;t with the de­<lb/>grees of Velocity repre&longs;ented by the Triangle A B C: <lb/>But along the Horizontal Plane the Motion is Equa­<lb/>ble, for that there is no cau&longs;e of Acceleration, or Re­<lb/>tardation: Therefore it may be concluded that the <lb/>Space C D, pa&longs;&longs;ed in a Time equall to the Time A C is double to the <lb/>Space A C: For this Motion is made<emph.end type="italics"/> ex quiete <emph type="italics"/>Accelerate according <lb/>to the Parallels of the Triangle; and that according to the Parallels <lb/>of the Parallelogram, which, becau&longs;e they are infinite, are donble to <lb/>the infinite Parallels of the Triangle.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Moreover it may farther be ob&longs;erved, that what ever degree of <lb/>&longs;wiftne&longs;s is to be found in the Moveable, is indelibly impre&longs;&longs;ed upon it <lb/>of its own nature, all external cau&longs;es of Acceleration or Retardation <lb/>being removed; which hapneth only in Horizontal Planes: for in de­<lb/>clining Planes there is cau&longs;e of greater Acceleration, and in the ri&longs;ing <lb/>Planes of greater Retardation. </s> <s>From whence in like manner it fol­<lb/>loweth that the Motion along the Horizontal Plane is al&longs;o Perpetual: <lb/>for if it be Equable, it can neither be weakned nor retarded, nor much <lb/>le&longs;&longs;e de&longs;troyed. </s> <s>Farthermore, the degree of Celerity acquired by the <lb/>Moveable in a Natural De&longs;cent, being of its own Nature Indelible and <lb/>Penpetual, it is worthy con&longs;ideration, that if after the De&longs;cent along a <lb/>declining Plane a Reflexion be made along another Plane that is ri&longs;ing, <lb/>in this latter there is cau&longs;e of Retardation, for in the&longs;e kind of Planes<emph.end type="italics"/><pb xlink:href="069/01/184.jpg" pagenum="181"/><emph type="italics"/>the &longs;aid Moveable doth naturally de&longs;cend; whereupon there re&longs;ults a <lb/>mixture of certain contrary Affections, to wit, that degree of Celerity <lb/>acquired in the precedent De&longs;cent, which would of it &longs;elf carry the Move­<lb/>able uniformly<emph.end type="italics"/> in infinitum, <emph type="italics"/>and of Natural Propen&longs;ion to the Motion of <lb/>De&longs;cent according to that &longs;ame proportion of Acceleration wherewith it <lb/>alwaies moveth. </s> <s>So that it will be but rea&longs;onable, if, enquiring what <lb/>accidents happen when the Moveable after the De&longs;cent along any incli­<lb/>ned Plane is Reflected along &longs;ome ri&longs;ing Plane, we take that greate&longs;t de­<lb/>gree acquired in the De&longs;cent to keep it &longs;elf perpetually the &longs;ame in the <lb/>A&longs;cending Plane; But that there is &longs;uperadded to it in the A&longs;cent the <lb/>Natural Inclination downwards, that is the Motion from Re&longs;t Accelerate <lb/>according to the received proportion: And le&longs;t this &longs;hould, perchance, be <lb/>&longs;omewhat intricate to be under&longs;tood, it &longs;hall be more clearly explained by a <lb/>Scheme.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Let the De&longs;cent therefore be &longs;uppo&longs;ed to be made along the Declining <lb/>Plane A B, from which let the Reflex Motion be continued along another <lb/>Ri&longs;ing Plane B C: And in the fir&longs;t place let the Planes be equal, and <lb/>elevated at equal Angles to the Horizon G H. </s> <s>Now it is manife&longs;t, that <lb/>the Moveable<emph.end type="italics"/> ex quiete <emph type="italics"/>in A de&longs;cending along A B acquireth degrees of <lb/>Velocity according to the increa&longs;e of its Time, and that the degree in B <lb/>is the greate&longs;t of tho&longs;e acquired and by Nature immutably impre&longs;&longs;ed, I <lb/>mean the Cau&longs;es of new Acceleration or Retardation being removed: <lb/>of Acceleration, I &longs;ay, if it &longs;hould pa&longs;&longs;e any farther along the extended <lb/>Plane; and of Retardation, whil&longs;t the Reflection is making along the <lb/>Acclivity B C: But along the Horizontal Plane G H the Equable Mo­<lb/>tion according to the de-<emph.end type="italics"/><lb/><figure id="id.069.01.184.1.jpg" xlink:href="069/01/184/1.jpg"/><lb/><emph type="italics"/>gree of Velocity acquired <lb/>from A unto B would ex­<lb/>tend<emph.end type="italics"/> in infinitum. <emph type="italics"/>And <lb/>&longs;uch a Velocity would <lb/>that be which in a Time <lb/>equal to the Time of the <lb/>De&longs;cent along A B would pa&longs;&longs;e a Space in double the Horizon to the &longs;aid <lb/>A B. </s> <s>Now let us &longs;uppo&longs;e the &longs;ame Moveable to be Equably moved with <lb/>the &longs;ame degree of Swiftne&longs;&longs;e along the Plane B C, in &longs;uch &longs;ort that al&longs;o <lb/>in this Time equal to the Time of the De&longs;cent along A B a Space may be <lb/>pa&longs;&longs;ed a long B C extended double to the &longs;aid A B. </s> <s>And let us under­<lb/>&longs;tand that as &longs;oon as it beginneth to a&longs;cend there naturally befalleth the <lb/>&longs;ame that hapneth to it from A along the Plane A B, to wit, a certain <lb/>De&longs;cent<emph.end type="italics"/> ex quiete <emph type="italics"/>according to tho&longs;e degrees of Acceleration, by vertue <lb/>of which, as it befalleth in A B, it may de&longs;cend as much in the &longs;ame <lb/>Time along the Reflected Plane as it doth along A B: It is manife&longs;t, that <lb/>by this &longs;ame Mixture of the Equable Motion of A&longs;cent, and the Acce­<lb/>lerate of De&longs;cent the Moveable may be carried up to the Term C along <lb/>the Plane B C according to tho&longs;e degrees of Velocity, which &longs;hall be<emph.end type="italics"/><pb xlink:href="069/01/185.jpg" pagenum="182"/><emph type="italics"/>equal. </s> <s>And that two points at plea&longs;ure D and E being taken, equally <lb/>remote from the Angle B, the Tran&longs;ition along D B is made in a Time <lb/>equal to the Time of the Reflection along B E, we may collect from hence: <lb/>Draw D F, which &longs;hall be Parallel to B C; for it is manife&longs;t that the <lb/>De&longs;cent along A D is reflected along D F: And if after D the Move­<lb/>able pa&longs;&longs;e along the Horizontal Plane D E, the<emph.end type="italics"/> Impetus <emph type="italics"/>in E &longs;hall be <lb/>the &longs;ame as the<emph.end type="italics"/> Impetus <emph type="italics"/>in D: Therefore it will a&longs;cend from E to C: <lb/>And therefore the degree of Velocity in D is equal to the degree in E. <lb/></s> <s>From the&longs;e things, therefore, we may rationally affirm, that, if a de­<lb/>&longs;cent be made along any inclined Plane, after which a Reflection may <lb/>follow along an elevated Plane, the Moveable may by the conceived<emph.end type="italics"/><lb/>Impetus <emph type="italics"/>a&longs;cend untill it attain the &longs;ame beight, or Elevation from the <lb/>Horizon. </s> <s>As if a De&longs;cent be made along A B, the Moveable would <lb/>pa&longs;&longs;e along the Reflected Plane B C, untill it arrive at the Horizon <lb/>A C D; and that not only when the Inclinations of the Planes are <lb/>equal, but al&longs;o when they are unequal, as is the Plane B D: For it was <lb/>first &longs;uppo&longs;ed, that the degrees of Velocity are equal, which are acqui­<lb/>red upon Planes unequally inclined, &longs;o long as the Elevation of tho&longs;e <lb/>Planes above the Horizon was the &longs;ame: But, if there being the &longs;ame <lb/>Inclination of the Planes E B and B D, the De&longs;cent along E B &longs;ufficeth <lb/>to drive the Moveable along the Plane BD as far as D, &longs;eeing this Impul&longs;e<emph.end type="italics"/><lb/><figure id="id.069.01.185.1.jpg" xlink:href="069/01/185/1.jpg"/><lb/><emph type="italics"/>is made by the<emph.end type="italics"/> Impe­<lb/>tus <emph type="italics"/>of Velocity in the <lb/>point B; and if the<emph.end type="italics"/><lb/>Impetus <emph type="italics"/>be the &longs;ame <lb/>in B, whether the <lb/>Moveable de&longs;cend a­<lb/>long A B, or along E B: It is manife&longs;t, that the Moveable &longs;hall be in <lb/>the &longs;ame manner driven along B D, after the De&longs;cent along A B, and <lb/>after that along E B: But it will happen that the Time of the A&longs;cent <lb/>along B D &longs;hall be longer than along B C, like as the De&longs;cent along <lb/>E B is made in a longer time than along A B: But the Proportion of <lb/>tho&longs;e Times was before demon&longs;trated to be the &longs;ame as the Lengths of <lb/>tho&longs;e Planes. </s> <s>Now it follows, that we &longs;eek the proportion of the Spaces <lb/>pa&longs;t in equal Times along Planes, who&longs;e Inclinations are different, but <lb/>their Elevations the &longs;ame; that is, which are comprehended between <lb/>the &longs;ame Horizontal Parallels. </s> <s>And this hapneth according to the fol­<lb/>lowing Propo&longs;ition.<emph.end type="italics"/></s></p><pb xlink:href="069/01/186.jpg" pagenum="183"/><p type="head"> <s>THEOR. XV. PROP. XXIV.</s></p><p type="main"> <s>There being given between the &longs;ame Horizontal <lb/>Parallels a Perpendicular and a <emph type="italics"/>P<emph.end type="italics"/>lane eleva­<lb/>ted from its lowe&longs;t term, the Space that a <lb/>Moveable after the Fall along the <emph type="italics"/>P<emph.end type="italics"/>erpendi­<lb/>cular pa&longs;&longs;eth along the Elevated <emph type="italics"/>P<emph.end type="italics"/>lane in a <lb/>Time equal to the Time of the Fall, is greater <lb/>than that <emph type="italics"/>P<emph.end type="italics"/>erpendicular, but le&longs;&longs;e than double <lb/>the &longs;ame.</s></p><p type="main"> <s><emph type="italics"/>Between the &longs;ame Horizontal Parallels B C and H G let there <lb/>be the Perpendicular A E; and let the Elevated Plane be E B, <lb/>along which after the Fall along the Perpendicular A E out of <lb/>the Term E let a Reflexion be made towards B. </s> <s>I &longs;ay, that the Space, <lb/>along which the Moveable a&longs;cendeth in a Time equal to the Time of the <lb/>De&longs;cent A E, is greater than A E, but le&longs;&longs;e than double the &longs;ame A E. <lb/></s> <s>Let E D be equal to A E, and as E B is to B D, &longs;o let D B be to B F. </s> <s>It <lb/>&longs;hall be proved, fir&longs;t that the point F is the Term at which the Moveable <lb/>with a Reflex Motion along E B arriveth in a Time equal to the Time <lb/>A E: And then, that E F is greater than E A, but le&longs;&longs;e than double the <lb/>&longs;ame. </s> <s>If we &longs;uppo&longs;e the Time of the De&longs;cent along A E to be as A E, <lb/>the Time of the De&longs;cent along B E, or A&longs;cent along E B &longs;hall be as the <lb/>&longs;ame Line B E: And D B being a Mean-Proportional betwixt E B <lb/>and B F, and B E being the Time of De&longs;cent along the whole B E, B D <lb/>&longs;hall be the Time of the De&longs;cent along B F, and the Remaining part <lb/>D E the Time of the<emph.end type="italics"/><lb/><figure id="id.069.01.186.1.jpg" xlink:href="069/01/186/1.jpg"/><lb/><emph type="italics"/>De&longs;cent along the Re­<lb/>maining part F E: But <lb/>the Time along F E<emph.end type="italics"/> ex <lb/>quiete <emph type="italics"/>in B, and the <lb/>Time of the A&longs;cent a­<lb/>long E F is the &longs;ame, &longs;ince that the Degree of Velocity in E was acqui­<lb/>red along the De&longs;cent B E, or A E: Therefore the &longs;ame Time D E &longs;hall <lb/>be that in which the Moveable after the Fall out of A along A E, <lb/>with a Reflex Motion along E B &longs;hall reach to the Mark F: But it hath <lb/>been &longs;uppo&longs;ed that E D is equal to the &longs;aid A E: Which was fir&longs;t to be <lb/>proved. </s> <s>And becau&longs;e that as the whole E B is to the whole B D, &longs;o is the <lb/>part taken away D B to the part taken away B F, therefore, as the whole <lb/>E B is to the whole B D, &longs;o &longs;hall the Remainder E D be to D F: <lb/>But E B is greater than B D: Therefore E D is greater than D F, and <lb/>E F le&longs;&longs;e than double to D E or A E: Which was to be proved.<emph.end type="italics"/></s></p><pb xlink:href="069/01/187.jpg" pagenum="184"/><p type="main"> <s><emph type="italics"/>And the &longs;ame al&longs;o hapneth if the precedent Motion be not made <lb/>along the Perpendicular, but along an Inclined Plane; and the Demon­<lb/>&longs;tration is the &longs;ame, provided that the Reflex Plane be le&longs;&longs;e ri&longs;ing, that is, <lb/>longer than the declining Plane.<emph.end type="italics"/></s></p><p type="head"> <s>THEOR. XVI. <emph type="italics"/>P<emph.end type="italics"/>RO<emph type="italics"/>P.<emph.end type="italics"/> XXV.</s></p><p type="main"> <s>If after the De&longs;cent along any Inclined Plane a <lb/>Motion follow along the Plane of the Hori­<lb/>zon, the Time of the De&longs;cent along the Incli­<lb/>ned Plane &longs;hall be to the Time of the Motion <lb/>along any Horizontal Line; as the double <lb/>Length of the Inclined Plane is to the Line ta­<lb/>ken in the Horizon.</s></p><p type="main"> <s><emph type="italics"/>Let the Horizontal Line be C B, the inclined Plane A B, and after <lb/>the De&longs;cent along A B let a Motion follow along the Horizon, in <lb/>which take any Space B D. </s> <s>I &longs;ay, that the Time of the De&longs;cent <lb/>along A B to the Time of the Motion along B D is as the double of A B <lb/>to B D. </s> <s>For B C being &longs;uppo&longs;ed <lb/>the double of A B, it is manife&longs;t by<emph.end type="italics"/><lb/><figure id="id.069.01.187.1.jpg" xlink:href="069/01/187/1.jpg"/><lb/><emph type="italics"/>what hath already been demon&longs;tra­<lb/>ted that the Time of the De&longs;cent <lb/>along A B is equal to the Time of <lb/>the Motion along B C: But the <lb/>Time of the Motion along B C is to <lb/>the Time of the Motion along B D, as the Line C B is to the Line B D: <lb/>Therefore the Time of the Motion along A B is the Time along B D, as <lb/>the Double of A B is to B D: Which was to be proved.<emph.end type="italics"/></s></p><p type="head"> <s>PROBL X. PROP. XXVI.</s></p><p type="main"> <s>A Perpendicular between two Horizontal <emph type="italics"/>P<emph.end type="italics"/>aral­<lb/>lel Lines, as al&longs;o a Space greater than the &longs;aid <lb/><emph type="italics"/>P<emph.end type="italics"/>erpendicular, but le&longs;&longs;e than double the &longs;ame, <lb/>being given, to rai&longs;e a <emph type="italics"/>P<emph.end type="italics"/>lane between the &longs;aid <lb/><emph type="italics"/>P<emph.end type="italics"/>arallels from the lowe&longs;t Term of the <emph type="italics"/>P<emph.end type="italics"/>er­<lb/>pendicular, along which the Moveable may <lb/>with a Reflex Motion after the Fall along the <lb/><emph type="italics"/>P<emph.end type="italics"/>erpendicular pa&longs;&longs;e a Space equal to the Space <lb/>given, and in a Time equal to the Time of the <lb/>Fall along the <emph type="italics"/>P<emph.end type="italics"/>erpendicular.</s></p><pb xlink:href="069/01/188.jpg" pagenum="185"/><p type="main"> <s><emph type="italics"/>Let A B be a Perpendicular between the Horizontal Parallels A O <lb/>and B C; and let F E be greater than B A, but le&longs;&longs;e than double <lb/>the &longs;ame. </s> <s>It is required between the &longs;aid Parallels from the point <lb/>B to rai&longs;e a Plane, along which the Moveable after the Fall from A to <lb/>B may with a Reflex Motion in a Time equal to the Time of the Fall <lb/>along A B pa&longs;&longs;e a Space a&longs;cending equal to the &longs;aid E F. </s> <s>Suppo&longs;e E D <lb/>equall to A B, the Remaining Part D F &longs;hall be le&longs;&longs;e, for that the whole <lb/>E F is le&longs;&longs;e than double to A B: Let D I be equal to D F, and as E I is <lb/>to I D, &longs;o let D F be to another Space F X, and out of B let the Right-<emph.end type="italics"/><lb/><figure id="id.069.01.188.1.jpg" xlink:href="069/01/188/1.jpg"/><lb/><emph type="italics"/>Line B O be reflected, equal to E X. </s> <s>I &longs;ay, that the Plane along B O <lb/>is that along which after the Fall A B a Moveable in a Time equal <lb/>to the Time of the Fall along A B pa&longs;&longs;eth a&longs;cending a Space equal to <lb/>the given Space E F. </s> <s>Suppo&longs;e B R and R S equal to the &longs;aid E D and <lb/>D F. </s> <s>And becau&longs;e that as E I is to I D, &longs;o is D F to F X; therefore, <lb/>by Compo&longs;ition, as E D is to D I, &longs;o &longs;hall D X be to X F; that is, as <lb/>E D is to D F, &longs;o &longs;hall D X be to X F, and E X to X D; that is, as <lb/>B O is to O R, &longs;o &longs;hall R O be to O S: And if we &longs;uppo&longs;e the Time <lb/>along A B to be A B, the Time along O B &longs;hall be the &longs;ame O B, and <lb/>R O the Time along O S, and the Remaining Part B R the Time along <lb/>the Remaining Part S B, de&longs;cending from O to B: But the Time of <lb/>the De&longs;cent along S B from Rest in O, is equal to the Time of the <lb/>A&longs;cent from B to S after the Fall A B: Therefore B O is the Plane ele­<lb/>vated from B, along which after the Fall along A B the Space B S <lb/>equal to the given Space E F is pa&longs;&longs;ed in the Time B R or B A: Which <lb/>was required to be done.<emph.end type="italics"/></s></p><pb xlink:href="069/01/189.jpg" pagenum="186"/><p type="head"> <s>THEOR. XVII. PROP. XXVII.</s></p><p type="main"> <s>If a Moveable de&longs;cend along unequal <emph type="italics"/>P<emph.end type="italics"/>lanes, <lb/>who&longs;e Elevation is the &longs;ame, the Space that <lb/>&longs;hall be pa&longs;t along the lower part of the longe&longs;t <lb/>in a Time equal to that in which the whole <lb/>&longs;horter <emph type="italics"/>P<emph.end type="italics"/>lane is pa&longs;&longs;ed, is equal to the Space <lb/>that is compounded of the &longs;aid &longs;horter <emph type="italics"/>P<emph.end type="italics"/>lane <lb/>and of the part to which that &longs;horter <emph type="italics"/>P<emph.end type="italics"/>lane <lb/>hath the &longs;ame <emph type="italics"/>P<emph.end type="italics"/>roportion that the longer <lb/><emph type="italics"/>P<emph.end type="italics"/>lane hath to the Exce&longs;&longs;e by which the longe&longs;t <lb/>exceedeth the &longs;horte&longs;t.</s></p><p type="main"> <s><emph type="italics"/>Let A C be the longer Plane, and A B the &longs;horter, who&longs;e Elevation <lb/>A D is the &longs;ame; and in the lower part of A C take the Space <lb/>C E, equal to the &longs;aid A B; and as C A is to A E, (that is to <lb/>the exce&longs;&longs;e of the Plane C A above A B) &longs;o let C E be to E F. </s> <s>I &longs;ay, <lb/>that the Space F C is that which is pa&longs;t after the De&longs;cent out of A in <lb/>a Time equal to the Time of<emph.end type="italics"/><lb/><figure id="id.069.01.189.1.jpg" xlink:href="069/01/189/1.jpg"/><lb/><emph type="italics"/>the De&longs;cent along A B. </s> <s>For <lb/>the whole C A, being to the <lb/>whole A E, as the part taken <lb/>away C E is to the part taken <lb/>away E F, therefore the re­<lb/>maining part E A &longs;hall be to <lb/>the remaining part A F, as the <lb/>whole C A is to the whole A E: Therefore the three Spaces C A, <lb/>A E, and A F are three Continual proportionals. </s> <s>And if the Time <lb/>along A B be &longs;uppo&longs;ed to be as A B, the Time along A C &longs;hall be as <lb/>A C, and the Time along A F &longs;hall be as A E, and along the remain­<lb/>ing part F C &longs;hall be as E C: But E C is equal to the &longs;aid A B: There­<lb/>fore the Propo&longs;ition is manife&longs;t.<emph.end type="italics"/></s></p><p type="head"> <s>THEOR. XVIII. PROP. XXVIII.</s></p><p type="main"> <s><emph type="italics"/>Let the Horizontal Line A G be Tangent to a Circle, and from the <lb/>point of Contact let A B be the Diameter, and A E B two Chords <lb/>at plea&longs;ure: We are to a&longs;&longs;ign the proportion of the Time of the <lb/>Fall along A B to the Time of the De&longs;cent along both the Chords <lb/>A E B. </s> <s>Let B E be continued out till it meet the Tangent in G, and<emph.end type="italics"/><pb xlink:href="069/01/190.jpg" pagenum="187"/><emph type="italics"/>let the Angle B A E be cut in two equal parts, and draw A F. </s> <s>I &longs;ay, <lb/>that the Time along A B is to the Time along A E B, as A E is to A E F. <lb/></s> <s>For in regard the Angle F A B is equal to the Angle F A E, and the An­<lb/>gle E A G to the Angle A B F, the whole Angle G A F &longs;hall be equal to <lb/>the two Angles F A B, and A B F; <lb/>to which al&longs;o the Angle G F A<emph.end type="italics"/><lb/><figure id="id.069.01.190.1.jpg" xlink:href="069/01/190/1.jpg"/><lb/><emph type="italics"/>is equal: Therefore the Line G F <lb/>is equal to G A. </s> <s>And becau&longs;e the <lb/>Rectangle B G E is equal to the <lb/>Square of G A, it &longs;hall likewi&longs;e <lb/>be equal to the Square of G F, and <lb/>the three Lines B G, G F, and <lb/>G E &longs;hall be proportionals. </s> <s>And <lb/>if we &longs;uppo&longs;e A E to be the Time <lb/>along A E, G E &longs;hall be the Time <lb/>along G E, and G F the Time along the whole G B, and E F the Time <lb/>along E B, after the De&longs;cent out of G, or out of A, along A E: The Time, <lb/>therefore, along A E, or along A B &longs;hall be to the Time along A E B, as <lb/>A E is to A E F: Which was to be determined.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>More briefly thus. </s> <s>Let G F be cut equal to G A: It is manife&longs;t <lb/>that G F is the Mean-proportional between B G, and G E. </s> <s>The re&longs;t as <lb/>before.<emph.end type="italics"/></s></p><p type="head"> <s>PROBL. <emph type="italics"/>XI. P<emph.end type="italics"/>RO<emph type="italics"/>P. XXIX.<emph.end type="italics"/></s></p><p type="main"> <s>Any Horizontal Space being given upon the <lb/>end of which a Perpendicular is erected, <lb/>in which a part is taken equal to half of the <lb/>Space given in the Horizontal a Moveable fal­<lb/>ling from that height, and turned along the <lb/>Horizon, &longs;hall pa&longs;&longs;e the Horizontal Space to­<lb/>gether with the Perpendicular in a &longs;horter <lb/>Time than any other Space of the <emph type="italics"/>P<emph.end type="italics"/>erpendi­<lb/>cular with the &longs;ame Horizontal Space.</s></p><p type="main"> <s><emph type="italics"/>Let there be an Horizontal Space in which let any Space be given <lb/>B C, and on B let there be a Perpendicular erected, in which let <lb/>B A be the half of the fore&longs;aid B C. </s> <s>I &longs;ay, that the Time in which <lb/>a Moveable let fall out of A pa&longs;&longs;eth both the Spaces A B and B C is the <lb/>&longs;horte&longs;t of all Times in which the &longs;aid Space B C with a part of the <lb/>Perpendicular, whether greater or le&longs;&longs;er than the part A B, &longs;hall be pa&longs;­<lb/>&longs;ed. </s> <s>Let a greater be taken, as in the &longs;ir&longs;t Figure, or le&longs;&longs;er, as in the<emph.end type="italics"/><pb xlink:href="069/01/191.jpg" pagenum="188"/><emph type="italics"/>&longs;econd, which let be E B. </s> <s>It is to be proved that the Time in which the <lb/>Spaces E B and B C are pa&longs;&longs;ed is longer than the Time in which A B <lb/>and B C are pa&longs;&longs;ed. </s> <s>Let the Time along A B be as A B; the &longs;ame &longs;hall <lb/>be the Time of the Motion along the Horizontal Space B G; becau&longs;e <lb/>B C is double to A B, and the Time along both the Spaces A B C &longs;hall <lb/>be double of O B A. </s> <s>Let B O<emph.end type="italics"/><lb/><figure id="id.069.01.191.1.jpg" xlink:href="069/01/191/1.jpg"/><lb/><emph type="italics"/>be a Mean-proportional between <lb/>E B and B A. </s> <s>B O &longs;hall be the <lb/>Time of the Fall along E B. <lb/>Again, let the Horizontal Space <lb/>B D be double to the &longs;aid B E: <lb/>It is manife&longs;t that the Time of it <lb/>after the Fall E B is the &longs;ame <lb/>B O. </s> <s>As D B is to B C, or as <lb/>E B is to B A, &longs;o let O B be to <lb/>B N: and in regard the Motion <lb/>along the Horizontal Plane is Equable, and O B being the Time along <lb/>B D after the Fall out of E, therefore N B &longs;hall be the Time along B C <lb/>after the Fall from the &longs;ame Altitude E. </s> <s>Hence it is manife&longs;t, that O B, <lb/>together with B N is the Time along E B C; and becau&longs;e the double of <lb/>B A is the Time along A B C; it remains to be proved, that O B, to­<lb/>gether with B N is more than double B A. </s> <s>Now becau&longs;e O B is a Mean <lb/>between E B and B A, the proportion of E B to B A is double the pro­<lb/>portion of O B to B A: and, in regard that E B is to B A, as O B is to <lb/>B N, the proportion of O B to B N &longs;hall al&longs;o be double the proportion of <lb/>O B to B A: But that proportion of O B to B N is compounded of the <lb/>proportions of O B to B A, and of A B to B N: therefore the proportion <lb/>of A B to B N is the &longs;ame with that of O B to B A. </s> <s>Therefore B O, <lb/>B A, and B N are three continual Proportionals, and O B, together with <lb/>B N, are greater than double B A: Whereupon the Propo&longs;ition is ma­<lb/>nife&longs;t.<emph.end type="italics"/></s></p><pb xlink:href="069/01/192.jpg" pagenum="189"/><p type="head"> <s>THEOR. <emph type="italics"/>XIX.<emph.end type="italics"/> PROP. <emph type="italics"/>XXX.<emph.end type="italics"/></s></p><p type="main"> <s>If a Perpendicular be let fall from any point of the <lb/>Horizontal Line, and out of another point in <lb/>the &longs;ame Horizontal Line a Plane be drawn <lb/>forth untill it meet the Perpendicular, along <lb/>which a Moveable de&longs;cendeth in the &longs;horte&longs;t <lb/>time unto the &longs;aid Perpendicular, this Plane <lb/>&longs;hall be that which cutteth off a part equall to <lb/>the di&longs;tance of the a&longs;&longs;igned point from the end <lb/>of the Perpendicular.</s></p><p type="main"> <s><emph type="italics"/>Let the Perpendicular B D be let fall from the point B of the Ho­<lb/>rizontal Line A C, in which let there be any point C, and in the <lb/>Perpendicular let the Di&longs;tance B E be &longs;uppo&longs;ed equal to the Di­<lb/>&longs;tance B C, and draw C E. </s> <s>I &longs;ay, that of all Planes inclined out of <lb/>the point C till they meet the Perpendicular C E is that, along which <lb/>in the &longs;horte&longs;t of all Times the De&longs;cent<emph.end type="italics"/><lb/><figure id="id.069.01.192.1.jpg" xlink:href="069/01/192/1.jpg"/><lb/><emph type="italics"/>is made unto the Perpendicular. </s> <s>For <lb/>let the Planes C F and C G be inclined <lb/>above and below, and draw I K a Tan­<lb/>gent unto the Semidiameter B C of the <lb/>de&longs;cribed Circle in C, which &longs;hall be <lb/>equidi&longs;tant from the Perpendicular; <lb/>and unto the &longs;aid C F let E K be Paral­<lb/>lel cutting the Circumference of the Cir­<lb/>cle in L: It is manife&longs;t that the Time of <lb/>the De&longs;cent along L E is equal to the <lb/>Time of the De&longs;cent along C E: But <lb/>the Time along K E is longer than along <lb/>L E: Therefore the Time along K E is <lb/>longer than that along C E: But the <lb/>Time along K E is equal to the Time a­<lb/>long C F, they being equal, and drawn <lb/>according to the &longs;ame Inclination: Likewi&longs;e &longs;ince C G, and I E are <lb/>equal, and inclined according to the &longs;ame Inclination, the Times of the <lb/>Motions along them &longs;hall be equal: But H E being &longs;horter than I E, the <lb/>Time along it is al&longs;o &longs;horter than I E: Therefore the Time al&longs;o along <lb/>C E, (which is equal to the Time along H E) &longs;hall be &longs;horter than the <lb/>Time along I E: The Propo&longs;ition, therefore, is manife&longs;t.<emph.end type="italics"/></s></p><pb xlink:href="069/01/193.jpg" pagenum="190"/><p type="head"> <s>THEOR. <emph type="italics"/>XX.<emph.end type="italics"/> PROP. <emph type="italics"/>XXXI.<emph.end type="italics"/></s></p><p type="main"> <s>If a Right-Line &longs;hall be in any manner inclined <lb/>upon the Horizontal Line, the Plane produced <lb/>from a given point in the Horizon untill it <lb/>meet with the Inclined Plane, along which <lb/>the De&longs;cent is made in the &longs;horte&longs;t of all <lb/>Times, is that which &longs;hall divide the Angle <lb/>contained between the two <emph type="italics"/>P<emph.end type="italics"/>erpendiculars <lb/>drawn from the given <emph type="italics"/>P<emph.end type="italics"/>oint, the one unto the <lb/>Horizontal Line, the other to the Inclined <lb/>Line, into two equal parts.</s></p><p type="main"> <s><emph type="italics"/>Let C D be a Line inclined in any manner upon the Hori­<lb/>zontal Line A B, and let any point A be given in the Hori­<lb/>zon, and from it let A C be drawn Perpendicular to A B, <lb/>and A E Perpendicular to C D, and let the Line F A divide the <lb/>Angle C A E into two equal parts. </s> <s>I &longs;ay, that of all Planes incli­<lb/>ned out of any point of the Line C D to the point A that &longs;ame pro­<lb/>duced along F A is it along<emph.end type="italics"/><lb/><figure id="id.069.01.193.1.jpg" xlink:href="069/01/193/1.jpg"/><lb/><emph type="italics"/>which the De&longs;cent is made in <lb/>the &longs;horte&longs;t of all Times. </s> <s>Let <lb/>F G be drawn Parallel to AE; <lb/>the alternate Angles G F A <lb/>and F A E &longs;hall be equal: But <lb/>E A F is equal to that other <lb/>F A G: Therefore of the Tri­<lb/>angle the Sides F G and G A <lb/>&longs;hall be equal. </s> <s>If therefore <lb/>about the Center G, at the di­<lb/>&longs;tance G A, a Circle be de&longs;cri­<lb/>bed it &longs;hall pa&longs;&longs;e by F, and &longs;hall <lb/>touch the Horizontal, and the Inclined Lines in the points A and F: <lb/>For the Angle G F C is a Right Angle, and likewi&longs;e G F is equidi&longs;tant <lb/>to A E: Whence it is manife&longs;t that all Lines produced from the point <lb/>A unto the inclined Plane do extend beyond the Circumference, and, <lb/>which followeth of con&longs;equence, that the Motions along the &longs;ame do <lb/>take up more Time than along F A. </s> <s>Which was to be demon&longs;trated.<emph.end type="italics"/></s></p><pb xlink:href="069/01/194.jpg" pagenum="191"/><p type="head"> <s>LEMMA.</s></p><p type="main"> <s>If two Circles touch one another within, the innermo&longs;t of which <lb/>toucheth &longs;ome Right Line, and the exteriour one cutteth it, <lb/>three Lines produced from the Contact of the Circles unto <lb/>three points of the Tangent Right-Line, that is, to the Con­<lb/>tact of the interiour Circle, and to the Sections of the exte­<lb/>riour &longs;hall contain equall Angles in the Contact of the <lb/>Circles.</s></p><p type="main"> <s><emph type="italics"/>Let two Circles touch one another in the point A, of which let the <lb/>Centers be B, that of the le&longs;&longs;er, and C that of the greater; and let <lb/>the interiour Circle touch any Line F G in the point H, and let the grea­<lb/>ter cut it in the points F and G, and connect the three Lines A F, A H, <lb/>and A G. </s> <s>I &longs;ay, that the Angles by<emph.end type="italics"/><lb/><figure id="id.069.01.194.1.jpg" xlink:href="069/01/194/1.jpg"/><lb/><emph type="italics"/>them contained F A H and G A H are <lb/>equal. </s> <s>Produce A H untill it meeteth <lb/>the Circumference in I, and from the <lb/>Centers draw B H and C I, and thorow <lb/>the &longs;aid Centers let B C be drawn, <lb/>which continued forth &longs;hall meet with <lb/>the Contact A, and with the Circum­<lb/>ferences of the Circles in O and N. <lb/></s> <s>And becau&longs;e the Angles I C N and <lb/>H O B are equal, for as much as either <lb/>of them is double to the Angle I A N, <lb/>the Lines B H and C I &longs;hall be Parallels: And becau&longs;e B H drawn <lb/>from the Center to the Contact is Perpendicular to F G; C I &longs;hall al&longs;o be <lb/>Perpendicular to the &longs;ame, and the Arch F I equal to the Arch I G, and, <lb/>which followeth of con&longs;equence, the Angle F A I to the Angle I A G: <lb/>Which was to be demon&longs;trated.<emph.end type="italics"/></s></p><pb xlink:href="069/01/195.jpg" pagenum="192"/><p type="head"> <s>THEOR. <emph type="italics"/>XXI.<emph.end type="italics"/> PROP. <emph type="italics"/>XXXII.<emph.end type="italics"/></s></p><p type="main"> <s>If two points be taken in the Horizon, and any <lb/>Line &longs;hould be inclined from one of them to­<lb/>wards the other, out of which a Right-Line is <lb/>drawn unto the Inclined Line, cutting off a <lb/>part thereof equal to that which is included <lb/>between the points of the Horizon, the De­<lb/>&longs;cent along this la&longs;t drawn &longs;hall be &longs;ooner per­<lb/>formed, than along any other Right Lines pro­<lb/>duced from the &longs;ame point unto the &longs;aid Incli­<lb/>ned Line. </s> <s>And along other Lines which are <lb/>on each hand of this by equal Angles a De­<lb/>&longs;cent &longs;hall be made in equal Times.</s></p><p type="main"> <s><emph type="italics"/>In the Horizon let there be two points A and B, and from B incline <lb/>the Right Line B C, in which from the Term B take B D equal to <lb/>the &longs;aid B A, and draw a Line from A to D. </s> <s>I &longs;ay, that the De­<lb/>&longs;cent along A D is more &longs;wiftly made, than along any other what&longs;oever <lb/>drawn from the point A unto the inclined Line B C. </s> <s>For out of the <lb/>points A and D unto B A and<emph.end type="italics"/><lb/><figure id="id.069.01.195.1.jpg" xlink:href="069/01/195/1.jpg"/><lb/><emph type="italics"/>B D draw the Perpendiculars <lb/>A E and D E, inter&longs;ecting one <lb/>another in E: and fora&longs;much as <lb/>in the equicrural Triangle A B D <lb/>the Angles B A D and B D A <lb/>are equal, the remainders to the <lb/>Right-Angles D A E and E D A <lb/>&longs;hall be equal. </s> <s>Therefore a Circle <lb/>de&longs;cribed about the Center E at <lb/>the di&longs;tance A E &longs;hall al&longs;o pa&longs;&longs;e <lb/>by D; and the Lines B A and <lb/>B D will touch it in the points A <lb/>and D. </s> <s>And &longs;ince A is the end of the Perpendicular A E, the De&longs;cent <lb/>along A D &longs;hall be &longs;ooner performed, than along any other produced from <lb/>the &longs;ame Term A unto the Line B C beyond the Circumference of the <lb/>Circle: Which was fir&longs;t to be proved.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>But if in the Perpendicular A E being prolonged any Center be taken as <lb/>F, and at the di&longs;tance F A the Circle A G C be de&longs;cribed cutting the <lb/>Tangent Line in the points G and C; drawing A G and A C they &longs;hall <lb/>make equal Angles with the middle Line A D by what hath been afore<emph.end type="italics"/><pb xlink:href="069/01/196.jpg" pagenum="193"/><emph type="italics"/>demon&longs;trated, and the Motions thorow them &longs;hall be performed in equal <lb/>Times &longs;eeing that they terminate in A unto the Circumference of the <lb/>Circle A G O from the highe&longs;t point of it A.<emph.end type="italics"/></s></p><p type="head"> <s>PROBL. XII. PROP. <emph type="italics"/>XXXIII.<emph.end type="italics"/></s></p><p type="main"> <s>A Perpendicular and Plane inclined to it being <lb/>given, who&longs;e height is one and the &longs;ame, as al­<lb/>&longs;o the highe&longs;t term, to find a point in the Per­<lb/>pendicular above the common term, out of <lb/>which if a Moveable be demitted that &longs;hall <lb/>afterwards turn along the inclined Plane, the <lb/>&longs;aid Plane may be pa&longs;t in the &longs;ame Time in <lb/>which the <emph type="italics"/>P<emph.end type="italics"/>erpendicular <emph type="italics"/>ex quiete<emph.end type="italics"/> would be <lb/>pa&longs;&longs;ed.</s></p><p type="main"> <s><emph type="italics"/>Let the Perpendicular and inclined Plane, who&longs;e Altitude is the <lb/>&longs;ame, be A B and A C. </s> <s>It is required in the Perpendicular B A, <lb/>continued out from the point A to find a Point out of which a <lb/>Moveable de&longs;cending may pa&longs;&longs;e the Space A C in the &longs;ame Time in <lb/>which it will pa&longs;&longs;e the &longs;aid Perpendicular A B out of Re&longs;t in A. </s> <s>Draw <lb/>D C E at Right-Angles to A C, and let C D be cut equal to A B, and <lb/>draw a Line from A to D: The Angle A D C &longs;hall be greater than the <lb/>Angles C A D: (for C A is greater than A B or C D:) Let the <lb/>Angle D A E be equal to the Angle A D E; and to A E let E F an in­<lb/>clined Plane be Perpen-<emph.end type="italics"/><lb/><figure id="id.069.01.196.1.jpg" xlink:href="069/01/196/1.jpg"/><lb/><emph type="italics"/>dicular, and let both be­<lb/>ing prolonged meet in F, <lb/>and unto both A I and <lb/>A G &longs;uppo&longs;e C F to be <lb/>equal, and by G draw <lb/>G H equidi&longs;tant to the <lb/>Horizon. </s> <s>I &longs;ay, that H <lb/>is the point which is <lb/>&longs;ought. </s> <s>For &longs;uppo&longs;ing the <lb/>Time of the Fall along <lb/>the Perpendicular A B <lb/>to be A B, the Time along <lb/>A C ex quiete in A &longs;hall be the &longs;ame A C. </s> <s>And becau&longs;e in the Right­<lb/>angled Triangle A E F, from the Right Angle E unto the Ba&longs;e A F, <lb/>E C is a Perpendicular, A E &longs;hall be a Mean-Proportional betwixt F A <lb/>and A C, and C E a Mean betwixt A C and C F, that is, betwixt C A <lb/>and A I: and fora&longs;much as the Time of A C out of A is A C, A E<emph.end type="italics"/><pb xlink:href="069/01/197.jpg" pagenum="194"/><emph type="italics"/>&longs;hall be the Time of the whole A F, and E C the Time of A I: And be­<lb/>cau&longs;e in the Equicrural Triangle A E D the Side A E is equal to the <lb/>Side E D, E D &longs;hall be the Time along A F, and E C is the Time along <lb/>A I: Therefore C D, that is A B &longs;hall be the Time along A F<emph.end type="italics"/> ex qui­<lb/>ete <emph type="italics"/>in A; which is the &longs;ame as if we &longs;aid, that A B is the Time along <lb/>A G out of G, or out of H: Which was to be done.