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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
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| date | Thu, 02 May 2013 11:08:12 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <archimedes xmlns:xlink="http://www.w3.org/1999/xlink" > <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"></pb><p type="head"> <s>MATHEMATICAL <lb></lb>DISCOURSES <lb></lb>AND <lb></lb>DEMONSTRATIONS, <lb></lb>TOVCHING <lb></lb>Two <emph type="italics"></emph>NEW SCIENCES<emph.end type="italics"></emph.end>; pertaining to <lb></lb>THE <lb></lb>MECHANICKS <lb></lb>AND <lb></lb>LOCAL MOTION:</s></p><p type="head"> <s>BY <lb></lb><emph type="italics"></emph>GALILÆVS GALILÆVS LYNCEVS,<emph.end type="italics"></emph.end><lb></lb>Chiefe <emph type="italics"></emph>Phyloſopher<emph.end type="italics"></emph.end> and <emph type="italics"></emph>Mathematitian<emph.end type="italics"></emph.end> to the moſt <lb></lb>Serene <emph type="italics"></emph>GRAND DVKE<emph.end type="italics"></emph.end> of <emph type="italics"></emph>TVSCANY.<emph.end type="italics"></emph.end><lb></lb>WITH <lb></lb><emph type="italics"></emph>AN APPENDIX OF THE<emph.end type="italics"></emph.end><lb></lb>Centre of Gravity <lb></lb>Of ſome <emph type="italics"></emph>SOLIDS.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>Engliſhed from the Originall <emph type="italics"></emph>Latine<emph.end type="italics"></emph.end> and <emph type="italics"></emph>Italian, <lb></lb>By THOMAS SALUSBURY, Eſq<emph.end type="italics"></emph.end>;</s></p><p type="head"> <s><emph type="italics"></emph>LONDON,<emph.end type="italics"></emph.end><lb></lb>Printed by WILLIAM LEYBOURN, <emph type="italics"></emph>Anno Dom. <lb></lb> MDCLXV.<emph.end type="italics"></emph.end></s></p> <pb xlink:href="069/01/002.jpg" pagenum="1"></pb><p type="head"> <s>GALILEUS, <lb></lb>HIS <lb></lb>DIALOGUES <lb></lb>OF <lb></lb>MOTION.</s></p> </section> </front> <body> <chap> <pb xlink:href="069/01/003.jpg"></pb><p type="head"> <s>The Firſt Dialogue.</s></p><p type="head"> <s><emph type="italics"></emph>INTERLOCUTORS,<emph.end type="italics"></emph.end></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 reſort (Gentlemen) to <lb></lb><arrow.to.target n="marg984"></arrow.to.target><lb></lb>your Famous Arſenal of <emph type="italics"></emph>Venice,<emph.end type="italics"></emph.end> preſen<lb></lb>teth, in my thinking, to your Speculative <lb></lb><arrow.to.target n="marg985"></arrow.to.target><lb></lb>Wits, a large field to Philoſophate in: <lb></lb>and more particularly, as to that part <lb></lb>which is called the <emph type="italics"></emph>Mechanicks:<emph.end type="italics"></emph.end> in re<lb></lb>gard that there all kinds of Engines, and <lb></lb>Machines are continually put in uſe, by a <lb></lb>huge number of Artificers of all ſorts; <lb></lb>amongſt whom, as well through the obſervations of their Prede<lb></lb>ceſſors, as thoſe, which through their own care they continually <lb></lb>are making, it's probable, that there are ſome very learned, and <lb></lb>bravely diſcours'd Men.</s></p><p type="margin"> <s><margin.target id="marg984"></margin.target><emph type="italics"></emph>A Deſcription of <lb></lb>the Arſenal of<emph.end type="italics"></emph.end><lb></lb>Venice.</s></p><p type="margin"> <s><margin.target id="marg985"></margin.target><emph type="italics"></emph>It is a large field <lb></lb>for Wits to Philo<lb></lb>ſophate in.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. Sir, you are not therein miſtaken: and I my ſelf, out of <pb xlink:href="069/01/004.jpg" pagenum="2"></pb>a natural Curioſitie, do frequentlie for my Recreation viſit that <lb></lb>place, and confer with theſe perſons; which for a certain prehe<lb></lb><arrow.to.target n="marg986"></arrow.to.target><lb></lb>minence that they have above the reſt we call ^{*} <emph type="italics"></emph>Overſeers<emph.end type="italics"></emph.end>: whoſe <lb></lb>diſcourſe hath oft helped me in the inveſtigation of not only won<lb></lb>derful, but abſtruce, and incredible Effects: and indeed I have been <lb></lb>at a loſſe ſometimes, and deſpaired to penetrate how that could <lb></lb>poſſibly come to paſſe, which far from all expectation my ſenſes <lb></lb>demonſtrated to be true; and yet that which not long ſince that <lb></lb>good Old man told us, is a ſaying and propoſition, though com<lb></lb><arrow.to.target n="marg987"></arrow.to.target><lb></lb>mon enough, yet in my opinion wholly vain, as are many others, <lb></lb>often in the mouths of unskilful perſons; introduced by them, as <lb></lb>I ſuppoſe, to ſhew that they underſtand how to ſpeak ſomething <lb></lb>about that, of which nevertheleſſe 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"></emph>The Opinion of <lb></lb>Common Artificers <lb></lb>are often falſe.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>It may be Sir, you ſpeak of that laſt propoſition which <lb></lb>he affirmed, when we deſired to underſtand, why they made <lb></lb><arrow.to.target n="marg988"></arrow.to.target><lb></lb>ſo much greater proviſion of ſupporters, and other proviſions, <lb></lb>and reinforcements about that Galeaſſe, which was to be launcht <lb></lb>than is made about leſſer Veſſels, and he anſwered us, that they did <lb></lb>ſo to avoid the peril of breaking its Keel, through the mighty <lb></lb>weight of its vaſt bulk, an inconvenience to which leſſer ſhips are <lb></lb>not subject.</s></p><p type="margin"> <s><margin.target id="marg988"></margin.target><emph type="italics"></emph>Great Ships apter <lb></lb>than others to break <lb></lb>their Keels in <lb></lb>Launching, accor<lb></lb>ding to ſome.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>I do intend the ſame, and chiefly that laſt concluſion, <lb></lb>which he added to his others, and which I alwaies eſteemed a vain <lb></lb>conceit of the Vulgar, namely, That in theſe and other Machines <lb></lb>we muſt not argue from the leſſe to the greater, becauſe many <lb></lb>Mechanical Inventions take in little, which hold not in great. </s> <s>But <lb></lb>being that all the Reaſons of the Mechanicks, have their founda<lb></lb>tions from Geometry; in which I ſee not that greatneſſe and <lb></lb>ſmalneſſe make Circles, Triangles, Cilinders, Cones, or any other <lb></lb>ſolid Figures ſubject to different paſſions: when the great Ma<lb></lb>chine is conformed in all its members to the proportions of the <lb></lb>leſſe that is uſeful, and fit for exerciſe to which it is deſigned; I <lb></lb>cannot ſee why it alſo ſhould not be exempt from the unlucky, <lb></lb>ſiniſter, and deſtructive accidents that may befall it.</s></p><p type="main"> <s>SALV The ſaying of the Vulgar is abſolutely vain, and ſo <lb></lb>falſe, that its contrary may be affirmed with equal truth, ſaying, <lb></lb><arrow.to.target n="marg989"></arrow.to.target><lb></lb>That many Machines may be made more perfect in great than lit<lb></lb>tle: As for inſtance, a Clock that ſhews and ſtrikes the Houres, <lb></lb>may be made more exact in one certain ſize, than in another leſſe. <lb></lb></s> <s>With better ground is that ſame concluſion uſurped by other more <lb></lb>intelligent perſons, who refer the cauſe of ſuch effects in theſe <lb></lb>great Machines different from what is collected from the pure, and <lb></lb>abſtracted Demonſtrations of Geometry, to the imperfection of <lb></lb>the matter, which is ſubject to many alterations, and defects. <lb></lb></s> <s>But here, I know not whether I may without contracting ſome <pb xlink:href="069/01/005.jpg" pagenum="3"></pb>ſuſpition of Arrogance ſay, that thither alſo doth the recourſe to <lb></lb>the defects of the matter (able to blemiſh the perfecteſt Mathe<lb></lb>matical Demonſtrations) ſuffice to excuſe the diſobedience of <lb></lb><arrow.to.target n="marg990"></arrow.to.target><lb></lb>Machines in concrete, to the ſame abſtracted and Ideal: yet not<lb></lb>withſtanding I will ſpeak it, affirming, That abſtracting all imper<lb></lb>fections from the Matter, and ſuppoſing it moſt perfect, and unal<lb></lb>terable, and from all accidental mutation exempt, yet neverthe<lb></lb>leſſe its only being Material, cauſeth, that the greater Machine, <lb></lb>made of the ſame matter, and with the ſame proportions, as the <lb></lb>leſſer; ſhall anſwer in all other conditions to the leſſer in exact <lb></lb>Symetry, except in ſtrength, and reſiſtance againſt violent invaſi<lb></lb>ons: but the greater it is, ſo much in proportion ſhall it be wea<lb></lb>ker. </s> <s>And becauſe I ſuppoſe the Matter to be unalterable, that is <lb></lb>alwaies the ſame, it is manifeſt, that one may produce Demonſtra<lb></lb>tions of it, no leſſe ſimply and purely Mathematical, then of eter<lb></lb>nal, and neceſſary Affections: Therefore, <emph type="italics"></emph>Sagredus,<emph.end type="italics"></emph.end> Revoke the <lb></lb>opinion which you, and, it may be, all the reſt hold, that have ſtu<lb></lb>died the Mechanicks; that Machines, and Frames compoſed of the <lb></lb>ſame Matter, with punctual obſervation of the ſelf ſame proporti<lb></lb>on between their parts, ought to be equally, or to ſay better, pro<lb></lb>portionally diſpoſed to Reſiſt; and to yield to External injuries <lb></lb>and aſſaults: For if it may be Geometrically demonſtrated, that <lb></lb>the greater are alwaies in proportion leſs able to reſiſt, than the <lb></lb>leſſe; ſo that in fine there is not only in all Machines & Fabricks <lb></lb>Artiſicial, but Natural alſo, a term neceſſarily aſcribed, beyond <lb></lb>which neither Art, nor Nature may paſſe; may paſſe, I ſay, al<lb></lb>waies obſerving the ſame proportions with the Identity of the <lb></lb>Matter.</s></p><p type="margin"> <s><margin.target id="marg989"></margin.target><emph type="italics"></emph>Many Machines <lb></lb>may be made more <lb></lb>exact in great than <lb></lb>in little.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg990"></margin.target><emph type="italics"></emph>Great Material <lb></lb>Machines, al<lb></lb>though framed In <lb></lb>the ſame proportion <lb></lb>as others of the <lb></lb>ſame Matter that <lb></lb>are leſſer, are leſſe <lb></lb>ſtrong and able to <lb></lb>reſiſt external Im<lb></lb>petuſs's than the <lb></lb>leſſer.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>I already feel my Brains to turn round, and my Mind, <lb></lb>(like a Cloud unwillingly opened by the Lightning,) I perceive <lb></lb>to be ſurprized with a tranſcient, and unuſual Light, which from <lb></lb>affar off twinkleth, and ſuddenly aſtoniſheth me; and with ab<lb></lb>ſtruce, ſtrange, and indigeſted imaginations. </s> <s>And from what hath <lb></lb>been ſpoken, it ſeems to follow, that, it is a thing impoſſible to <lb></lb>frame two Fabricks of the ſame Matter, alike, and unequal, and <lb></lb>between themſelves in proportion equally able to Reſiſt; and <lb></lb>were it to be done, yet it would be impoſſible to find two only <lb></lb>Launces of the ſame wood, alike between themſelves in ſtrength, <lb></lb><arrow.to.target n="marg991"></arrow.to.target><lb></lb>and toughneſſe, but unequal in bigneſſe.</s></p><p type="margin"> <s><margin.target id="marg991"></margin.target><emph type="italics"></emph>A Wooden Launce <lb></lb>fixed in a Wall at <lb></lb>Right-Angles, and <lb></lb>reduced to ſuch a <lb></lb>length and thick<lb></lb>neſſe as that it may <lb></lb>endure, but made a <lb></lb>hairs breadth big<lb></lb>ger, breaketh with <lb></lb>its own weight, is <lb></lb>ſingly one and no <lb></lb>more.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>So it is Sir; and the better to aſſure you that we con<lb></lb>cur in opinion, I ſay, that if we take a Launce of wood of ſuch a <lb></lb>length and thickneſſe, that being fixed faſt <emph type="italics"></emph>(v. </s> <s>g.)<emph.end type="italics"></emph.end> in a Wall at <lb></lb>Right Angles, that is parallel to the Horizon, it is reduced to the <lb></lb>utmoſt length, that it will hold at, ſo that lengthened never<lb></lb>ſo-little more, it would break, being over-burthened with its own <pb xlink:href="069/01/006.jpg" pagenum="4"></pb>weight, there could not be another ſuch-a-one in the World: So <lb></lb>that if its length (for example) were Centuple to its thickneſſe, <lb></lb>there cannot be found another Launce of the ſame Matter, that <lb></lb>being in length Centuple to its thickneſſe, ſhall be able to ſuſtain <lb></lb>it ſelf preciſely, as that did, and no more: for all that are bigger <lb></lb>ſhall break, and the leſſer ſhall be able, beſides their own, to ſuſtain <lb></lb>ſome additional weight. </s> <s>And this that I ſay of the <emph type="italics"></emph>State of bear<lb></lb>ing it ſelf,<emph.end type="italics"></emph.end> I would have underſtood to be ſpoken of every other <lb></lb>Conſtitution, and thus if one Tranſome bear or ſuſtain the force <lb></lb>often Tranſomes equal to it, ſuch another Beam cannot bear the <lb></lb>weight of ten that are equal to it. </s> <s>Now be pleaſed, Sir, and you <lb></lb><arrow.to.target n="marg992"></arrow.to.target><lb></lb><emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> to obſerve, how true Concluſions, though at the firſt <lb></lb>ſight they ſeem improbable, yet never ſo little glanced at, do depoſe <lb></lb>the Vailes which obſcure them, and make a voluntary ſhew of their <lb></lb>ſecrets nakedly, and ſimply. </s> <s>Who ſees not, that a Horſe falling <lb></lb><arrow.to.target n="marg993"></arrow.to.target><lb></lb>from a height of three or four yards, will break his bones, but a <lb></lb>Dog falling ſo many yards, or a Cat eight or ten, will receive no <lb></lb>hurt; nor likewiſe a Graſhopper from a Tower, nor an Ant thrown <lb></lb>from the Orbe of the Moon? </s> <s>Little Children eſcape all harm in <lb></lb>their falls, whereas perſons grown up break either their ſhins or <lb></lb>faces. </s> <s>And as leſſer Animals are in proportion more robuſtious, <lb></lb>and ſtrong than greater, ſo the leſſer Plants better ſupport them<lb></lb>ſelves: and I already believe, that both of you think, that an Oake <lb></lb>two hundred foot high could not ſupport its branches ſpread like <lb></lb><arrow.to.target n="marg994"></arrow.to.target><lb></lb>one of an indifferent ſize; and that Nature could not have made <lb></lb>an Horſe as big as twenty Horſes, nor a Giant ten times as tall as a <lb></lb>Man, unleſſe ſhe did it either miraculouſly, or elſe by much alte<lb></lb>ring the proportion of the Members, and particularly of the Bones, <lb></lb>enlarging them very much above the Symetry of common Bones. <lb></lb></s> <s>To ſuppoſe likewiſe, that in Artificial Machines, the greater and <lb></lb>leſſer are with equal facility made, and preſerved, is a manifeſt Er<lb></lb>rour: and thus for inſtance, ſmall Spires, Pillars, and other ſolid <lb></lb>figures may be ſafely moved, laid along, and reared upright, with<lb></lb>out danger of breaking them; but the very great upon every ſini<lb></lb>ſter accident fall in pieces, and for no other reaſon but their own <lb></lb>weight. </s> <s>And here it is neceſſary that I relate an accident, worthy <lb></lb>of notice, as are all thoſe events that occur unexpectedly, eſpecial<lb></lb>ly when the means uſed to prevent an inconvenience, proveth in <lb></lb><arrow.to.target n="marg995"></arrow.to.target><lb></lb>fine the moſt potent cauſe of the diſorder. </s> <s>There was a very great <lb></lb>Pillar of Marble laid along, and two Rowlers were put under the <lb></lb>ſame neer to the ends of it; it came into the mind of a certain In<lb></lb>gineer ſome time after, that it would be expedient, the better to <lb></lb>ſecure it from breaking in the midſt through its own weight, to <lb></lb>put under it in that part yet another Rowler: the counſel ſeemed <lb></lb>generally very ſeaſonable, but the ſucceſſe demonſtrated it to be <pb xlink:href="069/01/007.jpg" pagenum="5"></pb>wholly contrary: for many moneths had not paſt, before the Pil<lb></lb>lar crackt, and broke in the middle juſt upon the new Rowler.</s></p><p type="margin"> <s><margin.target id="marg992"></margin.target><emph type="italics"></emph>Truth upon a little <lb></lb>Courting, throweth <lb></lb>off her Vail, and <lb></lb>ſhews her Secrets <lb></lb>maked.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg993"></margin.target><emph type="italics"></emph>Great Animals <lb></lb>receive more harm <lb></lb>by a fall than leſ<lb></lb>ſer.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg994"></margin.target><emph type="italics"></emph>Nature could not <lb></lb>have made of mea<lb></lb>ner Horſes bigger, <lb></lb>and have retained <lb></lb>the ſame ſtrength, <lb></lb>but by altering <lb></lb>their Symetry.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg995"></margin.target><emph type="italics"></emph>A great Marble <lb></lb>Pillar broken by <lb></lb>its own weight, <lb></lb>and why.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. </s> <s>This was an accident truly ſtrange, and indeed <emph type="italics"></emph>preter <lb></lb>ſpem,<emph.end type="italics"></emph.end> eſpecially if it were derived from the addition of new ſup<lb></lb>port in the middle.</s></p><p type="main"> <s>SALV. </s> <s>From that doubtleſs it did proceed; and the known cauſe <lb></lb>of the Effect removeth the wonder: for the two pieces of the Pillar <lb></lb>being taken from off the Rowlers, one of thoſe bearers on which <lb></lb>one end of the Column had reſted, was by length of time rotten, and <lb></lb>ſunk away; and that in the midſt continuing ſound, and ſtrong, <lb></lb>occaſioned that half the Column lay hollow in the air without any <lb></lb>ſupport at the end; ſo that its own unweildy weight, made it do <lb></lb>that, which it would not have done, if it had reſted only upon the <lb></lb>two firſt Bearers, for as they had ſhrunk away it would have fol<lb></lb>lowed. </s> <s>And here none can think that this would have faln out in <lb></lb>a little Column, though of the ſame ſtone, and of a length anſwe<lb></lb>rable to its thickneſſe, in the very ſame proportion as the thick<lb></lb>neſs, and length of the great Pillar.</s></p><p type="main"> <s>SAGR. </s> <s>I am now aſſured of the effect, but do not yet compre<lb></lb>hend the cauſe, how in the augmentation of Matter, the Reſiſtance <lb></lb>and Strength ought not alſo to multiply at the ſame rate. </s> <s>And I <lb></lb>admire at it ſo much the more, in regard I ſee, on the contrary, in <lb></lb>other caſes the ſtrength of Reſiſtance againſt Fraction to encreaſe <lb></lb>much more than the enlargement of the matter encreaſeth. </s> <s>For if <lb></lb>(for example) there be two Nailes faſtned in a Wall, the one twice <lb></lb>asthick as the other, that ſhall bear a weight not only double to this, <lb></lb>but triple, and quadruple.</s></p><p type="main"> <s>SALV. </s> <s>You may ſay octuple, and not be wide of the truth: <lb></lb><arrow.to.target n="marg996"></arrow.to.target><lb></lb>nor is this effect contrary to the former, though in appearance it <lb></lb>ſeemeth ſo different.</s></p><p type="margin"> <s><margin.target id="marg996"></margin.target><emph type="italics"></emph>A Naile double <lb></lb>in thickneſſe to <lb></lb>another being faſt<lb></lb>ned in a Wall, ſu<lb></lb>ſtains a Weight <lb></lb>octuple to that of <lb></lb>the leſſer.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>Therefore <emph type="italics"></emph>Salviatus,<emph.end type="italics"></emph.end> explain unto us theſe Riddles, and <lb></lb>level us theſe Rocks, if you can do it: for indeed I gueſſe this mat<lb></lb>ter of Reſiſtance to be a field repleniſhed with rare, and uſeful Con<lb></lb>templations, and if you be content that this be the ſubject of our <lb></lb>this-daies diſcourſe, it will be to me, and I believe to <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end><lb></lb>very acceptable.