<emph.end type="italics"/></s></p><p type="head"> <s>PROBL. <emph type="italics"/>XIII. P<emph.end type="italics"/>RO<emph type="italics"/>P. XXXIV.<emph.end type="italics"/></s></p><p type="main"> <s>An inclined <emph type="italics"/>P<emph.end type="italics"/>lane and Perpendicular who&longs;e &longs;ub­<lb/>lime term is the &longs;ame being given, to find a <lb/>more &longs;ublime point in the Perpendicular pro­<lb/>longed out of which a Moveable falling, and <lb/>being turned along the inclined <emph type="italics"/>P<emph.end type="italics"/>lane, may <lb/>pa&longs;&longs;e them both in the &longs;ame Time, as it doth <lb/>the &longs;ole inclined <emph type="italics"/>P<emph.end type="italics"/>lane <emph type="italics"/>ex quiete<emph.end type="italics"/> in its &longs;uperi­<lb/>our Term.</s></p><p type="main"> <s><emph type="italics"/>Let the inclined Plane and Perpendicular be A B and A C, who&longs;e <lb/>Term A is the &longs;ame. </s> <s>It is required in the Perpendicular prolonged <lb/>from A to find a &longs;ublime point, out of which the Moveable de&longs;cen­<lb/>ding, and being turned along the Plane A B, may pa&longs;&longs;e the a&longs;&longs;igned part <lb/>of the Perpendicular and the Plane A B in the &longs;ame Time, as it would the <lb/>&longs;ole Plane A B out of Re&longs;t in A.<emph.end type="italics"/></s></p><figure id="id.069.01.197.1.jpg" xlink:href="069/01/197/1.jpg"/><p type="main"> <s><emph type="italics"/>Let the Ho­<lb/>rizontal Line <lb/>be B C, and <lb/>let A N be <lb/>cut equal to <lb/>A C; and as <lb/>A B is to B N, <lb/>&longs;o let A L be <lb/>to L C: and <lb/>unto A L let <lb/>A I be equal, <lb/>and unto A C <lb/>and B I let C <lb/>E be a third <lb/>proportional, <lb/>marked in the <lb/>Perpendicular A C produced. </s> <s>I &longs;ay, that C E is the Space acquired; <lb/>&longs;o that the Perpendicular being extended above A, and the part A X <lb/>equal to C E being taken, a Moveable out of X will pa&longs;&longs;e both the<emph.end type="italics"/><pb xlink:href="069/01/198.jpg" pagenum="195"/><emph type="italics"/>Spaces X A B in the &longs;ame Time as it would the &longs;ole Space A B out of A. <lb/></s> <s>Draw the Horizontal Line X R Parallel to B C, with which let B A <lb/>being prolonged meet in R, and then A B being continued out unto D <lb/>draw E D Parallel to C B, and upon A D de&longs;cribe a Semicircle, and <lb/>from B, and Perpendicular to D A, erect B F till it meet with the Cir­<lb/>cumference. </s> <s>It is manife&longs;t that F B is a Mean-proportional betwixt <lb/>A B and B D, and that the Line drawn from F to A is a Mean-propor­<lb/>tional betwixt D A and A B. </s> <s>Suppo&longs;e B S equal to B I, and F H equal <lb/>to F B: And becau&longs;e, as A B is to B D, &longs;o is A C to C E, and becau&longs;e <lb/>B F is a Mean-proportional betwixt A B and B D, and becau&longs;e B I is a <lb/>Mean-proportional betwixt A C and C E; therefore as B A is to A C, <lb/>&longs;o is F B to B S. </s> <s>And becau&longs;e as B A is to A C, or A N, &longs;o is F B to <lb/>B S, therefore, by Conver&longs;ion of the proportion, B F is to F S, as A B is <lb/>to B N, that is, A L to L C; therefore the Rectangle under F B and <lb/>C L, is equal to the Rectangle under A L, and S F: But this Rectangle <lb/>A L, and S F, is the exce&longs;&longs;e of the Rectangle under A L and F B, or A I <lb/>and B F, over and above the Triangle A I and B S, or A I B; and the <lb/>Rectangle F B and L C is the exce&longs;&longs;e of the Rectangle A C and B F <lb/>over and above the Rectangle A L and B F: But the Rectangle A C and <lb/>B F is equal to the Rectangle A B I; (for as B A is to A C, &longs;o is F B to <lb/>B I:) The exce&longs;&longs;e, therefore, of the Rectangle A B I above the Rectan­<lb/>gle A I and B F, or A I and F H, is equal to the exce&longs;&longs;e of the Rectangle <lb/>A I and F H above the Rectangle A I B: Therefore twice the Rectan­<lb/>gle A I and F H is equal to the two Rectangles A B I and A I B; that <lb/>is twice A I B with the Square of B I. </s> <s>Let the Square A I be common <lb/>to both, and twice the Rectangle A I B with the two Squares A I, and <lb/>I B, (that is, the Square A B) &longs;hall be equal to twice the Rectangle <lb/>A I and F H, with the Square A I: Again, taking in commonly the <lb/>Square B F; the two Squares A B and B F, that is the &longs;ole Square A F <lb/>&longs;hall be equal to twice the Rectangle A I and F H, with the two Squares <lb/>A I and F B, that is A I and F H: But the &longs;ame Square A F is equal <lb/>to twice the Rectangle A H F, with the two Squares A H and H F: <lb/>Therefore twice the Rectangle A I and F H, with the Squares A I and <lb/>F H, are equal to twice the Rectangle A H F, with the Squares A H <lb/>and H F: And, the Common Square H F being taken away, twice the <lb/>Rectangle A I and F H, with the Square A I, &longs;hall be equal to twice the <lb/>Rectangle A H F, with the Square A H. </s> <s>And becau&longs;e that in all the <lb/>Rectangles F H is the Common Side, the Line A H &longs;hall be equal to A I: <lb/>For if it &longs;hould be greater or le&longs;&longs;er, then the Rectangles F H A and the <lb/>Square H A would al&longs;o be greater or le&longs;&longs;er than the Rectangles F H and <lb/>I A, and the Square I A: Contrary to what hath been demon&longs;trated.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Now if we &longs;uppo&longs;e the Time of the De&longs;cent along A B to be as A B, <lb/>the Time along A C &longs;hall be as A C, and I B the Mean-proportional be­<lb/>twixt A C and C E &longs;hall be the Time along C E, or along X A from <lb/>Re&longs;t in X: And becau&longs;e betwixt D A and A B, or R B and B A the<emph.end type="italics"/><pb xlink:href="069/01/199.jpg" pagenum="196"/><emph type="italics"/>Mean-proportional is A F, and between A B and B D, that is, R A and <lb/>A B the Mean is B F, to which F H is equal; Therefore,<emph.end type="italics"/> exprædemon­<lb/>&longs;tratis, <emph type="italics"/>the exce&longs;&longs;e A H &longs;hall be the Time along A B<emph.end type="italics"/> ex quiete <emph type="italics"/>in R, or <lb/>after the Fall out of X; &longs;ince the Time along the &longs;aid A B<emph.end type="italics"/> ex quiete <emph type="italics"/>in <lb/>A, &longs;hall be A B. </s> <s>Therefore the Time along X A is I B; and along A B <lb/>after R A, or after X A, is A I: Therefore the Time along X A B &longs;hall <lb/>be as A B, namely the &longs;elf-&longs;ame with the Time along the &longs;ole A B<emph.end type="italics"/> ex qui­<lb/>ete <emph type="italics"/>in A. </s> <s>Which was the Propo&longs;ition.<emph.end type="italics"/></s></p><p type="head"> <s>PROBL. XIV. PROP. XXXV.</s></p><p type="main"> <s>An Inflected Line unto a given <emph type="italics"/>P<emph.end type="italics"/>erpendicular be­<lb/>ing a&longs;&longs;igned, to take part in the Inflected Line, <lb/>along which alone <emph type="italics"/>ex quiete<emph.end type="italics"/> a Motion may be <lb/>made in the &longs;ame Time, as it would be along <lb/>the &longs;ame together with the Perpendicular.</s></p><p type="main"> <s><emph type="italics"/>Let the Perpendicular be A B, and a Line inflected to it B C. </s> <s>It is <lb/>required in B C to take a part, along which alone out of Re&longs;t a <lb/>Motion may be made in the &longs;ame Time as it would along the &longs;ame <lb/>together with the Perpendicular A B. </s> <s>Draw the Horizon A D, with <lb/>which let the Inclined Line C B prolonged meet in E; and &longs;uppo&longs;e B F <lb/>equal to B A, and on the Center E at the di&longs;tance E F de&longs;cribe the Circle <lb/>F I G; and continue out F E unto the Circumference in G; and as G B <lb/>is to B F, &longs;o let B H be to H F; and let H I touch the Circle in I. </s> <s>Then <lb/>out of B erect B K<emph.end type="italics"/><lb/><figure id="id.069.01.199.1.jpg" xlink:href="069/01/199/1.jpg"/><lb/><emph type="italics"/>Perpendicular to <lb/>F C, with which <lb/>let the Line E I L <lb/>meet in L; and la&longs;t <lb/>of all let fall L M <lb/>Perpendicular to E <lb/>L, meeting B C in <lb/>M. </s> <s>I &longs;ay, that along <lb/>the Line B M from <lb/>Rest in B a Motion <lb/>may be made in the <lb/>&longs;ame Time, as it <lb/>would be<emph.end type="italics"/> ex quiete <emph type="italics"/>in A along both A B and B M. </s> <s>Let E N be made <lb/>equal to E L. </s> <s>And becau&longs;e as G B is to B F, &longs;o is B H to H F; there­<lb/>fore, by Permutation as G B is to B H, &longs;o will B F be to F H; and, by <lb/>Divi&longs;ion, G H &longs;hall be to H B, as B H is to H F: Wherefore the Rect­<lb/>angle G H F &longs;hall be equal to the Square H B: But the &longs;aid Rectangle <lb/>is al&longs;o equal to the Square H I: Therefore B H is equal to the &longs;ame H I.<emph.end type="italics"/><pb xlink:href="069/01/200.jpg" pagenum="197"/><emph type="italics"/>And becau&longs;e in the Quadrilateral Figure I L B H the Sides H B and <lb/>H I are equal, and the Angles B and I Right Angles, the Side B L &longs;hall <lb/>likewi&longs;e be equal to the Side L I: But E I is equal to E F: Therefore the <lb/>whole Line L E, or N E is equal to the two Lines L B and E F: Let <lb/>the Common Line E F be taken away, and the remainder F N &longs;hall be <lb/>equal to L B: And F B was &longs;uppo&longs;ed equal to B A: Therefore L B &longs;hall <lb/>be equal to the two Lines A B and B N. Again, if we &longs;uppo&longs;e the <lb/>Time along A B to be the &longs;aid A B, the Time along E B &longs;hall be equal to <lb/>E B; and the Time along the whole E M &longs;hall be E N, namely, the <lb/>Mean-proportional betwixt M E and E B: I berefore the Time of the <lb/>De&longs;cent of the remaining part B M after E B, or after A B, &longs;hall be the <lb/>&longs;aid B N: But it hath been &longs;uppo&longs;ed, that the Time along A B is A B: <lb/>Therefore the Time of the Fall along both A B and B M is A B N: <lb/>And becau&longs;e the Time along E B<emph.end type="italics"/> ex quiete <emph type="italics"/>in E is E B, the Time along <lb/>B M<emph.end type="italics"/> ex quiete <emph type="italics"/>in B &longs;hall be the Mean-proportional between B E and <lb/>B M; and this is B L: The Time, therefore, along both A B M<emph.end type="italics"/> ex quiete <lb/><emph type="italics"/>in A is A B N: And the Time along B M only<emph.end type="italics"/> ex quiete <emph type="italics"/>in B is B L: <lb/>But it was proved that B L is equal to the two A B and B N: Therefore <lb/>the Propo&longs;ition is manife&longs;t.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Otherwi&longs;e with more expedition.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Let B C be the Inclined Plane, and B A the Perpendicular. </s> <s>Continue <lb/>out C B to E, and unto E C erect a Perpendicular at B, which being <lb/>prolonged &longs;uppo&longs;e B H equal to the exce&longs;&longs;e of B E above B A; and to the <lb/>Angle B H E let the Angle H E L be equal; and let E L continued out <lb/>meet with B K in L; and from L erect the Perpendicular L M unto E L <lb/>meeting B C in M. </s> <s>I &longs;ay, that<emph.end type="italics"/><lb/><figure id="id.069.01.200.1.jpg" xlink:href="069/01/200/1.jpg"/><lb/><emph type="italics"/>B M is the Space acquired in <lb/>the Plane B C. </s> <s>For becau&longs;e <lb/>the Angle M L E is a Right­<lb/>Angle, therefore B L &longs;hall be <lb/>a Mean-proportional betwixt <lb/>M B and B E; and L E a <lb/>Mean proportional betwixt M <lb/>E and E B; to which E L let <lb/>E N be cut equal: And the <lb/>three Lines N E, E L, and <lb/>L H &longs;hall be equal; and H B &longs;hall be the exce&longs;&longs;e of N E above B L: But <lb/>the &longs;aid H B is al&longs;o the exce&longs;&longs;e of N E above N B and B A: Therefore <lb/>the two Lines N B and B A are equal to B L. </s> <s>And if we &longs;uppo&longs;e E B <lb/>to be the Time along E B, B L &longs;hall be the Time along B M<emph.end type="italics"/> ex quiete <emph type="italics"/>in <lb/>B; and B N &longs;hall be the Time of the &longs;ame B M after E B or after A B; <lb/>and A B &longs;hall be the Time along A B: Therefore the Times along A B M, <lb/>namely, A B N, are equal to the Times along the &longs;ole Line B M<emph.end type="italics"/> ex quiete <lb/><emph type="italics"/>in B: Which was intended.<emph.end type="italics"/></s></p><pb xlink:href="069/01/201.jpg" pagenum="198"/><p type="head"> <s>LEMMAI.</s></p><p type="main"> <s><emph type="italics"/>Let D C be Perpendicular to the Diameter B A; and from the Term <lb/>B continue forth B E D at plea&longs;ure, and draw a Line from F to B. </s> <s>I <lb/>&longs;ay, that F B is a Mean-proportional be-<emph.end type="italics"/><lb/><figure id="id.069.01.201.1.jpg" xlink:href="069/01/201/1.jpg"/><lb/><emph type="italics"/>twixt D B and B E. </s> <s>Draw a Line from E <lb/>to F, and by B draw the Tangent B G; <lb/>which &longs;hall be Parallel to the former C D: <lb/>Wherefore the Angle D B G &longs;hall be equal <lb/>to the Angle F D B, like as the &longs;ame G B D <lb/>is equal al&longs;o to the Angle E F B in the al­<lb/>tern Portion or Segment: Therefore the <lb/>Triangles F B D and F E B are alike: And, <lb/>as B D is to B F, &longs;o is F B to B E.<emph.end type="italics"/></s></p><p type="head"> <s>LEMMA II.</s></p><p type="main"> <s><emph type="italics"/>Let the Line A C be greater than D F; and let A B have greater <lb/>proportion to B C, than D E hath to E F. </s> <s>I &longs;ay, that A B is greater <lb/>than D E. </s> <s>For becau&longs;e A B hath to B C<emph.end type="italics"/><lb/><figure id="id.069.01.201.2.jpg" xlink:href="069/01/201/2.jpg"/><lb/><emph type="italics"/>greater proportion than D E hath to D F, <lb/>therefore look what proportion A B hath to <lb/>B C, the &longs;ame &longs;hall D E have to a Line le&longs;­<lb/>&longs;er than E F; let it have it to E G: And <lb/>becau&longs;e A B to B C, is as D E, to E G, there­<lb/>fore, by Compo&longs;ition, and by converting the Proportion, as C A is to A B, <lb/>&longs;o is G D to D E: But C A is greater than G D: Therefore B A &longs;hall <lb/>be greater than D E.<emph.end type="italics"/></s></p><p type="head"> <s>LEMMA III.</s></p><figure id="id.069.01.201.3.jpg" xlink:href="069/01/201/3.jpg"/><p type="main"> <s><emph type="italics"/>Let A C I B be the Quadrant of a Circle: <lb/>and to A C let B E be drawn from B Pa­<lb/>rallel: And out of any Center taken in the <lb/>&longs;ame de&longs;cribe the Circle B O E S, touching <lb/>A B in B, and cutting the Circumference of <lb/>the Quadrant in I; and draw a Line from <lb/>C to B, and another from C to I continued <lb/>out to S. </s> <s>I &longs;ay, that the Line C I is alwaies <lb/>le&longs;&longs;e than C O. </s> <s>Draw a Line from A to I; <lb/>which toucheth the Circle B O E. </s> <s>And if <lb/>D I be drawn it &longs;hall be equal to D B: And <lb/>becau&longs;é D B toucheth the Quadrant, the &longs;aid <lb/>D I &longs;hall likewi&longs;e touch it; and &longs;hall be Per-<emph.end type="italics"/><pb xlink:href="069/01/202.jpg" pagenum="199"/><emph type="italics"/>pendicular to the Diameter A I: Wherefore al&longs;o A I toucheth the Cir­<lb/>cle B O E in I. And, becau&longs;e the Angle A I C is greater than the An­<lb/>gle A B C, as in&longs;i&longs;ting on a larger Periphery: Therefore the Angle <lb/>S I N &longs;hall be al&longs;o greater than the &longs;ame A B C: Therefore the Portion <lb/>I E S is greater than the Portion B O; and the Line C S, nearer to the <lb/>Center, greater than C B: Therefore al&longs;o C O is greater than C I; <lb/>for that S C is to C B, as O C is to C I.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>And the &longs;ame al&longs;o would happen to be greater, if (as in the other <lb/>Figure) the Quadrant B I C were<emph.end type="italics"/><lb/><figure id="id.069.01.202.1.jpg" xlink:href="069/01/202/1.jpg"/><lb/><emph type="italics"/>le&longs;&longs;er: For the Perpendicular D B <lb/>will cut the Circle C I B: Wherefore <lb/>D I al&longs;o is equal to the &longs;aid D B; and <lb/>the Angle D I A &longs;hall be Obtu&longs;e, and <lb/>therefore A I N will al&longs;o cut B I N: <lb/>And becau&longs;e the Angle A B C is le&longs;&longs;e <lb/>than the Angle A I C, which is equal <lb/>to S I N; and this now is le&longs;&longs;e than that <lb/>which would be made at the Contact in <lb/>I by the Line S I: Therefore the Porti­<lb/>on S E I is much greater than the Por­<lb/>tion B O: Wherefore,<emph.end type="italics"/> &c. <emph type="italics"/>Which was <lb/>to be demon&longs;trated.<emph.end type="italics"/></s></p><p type="head"> <s>THEOR. <emph type="italics"/>XXII.<emph.end type="italics"/> PROP. <emph type="italics"/>XXXVI.<emph.end type="italics"/></s></p><p type="main"> <s>If from the lowe&longs;t point of a Circle erect unto <lb/>the Horizon a Plane &longs;hould be elevated &longs;ub­<lb/>tending a Circumference not greater than a <lb/>Quadrant, from who&longs;e Terms two other <lb/>Planes are Inflected to any point of the Cir­<lb/>cumference, the De&longs;cent along both the Infle­<lb/>cted Planes would be performed in a &longs;horter <lb/>Time than along the former elevated Plane <lb/>alone, or than along but one of the other two, <lb/>namely, along the lower.</s></p><p type="main"> <s><emph type="italics"/>Let C B D be the Circumference not greater than a Quadrant of a <lb/>Circle erect unto the Horizon on the lower point C, in which let <lb/>C D be an elevated Plane; and let two Planes be inflected from the <lb/>Terms D and C to any point in the Circumference taken at plea&longs;ure, <lb/>as B. </s> <s>I &longs;ay, that the Time of the De&longs;cent along both tho&longs;e Planes D B C <lb/>is &longs;horter than the Time of the De&longs;cent along the &longs;ole Plane D C, or <lb/>along the other only B C<emph.end type="italics"/> ex quiete <emph type="italics"/>in B. </s> <s>Let the Horizontal Line M D A<emph.end type="italics"/><pb xlink:href="069/01/203.jpg" pagenum="200"/><emph type="italics"/>be drawn by D, with which let C B prolonged meet in A; and let fall <lb/>the Perpendiculars D N and M C to M D, and B N to B D; and about <lb/>the Right-angled Triangle D B N de&longs;cribe the Semicircle D F B N, <lb/>cutting D C in F; and let D O be a Mean-proportional betwixt C D <lb/>and D F; and A V a Mean-proportional betwixt C A and A B: And <lb/>let P S be the time in which the whole D C, or B C, &longs;hall be pa&longs;&longs;ed; <lb/>(for it is manife&longs;t that they &longs;hall be both pa&longs;t in the &longs;ame Time;) And <lb/>look what proportion C D hath to D O, the &longs;ame &longs;hall the Time S P <lb/>have to the Time P R: the Time P R &longs;hall be that in which a Movea­<lb/>ble out of D will pa&longs;&longs;e D F; and R S that in which it &longs;hall pa&longs;&longs;e the re­<lb/>mainder F C. </s> <s>And becau&longs;e P S is al&longs;o the Time in which the Movea­<lb/>ble out of B &longs;hall pa&longs;&longs;e B C; if it be &longs;uppo&longs;ed that as B C is to C D, &longs;o is <lb/>S P to P T, P T &longs;hall be the Time of the De&longs;cent out of A to C: by <lb/>rea&longs;on D C is a Mean-proportional betwixt A C and C B, by what was <lb/>before demon&longs;trated: La&longs;t of all, as C A is to A V, &longs;o let T P be to<emph.end type="italics"/><lb/><figure id="id.069.01.203.1.jpg" xlink:href="069/01/203/1.jpg"/><lb/><emph type="italics"/>P G: P G &longs;hall be the Time, <lb/>in which thé Moveable out <lb/>of A de&longs;cendeth to B. </s> <s>And <lb/>becau&longs;e of the Circle D F N <lb/>the Diameter erect to the <lb/>Horizon is D N, the Lines <lb/>D F and D B &longs;hall be pa&longs;­<lb/>&longs;ed in equal Times. </s> <s>So that <lb/>if it &longs;hould be demon&longs;tra­<lb/>ted that the Moveable would <lb/>&longs;ooner pa&longs;&longs;e B C after the <lb/>De&longs;cent D B, than F C after the Lation D F; we &longs;hould have our in­<lb/>tent. </s> <s>But the Moveable will with the &longs;ame Celerity of Time pa&longs;&longs;e B C <lb/>coming out of D along D B, as if it came out of A along A B: for that <lb/>in both the De&longs;cents D B and A B it acquireth equal Moments of Velo­<lb/>city: Therefore it &longs;hall re&longs;t to be demon&longs;trated that the Time is &longs;horter <lb/>in which B C is pa&longs;&longs;ed after A B, than that in which F C is pa&longs;t after <lb/>D F. </s> <s>But it hath been demon&longs;trated, that the Time in which B C is <lb/>pa&longs;&longs;ed after A B is G T; and the Time of F C after D F is R S. </s> <s>It is <lb/>to be proved therefore, that R S is greater than G T: Which is thus <lb/>done. </s> <s>Becau&longs;e as S P is to P R, &longs;o is C D to D O, therefore, by Conver­<lb/>&longs;ion of proportion, and by Inver&longs;ion, as R S is to S P, &longs;o is O C to C D: <lb/>and as S P is to P T, &longs;o is D C to C A: And, becau&longs;e as T P is to PG, <lb/>&longs;o is C A to A V: Therefore al&longs;o, by Conver&longs;ion of the proportion, as <lb/>P T is to T G, &longs;o is A C to C V: therefore, ex equali, as R S is to G T, <lb/>&longs;o is O C to C V. </s> <s>But O C is greater than C V, as &longs;hall anon be de­<lb/>mon&longs;trated: Therefore the Time R S is greater than the Time G T: <lb/>Which it was required to demon&longs;trate. </s> <s>And becau&longs;e C F is greater than <lb/>C B, and F D le&longs;&longs;e than B A, therefore C D &longs;hall have greater propor­<lb/>tion to D F than C A to A B: And as C D is to D F, &longs;o is the Square<emph.end type="italics"/><pb xlink:href="069/01/204.jpg" pagenum="201"/><emph type="italics"/>C O to the Square O F; fora&longs;much as C D, D O, and O F are Propor­<lb/>tionals: And as C A is to A B, &longs;o is the Square C V to the Square <lb/>V B: Therefore C O hath greater proportion to O F, than C V to V B: <lb/>Therefore, by the foregoing Lemma, C O is greater than C V. </s> <s>It is <lb/>manife&longs;t moreover, that the Time along D C is to the Time along <lb/>D B C, as D O C is to D O together with C V.<emph.end type="italics"/></s></p><p type="head"> <s>SCHOLIUM.</s></p><p type="main"> <s>From the&longs;e things that have been demon&longs;trated may evidently <lb/>be gathered, that the &longs;wifte&longs;t of all Motions betwixt Term <lb/>and Term is not made along the &longs;horte&longs;t Line, that is by the <lb/>Right, but along a portion of a Circle.</s></p><p type="main"> <s><emph type="italics"/>For in the Quadrat B A E C, who&longs;e Side B C is erect to the Hori­<lb/>zon, let the Arch A C be divided into any number of equal parts, <lb/>A D, D E, E F, F G, G C; and let Right-lines be drawn from C to <lb/>the Points A, D, E, F, G, H; and al&longs;o by Lines joyn A D, D E, E F, <lb/>F G. and G C. </s> <s>It is manifest, that the Motion along the two Lines <lb/>A D C is &longs;ooner performed than along the<emph.end type="italics"/><lb/><figure id="id.069.01.204.1.jpg" xlink:href="069/01/204/1.jpg"/><lb/><emph type="italics"/>&longs;ole Line A C, or D C out of Re&longs;t in D: <lb/>But out of Re&longs;t in A, D C is &longs;ooner pa&longs;t <lb/>than the two A D C: But along the two <lb/>D E C out of Re&longs;t in A the De&longs;cent is <lb/>likewi&longs;e &longs;ooner made than along the &longs;ole <lb/>C D: Therefore the De&longs;cent along the <lb/>three Lines A D E C &longs;hall be performed <lb/>&longs;ooner than along the two A D C. </s> <s>And <lb/>in like manner the De&longs;cent along A D E <lb/>preceding, the Motion is more &longs;peedily con­<lb/>&longs;ummated along the two EFC than along the &longs;ole FC: Therfore along the <lb/>four A D E F C the Motion is quicklier accompli&longs;hed than along the <lb/>three A D E C: And &longs;o, in the la&longs;t place, along the two F G C after the <lb/>precedent De&longs;cent along A D E F the Motion will be &longs;ooner con&longs;umma­<lb/>ted than along the &longs;ole F C: Therefore along the five A D E F G C <lb/>the De&longs;cent &longs;hall be effected in a yet &longs;horter Time than along the four <lb/>A D E F C: Whereupon the nearer by in&longs;cribed Poligons we approach <lb/>the Circumference, the &longs;ooner will the Motion be performed between the <lb/>two a&longs;&longs;igned points A C.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>And that which is explained in a Quadrant, holdeth true likewi&longs;e <lb/>in a Circumference le&longs;&longs;e than the Quadrant: and the Ratiocination is <lb/>the &longs;ame.<emph.end type="italics"/></s></p><pb xlink:href="069/01/205.jpg" pagenum="202"/><p type="head"> <s>PROBL.XV. PROP. XXXVII.</s></p><p type="main"> <s>A Perpendicular and Inclined Plane of the &longs;ame <lb/>Elevation being given, to find a part in the In­<lb/>clined Plane that is equal to the Perpendicu­<lb/>lar, and pa&longs;&longs;ed in the &longs;ame Time as the &longs;aid <lb/>Perpendicular.</s></p><p type="main"> <s><emph type="italics"/>LET A B be the Perpendicular, and A C the Inclined Plane. </s> <s>It is <lb/>required in the Inclined to find a part equal to the Perpendicular <lb/>A B, that after Re&longs;t in A may be pa&longs;&longs;ed in a Time equal to the <lb/>Time in which the Perpendicular is pa&longs;&longs;ed. </s> <s>Let A D be equal to A B, <lb/>and cut the Remainder B C in two equal parts in I; and as A C is to<emph.end type="italics"/><lb/><figure id="id.069.01.205.1.jpg" xlink:href="069/01/205/1.jpg"/><lb/><emph type="italics"/>C I, &longs;o let C I be to another Line <lb/>A E; to which let D G be equal: It <lb/>is manife&longs;t that E G is equal to A D <lb/>and to A B. </s> <s>I &longs;ay moreover, that <lb/>this &longs;ame E G is the &longs;ame that is <lb/>pa&longs;&longs;ed by the Moveable coming out <lb/>of Re&longs;t in A in a Time equal to the <lb/>Time in which the Moveable fall eth along A B. </s> <s>For becau&longs;e that as <lb/>A C is to C I, &longs;o is C I to A E, or I D to D G; Therefore by Conver&longs;ion <lb/>of the proportion, as C A is to A I, &longs;o is D I to I G. </s> <s>And becau&longs;e as the <lb/>whole C A is to the whole A I, &longs;o is the part taken away C I to the part <lb/>I G; therefore the Remaining part I A &longs;hall be to the Remainder A G, <lb/>as the whole C A is to the whole A I: Therefore A I is a Mean-propor­<lb/>tional betwixt C A and A G; and C I a Mean-proportional betwixt <lb/>C A and A E: If therefore we &longs;uppo&longs;e the Time along A B to be as A B; <lb/>A C &longs;hall be the Time along A C, and C I or I D the Time along A E: <lb/>And becau&longs;e A I is a Mean-proportional betwixt C A and A G; and <lb/>C A is the Time along the whole A C: Therefore A I &longs;hall be the Time <lb/>along. </s> <s>A G; and the Remainder I C that along the Remainder G C: But <lb/>D I was the Time along A E: Therefore D I and I C are the Times <lb/>along both the Spaces A E and C G: Therefore the Remainder D A &longs;hall <lb/>be the Time along E G, to wit, equal to the Time along A B. </s> <s>Which was <lb/>to be done.<emph.end type="italics"/></s></p><p type="head"> <s>COROLLARIE.</s></p><p type="main"> <s>Hence it is manife&longs;t, that the Space required is an intermedial be­<lb/>tween the upper and lower parts that are pa&longs;t in equal <lb/>Times.</s></p><pb xlink:href="069/01/206.jpg" pagenum="203"/><p type="head"> <s><emph type="italics"/>P<emph.end type="italics"/>ROBL. XVI. <emph type="italics"/>P<emph.end type="italics"/>RO<emph type="italics"/>P.<emph.end type="italics"/> XXXVIII.</s></p><p type="main"> <s>Two Horizontal Planes cut by the Perpendicular <lb/>being given, to find a &longs;ublime point in the <emph type="italics"/>P<emph.end type="italics"/>er­<lb/>pendicular, out of which Moveables falling <lb/>and being reflected along the Horizontal <lb/><emph type="italics"/>P<emph.end type="italics"/>lanes may in Times equal to the Times of <lb/>the De&longs;cents along the &longs;aid Horizontal <emph type="italics"/>P<emph.end type="italics"/>lanes, <lb/>namely, along the upper and along the lower, <lb/>pa&longs;&longs;e Spaces that have to each other any given <lb/>proportion of the le&longs;&longs;er to the greater.</s></p><p type="main"> <s><emph type="italics"/>LET the Planes C D and B E be inter&longs;ected by the Perpendicular <lb/>A C B, and let the given proportion of the le&longs;&longs;e to the greater be <lb/>N to F G. </s> <s>It is required in the Perpendicular A B to find a point <lb/>on high, out of which a Moveable falling, and reflected along C D may <lb/>in a Time equal to the Time of its Fall, pa&longs;&longs;e a Space, that &longs;hall have <lb/>unto the Space pa&longs;&longs;ed by the other Moveable coming out of the &longs;ame &longs;ub­<lb/>lime point in a Time equal to the Time of its Fall with a Reflex Motion <lb/>along the Plane B E the &longs;ame proportion as the given Line N batb to<emph.end type="italics"/><lb/><figure id="id.069.01.206.1.jpg" xlink:href="069/01/206/1.jpg"/><lb/><emph type="italics"/>F G. </s> <s>Let G H be <lb/>made equal to the <lb/>&longs;aid N; and as F H <lb/>is to H G, &longs;o let <lb/>B C be to C L. </s> <s>I &longs;ay, <lb/>L is the &longs;ublime <lb/>point required. </s> <s>For <lb/>taking C M double <lb/>to C L, draw L M <lb/>meeting the Plane <lb/>B E in O; B O <lb/>&longs;hall be double to <lb/>B L: And becau&longs;e, <lb/>as F H is to H G, &longs;o is B C to C L; therefore, by Compo&longs;ition and In­<lb/>ver&longs;ion, as H G, that is, N is to G F, &longs;o is C L to L B, that is, C M to <lb/>B O: But becau&longs;e C M is double to L C; let the Space C M be that <lb/>which by the Moveable coming from L after the Fall L C is pa&longs;&longs;ed along <lb/>the Plane C D; and by the &longs;ame rea&longs;on B O is that which is pa&longs;&longs;ed after <lb/>the Fall L B in a Time equal to the Time of the Fall along L B; fora&longs;­<lb/>much as B O is double to B L: Therefore the Propo&longs;ition is manife&longs;t.<emph.end type="italics"/></s></p><pb xlink:href="069/01/207.jpg" pagenum="204"/><p type="main"> <s>SAGR. </s> <s>Really me thinks that we may ju&longs;tly grant our <emph type="italics"/>Acade­<lb/>mian<emph.end type="italics"/> what he without arrogance a&longs;&longs;umed to him&longs;elf in the begining <lb/>of this his Treati&longs;e of &longs;hewing us a <emph type="italics"/>New Science<emph.end type="italics"/> about <emph type="italics"/>a very old <lb/>Subject.<emph.end type="italics"/> And to &longs;ee with what Facility and Per&longs;picuity he deduceth <lb/>from one &longs;ole Principle the Demon&longs;trations of &longs;o many Propo&longs;iti­<lb/>ons, maketh me not a little to wonder how this bu&longs;ine&longs;s e&longs;caped <lb/>unhandled by <emph type="italics"/>Archimedes, Apollonius, Euclid,<emph.end type="italics"/> and &longs;o many other <lb/><emph type="italics"/>I<emph.end type="italics"/>llu&longs;trious Mathematicians and Phylo&longs;ophers: e&longs;pecially &longs;ince <lb/>there are found many great Volumns of <emph type="italics"/>Motion.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>There is extant a &longs;mall Fragment of <emph type="italics"/>Euclid<emph.end type="italics"/> touching <lb/><emph type="italics"/>Motion,<emph.end type="italics"/> but there are no marks to be &longs;een therein of any &longs;teps that he <lb/>took towards the di&longs;covery of the Proportion of <emph type="italics"/>Acceleration,<emph.end type="italics"/> and <lb/>of its Varieties along different <emph type="italics"/>I<emph.end type="italics"/>nclinations. </s> <s>So that indeed one <lb/>may &longs;ay, that never till now was the door opened to a new Con­<lb/>templation fraught with infinite and admirable Conclu&longs;ions, which <lb/>in times to come may bu&longs;ie other Wits.</s></p><p type="main"> <s>SAGR. <emph type="italics"/>I<emph.end type="italics"/> verily believe, that as tho&longs;e few Pa&longs;&longs;ions (<emph type="italics"/>I<emph.end type="italics"/> will &longs;ay <lb/>for example) of the Circle demon&longs;trated by <emph type="italics"/>Euclid<emph.end type="italics"/> in the third of <lb/>his <emph type="italics"/>Elements<emph.end type="italics"/> are an introduction to innumerable others more ab­<lb/>&longs;truce, &longs;o tho&longs;e produced and demon&longs;trated in this &longs;hort Tractate, <lb/>when they &longs;hall come to the hands of other Speculative Wits, &longs;hall <lb/>be a manuduction unto infinite others mote admirable: and it is to <lb/>be believed that thus it will happen by rea&longs;on of the Nobility of <lb/>the Argument above all others Phy&longs;ical.</s></p><p type="main"> <s>This daies Conference hath been very long and laborious; in <lb/>which <emph type="italics"/>I<emph.end type="italics"/> have ta&longs;ted more of the &longs;imple Propo&longs;itions than of their <lb/>Demon&longs;trations; many of which, <emph type="italics"/>I<emph.end type="italics"/> believe, will co&longs;t me more than <lb/>an hour a piece well to comprehend them: a task that <emph type="italics"/>I<emph.end type="italics"/> re&longs;erve to <lb/>my &longs;elf to perform at lea&longs;ure, you leaving the Book in my hands &longs;o <lb/>&longs;oon as we &longs;hall have heard this part that remains about the Moti­<lb/>on of Projects: which &longs;hall, if you &longs;o plea&longs;e, be to morrow.</s></p><p type="main"> <s>SALV. <emph type="italics"/>I<emph.end type="italics"/> &longs;hall not fail to be with you.</s></p><p type="head"> <s><emph type="italics"/>The End of the Third Dialogue.<emph.end type="italics"/></s></p></chap><chap><pb xlink:href="069/01/208.jpg" pagenum="205"/><p type="head"> <s>GALILEUS, <lb/>HIS <lb/>DIALOGUES <lb/>OF <lb/>MOTION.</s></p><p type="head"> <s>The Fourth Dialogue.</s></p><p type="head"> <s><emph type="italics"/>INTERLOCUTORS,<emph.end type="italics"/></s></p><p type="head"> <s>SALVIATUS, SAGREDUS, and SIMPLICIUS.</s></p><p type="main"> <s>SALVIATUS.</s></p><p type="main"> <s><emph type="italics"/>Simplicius<emph.end type="italics"/> likewi&longs;e cometh in the nick of time, therefore <lb/>without interpo&longs;ing any <emph type="italics"/>Re&longs;t<emph.end type="italics"/> let us proceed to <emph type="italics"/>Motion<emph.end type="italics"/>; <lb/>and &longs;ee here the <emph type="italics"/>Text<emph.end type="italics"/> of our <emph type="italics"/>Author.<emph.end type="italics"/></s></p><p type="head"> <s>OF THE MOTION OF <lb/>PROJECTS.</s></p><p type="main"> <s><emph type="italics"/>What accidents belong to<emph.end type="italics"/> Equable Motion, <emph type="italics"/>as al&longs;o to the<emph.end type="italics"/> Na­<lb/>turally Accelerate <emph type="italics"/>along all whatever Inclinations of Planes, <lb/>we have con&longs;idered above. </s> <s>In this Contemplation which we are now <lb/>entering upon, I will attempt to declare, and with &longs;olid Demon&longs;trations<emph.end type="italics"/><pb xlink:href="069/01/209.jpg" pagenum="206"/><emph type="italics"/>to e&longs;tabli&longs;h &longs;ome of the principal Symptomes, and tho&longs;e worthy of know­<lb/>ledge, which befall a Moveable whil&longs;t it is moved with a Motion com­<lb/>pounded of a twofold Lation, to wit, of the Equable and Naturally­<lb/>Accelerate: and this is that Motion, which we call the Motion of Pro­<lb/>jects: who&longs;e Generation I constitute to be in this manner.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>I fancy in my mind a certain Moveable projected or thrown along <lb/>an Horizontal Plane, all impediment &longs;ecluded: Now it is manife&longs;t by <lb/>what we have el&longs;ewhere &longs;poken at large, that that Motion will be Equa­<lb/>ble and Perpetual along the &longs;aid Plane, if the Plane be extended<emph.end type="italics"/> in in­<lb/>finitum<emph type="italics"/>: but if we &longs;uppo&longs;e it terminate, and placed on high, the Move­<lb/>able, which I conceive to be endued with Gravity, being come to the end <lb/>of the Plane, proceeding forward, it addeth to the Equable and Indeli­<lb/>ble fir&longs;t Lation that propen&longs;ion downwards which it receiveth from its <lb/>Gravity, and from thence a certain Motion doth re&longs;ult compounded of <lb/>the Equable Horizontal, and of the De&longs;cending naturally. </s> <s>Accellerate <lb/>Lations: which I call<emph.end type="italics"/> Projection. <emph type="italics"/>Some of who&longs;e Accidents we will de­<lb/>mon&longs;trate; the fir&longs;t of which &longs;hall be this.<emph.end type="italics"/></s></p><p type="head"> <s>THEOR.I. PROP.I.</s></p><p type="main"> <s><emph type="italics"/>A Project, when it is moved with a Motion compounded <lb/>of the Horizontal Equable, and of the Naturally­<lb/>Accelerate downwards, &longs;hall de&longs;cribe a Semipara­<lb/>bolical Line in its Lation.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>It is requi&longs;ite, <emph type="italics"/>Salviatus,<emph.end type="italics"/> in favour of my &longs;elf, and, as I <lb/>believe, al&longs;o of <emph type="italics"/>Simplicius,<emph.end type="italics"/> here to make a pau&longs;e; for I <lb/>am not &longs;o far gone in Geometry as to have &longs;tudied <emph type="italics"/>Apol­<lb/>lonius,<emph.end type="italics"/> &longs;ave only &longs;o far as to know that he treateth of the&longs;e Para­<lb/>bola's, and of the other Conick Sections, without the knowledge <lb/>of which, and of their Pa&longs;&longs;ions, I do not think that one can under­<lb/>&longs;tand the Demon&longs;trations of other Propo&longs;itions depending on <lb/>them. </s> <s>And becau&longs;e already in the very fir&longs;t Propo&longs;ition it is pro­<lb/>po&longs;ed by the Author to prove the Line de&longs;cribed by the Project to <lb/>be Parabolical, I imagine to my &longs;elf, that being to treat of none <lb/>but &longs;uch Lines, it is ab&longs;olutely nece&longs;&longs;ary to have a perfect know­<lb/>ledge, if not of all the Pa&longs;&longs;ions of tho&longs;e Figures that are demon­<lb/>&longs;trated by <emph type="italics"/>Apollonius,<emph.end type="italics"/> at lea&longs;t of tho&longs;e that are nece&longs;&longs;ary for the Sci­<lb/>ence in hand.</s></p><p type="main"> <s>SALV. </s> <s>You undervalue your &longs;elf very much, to make &longs;trange <lb/>of tho&longs;e Notions, which but even now you admitted as very well <lb/>under&longs;tood: I told you heretofore, that in the Treati&longs;e of Re&longs;i­<lb/>&longs;tances we had need of the knowledge of certain Propo&longs;itions of <pb xlink:href="069/01/210.jpg" pagenum="207"/><emph type="italics"/>Apollonius,<emph.end type="italics"/> at which you made no &longs;eruple.</s></p><p type="main"> <s>SAGR. </s> <s>It may be either that I knew them by chance, or that I <lb/>might for once gue&longs;&longs;e at, and take for granted &longs;o much as &longs;erved my <lb/>turn in that Tractate: but here where I imagine that we are to <lb/>hear all the Demon&longs;trations that concern tho&longs;e Lines, it is not con­<lb/>venient, as we &longs;ay, to &longs;wallow things whole, lo&longs;ing our time and <lb/>pains.</s></p><p type="main"> <s>SIMP. </s> <s>But as to what concerns me, although <emph type="italics"/>Sagredus<emph.end type="italics"/> were, <lb/>as I believe he is, well provided for his occa&longs;ions, the very fir&longs;t <lb/>Terms already are new to me: for though our Philo&longs;ophers have <lb/>handled this Argument of the Motion of Projects, I do not remem­<lb/>ber that they have confined them&longs;elves to de&longs;ine what the Lines <lb/>are which they de&longs;cribe, &longs;ave only in general that they are alwaies <lb/>Curved Lines, except it be in Projections Perpendicularly upwards. <lb/></s> <s>Therefore in ca&longs;e that little Geometry that I have learnt from <emph type="italics"/>Eu­<lb/>clid<emph.end type="italics"/> &longs;ince the Time that we have had other Conferences, be not &longs;uf­<lb/>ficient to render me capable of the Notions requi&longs;ite for the under­<lb/>&longs;tanding of the following Demon&longs;trations, I mu&longs;t content my &longs;elf <lb/>with bare Propo&longs;itions believed, but not under&longs;tood.</s></p><p type="main"> <s>SALV. </s> <s>But I will have you to know them by help of the Au­<lb/>thor of this Book him&longs;elf, who when he heretofore granted me a <lb/>&longs;ight of this his Work, becau&longs;e I al&longs;o at that time was not perfect <lb/>in the Books of <emph type="italics"/>Apollonius,<emph.end type="italics"/> took the pains to demon&longs;trate to me <lb/>two mo&longs;t principal Pa&longs;&longs;ions of the Parabola without any other Pre­<lb/>cognition, of which two, and no more, we &longs;hall &longs;tand in need in <lb/>the pre&longs;ent Treati&longs;e; which are both likewi&longs;e proved by <emph type="italics"/>Apollonius,<emph.end type="italics"/><lb/>but after many others, which it would take up a long time to look <lb/>over, and I am de&longs;irous that we may much &longs;horten the Journey, ta­<lb/>king the fir&longs;t immediately from the pure and &longs;imple generation of <lb/>the &longs;aid Parabola, and from this al&longs;o immediately &longs;hall be deduced <lb/>the Demon&longs;tration of the &longs;econd. </s> <s>Coming therefore to the fir&longs;t;</s></p><p type="main"> <s>De&longs;cribe the Right Cone, who&longs;e Ba&longs;e let be the Circle I B K C, <lb/>and Vertex the point L, in which, cut by a Plane parallel to the <lb/><figure id="id.069.01.210.1.jpg" xlink:href="069/01/210/1.jpg"/><lb/>Side L K, ari&longs;eth the Section B A C <lb/>called a Parabola; and let its Ba&longs;e <lb/>B C cut the Diameter I K of the <lb/>Circle I B K C at Right-Angles; <lb/>and let the Axis of the Parabola <lb/>A D be Parallel to the &longs;ide L K; <lb/>and taking any point F in the Line <lb/>B F A, draw the Right-Line F E <lb/>parallel to B D. </s> <s>I &longs;ay, that the Square <lb/>of B D hath to the Square of F E <lb/>the &longs;ame proportion that the Axis <lb/>D A hath to the part A E. </s> <s>Let a Plane parallel to the Circle I B K C <pb xlink:href="069/01/211.jpg" pagenum="208"/>be &longs;uppo&longs;ed to pa&longs;&longs;e by the Point E, which &longs;hall make in the Cone <lb/>a Circular Section, who&longs;e Diameter is G E H. </s> <s>And becau&longs;e upon <lb/>the Diameter I K of the Circle I B K, B D is a Perpendicular, the <lb/>Square of B D &longs;hall be equal to the Rectangle made by the parts <lb/>I D and D K: And likewi&longs;e in the upper Circle which is under&longs;tood <lb/>to pa&longs;&longs;e by the points G F H, the Square of the Line F E is equal <lb/>to the Rectangle of the parts G E H: Therefore the Square of B D <lb/>hath the &longs;ame proportion to the Square of F E, that the Rectangle <lb/>I D K hath to the Rectangle G E H. </s> <s>And becau&longs;e the Line E D is <lb/>Parallel to H K, E H &longs;hall be equal to D K, which al&longs;o are Parallels: <lb/>And therefore the Rectangle I D K &longs;hall have the &longs;ame proportion <lb/>to the Rectangle G E H, as I D hath to G E; that is, that D A hath <lb/>to A E: Therefore the Rectangle I D K to the Rectangle G E H, <lb/>that is, the Square B D to the Square F E, hath the &longs;ame proportion <lb/>that the Axis D A hath to the part A E: Which was to be de­<lb/>mon&longs;trated.</s></p><p type="main"> <s>The other Propo&longs;ition, likewi&longs;e nece&longs;&longs;ary to the pre&longs;ent Tract, <lb/>we will thus make out. </s> <s>Let us de&longs;cribe the Parabola, of which let the <lb/>Axis C A be prolonged out unto D; and taking any point B, let the <lb/>Line B C be &longs;uppo&longs;ed to be continued out by the &longs;ame Parallel un­<lb/><figure id="id.069.01.211.1.jpg" xlink:href="069/01/211/1.jpg"/><lb/>to the Ba&longs;e of the &longs;aid Parabola; <lb/>and let D A be &longs;uppo&longs;ed equal <lb/>to the part of the Axis C A. </s> <s>I &longs;ay, <lb/>that the Right-Line drawn by <lb/>the points D and B, falleth not <lb/>within the Parabola, but without, <lb/>&longs;o as that it only toucheth the <lb/>&longs;ame in the &longs;aid point B: For, if <lb/>it be po&longs;&longs;ible for it to fall within, <lb/>it cutteth it above, or being pro­<lb/>longed, it cutteth it below. </s> <s>And <lb/>in that Line let any point G be <lb/>taken, by which pa&longs;&longs;eth the Right <lb/>Line F G E. </s> <s>And becau&longs;e the <lb/>Square F E is greater than the <lb/>Square G E, the &longs;aid Square F E <lb/>&longs;hall have greater proportion to <lb/>the Square B C, than the &longs;aid Square G E hath to the &longs;aid B C. </s> <s>And <lb/>becau&longs;e, by the precedent, the Square F E is to the Square B C as <lb/>E A is to A C; therefore E A hath greater proportion to A C, than <lb/>the Square G E hath to the Square B C; that is, than the Square <lb/>E D hath to the Square D C: (becau&longs;e in the Triangle D G E as <lb/>G E is to the Parallel B C, &longs;o is E <emph type="italics"/>D<emph.end type="italics"/> to <emph type="italics"/>D<emph.end type="italics"/> C:) But the Line E A to <lb/>A C, that is, to A <emph type="italics"/>D<emph.end type="italics"/> hath the &longs;ame proportion that four Rectangles <lb/>E A <emph type="italics"/>D<emph.end type="italics"/> hath to four Squares of A <emph type="italics"/>D,<emph.end type="italics"/> that is, to the Square C <emph type="italics"/>D,<emph.end type="italics"/><pb xlink:href="069/01/212.jpg" pagenum="209"/>(which is equal to four Squares of A D:) Therefore four Rectan­<lb/>gles E A D &longs;hall have greater proportion to the Square C D, than <lb/>the Square E D hath to the Square D C: Therefore four Rectan­<lb/>gles E A D &longs;hall be greater than the Square E D: which is fal&longs;e, <lb/>for they are le&longs;&longs;e; becau&longs;e the parts E A and A D of the Line E D <lb/>are not equal: Therefore the Line D B toucheth the Parabola in B, <lb/>and doth not cut it: Which was to be demon&longs;trated.</s></p><p type="main"> <s>SIMP. </s> <s>You proceed in your Demon&longs;trations too &longs;ublimely, <lb/>and &longs;till, as far as I can perceive, &longs;uppo&longs;e that the Propo&longs;itions of <lb/><emph type="italics"/>Euclid<emph.end type="italics"/> are as familiar and ready with me, as the fir&longs;t Axioms them­<lb/>&longs;elves, which is not &longs;o. </s> <s>And the impo&longs;ing upon me, ju&longs;t now, that <lb/>four Rectangles E A <emph type="italics"/>D<emph.end type="italics"/> are le&longs;s than the Square <emph type="italics"/>D<emph.end type="italics"/> E becau&longs;e the <lb/>parts E A and A <emph type="italics"/>D<emph.end type="italics"/> of the Line E <emph type="italics"/>D<emph.end type="italics"/> are not equal, doth not &longs;atis&longs;ie <lb/>me, but leaveth me in doubt.</s></p><p type="main"> <s>SALV. </s> <s>The truth is, all the Mathematicians that are not vulgar <lb/>&longs;uppo&longs;e that the Reader hath ready by heart the Elements of <lb/><emph type="italics"/>Euclid<emph.end type="italics"/>: And here to &longs;upply your want, it &longs;hall &longs;u&longs;fice to remember <lb/>you of a Propo&longs;ition in the &longs;econd Book, in which it is demon&longs;trated <lb/>that when a Line is cut into equal parts, and into unequal, the <lb/>Rectangle of the unequal parts is le&longs;s than the Rectangle of the <lb/>equal, (that is, than the Square of the half) by &longs;o much as is the <lb/>Square of the Line comprized between the Sections. </s> <s>Whence it is <lb/>manife&longs;t, that the Square of the whole, which continueth four <lb/>Squares of the Half, is greater than four Rectangles of the unequal <lb/>parts. </s> <s>Now it is nece&longs;&longs;ary that we bear in mind the&longs;e two Propo&longs;i­<lb/>tions which have been demon&longs;trated, taken from the Conick Ele­<lb/>ments, for the better under&longs;tanding the things that follow in the <lb/>pre&longs;ent Treati&longs;e: for of the&longs;e two, and no more, the Author <lb/>makes u&longs;e. </s> <s>Now we may rea&longs;&longs;ume the Text to &longs;ee in what manner <lb/>he doth demon&longs;trate his fir&longs;t Propo&longs;ition, in which he intendeth to <lb/>prove unto us, That the Line de&longs;cribed by the Grave Moveable, <lb/>when it de&longs;cends with a Motion compounded of the Equable <lb/>Horizontal, and of the Natural <emph type="italics"/>D<emph.end type="italics"/>e&longs;cending is a Semiparabola.</s></p><p type="main"> <s><emph type="italics"/>Suppo&longs;e the Horizontal Line or Plane A B placed on high; upon<emph.end type="italics"/><lb/>[or along] <emph type="italics"/>which let the Moveable pa&longs;&longs;e with an Equable Motion out <lb/>of A unto B: and the &longs;upport of the Plane failing in B let there be <lb/>derived upon the Moveable from its own Gravity a Motion naturally <lb/>downwards according to the Perpendicular B N. </s> <s>Let the Line B E be <lb/>&longs;uppo&longs;ed applyed unto the Plane A B right out, as if it were the Efflux <lb/>or mea&longs;ure of the Time, on which at plea&longs;ure note any equal parts of <lb/>Time, B C, C D, D E: And out of the points B C D E &longs;uppo&longs;e Per­<lb/>pendicular Lines to be let fall equidi&longs;tant or parallel to B N: In the fir&longs;t <lb/>of which take any part C I, who&longs;e quadruple take in the following one <lb/>D F, nonuple E H, and &longs;o in the re&longs;t that follow according to the propor-<emph.end type="italics"/><pb xlink:href="069/01/213.jpg" pagenum="210"/><emph type="italics"/>tion of the Squares of C B, D B, E B, or, if you will, in the doubled <lb/>proportion of the Lines. </s> <s>And if unto the Moveable moved beyond B <lb/>towards C with the Equable Lation we &longs;uppo&longs;e the Perpendicular <lb/>De&longs;cent to be &longs;uperadded according to the quantity C I, in the Time <lb/>B C it &longs;hall be found con&longs;tituted in the Term I. </s> <s>And proceeding farther,<emph.end type="italics"/><lb/><figure id="id.069.01.213.1.jpg" xlink:href="069/01/213/1.jpg"/><lb/><emph type="italics"/>in the Time D B, namely, <lb/>in the double of B C, the <lb/>Space of the De&longs;cent down­<lb/>wards &longs;hall be quadruple to <lb/>the fir&longs;t Space C I: For <lb/>it hath beendemon&longs;trated in <lb/>the fir&longs;t Trastate, that the <lb/>Spaces pa&longs;&longs;ed by GraveBo­<lb/>dies with a Motion Natu­<lb/>rally Accelerate are in du­<lb/>plicate proportion of their Times. </s> <s>And it likewi&longs;e followeth, that the <lb/>Space E H pa&longs;&longs;ed in the Time B E, &longs;hall be as G. </s> <s>So that it is manife&longs;tly <lb/>proved, that the Spaces E H, D F, C I, are to one another as the Squares <lb/>of the Lines E B, D B, C B. </s> <s>Now from the points I, F, and H draw <lb/>the Right Lines I O, F G, H L, Parallel to the &longs;aid E B; and each of <lb/>the Lines H L, F G, and I O &longs;hall be equal to each of the other Lines <lb/>E B, D B, and C B; as al&longs;o each of tho&longs;e B O, B G, and B L, &longs;hall be <lb/>equal to each of tho&longs;e C I, D F, and E H: And the Square H L &longs;hall <lb/>be to the Square F G, as the Line L B to B G: And the Square F G <lb/>&longs;hall be to the Square I O, as G B to B O: Therefore the Points I, F, <lb/>and H are in one and the &longs;ame Parabolical Line. </s> <s>And in like manner <lb/>it &longs;hall be demon&longs;trated, any equalparticles of Time of what&longs;oever Mag­<lb/>nitude being taken, that the place of the Moveable who&longs;e Motion is <lb/>compounded of the like Lations, is in the &longs;ame Times to be found in the <lb/>&longs;ame Parabolick Line: Therefore the Propo&longs;ition is manife&longs;t.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>This Conclu&longs;ion is gathered from the Conver&longs;ion of the <lb/>fir&longs;t of tho&longs;e two Propo&longs;itions that went before, for the Parabola <lb/>being, for example, de&longs;cribed by the points B H, if either of the <lb/>two F or I were not in the de&longs;cribed Parabolick Line, it would be <lb/>within, or without; and by con&longs;equence the Line F G would be <lb/>either greater or le&longs;&longs;er than that which &longs;hould determine in the Pa­<lb/>rabolick Line; Wherefore the Square of HL would have, not to <lb/>the Square of F G, but to another greater or le&longs;&longs;er, the &longs;ame pro­<lb/>portion that the Line L B hath to BG, but it hath the &longs;ame propor­<lb/>tion to the Square of F G: Therefore the point F is in the Parabo­<lb/>lick Line: And &longs;o all the re&longs;t, <emph type="italics"/>&c.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>It cannot be denied but that the Di&longs;cour&longs;e is new, in­<lb/>genious and concludent, arguing <emph type="italics"/>ex &longs;uppo&longs;itione,<emph.end type="italics"/> that is, &longs;uppo&longs;ing <lb/>that the Tran&longs;ver&longs;e Motion doth continue alwaies Equable, and <pb xlink:href="069/01/214.jpg" pagenum="211"/>that the Natural <emph type="italics"/>Dcor&longs;um<emph.end type="italics"/> do likewi&longs;e keep its tenour of continu­<lb/>ally Accelerating according to a proportion double to the Times; <lb/>and that tho&longs;e Motions and their Velocities in mingling be not al­<lb/>tered, di&longs;turbed, and impeded, &longs;o that finally the Line of the Pro­<lb/>ject do not in the continuation of the Motion degenerate into an­<lb/>other kind; a thing which &longs;eemeth to me to be impo&longs;&longs;ible. </s> <s>For, in <lb/>regard that the Axis of our Parabola, according to which we &longs;up­<lb/>po&longs;e the Natural Motion of Graves to be made, being Perpendicu­<lb/>lar to the Horizon, doth terminate in the Center of the Earth; and <lb/>in regard that the Parabolical Line doth &longs;ucce&longs;&longs;ively enlarge from <lb/>its Axis, no Project would ever come to terminate in the Center, or <lb/>if it &longs;hould come thitherwards, as it &longs;eemeth nece&longs;&longs;ary that it mu&longs;t, <lb/>the Line of the Project &longs;hould de&longs;cribe another mo&longs;t different from <lb/>that of the Parabola.</s></p><p type="main"> <s>SIMP. </s> <s>I add to the&longs;e difficulties &longs;everal others; one of which is <lb/>that we &longs;uppo&longs;e, that the Horizontal Plane which hath neither accli­<lb/>vity or declivity is a Right Line; as if that &longs;uch a Line were in all <lb/>its parts equidi&longs;tant from the Center, which is not true: for depart­<lb/>ing from its middle it goeth towards the extreams, alwaies more and <lb/>more receding from the Center, and therefore alwaies a&longs;cending: <lb/>which of con&longs;equence rendereth it Impo&longs;&longs;ible that its Motion <lb/>&longs;hould be perpetual, or that it &longs;hould for any time continue Equa­<lb/>ble, and nece&longs;&longs;itates it to grow continually more and more weak. <lb/></s> <s>Moreover, it is, in my Opinion, impo&longs;&longs;ible to avoid the Impedi­<lb/>ment of the <emph type="italics"/>Medium,<emph.end type="italics"/> but that it will take away the Equability of <lb/>the Tran&longs;ver&longs;e Motion, and the Rule of the Acceleration in falling <lb/>Grave Bodies. </s> <s>By all which difficulties it is rendred very improba­<lb/>ble that the things demon&longs;trated with &longs;uch incon&longs;tant Suppo&longs;i­<lb/>tions &longs;hould afterwards hold true in the practical Experiments.</s></p><p type="main"> <s>SALV. </s> <s>All the Objections and Difficulties alledged are &longs;o <lb/>well grounded, that I e&longs;teem it impo&longs;&longs;ible to remove them; and <lb/>for my own part I admit them all, as al&longs;o I believe the Author <lb/>him&longs;elf would do. </s> <s>And I grant that the Conclu&longs;ions thus demon­<lb/>&longs;trated in Ab&longs;tract, do alter and prove fal&longs;e, and that &longs;o egregiou&longs;­<lb/>ly, in Concrete, that neither is the Tran&longs;ver&longs;e Motion Equable, <lb/>nor is the Acceleration of the Natural in the proportion &longs;uppo&longs;e, <lb/>nor is the Line of the Project Parabolical, <emph type="italics"/>&c. </s> <s>B<emph.end type="italics"/>ut yet on the <lb/>contrary, I de&longs;ire that you would not &longs;cruple to grant to this our <lb/>Author that which other famous Men have &longs;uppo&longs;ed, although <lb/>fal&longs;e. </s> <s>And the &longs;ingle Authority of <emph type="italics"/>Archimedes<emph.end type="italics"/> may &longs;atisfie every <lb/>one: who in his Mechanicks, and in the fir&longs;t Quadrature of the <lb/>Parabola, taketh it as a true Principle, that the <emph type="italics"/>B<emph.end type="italics"/>eam of the <emph type="italics"/>B<emph.end type="italics"/>allance <lb/>or Stilliard is a Right Line in all its points equidi&longs;tant from the <lb/>Common Center of Grave <emph type="italics"/>B<emph.end type="italics"/>odies, and that the Scale-ropes, to <lb/>which the Weights are hanged, are parallel to one another. </s> <s>Which <pb xlink:href="069/01/215.jpg" pagenum="212"/>Liberty of his hath been excu&longs;ed by &longs;ome, for that in our practices <lb/>the In&longs;truments we u&longs;e, and the Di&longs;tances which we take are &longs;o <lb/>&longs;mall in compari&longs;on of our great remotene&longs;s from the Center of <lb/>the Terre&longs;trial Globe, that we may very well take a Minute of a <lb/>degree of the great Circle as if it were a Right Line, and two Per­<lb/>pendiculars that &longs;hould hang at its extreams as if they were Paral­<lb/>lels. </s> <s>For if we were in practical Operations to keep account of <lb/>&longs;uch like Minutes, we &longs;hould begin to reprove the Architects, who <lb/>with the Plumb Line &longs;uppo&longs;e that they rai&longs;e very high Towers <lb/>between Lines equidi&longs;tant. </s> <s>And I here add, that we may &longs;ay that <lb/><emph type="italics"/>Archimedes,<emph.end type="italics"/> and others &longs;uppo&longs;e in their Contemplations that they <lb/>were con&longs;tituted remote at an infinite di&longs;tance from the Center; <lb/>in which ca&longs;e their A&longs;&longs;umptions were not fal&longs;e: And that therefore <lb/>they did conclude by Ab&longs;olute Demon&longs;tration. </s> <s>Again, if we will <lb/>practice the demon&longs;trated Conclu&longs;ions in terminate Di&longs;tances, by <lb/>&longs;uppo&longs;ing an immen&longs;e Di&longs;tance, we ought to defalk from the <lb/>truth demon&longs;trated that which our Di&longs;tance from the Center doth <lb/>import, not being really infinite, but yet &longs;uch as that it may be <lb/>termed Immen&longs;e in compari&longs;on of the Artifices that we make u&longs;e <lb/>of, the greate&longs;t of which will be the Ranges of Projects, and among&longs;t <lb/>the&longs;e that only of Canon &longs;hot; which though it be great, yet &longs;hall <lb/>it not exceed four of tho&longs;e Miles of which we are remote from the <lb/>Center well-nigh &longs;o many thou&longs;ands: and the&longs;e coming to deter­<lb/>mine in the Surface of the Terre&longs;trial Globe may very well only in­<lb/>&longs;en&longs;ibly alter that Parabolick Figure, which we grant would be <lb/>extreamly transformed in going to determine in the Center. </s> <s>In <lb/>the next place as to the perturbation proceeding from the Impedi­<lb/>ment of the <emph type="italics"/>Medium,<emph.end type="italics"/> this is more con&longs;iderable, and, by rea&longs;on of <lb/>its &longs;o great multiplicity of Varieties, incapable of being brought <lb/>under any certain Rules, and reduced to a Science: for if we <lb/>&longs;hould propo&longs;e to con&longs;ideration no more but the Impediment which <lb/>the Air procureth to the Motions con&longs;idered by us, this alone &longs;hall <lb/>be found to di&longs;turb all, and that infinite waies, according as we <lb/>infinite waies vary the Figures, Gravities, and Velocities of the <lb/>Moveables. </s> <s>For as to the Velocity, according as this &longs;hall be grea­<lb/>ter, the greater &longs;hall the oppo&longs;ition be that the Air makes again&longs;t <lb/>them, which &longs;hall yet more impede the &longs;aid Moveable according as <lb/>they are le&longs;s Grave: &longs;o that although the de&longs;cending Grave Body <lb/>ought to go Accelerating in a duplicate proportion to the Duration <lb/>of its Motion, yet neverthele&longs;s, albeit the Moveable were very <lb/>Grave, in coming from very great heights, the Impediment of the <lb/>Air &longs;hall be &longs;o great, as that it will take from it all power of far­<lb/>ther encrea&longs;ing its Velocity, and will reduce it to an Uniform and <lb/>Equable Motion: And this Adequation &longs;hall be &longs;o much the &longs;ooner <lb/>obtained, and in &longs;o much le&longs;&longs;er heights, by how much the Moveable <pb xlink:href="069/01/216.jpg" pagenum="213"/>&longs;hall be le&longs;s Grave. </s> <s>That Motion al&longs;o which along the Horizontal <lb/>Plane, all other Ob&longs;tacles being removed, ought to be Equable <lb/>and perpetual, &longs;hall come to be altered, and in the end arre&longs;ted by <lb/>the Impediment of the Air: and here likewi&longs;e &longs;o much the &longs;ooner, <lb/>by how much the Moveable &longs;hall be Lighter. </s> <s>Of which Accidents <lb/>of Gravity, of Velocity, and al&longs;o of Figure, as being varied &longs;eve­<lb/>ral waies, there can no fixed Science be given. </s> <s>And therefore that <lb/>we may be able Scientifically to treat of this Matter it is requi&longs;ite <lb/>that we ab&longs;tract from them; and, having found and demon&longs;trated <lb/>the Conclu&longs;ions ab&longs;tracted from the Impediments, that we make <lb/>u&longs;e of them in practice with tho&longs;e Limitations that Experience &longs;hall <lb/>from time to time &longs;hew us. </s> <s>And yet neverthele&longs;s the benefit &longs;hall <lb/>not be &longs;mall, becau&longs;e &longs;uch Matters, and their Figures &longs;hall be made <lb/>choice of as are le&longs;s &longs;ubject to the Impediments of the <emph type="italics"/>Medium<emph.end type="italics"/>; <lb/>&longs;uch are the very Grave, the Rotund: and the Spaces, and the <lb/>Velocities for the mo&longs;t part will not be &longs;o great, but that their ex­<lb/>orbitances may with ea&longs;ie ^{*} Allowance be reduced to a certainty. <lb/><arrow.to.target n="marg1095"></arrow.to.target><lb/>Yea more, in Projects practicable by us, that are of Grave Matters, <lb/>and of Round Figure, and al&longs;o that are of Matters le&longs;&longs;e Grave, <lb/>and of Cylindrical Figure, as Arrows, &longs;hot from Slings or Bows, <lb/>the variation of their Motion from the exact Parabolical Figure <lb/>&longs;hall be altogether in&longs;en&longs;ible. </s> <s>Nay, (and I will a&longs;&longs;ume to my &longs;elf <lb/>a little more freedom) that in ^{*} In&longs;truments that are practicable by <lb/><arrow.to.target n="marg1096"></arrow.to.target><lb/>us, their &longs;malne&longs;s rendreth the extern and accidental Impediments, <lb/>of which that of the <emph type="italics"/>Medium<emph.end type="italics"/> is mo&longs;t con&longs;iderable, to be but of <lb/>very &longs;mall note, I am able by two experiments to make manife&longs;t. <lb/></s> <s>I will con&longs;ider the Motions made thorow the Air, for &longs;uch are tho&longs;e <lb/>chiefly of which we &longs;peak: again&longs;t which the &longs;aid Air in two man­<lb/>ners exerci&longs;eth its power. </s> <s>The one is by more impeding the Movea­<lb/>bles le&longs;s Grave, than tho&longs;e very Grave. </s> <s>The other is in more oppo­<lb/>&longs;ing the greater than the le&longs;s Velocity of the &longs;ame Moveable. </s> <s>As <lb/>to the fir&longs;t; Experience &longs;hewing us that two Balls of equal <lb/>bigne&longs;s, but in weight one ten or twelve times more Grave than the <lb/>other, as, for example, one of Lead and another of Oak would <lb/>be, de&longs;cending from an height of 150, or 200 Yards, arrive to the <lb/>Earth with Velocity very little different, it a&longs;&longs;ureth us that the Im­<lb/>pediment or Retardment of the Air in both is very &longs;mall: for if <lb/>the <emph type="italics"/>B<emph.end type="italics"/>all of Lead departing from on high in the &longs;ame Moment with <lb/>that of Wood, were but little retarded, and this much, the Lead at <lb/>its coming to the ground &longs;hould leave the Wood a very con&longs;idera­<lb/>ble Space behind, &longs;ince it is ten times more Grave; which never­<lb/>thele&longs;s doth not happen: nay, its Anticipation &longs;hall not be &longs;o <lb/>much as the hundredth part of the whole height. </s> <s>And between a <lb/><emph type="italics"/>B<emph.end type="italics"/>all of Lead, and another of Stone which weighs a third part, or <lb/>half &longs;o much as it, the difference of the Times of their coming to <pb xlink:href="069/01/217.jpg" pagenum="214"/>the ground would be hardly ob&longs;ervable. </s> <s>Now becau&longs;e the <emph type="italics"/>Impe­<lb/>tus<emph.end type="italics"/> that a <emph type="italics"/>B<emph.end type="italics"/>all of Lead acquireth in falling from an height of 200 <lb/>Yards (which is &longs;o much that continuing it in an Equable Moti­<lb/>on it would in a like Time run 400 Yards) is very con&longs;iderable in <lb/>compari&longs;on of the Velocity that we confer with <emph type="italics"/>B<emph.end type="italics"/>ows or other Ma­<lb/>chines, upon our Projects (excepting the <emph type="italics"/>Impetus's<emph.end type="italics"/> that depend <lb/>on the Fire) we may without any notable Errour conclude and <lb/>account the Propo&longs;itions to be ab&longs;olutely true that are demon&longs;tra­<lb/>ted without any regard had to the alteration of the <emph type="italics"/>Medium.<emph.end type="italics"/> In <lb/>the next place as touching the other part, that is to &longs;hew, that the <lb/>Impediment that the &longs;aid Moveable receiveth from the Air whil&longs;t <lb/>it moveth with great Velocity is not much greater than that which <lb/>oppo&longs;eth it in moving &longs;lowly, the en&longs;uing Experiment giveth us <lb/>full a&longs;&longs;urance of it. </s> <s>Su&longs;pend by two threads both of the &longs;ame <lb/>length, <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> four or five Yards, two equal <emph type="italics"/>B<emph.end type="italics"/>alls of Lead: and <lb/>having fa&longs;tned the &longs;aid threads on high, let both the <emph type="italics"/>B<emph.end type="italics"/>alls be re­<lb/>moved from the &longs;tate of Perpendicularity; but let the one be re­<lb/>moved 80. or more degrees, and the other not above 4 or 5: &longs;o <lb/>that one of them being left at liberty de&longs;cendeth, and pa&longs;&longs;ing be­<lb/>yond the Perpendicular, de&longs;cribeth very great Arches of 160, 150, <lb/>140, <emph type="italics"/>&c.<emph.end type="italics"/> degrees, dimini&longs;hing them by little and little: but the <lb/>other &longs;winging freely pa&longs;&longs;eth little Arches of 10, 8, 6, <emph type="italics"/>&c.<emph.end type="italics"/> this <lb/>al&longs;o dimini&longs;hing them in like manner by little and little. </s> <s>Here I <lb/>&longs;ay, in the fir&longs;t place, that the fir&longs;t <emph type="italics"/>B<emph.end type="italics"/>all &longs;hall pa&longs;s its 180, 160, <emph type="italics"/>&c.<emph.end type="italics"/><lb/>degrees in as much Time as the other doth its 10, 8, <emph type="italics"/>&c.<emph.end type="italics"/> From <lb/>whence it is manife&longs;t, that the Velocity of the fir&longs;t <emph type="italics"/>B<emph.end type="italics"/>all &longs;hall be 16 <lb/>and 18 times greater than the Velocity of the &longs;econd: &longs;o that in <lb/>ca&longs;e the greater Velocity were to be more impeded by the Air than <lb/><arrow.to.target n="marg1097"></arrow.to.target><lb/>the le&longs;&longs;er, the Vibrations &longs;hould be more ^{*} rare in the greate&longs;t <lb/>Arches of 180, or 160 degrees, <emph type="italics"/>&c.<emph.end type="italics"/> than in the lea&longs;t of 10, 8, 4, <lb/>and al&longs;o of 2, and of 1; but this is contradicted by Experience: <lb/>for if two A&longs;&longs;i&longs;tants &longs;hall &longs;et them&longs;elves to count the Vibrations, <lb/>one the greate&longs;t, the other the lea&longs;t, they will find that they &longs;hall <lb/>number not only tens, but hundreds al&longs;o, without di&longs;agreeing one <lb/>&longs;ingle Vibration, yea, or one &longs;ole point. </s> <s>And this ob&longs;ervati­<lb/>on joyntly a&longs;&longs;ureth us of the two Propo&longs;itions, namely, that the <lb/>greate&longs;t and lea&longs;t Vibrations are all made one after another under <lb/>equal Times, and that the Impediment and Retardment of the Air <lb/>operates no more in the &longs;wifte&longs;t Motion, than in the &longs;lowe&longs;t: <lb/>contrary to that which before it &longs;eemed that we our &longs;elves al&longs;o <lb/>would have judged for company.</s></p><p type="margin"> <s><margin.target id="marg1095"></margin.target>* Tarra.</s></p><p type="margin"> <s><margin.target id="marg1096"></margin.target>* Artifizii.</s></p><p type="margin"> <s><margin.target id="marg1097"></margin.target>Or &longs;ewer.</s></p><p type="main"> <s>SAGR. Rather, becau&longs;e it cannot be denied but that the Air <lb/>impedeth both tho&longs;e and the&longs;e, &longs;ince they both continually grow <lb/>more languid, and at la&longs;t cea&longs;e, it is requi&longs;ite to &longs;ay that tho&longs;e Re­<lb/>tardations are made with the &longs;ame proportion in the one and in the <pb xlink:href="069/01/218.jpg" pagenum="215"/>other Operation. </s> <s>And then, the being to make greater Re&longs;i&longs;tance <lb/>at one time than at another, from what other doth it proceed, but <lb/>only from its being a&longs;&longs;ailed at one time with a greater <emph type="italics"/>Impetus<emph.end type="italics"/> and <lb/>Velocity, and at another time with le&longs;&longs;er? </s> <s>And if this be &longs;o then the <lb/>&longs;ame quantity of the Velocity of the Moveable is at once the Cau&longs;e <lb/>and the Mealure of the quantity of the Re&longs;i&longs;tance. </s> <s>Therefore all <lb/>Motions, whether they be &longs;low or &longs;wift, are retarded and impe­<lb/>ded in the &longs;ame proportion: a Notion in my judgment not con­<lb/>temptible.</s></p><p type="main"> <s>SALV. </s> <s>We may al&longs;o in this &longs;econd ca&longs;e conclude, That the <lb/>Fallacies in the Conclu&longs;ions, which are demon&longs;trated, ab&longs;tracting <lb/>from the extern Accidents, are in our In&longs;truments of very &longs;mall <lb/>con&longs;ideration, in re&longs;pect of the Motions of great Velocities of <lb/>which for the mo&longs;t part we &longs;peak, and of the Di&longs;tances which are <lb/>but very &longs;mall in relation to the Semidiameter and great Circles of <lb/>the Terre&longs;trial Globe.</s></p><p type="main"> <s>SIMP. </s> <s>I would gladly hear the rea&longs;on why you &longs;eque&longs;trate <lb/>the Projects from the <emph type="italics"/>Impetus<emph.end type="italics"/> of the Fire, that is, as I conceive from <lb/>the force of the Powder, from the other Projects made by Slings, <lb/>Bows, or Cro&longs;s-bows, touching their not being in the &longs;ame manner <lb/>&longs;ubject to the Acceleration and Impediment of the Air.</s></p><p type="main"> <s>SALV. </s> <s>I am induced thereto by the exce&longs;&longs;ive, and, as I may &longs;ay, <lb/>Supernatural Fury or Impetuou&longs;ne&longs;s with which tho&longs;e Projects are <lb/>driven out: For indeed I think that the Velocity with which a <emph type="italics"/>B<emph.end type="italics"/>ul­<lb/>let is &longs;hot out of a Musket or Piece of Ordinance may without any <lb/>Hyperbole be called Supernatural. </s> <s>For one of tho&longs;e <emph type="italics"/>B<emph.end type="italics"/>ullets de­<lb/>&longs;cending naturally thorow the Air from &longs;ome immen&longs;e height, its <lb/>Velocity, by rea&longs;on of the Re&longs;i&longs;tance of the Air will not go in­<lb/>crea&longs;ing perpetually: but that which in Cadent <emph type="italics"/>B<emph.end type="italics"/>odies of &longs;mall <lb/>Gravity is &longs;een to happen in no very great ^{*} Space, I mean their <lb/><arrow.to.target n="marg1098"></arrow.to.target><lb/>being reduced in the end to an Equable Motion, &longs;hall al&longs;o happen <lb/>after a De&longs;cent of thou&longs;ands of yards, in a <emph type="italics"/>B<emph.end type="italics"/>all of Iron or Lead: <lb/>and this determinate and ultimate Velocity may be &longs;aid to be the <lb/>greate&longs;t that &longs;uch a <emph type="italics"/>B<emph.end type="italics"/>ody can obtain or acquire thorow the Air: <lb/>which Velocity I account to be much le&longs;&longs;er than that which cometh <lb/>to be impre&longs;&longs;ed on the &longs;ame <emph type="italics"/>B<emph.end type="italics"/>all by the fired Powder. </s> <s>And of this <lb/>a very appo&longs;ite Experiment may adverti&longs;e us. </s> <s>At an height of an <lb/>hundred or more yards let off a Musket charged with a Leaden <lb/><emph type="italics"/>B<emph.end type="italics"/>ullet perpendicularly downwards upon a Pavement of Stone; and <lb/>with the &longs;ame Musket &longs;hoot again&longs;t &longs;uch another Stone at the Di­<lb/>&longs;tance of a yard or two, and then &longs;ee which of the two <emph type="italics"/>B<emph.end type="italics"/>ullets is <lb/>more flatted: for if that coming from on high be le&longs;s ^{*} dented than <lb/><arrow.to.target n="marg1099"></arrow.to.target><lb/>the other, it &longs;hall be a &longs;ign that the Air hath impeded it, and dimi­<lb/>ni&longs;hed the Velocity conferred upon it by the Fire in the beginning <lb/>of the Motion: and that, con&longs;equently, &longs;o great a Velocity the Air <pb xlink:href="069/01/219.jpg" pagenum="216"/>would not &longs;uffer it to gain coming from never &longs;o great an height: <lb/>for in ca&longs;e the Velocity impre&longs;&longs;ed upon it by the Fire &longs;hould not <lb/>exceed that which it might acquire of its &longs;elf de&longs;cending naturally, <lb/>the battery downwards ought rather to be more valid than le&longs;s. <lb/></s> <s>I have not made &longs;uch an Experiment, but incline to think that a <lb/>Musket or Cannon Bullet falling from never &longs;o great an height, <lb/>will not make that percu&longs;&longs;ion which it maketh in a Wall at a Di­<lb/>&longs;tance of a few yards, that is of &longs;o few that the &longs;hort perforation, <lb/>or, if you will, Sci&longs;&longs;ure to be made in the Air &longs;ufficeth not to ob­<lb/>viate the exce&longs;s of the &longs;upernatural impetuo&longs;ity impre&longs;&longs;ed on it by <lb/>the Fire. </s> <s>This exce&longs;&longs;ive <emph type="italics"/>Impetus<emph.end type="italics"/> of &longs;uch like forced &longs;hots may <lb/>cau&longs;e &longs;ome deformity in the Line of the Projection; making <lb/>the beginning of the Parabola le&longs;s inclined or curved than the end. <lb/><emph type="italics"/>B<emph.end type="italics"/>ut this can be but of little or no prejudice to our Author in <lb/>practical Operations: among&longs;t the which the principal is the com­<lb/>po&longs;ition of a Table for the Ranges, or Flights, which containeth <lb/>the di&longs;tances of the Falls of <emph type="italics"/>B<emph.end type="italics"/>alls &longs;hot according to all Elevations. <lb/></s> <s>And becau&longs;e the&longs;e kinds of Projections are made with Mortar­<lb/>Pieces, and with no great charge; in the&longs;e the <emph type="italics"/>Impetus<emph.end type="italics"/> not being <lb/>&longs;upernatural, the Ranges de&longs;cribe their Lines very exactly.</s></p><p type="margin"> <s><margin.target id="marg1098"></margin.target>* Or Way.