</s></p><p type="main"> <s>SALV. </s> <s>I cannot refuſe to ſerve you, ſince my Memory ſerveth <lb></lb><arrow.to.target n="marg997"></arrow.to.target><lb></lb>me, in minding me of that which I formerly learnt of our <emph type="italics"></emph>Accade<lb></lb>mick,<emph.end type="italics"></emph.end> who hath made many Speculations on this ſubject, and all <lb></lb>conformable (as his manner is) to Geometrical Demonſtration: <lb></lb>inſomuch that, not without reaſon, this of his may be called a <emph type="italics"></emph>New <lb></lb>Science<emph.end type="italics"></emph.end>; for though ſome of the Concluſions have been obſerved <lb></lb><arrow.to.target n="marg998"></arrow.to.target><lb></lb>by others, and in the firſt place by <emph type="italics"></emph>Ariſtotle,<emph.end type="italics"></emph.end> yet nevertheleſſe are <lb></lb>they not any of the moſt curious, or (which more importeth) <lb></lb>proved by neceſſary Demonſtrations deduced from their primary, <pb xlink:href="069/01/008.jpg" pagenum="6"></pb>and indubitable fundamentals. </s> <s>And becauſe, as I ſay, I deſire de<lb></lb>monſtratively to aſſure you, and not with only probable diſcour<lb></lb>ſes to perſwade you; preſuppoſing, that you have ſo much know<lb></lb>ledge of the Mechanical Concluſions, by others heretofore funda<lb></lb>mentally handled, as ſufficeth for our purpoſe; it is requiſite, that <lb></lb>before we proceed any further, we conſider what effect that is which <lb></lb>opperates in the Fraction of a Beam of Wood, or other Solid, whoſe <lb></lb>parts are firmly connected; becauſe this is the firſt <emph type="italics"></emph>Notion,<emph.end type="italics"></emph.end> where<lb></lb>on the firſt and ſimple principle dependeth, which as familiarly <lb></lb>known, we may take for granted. </s> <s>For the clearer explanation <lb></lb>whereof; let us take the Cilinder, or Priſme, <emph type="italics"></emph>A. B.<emph.end type="italics"></emph.end> of Wood, or <lb></lb>other ſolid and coherent matter, faſtned above in <emph type="italics"></emph>A,<emph.end type="italics"></emph.end> and hanging <lb></lb>perpendicular; to which, at the other end <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> let there hang the <lb></lb>Weight <emph type="italics"></emph>C<emph.end type="italics"></emph.end>: It is manifeſt, that how great ſoever the Tenacity and <lb></lb>coherence of the parts of the ſaid Solid to one another be, ſo it be <lb></lb>not infinite, it may be overcome by the <lb></lb>Force of the drawing Weight C: whoſe <lb></lb>Gravity I ſuppoſe may be encreaſed as much <lb></lb><figure id="id.069.01.008.1.jpg" xlink:href="069/01/008/1.jpg"></figure><lb></lb>as we pleaſe; by the encreaſe whereof the <lb></lb>ſaid Solid in fine ſhall break, like as if it had <lb></lb>been a Cord. </s> <s>And, as in a Cord, we under<lb></lb>ſtand its reſiſtance to proceed from the mul<lb></lb>titude of the ſtrings or threads in the Hemp <lb></lb>that compoſe it, ſo in Wood we ſee its veins, <lb></lb>and grain diſtended lengthwaies, that render <lb></lb>it far more reſiſting againſt Fraction, then any <lb></lb>Rope would be, of the ſame thickneſſe: but <lb></lb>in a Cylinder of ſtone or Metal the Tenacity <lb></lb>of its parts, (which yet ſeemeth greater) de<lb></lb>pendeth on another kind of Cement, <lb></lb>than of ſtrings, or grains, and yet they alſo <lb></lb>being drawn with equivalent force, break.</s></p><p type="margin"> <s><margin.target id="marg997"></margin.target><emph type="italics"></emph>By Accademick <lb></lb>here, as in his <lb></lb>Dialogues of the <lb></lb>Syſteme,<emph.end type="italics"></emph.end> Galile<lb></lb>us <emph type="italics"></emph>meaneth him<lb></lb>ſelf.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg998"></margin.target>Ariſtotle <emph type="italics"></emph>the firſt <lb></lb>Obſerver of Me<lb></lb>chanical Concluſi<lb></lb>ons, but they nei<lb></lb>ther not the moſt <lb></lb>curious nor ſolidly <lb></lb>demonſtrated.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. </s> <s>If the thing ſucceed as you ſay, I underſtand well <lb></lb>enough, that the thread or grain of the Wood which is as long as <lb></lb>the ſaid Wood may make it ſtrong and able to Reſiſt a great vio<lb></lb>lence done to it to break it: But a Cord compoſed of ſtrings of <lb></lb>Hemp, no longer than two, or three foot a piece, how can it be ſo <lb></lb>ſtrong when it is ſpun out, it may be, to a hundred times that <lb></lb>length? </s> <s>Now I would farther underſtand your opinion concern<lb></lb>ing the Connection of the parts of Metals, Stones, and other mat<lb></lb>ters deprived of ſuch Ligatures, which nevertheleſſe, if I be not <lb></lb>deceived, are yet more tenacious.</s></p><p type="main"> <s>SALV. </s> <s>We muſt be neceſſitated to digreſſe into new Specu<lb></lb>lations, and not much to our purpoſe, if we ſhould reſolve thoſe <lb></lb>difficulties you ſtart.</s></p><pb xlink:href="069/01/009.jpg" pagenum="7"></pb><p type="main"> <s>SAGR. </s> <s>But if Digreſſions may lead us to the knowledge of <lb></lb>new Truths, what prejudice is it to us, that are not obliged to a <lb></lb>ſtrict and conciſe method, but that make our Congreſſions only <lb></lb>for our divertiſement to digreſſe ſometimes, leſt we let ſlip thoſe <lb></lb>Notions, which perhaps the offered occaſion being paſt, may never <lb></lb>meet with another opportunity of remembrance? </s> <s>Nay, who knows <lb></lb>not, that many times curioſity may thereby diſcover hints of more <lb></lb>worth, than the primarily intended Concluſions? </s> <s>Therefore I <lb></lb>entreat you to give ſatisfaction to <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> and my ſelf alſo, <lb></lb>no leſſe curious than he, and deſirous to underſtand what that <lb></lb>Cement is, that holdeth the parts of thoſe Solids ſo tenaciouſly <lb></lb>conjoyned, which yet nevertheleſſe in concluſion are diſſoluble: <lb></lb>a knowledge which furthermore is neceſſary for the underſtanding <lb></lb>of the coherence of the parts of thoſe very ligaments, whereof <lb></lb>ſome Solids are compoſed.</s></p><p type="main"> <s>SALV. Well, ſince it is your pleaſure, I will herein ſerve you. <lb></lb><arrow.to.target n="marg999"></arrow.to.target><lb></lb>And the firſt difficulty is, how the threads of a Cord or Rope <lb></lb>an hundred foot long ſhould ſo cloſely connect together (none <lb></lb>of them exceeding two or three foot) that it requireth a great <lb></lb>violence to break them. </s> <s>But tell me, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> cannot you hold <lb></lb>one ſingle ſtring of Hemp ſo faſt between your fingers by one <lb></lb><arrow.to.target n="marg1000"></arrow.to.target><lb></lb>end, that I pulling by the other end ſhould break it ſooner than <lb></lb>get it from you? </s> <s>Queſtionleſſe you might: when then, thoſe <lb></lb>threads are not only at the end, but alſo in every part of their <lb></lb>length, held faſt with much ſtrength by him that graſpeth them, is <lb></lb>it not apparent, that it is a much harder matter to pluck them <lb></lb>from him that holds them, then to break them? </s> <s>Now in the Cord, <lb></lb><arrow.to.target n="marg1001"></arrow.to.target><lb></lb>the ſame act of twiſting, binds the threads mutually within one <lb></lb>another, in ſuch ſort, that pulling the Cord with great force, the <lb></lb>threads of it break inſunder, but ſeparate and part not from one <lb></lb>another; as is plainly ſeen by viewing the ſhort ends of the ſaid <lb></lb>threads in the broken place, that are not a ſpan long; as they <lb></lb>would be, if the diviſion of the Cord had been made by the ſole <lb></lb>ſeperating of them in drawing the Cord, and not by breaking <lb></lb>them.</s></p><p type="margin"> <s><margin.target id="marg999"></margin.target><emph type="italics"></emph>What that Cement <lb></lb>is that Connecteth <lb></lb>the parts of Solids.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1000"></margin.target><emph type="italics"></emph>How a Rope or <lb></lb>Cord reſiſteth Fra<lb></lb>ction.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1001"></margin.target><emph type="italics"></emph>In breaking a Rope <lb></lb>the parts are not <lb></lb>ſeparated, but bro<lb></lb>kon.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>For confirmation of this, let me add, that the Cord is <lb></lb>ſometimes ſeen to break, not by pulling it length-waies, but by <lb></lb>over-twiſting it: an argument, in my judgment, concluding that <lb></lb>the threads are ſo enterchangeably compreſt by one another, that <lb></lb>thoſe compreſſings permit not the compreſſed to ſlip ſo very little, <lb></lb>as is requiſite to lengthen it out that it wind about the Cord, <lb></lb>which in the twining breaketh, and conſequently in ſome ſinall <lb></lb>meaſure ſwels in thickneſſe.</s></p><p type="main"> <s>SALV. </s> <s>You ſay very well; but conſider by the way, how one <lb></lb>truth draweth on another. </s> <s>That thread, which griped between the <pb xlink:href="069/01/010.jpg" pagenum="8"></pb>fingers, did not yield to follow him that would have forceably <lb></lb>drawn it from between them, reſiſted, becauſe it was ſtayed by a <lb></lb>double compreſſion, ſince the upper finger preſt no leſſe againſt <lb></lb>the nether, than it preſſed againſt that. </s> <s>And there is no queſtion, <lb></lb>that if of theſe two preſſures, one alone might be retained, there <lb></lb>would remain half of that Reſiſtance, which depended conjunctive<lb></lb>ly on them both: but becauſe you cannot with removing, <emph type="italics"></emph>v.g.<emph.end type="italics"></emph.end> the <lb></lb>upper finger take away its preſſion, without taking away the other <lb></lb>part alſo; it will be neceſſary by ſome new Artifice to retain one <lb></lb>of them, and to find a way that the ſame thread may compreſſe it <lb></lb>ſelf againſt the finger or other ſolid body upon which it is put; and <lb></lb>this is done by winding the ſame thread about the Solid. </s> <s>For the <lb></lb>better underſtanding whereof, I will briefly give it you in Figure; <lb></lb>and let <emph type="italics"></emph>A B<emph.end type="italics"></emph.end> and C<emph type="italics"></emph>D<emph.end type="italics"></emph.end> be two Cilinders, and between them let there <lb></lb>be diſtended the thread <emph type="italics"></emph>E F,<emph.end type="italics"></emph.end> which for greater plainneſſe I will <lb></lb>repreſent to be a ſmall Cord: there is no doubt but that the two <lb></lb>Cylinders being preſſed hard one againſt the other, the Cord <lb></lb><emph type="italics"></emph>E F<emph.end type="italics"></emph.end> pulled by the end <emph type="italics"></emph>F<emph.end type="italics"></emph.end> will Reſiſt no ſmal force before <lb></lb>it will ſlip from between the two Solids compreſſing it: but if <lb></lb>we remove one of them, though the Cord <lb></lb><figure id="id.069.01.010.1.jpg" xlink:href="069/01/010/1.jpg"></figure><lb></lb>continue touching the other, yet ſhall it not <lb></lb>by ſuch contact be hindered from ſlipping <lb></lb>away. </s> <s>But if holding it faſt, though but <lb></lb>gently in the point A, towards the top of the <lb></lb>Cylinder, we wind, or belay it about the <lb></lb>ſame ſpirally in A F L O T R, and pull it by <lb></lb>the end R: it is manifeſt, that it will begin <lb></lb>to preſſe the Cylinder, and if the windings <lb></lb>and wreathes be many, it ſhall in its effectual <lb></lb>drawing alwaies preſſe it ſo much the ſtrai<lb></lb>ter about the Cylinder: and by multiplying <lb></lb>the wreathes if you make the contact longer, <lb></lb>and conſequently more invincible, the more <lb></lb>difficult ſtill ſhall it be to withdraw the <lb></lb>Cord, and make it yield to the force that <lb></lb>pulls it. </s> <s>Now who ſeeth not, that the ſame <lb></lb>Reſiſtance is in the threads, which with many thouſand ſuch <lb></lb>twinings ſpin the thick Cord? </s> <s>Yea, the ſtreſſe of ſuch twiſting <lb></lb>bindeth with ſuch Tenacity, that a few Ruſhes, and of no great <lb></lb>length, (ſo that the wreaths and windings are but few where<lb></lb>with they entertwine) make very ſtrong bands, called, as I take it, <lb></lb><arrow.to.target n="marg1002"></arrow.to.target><lb></lb>^{*} Thum-ropes.</s></p><p type="margin"> <s><margin.target id="marg1002"></margin.target>* Fuſta.</s></p><p type="main"> <s>SAGR. </s> <s>Your Diſcourſe hath removed the wonder out of my <lb></lb>mind at two effects, whereof I did not well underſtand the rea<lb></lb>ſon; One was to ſee, how two, or at the moſt three twines of the <pb xlink:href="069/01/011.jpg" pagenum="9"></pb>Rope about the Axis of a Crane did not only hold it, that be<lb></lb>ing drawn by the immenſe force of the weight, which it held, it <lb></lb>ſlipt nor ſhrunk not; but that moreover turning the Crane about, <lb></lb>the ſaid Axis with the ſole touch of the Rope which begirteth it, <lb></lb>did in the after-turnings, draw and raiſe up vaſt ſtones, whilſt the <lb></lb>ſtrength of a little Boy ſufficed to hold and ſtay the other end of <lb></lb>the ſame Cord. </s> <s>The other is at a plain, but cunning, Inſtrument found <lb></lb>out by a young Kinſman of mine, by which with a Cord he could <lb></lb>let himſelf down from a window without much gauling the palmes <lb></lb>of his hands, as to his great ſmart not long before he had done. </s> <s>For <lb></lb><arrow.to.target n="marg1003"></arrow.to.target><lb></lb>the better underſtanding whereof, rake this Scheame: About ſuch <lb></lb>a Cylinder of Wood as A B, two Inches <lb></lb>thick, and ſix or eight Inches long, he cut a <lb></lb>hollow notch ſpirally, for one turn and a <lb></lb><figure id="id.069.01.011.1.jpg" xlink:href="069/01/011/1.jpg"></figure><lb></lb>half and no more, and of wideneſſe fit for <lb></lb>the Cord he would uſe; which he made to <lb></lb>enter through the notch at the end A, and <lb></lb>to come out at the other B, incircling after<lb></lb>wards the Cylinder in a barrel or ſocket of <lb></lb>Wood, or rather Tin, but divided length<lb></lb>waies, and made with Claſpes or Hinges to <lb></lb>open and ſhut at pleaſure: and then graſp<lb></lb>ing and holding the ſaid Barrel or Caſe with <lb></lb>both his hands, the rope being made faſt <lb></lb>above, he hung by his arms; and ſuch was <lb></lb>the compreſſion of the Cord between the <lb></lb>moving Socket and the Cylinder, that at <lb></lb>pleaſure griping his hands cloſer he could <lb></lb>ſtay himſelf without deſcending, and ſlacking his hold a little, he <lb></lb>could let himſelf down as he pleaſed.</s></p><p type="margin"> <s><margin.target id="marg1003"></margin.target><emph type="italics"></emph>An Hand-Pully <lb></lb>or Inſtrument in<lb></lb>vented by an ama<lb></lb>rous perſon to let <lb></lb>himſelf down from <lb></lb>any great height <lb></lb>with a Cord with<lb></lb>out gauling his <lb></lb>hands.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>Aningenious invention verily, and for a full explanati<lb></lb>on of its nature, me-thinks I diſcover, as it were by a ſhadow, the <lb></lb>light of ſome other additional diſcoveries: but I will not at this <lb></lb>time deviate any more from my purpoſe upon this particular: and <lb></lb>the rather in regard you are deſirous to hear my opinion of the <lb></lb>Reſiſtance of other Bodies againſt Fraction, whoſe texture is not <lb></lb><arrow.to.target n="marg1004"></arrow.to.target><lb></lb>with threads, and fibrous ſtrings, as is that of Ropes, and moſt <lb></lb>kinds of Wood: but the connection of their parts ſeem to de<lb></lb>pend on other Cauſes; which in my judgment may be reduced to <lb></lb>two heads; one is the much talked-of Repugnance that Nature <lb></lb>hath againſt the admiſſion of Vacuity: for another (this of Va<lb></lb>cuity not ſufficing) there muſt be introduced ſome glue, viſcous <lb></lb>matter, or Cement, that tenaciouſly connecteth the Corpuſcles of <lb></lb>which the ſaid Body is compacted.</s></p><p type="margin"> <s><margin.target id="marg1004"></margin.target><emph type="italics"></emph>Why ſuch Bodies <lb></lb>reſiſt Fraction that <lb></lb>are not connected <lb></lb>with Fibrous fila<lb></lb>ments.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>I will firſt ſpeak of <emph type="italics"></emph>Vacuity,<emph.end type="italics"></emph.end> ſhewing by plain experiments, <pb xlink:href="069/01/012.jpg" pagenum="10"></pb><arrow.to.target n="marg1005"></arrow.to.target><lb></lb>what and how great its virtue is. </s> <s>And firſt of all the ſeeing at <lb></lb>pleaſure two flat pieces of either Marble, Metal, or Glaſſe, exqui<lb></lb>ſitely planed, ſlickt, and poliſhed, that being laid upon one the <lb></lb>other, without any difficulty ſlide along upon each other, if drawn <lb></lb><arrow.to.target n="marg1006"></arrow.to.target><lb></lb>ſidewaies, (a certain argument that no glue connects them,) but <lb></lb>that going about to ſeperate them, keeping them equidiſtant, <lb></lb>there is found ſuch repugnance, that the uppermoſt will be lif<lb></lb>ted up, and will draw the other after it, and keep it perperually <lb></lb>raiſed, though it be pretty thick, and heavy, evidently proveth to <lb></lb>us, how much Nature abhorreth to admit, though for a ſhort mo<lb></lb>ment of time, the void ſpace, that would be between them, till <lb></lb>the concourſe of the parts of the Circum-Ambient Air ſhould have <lb></lb>poſſeſt, and repleated it. </s> <s>We ſee likewiſe, that if thoſe two Plates <lb></lb>be not exactly poliſhed, and conſequently their contact not every <lb></lb>where exquiſite; in going about to ſeparate them gently, there will <lb></lb>be found no Renitence more than that of their meer weight, but in <lb></lb>the ſudden raiſing, the nether Stone will riſe, and inſtantly fall <lb></lb>down again, following the upper only for that very ſmall time <lb></lb>which ſerveth for the expanſion of that little Air which interpo<lb></lb>ſeth betwixt the Plates, that did not every where touch, and for <lb></lb>the ingreſſion of the other circumfuſed. </s> <s>The like Reſiſtance, which <lb></lb>ſo ſenſibly diſcovers it ſelf betwixt the two Plates, cannot be <lb></lb>doubted to reſide alſo between the parts of a Solid, and that it en<lb></lb>tereth into their connection, at leaſt in part, and as their Concomi<lb></lb>tant Cauſe.</s></p><p type="margin"> <s><margin.target id="marg1005"></margin.target><emph type="italics"></emph>The firſt Cauſe of <lb></lb>the Cohorence of <lb></lb>Bodies is their Re<lb></lb>pugnance to Vacu<lb></lb>ity.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1006"></margin.target><emph type="italics"></emph>This is proved by <lb></lb>the Coherence of <lb></lb>two poliſhed Mar<lb></lb>bles.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. Hold, I pray you, and permit me to impart unto you a <lb></lb>particular Conſideration, juſt now come into my Mind, and this it <lb></lb>is; That ſeeing how the lower Plate followeth the upper, and is <lb></lb>by a ſpeedy motion raiſed, we are thereby aſcertained that (con<lb></lb>trary to the ſaying of many Philoſophers, and perchance of <emph type="italics"></emph>Ari<lb></lb>ſtotle<emph.end type="italics"></emph.end> himſelf) the Motion in <emph type="italics"></emph>Vacuity<emph.end type="italics"></emph.end> would not be Inſtantaneous; <lb></lb>for ſhould in be ſuch, the propoſed Plates without the leaſt repug<lb></lb>nance would Seperate; ſince the ſelf ſame inſtant of time would <lb></lb>ſuffice for their ſeparation, and for the concourſe of the Ambient <lb></lb>Air to repleat that <emph type="italics"></emph>Vacuity,<emph.end type="italics"></emph.end> which might remain between them. <lb></lb></s> <s>By the Inferiour Plates following the Superiour therefore may be <lb></lb>gathered, that in the <emph type="italics"></emph>Vacuity<emph.end type="italics"></emph.end> the Motion would not be Inſtanta<lb></lb><arrow.to.target n="marg1007"></arrow.to.target><lb></lb>neous. </s> <s>And alſo it may be inferred, that even betwixt thoſe Plates <lb></lb>there reſteth ſome <emph type="italics"></emph>Vacuity,<emph.end type="italics"></emph.end> at leaſt for ſome very ſhort time; that <lb></lb>is, for ſo long as the Ambient Air is moving whilſt it concurreth to <lb></lb>replete the <emph type="italics"></emph>Vacuum:<emph.end type="italics"></emph.end> for if there did no <emph type="italics"></emph>Vacuity<emph.end type="italics"></emph.end> remain, there <lb></lb>would be no need either of the Concourſe, or Motion of the Am<lb></lb>bient We muſt therefore ſay that <emph type="italics"></emph>Vacuity<emph.end type="italics"></emph.end> ſometimes is admit<lb></lb>ted, though by Violence or againſt Nature, (albeit it is my opi<lb></lb>nion, that nothing is contrary to Nature, but that which is im<pb xlink:href="069/01/013.jpg" pagenum="11"></pb>poſſible, which again never is.) But here ſtarts up another diffi<lb></lb>culty, and it is, That though Experience aſſures me of the truth of <lb></lb>the Concluſion, yet my Judgment is not thorowly ſatisfied of the <lb></lb>Cauſe, to which ſuch an effect may be aſcribed. </s> <s>For as much as <lb></lb>the effect of the Seperation of the two Plates, is in time before the <lb></lb>Vacuity which ſhould ſucceed by conſequence upon the Separa<lb></lb>tion. </s> <s>And becauſe, in my opinion, the Cauſe ought, if not in <lb></lb><arrow.to.target n="marg1008"></arrow.to.target><lb></lb>Time, at leaſt in Nature, to precede the Effect: and that of a Po<lb></lb>ſitive Effect, the Cauſe ought alſo to be Poſitive; I cannot con<lb></lb>ceive, how the Cauſe of the Adheſion of the two Plates, and of <lb></lb>their Repugnance to Separation, (Effects that are already in <lb></lb>Act) ſhould be aſſigned to Vacuity, which yet is not, but ſhould <lb></lb>follow. </s> <s>And of things that are not in being, there can be no Ope<lb></lb><arrow.to.target n="marg1009"></arrow.to.target><lb></lb>ration; according to the infallible Maxime of Philoſophy.</s></p><p type="margin"> <s><margin.target id="marg1007"></margin.target><emph type="italics"></emph>Vacuity partly the <lb></lb>cauſe of the Cohe<lb></lb>rence between the <lb></lb>parts of Solids.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1008"></margin.target><emph type="italics"></emph>Of a Poſitive Ef<lb></lb>fect the Cauſe is <lb></lb>Poſitive.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1009"></margin.target><emph type="italics"></emph>Non-entity is at<lb></lb>tended with Non<lb></lb>operation.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. </s> <s>But ſince you grant <emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end> this Axiome, I do not <lb></lb>think you will deny another that is moſt excellent, and true; to <lb></lb><arrow.to.target n="marg1010"></arrow.to.target><lb></lb>wit, That Nature doth not attempt Impoſſibilities: Upon which <lb></lb>Axiom I think the Solution of our doubt depends: becauſe there<lb></lb>fore a void ſpace is of it ſelf impoſſible, Nature forbids the doing <lb></lb>that, in conſequence of which Vacuity would neceſſarily ſucceed; <lb></lb>and ſuch an act is the ſeparation of the two Plates.</s></p><p type="margin"> <s><margin.target id="marg1010"></margin.target><emph type="italics"></emph>Nature doth not <lb></lb>attempt Impoſſibi<lb></lb>lities.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. Now, (admitting this which <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> alledgeth is a <lb></lb>ſufficient Solution of my Doubt) in perſuance of the diſcourſe <lb></lb>with which I began, it ſeemeth to me, that this ſame Repugnance <lb></lb>to Vacuity ſhould be a ſufficient Cement in the parts of a Solid of <lb></lb>Stone, Metal, or what other ſubſtance is more firmly conjoyned, <lb></lb>and averſe to Diviſion. </s> <s>For if a ſingle Effect, hath but one ſole <lb></lb>Cauſe, as I underſtand, and think; or if many be aſſigned, they <lb></lb>are reducible to one alone: why ſhould not this of Vacuity, which <lb></lb>certainly is one, be ſufficient to anſwer all Reſiſtances?</s></p><p type="main"> <s>SALV. </s> <s>I will not at this time enter upon this conteſt, whether <lb></lb>Vacuity, without other Cement, be in it ſelf alone ſufficient to <lb></lb>keep together the ſeparable parts of firm Bodies; but yet this I <lb></lb>ſay, that the Reaſon of the Vacuity, which is of force, and con<lb></lb>oluding in the two Plates, ſufficeth not of it ſelf alone for the <lb></lb>firm connection of the parts of a ſolid Cylinder of Marble, or <lb></lb>Metal, the which forced with great violence, pulling them ſtreight <lb></lb>out, in fine, divide and ſeparate. </s> <s>And in caſe I have found a way <lb></lb>to diſtinguiſh this already-known Reſiſtance dependent on Va<lb></lb>ouity, from all others whatſoever that may concur with it in <lb></lb>ſtrengthening the Connection, and make you ſee how that it alone <lb></lb>is not neer ſufficient for ſuch an Effect, would not you grant that <lb></lb>it would be neceſſary to introduce ſome other? </s> <s>Help him out, <emph type="italics"></emph>Sim<lb></lb>plicius,<emph.end type="italics"></emph.end> for he ſtands ſtudying what to anſwer.</s></p><p type="main"> <s>SIMP. </s> <s>The Suſpenſion of <emph type="italics"></emph>Sagredus<emph.end type="italics"></emph.end> muſt needs be upon ano<pb xlink:href="069/01/014.jpg" pagenum="12"></pb>ther account, there being no place left for doubting of ſo clear, and <lb></lb>neceſſary a Conſequence.</s></p><p type="main"> <s>SAGR. </s> <s>You Divine <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> I was thinking if a Million of <lb></lb>Gold <emph type="italics"></emph>per annum,<emph.end type="italics"></emph.end> coming from <emph type="italics"></emph>Spaine,<emph.end type="italics"></emph.end> not being ſufficient to pay <lb></lb>the Army, whether it was neceſſary to make any other proviſion <lb></lb>than of Money to pay the Souldiers. </s> <s>But proceed, <emph type="italics"></emph>Salviatus,<emph.end type="italics"></emph.end> and <lb></lb>ſuppoſing that I admit of your Conſequence, ſhew us how to ſe<lb></lb>parate the opperation of Vacuity from the other, that meaſuring <lb></lb>it we may ſee how it's inſufficient for the Effect of which we ſpeak.</s></p><p type="main"> <s>SALV. </s> <s>Your Genius hath prompted you. </s> <s>Well, I will tell you <lb></lb>the way to part the Virtue of Vacuity from the reſt, and then how <lb></lb>to meaſure it. </s> <s>And to ſever it, we will take a continuate matter, <lb></lb><arrow.to.target n="marg1011"></arrow.to.target><lb></lb>whoſe parts are deſtitute of all other Reſiſtance to Separation, ſave <lb></lb>only that of Vacuity, ſuch as Water at large hath been demon<lb></lb>ſtrated to be in a certain Tractate of our <emph type="italics"></emph>Accademick.<emph.end type="italics"></emph.end> So that <lb></lb>when ever a Cylinder of Water is ſo diſpoſed, that being drawn <lb></lb>we find a Reſiſtance againſt the ſeparation of its parts, this muſt <lb></lb>be acknowledged to proceed from no other cauſe, but from re<lb></lb>pugnance to Vacuity. </s> <s>But to make ſuch an experiment, I have <lb></lb>imagined a device, which with the help of a ſmall Diagram, may <lb></lb>be better expreſt than by my bare words. </s> <s>Let this Figure C A B D <lb></lb>be the Profile of a Cylinder of Metal, or of Glaſs, which muſt <lb></lb>be made hollow within, but turned exactly round; into whoſe <lb></lb>Concave muſt enter a Cylinder of Wood, exquiſitely fitted to <lb></lb>touch every where, whoſe Profile is noted by <lb></lb>E G H F, which Cylinder may be thruſt up<lb></lb><figure id="id.069.01.014.1.jpg" xlink:href="069/01/014/1.jpg"></figure><lb></lb>wards, and downwards: and this I would <lb></lb>have bored in the middle, ſo that there may <lb></lb>a rod of Iron paſs thorow, hooked in the end <lb></lb>K, and the other end I, ſhall grow thicker in <lb></lb>faſhion of a Cone, or Top; and let the <lb></lb>hole made for the ſame thorow the Cylinder <lb></lb>of Wood be alſo cut hollow in the upper <lb></lb>part, like a Conical Superficies, and exactly <lb></lb>fitted to receive the Conick end I, of the <lb></lb>Iron I K, as oft as it is drawn down by the <lb></lb>part K. </s> <s>Then I put the Cylinder of Wood <lb></lb>E H into the Concave Cylinder A D, and <lb></lb>would not have it come to touch the upper<lb></lb>moſt Superficies of the ſaid hollow Cylinder, <lb></lb>but that it ſtay two or three fingers breadth <lb></lb>from it: and I would have that ſpace filled with Water; which <lb></lb>ſhould be put therein, holding the Veſſel with the mouth C D up<lb></lb>wards; and thereupon preſs down the Stopper E H, holding the <lb></lb>Conical part I ſomewhat diſtant from the hollow that was made <pb xlink:href="069/01/015.jpg" pagenum="13"></pb>for it in the Wood, to leave way for the Air to go out, which in <lb></lb>thruſting down the Stopper will iſſue out by the hole of the <lb></lb>Wood, which therefore ſhould be made a little wider than the <lb></lb>thickneſs of the Hook of Iron I K. </s> <s>The Air being let out, and the <lb></lb>Iron pull'd back, which cloſe ſtoppeth the wood with its Conick <lb></lb>part I, then turn the veſſel with its mouth downwards, and faſten to <lb></lb>the hook K a Bucket that may receive into it ſand, or other weigh<lb></lb>ty matter, and you may hang ſo much weight thereat, that at length <lb></lb>the Superiour ſurface of the Stopper E F will ſeparate and forſake <lb></lb>the inferiour part of the Water; to which nothing elſe held it con<lb></lb>nected but the Repugnance againſt Vacuity: afterwards weighing <lb></lb>the Stopper with the Iron, the Bucket, and all that was in it, you <lb></lb>will have the quantity of the Force of the Vacuity. </s> <s>And if affixing <lb></lb>to a Cylinder of Marble, or Chriſtal, as thick as the Cylinder of <lb></lb>Water, ſuch a weight, that together with the proper weight of the <lb></lb>Marble or Chriſtal it ſelf, equalleth the gravity of all thoſe fore<lb></lb>named things, a Rupture follow thereupon; we may without <lb></lb>doubt affirm, that the only reaſon of Vacuity holdeth the parts of <lb></lb>Marble and Chriſtal conjoyned: but not ſufficing; and ſeeing <lb></lb>that to break it there muſt be added four times as much weight, <lb></lb>it muſt be confeſſed, that the Reſiſtance of Vacuity is one part of <lb></lb>ſive, and that the other Reſiſtance is quadruple to that of Vacuity.</s></p><p type="margin"> <s><margin.target id="marg1011"></margin.target><emph type="italics"></emph>How to meaſure <lb></lb>the Virtue of Va<lb></lb>cuity in Solids di<lb></lb>ſtinct from other <lb></lb>convenient Cauſes <lb></lb>of their Coherence. <lb></lb></s> <s>Water a Continu<lb></lb>ate Matter, and <lb></lb>void of all other a<lb></lb>verſion to ſeparati<lb></lb>on, ſave that of Va<lb></lb>cuity.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. </s> <s>It cannot be denied, but that the Invention is Ingen<lb></lb>ous: but I hold it to be ſubject to many difficulties, which makes <lb></lb>me queſtion it; for who ſhall aſſure us, that the Air cannot pene<lb></lb>trate between the Glaſs, and the Stopper, though it be cloſe ſtopt <lb></lb>with Flax, or other pliant matter? </s> <s>And alſo it's a Queſtion, whe<lb></lb>ther Wax or Turpentine will ſerve to make the Cone I, ſtop the <lb></lb>hole cloſe: Again, Why may not the parts of the Water with<lb></lb>draw and rarefie themſelves? </s> <s>Why may not the Air, or Exhalati<lb></lb>ons, or other more ſubtil Subſtances penetrate through the Poroſi<lb></lb>ties of the Wood, or Glaſs it ſelf?</s></p><p type="main"> <s>SALV. <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> is very nimble at raiſing doubts, and, in part, <lb></lb>helping us to reſolve them, as to the Penetration of the Air through <lb></lb>the Wood, or between the Wood and Glaſs. </s> <s>But I moreover <lb></lb>obſerve, that we may at the ſame time ſecure our ſelves, and with<lb></lb>all acquire new Notions, if the fore-named doubts take place; for <lb></lb>if the Water be by Nature, howbeit with violence, capable of ex<lb></lb>tention, as it falleth out in Air, you ſhall ſee the Stopper to de<lb></lb>ſcend: and if in the upper part of the Glaſs we make a ſmall pro<lb></lb>minent Boſs, as this V; in caſe any Air, or other more Tenuous or <lb></lb>Spirituous Matter ſhould penetrate thorow the Subſtance, or Poroſi<lb></lb>ty of the Glaſs, or Wood, it would be ſeen to reunite (the water <lb></lb>giving place) in the eminence V: which things not being percei<lb></lb>ved, we reſt aſſured that the Experiment was made with due <pb xlink:href="069/01/016.jpg" pagenum="14"></pb>caution: and ſee that the Water is not capable oſ extenſion, nor <lb></lb>the Glaſs permeable by any matter, though never ſo ſubtil.</s></p><p type="main"> <s>SAGR. </s> <s>And I, by means of theſe Diſcourſes have found the <lb></lb>Cauſe of an Effect, that hath for a long time puzled my mind <lb></lb><arrow.to.target n="marg1012"></arrow.to.target><lb></lb>with wonder, and kept it in Ignorance. </s> <s>I have heretofore ob<lb></lb>ſerved a Ciſtern, wherein, for the drawing thence of Water, there <lb></lb>was made a Pump, by ſome one that thought, perhaps, (but in <lb></lb>vain) to be thereby able to draw, with leſs labour, the ſame, or <lb></lb>greater quantity of Water, than with the ordinary Buckets; and <lb></lb>this Pump had its Sucker and Value on high, ſo that the Water <lb></lb>was made to aſcend by Attraction, and not by Impulſe, as do the <lb></lb>Pumps that work below. </s> <s>This, whilſt there is any Water in the <lb></lb>Ciſtern to ſuch a determinate height, will draw it plentifully; but <lb></lb>when the Water ebbeth below a certain Mark, the Pump will <lb></lb>work no more. </s> <s>I conceited, the firſt time that I obſerved this ac<lb></lb>cident, that the Engine ____ had been ſpoyled, and looking for <lb></lb>the Workman, that he might amend it; he told me, that there was <lb></lb>no defect at all, other than what was in the Water, which being <lb></lb>fallen too low, permitted not it ſelf to be raiſed to ſuch a height; <lb></lb><arrow.to.target n="marg1013"></arrow.to.target><lb></lb>and farther ſaid, that neither Pump, or other Machine, that raiſeth <lb></lb>the water by Attraction, was poſſibly able to make it riſe a hair <lb></lb>more than eighteen Braces, and be the Pumps wide or narrow, this <lb></lb>is the utmoſt limited meaſure of their height. </s> <s>And I have hitherto <lb></lb>been ſo dull of apprehenſion, that though I knew that a Rope, a <lb></lb>Stick, and a Rod of Iron might be ſo and ſo lengthened, that at <lb></lb>laſt, holding it up on high in the Air, its own weight would break <lb></lb>it, yet I never remembred, that the ſame would much more eaſily <lb></lb>happen in a Rope, or Thread of Water. </s> <s>And what other is that <lb></lb>which is attracted in the Pump than a Cylinder of Water, which <lb></lb>having its contraction above, prolonged more and more, in the end <lb></lb>arriveth to that term, beyond which being drawn, it breaketh by <lb></lb>its foregoing over-weight, juſt as if it was a Rope.</s></p><p type="margin"> <s><margin.target id="marg1012"></margin.target><emph type="italics"></emph>The Nature of the <lb></lb>attraction of Wa<lb></lb>ter by Pumps.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1013"></margin.target><emph type="italics"></emph>Water raiſed or at<lb></lb>tracted by a Pump <lb></lb>riſeth not above <lb></lb>eleven yards.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>It is even ſo as you ſay; and becauſe the ſaid height of <lb></lb>eighteen Braces is the prefixed term of the Elevation, to which any <lb></lb>quantity of Water, be it (that is to ſay, be the Pump) broad, <lb></lb>narrow, or even, ſo narrow as to the thickneſs of a ſtraw, can ſu<lb></lb>ſtain it ſelf; when ever we weigh the water contained in eighteen <lb></lb>Braces of Pipe, be it broad or narrow, we have the value of Reſi<lb></lb>ſtance of Vacuity in Cylinders of whatſoever ſolid matter, of the <lb></lb>thickneſs of the propoſed Pipes. </s> <s>And ſince I have ſaid ſo much, <lb></lb><arrow.to.target n="marg1014"></arrow.to.target><lb></lb>we will ſhew, that a man may eaſily find in all Metals, Stones, Tim<lb></lb>bers, Glaſſes, <emph type="italics"></emph>&c.<emph.end type="italics"></emph.end> How far one may lengthen out Cylinders, <lb></lb>ſtrings, or rods of any thickneſs, beyond which, being oppreſt with <lb></lb>their own weight, they can no longer hold, but break in pieces. <lb></lb></s> <s>Take for example a Braſs wyer of any certain thickneſs, and length, <pb xlink:href="069/01/017.jpg" pagenum="15"></pb>and fixing one of its ends on high, add gradually more and more <lb></lb>weight to the other, till at laſt it break, and let the greateſt weight <lb></lb>that it can bear be <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> fifty pounds. </s> <s>It is manifeſt that fifty <lb></lb>pound of Braſs more than its own weight, which let us ſuppoſe, <lb></lb>for example, to be one eighth of an Ounce, drawn out into a <lb></lb>Wyer of the like thickneſs, would be the greateſt length of the <lb></lb>Wyer that could bear it ſelf. </s> <s>Then meaſure how long the Wyer <lb></lb>was which brake, and let it be for inſtance a y ard; and becauſe it <lb></lb>weighed one eighth of an Ounce; and poiſed, or bore it ſelf, and <lb></lb>fifty pounds more; which are Four Thouſand Eight Hundred <lb></lb>eighths of Ounces; we ſay, that all Wyers of Braſs, whatever <lb></lb>thickneſs they be of, can hold, at the length of Four Thouſand <lb></lb>Eight Hundred and one yards, and no more: and ſo, a Braſs Wyer <lb></lb>being able to hold to the length of 4801 yards; the Reſiſtance it <lb></lb>findeth dependent on Vacuity, in reſpect of the remainder, is as <lb></lb>much as is equivalent to the weight of a Rope of Water eighteen <lb></lb>Braces long, and of the ſame thickneſs with the ſaid Braſs Wyer: <lb></lb>and finding Braſs to be <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> nine times heavier than Water, in <lb></lb>any Wyer of Braſs, the Reſiſtance againſt Fraction dependent on <lb></lb>the reaſon of Vacuity, importeth as much as two Braces of the <lb></lb>ſame Wyer weigheth. </s> <s>And thus arguing, and operating, we may <lb></lb>find the length of the Wyers, or Threads of all Solid Matters re<lb></lb>duced to the utmoſt length that they can ſubſiſt of, and alſo what <lb></lb>part Vacuity hath in their Reſiſtance.</s></p><p type="margin"> <s><margin.target id="marg1014"></margin.target><emph type="italics"></emph>To what length Cy<lb></lb>linders or Ropes of <lb></lb>any Matter may <lb></lb>be prolonged, be<lb></lb>yond which being <lb></lb>charged they break <lb></lb>by their own weight<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>It reſteth now, that you declare to us wherein conſiſts <lb></lb>the remainder of that Tenacity, that is, what that Glue or Reni<lb></lb>tence is, which connecteth together the parts of a Solid, beſides <lb></lb>that which is derived from Vacuity; becauſe I cannot imagine <lb></lb>what that Cement is, that cannot be burnt, or conſumed in a ve<lb></lb>ry hot Furnace in two, three, or four Moneths, nor ten, nor an hun<lb></lb>dred; and yet Gold, Silver, and Glaſs, ſtanding ſo long Liquiſi<lb></lb>ed, when it is taken out, its parts return, upon cooling, to reunite, <lb></lb>and conjoyn, as before. </s> <s>And again, becauſe the ſame difficulty <lb></lb>which I meet within the Connection of the parts of the Glaſs, I <lb></lb>find alſo in the parts of the Cement, that is, what thing that <lb></lb>ſhould be which maketh them cleave ſo cloſs together.</s></p><p type="main"> <s>SALV. </s> <s>I told you but even now, that your Genius prompted <lb></lb>you: I am alſo in the ſame ſtrait: and alſo whereas I have in gene<lb></lb>ral told you, how that Repugnance againſt Vacuity is unqueſti<lb></lb>onably that which permits not, nnleſs with great violence, the ſe<lb></lb>paration of the two Plates, and moreover of the two great pieces of <lb></lb>the Pillar of Marble, or Braſs, I cannot ſee why it ſhould not alſo <lb></lb>take place, and be likewiſe the Cauſe of the Coherence of the leſ<lb></lb>ſer parts, and even of the very leaſt and laſt, of the ſame Matters: <lb></lb><arrow.to.target n="marg1015"></arrow.to.target><lb></lb>and being that of one ſole Effect, there is but one only true, and <pb xlink:href="069/01/018.jpg" pagenum="16"></pb>moſt potent Cauſe; if I can find no other Cement, why may I not <lb></lb>try whether this of Vacuity, which I have already found, may be <lb></lb>ſufficient?</s></p><p type="margin"> <s><margin.target id="marg1015"></margin.target><emph type="italics"></emph>There is but one <lb></lb>ſole Cauſe of one <lb></lb>ſole Effect.