</s></p><p type="margin"> <s><margin.target id="marg1099"></margin.target>* Or battered.</s></p><p type="main"> <s><emph type="italics"/>B<emph.end type="italics"/>ut for the pre&longs;ent let us proceed forwards in the Treati&longs;e, <lb/>where the Author de&longs;ireth to lead us to the Contemplation and <lb/>Inve&longs;tigation of the <emph type="italics"/>Impetus<emph.end type="italics"/> of the Moveable whil&longs;t it moveth <lb/>with a Motion compounded of two. </s> <s>And fir&longs;t of that compoun­<lb/>ded of two Equable Motions; the one Horizontal, and the other <lb/>Perpendicular.</s></p><p type="head"> <s>THEOR. II. PROP. II.</s></p><p type="main"> <s>If any Moveable be moved with a twofold Equa­<lb/>ble Motion, that is, Horizontal and Perpen­<lb/>dicular, the <emph type="italics"/>Impetus<emph.end type="italics"/> or Moment of the Lation <lb/>compounded of both the Motions &longs;hall be <emph type="italics"/>po­<lb/>tentia<emph.end type="italics"/> equal to both the Moments of the fir&longs;t <lb/>Motions.</s></p><p type="main"> <s><emph type="italics"/>For let any Moveable be moved Equably with a double Lation, <lb/>and let the Mutations of the Perpendicular an&longs;wer to the Space <lb/>A B, and let B C an&longs;wer to the Horizontal Lation pa&longs;&longs;ed in <lb/>the &longs;ame Time. </s> <s>Fora&longs;much therefore as the Spa-<emph.end type="italics"/><lb/><figure id="id.069.01.219.1.jpg" xlink:href="069/01/219/1.jpg"/><lb/><emph type="italics"/>ces A B, and B C are pa&longs;&longs;ed by the Equable Mo­<lb/>tion in the &longs;ame Time, their Moments &longs;hall be to <lb/>cach other as the &longs;aid A B and B C. </s> <s>But the <lb/>Moveable which is moved according to the&longs;e two Mutations &longs;hall de-<emph.end type="italics"/><pb xlink:href="069/01/220.jpg" pagenum="217"/><emph type="italics"/>&longs;cribe the Diagonal A C, and its Moment &longs;hall be as A C. </s> <s>But A C is<emph.end type="italics"/><lb/>potentia <emph type="italics"/>equal to the &longs;aid A B and B C: therefore the Moment com­<lb/>pounded of both the Moments A B and B C, is<emph.end type="italics"/> potentia <emph type="italics"/>equal to them <lb/>both taken together: Which was to be demon&longs;trated.<emph.end type="italics"/></s></p><p type="main"> <s>SIMP. </s> <s>It is nece&longs;&longs;ary that you ea&longs;e me of one Scruple that <lb/>cometh into my mind, it &longs;eemeth to me that this which is now con­<lb/>cluded oppugneth another Propo&longs;ition of the former Tractate: in <lb/>which it is affirmed, That the <emph type="italics"/>Impetus<emph.end type="italics"/> of the Moveable coming <lb/>from A into B is equal to that coming from A into C; and now it is <lb/>concluded, that the <emph type="italics"/>Impetus<emph.end type="italics"/> in C is greater than that in B.</s></p><p type="main"> <s>SALV. </s> <s>The Propo&longs;itions, <emph type="italics"/>Simplicius,<emph.end type="italics"/> are both true, but very <lb/>different from one another. </s> <s>Here the Author &longs;peaks of one &longs;ole <lb/>Moveable moved with one &longs;ole Motion, but compounded of two, <lb/>both Equable; and there he &longs;peaks of two Moveables moved <lb/>with Motions Naturally Accelerated, one along the Perpendicular <lb/>A B, and the other along the Inclined Plane A C: and moreover, <lb/>the Times there are not &longs;uppo&longs;ed equal, but the Time along <lb/>the Inclined Plane A C is greater than the Time along the Perpen­<lb/>dicular A B: but in the Motion &longs;poken of at pre&longs;ent, the Motions <lb/>along A B, B C and A C are under&longs;tood to be Equable, and made <lb/>in the &longs;ame Time.</s></p><p type="main"> <s>SIMP. </s> <s>Excu&longs;e me, and go on, for I am &longs;atisfied.</s></p><p type="main"> <s>SALV. </s> <s>The Author proceeds to &longs;hew us that which hapneth <lb/>concerning the <emph type="italics"/>Impetus<emph.end type="italics"/> of a Moveable moved in like manner with <lb/>one Motion compounded of two, that is to &longs;ay, the one Horizon­<lb/>tal and Equable, and the other Perpendicular but Naturally-Acce­<lb/>lerate, of which in fine the Motion of the Project is compounded, <lb/>and by which the Parabolick Line is de&longs;cribed; in each point of <lb/>which the Author endeavours to determine what the <emph type="italics"/>Impetus<emph.end type="italics"/> of the <lb/>Project is; for under&longs;tanding of which he &longs;heweth us the manner, <lb/>or, if you will, Method of regulating and mea&longs;uring that &longs;ame <emph type="italics"/>Im­<lb/>petus<emph.end type="italics"/> upon the &longs;aid Line, along which the Motion of the Grave <lb/>Moveable de&longs;cending with a Natural-Accelerate Motion departing <lb/>from Re&longs;t is made, &longs;aying:</s></p><p type="head"> <s>THEOR. III. PROP. III.</s></p><p type="main"> <s><emph type="italics"/>Let a Motion be made along the Line A B out of Re&longs;t in A, and <lb/>take in &longs;ome point C; and &longs;uppo&longs;e the &longs;aid A C to be the Time or <lb/>Mea&longs;ure of the Time of the &longs;aid Fall along the Space A C, as al&longs;o <lb/>the Mea&longs;ure of the<emph.end type="italics"/> Impetus <emph type="italics"/>or Moment in the Point C acquired by <lb/>the De&longs;cent along A C. </s> <s>Now let there be taken in the &longs;aid Line <lb/>A B any other Point, as &longs;uppo&longs;e B, in which we are to determine of the<emph.end type="italics"/><lb/>Impetus <emph type="italics"/>acquired by the Moveable along the Fall A B, in proportion to<emph.end type="italics"/><pb xlink:href="069/01/221.jpg" pagenum="218"/><emph type="italics"/>the<emph.end type="italics"/> Impetus, <emph type="italics"/>which it obtaineth in C, who&longs;e Mea&longs;ure is &longs;uppo&longs;ed to be <lb/>A C, Let A S be a Mean-proportional betwixt B A and A C. </s> <s>We will <lb/>demon&longs;trate that the<emph.end type="italics"/> Impetus <emph type="italics"/>in B is to the<emph.end type="italics"/> Impetus <emph type="italics"/>in C, as S A is to <lb/>A C. </s> <s>Let the Horizontal Line C D be double to the &longs;aid A C; and B E <lb/>double to B A. </s> <s>It appeareth by what hath been demon&longs;trated, That the <lb/>Cadent along A C being turned along the Horizon C D, and according <lb/>to the<emph.end type="italics"/> Impetus <emph type="italics"/>acquired in C, with an Equable Motion, &longs;hall pa&longs;s the <lb/>Space C D in a Time equal to that <lb/>in which the &longs;aid A C is pa&longs;&longs;ed<emph.end type="italics"/><lb/><figure id="id.069.01.221.1.jpg" xlink:href="069/01/221/1.jpg"/><lb/><emph type="italics"/>with an Accelerate Motion; and <lb/>likewi&longs;e that B E is pa&longs;&longs;ed in the <lb/>&longs;ame time as A B: But the Time of <lb/>the De&longs;cent along A B is A S: There­<lb/>fore the Horizontal Line B E is <lb/>pa&longs;&longs;ed in A S. </s> <s>As the Time S A is <lb/>to the Time A C, &longs;o let E B be to <lb/>B L. </s> <s>And becau&longs;e the Motion by <lb/>B E is Equable, the Space B L &longs;hall be pa&longs;&longs;ed in the Time A C ac­<lb/>cording to the Moment of Celerity in B: But in the &longs;ame Time A C <lb/>the Space C D is pa&longs;&longs;ed, according to the Moment of Velocity in C: <lb/>the Moments of Velocity therefore are to one another as the Spaces <lb/>which according to the &longs;ame Moments are pa&longs;&longs;ed in the &longs;ame Time: <lb/>Therefore the Moment of Velocity in C is to the Moment of Celerity in <lb/>B, as D C is to B L. </s> <s>And becau&longs;e as D C is to B E, &longs;o are their halfs, <lb/>to wit, C A to A B: but as E B is to B L, &longs;o is B A to A S: Therefore,<emph.end type="italics"/><lb/>exæquali, <emph type="italics"/>as D C is to B L, &longs;o is C A to A S: that is, as the Moment <lb/>of Velocity in C is to the Moment of Velocity in B, &longs;o is C A to A S; that <lb/>is, the Time along C A to the Time along A B. </s> <s>I he manner of Mea&longs;u­<lb/>ring the<emph.end type="italics"/> Impetus, <emph type="italics"/>or the Moment of Velocity upon a Line along which it <lb/>makes a Motion of De&longs;cent is therefore manife&longs;t; which<emph.end type="italics"/> Impetus <lb/><emph type="italics"/>is indeed &longs;uppo&longs;ed to encrea&longs;e according to the Proportion of the <lb/>Time.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>But this, before we proceed any farther, is to be premoni&longs;hed, that in <lb/>regard we are to &longs;peak for the future of the Motion compounded of the <lb/>Equable Horizontal, and of the Naturally Accelerate downwards, (for <lb/>from this Mixtion re&longs;ults, and by it is de&longs;igned the Line of the Project, <lb/>that is a Parabola;) it is nece&longs;&longs;ary that we define &longs;ome common mea&longs;ure <lb/>according to which we may mea&longs;ure the Velocity,<emph.end type="italics"/> Impetus, <emph type="italics"/>or Moment <lb/>of both the Motions. </s> <s>And &longs;eeing that of the Equable Motion the de­<lb/>grees of Velocity are innumerable, of which you may not take any <lb/>promi&longs;cuou&longs;ly, but one certain one which may be be compared and con­<lb/>joyned with the Degree of Velocity naturally Accelerate. </s> <s>I can think of <lb/>no more ea&longs;ie way for the electing and determining of that, than by a&longs;­<lb/>&longs;uming another of the &longs;ame kind. </s> <s>And that I may the better expre&longs;s <lb/>my meaning; Let A C be Perpendicular to the Horizon C B; and A C<emph.end type="italics"/><pb xlink:href="069/01/222.jpg" pagenum="219"/><emph type="italics"/>to be the Altitude, and C B the Amplitude of the Semiparabola A B; <lb/>which is de&longs;cribed by the Compo&longs;ition of two Lations; of which one is <lb/>that of the Moveable de&longs;cending along A C with a Motion Naturally <lb/>Acceler ate<emph.end type="italics"/> ex quiete <emph type="italics"/>in A; the other is the Equable Tran&longs;ver&longs;al Moti­<lb/>on according to the Horizontal Line A D. The<emph.end type="italics"/> Impetus <emph type="italics"/>acquired in C <lb/>along the De&longs;cent A C is determined by the quantity of the &longs;aid height <lb/>A C; for the<emph.end type="italics"/> Impetus <emph type="italics"/>of a Moveable<emph.end type="italics"/><lb/><figure id="id.069.01.222.1.jpg" xlink:href="069/01/222/1.jpg"/><lb/><emph type="italics"/>falling from the &longs;ame height is alwaies <lb/>one and the &longs;ame: but in the Horizontal <lb/>Line one may a&longs;&longs;ign not one, but innume­<lb/>rable Degrees of Velocities of Equable <lb/>Motions: out of which multitude that I <lb/>may &longs;ingle out, and as it were point with <lb/>the finger to that which I make choice of, <lb/>I extend or prolong the Altitude C A<emph.end type="italics"/> in <lb/>&longs;ublimi, <emph type="italics"/>in which, as was done before, I <lb/>will pitch upon A E; from which if I <lb/>conceive in my mind a Moveable to fall<emph.end type="italics"/><lb/>ex quiete <emph type="italics"/>in E, it appeareth that its<emph.end type="italics"/> Im­<lb/>petus <emph type="italics"/>acquired in the Time A, is one with which I conceive the &longs;ame <lb/>Moveable being turned along A D to be moved; and its degree of <lb/>Vclocity to be that, which in the Time of the De&longs;cent along E A pa&longs;&longs;eth <lb/>a Space in the Horizon double to the &longs;aid E A. </s> <s>This Præmonition I <lb/>judged nece&longs;&longs;ary.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>It is moreover to be advertized that the Amplitude of the Semi­<lb/>parabola A B &longs;hall be called by me the Horizontal Line<emph.end type="italics"/> [or Plane] <lb/><emph type="italics"/>C B.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>The Altitude, to with A C, the Axis of the &longs;aid Parabola.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>And the Line E A, by who&longs;e De&longs;cent the Horizontal<emph.end type="italics"/> Impetus <emph type="italics"/>is de­<lb/>termined, I call the Sublimity, or height.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>The&longs;e things being declared and defined, I proceed to Demon&longs;tra­<lb/>tion.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. Stay, I pray you, for here me thinks it is convenient to <lb/>adorn this Opinion of our Author with the conformity of it to <lb/>the Conceit of <emph type="italics"/>Plato<emph.end type="italics"/> about the determining the different Veloci­<lb/>ties of the Equable Motions of the Revolutions of the Cœle&longs;tial <lb/>Bodies; who, having perhaps had a conjecture that no Moveable <lb/>could pa&longs;&longs;e from Re&longs;t into any determinate degree of Velocity in <lb/>which it ought afterwards to be perpetuated, unle&longs;s by pa&longs;&longs;ing <lb/>thorow all the other le&longs;&longs;er degrees of Velocity, or, if you will, <lb/>greater degrees of Tardity, which interpo&longs;e between the a&longs;&longs;igned <lb/>degree, and the highe&longs;t degree of Tardity, that is of Re&longs;t, &longs;aid that <lb/>God after he had created the Moveable Cœle&longs;tial <emph type="italics"/>B<emph.end type="italics"/>odies that he <lb/>might a&longs;&longs;ign them tho&longs;e Velocities wherewith they were afterwards <pb xlink:href="069/01/223.jpg" pagenum="220"/>to be perpetually moved with an Equable Circular Motion, made <lb/>them, they departing from Re&longs;t, to move along determinate Spaces <lb/>with that Natural Motion in a Right Line, according to which we <lb/>&longs;en&longs;ibly &longs;ee our Moveables to move from the &longs;tate of Re&longs;t &longs;ucce&longs;­<lb/>&longs;ively Accelerating. </s> <s>And he addeth, that having made them to <lb/>acquire that degree in which it plea&longs;ed him that they &longs;hould after­<lb/>wards be perpetually con&longs;erved, he converted their Right or direct <lb/>Motion into Circular; which only is apt to con&longs;erve it &longs;elf Equa­<lb/>ble, alwaies revolving without receding from, or approaching to <lb/>any prefixed term by them de&longs;ired. </s> <s>The Conceit is truly worthy <lb/>of <emph type="italics"/>Plato<emph.end type="italics"/>; and is the more to be e&longs;teemed in that the grounds there­<lb/>of pa&longs;&longs;ed over in &longs;ilence by him, and di&longs;covered by our Author by <lb/>taking off the Mask or Poetick Repre&longs;entation, do &longs;hew it to be <lb/>in its native a&longs;pect a true Hi&longs;tory. </s> <s>And I think it very credible that <lb/>we having by the Doctrine of A&longs;tronomy &longs;ufficiently competent <lb/>Knowledge of the Magnitudes of the Orbes of the Planets, and of <lb/>their Di&longs;tances from the Center about which they move, as al&longs;o <lb/>of their Velocities, our Author (to whom <emph type="italics"/>Plato's<emph.end type="italics"/> Conjecture was <lb/>not unknown) may &longs;ometime for his curio&longs;ity have had &longs;ome <lb/>thought of attempting to inve&longs;tigate whether one might a&longs;&longs;ign a <lb/>determinate Sublimity from which the <emph type="italics"/>B<emph.end type="italics"/>odies of the Planets depar­<lb/>ting, as from a &longs;tate of Re&longs;t, and moved for certain Spaces with a <lb/>Right and Naturally Accelerate Motion, afterwards converting <lb/>the Acquired Velocity into Equable Motions, they might be found <lb/>to corre&longs;pond with the greatne&longs;s of their Orbes, and with the Times <lb/>of their Revolutions.</s></p><p type="main"> <s><emph type="italics"/>S<emph.end type="italics"/>ALV. </s> <s>I think I do remember that he hath heretofore told me, <lb/>that he had once made the Computation, and al&longs;o that he found <lb/>it exactly to an&longs;wer the Ob&longs;ervations; but that he had no mind to <lb/>&longs;peak of them, doubting le&longs;t the two many Novelties by him di&longs;­<lb/>covered, which had provoked the di&longs;plea&longs;ure of many again&longs;t him, <lb/>might blow up new &longs;parks. <emph type="italics"/>B<emph.end type="italics"/>ut if any one &longs;hall have the like de­<lb/>&longs;ire he may of him&longs;elf by the Doctrine of the pre&longs;ent Tract give <lb/>him&longs;elf content. <emph type="italics"/>B<emph.end type="italics"/>ut let us pur&longs;ue our bu&longs;ine&longs;s, which is to <lb/>&longs;hew;</s></p><p type="head"> <s>PROBL. I. PROP. IV.</s></p><p type="main"> <s>How in a Parabola given, de&longs;cribed by the Pro­<lb/>ject, the <emph type="italics"/>Impetus<emph.end type="italics"/> of each &longs;everal point may be <lb/>determined.</s></p><p type="main"> <s><emph type="italics"/>Let the Semiparabola be B E C, who&longs;e Amplitude is C D and Al­<lb/>titude D B, with which continued out on high the Tangent of the <lb/>Parabola C A meeteth in A; and along the<emph.end type="italics"/> Vertex <emph type="italics"/>B let B I be<emph.end type="italics"/><pb xlink:href="069/01/224.jpg" pagenum="221"/><emph type="italics"/>an Horizontal Line, and parallel to C D. </s> <s>And if the Amplitude C D <lb/>be equal to the whole Altitude D A, B I &longs;hall be equal to B A and B D. <lb/></s> <s>And if the Time of the Fall along A B, and the Moment of Velocity <lb/>acquired in B along the De&longs;cent A B<emph.end type="italics"/> ex quiete <emph type="italics"/>in A be &longs;uppo&longs;ed to be <lb/>mea&longs;ured by the &longs;aid A B, then D C (that is twice B I) &longs;hall be the <lb/>Space which &longs;hall be pa&longs;&longs;ed by the<emph.end type="italics"/> Impetus A <emph type="italics"/>B turned along the Hori­<lb/>zontal Line in the &longs;ame Time: But in the &longs;ame Time falling along B D <lb/>out of Re&longs;t in B, it &longs;hall pa&longs;s the Altitude B D: Therefore the Movea­<lb/>ble falling out of Re&longs;t in A along A B, <lb/>being converted with the<emph.end type="italics"/> Impetus <emph type="italics"/>A B<emph.end type="italics"/><lb/><figure id="id.069.01.224.1.jpg" xlink:href="069/01/224/1.jpg"/><lb/><emph type="italics"/>along the Horizontal Parallel &longs;hall <lb/>pa&longs;s a Space equal to D C. </s> <s>And the <lb/>Fall along B D &longs;upervening, it pa&longs;&longs;eth <lb/>the Altitude B D, and de&longs;cribes the <lb/>Parabola B C; who&longs;e<emph.end type="italics"/> Impetus <emph type="italics"/>in the <lb/>Term C is compounded of the Equable <lb/>Tran&longs;ver&longs;al who&longs;e Moment is as A B, <lb/>and of another Moment acquired in the <lb/>Fall B D in the Term D or C; which <lb/>Moments are Equal. </s> <s>If therefore we <lb/>&longs;uppo&longs;e A B to be the Mea&longs;ure of one of them, as &longs;uppo&longs;e of the Equa­<lb/>ble Tran&longs;ver&longs;al; and B I, which is equal to B D, to be the Mea&longs;ure of <lb/>the<emph.end type="italics"/> Impetus <emph type="italics"/>acquired in D or C; then the Subten&longs;e I A &longs;hall be the <lb/>quantity of the Moment compound of them both: Therefore it &longs;hall be <lb/>the quantity or Mea&longs;ure of the whole Moment which the Project de&longs;cend­<lb/>ing along the Parabola B C &longs;hall acquire of<emph.end type="italics"/> Impetus <emph type="italics"/>in C. </s> <s>This pre­<lb/>mi&longs;ed, take in the Parabola any point E, in which we are to determine <lb/>of the<emph.end type="italics"/> Impetus <emph type="italics"/>of the Project. </s> <s>Draw the Horizontal Parallel E F, <lb/>and let B G be a Mean-proportional between B D and B F. </s> <s>And fora&longs;­<lb/>much as A B or B D is &longs;uppo&longs;ed to be the Mea&longs;ure of the Time, and of <lb/>the Moment of the Velocity in the Fall B D<emph.end type="italics"/> ex quiete <emph type="italics"/>in B: B G &longs;hall <lb/>be the Time, or the Mea&longs;ure of the Time, and of the<emph.end type="italics"/> Impetus <emph type="italics"/>in F, coming <lb/>out of B. </s> <s>If therefore B O be &longs;uppo&longs;ed equal to B G, the Diagonal <lb/>drawn from A to O &longs;hall be the quantity of the<emph.end type="italics"/> Impetus <emph type="italics"/>in E; for <lb/>A B hath been &longs;uppo&longs;ed the determinator of the Time, and of the<emph.end type="italics"/> Impe­<lb/>tus <emph type="italics"/>in B, which turned along the Horizontal Parallel doth alwaies <lb/>continue the &longs;ame: And B O determineth the<emph.end type="italics"/> Impetus <emph type="italics"/>in F or in E <lb/>along the De&longs;cent<emph.end type="italics"/> ex quiete <emph type="italics"/>in B in the Altitude B F: But the&longs;e two <lb/>A B and B O are<emph.end type="italics"/> potentia <emph type="italics"/>equal to the Power A O. </s> <s>Therefore that is <lb/>manife&longs;t which was &longs;ought.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>The Contemplation of the Compo&longs;ition of the&longs;e diffe­<lb/>rent <emph type="italics"/>Impetus's,<emph.end type="italics"/> and of the quantity of that <emph type="italics"/>Impetus<emph.end type="italics"/> which re&longs;ults <lb/>from this mixture, is &longs;o new to me, that it leaveth my mind in no <lb/>&longs;mall confu&longs;ion. </s> <s>I do not &longs;peak of the mixtion of two Motions <pb xlink:href="069/01/225.jpg" pagenum="222"/>Equable, though unequal to one another, made the one along the <lb/>Horizontal Line, and the other along the Perpendicular, for I very <lb/>well comprehend that there is made a Motion of the&longs;e two <emph type="italics"/>poten­<lb/>tia<emph.end type="italics"/> equal to both the Compounding Motions, but my confu&longs;ion <lb/>ari&longs;eth upon the mixing of the Equable-Horizontal and Perpendi­<lb/>cular-Naturally-Accelerate Motion. </s> <s>Therefore I could wi&longs;h we <lb/>might toge ther a little better con&longs;ider this bu&longs;ine&longs;s.</s></p><p type="main"> <s>SIMP. </s> <s>And I &longs;tand the more in need thereof in that I am not <lb/>yet &longs;o well &longs;atisfied in Mind as I &longs;hould be, in the Propo&longs;itions that <lb/>are the fir&longs;t foundations of the others that follow upon them. </s> <s>I <lb/>will add, that al&longs;o in the Mixtion of the two Motions Equable <lb/>Horizontal, and Perpendicular, I would better under&longs;tand that <lb/><emph type="italics"/>Potentia<emph.end type="italics"/> of their Compound. </s> <s>Now, <emph type="italics"/>Salviatus,<emph.end type="italics"/> you &longs;ee what we <lb/>want and de&longs;ire.</s></p><p type="main"> <s>SALV. </s> <s>Your de&longs;ire is very rea&longs;onable: and I will e&longs;&longs;ay whe­<lb/>ther my having had a longer time to think thereon may facilitate <lb/>your &longs;atisfaction. </s> <s>But you mu&longs;t bear with and excu&longs;e me if in di&longs;­<lb/>cour&longs;ing I &longs;hall repeat a great part of the things hitherto delivered <lb/>by our Author.</s></p><p type="main"> <s>It is not po&longs;&longs;ible for us to &longs;peak po&longs;itively touching Motions and <lb/>their Velocities or <emph type="italics"/>Impetus's,<emph.end type="italics"/> be they Equable, or be they Naturally <lb/>Accelerate, unle&longs;s we fir&longs;t agree upon the Mea&longs;ure that we are to <lb/>u&longs;e in the commen&longs;uration of tho&longs;e Velocities, as al&longs;o of the Time. <lb/></s> <s>As to the Mea&longs;ure of the Time, we have already that which is <lb/>commonly received by all of Hours, Prime-Minutes, and Se­<lb/>conds, <emph type="italics"/>&c.<emph.end type="italics"/> and as for the mea&longs;uring of Time we have that com­<lb/>mon Mea&longs;ure received by all, &longs;o it is requi&longs;ite to a&longs;&longs;ign another <lb/>Mea&longs;ure for the Velocities that is commonly under&longs;tood and re­<lb/>ceived by every one; that is, which every where is the &longs;ame. </s> <s>The <lb/>Author, as hath been declared, adjudged the Velocity of Naturally <lb/>de&longs;cending Grave-Bodies to be fit for this purpo&longs;e; the encrea&longs;ing <lb/>Velocities of which are the &longs;ame in all parts of the World. </s> <s>So that <lb/>that &longs;ame degree of Velocity which (for example) a Ball of Lead of <lb/>a pound acquireth in having, departing from Re&longs;t, de&longs;cended Per­<lb/>pendicularly as much as the height of a Pike, is alwaies, and in all <lb/>places the &longs;ame, and therefore mo&longs;t commodious for explicating <lb/>the quantity of the <emph type="italics"/>Impetus<emph.end type="italics"/> that is derived from the Natural De­<lb/>&longs;cent. </s> <s>Now it remains to find a way to determine likewi&longs;e the <lb/>Quantity of the <emph type="italics"/>Impetus<emph.end type="italics"/> in an Equable Motion in &longs;uch a manner, <lb/>that all tho&longs;e which di&longs;cour&longs;e about it may form the &longs;ame conceit <lb/>of its greatne&longs;s and Velocity; &longs;o that one may not imagine it more <lb/>&longs;wift, and another le&longs;s; whereupon afterwards in conjoyning and <lb/>mingling this Equable Motion imagined by them with the e&longs;tabli­<lb/>&longs;hed Accelerate Motion &longs;everal men may form &longs;everal Conceits of <lb/>&longs;everal greatne&longs;&longs;es of <emph type="italics"/>Impetus's.<emph.end type="italics"/> To determine and repre&longs;ent this <pb xlink:href="069/01/226.jpg" pagenum="223"/><emph type="italics"/>Impetus,<emph.end type="italics"/> and particular Velocity our Author hath not found any <lb/>way more commodious, than the making u&longs;e of the <emph type="italics"/>Impetus<emph.end type="italics"/> which <lb/>the Moveable from time to time acquires in the Naturally-Accele­<lb/>rate Motion, any acquired Moment of which being reduced into <lb/>an Equable Motion retaineth its Velocity preci&longs;ely limited, and <lb/>&longs;uch, that in &longs;uch another Time as that wherein it did De&longs;cend, it <lb/>pa&longs;&longs;eth double the Space of the Height from whence it fell. </s> <s>But <lb/>becau&longs;e this is the principal point in the bu&longs;ine&longs;s that we are upon, <lb/>it is good to make it to be perfectly under&longs;tood by &longs;ome particular <lb/>Example. </s> <s>Rea&longs;&longs;uming therefore the Velocity and <emph type="italics"/>Impetus<emph.end type="italics"/> acqui­<lb/>red by the Cadent Moveable, as we &longs;aid before, from the height <lb/>of a Pike, of which Velocity we will make u&longs;e for a Mea&longs;ure of <lb/>other Velocities and <emph type="italics"/>Impetu&longs;&longs;es<emph.end type="italics"/> upon other occa&longs;ions, and &longs;uppo­<lb/>&longs;ing, for example, that the Time of that Fall be four &longs;econd Mi­<lb/>nutes of an hour, to find by this &longs;ame Mea&longs;ure how great the <emph type="italics"/>Im­<lb/>petus<emph.end type="italics"/> of the Moveable would be falling from any other height <lb/>greater, or le&longs;&longs;er, we ought not from the proportion that this other <lb/>height hath to the height of a Pike to argue and conclude the quan­<lb/>tity of the <emph type="italics"/>Impetus<emph.end type="italics"/> acquired in this &longs;econd height, thinking, for <lb/>example, that the Moveable falling from quadruple the height <lb/>hath acquired quadruple Velocity, for that it is fal&longs;e: for that the <lb/>Velocity of the Naturally-Accelerate Motion doth not increa&longs;e or <lb/>decrea&longs;e according to the proportion of the Spaces, but according <lb/>to that of the Times, than which that of the Spaces is greater in a <lb/>duplicate proportion, as was heretofore demon&longs;trated. </s> <s>Therefore <lb/>when in a Right Line we have a&longs;&longs;igned a part for the Mea&longs;ure of <lb/>the Velocity, and al&longs;o of the Time, and of the Space in that Time <lb/>pa&longs;&longs;ed (for that for brevity &longs;ake all the&longs;e three Magnitudes are <lb/>often repre&longs;ented by one &longs;ole Line,) to find the quantity of the <lb/>Time, and the degree of Velocity that the &longs;ame Moveable would <lb/>have acquired in another Di&longs;tance we &longs;hall obtain the &longs;ame, not <lb/>immediataly by this &longs;econd Di&longs;tance, but by the Line which &longs;hall <lb/>be a Mean-proportional betwixt the two Di&longs;tances. </s> <s>But I will <lb/>better declare my &longs;elf by an Example. </s> <s>In the Line A C Perpendi­<lb/>cular to the Horizon let the part A B be under&longs;tood to <lb/>be a Space pa&longs;&longs;ed by a Moveable naturally de&longs;cending <lb/><figure id="id.069.01.226.1.jpg" xlink:href="069/01/226/1.jpg"/><lb/>with an Accelerate Motion: the Time of which pa&longs;­<lb/>&longs;age, in regard I may repre&longs;ent it by any Line, I will, for <lb/>brevity, imagine it to be as much as the &longs;ame Line A B <lb/>and likewi&longs;e for a Mea&longs;ure of the <emph type="italics"/>Impetus<emph.end type="italics"/> and Velocity <lb/>acquired by that Motion, I again take the &longs;ame Line <lb/>A B; &longs;o that of all the Spaces that are in the progre&longs;s of <lb/>the Di&longs;cour&longs;e to be con&longs;idered the part A B may be the <lb/>Mea&longs;ure. </s> <s>Having all our plea&longs;ure e&longs;tabli&longs;hed under one <lb/>&longs;ole Magnitude A B the&longs;e three Mea&longs;ures of different kinds of <pb xlink:href="069/01/227.jpg" pagenum="224"/>Quantities, that is to &longs;ay, of Spaces, of Times, and of <emph type="italics"/>Impetus's,<emph.end type="italics"/> let <lb/>it be required to determine in the a&longs;&longs;igned Space, and at the height <lb/>A C, how much the Time of the Fall of the Moveable from A to <lb/>C is to be, and what the <emph type="italics"/>Impetus<emph.end type="italics"/> is that &longs;hall be found to have been <lb/>acquired in the &longs;aid Term C, in relation to the Time and to the <lb/><emph type="italics"/>Impetus<emph.end type="italics"/> mea&longs;ured by A B. </s> <s>Both the&longs;e que&longs;tions &longs;hall be re&longs;olved <lb/>taking A D the Mean-proportional betwixt the two Lines A C <lb/>and A B; affirming the Time of the Fall along the whole Space <lb/>A C to be as the Time A D is in relation to A B, a&longs;&longs;igned in the <lb/>beginning for the Quantity of the Time in the Fall A B. </s> <s>And like­<lb/>wi&longs;e we will &longs;ay that the <emph type="italics"/>Impetus,<emph.end type="italics"/> or degree of Velocity that the <lb/>Cadent Moveable &longs;hall obtain in the Term C, in relation to the <lb/><emph type="italics"/>Impetus<emph.end type="italics"/> that it had in B, is as the &longs;ame Line A D is in relation to <lb/>A B, being that the Velocity encrea&longs;eth with the &longs;ame proportion <lb/>as the Time doth: Which Conclu&longs;ion although it was a&longs;&longs;umed as <lb/>a <emph type="italics"/>Po&longs;tulatum,<emph.end type="italics"/> yet the Author was plea&longs;ed to explain the Applicati­<lb/>on thereof above in the third Propo&longs;ition.</s></p><p type="main"> <s>This point being well under&longs;tood and proved, we come to the <lb/>Con&longs;ideration of the <emph type="italics"/>Impetus<emph.end type="italics"/> derived from two compound Moti­<lb/>ons: whereof let one be compounded of the Horizontal and alwaies <lb/>Equable, and of the Perpendicular unto the Horizon, and it al&longs;o <lb/>Equable: but let the other be compounded of the Horizontal like­<lb/>wi&longs;e alwaies Equable, and of the Perpendicular Naturally-Accele­<lb/>rate. </s> <s>If both &longs;hall be Equable, it hath been &longs;een already that the <lb/><emph type="italics"/>Impetus<emph.end type="italics"/> emerging from the compo&longs;ition of both is <emph type="italics"/>potentia<emph.end type="italics"/> equal to <lb/>both, as for more plainne&longs;s we will thus Exemplifie. </s> <s>Let the Move­<lb/>able de&longs;cending along the Perpendicular A B be &longs;uppo&longs;ed to have, <lb/>for example, three degrees of Equable <emph type="italics"/>Impetus,<emph.end type="italics"/> but being tran&longs;­<lb/>ported along A B towards C, let the &longs;aid Velocity and <emph type="italics"/>Impetus<emph.end type="italics"/> be <lb/>&longs;uppo&longs;ed four degrees, &longs;o that in the &longs;ame Time that falling it would <lb/>pa&longs;s along the Perpendicular, <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> three yards, <lb/><figure id="id.069.01.227.1.jpg" xlink:href="069/01/227/1.jpg"/><lb/>it would in the Horizontal pa&longs;s four, but in <lb/>that compounded of both the Velocities it <lb/>cometh in the &longs;ame Timefrom the point A un­<lb/>to the Term C, de&longs;cending all the way along the Diagonal Line <lb/>A C, which is not &longs;even yards long, as that &longs;hould be which is com­<lb/>pounded of the two Lines A B, 3, and B C, 4, but is 5; which 5 is <lb/><emph type="italics"/>potentia<emph.end type="italics"/> equal to the two others, 3 and 4: For having found the <lb/>Squares of 3 and 4, which are 9 and 16, and joyning the&longs;e together, <lb/>they make 25 for the Square of A C, which is equal to the two <lb/>Squares of A B and B C: whereupon A C &longs;hall be as much as is the <lb/>Side, or, if you will, Root of the Square 25, which is 5. For a con&longs;tant <lb/>and certain Rule therefore, when it is required to a&longs;&longs;ign the <lb/>Quantity of the <emph type="italics"/>Impetus<emph.end type="italics"/> re&longs;ulting from two <emph type="italics"/>Impetus's<emph.end type="italics"/> given, the <lb/>one Horizontal, and the other Perpendicular, and both Equable, <pb xlink:href="069/01/228.jpg" pagenum="225"/>they are each of them to be &longs;quared, and their Squares being put <lb/>together the Root of the Aggregate is to be extracted, which &longs;hall <lb/>give us the quantity of the <emph type="italics"/>Impetus<emph.end type="italics"/> compounded of them both. <lb/></s> <s>And thus in the foregoing example, that Moveable that by vertue <lb/>of the Perpendicular Motion would have percu&longs;&longs;ed upon the Hori­<lb/>zon with three degrees of Force, and with only the Horizontal Mo­<lb/>tion would have percu&longs;&longs;ed in C with four degrees, percu&longs;&longs;ing with <lb/>both the <emph type="italics"/>Impetus's<emph.end type="italics"/> conjoyned, the blow &longs;hall be like to that of the <lb/>Percutient moved with five degrees of Velocity and Force. </s> <s>And <lb/>this &longs;ame Percu&longs;&longs;ion would be of the &longs;ame Impetuo&longs;ity in all the <lb/>points of the Diagonal A C, for that the compounded <emph type="italics"/>Impetus's<emph.end type="italics"/><lb/>are alwaies the &longs;ame, never encrea&longs;ing or dimini&longs;hing.</s></p><p type="main"> <s>Let us now &longs;ee what befalls in compounding the Equable Hori­<lb/>zontal Motion with another Perpendicular to the Horizon which <lb/>beginning from Re&longs;t goeth Naturally Accelerating. </s> <s>It is already <lb/>manife&longs;t, that the Diagonal, which is the Line of the Motion com­<lb/>pounded of the&longs;e two, is not a Right Line, but Semiparabolical, <lb/>as hath been demon&longs;trated; ^{*} in which the <emph type="italics"/>Impetus<emph.end type="italics"/> doth go con­<lb/><arrow.to.target n="marg1100"></arrow.to.target><lb/>tinually encrea&longs;ing by means of the continual encrea&longs;e of the Ve­<lb/>locity of the Perpendicular Motion: Wherefore, to determine what <lb/>the <emph type="italics"/>Impetus<emph.end type="italics"/> is in an a&longs;&longs;igned point of that Parabolical Diagonal, it <lb/>is requi&longs;ite fir&longs;t to a&longs;&longs;ign the Quantity of the Uniform Horizontal <lb/><emph type="italics"/>Impetus,<emph.end type="italics"/> and then to find what is the <emph type="italics"/>Impetus<emph.end type="italics"/> of the falling Movea­<lb/>ble in the point a&longs;&longs;igned: the which cannot be determined without <lb/>the con&longs;ideration of the Time &longs;pent from the beginning of the <lb/>Compo&longs;ition of the two Motions: which Con&longs;ideration of the <lb/>Time is not required in the Compo&longs;ition of Equable Motions, the <lb/>Velocities and <emph type="italics"/>Impetus's<emph.end type="italics"/> of which are alwaies the &longs;ame: but here <lb/>where there is in&longs;erted into the mixture a Motion which beginning <lb/>from extream Tardity goeth encrea&longs;ing in Velocity according to <lb/>the continuation of the Time, it is nece&longs;&longs;ary that the quantity of <lb/>the Time do &longs;hew us the quantity of the degree of Velocity in the <lb/>a&longs;&longs;igned point: for, as to the re&longs;t, the <emph type="italics"/>Impetus<emph.end type="italics"/> compounded of the&longs;e <lb/>two (as in Uniform Motions) is <emph type="italics"/>potentia<emph.end type="italics"/> equal to both the others <lb/>compounding. </s> <s>But here again I will better explain my meaning by <lb/>an example. </s> <s>In A C the Perpendicular to the Horizon let any part <lb/>be taken A B; the which I will &longs;uppo&longs;e to &longs;tand for the Mea&longs;ure <lb/>of the Space of the Natural Motion made along the &longs;aid Perpen­<lb/>dicular, and likewi&longs;e let it be the Mea&longs;ure of the Time, and al&longs;o of <lb/>the degree of Velocity, or, if you will, of the <emph type="italics"/>Impetus's.<emph.end type="italics"/> It is ma­<lb/>nife&longs;t in the fir&longs;t place, that if the <emph type="italics"/>Impetus<emph.end type="italics"/> of the Moveable in B <lb/><emph type="italics"/>ex quiete<emph.end type="italics"/> in A &longs;hall be turned along B D parallel to the Horizon in <lb/>an Equable Motion, the quantity of its Velocity &longs;hall be &longs;uch that <lb/>in the Time A B it &longs;hall pa&longs;s a Space double to the Space A B, which <lb/>let be the Line B D. </s> <s>Then let B C be &longs;uppo&longs;ed equal to B A, and <pb xlink:href="069/01/229.jpg" pagenum="226"/>let C E be drawn parallel and equal to B D, and thus by the Points <lb/>B and E we &longs;hall de&longs;cribe the Parabolick Line B E I. </s> <s>And becau&longs;e <lb/>that in the Time A B with the <emph type="italics"/>Impetus<emph.end type="italics"/> A B the Horizontal Line B D <lb/>or C E is pa&longs;&longs;ed, double to A B, and in &longs;uch another Time the Per­<lb/>pendicular B C is pa&longs;&longs;ed with an acqui&longs;t of <emph type="italics"/>Impetus<emph.end type="italics"/> in C equal to <lb/>the &longs;aid Horizontal Line; therefore the Moveable in &longs;uch another <lb/>Time as A B &longs;hall be found to have pa&longs;&longs;ed from B to E along the <lb/>Parabola B E with an <emph type="italics"/>Impetus<emph.end type="italics"/> compounded of two, each equal to <lb/>the <emph type="italics"/>Impetus<emph.end type="italics"/> A B. </s> <s>And becau&longs;e one of them is Horizontal, and the <lb/>other Perpendicular, the <emph type="italics"/>Impetus<emph.end type="italics"/> compound of them &longs;hall be equal <lb/>in Power to them both, that is <lb/><figure id="id.069.01.229.1.jpg" xlink:href="069/01/229/1.jpg"/><lb/>double to one of them. </s> <s>So that <lb/>&longs;uppo&longs;ing B F equal to B A, and <lb/>drawing the Diagonal A F, the <lb/><emph type="italics"/>Impetus<emph.end type="italics"/> or the Percu&longs;&longs;ion in E <lb/>&longs;hall be greater than the Percu&longs;­<lb/>&longs;ion in B of the Moveable fal­<lb/>ling from the Height A, or than <lb/>the Percu&longs;&longs;ion of the Horizon­<lb/>tal <emph type="italics"/>Impetus<emph.end type="italics"/> along B D, according <lb/>to the proportion of A F to <lb/>A B. </s> <s>But in ca&longs;e, &longs;till retaining <lb/>B A for the Mea&longs;ure of the <lb/>Space of the Fall from Re&longs;t in <lb/>A unto B, and for the Mea&longs;ure of the Time and of the <emph type="italics"/>Impetus<emph.end type="italics"/> of <lb/>the falling Moveable acquired in B, the Altitude B O &longs;hould not be <lb/>equal to, but greater than A B, taking B G to be a Mean-propor­<lb/>tional betwixt the &longs;aid A B and B O, the &longs;aid B G would be the <lb/>Mea&longs;ure of the Time and of the <emph type="italics"/>Impetus<emph.end type="italics"/> in O, acquired in O by the <lb/>Fall from the height B O; and the Space along the Horizontal <lb/>Line, which being pa&longs;&longs;ed with the <emph type="italics"/>Impetus<emph.end type="italics"/> A B in the Time A B <lb/>would be double to A B, &longs;hall, in the whole duration of the Time <lb/>B G, be &longs;o much the greater, by how much in proportion B G is <lb/>greater than B A. </s> <s>Suppo&longs;ing therefore L B equal to B G, and draw­<lb/>ing the Diagonal A L, it &longs;hall give us the quantity compounded of <lb/>the two <emph type="italics"/>Impetus's<emph.end type="italics"/> Horizontal and Perpendicular, by which the <lb/>Parabola is de&longs;cribed; and of which the Horizontal and Equable is <lb/>that acquired in B by the fall of A B, and the other is that acquired <lb/>in O, or, if you will, in I by the De&longs;cent B O, who&longs;e Time, as al&longs;o <lb/>the quantity of its Moment was B G. </s> <s>And in this Method we &longs;hall <lb/>inve&longs;tigate the <emph type="italics"/>Impetus<emph.end type="italics"/> in the extream term of the Parabola, in ca&longs;e <lb/>its Altitude were le&longs;&longs;er than the Sublimity A B, taking the Mean­<lb/>proportional betwixt them both: which being &longs;et off upon the Ho­<lb/>rizontal Line in the place of B F, and the Diagonal drawn, as A F, <lb/>we &longs;hall hereby have the quantity of the <emph type="italics"/>Impetus<emph.end type="italics"/> in the extream <lb/>term of the Parabola.</s></p><pb xlink:href="069/01/230.jpg" pagenum="227"/><p type="margin"> <s><margin.target id="marg1100"></margin.target>* Or along <lb/>which.</s></p><p type="main"> <s>And to what hath hitherto been propo&longs;ed touching <emph type="italics"/>Impetus's,<emph.end type="italics"/><lb/>Blows, or if you plea&longs;e, Percu&longs;&longs;ions of &longs;uch like Projects, it is ne­<lb/>ce&longs;&longs;ary to add another very nece&longs;&longs;ary Con&longs;ideration; and this it is: <lb/>That it doth not &longs;uffice to have regard to the Velocity only of the <lb/>Project for the determining rightly of the Force and Violence of the <lb/>Percu&longs;&longs;ion, but it is requi&longs;ite likewi&longs;e to examine apart the State <lb/>and Condition of that which receiveth the Percu&longs;&longs;ion, in the effica­<lb/>cy of which it hath for many re&longs;pects a great &longs;hare and intere&longs;t. <lb/></s> <s>And fir&longs;t there is no man but knows that the thing &longs;mitten doth &longs;o <lb/>much &longs;uffer violence from the Velocity of the Percutient by how <lb/>much it oppo&longs;eth it, and either totally or partially checketh its <lb/>Motion: For if the Blow &longs;hall light upon &longs;uch an one as yieldeth to <lb/>the Velocity of the Percutient without any Re&longs;i&longs;tance, that Blow <lb/>&longs;hall be nullified: And he that runneth to hit his Enemy with his <lb/>Launce, if at the overtaking of him it &longs;hall fall out that he moveth, <lb/>giving back with the like Velocity, he &longs;hall make no thru&longs;t, and the <lb/>Action &longs;hall be a meer touch without doing any harm.</s></p><p type="main"> <s>But if the Percu&longs;&longs;ion &longs;hall happen to be received upon an Object <lb/>which doth not wholly yield to the Percutient, but only partially, <lb/>the Percu&longs;&longs;ion &longs;hall do hurt, though not with its whole <emph type="italics"/>Impetus,<emph.end type="italics"/> but <lb/>only with the exce&longs;s of the Velocity of the &longs;aid Percutient above <lb/>the Velocity of the recoile and rece&longs;&longs;ion of the Object percu&longs;&longs;ed: <lb/>&longs;o that, if <emph type="italics"/>v. </s> <s>g.<emph.end type="italics"/> the Percutient &longs;hall come with 10 degrees of Velo­<lb/>city upon the Percu&longs;&longs;ed Body, which giving back in part retireth <lb/>with 4 degrees, the <emph type="italics"/>Impetus<emph.end type="italics"/> and Percu&longs;&longs;ion &longs;hall be as if it were of <lb/>6 degrees. </s> <s>And la&longs;tly, the Percu&longs;&longs;ion &longs;hall be entire and perfect on <lb/>the part of the Percutient when the thing percu&longs;&longs;ed yieldeth not, <lb/>but wholly oppo&longs;eth and &longs;toppeth the whole Motion of the Percu­<lb/>tient; if haply there can be &longs;uch a ca&longs;e. </s> <s>And I &longs;ay on the part of <lb/>the Percutient, for when the Body percu&longs;&longs;ed moveth with a contra­<lb/>ry Motion towards the Percutient, the Blow and Shock &longs;hall be <lb/>&longs;o much the more Impetuous by how much the two Velocities uni­<lb/>ted are greater than the &longs;ole Velocity of the Percutient. </s> <s>More­<lb/>over, you are likewi&longs;e to take notice, that the more or le&longs;s yielding <lb/>may proceed not only from the quality of the Matter more or le&longs;s <lb/>hard, as if it be of Iron, of Lead, or of Wooll, <emph type="italics"/>&c.<emph.end type="italics"/> but al&longs;o from <lb/>the Po&longs;ition of the Body that receiveth the Percu&longs;&longs;ion. </s> <s>Which Po­<lb/>&longs;ition if it &longs;hall be &longs;uch as that the Motion of the Percutient hap­<lb/>neth to hit it at Right-Angles, the <emph type="italics"/>Impetus<emph.end type="italics"/> of the Percu&longs;&longs;ion &longs;hall <lb/>be the greate&longs;t: but if the Motion &longs;hall proceed obliquely, and, as <lb/>we &longs;ay, a&longs;lant, the Percu&longs;&longs;ion &longs;hall be weaker; and that more, and <lb/>more according to its greater and greater Obliquity: for an Ob­<lb/>ject in that manner &longs;cituate, albeit of very &longs;olid matter, doth not <lb/>damp or arre&longs;t the whole <emph type="italics"/>Impetus<emph.end type="italics"/> and Motion of the Percutient, <lb/>which &longs;lanting pa&longs;&longs;eth farther, continuing at lea&longs;t in &longs;ome part to <pb xlink:href="069/01/231.jpg" pagenum="228"/>move along the Surface of the oppo&longs;ed Body Re&longs;i&longs;ting. </s> <s>When <lb/>therefore we have even now determined of the greatne&longs;s of the <lb/><emph type="italics"/>Impetus<emph.end type="italics"/> of the Project in the end of the Parabolicall Line, it ought <lb/>to be under&longs;tood to be meant of the Percu&longs;&longs;ion received upon a <lb/>Line at Right Angles with the &longs;ame Parabolick Line, or with the <lb/>Line that is Tangent to the Parabola in the fore&longs;aid point: for <lb/>although that &longs;ame Motion be compounded of an Horizontal and <lb/>a Perpendicular Motion, the <emph type="italics"/>Impetus<emph.end type="italics"/> is not at the greate&longs;t either <lb/>upon the Horizontal Plane, or upon that erect to the Horizon, be­<lb/>ing received upon them both obliquely.</s></p><p type="main"> <s>SAGR. </s> <s>Your &longs;peaking of the&longs;e Blows, and the&longs;e Percu&longs;&longs;ions <lb/>hath brought into my mind a Problem, or, if you will, Que&longs;tion <lb/>in the Mechanicks, the &longs;olution whereof I could never find in any <lb/>Author, nor any thing that doth dimini&longs;h my admiration, or &longs;o <lb/>much as in the lea&longs;t afford my judgment &longs;atisfaction. </s> <s>And my <lb/>doubt and wonder lyeth in my not being able to comprehend <lb/>whence that Immen&longs;e Force and Violence &longs;hould proceed, and on <lb/>what Principle it &longs;hould depend, which we &longs;ee to con&longs;i&longs;t in Per­<lb/>cu&longs;&longs;ion, in that with the &longs;imple &longs;troke of an Hammer, that doth <lb/>not weigh above eight or ten pounds, we &longs;ee &longs;uch Re&longs;i&longs;tances to be <lb/>overcome as would not yield to the weight of a Grave Body that <lb/>without Percu&longs;&longs;ion hath an <emph type="italics"/>Impetus<emph.end type="italics"/> only by pre&longs;&longs;ing and bearing <lb/>upon it, albeit the weight of this be many hundreds of pounds <lb/>more. </s> <s>I would likewi&longs;e find out a way to mea&longs;ure the Force of this <lb/>Percu&longs;&longs;ion, which I do not think to be infinite, but rather hold <lb/>that it hath its Term in which it may be compared, and in the end <lb/>Regulated with other Forces of pre&longs;&longs;ing Gravities, either of Lea­<lb/>vers, or of Screws, or of other Mechanick In&longs;truments, of who&longs;e <lb/>multiplication of Force I am thorowly &longs;atisfied.</s></p><p type="main"> <s>SALV. </s> <s>You are not alone in the admirablene&longs;s of the effect, <lb/>and the ob&longs;curity of the cau&longs;e of &longs;o &longs;tupendious an Accident. </s> <s>I <lb/>ruminated a long time upon it in vain, my &longs;tupifaction &longs;till encrea­<lb/>&longs;ing; till in the end meeting with our <emph type="italics"/>Academian,<emph.end type="italics"/> I received from <lb/>him a double &longs;atisfaction: fir&longs;t in hearing that he al&longs;o had been a <lb/>long time at the &longs;ame lo&longs;s; and next in under&longs;tanding that after he <lb/>had at times &longs;pent many thou&longs;ands of hours in &longs;tudying and con­<lb/>templating thereon, he had light upon certain Notions far from <lb/>our fir&longs;t conceptions, and therefore new, and for their Novelty to <lb/>be admired. </s> <s>And becau&longs;e that I already &longs;ee that your Curio&longs;ity <lb/>would gladly hear tho&longs;e Conceits which are Remote from common <lb/>Conjecture, I &longs;hall not &longs;tay for your entreaty, but I give you my <lb/>word that &longs;o &longs;oon as we &longs;hall have fini&longs;hed the Reading of this <lb/>Treati&longs;e of Projects, I will &longs;et before you all tho&longs;e Fancies, or, I <lb/>might &longs;ay, Extravagancies that are yet left in my memory of the <lb/>Di&longs;cour&longs;es of the Academick. </s> <s>In the mean time let us pro&longs;ecute <lb/>the Propo&longs;itions of our Author.</s></p><pb xlink:href="069/01/232.jpg" pagenum="229"/><p type="head"> <s>PROBL. II. PROP. V.</s></p><p type="main"> <s>In the Axis of a given Parabola prolonged to find <lb/>a &longs;ublime point out of which the Moveable <lb/>falling &longs;hall de&longs;cribe the &longs;aid Parabola.</s></p><p type="main"> <s><emph type="italics"/>Let the Parabola be A B, its Amplitude H B, and its prolonged <lb/>Axis H E; in which a Sublimity is to be found, out of which the <lb/>Moveable falling, and converting the<emph.end type="italics"/> Impetus <emph type="italics"/>conceived in A <lb/>along the Horizontal Line, de&longs;cribeth the Parabola A B. </s> <s>Draw the <lb/>Horizontal Line A G, which &longs;hall be Parallel to B H, and &longs;uppo&longs;ing A F <lb/>equal to A H draw the Right Line F B, which toucheth the Parabola in <lb/>B, and cutteth the Horizontal Line A G in G; and unto F A and A G <lb/>let A E be a third Proportional. </s> <s>I &longs;ay, that E is the &longs;ublime Point re­<lb/>quired, out of which the Moveable falling<emph.end type="italics"/> ex quiete <emph type="italics"/>in E, and the<emph.end type="italics"/> Im­<lb/>petus <emph type="italics"/>conceived in A being converted along the Horizontal Line over­<lb/>taking the<emph.end type="italics"/> Impetus <emph type="italics"/>of the De&longs;cent<emph.end type="italics"/><lb/><figure id="id.069.01.232.1.jpg" xlink:href="069/01/232/1.jpg"/><lb/><emph type="italics"/>in H<emph.end type="italics"/> ex quiete <emph type="italics"/>in A, de&longs;cribeth the <lb/>Parabola A B. </s> <s>For if we &longs;uppo&longs;e <lb/>E A to be the Mea&longs;ure of the Time <lb/>of the Fall from E to A, and of <lb/>the<emph.end type="italics"/> Impetus <emph type="italics"/>acquired in A, A G <lb/>(that is a Mean-proportional be­<lb/>tween E A and A F) &longs;hall be the <lb/>Time and the<emph.end type="italics"/> Impetus <emph type="italics"/>coming <lb/>from F to A, or from A to H. </s> <s>And <lb/>becau&longs;e the Moveable coming out of <lb/>E in the Time E A with the<emph.end type="italics"/> Impetus <emph type="italics"/>acquired in A pa&longs;&longs;eth in the Ho­<lb/>rizontal Lation with an Equable Motion the double of E A; There­<lb/>fore likewi&longs;e moving with the &longs;ame<emph.end type="italics"/> Impetus <emph type="italics"/>it &longs;hall in the Time A G <lb/>pa&longs;s the double of G A, to wit, the Mean-proportional B H (for the <lb/>Spaces pa&longs;&longs;ed with the &longs;ame Equable Motion are to one another as the <lb/>Times of the &longs;aid Motions:) And along the Perpendicular A H &longs;hall <lb/>be pa&longs;&longs;ed with a Motion<emph.end type="italics"/> ex quiete <emph type="italics"/>in the &longs;ame Time G A: Therefore <lb/>the Amplitude H B, and Altitude A H are pa&longs;&longs;ed by the Moveable in the <lb/>&longs;ame Time: Therefore the Parabola A B &longs;hall be de&longs;cribed by the <lb/>De&longs;cent of the Project coming from the Sublimity E: Which was re­<lb/>quired.<emph.end type="italics"/></s></p><p type="head"> <s>COROLLARY.</s></p><p type="main"> <s>Hence it appeareth that the half of the <emph type="italics"/>B<emph.end type="italics"/>a&longs;e or Amplitude of the <lb/>Semiparabola (which is the fourth part of the Amplitude of <lb/>the whole Parabola) is a Mean-proportional betwixt its Al­<lb/>titude and the Sublimity out of which the Moveable falling <lb/>de&longs;cribeth it.</s></p><pb xlink:href="069/01/233.jpg" pagenum="230"/><p type="head"> <s>PROBL. III. <emph type="italics"/>P<emph.end type="italics"/>RO<emph type="italics"/>P.<emph.end type="italics"/> VI.</s></p><p type="main"> <s>The Sublimity and Altitude of a Semiparabola <lb/>being given to find its Amplitude.</s></p><p type="main"> <s><emph type="italics"/>Let A C be perpendicular to the<emph.end type="italics"/><lb/><figure id="id.069.01.233.1.jpg" xlink:href="069/01/233/1.jpg"/><lb/><emph type="italics"/>Horizontal Line D C, in <lb/>which let the Altitude C B and <lb/>the Sublimity B A be given: It is <lb/>required in the Horizontal Line <lb/>D C to find the Amplitude of the <lb/>Semiparabola that is de&longs;cribed out of <lb/>the Sublimity B A with the Alti­<lb/>tude B C. </s> <s>Take a Mean proportional <lb/>between C B and B A, to which let <lb/>C D be double, I &longs;ay, that C D is <lb/>the Amplitude required. </s> <s>The which <lb/>is manife&longs;t by the precedent Propo&longs;ition.<emph.end type="italics"/></s></p><p type="head"> <s>THEOR. IV. PROP. VII.</s></p><p type="main"> <s>In Projects which de&longs;cribe Semiparabola's of the <lb/>&longs;ame Amplitude, there is le&longs;s <emph type="italics"/>Impetus<emph.end type="italics"/> required <lb/>in that which de&longs;cribeth that who&longs;e Ampli­<lb/>tude is double to its Altitude, than in any <lb/>other.</s></p><p type="main"> <s><emph type="italics"/>For let the Semiparabola be B D, who&longs;e Amplitude C D is dou­<lb/>ble to its Altitude C B; and in its Axis extended on high let B A <lb/>be &longs;uppo&longs;ed equal to the Altitude B C; and draw a Line from <lb/>A to D which toucheth the Semiparabola in D, and &longs;hall cut the Hori­<lb/>zontal Line B E in E; and B E &longs;hall be equal to B C or to B A: It is <lb/>manife&longs;t that it is de&longs;cribed by the Project who&longs;e Equable Horizontal<emph.end type="italics"/><lb/>Impetus <emph type="italics"/>is &longs;uch as is that gained in B of a thing falling from Re&longs;t in A, <lb/>and the<emph.end type="italics"/> Impetus <emph type="italics"/>of the Natural Motion downwards, &longs;uch as is that of <lb/>a thing coming to C<emph.end type="italics"/> ex quiete <emph type="italics"/>in B. </s> <s>Whence it is manife&longs;t, that the<emph.end type="italics"/><lb/>Impetus <emph type="italics"/>compounded of them, and that &longs;triketh in the Term D is as the <lb/>Diagonal A E, that is<emph.end type="italics"/> potentia <emph type="italics"/>equal to them both. </s> <s>Now let there be <lb/>another Semiparabola G D, who&longs;e Amplitude is the &longs;ame C D, and the <lb/>Altitude C G le&longs;s, or greater than the Altitude B C, and let H D touch <lb/>the &longs;ame, cutting the Horizontal Line drawn by G in the point K; and <lb/>as H G is to G K, &longs;o let K G be to G L: by what hath been demon&longs;trated <lb/>G L &longs;hall be the Altitude from which the Project falling de&longs;cribeth the<emph.end type="italics"/><pb xlink:href="069/01/234.jpg" pagenum="231"/><emph type="italics"/>Parabola G D. </s> <s>Let G M be a Mean-proportional betwixt A B and <lb/>G L; G M &longs;hall be the Time, and the Moment or<emph.end type="italics"/> Impetus <emph type="italics"/>in G of the <lb/>Project falling from L, (for it hath been &longs;uppo&longs;ed that A B is the Mea­<lb/>&longs;ure of the Time and<emph.end type="italics"/> Impetus.) <emph type="italics"/>Again, let G N be a Mean-propor­<lb/>tional betwixt B C and C G: this G N &longs;hall be the Mea&longs;ure of the <lb/>Time and the<emph.end type="italics"/><lb/>Impetus <emph type="italics"/>of the<emph.end type="italics"/><lb/><figure id="id.069.01.234.1.jpg" xlink:href="069/01/234/1.jpg"/><lb/><emph type="italics"/>Project falling <lb/>from G to C. <lb/></s> <s>If therefore a <lb/>Line be drawn <lb/>from M to N <lb/>it &longs;hall be the <lb/>the Mea&longs;ure of <lb/>the<emph.end type="italics"/> Impetus <emph type="italics"/>of <lb/>the Project a­<lb/>long the Para­<lb/>bola B D, &longs;cri­<lb/>king in the <lb/>term D. Which<emph.end type="italics"/><lb/>Impetus, <emph type="italics"/>I &longs;ay, <lb/>is greater than the<emph.end type="italics"/> Impetus <emph type="italics"/>of the Project along the Parabola B D, <lb/>who&longs;e quantity was A E. </s> <s>For becau&longs;e G N is &longs;uppo&longs;ed the Mean-pro­<lb/>portional betwixt B C and C G, and B C is equal to B E, that is to H G; <lb/>(for they are each of them &longs;ubduple to D C:) Therefore as C G is to <lb/>G N, &longs;o &longs;hall N G be to G K: and, as C G or H G is to G K, &longs;o &longs;hall the <lb/>Square N G be to the Square of G K: But as H G is to G K, &longs;o was <lb/>K G &longs;uppo&longs;ed to be to G L: Therefore as N G is to the Square G K, &longs;o <lb/>is K G to G L: But as K G is to G L, &longs;o is the Square K G unto the <lb/>Square G M, (for G M is the Mean between K G and G L:) Therefore <lb/>the three Squares N G, K G, and G M are continual proportionals: And <lb/>the two extream ones N G and G M taken together, that is the Square <lb/>M N is greater than double the Square K G, to which the Square A E <lb/>is double: Therefore the Square M N is greater than the Square A E: <lb/>and the Line M N greater than the Line A E: Which was to be de­<lb/>mon&longs;trated.<emph.end type="italics"/></s></p><p type="head"> <s>CORROLLARY I.</s></p><p type="main"> <s>Hence it appeareth, that on the contrary, in the Project out of D <lb/>along the Semiparabola D B, le&longs;s <emph type="italics"/>Impetus<emph.end type="italics"/> is required than <lb/>along any other according to the greater or le&longs;&longs;er Elevation <lb/>of the Semiparabola B D, which is according to the Tan­<lb/>gent A D, containing half a Right-Angle upon the Hori­<lb/>zon.</s></p><pb xlink:href="069/01/235.jpg" pagenum="232"/><p type="head"> <s>COROLLARRY II.</s></p><p type="main"> <s>And that being &longs;o, it followeth, that if Projections be made with <lb/>the &longs;ame <emph type="italics"/>Impetus<emph.end type="italics"/> out of the Term D, according to &longs;everal <lb/>Elevations, that &longs;hall be the greate&longs;t Projection or Amplitude <lb/>of the Semiparabola or whole Parabola which followeth at <lb/>the Elevation of a ^{*} Semi-Right-Angle; and the re&longs;t, made <lb/><arrow.to.target n="marg1101"></arrow.to.target><lb/>according to greater or le&longs;&longs;er Angles, &longs;hall be greater or <lb/>le&longs;&longs;er.</s></p><p type="margin"> <s><margin.target id="marg1101"></margin.target>* Or, at the Ele­<lb/>vation of 45 de­<lb/>grees.</s></p><p type="main"> <s>SAGR. </s> <s>The &longs;trength of Nece&longs;&longs;ary Demon&longs;trations are full of <lb/>plea&longs;ure and wonder; and &longs;uch are only the Mathematical. </s> <s>I un­<lb/>der&longs;tood before upon tru&longs;t from the Relations of &longs;undry Gunners, <lb/>that of all the Ranges of a Cannon, or of a Mortar-piece, the grea­<lb/>te&longs;t, <emph type="italics"/>&longs;cilicet<emph.end type="italics"/> that which carryeth the Ball farthe&longs;t was that made at <lb/>the Elevation of a Semi-Right-Angle, which they call, of the Sixth <lb/>point of the Square: but the knowledge of the Cau&longs;e whence it <lb/>hapneth infinitely &longs;urpa&longs;&longs;eth the bare Notion that I received upon <lb/>their atte&longs;tation, and al&longs;o from many repeated Experiments.</s></p><p type="main"> <s>SALV. </s> <s>You &longs;ay very right: and the knowledge of one &longs;ingle <lb/>Effect acquired by its Cau&longs;es openeth the Intellect to under&longs;tand <lb/>and a&longs;certain our &longs;elves of other effects, without need of repairing <lb/>unto Experiments, ju&longs;t as it hapneth in the pre&longs;ent Ca&longs;e; in which <lb/>having found by demon&longs;trative Di&longs;cour&longs;e the certainty of this, <lb/>That the greate&longs;t of all Ranges is that of the Elevation of a Semi­<lb/>Right-Angle, the Author demon&longs;trates unto us that which po&longs;&longs;ibly <lb/>hath not been ob&longs;erved by Experience: and that is, that of the <lb/>other Ranges tho&longs;e are equal to one another who&longs;e Elevations ex­<lb/>ceed or fall &longs;hort by equal Angles of the Semi-right: &longs;o that the <lb/>Balls &longs;hot from the Horizon, one according to the Elevation of &longs;e­<lb/>ven Points, and the other of 5, &longs;hall light upon the Horizon at <lb/>equal Di&longs;tances: and &longs;o the Ranges of 8 and of 4 points, of 9 and <lb/>of 3, <emph type="italics"/>&c.<emph.end type="italics"/> &longs;hall be equal. </s> <s>Now hear the Demon&longs;tration of it.</s></p><p type="head"> <s>THEOR. V. PROP. VIII.</s></p><p type="main"> <s>The Amplitudes of Parabola's de&longs;cribed by Pro­<lb/>jects expul&longs;ed with the &longs;ame <emph type="italics"/>Impetus<emph.end type="italics"/> according <lb/>to the Elevations by Angles equidi&longs;tant above, <lb/><arrow.to.target n="marg1102"></arrow.to.target><lb/>and beneath from the ^{*} Semi-right, are equal to <lb/>each other.</s></p><pb xlink:href="069/01/236.jpg" pagenum="233"/><p type="margin"> <s><margin.target id="marg1102"></margin.target>* Or Angle of <lb/>45.</s></p><p type="main"> <s><emph type="italics"/>Of the Triangle M C B, about the Right-Angle C, let the Ho­<lb/>rizontal Line B C and the Perpendicular C M be equal; for <lb/>&longs;o the Angle M B C &longs;hall be Semi-right; and prolonging C M <lb/>to D, let there be con&longs;tituted in B two equal Angles above and below <lb/>the Diagonal M B,<emph.end type="italics"/> viz. <emph type="italics"/>M B E, and M B D. </s> <s>It is to be demon&longs;trated <lb/>that the Amplitudes of the Parabola's de&longs;cribed by the Projects be­<lb/>ing emitted<emph.end type="italics"/> [or &longs;hot off] <emph type="italics"/>with the &longs;ame<emph.end type="italics"/> Impetus <emph type="italics"/>out of the Term B, <lb/>according to the Elevations of the Angles E B C and D B C, are equal. <lb/></s> <s>For in regard that the extern Angle B M C, is equal to the two intern <lb/>M D B and M B D, the Angle M B C &longs;hall al&longs;o be equal to them. </s> <s>And if <lb/>we &longs;uppo&longs;e M B E in&longs;tead of the Angle M B D, <lb/>the &longs;aid Angle M B C &longs;hall be equal to the two<emph.end type="italics"/><lb/><figure id="id.069.01.236.1.jpg" xlink:href="069/01/236/1.jpg"/><lb/><emph type="italics"/>Angles M B E and B D C: And taking away <lb/>the common Angle M B E, the remaining An­<lb/>gle B D C &longs;hall be equal to the remaining An­<lb/>gle E B C: Therefore the Triangles D C B <lb/>and B C E are alike. </s> <s>Let the Right Lines <lb/>D C and E C be divided in the mid&longs;t in H and <lb/>F; and draw H I and F G parallel to the Ho­<lb/>rizontal Line C B; and as D H is to H I, &longs;o <lb/>let I H be to H L: the Triangle I H L &longs;hall be <lb/>like to the Triangle I H D, like to which al&longs;o is E G F. </s> <s>And &longs;eeing <lb/>that I H and G F are equal (to wit, halves of the &longs;ame B C:) There­<lb/>fore F E, that is F C, &longs;hall be equal to H L: And, adding the common <lb/>Line F H, C H &longs;hall be equal to F L. </s> <s>If therefore we under&longs;tand the Se­<lb/>miparabola to be de&longs;cribed along by H and B, who&longs;e Altitude &longs;hall be <lb/>H C, and Sublimity H L, its Amplitude &longs;hall be C B, which is double <lb/>to HI, that is, the Mean betwixt D H, or C H, and HL: And D B <lb/>&longs;hall be a Tangent to it, the Lines C H and H D being equal. </s> <s>And if, <lb/>again, we conceive the Parabola to be de&longs;cribed along by F and B from <lb/>the Sublimity FL, with the Altitude F C, betwixt which the Mean­<lb/>proportional is F G, who&longs;e double is the Horizontal Line C B: C B, as <lb/>before, &longs;hall be its Amplitude; and E B a Tangent to it, &longs;ince E F and <lb/>F C are equal: But the Angles D B C and E B C<emph.end type="italics"/> (&longs;cilicet, <emph type="italics"/>their Eleva­<lb/>tions) &longs;hall be equidi&longs;tant from the Semi-Right Angle: Therefore the <lb/>Propo&longs;ition is demon&longs;trated.<emph.end type="italics"/></s></p><p type="head"> <s>THEOR. VI. <emph type="italics"/>P<emph.end type="italics"/>RO<emph type="italics"/>P.<emph.end type="italics"/> IX.</s></p><p type="main"> <s>The Amplitudes of Parabola's, who&longs;e Altitudes <lb/>and Sublimities an&longs;wer to each other <emph type="italics"/>è contra­<lb/>rio,<emph.end type="italics"/> are equall.</s></p><pb xlink:href="069/01/237.jpg" pagenum="234"/><p type="main"> <s><emph type="italics"/>Let the Altitude G F of the Parabola F H have the &longs;ame proporti­<lb/>on to the Altitude C B of the Parabola B D, as the Sublimity B A <lb/>hath to the Sublimity F E. </s> <s>I &longs;ay, that the Amplitude H G is equal <lb/>to the Amplitude D C. </s> <s>For &longs;ince the fir&longs;t G F hath the &longs;ame propor­<lb/>tion to the &longs;econd C B, as the third B A hath to the fourth F E; There­<lb/>fore, the Rectangle<emph.end type="italics"/><lb/><figure id="id.069.01.237.1.jpg" xlink:href="069/01/237/1.jpg"/><lb/><emph type="italics"/>G F E of the fir&longs;t and <lb/>fourth, &longs;hall be equal to <lb/>the Rectangle C B A <lb/>of the &longs;econd and <lb/>third: Therefore the <lb/>Squares that are equal <lb/>to the&longs;e Rectangles &longs;hall <lb/>be equal to one another: <lb/>But the Square of half of G H is equal to the Rectangle G F E; and <lb/>the Square of half of C D is equal to the Rectangle C B A: There­<lb/>fore the&longs;e Squares, and their Sides, and the doubles of their Sides &longs;hall <lb/>be equal: But the&longs;e are the Amplitudes G H and C D: Therefore the <lb/>Propo&longs;ition is manife&longs;t.<emph.end type="italics"/></s></p><p type="head"> <s>LEMMA <emph type="italics"/>pro &longs;equenti.<emph.end type="italics"/></s></p><p type="main"> <s>If a Right Line be cut according to any proportion, the Squares <lb/>of the Mean-proportionals between the whole and the two <lb/>parts are equal to the Square of the whole.</s></p><p type="main"> <s><emph type="italics"/>Let A B be cut according to any proportion in C. </s> <s>I &longs;ay, that the <lb/>Squares of the Mean-proportional Lines between the whole A B and <lb/>the parts A C and C B, being taken together are equal to the Square of <lb/>the whole A B. </s> <s>And this appeareth, a Semi-<emph.end type="italics"/><lb/><figure id="id.069.01.237.2.jpg" xlink:href="069/01/237/2.jpg"/><lb/><emph type="italics"/>circle being de&longs;cribed upon the whole Line <lb/>B A, and from C a Perpendicular being ere­<lb/>cted C D, and Lines being drawn from D to <lb/>A, and from D to B. </s> <s>For D A is the Mean­<lb/>proportional betwixt A B and A C; and D B is the Mean-proporti­<lb/>onal between A B and B C: And the Squares of the Lines D A and <lb/>D B taken together are equal to the Square of the whole Line A B, <lb/>the Angle A D B in the Semicircle being a Right-Angle: Therefore <lb/>the Propo&longs;ition is manifest.<emph.end type="italics"/></s></p><pb xlink:href="069/01/238.jpg" pagenum="235"/><p type="head"> <s>THEOR. VII. PROP. X.</s></p><p type="main"> <s>The <emph type="italics"/>Impetus<emph.end type="italics"/> or Moment of any Semiparabola is <lb/>equal to the Moment of any Moveable falling <lb/>naturally along the Perpendicular to the Ho­<lb/>rizon that is equal to the Line compounded of <lb/>the Sublimity and of the Altitude of the Se­<lb/>miparabola.</s></p><p type="main"> <s><emph type="italics"/>Let the Semiparabola be A B, its Sublimity D A, and Altitude <lb/>A C, of which the Perpendicular D C is compounded. </s> <s>I &longs;ay, that <lb/>the<emph.end type="italics"/> Impetus <emph type="italics"/>of the Semiparabola in B is equal to the Moment of <lb/>the Moveable Naturally falling from D to C. </s> <s>Suppo&longs;e D C it &longs;elf to be <lb/>the Mea&longs;ure of the Time and of the<emph.end type="italics"/> Impetus; <emph type="italics"/>and take a Mean-pro­<lb/>portional betwixt C D and D A, to which let<emph.end type="italics"/><lb/><figure id="id.069.01.238.1.jpg" xlink:href="069/01/238/1.jpg"/><lb/><emph type="italics"/>C F be equal; and withal let C E be a Mean­<lb/>proportional between D C and C A: Now C F <lb/>&longs;hall be the Mea&longs;ure of the Time and of the Mo­<lb/>ment of the Moveable &longs;alling along D A out of <lb/>Re&longs;t in D; and C E &longs;hall be the Time and Mo­<lb/>ment of the Moveable falling along A C, out of <lb/>Re&longs;t in A, and the Moment of the Diagonal E F <lb/>&longs;hall be that compounded of both the others,<emph.end type="italics"/> &longs;cil. <lb/><emph type="italics"/>that of the Semiparabola in B. </s> <s>And becau&longs;e <lb/>D C is cut according to any proportion in A, and becau&longs;e C F and C E <lb/>are Mean-Proportionals between C D and the parts D A and A C; the <lb/>Squares of them taken together &longs;hall be equal to the Square of the <lb/>whole; by the Lemma aforegoing: But the Squares of them are al&longs;o <lb/>equal to the Square of E F: Therefore D F is equal al&longs;o to the Line D C: <lb/>Whence it is manife&longs;t that the Moments along D C, and along the Se­<lb/>miparabola A B, are equal in C and B: Which was required.<emph.end type="italics"/></s></p><p type="head"> <s>COROLLARY.</s></p><p type="main"> <s>Hence it is manife&longs;t, that of all Parabola's who&longs;e Altitudes and <lb/>Sublimities being joyned together are equal, the <emph type="italics"/>Impetus's<emph.end type="italics"/> are <lb/>al&longs;o equal.</s></p><pb xlink:href="069/01/239.jpg" pagenum="236"/><p type="head"> <s>PROBL. IV. PROP. XI.</s></p><p type="main"> <s>The <emph type="italics"/>Impetus<emph.end type="italics"/> and Amplitude of a Semiparabola be­<lb/>ing given, to find its Altitude, and con&longs;equently <lb/>its Sublimity.</s></p><p type="main"> <s><emph type="italics"/>Let the<emph.end type="italics"/> Impetus <emph type="italics"/>given be defined by the Perpendicular to the Ho­<lb/>rizon A B; and let the Amplitude along the Horizontal Line be <lb/>B C. </s> <s>It is required to find the Altitude and Sublimity of the <lb/>Parabola who&longs;e<emph.end type="italics"/> Impetus <emph type="italics"/>is A B, and Amplitude B C. </s> <s>It is manife&longs;t, <lb/>from what hath been already demon&longs;trated, that half the Amplitude B C <lb/>will be a Mean-proportional betwixt the Altitude and the Sublimity of <lb/>the &longs;aid Semiparabola, who&longs;e<emph.end type="italics"/> Impetus, <emph type="italics"/>by the precedent Propo&longs;ition, is <lb/>the &longs;ame with the<emph.end type="italics"/> Impetus <emph type="italics"/>of the Moveable falling from Re&longs;t in A along <lb/>the whole Perpendicular A B: Wherefore B A is &longs;o to be cut that the <lb/>Rectangle contained by its parts may be equal to the Square of half of <lb/>B C, which let be B D. </s> <s>Hence it appeareth <lb/>to be nece&longs;&longs;ary that D B do not exceed the<emph.end type="italics"/><lb/><figure id="id.069.01.239.1.jpg" xlink:href="069/01/239/1.jpg"/><lb/><emph type="italics"/>half of B A; for of Rectangles contained by <lb/>the parts the greate&longs;t is when the whole <lb/>Line is cut into two equal parts. </s> <s>Therefore <lb/>let B A be divided into two equal parts in E. <lb/></s> <s>And if B D be equal to B E the work is <lb/>done; and the Altitude of the Semipara­<lb/>bola &longs;hall be B E, and its Sublimity E A: <lb/>(and &longs;ee here by the way that the Amplitude <lb/>of the Parabola of a Semi-right Elevation, <lb/>as was demon&longs;trated above, is the greate&longs;t of <lb/>all tho&longs;e de&longs;cribed with the &longs;ame<emph.end type="italics"/> Impetus.) <lb/><emph type="italics"/>But let B D be le&longs;s than the half of B A, <lb/>which is &longs;o to be cut that the Rectangle under the parts may be equal to <lb/>the Square B D. </s> <s>Upon E A de&longs;cribe a Semicircle, upon which out of A <lb/>&longs;et off A F equal to B D, and draw a Line from F to E, to which cut <lb/>a part equal E G. </s> <s>Now the Rectangle B G A, together with the Square <lb/>E G, &longs;hall be equal to the Square E A; to which the two Squares A F <lb/>and F E are al&longs;o equal: Therefore the equal Squares G E and F E be­<lb/>ing &longs;ub&longs;tracted, there remaineth the Rectangle B G A equal to the <lb/>Square A F,<emph.end type="italics"/> &longs;cilicet, <emph type="italics"/>to B D; and the Line B D is a Mean-proportional <lb/>betwixt B G and G A. </s> <s>Whence it appeareth, that of the Semipa­<lb/>rabola who&longs;e Amplitude is B C, and<emph.end type="italics"/> Impetus <emph type="italics"/>A B, the Altitude is <lb/>B G, and the Sublimity G A. </s> <s>And if we &longs;et off B I below equal to G A, <lb/>this &longs;hall be the Altitude, and I A the Sublimity of the Semiparabola <lb/>I C. </s> <s>From what hath been already demon&longs;trated we are able,<emph.end type="italics"/></s></p><pb xlink:href="069/01/240.jpg" pagenum="237"/><p type="head"> <s>PROBL. V. PROP. XII.</s></p><p type="main"> <s>To collect by Calculation of the Amplitudes of all <lb/>Semiparabola's that are de&longs;cribed by Projects <lb/>expul&longs;ed with the &longs;ame <emph type="italics"/>Impetus,<emph.end type="italics"/> and to make <lb/>Tables thereof.