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. </s> <s>But when you have already demonſtrated the Reſi<lb></lb>ſtance of the great Vacuity in the ſeparation of the two great <lb></lb>parts of a Solid to be very ſmall in compariſon of that which con<lb></lb>necteth, and conſolidates the little Particles, or Atomes, why will <lb></lb>you not ſtill hold, for certain, that this is extreamly differing from <lb></lb>that?</s></p><p type="main"> <s>SALV. </s> <s>To this <emph type="italics"></emph>Sagredus<emph.end type="italics"></emph.end> anſwereth, That every particular <lb></lb>Souldier is ſtill paid with money collected by the general Impoſi<lb></lb>tions of Shillings and Pence, although a Million of Gold ſufficeth <lb></lb>not to pay the whole Army. </s> <s>And who knows, but that other ex<lb></lb>ceeding ſmall Vacuities may operate amongſt thoſe ſmall Atomes, <lb></lb>(even like as that was of the ſelf-ſame money) wherewith all <lb></lb>the parts are connected? </s> <s>I will tell you what I have ſometimes <lb></lb>fancied: and I give it you, not as an unqueſtionable Truth, but as a <lb></lb>kind of Conjecture very undigeſted, ſubmitting it to exacter con<lb></lb>ſiderations: Pick out of it what pleaſeth you, and judge of the reſt <lb></lb><arrow.to.target n="marg1016"></arrow.to.target><lb></lb>as you think fit. </s> <s>Conſidering ſometimes how the Fire, penetra<lb></lb>ting and inſinuating between the ſmall Atomes of this or that Me<lb></lb>tal, which were before ſo cloſely conſolidated, in the end ſepa<lb></lb>rates, and diſunites them; and how, the Fire being gone, they re<lb></lb>turn with the ſame Tenacity as before to Conſolidation, without <lb></lb>diminiſhing in quantity, (at all in Gold, and very little in other <lb></lb>Metals,) though they continue a long time melted; I have thought <lb></lb>that that might happen, by reaſon the extream ſmall parts of the <lb></lb>Fire, penetrating through the narrow pores of the Metal (through <lb></lb>which the leaſt parts of Air, or of many other Fluids, could not <lb></lb>for their cloſeneſs perforate) by repleating the ſmall interpoſing <lb></lb>Vacuities might free the minute parts of the ſame from the vio<lb></lb>lence, wherewith the ſaid Vacuities attract them one to another, <lb></lb>prohibiting their ſeparation: and thus becoming able to move <lb></lb>freely, their Maſs might become fluid, and continue ſuch, as long <lb></lb>as the ſmall parts of the Fire ſhould abide betwixt them: and that <lb></lb>thoſe departing, and leaving the former Vacuities, their wonted <lb></lb>attractions might return, and conſequently the Coheſion of the <lb></lb>parts. </s> <s>And, as to the Allegation made by <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> it may, in <lb></lb>my opinion, be thus reſolved; That although ſuch Vacuities ſhould <lb></lb>be very ſmall, and conſequently each of them eaſie to be over<lb></lb>come, yet nevertheleſs their innumerable multitude innumerably <lb></lb><arrow.to.target n="marg1017"></arrow.to.target><lb></lb>(if it be proper ſo to ſpeak) multiplieth the Reſiſtances: and we <lb></lb>have an evident proof what, and how great is the Force that reſul<lb></lb>teth from the conjunction of an immenſe number of very weak <lb></lb>Moments, in ſeeing a Weight of many thouſands of pounds, held <pb xlink:href="069/01/019.jpg" pagenum="17"></pb>by mighty Cables, to yield, and ſuffer it ſelf at laſt to be over<lb></lb>come by the aſſault of the innumerable Atomes of Water; which, <lb></lb>either carryed by the South-wind, or elſe by being diſtended into <lb></lb>very thin Miſts that move to and fro in the Air, inſinuate them<lb></lb>ſelves between ſtring and ſtring of the Hemp of the hardeſt twi<lb></lb>ſted Cables; nor can the immenſe force of the pendent Weight <lb></lb>prohibit their enterance; ſo that perforating the ſtrict paſſages be<lb></lb>tween the Pores, they ſwell the Ropes, and by conſequence ſhor<lb></lb>ten them, whereupon that huge Maſs is forcibly raiſed.</s></p><p type="margin"> <s><margin.target id="marg1016"></margin.target><emph type="italics"></emph>Moſt ſmall Va<lb></lb>cuities diſſemina<lb></lb>ted and interpoſed <lb></lb>between the ſmall <lb></lb>Corpuſcles of So<lb></lb>lids the probable <lb></lb>cauſe of the conſi<lb></lb>ſtence or connecti<lb></lb>on of thoſe Corpuſ<lb></lb>cles to one another,<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1017"></margin.target><emph type="italics"></emph>Innumerable A<lb></lb>tomes of Water in<lb></lb>ſinuating into Ca<lb></lb>bles draw and raiſe <lb></lb>an immenſe weight<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. There's no doubt but that ſo long as a Reſiſtance is not <lb></lb><arrow.to.target n="marg1018"></arrow.to.target><lb></lb>infinite, it may by a multitude of moſt minute Forces be over<lb></lb>come; inſomuch that a competent number even of Ants would <lb></lb>be able to carry to ſhore a whole ſhips lading of Corn: for Senſe <lb></lb>giveth us quotidian examples, that an Ant carrieth a ſingle grain <lb></lb>with eaſe; and its cleer, that in the Ship there are not infinite <lb></lb>grains, but that they are compriſed in a certain number; and if you <lb></lb>take another number four or ſix times bigger than that, and take <lb></lb>alſo another of Ants equal to it, and ſet them to work, they ſhall <lb></lb>carry the Corn, and the Ship alſo. </s> <s>It is true indeed, that it will be <lb></lb>needful that the number be great, as alſo in my judgment that of <lb></lb>the <emph type="italics"></emph>Vacuities,<emph.end type="italics"></emph.end> which hold together the ſinall parts of the <lb></lb>Mettal.</s></p><p type="margin"> <s><margin.target id="marg1018"></margin.target><emph type="italics"></emph>Any finite Reſi<lb></lb>ſtance is ſuperable <lb></lb>by any the leaſt <lb></lb>Force, multiplied.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>But though they were required to be infinite, do you <lb></lb>think it impoſſible?</s></p><p type="main"> <s>SAGR. </s> <s>Not if the Mettal were of an infinite maſſe; other<lb></lb>wiſe ----</s></p><p type="main"> <s>SALV. </s> <s>Otherwiſe what? </s> <s>Go to, feeing we are faln upon <lb></lb>Paradoxes, let us ſee if we can any way demonſtrate, how that <lb></lb>in a continuate finite extenſion, it is not impoſſible to finde infi<lb></lb>nite <emph type="italics"></emph>Vacuities:<emph.end type="italics"></emph.end> and then, if we gain nothing elſe, yet at leaſt we <lb></lb><arrow.to.target n="marg1019"></arrow.to.target><lb></lb>ſhall finde a ſolution of that moſt admirable Problem propound<lb></lb>ed by <emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end> amongſt thoſe which he himſelf calleth admirable, <lb></lb>I mean amongſt his <emph type="italics"></emph>Mechanical Queſtions<emph.end type="italics"></emph.end>; and the Solution may <lb></lb>haply be no leſſe plain and concluding, than that which he himſelf <lb></lb>brings thereupon, and different alſo from that which Learned <lb></lb><arrow.to.target n="marg1020"></arrow.to.target><lb></lb><emph type="italics"></emph>Monſig. </s> <s>di Guevara<emph.end type="italics"></emph.end> very acutely diſcuſſeth. </s> <s>But it is firſt requiſite <lb></lb>to declare a Propoſition not toucht by others, on which the ſolution <lb></lb>of the queſtion dependeth, which afterwards, if I deceive not my <lb></lb>ſelf, will draw along with it other new and admirable Notions; for <lb></lb>underſtanding whereof the more exactly, we will give it you in <lb></lb>a Scheme: We ſuppoſe, therefore an equilateral, and equian<lb></lb>gled Poligon of any number of Sides at pleaſure, deſcribed <lb></lb>about this Center G; and in this example let it be a Hexagon <lb></lb>A B C D E F; like to which, and concentrick with the ſame <lb></lb>muſt be diſtributed another leſſer, which we mark H I K L M N; <pb xlink:href="069/01/020.jpg" pagenum="18"></pb>and let one Side of the greater A B be prolonged indeterminately <lb></lb>towards S, and of the leſſe the correſpondent Side H I is to be <lb></lb>produced in like manner towards the ſame part, repreſenting the <lb></lb>Line H T, parallel to A S; and let another paſſe by the Center <lb></lb>equidiſtant from the former, namely G V. </s> <s>This done, we ſuppoſe <lb></lb>the greater Poligon to turn about upon the Line A S, carrying <lb></lb>with it the other leſſer Poligon. </s> <s>It is manifeſt, that the point B, <lb></lb>the term of the Side A B, ſtanding ſtill, whilſt the Revolution <lb></lb>begins, the angle A riſeth, and the point C deſcendeth, deſcribing <lb></lb>the arch C <expan abbr="q;">que</expan> ſo that the Side B C is applyed to the line B Q, <lb></lb>equal to it ſelf: but in ſuch converſion the angle I of the leſſer <lb></lb>Poligon riſeth above the Line I T. for that I B is oblique upon <lb></lb>A S: nor will the point I fall upon the parallel I T, before the <lb></lb>point C come to Q: and by that time I ſhall be deſcended unto <lb></lb>O after it had deſcribed the Arch I O, without the Line H T: and <lb></lb>at the ſame time the Side I K ſhall have paſs'd to O P. </s> <s>But the Cen<lb></lb>ter G ſhall have gone all this time out of the Line G V, on which it <lb></lb>ſhal not fall, until it ſhall firſt have deſcribed the Arch G C. </s> <s>Having <lb></lb>made this firſt ſtep, the greater Poligon ſhall be tranſpoſed to reſt <lb></lb>with the Side B C upon the Line B <expan abbr="q;">que</expan> the Side I K of the leſſer <lb></lb>upon the Line O P, having skipt all the Line I O without touching <lb></lb><figure id="id.069.01.020.1.jpg" xlink:href="069/01/020/1.jpg"></figure><lb></lb>it; and the Center G ſhall be removed to C, making its whole <lb></lb>courſe without the Parallel G V: And in fine all the Figure ſhall <lb></lb>be remitted into a Poſition like the firſt; ſo that the Revolution <lb></lb>being continued, and coming to the ſecond ſtep, the Side of the <lb></lb>greater Poligon D C ſhall remove to Q X; K L of the leſſer (ha<lb></lb>ving firſt skipt the Arch P Y) ſhall fall upon Y Z, and the Center <lb></lb>proceeding evermore without G V ſhall fall on it in R, after the <lb></lb>great skip C R. </s> <s>And in the laſt place, having finiſhed an entire <lb></lb>Converſion, the greater Poligon will have impreſſed upon A S, ſix <pb xlink:href="069/01/021.jpg" pagenum="19"></pb>Lines equal to its Perimeter without any interpoſitions or skips: <lb></lb>the leſſer Poligon likewiſe ſhall have traced ſix Lines equal to its <lb></lb>Perimeter, but diſcontinued by the interpoſition of five Arches, <lb></lb>under which are the Chords, parts of the parallel H T not toucht <lb></lb>by the Poligon: And laſtly, the Center G never hath toucht the <lb></lb>Parallel G V except in ſix points. </s> <s>From hence you may compre<lb></lb>hend, how that the Space paſſed by the leſſer Poligon, is almoſt <lb></lb>equal to that paſſed by the greater, that is the Line H T is almoſt <lb></lb>equal to A S, then which it is leſſer only the quantity of one of <lb></lb>theſe Arches, taking the Line H T, together with all its Arches. <lb></lb></s> <s>Now, this which I have declared and explained to you in the exam<lb></lb>ple of theſe Hexagons, I would have you underſtand to hold true <lb></lb>in all other Poligons, of what number of Sides ſoever they be, ſo <lb></lb>that they be like Concentrick, and Conjoyned; and that at the <lb></lb>Converſion of the greater, the other, how much ſoever leſſer, be <lb></lb>ſuppoſed to revolve therewith: that is, you muſt underſtand, I ſay, <lb></lb>that the Lines by them paſſed are very near equal, computing in<lb></lb>to the Space paſt by the leſſer, the Intervals under the little Ar<lb></lb>ches not toucht by any part of the Perimeter of the ſaid leſſer Po<lb></lb>ligon. </s> <s>Let therefore the greater Poligon, of a thouſand Sides, paſs <lb></lb>round, and meaſure out a continued Line equal to its Perimeter; <lb></lb>and in the ſame time the leſs paſſeth a Line almoſt as long, but <lb></lb>compounded of a thouſand Particles equal to its thouſand Sides, <lb></lb>but diſcontinued with the interpoſition of a thouſand void Spaces: <lb></lb>for ſuch may we call them, in relation to the thouſand little Lines <lb></lb>toucht by the Sides of the Poligon. </s> <s>And what hath been ſpoken <lb></lb>hitherto admits of no doubt or ſcruple. </s> <s>But tell me, in caſe that <lb></lb>about a Center, as ſuppoſe the point A, (in the former Scheme) <lb></lb>we ſhould deſcribe two Circles concentrick, and united together; <lb></lb>and that from the points C and B of their Semi-Diameters, there <lb></lb>be drawn the Tangents C E, and B F, and by the Center A the Pa<lb></lb>rallel A D; ſuppoſing the greater Circle to be turned upon the <lb></lb>Line B F, (drawn equal to its Circumference, as likewiſe the other <lb></lb>two C E, and A D;) when it hath compleated one Revolution, <lb></lb>what ſhall the leſſer Circle, and Center have done? </s> <s>The Center <lb></lb>ſhall doubtleſs have run over, and touched the whole Line A D, <lb></lb>and the leſs Circumference ſhall with its touches have meaſured <lb></lb>all C E, doing the ſame as did the Poligons above; and different <lb></lb>only in this, that the Line H T was not touched in all its Parts by <lb></lb>the Perimeter of the leſſer Poligon, but there were as many parts <lb></lb>left untoucht with the interpoſition of ſalts, or skipped ſpaces; as <lb></lb>were theſe parts touched by the Sides: but here in the Circles, <lb></lb>the Circumference of the leſſer Circle, never ſeparates from the <lb></lb>Line C E, ſo as to leave any of its parts untou cht; nor is the parts <lb></lb>touching of the Circumference, leſs than the part toucht of the <pb xlink:href="069/01/022.jpg" pagenum="20"></pb>Right-line. </s> <s>Now how is it poſſible that the leſſer Circle ſhould <lb></lb>without skips run a Line ſo much bigger than its Circumfe<lb></lb>rence?</s></p><p type="margin"> <s><margin.target id="marg1019"></margin.target>Ariſtotles <emph type="italics"></emph>admi<lb></lb>rable Problem of <lb></lb>two Concentrick <lb></lb>Circles that turn <lb></lb>round, and its true <lb></lb>reſolution.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1020"></margin.target>Monſig. </s> <s>Gueva <lb></lb>ra <emph type="italics"></emph>honourably men<lb></lb>tioned.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>I was conſidering whether one might not ſay, that like <lb></lb>as the Center of the Circle trailed alone upon A D toucht, it all <lb></lb>being yet but one ſole Point; ſo likewiſe might the Points of the <lb></lb>leſſer Circumference, drawn by the revolution of the greater, go <lb></lb>gliding along ſome ſmall part of the Line C E.</s></p><p type="main"> <s><emph type="italics"></emph>S<emph.end type="italics"></emph.end>ALV. </s> <s>This cannot be, for two reaſons; firſt, becauſe there is <lb></lb>no reaſon why ſome of the touches like to C ſhould go gliding <lb></lb>along ſome part of the Line C E, more than others: and though <lb></lb>there ſhould; ſuch touches being (becauſe they are points) inſi<lb></lb>nite, the glidings along upon C E would be infinite; and ſo being, <lb></lb>they would make an infinite Line, but the Line C E is finite. </s> <s>The <lb></lb>other reaſon is, that the greater Circle, in its Revolution continu<lb></lb>ally changing contact, the leſſer Circle muſt of neceſſity do the <lb></lb>like; there being no other Point but B, by which a Right Line can <lb></lb>be drawn to the Center A, and paſſing through C; ſo that the <lb></lb>greater Circumference changing Contact, the leſs doth change it <lb></lb>alſo; nor doth any Point of the leſs touch more than one Point of <lb></lb>its Right-Line C E: beſides, that alſo in the converſion of the Po<lb></lb>ligons, no Point of the Perimeter of the leſs falls on more than one <lb></lb>Point of the Line, which was by the ſaid Perimeter traced, as may <lb></lb>be eaſily underſtood, conſidering the Line I K is parallel to B C, <lb></lb>whereupon, till juſt that B C fall on B R, I K continueth elevated <lb></lb>above I P, and toucheth it not before B C is on the very Point of <lb></lb>uniting with B Q, and then all in the ſame inſtant I K uniteth <lb></lb>with O P, and afterwards immediately riſeth above it again.</s></p><p type="main"> <s>SAGR. </s> <s>The buſineſs is really very intricate, nor can I think on <lb></lb>any Solution of it, therefore do you explain it to us as far as you <lb></lb>judge needful.</s></p><p type="main"> <s>SALV. </s> <s>I ſhould, for the evincing hereof, have recourſe to the <lb></lb>conſideration of the fore-deſcribed Poligons, the effect of which is <lb></lb>intelligible and already comprehended, and would ſay, that like as <lb></lb>in the Poligons of an hundred thouſand Sides, the Line paſſed and <lb></lb>meaſured by the Perimeter of the greater, that is by its hundred <lb></lb>thouſand Sides continually diſtended, is not conſiderably bigger <lb></lb>than that meaſured by the hundred thouſand Sides of the leſs, but <lb></lb>with the interpoſition of an hundred thouſand void ſpaces interve<lb></lb>ning; fo I would ſay in the Circles (which are Poligons of innu<lb></lb>merable Sides) that the Line meaſured by the infinite Sides of the <lb></lb>great Circle, lying continued one with another, to be equalled in <lb></lb>length by the Line traced by the infinite Sides of the leſs, but by <lb></lb>theſe including the interpoſition of the like number of intervening <lb></lb>Spaces: and like as the Sides are not quantitative, but yet infinite <pb xlink:href="069/01/023.jpg" pagenum="21"></pb>in number, ſo the interpoſing Vacuitics are not quantitative, but <lb></lb>infinite in number; that is, thoſe are infinite Points all filled, and <lb></lb>theſe are infinite points, part filled, and part empty. </s> <s>And here I <lb></lb>would have you note, that reſolving, and dividing a Line into quan<lb></lb>titative parts, and conſequently of a finite number, it is not poſſible <lb></lb>to diſpoſe them into a greater extention than that which they poſ<lb></lb>ſeſt whilſt they were continued, and connected, without the inter<lb></lb>poſition of a like number of void Spaces; but imagining it to be <lb></lb>reſolved into parts not quantitative, namely, into its infinite indivi<lb></lb>ſibles, we may conceive it produced to immenſity without the in<lb></lb>terpoſition of quantitative void ſpaces, but yet of infinite indiviſi<lb></lb>ble Vacuities. </s> <s>And this which is ſpoken of ſimple lines, ſhould alſo <lb></lb>be underſtood of Superficies, and Solid Bodies, conſidering that they <lb></lb>are compoſed of infinite Atomes not non-quantitative; if we would <lb></lb>divide them into certain quantitative parts, there's no queſtion, but <lb></lb>that we cannot diſpoſe them into Spaces more ample than the Solid <lb></lb>before occupied, unleſs with the interpoſition of a certain number <lb></lb>of quantitative void Spaces; void, I ſay, at leaſt of the matter of the <lb></lb>Solid: but if we ſhould propoſe the higheſt, and ultimate reſolution <lb></lb>made into the firſt, non-quantitative, but infinite firſt compoun<lb></lb>ding parts, we may be able to conceive ſuch compounding parts <lb></lb>extended unto an immenſe Space without the interpoſition of <lb></lb>quantitative void Spaces; but only of infinite non-quantitative Va<lb></lb>cuities: and in this manner a man may draw out, <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> a little Ball <lb></lb>of Gold into a very vaſt expanſion without admitting any quan<lb></lb>titative void Spaces; yet nevertheleſs we may admit the Gold to <lb></lb>be compounded of infinite induciſſible ones.</s></p><p type="main"> <s>SIMP. </s> <s>Me thinks that in this point you go the way of thoſe diſ<lb></lb>ſeminated Vacuities of a certain <emph type="italics"></emph>Ancient Philoſopher<emph.end type="italics"></emph.end> ------</s></p><p type="main"> <s>SALV. </s> <s>But you add not: [<emph type="italics"></emph>who denied Divine Providence:)<emph.end type="italics"></emph.end><lb></lb>as on ſuch another occaſion, ſufficiently beſides his purpoſe, a cer<lb></lb>tain Antagoniſt of our <emph type="italics"></emph>Accademick<emph.end type="italics"></emph.end> did ſubjoyn.