</s></p><p type="main"> <s><emph type="italics"/>It is obvious, from the things demon&longs;trated, that Parabola's are de­<lb/>&longs;cribed by Projects of the &longs;ame<emph.end type="italics"/> Impetus <emph type="italics"/>then, when their Subli­<lb/>mities together with their Altitudes do make up equal Perpendicu­<lb/>lars upon the Horizon. </s> <s>The&longs;e Perpendiculars therefore are to be com­<lb/>prehended between the &longs;ame Horizontal Parallels. </s> <s>Therefore let the <lb/>Horizontal Line C B be &longs;uppo&longs;ed equal to the Perpendicular B A, and <lb/>draw the Diagonal from A to C. </s> <s>The Angle A C B &longs;hall be Semi­<lb/>right, or 45 Degrees. </s> <s>And the Perpendicular B A being divided into <lb/>two equal parts in D, the Semiparabola D C &longs;hall be that which is de­<lb/>&longs;cribed from the Sublimity A D together with the Altitude D B: and <lb/>its<emph.end type="italics"/> Impetus <emph type="italics"/>in C &longs;hall be as great as that of the Moveable coming out of <lb/>Re&longs;t in A along the Perpendicular A B is in B. </s> <s>And if A G be drawn <lb/>parallel to B C, the united Altitudes and Sublimities of all other re­<lb/>maining Semiparabola's who&longs;e future<emph.end type="italics"/> Impetus's <emph type="italics"/>are the &longs;ame with tho&longs;e <lb/>now mentioned mu&longs;t be bounded by the Space between the Parallels<emph.end type="italics"/><lb/><figure id="id.069.01.240.1.jpg" xlink:href="069/01/240/1.jpg"/><lb/><emph type="italics"/>A G and B C. Farthermore, it having <lb/>been but now demon&longs;trated, that the Am­<lb/>plitudes of the Semiparabola's who&longs;e <lb/>Tangents are equidi&longs;tant either above or <lb/>below from the Semi right Elevation are <lb/>equal, the Calculations that we frame <lb/>for the greater Elevations will likewi&longs;e <lb/>&longs;erve for the le&longs;&longs;er. </s> <s>We choo&longs;e moreover <lb/>a number of ten thou&longs;and parts for the <lb/>greate&longs;t Amplitude of the Projection of <lb/>the Semiparabola made at the Elevation <lb/>of 45 degrees: &longs;o much therefore the Line <lb/>B A, and the Amplitude of the Semipa­<lb/>rabola B C, are to be &longs;uppo&longs;ed. </s> <s>And we <lb/>make choice of the number 10000, becau&longs;e we in our Calculation u&longs;e <lb/>the Table of Tangents, in which this number agreeth with the Tangent <lb/>of 45 degrees. </s> <s>Now, to come to the bu&longs;ine&longs;s, let C E be drawn, contain­<lb/>ing the Angle E C B greater (Acute neverthele&longs;s,) than the Angle <lb/>A C B; and let the Semiparabola be de&longs;cribed which is touched by the <lb/>Line E C, and who&longs;e Sublimity united with its Altitude is equal to <lb/>B A. </s> <s>In the Table of Tangents take the &longs;aid B E for the Tangent at the<emph.end type="italics"/><pb xlink:href="069/01/241.jpg" pagenum="238"/><emph type="italics"/>given Angle B C E, which divide into two equal parts at F. </s> <s>Then <lb/>find a third Proportional to B F and B C, (or to the half of B C,) <lb/>which &longs;hall of nece&longs;&longs;ity be greater than F A; therefore let it be F O: <lb/>Of the Semiparabola, therefore, in&longs;cribed in the Triangle E C B, ac­<lb/>cording to the Tangent C E, who&longs;e Amplitude is C B, the Altitude B F, <lb/>and the Sublimity F O is found: But the whole Line B O ri&longs;eth above <lb/>the Parallels A G and C B, whereas our work was to bound it between <lb/>them: For &longs;o both it and the Semiparabola D C &longs;hall be de&longs;cribed by <lb/>the Projects out of C expelled with the &longs;ame<emph.end type="italics"/> Impetus. <emph type="italics"/>Therefore we <lb/>are to &longs;eek another like to this, (for innumerable greater and &longs;maller, <lb/>like to one another, may be de&longs;cribed within the Angle B C E) to who&longs;e <lb/>united Sublimity and Altitude B A &longs;hall be equal. </s> <s>Therefore as O B is <lb/>to B A, &longs;o let the Amplitude B C be to C R: and C R &longs;hall be found,<emph.end type="italics"/><lb/>&longs;cilicet <emph type="italics"/>the Amplitude of the Semiparabola according to the Elevation <lb/>of the Angle B C E, who&longs;e conjoyned Sublimity and Altitude is equal <lb/>to the Space contained between the Parallels G A and C B: Which <lb/>was required. </s> <s>The work, therefore, &longs;hall be after this manner.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Take the Tangent of the given Angle B C E, to the half of which <lb/>add the third Proportional of it, and half of B C, which let be F O: <lb/>Then as O B is to B A, &longs;o let B C be to another, which let be C R, to wit, <lb/>the Amplitude &longs;ought. </s> <s>Let us give an Example.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Let the Angle E C B be 50 degrees, its Tangent &longs;hall be 11918, <lb/>who&longs;e half, to wit, B F, is 5959, and the half of B C is 5000, the third <lb/>proportional of the&longs;e halves is 4195, which added to the &longs;aid B F <lb/>maketh 10154: for the &longs;aid B O. Again, as O B is to B A, that is as <lb/>10154 is to 10000, &longs;o is B E, that is 10000 (for each of them is the <lb/>Tangent of 45 degrees) to another: and that &longs;hall give us the required <lb/>Altitude R C 9848, of &longs;uch as B C (the greate&longs;t Amplitude) is <lb/>10000. To the&longs;e the Amplitudes of the whole Parabola's are double,<emph.end type="italics"/><lb/>&longs;cilicet <emph type="italics"/>19696 and 20000. And &longs;o much likewi&longs;e is the Amplitude of <lb/>the Parabola according to the Elevation of 40 degrees, &longs;ince it is equal­<lb/>ly di&longs;tant from 45 degrees.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>For the perfect under&longs;tanding of this Demon&longs;tration I <lb/>mu&longs;t be informed how true it is, that the Third Proportional to <lb/>B F and B I, is (as the Author &longs;aith) nece&longs;&longs;arily greater than <lb/>F A.</s></p><p type="main"> <s>SALV. </s> <s>That inference, as I conceive, may be deduced thus. <lb/></s> <s>The Square of the Mean of three proportional Lines is equal to <lb/>the Rectangle of the other two: whence the Square of B I, or of <lb/>B D equal to it, ought to be equal to the Rectangle of the fir&longs;t F B <lb/>multiplied into the third to be found: which third is of nece&longs;&longs;ity to <lb/>be greater than F A, becau&longs;e the Rectangle of B F multiplied into <lb/>F A is le&longs;s than the Square B D: and the Defect is as much as the <lb/>Square of D F, as <emph type="italics"/>Euclid<emph.end type="italics"/> demon&longs;trates in a Propo&longs;ition of his <pb xlink:href="069/01/242.jpg" pagenum="239"/>Second Book. </s> <s>You mu&longs;t al&longs;o know, that the point F which divi­<lb/>deth the Tangent E B in the middle, will many other times fall <lb/>above the point A, and once al&longs;o in the &longs;aid A: In which ca&longs;es it is <lb/>evident of it &longs;elf, that the third proportional to the half of the Tan­<lb/>gent, and to B I (which giveth the Sublimity) is all above A. </s> <s>But <lb/>the Author hath taken a Ca&longs;e in which it was not manife&longs;t that the <lb/>&longs;aid third Proportional is alwaies greater than F A: and which <lb/>therefore being &longs;et off above the point F pa&longs;&longs;eth beyond the Paral­<lb/>lel A G. </s> <s>Now let us proceed.</s></p><p type="main"> <s><emph type="italics"/>It will not be unprofitable if by help of this Table we compo&longs;e ano­<lb/>ther, &longs;hewing the Altitudes of the &longs;ame Semiparabola's of Projects of <lb/>the &longs;ame<emph.end type="italics"/> Impetus. <emph type="italics"/>And the Con&longs;truction of it is in this manner.<emph.end type="italics"/></s></p><p type="head"> <s>PROBL. VI. PROP. XIII.</s></p><p type="main"> <s>From the given Amplitudes of Semiparabola's in <lb/>the following Table &longs;et down, keeping the <lb/>common <emph type="italics"/>Impeius<emph.end type="italics"/> with which every one of <lb/>them is de&longs;cribed, to compute the Altitudes of <lb/>each &longs;everal Semiparabola.</s></p><p type="main"> <s><emph type="italics"/>Let the Amplitude given be B C, and of the<emph.end type="italics"/> Impetus, <emph type="italics"/>which is <lb/>&longs;uppo&longs;ed to be alwaies the &longs;ame, let the Mea&longs;ure be O B, to wit, <lb/>the Aggregate of the Altitude and Sublimity. </s> <s>The &longs;aid Altitude <lb/>is required to be found and di&longs;tingui&longs;hed. </s> <s>Which &longs;hall then be done when <lb/>B O is &longs;o divided as that the Rectangle contained under its parts is <lb/>equal to the Square of half the Amplitude B C. </s> <s>Let that &longs;ame divi­<lb/>&longs;ion fall in F; and let both O B and B C be cut in the mid&longs;t at D and I.<emph.end type="italics"/><lb/><figure id="id.069.01.242.1.jpg" xlink:href="069/01/242/1.jpg"/><lb/><emph type="italics"/>The Square I B, therefore, is equal to the <lb/>Rectangle B F O: And the Square D O is <lb/>equal to the &longs;ame Rectangle together with the <lb/>Square F D. </s> <s>If therefore from the Square <lb/>D O we deduct the Square B I, which is equal <lb/>to the Rectangle B F O, there &longs;hall remain <lb/>the Square F D; to who&longs;e Side D F, B D be­<lb/>ing added it &longs;hall give the de&longs;ired Altitude <lb/>Altitude B F. </s> <s>And it is thus compounded<emph.end type="italics"/><lb/>ex datis. <emph type="italics"/>From half of the Square B O known <lb/>&longs;ub&longs;tract the Square B I al&longs;o known, of the remainder take the Square <lb/>Root, to which add D B known; and you &longs;hall have the Altitude &longs;ought <lb/>B F. </s> <s>For example. </s> <s>The Altitude of the Parabola de&longs;cribed at the <lb/>Elevation of 55 degrees is to be found. </s> <s>The Amplitude, by the follow­<lb/>ing Table is 9396, its half is 4698, the Square of that is 22071204,<emph.end type="italics"/><pb xlink:href="069/01/243.jpg" pagenum="240"/><emph type="italics"/>this &longs;ub&longs;tracted from the Square of the half B O, which is alwaies <lb/>the &longs;ame, to wit, 2500000, the remainder is 2928796, who&longs;e Square <lb/>Root is 1710 very near, this added to the half of B O, to wit, 5000, <lb/>gives 67101, and &longs;o much is the Altitude B F. </s> <s>It will not be unprofi­<lb/>table, to give the Third Table, containing the Altitudes and Sublimi­<lb/>ties of Semiparabola's, who&longs;e Amplitude &longs;hall be alwaies the &longs;ame.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>This I would very gladly &longs;ee &longs;ince by it I may come to <lb/>know the Difference of the <emph type="italics"/>Impetus's,<emph.end type="italics"/> and of the Forces that are <lb/>required for carrying the Project to the &longs;ame Di&longs;tance with Ranges <lb/>which are called at Random: which Difference I believe is very <lb/>great according to the different Elevations [<emph type="italics"/>or Mountures:<emph.end type="italics"/>] &longs;o that <lb/>if, for example, one would at the Elevation of 3 or 4 degrees, or of <lb/>87 or 88 make the Ball to fall where it did, being &longs;hot at the Ele­<lb/>vation of <emph type="italics"/>gr.<emph.end type="italics"/> 45. (where, as hath been &longs;hewn, the lea&longs;t <emph type="italics"/>Impetus<emph.end type="italics"/> is <lb/>required) I believe that it would require a very much greater <lb/>Force.</s></p><p type="main"> <s>SALV. </s> <s>You are in the right: and you will find that to do the <lb/>full execution in all the Elevations it is requi&longs;ite to make great Pro­<lb/>gre&longs;&longs;ions towards an infinite <emph type="italics"/>Impetus.<emph.end type="italics"/> Now let us &longs;ee the Con&longs;tru­<lb/>ction of the Table.<pb xlink:href="069/01/244.jpg" pagenum="241"/><arrow.to.target n="table75"></arrow.to.target><lb/><arrow.to.target n="table76"></arrow.to.target><lb/><arrow.to.target n="table77"></arrow.to.target></s></p><pb xlink:href="069/01/245.jpg" pagenum="242"/><table><table.target id="table75"></table.target><row><cell>Degrees of Elevation.</cell><cell></cell><cell></cell></row><row><cell>The Amplitudes of the Semipara-bola's, de&longs;cribed with the &longs;ame <emph type="italics"/>Impetus.<emph.end type="italics"/></cell><cell></cell><cell></cell></row><row><cell>Gr.</cell><cell></cell><cell>Gr.</cell></row><row><cell>45</cell><cell>10000</cell><cell></cell></row><row><cell>46</cell><cell>9994</cell><cell>44</cell></row><row><cell>47</cell><cell>9976</cell><cell>43</cell></row><row><cell>48</cell><cell>9945</cell><cell>42</cell></row><row><cell>49</cell><cell>9902</cell><cell>41</cell></row><row><cell>50</cell><cell>9848</cell><cell>40</cell></row><row><cell>51</cell><cell>9782</cell><cell>39</cell></row><row><cell>52</cell><cell>9704</cell><cell>38</cell></row><row><cell>53</cell><cell>9612</cell><cell>37</cell></row><row><cell>54</cell><cell>9511</cell><cell>36</cell></row><row><cell>55</cell><cell>9396</cell><cell>35</cell></row><row><cell>56</cell><cell>9272</cell><cell>34</cell></row><row><cell>57</cell><cell>9136</cell><cell>33</cell></row><row><cell>58</cell><cell>8989</cell><cell>32</cell></row><row><cell>59</cell><cell>8829</cell><cell>31</cell></row><row><cell>60</cell><cell>8659</cell><cell>30</cell></row><row><cell>61</cell><cell>8481</cell><cell>29</cell></row><row><cell>62</cell><cell>8290</cell><cell>28</cell></row><row><cell>63</cell><cell>8090</cell><cell>27</cell></row><row><cell>64</cell><cell>7880</cell><cell>26</cell></row><row><cell>65</cell><cell>7660</cell><cell>25</cell></row><row><cell>66</cell><cell>7431</cell><cell>24</cell></row><row><cell>67</cell><cell>7191</cell><cell>23</cell></row><row><cell>68</cell><cell>6944</cell><cell>22</cell></row><row><cell>69</cell><cell>6692</cell><cell>21</cell></row><row><cell>70</cell><cell>6428</cell><cell>20</cell></row><row><cell>71</cell><cell>6157</cell><cell>19</cell></row><row><cell>72</cell><cell>5878</cell><cell>18</cell></row><row><cell>73</cell><cell>5592</cell><cell>17</cell></row><row><cell>74</cell><cell>5300</cell><cell>16</cell></row><row><cell>75</cell><cell>5000</cell><cell>15</cell></row><row><cell>76</cell><cell>4694</cell><cell>14</cell></row><row><cell>77</cell><cell>4383</cell><cell>13</cell></row><row><cell>78</cell><cell>4067</cell><cell>12</cell></row><row><cell>79</cell><cell>3746</cell><cell>11</cell></row><row><cell>80</cell><cell>3420</cell><cell>10</cell></row><row><cell>81</cell><cell>3090</cell><cell>9</cell></row><row><cell>82</cell><cell>2756</cell><cell>8</cell></row><row><cell>83</cell><cell>2419</cell><cell>7</cell></row><row><cell>84</cell><cell>2079</cell><cell>6</cell></row><row><cell>85</cell><cell>1736</cell><cell>5</cell></row><row><cell>86</cell><cell>1391</cell><cell>4</cell></row><row><cell>87</cell><cell>1044</cell><cell>3</cell></row><row><cell>88</cell><cell>698</cell><cell>2</cell></row><row><cell>89</cell><cell>349</cell><cell>1</cell></row></table><table><table.target id="table76"></table.target><row><cell>Degrees of Elevation.</cell><cell></cell><cell></cell><cell></cell></row><row><cell>The Altitudes of the Se-miparabola's, who&longs;e <emph type="italics"/>Impetus<emph.end type="italics"/> is the &longs;ame.</cell><cell></cell><cell></cell><cell></cell></row><row><cell>Gr.</cell><cell></cell><cell>Gr.</cell><cell></cell></row><row><cell>1</cell><cell>3</cell><cell>46</cell><cell>5173</cell></row><row><cell>2</cell><cell>13</cell><cell>47</cell><cell>5346</cell></row><row><cell>3</cell><cell>28</cell><cell>48</cell><cell>5523</cell></row><row><cell>4</cell><cell>50</cell><cell>49</cell><cell>5698</cell></row><row><cell>5</cell><cell>76</cell><cell>50</cell><cell>5868</cell></row><row><cell>6</cell><cell>108</cell><cell>51</cell><cell>6038</cell></row><row><cell>7</cell><cell>150</cell><cell>52</cell><cell>6207</cell></row><row><cell>8</cell><cell>194</cell><cell>53</cell><cell>6379</cell></row><row><cell>9</cell><cell>245</cell><cell>54</cell><cell>6546</cell></row><row><cell>10</cell><cell>302</cell><cell>55</cell><cell>6710</cell></row><row><cell>17</cell><cell>365</cell><cell>56</cell><cell>6873</cell></row><row><cell>12</cell><cell>432</cell><cell>57</cell><cell>7033</cell></row><row><cell>13</cell><cell>506</cell><cell>58</cell><cell>7190</cell></row><row><cell>14</cell><cell>585</cell><cell>59</cell><cell>7348</cell></row><row><cell>15</cell><cell>670</cell><cell>60</cell><cell>7502</cell></row><row><cell>16</cell><cell>760</cell><cell>61</cell><cell>7649</cell></row><row><cell>17</cell><cell>855</cell><cell>62</cell><cell>7796</cell></row><row><cell>18</cell><cell>955</cell><cell>63</cell><cell>7939</cell></row><row><cell>19</cell><cell>1060</cell><cell>64</cell><cell>8078</cell></row><row><cell>20</cell><cell>1170</cell><cell>65</cell><cell>8214</cell></row><row><cell>21</cell><cell>1285</cell><cell>66</cell><cell>8346</cell></row><row><cell>22</cell><cell>1402</cell><cell>67</cell><cell>8474</cell></row><row><cell>23</cell><cell>1527</cell><cell>68</cell><cell>8597</cell></row><row><cell>24</cell><cell>1685</cell><cell>69</cell><cell>8715</cell></row><row><cell>25</cell><cell>1786</cell><cell>70</cell><cell>8830</cell></row><row><cell>26</cell><cell>1922</cell><cell>71</cell><cell>8940</cell></row><row><cell>27</cell><cell>2061</cell><cell>72</cell><cell>9045</cell></row><row><cell>28</cell><cell>2204</cell><cell>73</cell><cell>9144</cell></row><row><cell>29</cell><cell>2351</cell><cell>74</cell><cell>9240</cell></row><row><cell>30</cell><cell>2499</cell><cell>75</cell><cell>9330</cell></row><row><cell>31</cell><cell>2653</cell><cell>76</cell><cell>9415</cell></row><row><cell>32</cell><cell>2810</cell><cell>77</cell><cell>9493</cell></row><row><cell>33</cell><cell>2967</cell><cell>78</cell><cell>9567</cell></row><row><cell>34</cell><cell>3128</cell><cell>79</cell><cell>9636</cell></row><row><cell>35</cell><cell>3289</cell><cell>80</cell><cell>9698</cell></row><row><cell>36</cell><cell>3456</cell><cell>81</cell><cell>9755</cell></row><row><cell>37</cell><cell>3621</cell><cell>82</cell><cell>9806</cell></row><row><cell>38</cell><cell>3793</cell><cell>83</cell><cell>9851</cell></row><row><cell>39</cell><cell>3962</cell><cell>84</cell><cell>9890</cell></row><row><cell>40</cell><cell>4132</cell><cell>85</cell><cell>9924</cell></row><row><cell>41</cell><cell>4302</cell><cell>86</cell><cell>9951</cell></row><row><cell>42</cell><cell>4477</cell><cell>87</cell><cell>9972</cell></row><row><cell>43</cell><cell>4654</cell><cell>88</cell><cell>9987</cell></row><row><cell>44</cell><cell>4827</cell><cell>89</cell><cell>9998</cell></row><row><cell>45</cell><cell>5000</cell><cell>90</cell><cell>10000</cell></row></table><table><table.target id="table77"></table.target><row><cell>A Table containing the Altitudes and Subli-mities of the Semiparabola's, who&longs;e Am-plitudes are the &longs;ame, that is to &longs;ay, of 10000 parts, calculated to each Deg. of Elevation.</cell><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell></row><row><cell>Gr.</cell><cell>Altit.</cell><cell>Sublim.</cell><cell>Gr.</cell><cell>Altit.</cell><cell>Sublim.</cell></row><row><cell>1</cell><cell>87</cell><cell>286533</cell><cell>46</cell><cell>5177</cell><cell>4828</cell></row><row><cell>2</cell><cell>175</cell><cell>142450</cell><cell>47</cell><cell>5363</cell><cell>4662</cell></row><row><cell>3</cell><cell>262</cell><cell>95802</cell><cell>48</cell><cell>5553</cell><cell>4502</cell></row><row><cell>4</cell><cell>349</cell><cell>71531</cell><cell>49</cell><cell>5752</cell><cell>4345</cell></row><row><cell>5</cell><cell>437</cell><cell>57142</cell><cell>50</cell><cell>5959</cell><cell>4106</cell></row><row><cell>6</cell><cell>525</cell><cell>47573</cell><cell>51</cell><cell>6174</cell><cell>4048</cell></row><row><cell>7</cell><cell>614</cell><cell>40716</cell><cell>52</cell><cell>6399</cell><cell>3006</cell></row><row><cell>8</cell><cell>702</cell><cell>35587</cell><cell>53</cell><cell>6635</cell><cell>3765</cell></row><row><cell>9</cell><cell>792</cell><cell>31565</cell><cell>54</cell><cell>6882</cell><cell>3632</cell></row><row><cell>10</cell><cell>881</cell><cell>28367</cell><cell>55</cell><cell>7141</cell><cell>3500</cell></row><row><cell>11</cell><cell>972</cell><cell>25720</cell><cell>56</cell><cell>7413</cell><cell>3372</cell></row><row><cell>12</cell><cell>1063</cell><cell>23518</cell><cell>57</cell><cell>7699</cell><cell>3247</cell></row><row><cell>13</cell><cell>1154</cell><cell>21701</cell><cell>58</cell><cell>8002</cell><cell>3123</cell></row><row><cell>14</cell><cell>1246</cell><cell>20056</cell><cell>59</cell><cell>8332</cell><cell>3004</cell></row><row><cell>11</cell><cell>1339</cell><cell>18663</cell><cell>60</cell><cell>8600</cell><cell>2887</cell></row><row><cell>16</cell><cell>1434</cell><cell>17405</cell><cell>61</cell><cell>9020</cell><cell>2771</cell></row><row><cell>17</cell><cell>1529</cell><cell>16355</cell><cell>62</cell><cell>9403</cell><cell>2658</cell></row><row><cell>18</cell><cell>1624</cell><cell>15389</cell><cell>63</cell><cell>9813</cell><cell>2547</cell></row><row><cell>19</cell><cell>1722</cell><cell>14522</cell><cell>64</cell><cell>10251</cell><cell>2438</cell></row><row><cell>20</cell><cell>1820</cell><cell>13736</cell><cell>65</cell><cell>10722</cell><cell>2331</cell></row><row><cell>21</cell><cell>1919</cell><cell>13024</cell><cell>66</cell><cell>11220</cell><cell>2226</cell></row><row><cell>22</cell><cell>2020</cell><cell>12376</cell><cell>67</cell><cell>11779</cell><cell>2122</cell></row><row><cell>23</cell><cell>2123</cell><cell>11778</cell><cell>68</cell><cell>12375</cell><cell>2020</cell></row><row><cell>24</cell><cell>2226</cell><cell>11230</cell><cell>69</cell><cell>13025</cell><cell>1919</cell></row><row><cell>25</cell><cell>2332</cell><cell>10722</cell><cell>70</cell><cell>13237</cell><cell>1819</cell></row><row><cell>26</cell><cell>2439</cell><cell>10253</cell><cell>71</cell><cell>14521</cell><cell>1721</cell></row><row><cell>27</cell><cell>2547</cell><cell>9814</cell><cell>72</cell><cell>15388</cell><cell>1624</cell></row><row><cell>28</cell><cell>2658</cell><cell>9404</cell><cell>73</cell><cell>16354</cell><cell>1528</cell></row><row><cell>29</cell><cell>2772</cell><cell>9020</cell><cell>74</cell><cell>17437</cell><cell>1413</cell></row><row><cell>30</cell><cell>2887</cell><cell>8659</cell><cell>75</cell><cell>18660</cell><cell>1339</cell></row><row><cell>31</cell><cell>3008</cell><cell>8336</cell><cell>76</cell><cell>20054</cell><cell>1246</cell></row><row><cell>32</cell><cell>3124</cell><cell>8001</cell><cell>77</cell><cell>21657</cell><cell>1154</cell></row><row><cell>33</cell><cell>3247</cell><cell>7699</cell><cell>78</cell><cell>23523</cell><cell>1062</cell></row><row><cell>34</cell><cell>3373</cell><cell>7413</cell><cell>79</cell><cell>25723</cell><cell>972</cell></row><row><cell>35</cell><cell>3501</cell><cell>7141</cell><cell>80</cell><cell>28356</cell><cell>881</cell></row><row><cell>36</cell><cell>3633</cell><cell>6882</cell><cell>81</cell><cell>31560</cell><cell>792</cell></row><row><cell>37</cell><cell>3768</cell><cell>6635</cell><cell>82</cell><cell>35577</cell><cell>702</cell></row><row><cell>38</cell><cell>3906</cell><cell>6395</cell><cell>83</cell><cell>40222</cell><cell>613</cell></row><row><cell>39</cell><cell>4049</cell><cell>6174</cell><cell>84</cell><cell>47572</cell><cell>525</cell></row><row><cell>40</cell><cell>4196</cell><cell>5959</cell><cell>85</cell><cell>57150</cell><cell>437</cell></row><row><cell>41</cell><cell>4246</cell><cell>5752</cell><cell>86</cell><cell>71503</cell><cell>349</cell></row><row><cell>42</cell><cell>4502</cell><cell>5553</cell><cell>87</cell><cell>95405</cell><cell>262</cell></row><row><cell>43</cell><cell>4662</cell><cell>5362</cell><cell>88</cell><cell>143181</cell><cell>174</cell></row><row><cell>44</cell><cell>4828</cell><cell>5177</cell><cell>89</cell><cell>286499</cell><cell>87</cell></row><row><cell>45</cell><cell>5000</cell><cell>5000</cell><cell>90</cell><cell>Infinite</cell><cell></cell></row></table><p type="head"> <s>PROBL. VII. PROP. XIV.</s></p><p type="main"> <s>To find the Altitudes and Sublimities of Semipa­<lb/>rabola's who&longs;e Amplitudes &longs;hall be equal for <lb/>each degree of Elevation.</s></p><p type="main"> <s><emph type="italics"/>This we &longs;hall ea&longs;ily do. </s> <s>For &longs;uppo&longs;ing the Amplitude of the Semi­<lb/>par abola to be of 10000 parts, the half of the Tangent of each <lb/>degree of Elevation &longs;hews the Altitude. </s> <s>As for example, of the <lb/>Semiparabola who&longs;e Elevation is 30 degrees, and Amplitude, as is <lb/>&longs;uppo&longs;ed, 10000 parts, the Altitude &longs;hall be 2887, for &longs;o much, very <lb/>near, is the half of the Tangent. </s> <s>And having found the Altitude the <lb/>Sublimity is to be known in this manner. </s> <s>For a&longs;much as it hath been <lb/>demon&longs;trated that the half of the Amplitude of a Semiparabola is the <lb/>Mean proportional betwixt the Altitude and Sublimity, and the Alti­<lb/>tude being already found, and the half of the Amplitude being alwaies <lb/>the &longs;ame, to wit, 5000 parts, if we &longs;hall divide the Square thereof by <lb/>the Altitude found, the de&longs;ired Sublimity &longs;hall come forth. </s> <s>As in the <lb/>Example: The Altitude found was 2887; The Square of the 5000 <lb/>parts is 25000000; which being divided by 2887, giveth 8659, ve­<lb/>ry near, for the Sublimity &longs;ought.<emph.end type="italics"/></s></p><p type="main"> <s>SALV. </s> <s>Now here we &longs;ee, in the &longs;ir&longs;t place, that the Conje­<lb/>cture is very true which was mentioned afore, that in different <lb/>Elevations the farther one goeth from the middlemo&longs;t, whether it <lb/>be in the Higher, or in the Lower, &longs;o much greater <emph type="italics"/>Impetus<emph.end type="italics"/> and Vio­<lb/>lence is required to carry the Project to the &longs;ame Di&longs;tance. </s> <s>For the <lb/><emph type="italics"/>Impetus<emph.end type="italics"/> lying in the mixture of the two Motions, Equable, Hori­<lb/>zontal, and Perpendicular Naturally-Accelerate, of which <emph type="italics"/>Impetus<emph.end type="italics"/><lb/>the Aggregate of the Altitude and Sublimity is the Mea&longs;ure, we do <lb/>&longs;ee in the propounded Table that that &longs;ame Aggregate is lea&longs;t in <lb/>the Elevation of <emph type="italics"/>gr.<emph.end type="italics"/> 45, in which the Altitude and Sublimity are <lb/>equal, <emph type="italics"/>&longs;cilicet<emph.end type="italics"/> each 5000, and their Aggregate 10000. But if we <lb/>&longs;hould look on any greater Elevation, as, for example, of <emph type="italics"/>gr.<emph.end type="italics"/> 50, we <lb/>&longs;hould &longs;ind the Altitude to be 5959, and the Sublimity 4196, which <lb/>added together make 10155. And &longs;o much al&longs;o we &longs;hould find the <lb/><emph type="italics"/>Impetus<emph.end type="italics"/> of <emph type="italics"/>gr.<emph.end type="italics"/> 40 to be, this and that Elevation being equally re­<lb/>mote from the middlemo&longs;t. </s> <s>Where we are to note, in the &longs;econd <lb/>place, that it is true, That equal <emph type="italics"/>Impetus's<emph.end type="italics"/> are &longs;ought by two, and <lb/>two in the Elevations equidi&longs;tant from the middlemo&longs;t, with this <lb/>pretty variation over and above that the Altitudes and the Subli­<lb/>mities of the ^{*} &longs;uperiour Elevations an&longs;wer alternally to the Sub­<lb/><arrow.to.target n="marg1103"></arrow.to.target><lb/>limities and Altitudes of the Inferiour: &longs;o that whereas in the <pb xlink:href="069/01/246.jpg" pagenum="243"/>example propo&longs;ed, in the Elevation of <emph type="italics"/>gr.<emph.end type="italics"/> 50. the Altitude is 5959 <lb/>and the Sublimity 4196, in the Elevation of <emph type="italics"/>gr.<emph.end type="italics"/> 40. it falls out on <lb/>the contrary that the Altitude is 4196, and the Sublimity 5959: <lb/>And the &longs;ame happens in all others without any difference; &longs;ave <lb/>only that for the avoyding of tediou&longs;ne&longs;s in Calculations we have <lb/>kept no account of &longs;ome fractions, which in &longs;o great &longs;ums are of no <lb/>value, but may without any prejudice be omitted.</s></p><p type="margin"> <s><margin.target id="marg1103"></margin.target>* <emph type="italics"/>i.e.<emph.end type="italics"/> Tho&longs;e above <lb/>45 deg.</s></p><p type="main"> <s>SAGR. </s> <s>I am ob&longs;erving that of the two <emph type="italics"/>Impetus's<emph.end type="italics"/> Horizontal and <lb/>Perpendicular in Projections, the more Sublime they are, they need <lb/>&longs;o much the le&longs;s of the Horizontal, and the more of the Perpendi­<lb/>cular. </s> <s>Moreover in tho&longs;e of &longs;mall Elevation, great mu&longs;t be the <lb/>Force of the Horizontal <emph type="italics"/>Impetus,<emph.end type="italics"/> which is to carry the Project in a <lb/>little Altitude. </s> <s>But although I comprehend very well that in the <lb/>Total Elevation of <emph type="italics"/>gr.<emph.end type="italics"/> 90, all the force in the world &longs;ufficeth not <lb/>to drive the Project one &longs;ingle Inch from the Perpendicular, but <lb/>that it mu&longs;t of nece&longs;&longs;ity fall in the &longs;ame place whence it was expel­<lb/>led; yet dare I not with the like certainty affirm that likewi&longs;e in the <lb/>nullity of Elevation, that is in the Horizontal Line, the Project <lb/>cannot by any Force le&longs;s than infinite, be driven to any di­<lb/>&longs;tance: So, as that, for example, a Culverin it &longs;elf &longs;hould not be <lb/>able to carry a Ball of Iron Horizontally, or, as they &longs;ay, at Point <lb/>blank, that is at no point, which is when it hath no Elevation. </s> <s>I <lb/>&longs;ay, in this ca&longs;e I &longs;tand in &longs;ome doubt; and that I do not re&longs;olute­<lb/>ly deny the thing, the rea&longs;on depends on another Accident which <lb/>&longs;eems no le&longs;s &longs;trange, and yet I have a very nece&longs;&longs;ary Demon&longs;trati­<lb/>on for it. </s> <s>And the Accident is this, the Impo&longs;&longs;ibility of di&longs;tending <lb/>a Rope, &longs;o, as that it may be &longs;tretched right out, and parallel to the <lb/>Horizon, but that it alwaies &longs;wayes and bendeth, nor is there any <lb/>Force that can &longs;tretch it otherwi&longs;e.</s></p><p type="main"> <s>SALV. </s> <s>So then, <emph type="italics"/>Sagredus,<emph.end type="italics"/> your wonder cea&longs;eth in this ca&longs;e of <lb/>the Rope becau&longs;e you have the Demon&longs;tration of it. </s> <s>But if we <lb/>&longs;hall well con&longs;ider the matter, it may be we &longs;hall find &longs;ome corre­<lb/>&longs;pondence between the Accident of the Project and this of the <lb/>Rope. </s> <s>The Curvity of the Line of the Horizontal Projection &longs;eem­<lb/>eth to be derived from two Forces, of which one, (which is that of <lb/>the Projicient) driveth it Horizontally, and the other, (which <lb/>is the Gravity of the Project) draweth it downwards Perpendicu­<lb/>larly. </s> <s>Now &longs;o in the &longs;tretching of the Rope, there are the Forces <lb/>of tho&longs;e that pull it Horizontally, and there is al&longs;o the weight of <lb/>the Rope it &longs;elf, which naturally inclineth it downwards. </s> <s>The&longs;e <lb/>two effects are very much alike in the generation of them. </s> <s>And if <lb/>you allow the weight of the Rope &longs;o much &longs;trength and power as to <lb/>be able to oppo&longs;e and overcome any whatever Immen&longs;e Force, that <lb/>would di&longs;tend it right out, why will you deny the like to the weight <lb/>of the Bullet? </s> <s>But be&longs;ides, I &longs;hall tell you, and at once procure your <pb xlink:href="069/01/247.jpg" pagenum="244"/>wonder, and delight, that the Rope thus tentered, and &longs;tretcht little <lb/>or much, doth &longs;hape it &longs;elf into Lines that come very near to Para­<lb/>bolical, and the re&longs;emblance is &longs;o great, that if you draw a Para­<lb/>bolical Line upon a plain Superficies that is erect unto the Horizon, <lb/>and holding it rever&longs;ed, that is with the Vertex downwards and <lb/>with the Ba&longs;e Parallel to the Horizon, you cau&longs;e a Chain to be held <lb/>pendent, and &longs;u&longs;tained at the extreams of the Ba&longs;e of the De&longs;cribed <lb/>Parabola, you &longs;hall &longs;ee the &longs;aid Chain, as you &longs;laken it more or le&longs;s, <lb/>to incurvate and apply it &longs;elf to the &longs;ame Parabola, and this &longs;ame <lb/>Application &longs;hall be &longs;o much the more exact, when the de&longs;cribed <lb/>Parabola is le&longs;s curved, that is more di&longs;tended: So that in Parabola's <lb/>de&longs;cribed with Elevations under <emph type="italics"/>gr.<emph.end type="italics"/> 45, the Chain an&longs;wereth the <lb/>Parabola almo&longs;t to an hair.</s></p><p type="main"> <s>SAGR. </s> <s>It &longs;eems then that with &longs;uch a Chain wrought into &longs;mall <lb/>Links one might in an in&longs;tant trace out many Parabolick Lines up­<lb/>on a plain Superficies.</s></p><p type="main"> <s>SALV. </s> <s>One might, and that al&longs;o with no &longs;mall commodity, as I <lb/>&longs;hall tell you anon.</s></p><p type="main"> <s>SIMP. </s> <s>But before you pa&longs;s any farther, I al&longs;o would gladly be <lb/>a&longs;certained at lea&longs;t in that Propo&longs;ition of which you &longs;ay there is a <lb/>very nece&longs;&longs;ary Demon&longs;tration, I mean that of the Impo&longs;&longs;ibility of <lb/>di&longs;tending a Rope, by any whatever immen&longs;e Force, right out and <lb/>equidi&longs;tant from the Horizon.</s></p><p type="main"> <s>SAGR. </s> <s>I will &longs;ee if I remember the Demon&longs;tration, for under­<lb/>&longs;tanding of which it is nece&longs;&longs;ary, <emph type="italics"/>Simplicius,<emph.end type="italics"/> that you &longs;uppo&longs;e for <lb/>true, that which in all Mechanick In&longs;truments is confirmed, not on­<lb/>ly by Experience, but al&longs;o by Demon&longs;tration: and this it is, That <lb/>the Velocity of the Mover, though its Force be very &longs;mall, may <lb/>overcome the Re&longs;i&longs;tance, though very great, of a Re&longs;i&longs;ter, which <lb/>mu&longs;t be moved &longs;lowly when ever the Velocity of the Mover hath <lb/>greater proportion to the Tardity of the Re&longs;i&longs;ter, than the Re&longs;i­<lb/>&longs;tance of that which is to be moved hath to the Force of the Mo­<lb/>ver.</s></p><p type="main"> <s>SIMP. </s> <s>This I know very well, and it is demon&longs;trated by <emph type="italics"/>Ari­<lb/>&longs;totle<emph.end type="italics"/> in his Mechanical Que&longs;tions, and is manife&longs;tly &longs;een in the Lea­<lb/>ver and in the Stiliard, in which the Roman which weigheth not <lb/>above 4 pounds, will lift up a weight of 400 in ca&longs;e the di&longs;tance of <lb/>the &longs;aid Roman from the Center on which the Beam turneth be <lb/>more than an hundred times greater than the di&longs;tance of that point <lb/>at which the great weight hangeth from the &longs;ame Center: and this <lb/>cometh to pa&longs;s becau&longs;e in the de&longs;cent which the Roman maketh <lb/>pa&longs;&longs;eth a Space above an hundred times greater than the Space <lb/>which the great weight mounteth in the &longs;ame Time: Which is all <lb/>one as to &longs;ay, that the little Roman moveth with a Velocity above <lb/>an hundred times greater than the Velocity of the great Weight.</s></p><pb xlink:href="069/01/248.jpg" pagenum="245"/><p type="main"> <s>SAGR. </s> <s>You argue very well, and make no &longs;eruple at all of <lb/>granting, that be the Force of the Mover never &longs;o &longs;mall it &longs;hall &longs;u­<lb/>perate any what ever great Re&longs;i&longs;tance at all times when that &longs;hall <lb/>more exceed in Velocity than this doth in Force and Gravity. <lb/></s> <s>Now come we to the ca&longs;e of the Rope. </s> <s>And drawing a &longs;mall <lb/>Scheme be plea&longs;ed to under&longs;tand for once that this Line A B, re&longs;t­<lb/>ing upon the two fixed and &longs;tanding points A and B, to have hang­<lb/>ing at its ends, as you &longs;ee, two immen&longs;e Weights C and D, which <lb/>drawing it with great Force make it to &longs;tand directly di&longs;tended, it <lb/>being a &longs;imple Line without any gravity. </s> <s>And here I proceed, and <lb/>tell you, that if at the mid&longs;t of that which is the point E, you &longs;hould <lb/>hang any never &longs;o little a Weight, as is this H, the Line A B would <lb/>yield, and inclining towards the point F, and by con&longs;equence <lb/>lengthening, will con&longs;train the two great Weights C and D to <lb/>a&longs;cend upwards: which I demon&longs;trate to you in this manner: <lb/>About the two points A and B as Centers I de&longs;cribe two Quadrants <lb/>E F G, and E L M, and in regard that the two Semidiameters AI <lb/>and B L are equal to the two Semidiameters A E and E B, the exce&longs;­<lb/>&longs;es F I and F L &longs;hall be the quantity of the prolongations of the <lb/>parts A F and F B, above A E and E B; and of con&longs;equence &longs;hall <lb/><figure id="id.069.01.248.1.jpg" xlink:href="069/01/248/1.jpg"/><lb/>determine the A&longs;cents <lb/>of the Weights C and <lb/>D, in ca&longs;e that the <lb/>Weight H had had a <lb/>power to de&longs;cend to F: <lb/>which might then be <lb/>in ca&longs;e the Line E F, <lb/>which is the quantity <lb/>of the De&longs;cent of the <lb/>&longs;aid Weight H, had <lb/>greater proportion to <lb/>the Line F I which de­<lb/>termineth the A&longs;cent of <lb/>the two Weights C & <lb/>D, than the pondero­<lb/>&longs;ity of both tho&longs;e Weights hath to the pondero&longs;ity of the Weight <lb/>H. </s> <s>But this will nece&longs;&longs;arily happen, be the pondero&longs;ity of the <lb/>Weights C and D never &longs;o great, and that of H never &longs;o &longs;mall; for <lb/>the exce&longs;s of the Weights C and D above the Weight His not &longs;o <lb/>great, but that the exce&longs;s of the Tangent E F above the part of the <lb/>Secant F I may bear a greater proportion. </s> <s>Which we will prove <lb/>thus: Let there be a Circle who&longs;e Diameter is G A I; and look <lb/>what proportion the pondero&longs;ity of the Weights C and D have to <lb/>the pondero&longs;ity of H, let the Line B O have the &longs;ame proportion to <lb/>another, which let be C, than which let D be le&longs;&longs;er: So that B O <pb xlink:href="069/01/249.jpg" pagenum="246"/>&longs;hall have greater proportion to D, than to C. </s> <s>Unto O B and D <lb/>take a third proportional B E; and as O E is to E B, &longs;o let the Dia­<lb/>meter G I (prolonging it) be to I F: and from the Term F <lb/>draw the Tangent F N. </s> <s>And becau&longs;e it hath been pre&longs;uppo&longs;ed, <lb/>that as O E is to E B, &longs;o is G I to I F: therefore, by Compo&longs;ition, as <lb/>O B is to B E, &longs;o is G F to F I: But betwixt O B and B E the Mean­<lb/>proportional is D; and betwixt G F and F I the Mean-proporti­<lb/>onal is N F: Therefore N F hath the &longs;ame proportion to F I that <lb/>O B hath to D: which proportion is greater than that of the <lb/>Weights C and D to the Weight H. Therefore, the De&longs;cent or <lb/>Velocity of the Weight H having greater proportion to the A&longs;cent <lb/>or Velocity of the Weights C and D, than the pondero&longs;ity of the <lb/>&longs;aid Weights C and D hath to the pondero&longs;ity of the Weight H: <lb/>It is manife&longs;t, that the Weight H &longs;hall de&longs;cend, that is, that the <lb/>Line A B &longs;hall depart from Horizontal Rectitude. </s> <s>And that which <lb/>befalleth the right Line A B deprived of Gravity in ca&longs;e any &longs;mall <lb/>Weight H cometh to be hanged at the &longs;ame in E, happens al&longs;o to <lb/>the &longs;aid Rope A B, &longs;uppo&longs;ed to be of ponderous Matter, without <lb/>the addition of any other Grave Body; for that the Weight of <lb/>the Matter it &longs;elf compounding the &longs;aid Rope AB is &longs;u&longs;pended <lb/>thereat.