</s></p><p type="main"> <s>SIMP. </s> <s>I ſee very well, and not without indignation, the malice <lb></lb>of ſuch contradictors; but I ſhall forbear theſe Cenſures, not only <lb></lb>upon the ſcore of Good-Manners, but becauſe I know how diſa<lb></lb>greeing ſuch Tenets are to the well-tempered, and well-diſpoſed <lb></lb>mind of a perſon, ſo Religious and Pious, yea, Orthodox and Ho<lb></lb>ly, as you, Sir. </s> <s>But returning to my purpoſe; I find many ſcruples <lb></lb>to ariſe in my mind about your laſt Diſcourſe, which I know not <lb></lb>how to reſolve. </s> <s>And this preſents its ſelf for one, that if the Cir<lb></lb>cumferences of two Circles are equall to the two Right Lines <lb></lb>C E, and B F, this taken continually, and that, with the interpoſi<lb></lb>tion of infinite void Points; how can A D, deſcribed by the Center, <lb></lb>which is but one ſole Point, be ſaid to be equal to the ſame, it con<lb></lb>taining infinite of them? </s> <s>Again, that ſame compoſing the Line of <pb xlink:href="069/01/024.jpg" pagenum="22"></pb>Points, the diviſible of indiviſibles, the quantitative of non-quan<lb></lb>titative, is a rock very hard, in my judgment, to paſs over: And <lb></lb>the very admitting of Vacuity, ſo thorowly confuted by <emph type="italics"></emph>Ariſtotle,<emph.end type="italics"></emph.end><lb></lb>no leſs puzleth me than thoſe difficulties themſelves.</s></p><p type="main"> <s>SALV. </s> <s>There be, indeed, theſe and other difficulties; but re<lb></lb>member, that we are amongſt Infinites, and Indiviſibles: thoſe in<lb></lb>comprehenſible by our finite underſtanding for their Grandure; <lb></lb>and theſe for their minuteneſs: nevertheleſs we ſee that Humane <lb></lb>Diſcourſe will not be beat off from ruminating upon them, in <lb></lb>which regard, I alſo aſſuming ſome liberty, will produce ſome of <lb></lb>my conceits, if not neceſſarily concluding, yet for novelty ſake, <lb></lb>which is ever the meſſenger of ſome wonder: but perhaps the car<lb></lb>rying you ſo far out of your way begun, may ſeem to you imper<lb></lb>tinent, and conſequently little pleaſing.</s></p><p type="main"> <s>SAGR. </s> <s>Pray you let us enjoy the benefit, and priviledge, of free <lb></lb>ſpeaking which is allowed to the living, and amongſt friends; eſpe<lb></lb>cially, in things arbitrary, and not neceſſary; different from Diſcourſe <lb></lb>with dead Books, which ſtart us a thouſand doubts, and reſolve not <lb></lb>one of them. </s> <s>Make us therefore partakers of thoſe Conſiderations, <lb></lb>which the courſe of our Conferences ſuggeſt unto you; for we <lb></lb>want no time, ſeeing we are diſengaged from urgent buſineſſes, to <lb></lb>continue and diſcuſſe the other things mentioned; and particular<lb></lb>ly, the doubts, hinted by <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> muſt by no means eſcape us.</s></p><p type="main"> <s>SAIV. </s> <s>It ſhall be ſo, ſince it pleaſeth you: and beginning at <lb></lb>the firſt, which was, how it's poſſible to imagine that a ſingle Point <lb></lb>is equal to a Line; in regard I can do no more for the preſent, I <lb></lb>will attempt to ſatisfie, or, at leaſt, qualifie one improbability with <lb></lb>another like it, or greater; as ſome times a Wonder is ſwallowed <lb></lb>up in a Miracle. </s> <s>And this ſhall be by ſhewing you two equal Su<lb></lb>perficies, and at the ſame time two Bodies, likewiſe equal, and <lb></lb>placed upon thoſe Superficies as their Baſes; and that go (both <lb></lb>theſe and thoſe) continually and equally diminiſhing in the ſelf<lb></lb><arrow.to.target n="marg1021"></arrow.to.target><lb></lb>ſame time, and that in their remainders reſt alwaies equal between <lb></lb>themſelves, and (laſtly) that, as well Superſicies, as Solids, deter<lb></lb>mine their perpetual precedent equalities, one of the Solids with <lb></lb>one of the Superficies in a very long Line; and the other Solid <lb></lb>with the other Superficies in a ſingle Point: that is, the latter in <lb></lb>one Point alone, the other in infinite.</s></p><p type="margin"> <s><margin.target id="marg1021"></margin.target><emph type="italics"></emph>The equal Super<lb></lb>ficies of two Solids <lb></lb>continually ſub<lb></lb>ſtracting from <lb></lb>them both equal <lb></lb>parts, are reduced, <lb></lb>the one into the <lb></lb>Circumference of a <lb></lb>Circle, and the o<lb></lb>ther into a Point.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>S<emph.end type="italics"></emph.end>AGR. </s> <s>An admirable propoſal, really, yet let us hear you ex<lb></lb>plain and demonſtrate it.</s></p><p type="main"> <s>SALV. </s> <s>It is neceſſary to give you it in Figure, becauſe the proof <lb></lb>is purely Geometrical. </s> <s>Therefore ſuppoſe the Semicircle A F B, <lb></lb>and its Center to be C, and about it deſcribe the Rectangle <lb></lb>A D E B, and from the Center unto the Points D and E let there <lb></lb>be drawn the Lines C D, and C E; Then drawing the Semi-Dia<pb xlink:href="069/01/025.jpg" pagenum="23"></pb>meter C F, perpendicular to one of the two Lines A B, or D E <lb></lb>and immoveable; we ſuppoſe all this Figure to turn round about <lb></lb>that Perpendicular: It is manifeſt, that there will be deſcribed by <lb></lb>the Parallelogram A D E B, a Cylinder; by the Semi-circle A F B, <lb></lb>an Hemi-Sphære; and by the Triangle C D E a Cone. </s> <s>This pre<lb></lb>ſuppoſed, I would have you imagine the Hemiſphære to be taken <lb></lb>away, leaving behind the Cone, and that which ſhall remain of <lb></lb>the Cylinder; which for the Figure, which it ſhall retain like to a <lb></lb>Diſh, we will hereafter call a Diſh: touching which, and the <lb></lb>Cone, we will ſirſt demonſtrate that they are equal; and next <lb></lb>a Plain being drawn parallel to the Circle, which is the foot or <lb></lb>Baſe of the Diſh, whoſe Diameter is the Line D E, and its Center <lb></lb>F; we will demonſtrate, that ſhould the ſaid Plain paſs, <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> by <lb></lb>the Line G H, cutting the Diſh in the points G I, and O N; and <lb></lb>the Cone in the points H and L; it would cut the part of the <lb></lb>Cone C H L, equal alwaies to the part of the Diſh, whoſe Profile <lb></lb>is repreſented to us by the Triangles G A I, and B O N: and more<lb></lb>over we will prove the Baſe alſo of the ſame Cone, (that is the <lb></lb>Circle, whoſe Diameter is H L) to be equal to that circular Su<lb></lb>perficies, which is Baſe of the part of the Diſh; which is, as we <lb></lb>may ſay, a Rimme as broad as G I; (note here by the way what <lb></lb>Mathematical Definitions are: they be an impoſition of names, or, <lb></lb>we may ſay, abreviations of ſpeech, ordain'd and introduced to <lb></lb>prevent the trouble and pains, which you and I meet with, at pre<lb></lb>ſent, in that we have not agreed together to call <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> this Super<lb></lb>ficies a circular Rimme, and that very ſharp Solid of the Diſh a <lb></lb>round Razor:) now howſoever you pleaſe to call them, it ſufficeth <lb></lb>you to know, that the Plain produced to any diſtance at pleaſure, <lb></lb>ſo that it be parallel to the Baſe, <emph type="italics"></emph>viz.<emph.end type="italics"></emph.end> to the Circle whoſe Diame<lb></lb>ter D E cuts alwaies the two Solids, namely, the part of the Cone <lb></lb>C H L, and the upper part of the Diſh equal to one another: and <lb></lb>likewiſe the two Superficies, Baſis of the ſaid Solids, <emph type="italics"></emph>viz.<emph.end type="italics"></emph.end> the ſaid <lb></lb>Rimme, and the Circle H L, equal alſo to one another. </s> <s>Whence <lb></lb>followeth the forementioned Wonder; namely, that if we ſhould <lb></lb>ſuppoſe the cutting-plain to be <lb></lb>ſucceſſively raiſed towards the <lb></lb><figure id="id.069.01.025.1.jpg" xlink:href="069/01/025/1.jpg"></figure><lb></lb>Line A B, the parts of the Solid <lb></lb>cut are alwaies equall, as alſo the <lb></lb>Superficies, that are their Baſes, <lb></lb>are evermore equal; and, in <lb></lb>fine, raiſing the ſaid Plain higher <lb></lb>and higher, the two Solids (ever <lb></lb>equal) as alſo their Baſes, (Su<lb></lb>perficies ever equal) ſhall one couple of them terminate in a Cir<lb></lb>cumference of a Circle, and the other couple in one ſole point; <pb xlink:href="069/01/026.jpg" pagenum="24"></pb>for ſuch are the upper Verge or Rim of the Diſh, and the Vertex <lb></lb>of the Cone. </s> <s>Now whilſt that in the diminution of the two So<lb></lb>lids, they till the very laſt maintain their equality to one another, it <lb></lb>is, in my thoughts, proper to ſay, that the higheſt and ultimate terms <lb></lb>of ſuch Diminutions are equal, and not one infinitely bigger than <lb></lb>the other. </s> <s>It ſeemeth therefore, that the Circumference of an im<lb></lb>menſe Circle may be ſaid to be equal to one ſingle point; and <lb></lb>this that befalls in Solids, holdeth likewiſe in the Superficies their <lb></lb>Baſes; that they alſo in the common Diminution conſerving al<lb></lb>waies equality, in fine, determine at the inſtant of their ultimate <lb></lb>Diminution the one, (that is, that of the Diſh) in their Circum<lb></lb>ference of a Circle, the other (to wit, that of the Cone) in one <lb></lb>ſole point. </s> <s>And why may not theſe be called equal, if they be the <lb></lb>laſt remainders, and footſteps left by equal Magnitudes? </s> <s>And note <lb></lb>again, that were ſuch Veſſels capable of the immenſe Cœleſtial <lb></lb>Hemiſpheres: both their upper Rims, and the points of the contai<lb></lb>ned Cones (keeping evermore equally to one another) would fi<lb></lb>nally determine, thoſe, in Circumferences equal to thoſe of the <lb></lb>greateſt Circles of the Cœleſtial Orbes, and theſe in ſimplo points. <lb></lb></s> <s>Whence, according to that which ſuch Speculations perſwade us <lb></lb>to, all Circumferences of Circles, how unequal ſoever, may be <lb></lb>ſaid to be equal to one another, and each of them equal to one ſole <lb></lb>point.</s></p><p type="main"> <s>SAGR. </s> <s>The Speculation is, in my eſteem, ſo quaint and curi<lb></lb>ous, that, for my part, though I could, yet would I not oppoſe it, <lb></lb>for I take it for a piece of Sacriledge to deface ſo fine a Structure, <lb></lb>by ſpurning at it with any pedantick contradiction; yet for our en<lb></lb>tire ſatisfaction, give us the proof (which you ſay is Geometrical) <lb></lb>of the equality alwaies retained between thoſe Solids, and thoſe <lb></lb>their Baſes, which I think muſt needs be very ſubtil, the philoſo<lb></lb>phical Contemplation being ſo nice, which depends on the ſaid <lb></lb>Concluſion.</s></p><p type="main"> <s>SALV. </s> <s>The Demonſtration is but ſhort, and eaſie. </s> <s>Let us keep <lb></lb>to the former Figure, in which the Angle I P C being a Right An<lb></lb>gle, the Square of the Semi-Diameter I C is equal to the two <lb></lb>Squares of the Sides I P, and P C. </s> <s>But the Semi-Diameter I C, is <lb></lb>equal to A C, and this to G P; and C P is equal to P H; therefore <lb></lb>the Square of the Line G P is equal to the two Squares of I P, and <lb></lb>P H, and the Quadruple to the Quadruples; that is, the Quadrate <lb></lb>of the Diameter G N is equal to the two Quadrates I O, and H L: <lb></lb>and becauſe Circles are to each other, as the Squares of their Dia<lb></lb>meters; the Circle whoſe Diameter is G N, ſhall be equall to the <lb></lb>two Circles whoſe Diameters are I O, and H L; and taking away <lb></lb>the Common Circle, whoſe Diameter is I O; the reſidue of the <lb></lb>Circle G N ſhall be equal to the Circle, whoſe Diameter is H L. <pb xlink:href="069/01/027.jpg" pagenum="25"></pb>And this is as to the firſt part: Now as for the other part, we will, <lb></lb>for the preſent, omit its Demonſtration, as well becauſe that if you <lb></lb>would ſee it, you ſhall find it in the twelfth Propoſition of the Se<lb></lb><arrow.to.target n="marg1022"></arrow.to.target><lb></lb>cond Book <emph type="italics"></emph>De centro Gravitatis Solidorum,<emph.end type="italics"></emph.end> publiſhed by <emph type="italics"></emph>Signeur <lb></lb>Lucas Valerius,<emph.end type="italics"></emph.end> the new <emph type="italics"></emph>Archimedes<emph.end type="italics"></emph.end> of our Age; who upon ano<lb></lb>ther occaſion hath made uſe of it; as becauſe in our caſe it ſuffi<lb></lb>ceth to have ſeen, how the Superficies, already explained, are ever<lb></lb>more equal; and that alwaies diminiſhing equally, they in the end <lb></lb>determine, one in a ſingle point, and the other in the Circumfe<lb></lb>rence of a Circle, be it never-ſomuch bigger, for in this lyeth our <lb></lb>Wonder.</s></p><p type="margin"> <s><margin.target id="marg1022"></margin.target>Lucas Valerius, <lb></lb><emph type="italics"></emph>the other<emph.end type="italics"></emph.end> Archi<lb></lb>chimedes <emph type="italics"></emph>of our <lb></lb>Age, hath written <lb></lb>admirably,<emph.end type="italics"></emph.end> De <lb></lb>Centro Gravita<lb></lb>tis Solidorum.</s></p><p type="main"> <s>SAGR. </s> <s>The Demonſtration is as ingenious, as the reflection <lb></lb>grounded upon it is admirable. </s> <s>Now let us hear ſomewhat about <lb></lb>the other Doubt ſuggeſted by <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> if you have any particu<lb></lb>lars worth note to hint thereupon, but I ſhould incline to think it <lb></lb>impoſſible to be, in regard it is a Controverſie that hath been ſo <lb></lb>canvaſſed.</s></p><p type="main"> <s>SALV. </s> <s>You ſhall have ſome of my particular thoughts thereon; <lb></lb>firſt repeating what but even now I told you, namely, that Infini<lb></lb>ty alone, as alſo Indiviſibility, are things incompre henſible to us: <lb></lb>now think how they will be conjoyned together: and yet if you <lb></lb>would compound the Line of indiviſible points, you muſt make <lb></lb>them infinite; and thus it will be requiſite to apprehend in the <lb></lb>ſame inſtant both Infinite, and Indiviſible. </s> <s>The things that ar ſe<lb></lb>veral times have come into my mind, on this occaſion, are many; <lb></lb>part whereof, and the more conſiderable, it may be, I cannot upon <lb></lb>ſuch a ſudden remember; but it may happen, that in the ſequal <lb></lb>of the Diſcourſe, coming to put queſtions and doubts to you, and <lb></lb>particularly to <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> they may, on the other ſide, re-mind <lb></lb>me of that, which without ſuch excitement would have lain dor<lb></lb>mant in my Fancy: and therefore, with my wonted freedom, per<lb></lb>mit me that I produce any wild conjectures, for ſuch may we fitly <lb></lb>call them in compariſon of ſupernatural Doctrines, the only true <lb></lb>and certain determiners of our Controverſies, and unerring guides <lb></lb>in our obſcure, and dubious paths, or rather Laberinths.</s></p><p type="main"> <s>Amongſt the firſt Inſtances that are wont to be produced <lb></lb><arrow.to.target n="marg1023"></arrow.to.target><lb></lb>againſt thoſe that compound <emph type="italics"></emph>Continuum<emph.end type="italics"></emph.end> of Indiviſibles, this is uſu<lb></lb>ally one; That an Indiviſible, added to another Indiviſible, produ<lb></lb>ceth not a thing diviſible; for if that were ſo, it would follow, that <lb></lb>even the Indiviſibles were diviſible: for if two Indiviſibles, as for <lb></lb>example, two Points conjoyned, ſhould make a Quantity that <lb></lb>ſhould be a diviſible Line, much more ſuch ſhould one be that is <lb></lb>compounded of three, five, ſeven, or others, that are odd num<lb></lb>bers; the which Lines, being to be cut in two equal parts, render <lb></lb>diviſible that Indiviſible which was placed in the middle. </s> <s>In this <pb xlink:href="069/01/028.jpg" pagenum="26"></pb>and other Objections of this kind you may ſatisfie the propoſer of <lb></lb>them, telling him, that neither two Indiviſibles, nor ten, nor an <lb></lb>hundred, no, nor a thouſand can compound a Magnitude diviſible, <lb></lb>and quantitative, but being infinite they may.</s></p><p type="margin"> <s><margin.target id="marg1023"></margin.target>Continuum <emph type="italics"></emph>com<lb></lb>pounded of Indivi<lb></lb>ſibles.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. </s> <s>Here already riſeth a doubt, which I think unreſolvable; <lb></lb>and it is, that we being certain to find Lines one bigger than ano<lb></lb>ther, although both contain infinite Points, we muſt of neceſſity <lb></lb>confeſs, that we have found in the ſame Species a thing bigger than <lb></lb>infinite; becauſe the Infinity of the Points of the greater Line, ſhall <lb></lb>exceed the Infinity of the Points of the leſſer. </s> <s>Now this aſſigning <lb></lb>of an Infinite bigger than an Infinite is, in my opinion, a conceit <lb></lb>that can never by any means be apprehended.</s></p><p type="main"> <s>SALV. </s> <s>Theſe are ſome of thoſe difficulties, which reſult from <lb></lb>the Diſcourſes that our finite Judgments make about Infinites, gi<lb></lb>ving them thoſe attributes which we give to things finite and ter<lb></lb>minate; which I think is inconvenient; for I judge that theſe <lb></lb>terms of Majority, Minority, and Equality ſute not with Infinites, <lb></lb>of which we cannot ſay that one is greater, or leſs, or equal to ano<lb></lb>ther: for proof of which there cometh to my mind a Diſcourſe, <lb></lb>which, the better to explain, I will propound by way of Interroga<lb></lb>tories to <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> that ſtarted the queſtion.</s></p><p type="main"> <s>I ſuppoſe that you very well underſtand which are Square Num<lb></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></lb>proceeds from the multiplication of another Number into it ſelf; <lb></lb>and ſo four, and nine, are Square Numbers, that ariſing from two, <lb></lb>and this from three multiplied into themſelves.</s></p><p type="main"> <s>SALV. </s> <s>Very well; And you know alſo, that as the Products are <lb></lb>called Squares: the Produſors, that is, thoſe that are multiplied, are <lb></lb>called Sides, or Roots; and the others, which proceed not from <lb></lb>Numbers multiplied into themſelves, are not Squares. </s> <s>So that if I <lb></lb>ſhould ſay, all Numbers comprehending the Square, and the not <lb></lb>Square Numbers, are more than the Square alone, I ſhould ſpeak a <lb></lb>moſt unqueſtionable truth: Is it not ſo?</s></p><p type="main"> <s>SIMP. </s> <s>It cannot be denied.</s></p><p type="main"> <s>SALV. </s> <s>Farther queſtioning, if I ask you how many are the <lb></lb>Numbers Square, you can anſwer me truly, that they be as many, <lb></lb>as are their propper Roots; ſince every Square hath its Root, and <lb></lb>every Root its Square, nor hath any Square more than one ſole <lb></lb>Root, or any Root more than one ſole Square.</s></p><p type="main"> <s>SIMP. True.<lb></lb><arrow.to.target n="marg1024"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1024"></margin.target><emph type="italics"></emph>An Infinite Num<lb></lb>ber, as it contains <lb></lb>infinite Square <lb></lb>and Cupe Roots, ſo <lb></lb>it conta neth infi<lb></lb>nite Square and <lb></lb>Cube Numbers.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>But if I ſhall demand how many Roots there be, you <lb></lb>cannot deny but that they be as many as all Numbers, ſince there <lb></lb>is no Number that is not the Root of ſome Square: And this be<lb></lb>ing granted, it is requiſite to affirm, that Square Numbers are as <pb xlink:href="069/01/029.jpg" pagenum="27"></pb>many as their Roots, and Roots are all Numbers: and yet in the <lb></lb>beginning we ſaid, that all Numbers are far more than all Squares, <lb></lb>the greater part not being Squares: and yet nevertheleſs the num<lb></lb>ber of the Squares goeth diminiſhing alwaies with greater propor<lb></lb>tion, by how much the greater number it riſeth to; for in an hun<lb></lb>dred there are ten Squares, which is as much as to ſay, the tenth <lb></lb>part are Squares: in ten thouſand only the hundredth part are <lb></lb>Squares: in a Million only the thouſandth, and yet in an Infinite <lb></lb>Number, if we are able to comprehend it, we may ſay the Squares <lb></lb>are as many, as all Numbers put together.</s></p><p type="main"> <s>SAGR. </s> <s>What is to be reſolved then on this occaſion?</s></p><p type="main"> <s>SALV. </s> <s>I ſee no other deciſion that it may admit, but to ſay, <lb></lb>that all Numbers are infinite, Squares are infinite, their Roots are <lb></lb>infinite; and that neither is the multitude of Squares leſs than all <lb></lb>Numbers, nor this greater than that: and in concluſion, that the <lb></lb>Attributes of Equality, Majority, and Minority, have no place <lb></lb>in Infinites, but only in terminate quantities. </s> <s>And therefore when <lb></lb><emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> propoundeth to me many unequal <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ines, and demand<lb></lb>eth of me, how it can be, that in the greater there are no more <lb></lb>Points than in the leſs: I anſwer him, That there are neither more, <lb></lb>nor leſs, nor juſt ſo many; but in each of them infinite. </s> <s>Or if I <lb></lb>had anſwered him, that the Points in one, are as many as there are <lb></lb>Square Numbers; in another bigger, as many as all Numbers; in <lb></lb>a leſs, as many as the Cubick Numbers, might not I have given ſa<lb></lb>tisfaction, by aſſigning more to one, than to another, and yet to <lb></lb>every one infinite? </s> <s>And thus much as to the firſt difficulty.</s></p><p type="main"> <s>SAGR. Hold, I pray you, and give me leave to add unto what hath <lb></lb>been ſpoken hitherto, a thought which I juſt now light on, and it <lb></lb>is this, that granting what hath been ſaid, me-thinks, that not on<lb></lb>ly it's improper to ſay, one Infinite is greater than another Infinite, <lb></lb>but alſo, that it's greater than a Finite; for if an Infinite Number <lb></lb>were greater, <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> than a Million; it would thereupon follow, <lb></lb>that paſſing from the Million to others, and ſo to others continual<lb></lb>ly greater, one ſhould paſs on towards Infinity; which is not ſo: but <lb></lb>on the contrary, to how much the greater Numbers we go, ſo <lb></lb>much the more we depart from Infinite Number; becauſe in Num<lb></lb>bers, the greater you take, ſo much the rarer and rarer alwaies are <lb></lb>Square Numbers contained in them; but in Infinite Number the <lb></lb>Squares can be no leſs than all Numbers, as but juſt now was con<lb></lb>cluded: therefore the going towards Numbers alwaies greater, and <lb></lb>greater, is a departing farther from Infinite Number.</s></p><p type="main"> <s>SALV. </s> <s>And ſo by your ingenious Diſcourſe we may conclude, <lb></lb>that the Attributes of Greater, Leſſer, or Equal, have no place, <lb></lb>not only amongſt Infinites; but alſo betwixt Infinites, and Fi<lb></lb>nites.</s></p><pb xlink:href="069/01/030.jpg" pagenum="28"></pb><p type="main"> <s>I paſs now to another Conſideration; and it is, that in regard <lb></lb>that the Line, and every continued quantity are divideable conti<lb></lb>nually into diviſibles, I ſee not how we can avoid granting that the <lb></lb>compoſition is of infinite Indiviſibles: becauſe a diviſion and ſub<lb></lb>diviſion that may be proſecuted perpetually ſuppoſeth that the <lb></lb>parts are infinite; for otherwiſe the ſubdiviſion would be termina<lb></lb>ble: and the parts being Infinite, it followeth of conſequence <lb></lb>that they be non-quantitative; for infinite quantitative parts make <lb></lb>an infinite extenſion: and thus we have a <emph type="italics"></emph>Continuum<emph.end type="italics"></emph.end> compoun<lb></lb>ded of infinite Indiviſibles.</s></p><p type="main"> <s>SIMP. </s> <s>But if we may continually proſecute the diviſion in <lb></lb>quantitative parts, what need have we, for ſuch reſpect, to intro<lb></lb>duce the non-quantitative?</s></p><p type="main"> <s>SALV. </s> <s>The very poſſibility of perpetually proſecuting the di<lb></lb>viſion in quantitative parts induceth the neceſſity of the compoſiti<lb></lb>on of infinite non-quantitative. </s> <s>Therefore, coming cloſer to you, <lb></lb>I demand you to tell me reſolutely, whether the quantitative parts <lb></lb>in <emph type="italics"></emph>Continuum<emph.end type="italics"></emph.end> 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></lb>in Power, and Finite in Act. </s> <s>Infinite in Power, that is, before the <lb></lb>Diviſion; but Finite in Act, that is, after they are divided: for the <lb></lb>parts are not actually underſtood to be in the whole, till it is di<lb></lb>vided, or at leaſt marked; otherwiſe we ſay that they are in <lb></lb>Power.</s></p><p type="main"> <s>SALV. </s> <s>So that a Line <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> twenty foot long, is not ſaid to <lb></lb>contain twenty Lines of one foot a piece, actually, but only after <lb></lb>it is divided into twenty equal parts: but is till then ſaid to contain <lb></lb>them only in power. </s> <s>Now be it as you pleaſe; and tell me whe<lb></lb>ther, when the actual Diviſion of ſuch parts is made, that firſt <lb></lb>whole encreaſeth or diminiſheth, or elſe continueth of the ſame <lb></lb>bigneſs?</s></p><p type="main"> <s>SIMP. </s> <s>It neither encreaſeth, nor diminiſheth.</s></p><p type="main"> <s>SALV. </s> <s>So I think alſo. </s> <s>Therefore the quantitative parts in <emph type="italics"></emph>Con<lb></lb>tinuum<emph.end type="italics"></emph.end> quantity, be they in Act, or be they in Power, make not its <lb></lb>quantity bigger or leſſer: but it is very plain that theſe quantita<lb></lb>tive parts, actually contained in their whole, if they be infinite, <lb></lb>make it an infinite Magnitude; therefore quantitative parts, <lb></lb>though infinite only in power, cannot be contained, but only in an <lb></lb>infinite Magnitude: therefore in a finite Magnitude infinite quan<lb></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"></emph>Continuum<emph.end type="italics"></emph.end> may be <lb></lb>inceſſantly divided into parts ſtill capable of new diviſions?</s></p><p type="main"> <s>SALV. </s> <s>It ſeems that that diſtinction of Power, and Act, makes <lb></lb>that feaſible one way, which another way would be impoſſible. <lb></lb></s> <s>But I will ſee to adjuſt theſe matters by making another account: <pb xlink:href="069/01/031.jpg" pagenum="29"></pb>And to the Queſtion, which was put, Whether the quantitative <lb></lb>parts in a terminated <emph type="italics"></emph>Continuum<emph.end type="italics"></emph.end> be finite or infinite; I will anſwer <lb></lb>directly contrary to that which <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> replied, namely, that <lb></lb>they be neither finite, nor infinite.</s></p><p type="main"> <s>SIMP. </s> <s>I ſhould never have found ſuch an anſwer, not imagi<lb></lb>ning that there was any mean term between finite and infinite; <lb></lb>ſo that the diviſion or diſtinction which makes a thing to be either <lb></lb>Finite, or Infinite, is imperfect and deficient.</s></p><p type="main"> <s>SALV. </s> <s>In my opinion it is; and ſpeaking of ^{*} Diſcrete Quan</s></p><p type="main"> <s><arrow.to.target n="marg1025"></arrow.to.target><lb></lb>tities, me thinks that there is a third mean term between Finite and <lb></lb>Infinite, which is that which anſwereth to every aſſigned Number: <lb></lb>So that being demanded in our preſent caſe, Whether the quanti<lb></lb>tative parts in <emph type="italics"></emph>Continuum<emph.end type="italics"></emph.end> be Finite, or Infinite, the moſt congru<lb></lb>ous reply is to ſay, that they are neither Finite, nor Infinite, but ſo <lb></lb>many, as that they <emph type="italics"></emph>Anſwer<emph.end type="italics"></emph.end> to any number aſſigned: the which to <lb></lb>do, it is neceſſary that they be not comprehended in a limited <lb></lb>Number, for then they would not anſwer to a greater: nor, again, <lb></lb>is it neceſſary, that they be infinite, for no aſſigned Number is infi<lb></lb>nite. </s> <s>And thus at the pleaſure of the Demander, a Line being <lb></lb>propounded, we may be able to aſſign in it an hundred quantita<lb></lb>tive parts, or a thouſand, or an hundred thouſand, according to <lb></lb>the number which he beſt likes; ſo that it be not divided into in<lb></lb>finite. </s> <s>I grant therefore to the Philoſophers, that <emph type="italics"></emph>Continuum<emph.end type="italics"></emph.end> con<lb></lb>taineth as many quantitative parts as they pleaſe, and grant them <lb></lb>that it containeth the ſame either in Act, or in Power, which they <lb></lb>beſt like: but this I add again, that in like manner, as in a Line of <lb></lb>ten yards, there are contained ten Lines of one yard a piece, and <lb></lb>thirty Lines of a foot a piece, and three hundred and ſixty Lines <lb></lb>of an inch a piece, ſo it contains infinite Points; denominate them <lb></lb>in Act, or in Power, as you will: and I remit my ſelf in this matter <lb></lb>to your opinion and judgment, <emph type="italics"></emph>Simplicius.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1025"></margin.target><emph type="italics"></emph>Quantitative parts <lb></lb>in Diſcrete Quan<lb></lb>tity are neither fi<lb></lb>nite nor infinite, <lb></lb>but anſwerable to <lb></lb>every given Num<lb></lb>ber.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. </s> <s>I cannot but commend your Diſcourſe: but am great<lb></lb>ly afraid, that this parity of the Points, being contained in the like <lb></lb>manner as the quantitative parts, will not agree with abſolute ex<lb></lb>actneſs; nor ſhall it be ſo eaſie a matter for you to divide the gi<lb></lb>ven Line into infinite Points, as for thoſe Philoſophers to divide it <lb></lb>into ten yards, or thirty feet, nay, I hold it wholly impoſſible to <lb></lb>effect ſuch a diviſion: ſo that this will be one of thoſe Powers that <lb></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></lb>thing is not feaſible, render it not impoſſible; for I think alſo, that <lb></lb>you cannot ſo eaſily effect a diviſion to be made of a Line into a <lb></lb>thouſand parts; and much leſs being to divide it into 937, or ſome <lb></lb>other great Prime Number. </s> <s>But if I diſpatch this, which you, it may <lb></lb>be, judge an impoſſible diviſion, in as ſhort a time, as another <pb xlink:href="069/01/032.jpg" pagenum="30"></pb>would require to divide it into forty, you will be content more <lb></lb>willingly to admit of it in our future Diſcourſe?</s></p><p type="main"> <s>SIMP. </s> <s>I am pleaſed with your way of arguing, as you now do <lb></lb>mix it with ſome pleaſantneſs: and to your queſtion I reply, that <lb></lb>the facility would ſeem more than ſufficient, if the reſolving it into <lb></lb>Points were but as eaſie, as to divide it into a thouſand parts.</s></p><p type="main"> <s>SALV. </s> <s>Here I will tell you a thing, which haply will make you <lb></lb>wonder in this matter of going about, or being able to reſolve the <lb></lb>Line into its Infinites, keeping that order which others obſerve in <lb></lb>dividing it into forty, ſixty, or an hundred parts; namely, by di<lb></lb>viding it firſt into two, then into four: in which order he that <lb></lb>ſhould think to find its infinite Points would groſly delude himſelf; <lb></lb>for by that progreſſion, though continued to eternity, he ſhould <lb></lb>never arrive to the diviſion of all its quantitative parts: yea, he is <lb></lb>in that way ſo far from being able to arrive at the intended term <lb></lb>of Indiviſibility, that he rather goeth farther from it; and whilſt <lb></lb>he thinks by continuing the diviſion, and multiplying the multi<lb></lb>tudes of the parts, to approach to Infinite, I am of opinion, that he <lb></lb>more and more removes from it: and my reaſon is this; In the <lb></lb>Diſcourſe, we had even now, we concluded, that, in an infinite <lb></lb>Number, there was, of neceſſity, as many Square, or Cube Num<lb></lb>bers, as there were Numbers; ſince that thoſe and theſe were as ma<lb></lb>ny as their Roots, and Roots comprehend all Numbers: Next we <lb></lb>did ſee, that the greater the Numbers were that were taken, the <lb></lb>ſeldomer are their Squares to be found in them, and ſeldomer yet <lb></lb>their Cubes: Therefore it is manifeſt, that the greater the Number <lb></lb>is to which you paſs, the farther you remove from Infinite Num<lb></lb>ber: from whence it followeth, that turning backwards, (ſeeing <lb></lb>that ſuch a progreſſion more removes us from the deſired term) if <lb></lb><arrow.to.target n="marg1026"></arrow.to.target><lb></lb>any number may be ſaid to be infinite it is the Unite: and, indeed, <lb></lb>there are in it thoſe conditions, and neceſſary qualities of the Infi<lb></lb>nite Number, I mean, of containing in it as many Squares as Cubes, <lb></lb>and as Numbers.</s></p><p type="margin"> <s><margin.target id="marg1026"></margin.target><emph type="italics"></emph>The Unite of all <lb></lb>Numbers may <lb></lb>moſt properly be <lb></lb>ſaid to be Infinite.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. </s> <s>I do not apprehend very well, how this buſineſs ſhould <lb></lb>be underſtood.</s></p><p type="main"> <s>SALV. </s> <s>The thing hath no difficulty at all in it, for the Unite <lb></lb>is a Square, a Cube, a Squared Square, and all other Powers; nor <lb></lb>is there any particular whatſoever eſſential to the Square, or to the <lb></lb>Cube, which doth not agree with the Unite; as <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> one proper<lb></lb>ty of two Square-numbers is to have between them a Number <lb></lb>mean-proportional; take any Square number for one of the terms, <lb></lb>and the Unite for the other, and you ſhall likewiſe ever find be<lb></lb>tween them a Number Mean-proportional. </s> <s>Let the two Square <lb></lb>Numbers be 9 and 4, you ſee that between 9 and 1 the Mean<lb></lb>proportional is 3, and between 4 and 1 the Mean-proportional <pb xlink:href="069/01/033.jpg" pagenum="31"></pb>is 2, and between the two Squares 9 and 4, 6 is the Mean. </s> <s>The <lb></lb>property of Cubes is to have neceſſarily between them two Num<lb></lb>bers Mean-proportional. </s> <s>Suppoſe 8, and 27, the Means between <lb></lb>them are 12 and 18; and between the Unite and 8 the Means <lb></lb>are 2 and 4; betwixt the Unite and 27 there are 3, and 9. We <lb></lb>therefore conclude, <emph type="italics"></emph>That there is no other Infinite Number but the <lb></lb>Vnite.<emph.end type="italics"></emph.end> And theſe be ſome of thoſe Wonders, that ſurmount the <lb></lb>comprehenſion of our Imagination, and that advertize us how ex<lb></lb>ceedingly they err, who diſcourſe about Infinites with thoſe very <lb></lb>Attributes, that are uſed about Finites; the Natures of which have <lb></lb>no congruity with each other. </s> <s>In which affair I will not conceal <lb></lb>from you an admirable accident, that I met with ſome time ſince, <lb></lb>explaining the vaſt difference, yea, repugnance and contrariety of <lb></lb>Nature, that a terminate quantity would incur by changing or paſ<lb></lb>ſing into Infinite. </s> <s>We aſſign this Right Line A B, of any length at <lb></lb>pleaſure, and any point in the ſame, as C being taken, dividing it <lb></lb>into two unequal parts: I ſay, that many couples Lines, (hold<lb></lb>ing the ſame proportion between themſelves as have the parts <lb></lb>A C, and B C,) departing from the terms A and B to meet with <lb></lb>one another; the points of their Interſection ſhall all fall in the <lb></lb>Circumference of one and the ſame Circle: as for example, A L <lb></lb>and B L departing [or <emph type="italics"></emph>being drawn<emph.end type="italics"></emph.end>] from the Points A and B, and <lb></lb>having between themſelves the ſame proportion, as have the parts <lb></lb>A C and B C, and concurring in the point L: and the ſame pro<lb></lb>portion being between two others A K, and B K, concurring in K, <lb></lb>alſo others as A I, and B I; A H, and B H; A G, and B G; A F, <lb></lb>and B F; A E, and B E: I ſay, that the points of their Interſecti<lb></lb>on L, K, I, H, G, F, E, do all fall in the Circumference of one <lb></lb>and the ſame Semi-circle: ſo that we ſhould imagine the point <lb></lb>C to mve conti<lb></lb><figure id="id.069.01.033.1.jpg" xlink:href="069/01/033/1.jpg"></figure><lb></lb>nuallyafter ſuch <lb></lb>a ſort, that the <lb></lb>Lines produced <lb></lb>from it to the fix<lb></lb>ed terms A and <lb></lb>B retain alwaies <lb></lb>the ſame propor<lb></lb>tion that is be<lb></lb>tween the firſt <lb></lb>parts A C and C B, that point C ſhall decribe the Circumference <lb></lb>of a Circle, as we ſhall ſhew you preſently. </s> <s>And the Circle in ſuch <lb></lb>ſort deſcribed ſhall be alwaies greater and greater ſucceſſively, <lb></lb>according as the point C is taken nearer to the middle point <lb></lb>which is O; and the Circle ſhall be leſſer which ſhall be deſcribed <lb></lb>from a point nearer to the extremity B, inſomuch, that from the <pb xlink:href="069/01/034.jpg" pagenum="32"></pb>infinite Points which may be taken in the Line O B, there may be <lb></lb>deſcribed Circles (moving them in ſuch ſort as above is preſcri<lb></lb>bed) of any Magnitude; leſſer than the Pupil of the eye of a <lb></lb>Flea, and bigger than the Equinoctial of the <emph type="italics"></emph>Primum Mobile.<emph.end type="italics"></emph.end><lb></lb>Now, if raiſing any of the Points comprehended betwixt the terms <lb></lb>O and B, from every one we may deſcribe Circles, and vaſt ones <lb></lb>from the Points nearer to O; then if we raiſe the Point O it ſelf, <lb></lb>and continue to move it in ſuch ſort as aforeſaid, that is, that the <lb></lb>Lines drawn from it to the terms A and B keep the ſame proporti<lb></lb>on as have the firſt Lines A O, and O B, what Line ſhall be deſcri<lb></lb>bed? </s> <s>There would be deſcribed the Circumference of a Circle, <lb></lb>but of a Circle bigger than the biggeſt of all Circles, therefore of <lb></lb>a Circle that is infinite: but it doth alſo deſcribe a Right Line, and <lb></lb>perpendicular upon A B, erected from the Point O, and produced <lb></lb><emph type="italics"></emph>in infinitum<emph.end type="italics"></emph.end> without ever turning to reunite its laſt term with the <lb></lb>firſt, as the others did; for the limited motion of the Point C, after <lb></lb>it had deſigned the upper Semi-circle C H E, continued to de<lb></lb>ſcribe the Lower E M C, reuniting its extream terms in the point <lb></lb>C: But the Point O being moved to deſign (as all the other Points <lb></lb>of the Line A B, for the Points taken in the other part O A <lb></lb>ſhall deſign their Circles, and thoſe Points neareſt to O the <lb></lb>greateſt) its Circle; to make it the biggeſt of all, and conſe<lb></lb>quently infinite, it can never return any more to its firſt term, and <lb></lb><arrow.to.target n="marg1027"></arrow.to.target><lb></lb>in a word deſigneth an Infinite Right-Line for the Circumference <lb></lb>of its Infinite Circle. </s> <s>Conſider now, what difference there is be<lb></lb>tween a finite Circle, and an infinite; ſeeing that this in ſuch man<lb></lb>ner changeth its being that it wholly loſeth both its being, and <lb></lb>power of being; for we have already well comprehended, that <lb></lb>there cannot be aſſigned an infinite Circle; by which we may <lb></lb>conſequently know that there can be no infinite Sphære, or other <lb></lb>Body, or figured Superficies. </s> <s>Now what ſhall we ſay to this Meta<lb></lb>morphoſis in paſſing from Finite to Infinite? </s> <s>And why ſhould we <lb></lb>find greater repugnance, whilſt ſeeking Infinity in Numbers, we <lb></lb><arrow.to.target n="marg1028"></arrow.to.target><lb></lb>come to conclude it to be in the Unite? </s> <s>And whilſt that breaking <lb></lb>a Solid into many pieces, and purſuing to reduce it into very ſmall <lb></lb>powder, it were reſolved into its infinite Atomes, admitting no far<lb></lb>ther diviſion, why may we not ſay that it is returned into one ſole <lb></lb><emph type="italics"></emph>Continuum,<emph.end type="italics"></emph.end> but perhaps fluid, as the Water, or Quickſilver, or <lb></lb>other Metall melted? </s> <s>And do we not ſee Stones liquified into <lb></lb>Glaſs, and Glaſs it ſelf with much Fire to become more fluid than <lb></lb>Water?