</s></p><p type="main"> <s>SIMP. </s> <s>You have fully &longs;atisfied me; therefore <emph type="italics"/>Salviatus<emph.end type="italics"/> may ac­<lb/>cording to his promi&longs;e declare unto us, what the Commodity is that <lb/>may be drawn from &longs;uch like Chains, and after that relate unto us <lb/>tho&longs;e Speculations which have been made by our <emph type="italics"/>Accademian<emph.end type="italics"/><lb/>touching the Force of Percu&longs;&longs;ion.</s></p><p type="main"> <s><emph type="italics"/>S<emph.end type="italics"/>ALV. </s> <s>We are for this day &longs;ufficiently employed in the Con­<lb/>templations already delivered, and the Time, which is pretty late, <lb/>would not be enough to carry us through the matters you mention; <lb/>therefore we &longs;hall defer our Conference till &longs;ome more convenient <lb/>time.</s></p><p type="main"> <s>SAGR. </s> <s>I concur with you in opinion, for that by &longs;undry di&longs;­<lb/>cour&longs;es that I have had with the Friends of our <emph type="italics"/>Academick<emph.end type="italics"/> I have <lb/>learnt that this Argument of the Force of Percu&longs;&longs;ion is very ob­<lb/>&longs;cure, nor hath hitherto any one that hath treated thereof penetra­<lb/>ted its intricacies, full of darkne&longs;s, and altogether remote from <lb/>mans fir&longs;t imaginations: and among&longs;t the Conclu&longs;ions that I have <lb/>heard of, one runs in my mind that is very extravagant and odde, <lb/>namely, That the Force of Percu&longs;&longs;ion is Interminate, if not Infi­<lb/>nite. </s> <s>We will therefore attend the lea&longs;ure of <emph type="italics"/>Salviatus.<emph.end type="italics"/> But for <lb/>the pre&longs;ent, tell me what things are tho&longs;e which are written at the <lb/>end of the Treati&longs;e of Projects?</s></p><p type="main"> <s>SALV. </s> <s>The&longs;e are certain Propo&longs;itions touching the Center of <lb/>Gravity of Solids, which our <emph type="italics"/>Academick<emph.end type="italics"/> found out in his youth, <lb/><arrow.to.target n="marg1104"></arrow.to.target><lb/>conceiving that what ^{*} <emph type="italics"/>Frederico Comandino<emph.end type="italics"/> had writ touching the <pb xlink:href="069/01/250.jpg" pagenum="247"/>&longs;ame was not altogether without Imper&longs;ection. </s> <s>He therefore <lb/>thought that with the&longs;e Propo&longs;itions, which here you &longs;ee written, <lb/>he might &longs;upply that which is wanting in the Book of <emph type="italics"/>Comandine<emph.end type="italics"/>; <lb/>and he applyed him&longs;elf to the &longs;ame at the In&longs;tance of the mo&longs;t <lb/>Illu&longs;trious Lord Marque&longs;s <emph type="italics"/>Guid' Vbaldo dal Monte,<emph.end type="italics"/> the mo&longs;t ex­<lb/>cellent Mathematician of his Time, as his &longs;everal Printed Works <lb/>do &longs;peak him; and gave a Copy thereof to that Noble Lord with <lb/>thoughts to have pur&longs;ued the &longs;ame Argument in other Solids not <lb/>mentioned by <emph type="italics"/>Comandine:<emph.end type="italics"/> But he chanced after &longs;ome Time to <lb/>meet with the ^{*} Book of <emph type="italics"/>Signore Luca Valerio,<emph.end type="italics"/> a mo&longs;t famous <lb/><arrow.to.target n="marg1105"></arrow.to.target><lb/>Geometrician, and &longs;aw that he re&longs;olveth all the&longs;e matters with­<lb/>out omi&longs;&longs;ion of any thing, he proceeded no farther, although his <lb/>Agre&longs;&longs;ions were by methods very different from the&longs;e of <emph type="italics"/>Signore <lb/>Valerio.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1104"></margin.target>* <emph type="italics"/>Fredericus Co­<lb/>mandinus.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1105"></margin.target>* <emph type="italics"/>De.<emph.end type="italics"/></s></p><p type="main"> <s>SAGR. </s> <s>It would be a favour, therefore, if, for this time, which <lb/>interpo&longs;eth between this and our next Meeting, you would plea&longs;e <lb/>to leave the Book in my hands: for I &longs;hall all the while be read­<lb/>ing and &longs;tudying the Propo&longs;itions that are con&longs;equently therein <lb/>writ.</s></p><p type="main"> <s>SALV. </s> <s>I &longs;hall very willingly obey your Command; and hope <lb/>that you will take plea&longs;ure in the&longs;e Propo&longs;itions.</s></p></chap><chap><pb xlink:href="069/01/251.jpg" pagenum="248"/><p type="head"> <s>AN <lb/>APPENDIX, <lb/>In which is contained certain <lb/>THE OREMS and their DEMONSTRATIONS: <lb/>Formerly written by the &longs;ame Author, touching the <lb/><emph type="italics"/>CENTER<emph.end type="italics"/> of <emph type="italics"/>GRAVITY,<emph.end type="italics"/> of <lb/>SOLIDS.</s></p><p type="head"> <s>POSTVLATVM.</s></p><p type="main"> <s><emph type="italics"/>We pre&longs;uppo&longs;e equall Weights to be alike di&longs;po­<lb/>&longs;ed in &longs;ever all Ballances, if the Center of Gra­<lb/>vity of &longs;ome of tho&longs;e Compounds &longs;hall divide the Ballance <lb/>according to &longs;ome proportion, and the Ballance &longs;hall <lb/>al&longs;o divide their Center of Gravity according to the <lb/>&longs;ame proportion.<emph.end type="italics"/></s></p><p type="head"> <s>LEMMA.</s></p><p type="main"> <s><emph type="italics"/>Let the line A B be cut in two equall parts in C, <lb/>who&longs;e half A C let be divided in E, &longs;o that as B E is to <lb/>E A, &longs;o may A E be to E C. </s> <s>I &longs;ay that B E is double<emph.end type="italics"/><lb/><figure id="id.069.01.251.1.jpg" xlink:href="069/01/251/1.jpg"/><lb/><emph type="italics"/>to E A. </s> <s>For as B E is to E <lb/>A, &longs;o is E A to E C: there­<lb/>fore by Compo&longs;ition and by Permutation of Proportion, as <lb/>B A is to A C, &longs;o is A E to E C: But as A E is to E C, <lb/>that is, B A to A C, &longs;o is B E to E A: Wherefore B <lb/>E is double to E A.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>This &longs;uppo&longs;ed, we will Demon&longs;trate, That,<emph.end type="italics"/></s></p><pb xlink:href="069/01/252.jpg" pagenum="249"/><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>If certain Magnitudes at any Rate equally exceed­<lb/>ing one another, and who&longs;e exce&longs;s is equal to <lb/>the lea&longs;t of them, be &longs;o di&longs;po&longs;ed in the Balance, <lb/>as that they hang at equal di&longs;tances, to divide <lb/>the Center of Gravity of the whole Balance <lb/>&longs;o, that the part towards the le&longs;&longs;er Magnitudes <lb/>be double to the remainder.</s></p><p type="main"> <s><emph type="italics"/>In the ^{*} Ballance A B, therefore, let there be &longs;u&longs;pended at equal di-<emph.end type="italics"/><lb/><arrow.to.target n="marg1106"></arrow.to.target><lb/><emph type="italics"/>&longs;tances any number of Magnitudes, as hath been &longs;aid, F, G, H, K, <lb/>N; of which let the lea&longs;t be N, and let the points of the Su&longs;pen&longs;ions <lb/>be A, C, D, E, B, and let the Center of Gravity of all the Magnitudes <lb/>&longs;o di&longs;po&longs;ed be X. </s> <s>It is to be proved that the part of the Ballance B X <lb/>towards the le&longs;&longs;er Magnitudes is double to the remaining part X A.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1106"></margin.target>* Or Beam.</s></p><p type="main"> <s><emph type="italics"/>Let the Ballance be divided in two equal parts in D, for it mu&longs;t ei­<lb/>ther fall in &longs;ome point of the Su&longs;pen&longs;ions, or el&longs;e in the middle point be­<lb/>tween two of the points of the Su&longs;pen&longs;ions: and let the remaining di­<lb/>&longs;tances of the Su&longs;pen&longs;ions which fall between A and D, be all divided <lb/>into halves by the Points M and I; and let all the Magnitudes be divi-<emph.end type="italics"/><lb/><figure id="id.069.01.252.1.jpg" xlink:href="069/01/252/1.jpg"/><lb/><emph type="italics"/>ded into parts equal to <lb/>N: Now the parts of F <lb/>&longs;hall be &longs;o many in num­<lb/>ber, as tho&longs;e Magnitudes <lb/>be which are &longs;u&longs;pended <lb/>at the Ballance, and the <lb/>parts of G one fewer, <lb/>and &longs;o of the re&longs;t. </s> <s>Let <lb/>the parts of F therefore be N, O, R, S, T, and let tho&longs;e of G be N, O, <lb/>R, S, tho&longs;e of H al&longs;o N, O, R, then let tho&longs;e of K be N, O: and all the <lb/>Magnitudes in which are N &longs;hall be equal to F; and all the Magnitudes <lb/>in which are O &longs;hall be equal to G; and all the Magnitudes in which <lb/>are R &longs;hall be equal to H; and tho&longs;e in which S &longs;hall be equal to K; and <lb/>the Magnitude T is equal to N. </s> <s>Becau&longs;e therefore all the Magnitudes <lb/>in which are N are equal to one another, they &longs;hall equiponderate in <lb/>the point D, which divideth the Ballance into two equal parts; and for <lb/>the &longs;ame cau&longs;e all the Magnitudes in which are O do equiponderate in <lb/>I; and tho&longs;e in which are R in C; and in which are S in M do equi­<lb/>ponderate; and T is &longs;u&longs;pended in A. </s> <s>Therefore in the Ballance A D at <lb/>the equal di&longs;tances D, I, C, M, A, there are Magnitudes &longs;u&longs;pended ex­<lb/>ceeding one another equally, and who&longs;e exce&longs;s is equal to the lea&longs;t: and <lb/>the greate&longs;t, which is compounded of all the N N hangeth at D, the<emph.end type="italics"/><pb xlink:href="069/01/253.jpg" pagenum="250"/><emph type="italics"/>lea&longs;t which is T hangeth at A; and the re&longs;t are ordinately di&longs;po&longs;ed. <lb/></s> <s>And again there is another Ballance A B in which other Magnitudes <lb/>equal in number and Magnitude to the former are di&longs;po&longs;ed in the &longs;ame <lb/>order. </s> <s>Wherefore the Ballances A B and A D are divided by the Cen­<lb/>ter of all the Magnitudes according to the &longs;ame proportion: But the <lb/>Center of Gravity of the afore&longs;aid Magnitudes is X: Wherefore X <lb/>divideth the Ballances B A and A D according to the &longs;ame proportion; <lb/>&longs;o that as B X is to X A, &longs;o is X A to X D: Wherefore B X is double <lb/>to X A, by the Lemma aforegoing: Which was to be proved.<emph.end type="italics"/></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>If in a Parabolical Conoid Figure be de&longs;cribed, <lb/>and another circum&longs;cribed by Cylinders of <lb/>equal Altitude; and the Axis of the &longs;aid Co­<lb/>noid be divided in &longs;uch proportion that the <lb/>part towards the Vertex be double to that to­<lb/>wards the Ba&longs;e; the Center of Gravity of the <lb/>in&longs;cribed Figure of the Ba&longs;e portion &longs;hall be <lb/>neare&longs;t to the &longs;aid point of divi&longs;ion; and the <lb/>Center of Gravity of the circum&longs;cribed from <lb/>the Ba&longs;e of the Conoid &longs;hall be more remote: <lb/>and the di&longs;tance of either of tho&longs;e Centers <lb/>from that &longs;ame point &longs;hall be equal to the Line <lb/>that is the &longs;ixth part of the Altitude of one of <lb/>the Cylinders of which the Figures are com­<lb/>po&longs;ed.</s></p><p type="main"> <s><emph type="italics"/>Take therefore a Parabolical Conoid, and the Figures that have <lb/>been mentioned: let one of them be in&longs;cribed, the other circum­<lb/>&longs;cribed; and let the Axis of the Conoid, which let be A E, be di­<lb/>vided in N, in &longs;uch proportion as that A N be double to N E. </s> <s>It is to <lb/>be proved that the Center of Gravity of the in&longs;cribed Figure is in the <lb/>Line N E, but the Center of the circum&longs;cribed in the Line A N. </s> <s>Let <lb/>the Plane of the Figures &longs;o di&longs;po&longs;ed be cut through the Axis, and let <lb/>the Section be that of the Parabola B A C: and let the Section of the <lb/>cutting Plane, and of the Ba&longs;e of the Conoid be the Line B C; and <lb/>let the Sections of the Cylinders be the Rectangular Figures; as ap­<lb/>peareth in the de&longs;cription. </s> <s>Fir&longs;t, therefore, the Cylinder of the in&longs;cri­<lb/>bed who&longs;e Axis is D E, hath the &longs;ame proportion to the Cylinder who&longs;e <lb/>Axis is D Y, as the Quadrate I D hath to the Quadrate S Y; that is, <lb/>as D A hath to A Y: and the Cylinder who&longs;e Axis is D Y is<emph.end type="italics"/> potentia <pb xlink:href="069/01/254.jpg" pagenum="251"/><emph type="italics"/>to the Cylinder Y Z as S Y to R Z, that is, as Y A to A Z: and, by the <lb/>&longs;ame rea&longs;on, the Cylinder who&longs;e Axis is Z Y is to that who&longs;e Axis is <lb/>Z V, as Z A is to A V. </s> <s>The &longs;aid Cylinders, therefore, are to one ano­<lb/>ther as the Lines D A, A Y; Z A, A V: But the&longs;e are equally exceed­<lb/>ing to one another, and the exce&longs;s is equal to the lea&longs;t, &longs;o that A Z is <lb/>double to A V; and A Y is triple the<emph.end type="italics"/><lb/><figure id="id.069.01.254.1.jpg" xlink:href="069/01/254/1.jpg"/><lb/><emph type="italics"/>&longs;ame; and D A Quadruple. </s> <s>Tho&longs;e <lb/>Cylinders, therefore, are certain Mag­<lb/>nitudes in order equally exceeding one <lb/>another, who&longs;e exce&longs;s is equal to the <lb/>lea&longs;t of them, and is the Line X M, <lb/>in which they are &longs;u&longs;pended at equal <lb/>di&longs;tances (for that each of the Cy­<lb/>linders hath its Center of Gravity in <lb/>the mia&longs;t of the Axis.) Wherefore, <lb/>by what hath been above demon&longs;tra­<lb/>ted, the Center of Gravity of the Mag­<lb/>nitude compounded of them all divi­<lb/>deth the Line X M &longs;o, that the part <lb/>towards X is double to the re&longs;t. </s> <s>Divide it, therefore, and, let X<emph.end type="italics"/> <foreign lang="greek">a</foreign> <emph type="italics"/>be <lb/>double<emph.end type="italics"/> <foreign lang="greek">a</foreign> <emph type="italics"/>M: therefore is<emph.end type="italics"/> <foreign lang="greek">a</foreign> <emph type="italics"/>the Center of Gravity of the in&longs;cribed Fi­<lb/>gure. </s> <s>Divide A V in two equal parts in<emph.end type="italics"/> <foreign lang="greek">e</foreign>: <foreign lang="greek">e</foreign> <emph type="italics"/>X &longs;hall be double to <lb/>M E: But X<emph.end type="italics"/> <foreign lang="greek">a</foreign> <emph type="italics"/>is double to<emph.end type="italics"/> <foreign lang="greek">a</foreign> <emph type="italics"/>M: Wherefore<emph.end type="italics"/> <foreign lang="greek">e</foreign> <emph type="italics"/>E &longs;hall be triple E<emph.end type="italics"/> <foreign lang="greek">a.</foreign> <emph type="italics"/>But<emph.end type="italics"/><lb/><foreign lang="greek">a</foreign> <emph type="italics"/>E is triple E N: It is manife&longs;t, therefore, that E N is greater than <lb/>E X; and for that cau&longs;e<emph.end type="italics"/> <foreign lang="greek">a,</foreign> <emph type="italics"/>which is the Center of Gravity of the in­<lb/>&longs;cribed Figure, cometh nearer to the Ba&longs;e of the Conoid than N. </s> <s>And <lb/>becau&longs;e that as A E is to E N, &longs;o is the part taken away<emph.end type="italics"/> <foreign lang="greek">e</foreign> <emph type="italics"/>E to the part <lb/>taken away E<emph.end type="italics"/> <foreign lang="greek">a</foreign>: <emph type="italics"/>and the remaining part &longs;hall be to the remaming part, <lb/>that is, A<emph.end type="italics"/> <foreign lang="greek">e</foreign> <emph type="italics"/>to N<emph.end type="italics"/> <foreign lang="greek">a,</foreign> <emph type="italics"/>as A E to E N. Therefore<emph.end type="italics"/> <foreign lang="greek">a</foreign> <emph type="italics"/>N is the third part of <lb/>A<emph.end type="italics"/> <foreign lang="greek">e,</foreign> <emph type="italics"/>and the &longs;ixt part of A V. </s> <s>And in the &longs;ame manner the Cylinders of <lb/>the circum&longs;cribed Figure may be demon&longs;trated to be equally exceeding <lb/>one another, and the exce&longs;s to me equal to the least; and that they have <lb/>their Centers of Gravity at equal di&longs;tances in the Line<emph.end type="italics"/> <foreign lang="greek">e</foreign> <emph type="italics"/>M. </s> <s>If therefore<emph.end type="italics"/><lb/><foreign lang="greek">e</foreign> <emph type="italics"/>M be divided in<emph.end type="italics"/> <foreign lang="greek">p,</foreign> <emph type="italics"/>&longs;o as that<emph.end type="italics"/> <foreign lang="greek">e p</foreign> <emph type="italics"/>be double to the remaining part<emph.end type="italics"/> <foreign lang="greek">p</foreign> <emph type="italics"/>M;<emph.end type="italics"/><lb/><foreign lang="greek">p</foreign> <emph type="italics"/>&longs;hall be the Center of Gravity of the whole circum&longs;cribed Magnitude. <lb/></s> <s>And &longs;ince<emph.end type="italics"/> <foreign lang="greek">e p</foreign> <emph type="italics"/>is double to<emph.end type="italics"/> <foreign lang="greek">p</foreign> <emph type="italics"/>M; and A<emph.end type="italics"/> <foreign lang="greek">e</foreign> <emph type="italics"/>le&longs;s than double EM: (for <lb/>that they are equal:) the whole A E &longs;hall be le&longs;s than triple E<emph.end type="italics"/> <foreign lang="greek">p</foreign><emph type="italics"/>: Where­<lb/>fore E<emph.end type="italics"/> <foreign lang="greek">p</foreign> <emph type="italics"/>&longs;hall be greater than E N. And, &longs;ince<emph.end type="italics"/> <foreign lang="greek">e</foreign> <emph type="italics"/>M is triple to M<emph.end type="italics"/> <foreign lang="greek">p,</foreign><lb/><emph type="italics"/>and M E with twice<emph.end type="italics"/> <foreign lang="greek">e</foreign> <emph type="italics"/>A is likewi&longs;e triple to M E: the whole A E with <lb/>A<emph.end type="italics"/> <foreign lang="greek">e</foreign> <emph type="italics"/>&longs;hall be triple to E<emph.end type="italics"/> <foreign lang="greek">p</foreign><emph type="italics"/>: But A E is triple to E N: Wherefore the <lb/>remaining part A<emph.end type="italics"/> <foreign lang="greek">e</foreign> <emph type="italics"/>&longs;hall be triple to the remaining part<emph.end type="italics"/> <foreign lang="greek">p</foreign> <emph type="italics"/>N. </s> <s>Therefore <lb/>N<emph.end type="italics"/> <foreign lang="greek">p</foreign> <emph type="italics"/>is the &longs;ixth part of A V. </s> <s>And the&longs;e are the things that were to be <lb/>demon&longs;trated.<emph.end type="italics"/></s></p><pb xlink:href="069/01/255.jpg" pagenum="252"/><p type="head"> <s>COROLLARY.</s></p><p type="main"> <s>Hence it is manife&longs;t, that a Conoid may be in&longs;cribed in a Para­<lb/>bolical Figure, and another circum&longs;cribed, &longs;o, as that the <lb/>Centers of their Gravities may be di&longs;tant from the point N <lb/>le&longs;s than any Line given.</s></p><p type="main"> <s><emph type="italics"/>For if we a&longs;&longs;ume a Line &longs;excuple of the propo&longs;ed Line, and make the <lb/>Axis of the Cylinders, of which the Figures are compounded given <lb/>le&longs;&longs;er than this a&longs;&longs;umed Line, there &longs;hall fall Lines between the Centers <lb/>of Gravities of the&longs;e Figures and the mark N that are le&longs;s than the <lb/>Line propo&longs;ed.<emph.end type="italics"/></s></p><p type="main"> <s>The former Propo&longs;ition another way.</s></p><p type="main"> <s><emph type="italics"/>Let the Axis of the Conoid (which let be C D) be divided in <lb/>O, &longs;o, as that C O be double to O D. </s> <s>It is to be proved that the <lb/>Center of Gravity of the in&longs;cribed Figure is in the Line O D; <lb/>and the Center of the circum&longs;cribed in C O. </s> <s>Let the Plane of the Fi­<lb/>gures be cut through the Axis and C, as hath been &longs;aid. </s> <s>Becau&longs;e there­<lb/>fore the Cylinders S N, T M, V I,<emph.end type="italics"/><lb/><figure id="id.069.01.255.1.jpg" xlink:href="069/01/255/1.jpg"/><lb/><emph type="italics"/>X E are to one another as the Squares <lb/>of the Lines S D, T N, V M, X I; <lb/>and the&longs;e are to one another as the <lb/>Lines N C, C M, C I, C E: but <lb/>the&longs;e do exceed one another equally; <lb/>and the exce&longs;s is equal to the lea&longs;t, to <lb/>wit, C E: And the Cylinder T M is <lb/>equal to the Cylinder Q N; and the <lb/>Cylinder V I equal to P N; and X E <lb/>is equal to L N: Therefore the Cylin­<lb/>ders S N, Q N, P N, and L N do <lb/>equally exceed one another, and the <lb/>exce&longs;s is equal to the lea&longs;t of them, <lb/>namely, to the Cylinder L N. </s> <s>But <lb/>the exce&longs;s of the Cylinder S N, above <lb/>the Cylinder Q N is a Ring who&longs;e <lb/>height is Q T; that is, N D; and <lb/>its breadth S <expan abbr="q.">que</expan> And the exce&longs;s of the Cylinder Q N above P N, is a <lb/>Ring, who&longs;e breadth is Q P. </s> <s>And the exce&longs;s of the Cylinder P N above <lb/>L N is a Ring, who&longs;e breadth is P L. </s> <s>Wherefore the &longs;aid Rings S Q, <lb/>Q P, P L, are equal to another, and to the Cylinder L N. </s> <s>Therefore the <lb/>Ring S T equalleth the Cylinder X E: the Ring Q V, which is double <lb/>to S T, equalleth the Cylinder V I; which likewi&longs;e is double to the<emph.end type="italics"/><pb xlink:href="069/01/256.jpg" pagenum="253"/><emph type="italics"/>Cylinder X E: and for the &longs;ame cau&longs;e the Ring P X is equal to the <lb/>Cylinder T M; and the Cylinder L E &longs;hall be equal to the Cylinder S N. <lb/></s> <s>In the Beam or Ballance, therefore, K F connecting the middle points of <lb/>the Right-lines E I and D N, and cut into equal parts in the points H <lb/>and G, are certain Magnitudes &longs;u&longs;pended, to wit the Cylinders S N, <lb/>T M, V I, X E; and the Center of Gravity of the fir&longs;t Cylinder is K; <lb/>and of the &longs;econd H; of the third G; of the fourth F. </s> <s>And we have <lb/>another Ballance M K, which is the half of the &longs;aid F K, and a like <lb/>number of points di&longs;tributed into equal parts, to wit, M H, H N, N K, <lb/>and on it other Magnitudes, equal in number and bigne&longs;s to tho&longs;e which <lb/>are on the Beam F K, and having the Centers of Gravity in the points <lb/>M, H, N, and K, and di&longs;po&longs;ed in the &longs;ame order. </s> <s>For the Cylinder L E <lb/>hath its Center of Gravity in M; and is equal to the Cylinder S N that <lb/>hath its Center in K: And the Ring P X hath the Center H; and is <lb/>equal to the Cylinder T M, who&longs;e Center is H: And the Ring Q V ha­<lb/>ving the Center N is equal to the V I who&longs;e Center is G: And la&longs;tly, <lb/>the Ring S T having the Center K, is equal to the Cylinder X E who&longs;e <lb/>Center is F. </s> <s>Therefore the Center of Gravity of the &longs;aid Magnitudes <lb/>divideth the Beam in the &longs;ame proportion: But the Center of them is <lb/>one, and therefore &longs;ome point common to both the Beams or Ballance, <lb/>which let be Y. </s> <s>Therefore F Y and Y K &longs;hall be as K Y and Y M. </s> <s>F Y <lb/>therefore is double to Y K: and C E being divided into two equal parts <lb/>in Z, Z F, &longs;hall be double to K D: and for that cau&longs;e Z D triple to D Y: <lb/>But to the Right Line D O C D is triple: Therefore the Right Line <lb/>D O is greater than D Y: And for the like cau&longs;e Y the Center of the <lb/>in&longs;cribed Figure approacheth nearer the Ba&longs;e than the point O. </s> <s>And <lb/>becau&longs;e as C D is to D O, &longs;o is the part taken away Z D to the part ta­<lb/>ken away D Y; the remaining part C Z &longs;hall be to the remaining part <lb/>Y O, as C D is to D O; that is Y O &longs;hall be the third part of C Z; <lb/>that is, the &longs;ixth part of C E. </s> <s>Again we will, by the &longs;ame rea&longs;on, de­<lb/>mon&longs;trate the Cylinders of the circum&longs;cribed Figure to exceed one ano­<lb/>ther equally, and that the exce&longs;s is equal to the lea&longs;t, and that their <lb/>Centers of Gravity are con&longs;tituted in equal di&longs;tances upon the Beam <lb/>K Z: and likewi&longs;e that the Rings equal to tho&longs;e &longs;ame Cylinders are in <lb/>like manner di&longs;po&longs;ed on another Beam K G, the half of the &longs;aid K Z, <lb/>and that therefore the Center of Gravity of the circum&longs;cribed Figure, <lb/>which let be R, &longs;o divideth the Beam, as that Z R is to R K, as K R is to <lb/>R G. </s> <s>Therefore Z R &longs;hall be double to R K: But C Z is equal to the <lb/>Right Line K D, and not double to it. </s> <s>The whole C D &longs;hall be le&longs;&longs;er <lb/>than triple to D R: Wherefore the Right Line D R is greater than D O; <lb/>that is to &longs;ay, the Center of the circum&longs;cribed Figure recedeth from the <lb/>Ba&longs;e more than the point O. </s> <s>And becau&longs;e Z K is triple to K R; and <lb/>K D with twice Z C is triple to K D; the whole C D with C Z &longs;hall be <lb/>triple to D R: But C D is triple to D O: Wherefore the remaining <lb/>part C Z &longs;hall be triple to the remaining part R O; that is, O R<emph.end type="italics"/><pb xlink:href="069/01/257.jpg" pagenum="254"/><emph type="italics"/>is the &longs;ixth part of E C: Which was the Propo&longs;ition.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>This being pre-demon&longs;trated, we will prove that<emph.end type="italics"/></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>The Center of Gravity of the Parabolick <lb/>Conoid doth &longs;o divide the Axis, as that the <lb/>part towards the Vertex is double to the re­<lb/>maining part towards the Ba&longs;e.</s></p><p type="main"> <s><emph type="italics"/>Let there be a Parabolick Conoid who&longs;e Axis let be A B divided in <lb/>N &longs;o as that A N be double to N B. </s> <s>It is to be proved that the Cen­<lb/>ter of Gravity of the Conoid is the point N. </s> <s>For if it be not N, it <lb/>&longs;hall be either above or below it. </s> <s>Fir&longs;t let it be below; and let it be X: <lb/>And &longs;et off upon &longs;ome place by it &longs;elf the Line L O equal to N X; and let <lb/>L O be divided at plea&longs;ure in S: and look what proportion B X and <lb/>O S both together have to O S, and the &longs;ame &longs;hall the Conoid have to <lb/>the Solid R. </s> <s>And in the Conoid let Figures be de&longs;cribed by Cylinders <lb/>having equal Altitudes, &longs;o, as that that which lyeth between the Center <lb/>of Gravity and the point N be le&longs;s than L S: and let the exce&longs;s of the <lb/>Conoid above it be le&longs;s than the Solid R: and that this may be done is <lb/>clear. </s> <s>Take therefore the in&longs;cribed, who&longs;e Center of Gravity let be I: <lb/>now I X &longs;hall be greater than S O: And becau&longs;e that as X B with S O <lb/>is to S O, &longs;o is the Conoid to the Solid R: (and R is greater than the <lb/>exce&longs;s by which the Conoid exceeds the in&longs;cribed Figure:) the proporti­<lb/>on of the Conoid to the &longs;aid exce&longs;s &longs;hall be greater than both B X and <lb/>O S unto S O: And, by Divi&longs;ion, the in&longs;cribed Figure &longs;hall have grea­<lb/>ter proportion to the &longs;aid exce&longs;s than B X to S O: But B X hath to <lb/>X I a proportion yet le&longs;s than to S O: Therefore the in&longs;cribed Figure <lb/>&longs;hall have much greater proportion to the re&longs;t of the proportions than <lb/>B X to X I: Therefore what proportion the in&longs;cribed Figure hath to <lb/>there&longs;t of the portions, the &longs;ame &longs;hall a certain other Line have to X I: <lb/>which &longs;hall nece&longs;&longs;arily be greater than B X: Let it, therefore, be M X. <lb/></s> <s>We have therefore the Center of Gravity of the Conoid X: But the <lb/>Center of Gravity of the Figure in&longs;cribed in it is I: of the re&longs;t of the <lb/>portions by which the Conoid exceeds the in&longs;cribed Figure the Center of <lb/>Gravity &longs;hall be in the Line X M, and in it that point in which it &longs;hall <lb/>be &longs;o terminated, that look what proportion the in&longs;cribed Figure hath <lb/>to the exce&longs;s by which the Conoid exceeds it, the &longs;ame it &longs;hall have to <lb/>X I: But it hath been proved, that this proportion is that which M X <lb/>hath to X I: Therefore M &longs;hall be the Center of Gravity of tho&longs;e pro­<lb/>portions by which the Conoid exceeds the in&longs;cribed Figure: Which <lb/>certainly cannot be. </s> <s>For if along by M a Plane be drawn equidi&longs;tant to <lb/>the Ba&longs;e of the Conoid, all tho&longs;e proportions &longs;hall be towards one and<emph.end type="italics"/><pb xlink:href="069/01/258.jpg" pagenum="255"/><emph type="italics"/>the &longs;ame part, and not by it divided. </s> <s>Therefore the Center of Gravity <lb/>of the &longs;aid Conoid is not below the point N: Neither is it above. </s> <s>For, <lb/>if it may, let it be H: and again, as before, &longs;et the Line L O by it &longs;elf <lb/>equalto the &longs;aid H N, and divided at plea&longs;ure in S: and the &longs;ame pro­<lb/>portion that B N and S O both together have to S L, let the Conoid <lb/>have to R: and about the Conoid let a Figure be circum&longs;cribed con&longs;i­<lb/>&longs;ting of Cylinders, as hath been &longs;aid: by which let it be exceeded a le&longs;s <lb/>quantity than that of the Solid R: and let the Line betwixt the Center <lb/>of Gravity of the circum&longs;cribed Figure and the point N be le&longs;&longs;er than <lb/>S O: the remainder V H &longs;hall be greater than S L. </s> <s>And becau&longs;e that as <lb/>both B N and O S is to SL, &longs;o is the<emph.end type="italics"/><lb/><figure id="id.069.01.258.1.jpg" xlink:href="069/01/258/1.jpg"/><lb/><emph type="italics"/>Conoid to R: (and R is greater <lb/>than the exce&longs;s by which the circum­<lb/>&longs;cribed Figure exceeds the Conoid:) <lb/>Therefore B N and S O hath le&longs;s pro­<lb/>portion to S L than the Conoid to the <lb/>&longs;aid exce&longs;s. </s> <s>And B V is le&longs;&longs;er than <lb/>both B N and S O; and V H is grea­<lb/>ter than S L: much greater proporti­<lb/>on, therefore, hath the Conoid to the <lb/>&longs;aid proportions, than B V hath to <lb/>V H. </s> <s>Therefore whatever proporti­<lb/>on the Conoid hath to the &longs;aid pro­<lb/>portions, the &longs;ame &longs;hall a Line greater <lb/>than B V have to V H. </s> <s>Let the &longs;ame be M V: And becau&longs;e the Center <lb/>of Gravity of the circum&longs;cribed Figure is V, and the Center of the <lb/>Conoid is H. and &longs;ince that as the Conoid to the re&longs;t of the proportions, <lb/>&longs;ois M V to V H, M &longs;hall be the Center of Gravity of the remaining <lb/>proportions: which likewi&longs;e is impo&longs;&longs;ible: Therefore the Center of <lb/>Gravity of the Conoid is not above the point N: But it hath been de­<lb/>mon&longs;trated that neither is it beneath: It remains, therefore, that it ne­<lb/>ce&longs;&longs;arily be in the point N it &longs;elf. </s> <s>And the &longs;ame might be demon&longs;trated <lb/>of Conoidal Plane cut upon an Axis not erect. </s> <s>The &longs;ame in other terms, <lb/>as appears by what followeth:<emph.end type="italics"/></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>The Center of Gravity of the Parabolick Co­<lb/>noid falleth betwixt the Center of the cir­<lb/>cum&longs;cribed Figure and the Center of the in­<lb/>&longs;cribed.</s></p><pb xlink:href="069/01/259.jpg" pagenum="256"/><p type="main"> <s><emph type="italics"/>Let there be a Conoid who&longs;e Axis is A B, and the Center of the <lb/>circum&longs;cribed Figure C, and the Center of the in&longs;cribed O. </s> <s>I &longs;ay <lb/>the Center of the Conoid is betwixt the points C and O. </s> <s>For if <lb/>not, it &longs;hall be either above them, or below them, or in one of them. </s> <s>Let <lb/>it be below, as in R. </s> <s>And becau&longs;e R is the Center of Gravity of the <lb/>whole Conoid; and the Center of Gravity of the in&longs;cribed Figure is O: <lb/>Therefore of the remaining proportions by which the Conoid exceeds <lb/>the in&longs;cribed Figure the Center of Gravity &longs;hall be in the Line O R ex­<lb/>tended towards R, and in that point in which it is &longs;o determined, that, <lb/>what proportion the &longs;aid proportions have to the in&longs;cribed Figure, the <lb/>&longs;ame &longs;hall O R have to the Line falling betwixt R and that falling point. <lb/></s> <s>Let this proportion be that of O R to R X. </s> <s>Therefore X falleth either <lb/>without the Conoid or within, or in its<emph.end type="italics"/><lb/><figure id="id.069.01.259.1.jpg" xlink:href="069/01/259/1.jpg"/><lb/><emph type="italics"/>Ba&longs;e. </s> <s>That it falleth without, or in its <lb/>Ba&longs;e it is already manife&longs;t to be an ab&longs;ur­<lb/>dity. </s> <s>Let it fall within: and becau&longs;e X R <lb/>is to R O, as the in&longs;cribed Figure is to <lb/>the exce&longs;s by which the Conoid exceeds <lb/>it; the &longs;ame proportion that B R hath to <lb/>R O, the &longs;ame let the in&longs;cribed Figure <lb/>have to the Solid K: Which nece&longs;&longs;arily <lb/>&longs;hall be le&longs;&longs;er than the &longs;aid exce&longs;s. </s> <s>And let <lb/>another Figure be in&longs;cribed which may be <lb/>exceeded by the Conoid a le&longs;s quantity <lb/>than is K, who&longs;e Center of Gravity falleth betwixt O and C. </s> <s>Let it <lb/>be V. And, becau&longs;e the fir&longs;t Figure is to K as B R to R O, and the &longs;e­<lb/>cond Figure, who&longs;e Center V is greater than the fir&longs;t, and exceeded <lb/>by the Conoid a le&longs;s quantity than is K; what proportion the &longs;econd <lb/>Figure hath to the exce&longs;s by which the Conoid exceeds it, the &longs;ame <lb/>&longs;hall a Line greater than B R have to R V. </s> <s>But R is the Center of Gra­<lb/>vity of the Conoid; and the Center of the &longs;econd in&longs;cribed Figure V: <lb/>The Center therefore of the remaining proportions &longs;hall be without <lb/>the Conoid beneath B: Which is impo&longs;&longs;ible. </s> <s>And by the &longs;ame means <lb/>we might demon&longs;trate the Center of Gravity of the &longs;aid Conoid not to <lb/>be in the Line C A. </s> <s>And that it is none of the points betwixt C and <lb/>O is manife&longs;t. </s> <s>For &longs;ay, that there other Figures de&longs;cribed, greater <lb/>&longs;omething than the in&longs;cribed Figure who&longs;e Center is O, and le&longs;s than <lb/>that circum&longs;cribed Figure who&longs;e Center is C, the Center of the Conoid <lb/>would fall without the Center of the&longs;e Figures: Which but now was <lb/>concluded to be impo&longs;&longs;ible: It re&longs;ts therefore that it be betwixt the Cen­<lb/>ter of the circum&longs;cribed and in&longs;cribed Figure. </s> <s>And if &longs;o, it &longs;hall ne­<lb/>ce&longs;&longs;arily be in that point which divideth the Axis, &longs;o as that the part <lb/>towards the Vertex is double to the remainder; &longs;ince N may circum­<lb/>&longs;cribe and in&longs;cribe Figures, &longs;o, that tho&longs;e Lines which fall between<emph.end type="italics"/><pb xlink:href="069/01/260.jpg" pagenum="257"/><emph type="italics"/>their Centers and the &longs;aid points, may be le&longs;&longs;er than any other Lines. <lb/></s> <s>To expre&longs;s the &longs;ame in other terms, we have reduced it to an impo&longs;&longs;ibi­<lb/>lity, that the Center of the Conoid &longs;hould not fall betwixt the Centers of <lb/>the in&longs;cribed and circum&longs;cribed Figures.<emph.end type="italics"/></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>Suppo&longs;ing three proportional Lines, and that <lb/>what proportion the lea&longs;t hath to the exce&longs;s <lb/>by which the greate&longs;t exceeds the lea&longs;t, the <lb/>&longs;ame &longs;hould a Line given have to two thirds of <lb/>the exce&longs;s by which the greate&longs;t exceeds the <lb/>middlemo&longs;t: and moreover, that what pro­<lb/>portion that compounded of the greate&longs;t, and <lb/>of double the middlemo&longs;t, hath unto that com­<lb/>pounded of the triple of the greate&longs;t and mid­<lb/>dlemo&longs;t, the &longs;ame hath another Line given, to <lb/>the exce&longs;s by which the greate&longs;t exceeds the <lb/>middle one; both the given Lines taken toge­<lb/>ther &longs;hall be a third part of the greate&longs;t of the <lb/>proportional Lines.