</s></p><p type="margin"> <s><margin.target id="marg1027"></margin.target><emph type="italics"></emph>The difference be<lb></lb>twixt a finite and <lb></lb>infinite Circle.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1028"></margin.target><emph type="italics"></emph>Vnity participates <lb></lb>of Infinity.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>Should we therefore think Fluids to be ſo called, be<lb></lb>cauſe they are reſolved into their firſt, infinite, indiviſible com<lb></lb>pounding parts?</s></p><p type="main"> <s>SALV. </s> <s>I know not how to find a better anſwer to reſolve cer<pb xlink:href="069/01/035.jpg" pagenum="33"></pb>tain ſenſible appearances, amongſt which this is one: When I take <lb></lb>a hard Body, be it either Stone, or Metal, and with a Hammer, or <lb></lb>very fine File, endeavour to divide it, as much as is poſſible, into <lb></lb>its moſt minute and impalpable powder; it is very clear, that its <lb></lb>leaſt Atomes, albeit for their ſmalneſs they are imperceptible, one by <lb></lb>one, to our ſight and touch; yet are they quantitative, figured, and <lb></lb>numerable: and it happens in them, that being accumulated to<lb></lb>gether, they continue in heap; and being laid hollow, or with a <lb></lb>pit in the midſt, the hollowneſs or pit remains, the parts heaped <lb></lb>about it not returning to fill it up; and being ſtirr'd, or ſhaken, <lb></lb>they ſuddenly ſettle ſo ſoon as their external mover leaves them, <lb></lb>And the like effects are ſeen in all the Aggregates of ſmall Bodies, <lb></lb>bigger, and bigger, and of any kind of Figure, although Sphærical; <lb></lb>as we ſee in heaps of Peaſe, Wheat, Bird ſhot, and other matters. </s> <s>But <lb></lb>if we try to find the like accidents in Water, you will meet with <lb></lb>none of them; but, being raiſed, it inſtantly returns to a level, if <lb></lb>it be not by a veſſel, or ſome other external ſtay upheld; being <lb></lb>made hollow, it preſently diffuſeth to fill up the Cavity; and be<lb></lb>ing long moved, it continually undulates, and ſpreads its waves very <lb></lb>far. </s> <s>From this, I think, we may very rationally infer, that the minute <lb></lb><arrow.to.target n="marg1029"></arrow.to.target><lb></lb>parts of Water, into which it ſeemeth to be reſolved, (ſince it hath <lb></lb>leſs conſiſtence than any the fineſt powder, yea, hath no conſi<lb></lb>ſtence at all) are vaſtly differing from Atomes quantitative and <lb></lb>diviſible; nor know I how to find any other difference therein <lb></lb>than that of being indiviſible. </s> <s>Methinks, alſo, that its moſt exqui<lb></lb>ſite tranſparency, affords us ſufficient grounds to conjecture there<lb></lb>of; for if we take the moſt diaphanous Chriſtal that is, and begin <lb></lb>to break, and pound it to powder, when it is in powder it loſeth <lb></lb>its tranſparency, and ſo much the more, the ſmaller it is pounded; <lb></lb>but yet Water which is ground to the higheſt degree, hath alſo the <lb></lb>higheſt degree of Diaphaneity Gold and Silver, reduced by <emph type="italics"></emph>Aqua<lb></lb>fortis<emph.end type="italics"></emph.end> into a ſmaller Powder than any File can make, yet they con<lb></lb>tinue powder, and become not fluid; nor do they liquifie till the <lb></lb>Indiviſibles of the Fire, or of the Sun-beams diſſolve them, as, I be<lb></lb>lieve, into their firſt and higheſt infinite and indiviſible compoun<lb></lb>ding parts.</s></p><p type="margin"> <s><margin.target id="marg1029"></margin.target><emph type="italics"></emph>Fluid Bodies are <lb></lb>ſuch, for that they <lb></lb>are reſolved into <lb></lb>their firſt Indiviſi<lb></lb>ble Atomes.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>This which you have hinted of the Light I have many <lb></lb>times obſerved with admiration: I have ſeen, I ſay, a burning<lb></lb>Glaſs, of a foot Diameter, liquifie or melt lead in an inſtant; <lb></lb>whence I came to be of opinion, that if the Glaſſes were very big, <lb></lb>and very polite, and of Parabolical Figure, they would no leſs melt <lb></lb>every other Metal in a very ſhort time; ſeeing that that, not very <lb></lb>big, nor very clear, and of a Sphærical Concave, with ſuch force <lb></lb>melted Lead, and burnt every combuſtible matter: effects, that <lb></lb>make the wonders, reported of the Burning-glaſſes of <emph type="italics"></emph>Archimedes,<emph.end type="italics"></emph.end><lb></lb>credible to me.<pb xlink:href="069/01/036.jpg" pagenum="34"></pb><arrow.to.target n="marg1030"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1030"></margin.target>Archimedes <emph type="italics"></emph>his <lb></lb>Burning — Glaſſes <lb></lb>admirable.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>Touching the Effects of the Glaſſes, invented by <emph type="italics"></emph>Ar<lb></lb>chimedes,<emph.end type="italics"></emph.end> all the Miracles, that ſeveral Writers record of them, <lb></lb>are to me rendred credible by the reading of <emph type="italics"></emph>Archimedes<emph.end type="italics"></emph.end> his own <lb></lb>Books, which I have with infinite amazement peruſed and ſtudied: <lb></lb>and if any doubts had been left me; that which laſt of all Father </s></p><p type="main"> <s><arrow.to.target n="marg1031"></arrow.to.target><lb></lb><emph type="italics"></emph>Buonaventura Cavalieri<emph.end type="italics"></emph.end> hath publiſhed, touching <emph type="italics"></emph>Lo Specehio <lb></lb>Vſtorio,<emph.end type="italics"></emph.end> (or the Burning glaſs) and which I have read with ad<lb></lb>miration, is ſufficient to reſolve them all.</s></p><p type="margin"> <s><margin.target id="marg1031"></margin.target>Buonaventura <lb></lb>Cavalieri, <emph type="italics"></emph>the Je<lb></lb>ſuate, a famous <lb></lb>Mathematician, <lb></lb>and his Book en<lb></lb>titled,<emph.end type="italics"></emph.end> Lo Spec<lb></lb>chio Uſtorio.</s></p><p type="main"> <s>SAGR. </s> <s>I have alſo ſeen that Tract, and peruſed it with much <lb></lb>delight and wonder; and becauſe I formerly had knowledge of <lb></lb>the Author, I was confirmed in the opinion which I had conceived <lb></lb>of him, that he was like to prove one of the principal Mathemati<lb></lb>cians of our Age. </s> <s>But returning to the admirable effects of the <lb></lb>Sun-Beams in melting of Metals, are we to believe that ſuch, and <lb></lb>ſo violent an operation is without Motion, or elſe that it is with <lb></lb>Motion, but extream ſwift?<lb></lb><arrow.to.target n="marg1032"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1032"></margin.target><emph type="italics"></emph>Burnings are per<lb></lb>formed with a moſt <lb></lb>ſwift Motion.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>We ſee other burnings, and meltings to be performed <lb></lb>with Motion, and with a moſt ſwift Motion. </s> <s>Obſerve the ope<lb></lb>rations of Lightnings, of Powder in Mines, and in Petards, <lb></lb>and, in ſum, how by quickning the flame of Coles, mixt with <lb></lb>groſs and impure vapours, by Bellows, encreaſeth its force in <lb></lb>the melting of Metals: ſo that I cannot ſee how the Action of <lb></lb>Light, albeit moſt pure, can be without Motion, and that alſo ve<lb></lb>ry ſwift.</s></p><p type="main"> <s><emph type="italics"></emph>S<emph.end type="italics"></emph.end>AGR. </s> <s>But what and how great ought we to judge this Velo<lb></lb>city of the Light? </s> <s>Is it haply <emph type="italics"></emph>Inſtantaneous,<emph.end type="italics"></emph.end> and done in a moment, <lb></lb>or, as the reſt of Motions, performed in Time? </s> <s>May we not by <lb></lb>Experiment be aſſured what it is?</s></p><p type="main"> <s>SIMP. </s> <s>Quotidian experience ſhews the expanſion of Light to <lb></lb>be <emph type="italics"></emph>Inſtantaneous<emph.end type="italics"></emph.end>; in that beholding a Cannon, let off at a great <lb></lb>diſtance, the flaſh of the fire, without interpoſition of time, is tranſ<lb></lb>mitted to our eye, but ſo is not the Report to our ear untill a con<lb></lb>ſiderable time after.</s></p><p type="main"> <s>SAGR. True, but, I pray you, what doth this obvious experi<lb></lb>ment evince; but only this, that the Report is longer in arriving at <lb></lb>our Ear, than the Flaſh at our Eye; but it aſſures me not, that the <lb></lb>tranſmiſſion of the Light is therefore <emph type="italics"></emph>Inſtantaneous<emph.end type="italics"></emph.end> rather than in <lb></lb>Time, but only moſt ſwift. </s> <s>Nor doth ſuch an obſervation con<lb></lb>clude more than that other, of ſuch who ſay, that as ſoon as the <lb></lb>Sun cometh to the Horizon, its Light arriveth at our eye: for who <lb></lb>ſhall aſſure me, that its beams arrive not at the ſaid term, afore they <lb></lb>reach our ſight?</s></p><p type="main"> <s>SALV. </s> <s>The inconcludency of theſe, and other obſervations of <lb></lb>the like Nature, made me once think of ſome other way, whereby <lb></lb>we may without errour be aſcertained whether the illumination, <pb xlink:href="069/01/037.jpg" pagenum="35"></pb>that is, whether the expanſion of the Light were really <emph type="italics"></emph>Inſtantane<lb></lb>ous<emph.end type="italics"></emph.end>; ſeeing that the very ſwift Motion of Sound, aſſureth us, that <lb></lb>that of Light cannot but be extream ſwift. </s> <s>And the experiment I </s></p><p type="main"> <s><arrow.to.target n="marg1033"></arrow.to.target><lb></lb>hit upon, was this; I would have two perſons take each of them a <lb></lb>Light, which, by holding it in a Lanthorn, or other coverture, they <lb></lb>may cover, and diſcover at pleaſure by interpoſing their hand to the <lb></lb>fight of each other; and, that placing themſelvs againſt one another, <lb></lb>ſome few paces diſtance, they may practice the ſpeedy diſcovery, <lb></lb>and occultation of their Lights from the ſight of each other: So <lb></lb>that when one ſeeth the others Light, he immediatly diſcloſe his: <lb></lb>which correſpondence, after ſome Reſponſes mutually made, will <lb></lb>become ſo exactly Inſtantaneous, that, without ſenſible variation, <lb></lb>at the diſcovery of the one, the other ſhall at the ſame time ap<lb></lb>pear to the ſight of him that diſclos'd the firſt. </s> <s>Having adjuſted <lb></lb>this practice at this ſmall diſtance, let us place the two perſons with <lb></lb>two ſuch Lights at two or three miles diſtance; and by night re<lb></lb>newing the ſame experiment; Let them intenſely obſerve if the <lb></lb>Reſponſes of the diſcloſures, and occultations do follow the ſame <lb></lb>tenour which they did near hand: for if they keep the ſame pro<lb></lb>portion, it may be with certainty enough concluded, that the ex<lb></lb>panſion of Light is Inſtantaneous; but if it ſhould require time in <lb></lb>a diſtance of three miles, which importeth ſix for the going of <lb></lb>one, and return of the other, the ſtay would be ſufficiently obſer<lb></lb>vable. </s> <s>And if this Experiment be made at greater diſtances, <lb></lb>namely, at eight or ten miles, we may make uſe of the <emph type="italics"></emph>Teleſcope,<emph.end type="italics"></emph.end><lb></lb>the Obſervators accommodating each of them one at the places, <lb></lb>where by night the Lights are to be obſerved; which though not <lb></lb>very big, and ſo not viſible, at that great diſtance, to the eye at <lb></lb>large; (though eaſie to be diſcloſed, and hid) by help of the <lb></lb><emph type="italics"></emph>Teleſcopes<emph.end type="italics"></emph.end> before admitted, and fixed they may commodiouſly be <lb></lb>diſcerned.</s></p><p type="margin"> <s><margin.target id="marg1033"></margin.target><emph type="italics"></emph>The Velocity of <lb></lb>Light, how to find <lb></lb>by Experiment <lb></lb>whether it be In<lb></lb>ſtantaneosu or not.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>The Invention ſeems to me no leſs certain than ingenu<lb></lb>ous; but tell us what upon experimenting it you concluded.</s></p><p type="main"> <s><emph type="italics"></emph>S<emph.end type="italics"></emph.end>ALV. Really, I have not tryed it, ſave only at a ſmall diſtance, <lb></lb>namely, leſs than a Mile: whereby I could come to no certainty <lb></lb>whether the apparence of the oppoſite Light was truly Inſtantane<lb></lb>ous; But if not Inſtantaneous, yet it was of exceeding great Velo<lb></lb>city, and I may ſay Momentary: and for the preſent, I would re<lb></lb>ſemble it to that Motion which we ſee a flaſh of Lightning make <lb></lb>in the Clouds ten or more Miles off: of which Light we diſtin<lb></lb>guiſh the beginning, and, I may fay, the ſource and riſe of it, in a <lb></lb>particular place in thoſe Clouds; but yet its wide expanſion imme<lb></lb>diatly ſucceeds amongſt thoſe adjacent: which to me ſeems an ar<lb></lb>gument that it is ſome ſmall time in doing; becauſe had the illu<lb></lb>mination been made all at once, and not by degrees, it feems to <pb xlink:href="069/01/038.jpg" pagenum="36"></pb>me that we could not have diſtinguiſhed its original, or rather the <lb></lb>Center of its flake, and extream Dilatations. </s> <s>But into what Oceans <lb></lb>do we by degrees engage our ſelves? </s> <s>Amongſt <emph type="italics"></emph>Vacuities, Infinites, <lb></lb>Indiviſibles,<emph.end type="italics"></emph.end> and <emph type="italics"></emph>Instantaneous Motions<emph.end type="italics"></emph.end>; ſo that we ſhall not be <lb></lb>able by a thouſand Diſcourſes to recover the Shore?</s></p><p type="main"> <s>SAGR. </s> <s>They are things, indeed, very diſproportionate to our <lb></lb>underſtanding. </s> <s>Behold Infinite, ſought amongſt Numbers, ſeemeth <lb></lb>to determine in the Unite: From Indiviſibles ariſeth things that <lb></lb>are continually diviſible: Vacuity ſeems only to reſide indiviſibly <lb></lb>mixt with Repletion: and, in brief, theſe things ſo change the <lb></lb>nature of thoſe underſtood by us, that even the Circumference of <lb></lb>a Circle becometh an Infinite Right-Line; which, if I well re<lb></lb>member, is that Propoſition which you, <emph type="italics"></emph>Salviatus,<emph.end type="italics"></emph.end> are to mani<lb></lb>feſt by Geometrical Demonſtration. </s> <s>Therefore, if you think fit, <lb></lb>it would be well, without any more digreſſions, to make it out <lb></lb>to us.</s></p><p type="main"> <s>SALV. </s> <s>I am ready to ſerve you in demonſtrating the enſuing <lb></lb>Problem for your fuller information.</s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s><emph type="italics"></emph>A Right-Line being given, divided, according to any <lb></lb>proportion, into unequal parts, to deſcribe a Circle, to <lb></lb>the Circumference of which, at any point of the ſame, <lb></lb>two Right-Lines being produced from the terms of <lb></lb>the given Right Line, they may retain the ſame pro<lb></lb>portion that the parts of the ſaid Line given have to <lb></lb>one another, ſo that thoſe be Homologous which de<lb></lb>part ſrom the ſame terms.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let the given Right-Line be AB, unequally divided ac<lb></lb>cording to any proportion in the point C; it is required to <lb></lb>deſcribe a Circle at any point of whoſe Circumference two <lb></lb>Right Lines, produced from the terms A and B, concurring, have <lb></lb>the ſame proportion to each other, that A C, hath to B C, ſo that <lb></lb>thoſe be Homologous which depart from the ſame term. </s> <s>Upon <lb></lb>the Center C, at the diſtance of the leſſer part C B, let a Circle be <lb></lb>ſuppoſed to be deſcribed, to the Circumference of which from the <lb></lb>point A the Right-line A D is made a Tangent, and indetermi<lb></lb>nately prolonged towards E: and let the Contact be in D, and <lb></lb>draw a Line from C to D, which ſhall be perpendicular to A E; <lb></lb>and let B E be perpendicular to B A, which produced, ſhall inter<pb xlink:href="069/01/039.jpg" pagenum="37"></pb>ſect A E, the Angle A being acute: Let the interſection be in E, <lb></lb>from whence let fall a Perpendicular to A E, which produced, will <lb></lb>meet with A B infinitely prolonged in F. </s> <s>I ſay, firſt, that the <lb></lb>Right-lines F E, and F C are equal: ſo that drawing the Line <lb></lb>E C, we ſhall, in the <lb></lb><figure id="id.069.01.039.1.jpg" xlink:href="069/01/039/1.jpg"></figure><lb></lb>two Triangles D E C, <lb></lb>B E C, have the two <lb></lb>Sides of the one, D E, <lb></lb>and C E, equal to the <lb></lb>two Sides of the other <lb></lb>B E, and E C; the <lb></lb>two Sides, D E, and <lb></lb>E B, being Tangents <lb></lb>to the Circle D B, <lb></lb>and the Baſes D C, <lb></lb>and C B, are likewiſe <lb></lb>equal: wherefore the <lb></lb>two Angles D E C, <lb></lb>and B E C, ſhall be <lb></lb>equal. </s> <s>And becauſe the Angle B C E wanteth of being a Right<lb></lb>Angle, as much as the Angle B E C; and the Angle C E F, to <lb></lb>make it a Right-Angle, wants the Angle C E D, thoſe Supple<lb></lb>ments being equal, the Angles F C E, and F E C ſhall be equal, <lb></lb>and ſo conſequently the Sides F E, and F C; wherefore making <lb></lb>the point F a Center, and at the diſtance F E, deſcribing a Circle, <lb></lb>it ſhall paſs by the point C. </s> <s>Deſcribe it, and let it be C E G. </s> <s>I ſay, <lb></lb>that this is the Circle required, by any point of the Circumfe<lb></lb>rence of which, any two Lines that ſhall interſect, departing from <lb></lb>the terms A and B, ſhall be in proportion to each other, as are the <lb></lb>two parts A C, and B C, which beſore did concur in the point C. <lb></lb></s> <s>This is manifeſt in the two that concur or interſect in the point E, <lb></lb>that is A E, and B E; the Angle E of the Triangle A E B being <lb></lb>divided in the midſt by C E; ſo that as A C is to C B, ſo is A E <lb></lb>to B E. </s> <s>The ſame we prove in the two A G, and B G, determined <lb></lb>in the point G. </s> <s>Therefore being (by the Similitude of the Tri<lb></lb>angles A F E, and E F B) that as A F is to E F, ſo is E F to F B; <lb></lb>that is, as A F is to F C, ſo is C F to F B: So by Diviſion; as A C <lb></lb>is to C F, (that is, to F G) ſo is C B to B F; and the whole A B <lb></lb>is to the whole B G, as the part C B to the part B F: and by Com<lb></lb>poſition; as A G is to G B, ſo is C F to F B; that is, as E F to <lb></lb>F B, that is, as A E to E B, and A C to C B: Which was to be de<lb></lb>monſtrated. </s> <s>Again, let any other Point be taken in the Circum<lb></lb>ference, as H; in which the two Lines A H and B H concur. </s> <s>I ſay, in <lb></lb>like manner as before, that as A C is to C B, ſo is A H to B H. <lb></lb></s> <s>Continue H B untill it interſect the Circumference in I, and draw <pb xlink:href="069/01/040.jpg" pagenum="38"></pb>a Line joyning I to F. </s> <s>And becauſe it hath been proved already <lb></lb>that as A B is to B G, ſo is C B to B F, the Rectangle A B F ſhall be <lb></lb>equall to the Rectangle C B G, that is I B H: and therefore, as <lb></lb>A B is to B H, ſo is I B to B F, and the Angles at B are equal: <lb></lb>Therefore A H is to H B, as I F, that is E F, to F B, and as A E <lb></lb>to E B.</s></p><p type="main"> <s>I ſay moreover, that it is impoſſible, that the Lines, which have <lb></lb>this ſame proportion, departing from the terms A and B, ſhould <lb></lb>meet in any point, either within or without the ſaid Circle: For<lb></lb>aſmuch as if it be poſſible that two Lines ſhould concur in the <lb></lb>point L, placed without; let them be A L, and B L; and continue <lb></lb>L B to the Circumference in M, and conjoyn M to F. </s> <s>If therefore <lb></lb>A L is to B L, as A C to B C, that is, as M F to F B, we ſhall have <lb></lb>two Triangles A L B, and M F B, which about the two Angles <lb></lb>A L B and M F B have their Sides proportional, their upper Angles <lb></lb>in the point B equal, and the two remaining Angles F M B and <lb></lb>L A B leſs than Right Angles (for that the Right-angle at the <lb></lb>point M hath for its Baſe the whole Diameter C G, and not the <lb></lb>ſole part B F, and the other at the point A is acute by reaſon the <lb></lb>Line A L Homologous to A C, is greater than B L Homologous to <lb></lb>B C) Therefore the Triangles A B L, and M B F are like: and <lb></lb>therefore as A B is to B L, ſo is M B to B F; Wherefore the <lb></lb>Rectangle A B F ſhall be equall to the Rectangle M B L. </s> <s>But the <lb></lb>Rectangle A B F hath been demonſtrated to be equal to that of <lb></lb>C B G: Therefore the Rectangle M B L is equal to the Rectangle <lb></lb>C B G, which is impoſſible: Therefore the Concourſe of the Lines <lb></lb>cannot fall without the Circle. </s> <s>And in like manner it may be de<lb></lb>monſtrated that it cannot fall within; Therefore all the Concour<lb></lb>ſes fall in the Circumference it ſelf.