</s></p><p type="main"> <s><emph type="italics"/>Let A B, B C, and B F, be three proportional Lines; and what <lb/>proportion B F hath to F A, the &longs;ame let M S have to two thirds <lb/>of C A. </s> <s>And what proportion that compounded of A B and the <lb/>double of B C hath to that compounded of the triple of both A B and <lb/>B C, the &longs;ame let another, to wit S N, have to A C. </s> <s>Becau&longs;e therefore <lb/>that A B, B C, and C F,<emph.end type="italics"/><lb/><figure id="id.069.01.260.1.jpg" xlink:href="069/01/260/1.jpg"/><lb/><emph type="italics"/>are proportionals, A G <lb/>and C F &longs;hall, for the &longs;ame <lb/>rea&longs;on, be likewi&longs;e &longs;o. <lb/></s> <s>Therefore, as A B is to <lb/>B C, &longs;o is A C to C F: <lb/>and as the triple of A B is to the triple of B C, &longs;o is A C to C F: <lb/>Therefore, what proportion the triple of A B with the triple of B C <lb/>hath to the triple of C B, the &longs;ame &longs;hall A C have to a Line le&longs;s than <lb/>C F. </s> <s>Let it be C O. </s> <s>Wherefore by Compo&longs;ition and by Conver&longs;ion of <lb/>proportion, O A &longs;hall have to A C, the &longs;ame proportion, as triple A B <lb/>with Sextuple B C, hath to triple A B with triple B C. </s> <s>But A C hath <lb/>to S N the &longs;ame proportion, that triple A B with triple B C hath to A B <lb/>with double B C: Therefore,<emph.end type="italics"/> ex equali, <emph type="italics"/>O A to NS &longs;hall have the <lb/>&longs;ame proportion, as triple A B with Sexcuple B C hath to A B with<emph.end type="italics"/><pb xlink:href="069/01/261.jpg" pagenum="258"/><emph type="italics"/>double B C: But triple A B with &longs;excuple B C, are triple to A B with <lb/>double B C. </s> <s>Therefore A O is triple to S N.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Again, becau&longs;e O C is to C A as triple C B is to triple A B with tri­<lb/>ple C B: and becau&longs;e as C A is to A F, &longs;o is triple A B to triple B C: <lb/>Therefore,<emph.end type="italics"/> ex equali, <emph type="italics"/>by perturbed proportion, as O C is to C F, &longs;o &longs;hall <lb/>triple A B be to triple A B with treble B C: And, by Conver&longs;ion of <lb/>proportion, as O F is to F C, &longs;o is triple B C to triple A B with triple <lb/>B C: And as C F is to F B, &longs;o is A C to C B, and triple A C to triple <lb/>C B: Therefore,<emph.end type="italics"/> ex equali, <emph type="italics"/>by Perturbation of proportion, as O F is <lb/>to F B, &longs;o is triple A C to the triple of both A B and A C together. <lb/></s> <s>And becau&longs;e F C and C A are in the &longs;ame proportion as C B and B A; <lb/>it &longs;hall be that as F C is to C A, &longs;o &longs;hall B C be to B A. And, by Com­<lb/>po&longs;ition, as F A is to A C, &longs;o are both B A and B C to B A: and &longs;o the <lb/>triple to the triple: Therefore as F A is to A C, &longs;o the compound of tri­<lb/>ple B A and triple B C is to triple A B. Wherefore, as F A is to two <lb/>thirds of A C, &longs;o is the compound of triple B A and triple B C to two <lb/>thirds of triple B A; that is, to double B A: But as F A is to two thirds <lb/>of A C, &longs;o is F B to M S: Therefore, as F B is to M S, &longs;o is the compound <lb/>of triple B A and triple B C to double B A: But as O B is to F B, &longs;o <lb/>was Sexcuple A B to triple of both A B and B C: Therefore,<emph.end type="italics"/> ex equa­<lb/>li, <emph type="italics"/>O B &longs;hall have to M S the &longs;ame proportion as Sexcuple A B hath to <lb/>double B A. </s> <s>Wherefore M S &longs;hall be the third part of O B: And it <lb/>hath been demon&longs;trated, that S N is the third part of A O: It is mani­<lb/>fe&longs;t therefore, that MN is a third part likewi&longs;e of A B: And this is <lb/>that which was to be demon&longs;trated.<emph.end type="italics"/></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>Of any <emph type="italics"/>Fru&longs;tum<emph.end type="italics"/> or Segment cut off from a Para­<lb/>bolick Conoid the Center of Gravity is in the <lb/>Right Line that is Axis of the <emph type="italics"/>Fru&longs;tum<emph.end type="italics"/>; which <lb/>being divided into three equal parts the Cen­<lb/>ter of Gravity is in the middlemo&longs;t and &longs;o di­<lb/>vides it, as that the part towards the le&longs;&longs;er Ba&longs;e <lb/>hath to the part towards the greater Ba&longs;e, the <lb/>&longs;ame proportion that the greater Ba&longs;e hath to <lb/>the le&longs;&longs;er.</s></p><p type="main"> <s><emph type="italics"/>From the Conoid who&longs;e Axis is R B let there be cut off the Solid <lb/>who&longs;e Axis is B E; and let the cutting Plane be equidi&longs;taut to <lb/>the Ba&longs;e: and let it be cut in another Plane along the Axis erect <lb/>upon the Ba&longs;e, and let it be the Section of the Parabola V R C: R B <lb/>&longs;hall be the Diameter of the proportion, or the equidi&longs;tant Diameter<emph.end type="italics"/><pb xlink:href="069/01/262.jpg" pagenum="259"/><emph type="italics"/>L M, V C: they &longs;hall be ordinately applyed. </s> <s>Divide therefore E B in­<lb/>to three equal parts, of which let the middlemo&longs;t be Q Y: and divide <lb/>this &longs;o in the point I that Q I may have the &longs;ame proportion to I Y, as <lb/>the Ba&longs;e who&longs;e Diameter is V C hath to the Ba&longs;e who&longs;e Diameter is <lb/>L M; that is, that the Square V C hath to Square L M. </s> <s>It is to be de­<lb/>mon&longs;trated that I is the Center of Gravity of the Fru&longs;trum L M C. <lb/></s> <s>Draw the Line N S, by the by, equall to B R: and let S X be equal to <lb/>E R: and unto N S and S X a&longs;&longs;ume a third proportional S G: and as <lb/>N G is to G S, &longs;o let B Q be to I O. </s> <s>And it nothing matters whether <lb/>the point O fall above or below L M. </s> <s>And becau&longs;e in the Section V R C <lb/>the Lines L M and V C are ordinately<emph.end type="italics"/><lb/><figure id="id.069.01.262.1.jpg" xlink:href="069/01/262/1.jpg"/><lb/><emph type="italics"/>applyed, it &longs;hall be that as the Square <lb/>V C is to the Square L M, &longs;o is the Line <lb/>B R to R E: And as the Square V C is <lb/>to the Square L M, &longs;o is Q I to I Y: and <lb/>as B R is to R E, &longs;o is N S to S X: There­<lb/>fore Q I is to I Y, as R S is to S X. </s> <s>Where­<lb/>fore as G Y is to Y I, &longs;o &longs;hall both N S and <lb/>S X be to S X: and as E B is to Y I, &longs;o <lb/>&longs;hall the compound of triple N S and tri­<lb/>ple S X be to S X: But as E B is to B Y, <lb/>&longs;o is the compound of triple N S and S X <lb/>both together to the compound of N S and S X: Therefore, as E B is to <lb/>B I, &longs;o is the compound of triple N S and triple S X to the compound of <lb/>N S and double S X. </s> <s>Therefore N S, S X, and S G are three proporti­<lb/>onal Lines: And as S G is to G N, &longs;o is the a&longs;&longs;umed O I to two thirds <lb/>of E B; that is, to N X: And as the compound of N S and double <lb/>S X is to the compound of triple N S and triple S X, &longs;o is another a&longs;&longs;u­<lb/>med Line I B to B E; that is, to N X. </s> <s>By what therefore hath been <lb/>above demon&longs;trated, tho&longs;e Lines taken together are a third part of N S; <lb/>that is, of R B: Therefore R B is triple to B O: Wherefore O &longs;hall <lb/>be the Center of Gravity of the Conoid v R C. </s> <s>And let it be the Cen­<lb/>ter of Gravity of the<emph.end type="italics"/> Fru&longs;trum <emph type="italics"/>L R M of the Conoid: Therefore the <lb/>Center of Gravity of V L M C is in the Line O B, and in that point <lb/>which &longs;o terminates it, that as V L M C of the<emph.end type="italics"/> Fru&longs;trum <emph type="italics"/>is to the <lb/>proportion L R M, &longs;o is the Line A O to that which intervenes betwixt <lb/>O and the &longs;aid point. </s> <s>And becau&longs;e R O is two thirds of R B; and <lb/>R A two thirds of R E; the remaining part A O &longs;hall be two thirds <lb/>of the remaining part E B. </s> <s>And becau&longs;e that as the<emph.end type="italics"/> Fru&longs;tum <emph type="italics"/>V L M C <lb/>is to the proportion L R M, &longs;o is N G to G S: and as N G to G S, &longs;o is <lb/>two thirds of E B to O I: and two thirds of E B is equal to the Line <lb/>A O: it &longs;hall be that as the<emph.end type="italics"/> Fru&longs;tum <emph type="italics"/>V L M O is to the proportion <lb/>L R M, &longs;o is A O to O I. </s> <s>It is manife&longs;t therefore that of the<emph.end type="italics"/> Fru&longs;tum <lb/><emph type="italics"/>V L M C the Center of Gravity is the point I, and &longs;o divideth the Axis, <lb/>[as?] that the part towards the le&longs;&longs;er Ba&longs;e is to the part towards the grea-<emph.end type="italics"/><pb xlink:href="069/01/263.jpg" pagenum="260"/><emph type="italics"/>ter, as the double of the greater Ba&longs;e together with the Le&longs;&longs;er is to the <lb/>double of the le&longs;&longs;er together with the greater. </s> <s>Which is the Propo&longs;ition <lb/>more elegantly expre&longs;&longs;ed.<emph.end type="italics"/></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>If any number of Magnitudes &longs;o di&longs;po&longs;ed to one <lb/>another, as that the &longs;econd addeth unto the fir&longs;t <lb/>the double of the fir&longs;t, the third addeth unto <lb/>the &longs;econd the triple of the fir&longs;t, the fourth <lb/>addeth unto the third the quadruple of the <lb/>fir&longs;t, and &longs;o every one of the following ones <lb/>addeth unto the next unto it the magnitude of <lb/>the fir&longs;t multiplyed according to the number <lb/>which it &longs;hall hold in order; if, I &longs;ay, the&longs;e <lb/>Magnitudes be &longs;u&longs;pended ordinarily on the <lb/>Ballance at equal di&longs;tances; the Center of the <lb/><emph type="italics"/>Equilibrium<emph.end type="italics"/> of all the compounding Magni­<lb/>tudes &longs;hall &longs;o divide the Beam, as that the part <lb/>towards the le&longs;&longs;er Magnitudes is triple to the <lb/>remainder.</s></p><p type="main"> <s><emph type="italics"/>Let the Beam be L T, and let &longs;uch Magnitudes as were &longs;poken of <lb/>hang upon it; and let them be A, F, G, H, K; of which A is in <lb/>the fir&longs;t place &longs;u&longs;pended at T. </s> <s>I &longs;ay, that the Center of the<emph.end type="italics"/> Equi­<lb/>librium <emph type="italics"/>&longs;o cuts the Beam T L as that the part towards T is triple to the <lb/>re&longs;t. </s> <s>Let T L be triple to L I; and S L triple to L P: and Q L to L N,<emph.end type="italics"/><lb/><figure id="id.069.01.263.1.jpg" xlink:href="069/01/263/1.jpg"/><lb/><emph type="italics"/>and L P to L O: I P, <lb/>P N, N O, and O L <lb/>&longs;hall be equal. </s> <s>And <lb/>in F let a Magnitude <lb/>be placed double to A; <lb/>in G another trebble to <lb/>the &longs;ame; in H ano­<lb/>ther Quadruple; and <lb/>&longs;o of the re&longs;t: and let <lb/>tho&longs;e Magnitudes be <lb/>taken in which there <lb/>is A; and let the &longs;ame <lb/>be done in the Magni­<lb/>tudes F, G, H, K. </s> <s>And <lb/>becau&longs;e in F the remaining Magnitude, to wit B, is equal to A; take it<emph.end type="italics"/><pb xlink:href="069/01/264.jpg" pagenum="261"/><emph type="italics"/>double in G, triple in H, &c. </s> <s>and let tho&longs;e Magnitudes be taken in <lb/>which there is B: and in the &longs;ame manner let tho&longs;e be taken in which is <lb/>C, D, and E: now all tho&longs;e in which there is A &longs;hall be equal to K: and <lb/>the compound of all the B B &longs;hall equal H; and the compound of C C <lb/>&longs;hall equal G; and the compound of all the D D &longs;hall equal F; and <lb/>E &longs;hall equal A. </s> <s>And becau&longs;e T I is double to I L, I &longs;hall be the point <lb/>of the<emph.end type="italics"/> Equilibrium <emph type="italics"/>of the Magnitudes compo&longs;ed of all the A A: and <lb/>likewi&longs;e &longs;ince S P is double to P L, P &longs;hall be the point of the<emph.end type="italics"/> Equilibri­<lb/>um <emph type="italics"/>of the compost of B B: and for the &longs;ame cau&longs;e N &longs;hall be the point <lb/>of the<emph.end type="italics"/> Equilibrium <emph type="italics"/>of the compo&longs;t of C C: and O of the compound <lb/>of D D: and L that of E. </s> <s>Therefore T L is a Beam on which at <lb/>equal di&longs;tances certain Magnitudes K, H, G, F, A do hang. </s> <s>And again <lb/>L I is another Ballance, on which, at di&longs;tances in like manner equal, do <lb/>hang &longs;uch a number of Magnitudes, and in the &longs;ame order equal to the <lb/>former. </s> <s>For the compound of all the A A, which hang on I, is equal to <lb/>K hanging at L; and the compo&longs;t of all B B, which is &longs;u&longs;pended at P, is <lb/>equal to H hanging at P; and likewi&longs;e the compound of C C, which <lb/>hangeth at N do equal G; and the compo&longs;t of D, which hang on O, <lb/>are equal to F; and E, hanging on L, is equal to A. </s> <s>Wherefore the <lb/>Ballances are divided in the &longs;ame proportion by the Center of the com­<lb/>pounds of the Magnitudes And the Center of the compound of, the &longs;aid <lb/>Magnitudes is one. </s> <s>Therefore the common point of the Right Line T L, <lb/>and of the Right Line L I &longs;hall be the Center, which let be X. </s> <s>Therefore <lb/>as T X is to X L, &longs;o &longs;hall L X be to X I; and the whole T L to the whole <lb/>L I. </s> <s>But T L is triple to L I: Wherefore T X &longs;hall al&longs;o be triple to X L.<emph.end type="italics"/></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>If any number of Magnitudes be &longs;o taken, that the <lb/>&longs;econd addeth unto the fir&longs;t the triple of the <lb/>fir&longs;t, and the third addeth unto the &longs;econd the <lb/>quintuple of the fir&longs;t, and the fourth addeth <lb/>unto the third the &longs;eptuple of the fir&longs;t, and &longs;o <lb/>the re&longs;t, every one encrea&longs;ing above the next to <lb/>it, and proceedeth &longs;till to a new multiplex of <lb/>the fir&longs;t Magnitude according to the con&longs;e­<lb/>quent odd numbers, like as the Squares of <lb/>Lines equally exceeding one another do pro­<lb/>ceed, whereof the exce&longs;s is equal to the lea&longs;t, <lb/>and if they be &longs;u&longs;pended on a Ballance at equal <lb/>Di&longs;tances, the Center of <emph type="italics"/>Equilibrium<emph.end type="italics"/> of all the <lb/>compound Magnitudes &longs;o divideth the Beam <pb xlink:href="069/01/265.jpg" pagenum="262"/>that the part towards the le&longs;&longs;er Magnitudes is <lb/>more than triple the remaining part; and al&longs;o <lb/>one may take a di&longs;tance that is to the &longs;ame le&longs;s <lb/>than triple.</s></p><p type="main"> <s><emph type="italics"/>In the Ballance B E let there be Magnitudes, &longs;uch as were &longs;poken off, <lb/>from which let there be other Magnitudes taken away that were to <lb/>one another as they were di&longs;po&longs;ed in the precedent, and let it be of <lb/>the compound of all <lb/>the A A: the re&longs;t<emph.end type="italics"/><lb/><figure id="id.069.01.265.1.jpg" xlink:href="069/01/265/1.jpg"/><lb/><emph type="italics"/>in which are C <lb/>&longs;hall be di&longs;tributed <lb/>in the &longs;ame order, <lb/>but the greate&longs;t de­<lb/>ficient. </s> <s>Let E D be <lb/>triple to D B; and <lb/>G F triple to F B. <lb/></s> <s>D &longs;hall be the Center <lb/>of the<emph.end type="italics"/> Equilibrium <lb/><emph type="italics"/>of the compound con­<lb/>&longs;i&longs;ting of all the A A; <lb/>and F that of the <lb/>compound of all the <lb/>C C. </s> <s>Wherefore the <lb/>Center of the com­<lb/>pound of both A A <lb/>and C C falleth be­<lb/>tween D and F. </s> <s>Let <lb/>it be O. </s> <s>It is there­<lb/>fore manife&longs;t that <lb/>E O is more than triple to O B; but G O le&longs;s thantriple to the <lb/>&longs;ame O B: Which was to be demon&longs;trated.<emph.end type="italics"/></s></p><pb xlink:href="069/01/266.jpg" pagenum="263"/><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>If to any Cone or portion of a Cone a Eigure con­<lb/>&longs;i&longs;ting of Cylinders of equal heights be in&longs;cri­<lb/>bed and another circum&longs;cribed; and if its Axis <lb/>be &longs;o divided as that the part which lyeth be­<lb/>twixt the point of divi&longs;ion and the Vertex be <lb/>triple to the re&longs;t; the Center of Gravity of <lb/>the in&longs;cribed Figure &longs;hall be nearer to the Ba&longs;e <lb/>of the Cone than that point of divi&longs;ion: and <lb/>the Center of Gravity of the circum&longs;cribed <lb/>&longs;hall be nearer to the Vertex than that &longs;ame <lb/>point.</s></p><p type="main"> <s><emph type="italics"/>Take therefore a Cone, who&longs;e Axis is N M. </s> <s>Let it be divided <lb/>in S &longs;o, as that N S be triple to the remainder S M. </s> <s>I &longs;ay, that <lb/>the Center of Gravity of any Figure in&longs;cribed, as was &longs;aid, in <lb/>a Cone doth con&longs;i&longs;t in the Axis N M, and approacheth nearer to the Ba&longs;e <lb/>of the Cone than the point S: and that the Center of Gravity of the <lb/>Circum&longs;cribed is likewi&longs;e in the Axis N M, and nearer to the Vertex <lb/>than is S. </s> <s>Let a Figure therefore be &longs;uppo&longs;ed to be in&longs;cribed by the Cy­<lb/>linders who&longs;e Axis M C, C B, B E, E A are equal. </s> <s>Fir&longs;t therefore <lb/>the Cylinder who&longs;e Axis is M C hath<emph.end type="italics"/><lb/><figure id="id.069.01.266.1.jpg" xlink:href="069/01/266/1.jpg"/><lb/><emph type="italics"/>to the Cylinder who&longs;e Axis is C B the <lb/>&longs;ame proportion as its Ba&longs;e hath to <lb/>the Ba&longs;e of the other (for their Alti­<lb/>tudes are equal.) But this propor­<lb/>tion is the &longs;ame with that which the <lb/>Square C N hath to the Square N B. <lb/></s> <s>And &longs;o we might prove, that the Cy­<lb/>linder who&longs;e Axis is C B hath to the <lb/>Cylinder who&longs;e Axis is B E the &longs;ame <lb/>proportion, as the Square B N hath to <lb/>the Square N E: and the Cylinder <lb/>who&longs;e Axis is B E hath to the Cylin­<lb/>der who&longs;e Axis is E A the &longs;ame pro­<lb/>portion that the Square E N hath to <lb/>the Square N A. </s> <s>But the Lines N C, <lb/>N B, E N, and N A equally exceed one <lb/>another, and their exce&longs;s equalleth the <lb/>lea&longs;t, that is N A. </s> <s>Therefore they are certain Magnitudes, to wit, in­<lb/>&longs;cribed Cylinders having con&longs;equently to one another the &longs;ame proporti­<lb/>on as the Squares of Lines that equally exceed one another, and the ex-<emph.end type="italics"/><pb xlink:href="069/01/267.jpg" pagenum="264"/><emph type="italics"/>ce&longs;s of which is equal to the lea&longs;t: and they are &longs;o di&longs;po&longs;ed on the Beam <lb/>T I that their &longs;everal Centers of Gravity con&longs;i&longs;t in it, and that at equal <lb/>di&longs;tances. </s> <s>Therefore by the things above demon&longs;trated it appeareth that <lb/>the Center of Gravity of all &longs;o compo&longs;ed Magnitudes do &longs;o divide the <lb/>Balance T I, that the part to wards T is more than triple to the remain­<lb/>der. </s> <s>Let this Center be O. </s> <s>T O therefore is more than triple to O I. <lb/></s> <s>But T N is triple to I M. </s> <s>Therefore the whole M O will be le&longs;s than a <lb/>fourth part of the whole M N, who&longs;e fourth part was &longs;uppo&longs;ed to be <lb/>M S. </s> <s>It is manife&longs;t, therefore, that the point O doth nearer approach <lb/>the Ba&longs;e of the Cone than S. </s> <s>And let the circum&longs;cribed Figure be com­<lb/>po&longs;ed of the Cylinders who&longs;e Axis M C, C B, B E, E A and A N are <lb/>equal to each other, and, like as in tho&longs;e in&longs;cribed, let them be to one <lb/>another as the Squares of the Lines M N, N C, B N, N E, A N, <lb/>which equally exceed one another, and the exce&longs;s is equal to the lea&longs;t <lb/>A N. Wherefore, by the premi&longs;es, the Center of Gravity of all the Cy­<lb/>linders &longs;o di&longs;po&longs;ed, which let be V, doth &longs;o divide the Beam R I, that the <lb/>part towards R, to wit R V, is more than triple to the remaining part <lb/>V I: but T V &longs;hall be le&longs;s than triple to the &longs;ame. </s> <s>But N T is triple to <lb/>all I M: Therefore all V M is more than the fourth part of all M N, <lb/>who&longs;e fourth part was &longs;uppo&longs;ed to be M S. </s> <s>Therefore the point V is <lb/>nearer to the Vertex than the Point S. </s> <s>Which was to be demon&longs;tra­<lb/>ted.<emph.end type="italics"/></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>About a given Cone a Figure may be circum&longs;cri­<lb/>bed and another in&longs;cribed con&longs;i&longs;ting of Cylin­<lb/>ders of equal height, &longs;o, as that the Line which <lb/>lyeth betwixt the Center of Gravity of the <lb/>circum&longs;cribed, and the Center of Gravity of <lb/>the in&longs;cribed, may be le&longs;&longs;er than any Line <lb/>given.</s></p><p type="main"> <s><emph type="italics"/>Let a Cone be given, who&longs;e Axis is A B; and let the Right Line <lb/>given be K. </s> <s>I &longs;ay; Let there be placed by the Cylinder L <lb/>equal to that in&longs;cribed in the Cone, having for its Altitude half <lb/>of the Axis A B: and let A B be divided in C, &longs;o as that A C be tri­<lb/>ple to C B: And as A C is to K, &longs;o let the Cylinder L be to the Solid X. <lb/></s> <s>And about the Cone let there be a Figure circum&longs;cribed of Cylin­<lb/>ders that have equal Altitude, and let another be in&longs;cribed, &longs;o as that <lb/>the circum&longs;cribed exceed the in&longs;cribed a le&longs;s quantity than the Solid X. <lb/></s> <s>And let the Center of Gravity of the circum&longs;cribed be E; which falls <lb/>above C: and let the Center of the in&longs;cribed be S, falling beneath C.<emph.end type="italics"/><pb xlink:href="069/01/268.jpg" pagenum="265"/><emph type="italics"/>I &longs;ay now, that the Line E S is le&longs;&longs;er than K. </s> <s>For if not, then let C A <lb/>be &longs;uppo&longs;ed equal to E O. </s> <s>Becau&longs;e therefore O E hath to K the &longs;ame <lb/>proportion that L hath to X; and the in&longs;cribed Figure is not le&longs;s than <lb/>the Cylinder L; and the exce&longs;s with which the &longs;aid Figure is exceeded <lb/>by the circum&longs;cribed is le&longs;s than the Solid X: therefore the in&longs;cribed <lb/>Figure &longs;hall have to the &longs;aid exce&longs;s<emph.end type="italics"/><lb/><figure id="id.069.01.268.1.jpg" xlink:href="069/01/268/1.jpg"/><lb/><emph type="italics"/>greater proportion than O E hath to <lb/>K: But the proportion of O E to K is <lb/>not le&longs;s than that which O E hath to <lb/>E S with E S. </s> <s>Let it not be le&longs;s than <lb/>K. </s> <s>Therefore the in&longs;cribed Figure <lb/>hath to the exce&longs;s of the circum&longs;cri­<lb/>bed Figure above it greater propor­<lb/>tion than O E hath to E S. </s> <s>Therefore <lb/>as the in&longs;cribed is to the &longs;aid exce&longs;s, <lb/>&longs;o &longs;hall it be to the Line E S. </s> <s>Let E R <lb/>be a Line greater than E O; and the <lb/>Center of Gravity of the in&longs;cribed <lb/>Figure is S; and the Center of the cir­<lb/>cum&longs;cribed is E. </s> <s>It is manife&longs;t there­<lb/>fore, that the Center of Gravity of <lb/>the remaining proportions by which <lb/>the circum&longs;cribed exceedeth the in <lb/>&longs;cribed is in the Line R E, and in that point by which it is &longs;o termina­<lb/>ted, that as the in&longs;cribed Figure is to the &longs;aid proportions, &longs;o is the Line <lb/>included betwixt E and that point to the Line E S. </s> <s>And this propor­<lb/>tion hath R E to E S. </s> <s>Therefore the Center of Gravity of the remain­<lb/>ing proportions with which the circum&longs;cribed Figure exceeds the in­<lb/>&longs;cribed &longs;hall be R, which is impo&longs;&longs;ible. </s> <s>For the Plane drawn thorow <lb/>R equidi&longs;tant to the Ba&longs;e of the Cone doth not cut tho&longs;e proportions. </s> <s>It <lb/>is therefore fal&longs;e that the Line E S is not le&longs;&longs;er than K. </s> <s>It &longs;hall therefore <lb/>be le&longs;s. </s> <s>The &longs;ame al&longs;o may be done in a manner not unlike this in Pyra­<lb/>mides, as ne could demon&longs;trate.<emph.end type="italics"/></s></p><p type="head"> <s>COROLLARY.</s></p><p type="main"> <s>Hence it is manife&longs;t, that a given Cone may circum&longs;cribe one <lb/>Figure and in&longs;cribe another con&longs;i&longs;ting of Cylinders of equal <lb/>Altitudes &longs;o, as that the Lines which are intercepted betwixt <lb/>their Centers of Gravity and the point which &longs;o divides the <lb/>Axis of the Cone, as that the part towards the Vertex is tri­<lb/>ple to the le&longs;t, are le&longs;s than any given Line.</s></p><p type="main"> <s><emph type="italics"/>For, &longs;ince it hath been demon&longs;trated, that the &longs;aid point dividing the <lb/>Axis, as was &longs;aid, is alwaies found betwixt the Centers of Gravity<emph.end type="italics"/><pb xlink:href="069/01/269.jpg" pagenum="266"/><emph type="italics"/>of the Circum&longs;cribed and in&longs;cribed Figures: and that it's po&longs;&longs;ible, that <lb/>there be a Line in the middle betwixt tho&longs;e Centers that is le&longs;s than any <lb/>Line a&longs;&longs;igned; it followeth that the &longs;ame given Line be much le&longs;s that <lb/>lyeth betwixt one of the &longs;aid Centers and the &longs;aid point that divides <lb/>the Axis.<emph.end type="italics"/></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>The Center of Gravity divideth the Axis of any <lb/>Cone or Pyramid &longs;o, that the part next the <lb/>Vertex is triple to the remainder.</s></p><p type="main"> <s><emph type="italics"/>Let there be a Cone who&longs;e Axis is A B. </s> <s>And in C let it be divided, <lb/>&longs;o that A C be triple to the remaining part C B. </s> <s>It is to be proved, <lb/>that C is the Center of Gravity of the Cone. </s> <s>For if it be not, the <lb/>Cone's Center &longs;hall be either above or below the point C. </s> <s>Let it be fir&longs;t <lb/>beneath, and let it be E. </s> <s>And draw the Line L P, by it &longs;elf, equal to <lb/>C E; which divided at plea&longs;ure in N. </s> <s>And as both B E and P N to­<lb/>gether are to P N, &longs;o let the Cone be to the Solid X: and in&longs;cribe in the <lb/>Cone a Solid Figure of Cylinders that have equal Ba&longs;es, who&longs;e Center <lb/>of Gravity is le&longs;s di&longs;tant from the point C than is the Line L N, and <lb/>the exce&longs;s of the Cone above it le&longs;s than the Solid X. </s> <s>And that this <lb/>may be done is manife&longs;t from what hath been already demon&longs;trated. <lb/></s> <s>Now let the in&longs;cribed Figure be &longs;uch as<emph.end type="italics"/><lb/><figure id="id.069.01.269.1.jpg" xlink:href="069/01/269/1.jpg"/><lb/><emph type="italics"/>was required, who&longs;e Center of Gravity <lb/>let be I. </s> <s>The Line I E therefore &longs;hall be <lb/>greater than N P together with L P. </s> <s>Let <lb/>C E and I C le&longs;s L N be equal: And be­<lb/>cau&longs;e both together B E and N P is to N P <lb/>as the Cone to X: and the exce&longs;s by which <lb/>the Cone exceeds the in&longs;cribed Figure is <lb/>le&longs;s than the Solid X: Therefore the Cone <lb/>&longs;hall have greater proportion to the &longs;aid <lb/>X S than both B E and N P to N P: and, by <lb/>Divi&longs;ion, the in&longs;cribed Figure &longs;hall have <lb/>greater proportion to the exce&longs;s by which <lb/>the Cone exceeds it, than B E to N P: But B E hath le&longs;s proportion to <lb/>E I than to N P with I E. </s> <s>Let N P be greater. </s> <s>Then the in&longs;cribed Fi­<lb/>gure hath to the exce&longs;s of the Cone above it much greater proportion <lb/>than B E to E I. </s> <s>Therefore as the in&longs;cribed Figure is to the &longs;aid exce&longs;s, <lb/>&longs;o &longs;hall a Line bigger than B E be to E I. </s> <s>Let that Line be M E. Becau&longs;e, <lb/>therefore, M E is to E I as the in&longs;cribed Figure is to the exce&longs;s of the <lb/>Cone above the &longs;aid Figure, and D is the Center of Gravity of the <lb/>Cone, and I the Center of Gravity of the in&longs;cribed Figure: Therefore<emph.end type="italics"/><pb xlink:href="069/01/270.jpg" pagenum="267"/><emph type="italics"/>M &longs;hall be the Center of Gravity of the remaining proportions by which <lb/>the Cone exceeds the in&longs;cribed Figure. </s> <s>Which is impo&longs;&longs;ible. </s> <s>Therefore <lb/>the Center of Gravity of the Cone is not below the point C. </s> <s>Nor is it <lb/>above it. </s> <s>For if it may be, let it be R. </s> <s>And again a&longs;&longs;ume L P cut at <lb/>plea&longs;ure in N: And as both B C and N P together are to N L, &longs;o let the <lb/>Cone be to X. </s> <s>And let a Figure be, in like manner, circum&longs;cribed about <lb/>the Cone, which exceeds the &longs;aid Cone a le&longs;s quantity than the Solid X. <lb/></s> <s>And let the Line which intercepts bet wixt its Center of Gravity and C, <lb/>be le&longs;&longs;er than N P. </s> <s>Now take the circum&longs;cribed Figure, who&longs;e Center <lb/>let be O; the remainder O R &longs;hall be greater than the &longs;aid N L. </s> <s>And <lb/>becau&longs;e, as both together B C and P N is to N L, &longs;o is the Cone to X: <lb/>And the exce&longs;s by which the circum&longs;cribed exceeds the Cone is le&longs;&longs;er <lb/>than X: And B O is le&longs;&longs;er than B C and P N together: And O R grea­<lb/>ter than L N: The Cone therefore &longs;hall have much greater proportion to <lb/>the remaining proportions by which it was exceeded by the circum&longs;cribed <lb/>Figure, than B O to O R. </s> <s>Let it be as M O is to O R. </s> <s>M O &longs;hall <lb/>be greater than B C; and M &longs;hall be the Center of Gravity of the pro­<lb/>portions by which the Cone is exceeded by the circum&longs;cribed Figure. <lb/></s> <s>Which is inconvenient. </s> <s>Therefore the Center of Gravity of the Cone is <lb/>not above the point C. </s> <s>But neither is it below it; as hath been proved. <lb/></s> <s>Therefore it &longs;hall be C it &longs;elf. </s> <s>And &longs;o in like manner may it be demon­<lb/>&longs;trated in any Pyramid.<emph.end type="italics"/></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>If there were four Lines continual proportionals; <lb/>and as the lea&longs;t of them were to the exce&longs;s by <lb/>which the greate&longs;t exceeds the lea&longs;t, &longs;o a Line <lb/>taken at plea&longs;ure &longs;hould be to 3/4 the exce&longs;s by <lb/>which the greate&longs;t exceeds the &longs;econd; and as <lb/>the Line equal to the&longs;e (<emph type="italics"/>viz.<emph.end type="italics"/> to the greate&longs;t, <lb/>double of the &longs;econd, and triple of the third) <lb/>is to the Line equal to the quadruple of the <lb/>fourth, the quadruple of the &longs;econd, and the <lb/>quadruple of the third, &longs;o &longs;hould another Line <lb/>taken be to the exce&longs;s of the greate&longs;t above the <lb/>&longs;econd: the&longs;e two Lines taken together &longs;hall <lb/>be a fourth part of the greate&longs;t of the propor­<lb/>tionals.</s></p><pb xlink:href="069/01/271.jpg" pagenum="268"/><p type="main"> <s><emph type="italics"/>For let A B, B C, B D, and B E be four proportional Lines. </s> <s>And <lb/>as B E is to E A, &longs;o let F G be to 3/4 of A C. </s> <s>And as the Line equal <lb/>to A B and to double B C and to triple B D is to the Line equal <lb/>to the quadruples of A B, B C, and B D, &longs;o let H G be to A C. </s> <s>It is <lb/>to be proved, that H F is a fourth part of A B. </s> <s>Fora&longs;much therefore <lb/>as A B, B C, B D, and B E<emph.end type="italics"/><lb/><figure id="id.069.01.271.1.jpg" xlink:href="069/01/271/1.jpg"/><lb/><emph type="italics"/>are proportionals, A C, <lb/>C D, and D E &longs;hall be in <lb/>the &longs;ame proportion: And <lb/>as the quadruple of the &longs;aid <lb/>A B, B C, and B D is to <lb/>A B with the double of B C and triple of B D, &longs;o is the quadruple of <lb/>A C, C D, and D E; that is, the quadruple of A E; to A C with the <lb/>double of C D, and triple of D E. </s> <s>And &longs;o is A C to H G. </s> <s>Therefore <lb/>as the triple of A E is to A C, with the double of C D and triple of <lb/>D E, &longs;o is 3/4 of A C to H G. </s> <s>And as the triple of A E is to the triple of <lb/>E B, &longs;o is 3/4 A C to G F: Therefore, by the Conver&longs;e of the twenty <lb/>fourth of the fifth, As triple A E is to A C with double C D and tri­<lb/>ple D B, &longs;o is 3/4 of A C to H F: And as the quadruple of A E is to A C <lb/>with the double of C D and triple of D B; that is, to A B with C B and <lb/>B D, &longs;o is A C to H F. And, by Permutation, as the quadruple of A E <lb/>is to A C, &longs;o is A B with C B and B D to H F. </s> <s>And as A C is to A E, &longs;o <lb/>is A B to A B with C B and B D. Therefore,<emph.end type="italics"/> ex æquali, <emph type="italics"/>by Perturbed <lb/>proportion, as quadruple A E is to A E, &longs;o is A B to H F. </s> <s>Wherefore it <lb/>is manife&longs;t that H F is the fourth part of A B.<emph.end type="italics"/></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>The Center of Gravity of the <emph type="italics"/>Fru&longs;tum<emph.end type="italics"/> of any Py­<lb/>ramid or Cone, cut equidi&longs;tant to the Plane <lb/>of the Ba&longs;e, is in the Axis, and doth &longs;o divide <lb/>the &longs;ame, that the part towards the le&longs;&longs;er Ba&longs;e <lb/>is to the remainder, as the triple of the greater <lb/>Ba&longs;e, with the double of the mean Space be­<lb/>twixt the greater and le&longs;&longs;er Ba&longs;e, together <lb/>with the le&longs;&longs;er Ba&longs;e is to the triple of the le&longs;&longs;er <lb/>Ba&longs;e, together with the &longs;ame double of the <lb/>mean Space, as al&longs;o of the greater Ba&longs;e.</s></p><pb xlink:href="069/01/272.jpg" pagenum="269"/><p type="main"> <s><emph type="italics"/>From a Cone or Pyramid who&longs;e Axis is A D, and equidi&longs;tant to <lb/>the Plane of the Ba&longs;e, let a<emph.end type="italics"/> Fru&longs;tum <emph type="italics"/>be cut who&longs;e Axis is V D. <lb/></s> <s>And as the triple of the greate&longs;t Ba&longs;e with the double of the <lb/>mean and lea&longs;t is to the triple of the lea&longs;t and double of the mean and <lb/>greate&longs;t, &longs;o is \ O to O D. </s> <s>It is to be proved that the Center of Gra­<lb/>vity of the<emph.end type="italics"/> Fru&longs;tum <emph type="italics"/>is in O. </s> <s>Let V M be the fourth part of V D. <lb/></s> <s>Set the Line H X by the by, equal to A D: and let K X be equal to A V: <lb/>and unto H X K let X L be a third proportional, and X S a fourth. <lb/></s> <s>And as H S is to S X, &longs;o let M D be to the Line taken from O towards <lb/>A: which let be O N. </s> <s>And becau&longs;e the greater Ba&longs;e is in proportion <lb/>to that which is mean betwixt the <lb/>greater and le&longs;&longs;er as D A to A V; that<emph.end type="italics"/><lb/><figure id="id.069.01.272.1.jpg" xlink:href="069/01/272/1.jpg"/><lb/><emph type="italics"/>is, as H X, to X K, but the &longs;aid <lb/>mean is to the lea&longs;t as K X to X L; <lb/>the greater, mean, and le&longs;&longs;er Ba&longs;es <lb/>&longs;hall be in the &longs;ame proportion as <lb/>H X, X K, and X L. </s> <s>Wherefore as <lb/>triple the greater Ba&longs;e, with double <lb/>the mean and le&longs;&longs;er, is to triple the <lb/>lea&longs;t with double the mean and grea­<lb/>te&longs;t; that is, as V O is to O D; &longs;o is <lb/>triple H X with double X K and X L <lb/>to triple X L, with double X K and <lb/>X H: And by Compo&longs;ition and Converting the proportion, O D &longs;hall <lb/>be to V D, as H X, with double X K and triple X L, to quadruple H X, <lb/>X K, and X L. </s> <s>There are, therefore, four proportional Lines, H X, <lb/>X K, X L, and X S: And as X S is to S H, &longs;o is the Line taken N O <lb/>to 3/4 of D V, to wit, to D M; that is, to 3/4 of H K: And as H X <lb/>with double X K and triple X L is to quadruple H X, X K and X L; <lb/>&longs;o is another Line taken O D to D V; that is, to H K. Therefore, by <lb/>the things demon&longs;trated, D N &longs;hall be the fourth part of H X; that <lb/>is, of A D. </s> <s>Wherefore the point N &longs;hall be the Center of Gravity <lb/>of the Cone or Pyramid who&longs;e Axis is A D. </s> <s>Let the Center of Gra­<lb/>vity of the Pyramid or Cone who&longs;e Axis is A V be I. </s> <s>It is therefore <lb/>manife&longs;t that the Center of Gravity of the<emph.end type="italics"/> Fru&longs;tum <emph type="italics"/>is in the Line <lb/>I N inclining towards the part N, and in that point of it which with <lb/>the point N include a Line to which I M hath the &longs;ame proportion that <lb/>the<emph.end type="italics"/> Fru&longs;tum <emph type="italics"/>cut hath to the Pyramid or Cone who&longs;e Axis is A V. <lb/></s> <s>It remaineth therefore to prove that I N hath the &longs;ame proportion <lb/>to N O, that the<emph.end type="italics"/> Fru&longs;tum <emph type="italics"/>hath to the Cone who&longs;e Axis is A V. </s> <s>But <lb/>as the Cone who&longs;e Axis is D A is to the Cone who&longs;e Axis is A V, &longs;o <lb/>is the Cube D A to the Cube D V; that is, the Cube H X to the <lb/>Cube X K: But this is the &longs;ame proportion that H X hath to X S. <lb/>Wherefore, by Divi&longs;ion, as H S is to S X, &longs;o &longs;hall the<emph.end type="italics"/> Fru&longs;tum <emph type="italics"/>who&longs;e<emph.end type="italics"/><pb xlink:href="069/01/273.jpg" pagenum="270"/><emph type="italics"/>Axis is D V be to the Cone or Pyramid who&longs;e Axis is V A. </s> <s>And as <lb/>H S is to S X, &longs;o al&longs;o is M D to O N. </s> <s>Wherefore the<emph.end type="italics"/> Fru&longs;tum <emph type="italics"/>is to the <lb/>Pyramid who&longs;e Axis is A V, as M D to N O. </s> <s>And becau&longs;e A N <lb/>is 3/4 of A D; and A I is 3/4 of A V; the remainder I N &longs;hall be 3/4 of the <lb/>remainder V D. </s> <s>Wherefore I N &longs;hall be equal to M D. <lb/></s> <s>And it hath been demon&longs;trated that M D is to N O, <lb/>as the<emph.end type="italics"/> Fru&longs;tum <emph type="italics"/>to the Cone A V. </s> <s>It is mani­<lb/>fe&longs;t, therefore, that I N hath likewi&longs;e <lb/>the &longs;ame proportion to N O: <lb/>Wherefore the Propo­<lb/>&longs;ition is manife&longs;t.<emph.end type="italics"/></s></p><p type="head"> <s><emph type="italics"/>FINIS.<emph.end type="italics"/><lb/></s></p> </chap> </body> <back></back> </text></archimedes>