</s></p><p type="main"> <s>But it is time that we return to give ſatisfaction to the Intreaty <lb></lb>of <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> ſhewing him that the reſolving the Line into its in<lb></lb>finite Points is not only not impoſſible, but that it hath in it no <lb></lb>more difficulty than to diſtinguiſh its quantitative parts; preſup<lb></lb>poſing one thing (notwithſtanding) which I think, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end><lb></lb>you will not deny me, and that is this; that you will not require me <lb></lb>to ſever the Points one from another, and ſhew you them one by <lb></lb>one diſtinctly upon this paper: for I my ſelfe ſhould be content, <lb></lb>if without enjoyning to pull the four or ſix parts of a Line from <lb></lb>one another, you ſhould but ſhew me its diviſions marked, or at <lb></lb>moſt inclined to Angles, framing them into a Square, or a Hexa<lb></lb>gon; therefore I perſwade my ſelf, that for the preſent you will <lb></lb>grant them then ſufficiently, and actually diſtinguiſhed.</s></p><p type="main"> <s>SIMP. </s> <s>I ſhall indeed.</s></p><p type="main"> <s>SALV. </s> <s>Now if the inclining of a Line to Angles, framing <lb></lb><arrow.to.target n="marg1034"></arrow.to.target><lb></lb>therewith ſometimes a Square ſometimes an Octagon, ſometimes <pb xlink:href="069/01/041.jpg" pagenum="39"></pb>a Poligon of Forty, of an <emph type="italics"></emph>H<emph.end type="italics"></emph.end>undred, of a Thouſand Angles be a <lb></lb>mutation ſufficient to reduce into Act thoſe four, eight, forty, <lb></lb>hundred, or thouſand parts, which were, as you ſay, Potentially <lb></lb>in the ſaid Line at firſt: if I make thereof a Poligon of infinite <lb></lb>Sides, namely, when I bend it into the Circumference of a Circle, <lb></lb>may not I, with the like leave, ſay, that I have reduced thoſe infi<lb></lb>nite parts into Act, which you before, whilſt it was ſtraight, ſaid <lb></lb>were Potentially contained in it? </s> <s>Nor may ſuch a Reſolution be <lb></lb>denied to be made into its Infinite Points, as well as that of its four <lb></lb>parts in forming thereof a Square, or into its thouſand parts in <lb></lb>forming thereof a Mill-angular Figure; by reaſon that there wants <lb></lb>not in it any of the Conditions found in the Poligon of a thou<lb></lb>ſand, or of an hundred thouſand Sides. </s> <s>This applied or layed to a <lb></lb>Right-Line covereth it, touching it with one of its Sides, that is, <lb></lb>with one of its hundred thouſandth parts; the Circle, which is a <lb></lb>Poligon of infinite Sides, toucheth the ſaid Right-line with one of <lb></lb>its Sides, that is one ſingle Point divers from all its Colaterals, and <lb></lb>therefore divided, and diſtinct from them, no leſs than a Side of <lb></lb>the Poligon from its Conterminals. </s> <s>And as the Poligon turned <lb></lb>round upon a Plane deſcribes, with the conſequent tacts of its Sides, <lb></lb>a Right-line equal to its Perimeter: ſo the Circle, rowled upon <lb></lb>ſuch a Plane, deſcribes or ſtamps upon it, by its infinite ſucceſſive <lb></lb>Contacts, a Right-line, equall to its own Circumference. </s> <s>I know <lb></lb>not at preſent, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> whether or no the Peripateticks, (to <lb></lb>whom I grant, as a thing moſt certain, that <emph type="italics"></emph>Continuum<emph.end type="italics"></emph.end> may be di<lb></lb>vided into parts alwaies diviſible, ſo that continuing the diviſion <lb></lb>and ſubdiviſion there can be no end thereof) will be content to <lb></lb>yield to me, that none of thoſe diviſions are the ultimate, as in<lb></lb>deed they be not, ſince that there alwaies remains another; but <lb></lb>that only to be the laſt, which reſolves it into infinite Indiviſibles; <lb></lb>to which I yield we can never attain, dividing and ſubdividing it <lb></lb>ſucceſſively into a greater, and greater multitude of parts: but <lb></lb>making uſe of the way which I propound to diſtinguiſh and re<lb></lb>ſolve all the infinite parts at one only draught, (an Artifice which <lb></lb>ought not to be denied me) I could perſwade my ſelf they <lb></lb>would ſatisfie themſelves, and admit this compoſition of <emph type="italics"></emph>Continu-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1035"></arrow.to.target><lb></lb><emph type="italics"></emph>um<emph.end type="italics"></emph.end> to conſiſt of Atomes abſolutely indiviſible: And eſpecially, <lb></lb>this one path being more current than any other to extricate us <lb></lb>out of very intricate Laberinths; ſuch as are, (beſides that alrea<lb></lb>dy touched of the Coherence of the parts of Solids) the concei<lb></lb>ving the buſineſs of Rarefaction and Condenſation, without <lb></lb>running into the inconvenience of being forced to admit forth of <lb></lb>void Spaces or Vacuities; and for this a Penetration of Bodies: in<lb></lb>conveniences, which both, in my opinion, may eaſily be avoided, <lb></lb>by granting the foreſaid Compoſition of Indiviſibles.</s></p><pb xlink:href="069/01/042.jpg" pagenum="40"></pb><p type="margin"> <s><margin.target id="marg1034"></margin.target><emph type="italics"></emph>How infinite points <lb></lb>are aſſigned in a <lb></lb>finite Line.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1035"></margin.target>Continuum <emph type="italics"></emph>com<lb></lb>pounded of Indivi<lb></lb>ſibles.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. </s> <s>I know not what the Peripateticks would ſay, in regard <lb></lb>that the Conſiderations you have propoſed would be, for the moſt <lb></lb>part, new unto them, and as ſuch, it is requiſite that they be exa<lb></lb>mined: and it may be, that they would find you anſwers, and <lb></lb>powerful Solutions, to unty theſe knots, which I, by reaſon of the <lb></lb>want of time and ingenuity proportionate, cannot for the preſent <lb></lb>reſolve. </s> <s>Therefore, ſuſpending this particular for this time, I <lb></lb>would gladly underſtand how the introduction of theſe Indiviſi<lb></lb>bles facilitateth the knowledge of Condenſation, and Rarefa<lb></lb>ction, avoiding at the ſame time a <emph type="italics"></emph>Vacuum,<emph.end type="italics"></emph.end> and the Penetration of <lb></lb>Bodies.</s></p><p type="main"> <s>SAGR. </s> <s>I alſo much long to underſtand the ſame, it being to <lb></lb>my Capacity ſo obſcure: with this <emph type="italics"></emph>proviſo,<emph.end type="italics"></emph.end> that I be not couzen<lb></lb>ed of hearing (as <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> ſaid but even now) the Reaſons of <lb></lb><emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end> in confutation of a <emph type="italics"></emph>Vacuum,<emph.end type="italics"></emph.end> and conſequently the Solu<lb></lb>tions which you bring, as ought to be done, whilſt that you ad<lb></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 firſt <lb></lb>it's neceſſary, that like as in favour of Rarefaction, we make uſe of <lb></lb>the Line deſcribed by the leſſer Circle bigger than its own Cir<lb></lb>cumference, whilſt it was moved at the Revolution of the greater; <lb></lb>ſo, for the underſtanding of Condenſation, we ſhall ſhew, how that, <lb></lb>at the converſion made by the leſſer Circle, the greater deſcribeth <lb></lb>a Right-line leſs than its Circumference; for the clearer explicati<lb></lb>on of which, let us ſet before us the conſideration of that which <lb></lb>befalls in the Poligons. </s> <s>In a deſcription like to that other; ſup<lb></lb>poſe two Hexagons about the common Center L, which let be <lb></lb>A B C, and H I K, with the Parallel-lines H O M, and A B C, up<lb></lb>on which they are to make their Revolutions; and the Angle I, of <lb></lb>the leſſer Poligon, reſting at a ſtay, turn the ſaid Poligon till ſuch <lb></lb>time as I K fall upon the Parallel, in which motion the point K <lb></lb>ſhall deſcribe the Arch K M, and the Side K I, ſhall unite with the <lb></lb>part I M; while this is in doing, you muſt obſerve what the Side <lb></lb>C B of the greater Poligon will do. </s> <s>And becauſe the Revolution <lb></lb>is made upon the Point I, the Line I B with its term B ſhall de<lb></lb>ſcribe, turning backward the Arch B b, below the Parallel c A, ſo <lb></lb>that when the Side K I ſhall fall upon the Line M I, the Line B C <lb></lb>ſhall fall upon the Line b c, advancing forwards only ſo much as <lb></lb>is the Line B c, and retiring back the part ſubtended by the Arch <lb></lb>B b, which falls upon the Line B A, and intending to continue af<lb></lb>ter the ſame manner the Revolution of the leſſer Poligon, this will <lb></lb>deſcribe, and paſs upon its Parallel, a Line equal to its Perimeter; <lb></lb>but the greater ſhall paſs a Line leſs than its Perimeter, the quan<lb></lb>tity of ſo many of the lines <emph type="italics"></emph>B<emph.end type="italics"></emph.end> b as it hath Sides, wanting one; <lb></lb>and that ſame line ſhall be very near equal to that deſcribed by <pb xlink:href="069/01/043.jpg" pagenum="41"></pb>the leſſer Poligon, exceeding it only the quantity of b B. </s> <s>Here <lb></lb>then, without the leaſt repugnance the cauſe is ſeen, why the grea<lb></lb>ter Poligon paſſeth or moveth not (being carried by the leſs) <lb></lb>with its Sides a greater Line than that paſſed by the leſs; that is, <lb></lb>becauſe that one part of each of them falleth upon its next coter<lb></lb>minal and precedent.</s></p><p type="main"> <s>But if we ſhould conſider the two Circles about the Center A, <lb></lb>reſting upon their Parallels, the leſſer touching his in the point B, <lb></lb>and the greater his in the <lb></lb><figure id="id.069.01.043.1.jpg" xlink:href="069/01/043/1.jpg"></figure><lb></lb>point C; here, in begin<lb></lb>ning to make the Revolu<lb></lb>tion of the leſs, it ſhall not <lb></lb>occur as before, that the <lb></lb>point B reſt for ſome time <lb></lb>immoveable, ſo that the <lb></lb>Line B C giving back, <lb></lb>carry with it the point C, <lb></lb>as it befell in the Poligons, <lb></lb>which reſting fixed in the <lb></lb>point I till that the Side <lb></lb>K I falling upon the Line <lb></lb>I M, the Line I B carried <lb></lb>back B, the term of the <lb></lb>Side C B, as far as b, by <lb></lb>which means the Side B C <lb></lb>fell on b c, ſuper-poſing or <lb></lb>reſting the part B b upon <lb></lb>the Line B A, and advancing forwards only the part <emph type="italics"></emph>B<emph.end type="italics"></emph.end> c, equal to <lb></lb>I M, that is to one Side of the leſſer Poligon: by which ſuperpoſi<lb></lb>tions, which are the exceſſes of the greater Sides above the leſs, the <lb></lb>advancements which remain equal to the Sides of the leſſer Poli<lb></lb>gon come to compoſe in the whole Revolution the Right-line <lb></lb>equal to that traced, and meaſured by the leſſer Poligon. </s> <s>But <lb></lb><arrow.to.target n="marg1036"></arrow.to.target><lb></lb>now, I ſay, that if we would apply this ſame diſcourſe to the ef<lb></lb>fect of the Circles, it will be requiſite to confeſs, that whereas the <lb></lb>Sides of whatſoever Poligon are comprehended by ſome Number, <lb></lb>the Sides of the Circle are infinite; thoſe are quantitative and di<lb></lb>viſible, theſe non-quantitative and Indiviſible: the terms of the <lb></lb>Sides of a Poligon in the Revolution ſtand ſtill for ſome time, that <lb></lb>is, each ſuch part of the time of an entire Converſion, as it is of <lb></lb>the whole Perimeter: in the Circles likewiſe the ſtay oſ the terms <lb></lb><arrow.to.target n="marg1037"></arrow.to.target><lb></lb>of its infinite Sides are momentary, for a Moment is ſuch part of a <lb></lb>limited Time, as a Point is of a Line, which containeth infinite of <lb></lb>them; the regreſſions made by the Sides of the greater Poligon, are <lb></lb>not of the whole Side, but only of its exceſs above the Side of the <pb xlink:href="069/01/044.jpg" pagenum="42"></pb>leſſer, getting forwards as much ſpace as the ſaid leſſer Side: in <lb></lb>Circles, the Point, or Side C in the inſtantaneous reſt of B recedeth <lb></lb>as much as is its exceſs above the Side B, advancing forward as <lb></lb>much as the quantity of the ſame B: And in ſhort, the infinite <lb></lb>indiviſible Sides of the greater Circle with their infinite indiviſible <lb></lb>Regreſſions, made in the infinite inſtantaneous ſtaies of the infi<lb></lb>nite terms of the infinite Sides of the leſſer Circle, and with their <lb></lb>infinite Progreſſes, equal to the infinite Sides of the ſaid leſſer <lb></lb>Circle, they compoſe and meaſure a Line equall to that deſcribed <lb></lb>by the leſſer Circle, containing in it ſelf infinite ſuperpoſitious <lb></lb>non-quantitative, which make a Conſtipation and Condenſation <lb></lb>without any penctration of quantitative parts: which cannot be <lb></lb>contrived to be done in the Line divided into quantitative parts, <lb></lb>as is the Perimeter of any Poligon, which being diſtended in a <lb></lb>Right-line at length, cannot be reduced to a leſſer length, unleſs <lb></lb>the Sides fall upon and Penetrate one the other. </s> <s>This Conſtipati<lb></lb>on of parts non-quantitative, but infinite without Penetration of <lb></lb>quantitative parts, and the former Diſtraction above declared of <lb></lb><arrow.to.target n="marg1038"></arrow.to.target><lb></lb>infinite Indiviſibles by the interpoſition of indiviſible Vacui<lb></lb>ties, I believe, is the moſt that can be ſaid for the Condenſation <lb></lb>and Rarefaction of Bodies, without being driven to introduce Pe<lb></lb>netration of Bodies, or quantitative Void Spaces. </s> <s>If there be any <lb></lb>thing therein that pleaſeth you, make uſe of it, if not, account it <lb></lb><arrow.to.target n="marg1039"></arrow.to.target><lb></lb>vain, and my diſcourſe alſo; and ſeek out ſome other explanation <lb></lb>that may better ſatisfie your Judgment. </s> <s>Only theſe two words <lb></lb>by the way, let us remember that we are amongſt Infinites, and In<lb></lb>diviſibles.</s></p><p type="margin"> <s><margin.target id="marg1036"></margin.target><emph type="italics"></emph>A Circle is a Poli<lb></lb>gon of infinite in<lb></lb>diviſible quantita<lb></lb>tive Sides.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1037"></margin.target><emph type="italics"></emph>An Inſtant or Mo<lb></lb>ment of quantita<lb></lb>tive Time, is the <lb></lb>ſame as a Point of <lb></lb>a quantitative <lb></lb>Line.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1038"></margin.target><emph type="italics"></emph>Rarefaction is the <lb></lb>diſtraction of infi<lb></lb>nite Indiviſibles <lb></lb>by the interpoſition <lb></lb>of infinite indiviſi<lb></lb>ble Vaeuities.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1039"></margin.target><emph type="italics"></emph>Condenſation, ac<lb></lb>cording to the ope<lb></lb>ration of the Au<lb></lb>thor, proceeds from <lb></lb>the Conſtipation of <lb></lb>quantitative and <lb></lb>indiviſible parts.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>That the Conceit is ingenious, and to my eares wholly <lb></lb>new, and ſtrange, I freely confeſs, but whether or no Nature pro<lb></lb>ceed in this order, I know not how to reſolve; Truth is, that till <lb></lb>ſuch time as I hear ſomething that may better ſatisfie me, that I <lb></lb>may not ſtand ſilent, I will adhere to this. </s> <s>But haply <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end><lb></lb>may have ſomwhat, which I have not yet met with, to explicate <lb></lb>the explication, which is produced by Philoſophers in ſo abſtruce <lb></lb>a matter; for, indeed, what I have hitherto read about Condenſa<lb></lb>tion, is to me ſo denſe, and that of Rarefaction ſo ſubtill, that <lb></lb>my weak ſight neither penetrates the one, nor comprehends the <lb></lb>other.</s></p><p type="main"> <s>SIMP. </s> <s>I am full of confuſion, and find great Rubbs in the one <lb></lb>path, and in the other, and more particularly in this new one: for <lb></lb>according to this Rule, an Ounce of Gold might be rarefied and <lb></lb>drawn forth into a Maſs bigger than the whole Earth, and the <lb></lb>whole Earth condenſed and reduced into a leſs Maſs than a Nut; <lb></lb>which I neither believe, nor think that you your ſelf do believe: <lb></lb>and the Conſiderations and Demonſtrations by you hitherto de<pb xlink:href="069/01/045.jpg" pagenum="43"></pb>livered, as they are things Mathematical, abſtract and ſeparate <lb></lb>from Senſible Matter, I believe, that when they come to be apply<lb></lb>ed to Matters Phyſical and Natural, they will not exactly comply <lb></lb>with theſe Rules.</s></p><p type="main"> <s>SALV. </s> <s>It is not in my power, nor, as I believe, do you deſire, <lb></lb>that I ſhould make that viſible which is inviſible; but as to ſuch <lb></lb>things as may be comprehended by our Senſes, in regard that you <lb></lb><arrow.to.target n="marg1040"></arrow.to.target><lb></lb>have inſtanced in Gold, do we not ſee an immenſe extenſion to <lb></lb>be made of its parts? </s> <s>I know not whether you may have ſeen the <lb></lb>Method that Wyer-drawers obſerve in diſgroſſing Gold Wyer: <lb></lb>which in reality is not Gold, ſave only in the Superficies, for the <lb></lb>internal ſubſtance is Silver; and the way of diſgroſſing it is this. <lb></lb></s> <s>They take a Cylinder, or, if you will, Ingot of Silver, about half <lb></lb>a yard long, and about three or four Inches thick, and this they <lb></lb><arrow.to.target n="marg1041"></arrow.to.target><lb></lb>gild or over-lay with Leaves of beaten Gold, which, you know, <lb></lb>is ſo thin that the Wind will blow it to and again, and of theſe <lb></lb>Leaves they lay on eight or ten, and no more. </s> <s>So ſoon as it is <lb></lb>gilded, they begin to draw it forth with extraordinary force, ma<lb></lb>king it to paſs thorow the hole of the Drawing Iron, and then <lb></lb>reiterate this forceable diſgroſsment again and again thorow holes <lb></lb>ſucceſſively narrower, ſo that, after ſeveral of theſe diſgroſments, <lb></lb>they bring it to the ſmalneſs of the hair of a womans head, if not <lb></lb>ſmaller, and yet it ſtill continueth gilded in its Superficies or out<lb></lb>ſide: Now I leave you to conſider to what a fineneſs and diſtenſi<lb></lb>on the ſubſtance of the Gold is brought.</s></p><p type="margin"> <s><margin.target id="marg1040"></margin.target><emph type="italics"></emph>Gold in the gilding <lb></lb>of Silver is drawn <lb></lb>forth and diſgroſ<lb></lb>ſed immenſly.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1041"></margin.target>* Or Thumb<lb></lb>breadths.</s></p><p type="main"> <s>SIMP. </s> <s>I do not ſee how it can be inferred from this Experi<lb></lb>ment, that there may be a diſgroſment of the matter of the Gold <lb></lb>ſufficient to effect thoſe wonders which you ſpeak of: Firſt, For <lb></lb>that the firſt gilding was with ten Leaves of Gold, which make a <lb></lb>conſiderable thickneſs: Secondly, howbeit in the extenſion and <lb></lb>diſgroſment that Silver encreaſeth in length, it yet withall dimi<lb></lb>niſheth ſo much in thickneſs, that compenſating the one dimenſi<lb></lb>on with the other, the Superficies doth not ſo enlarge, as that for <lb></lb>overlaying the Silver with Gold, the ſaid Gold need to be reduced <lb></lb>to a greater thinneſs than that of its firſt Leaves.</s></p><p type="main"> <s>SALV. </s> <s>You much deceive your ſelf, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> for the en<lb></lb>creaſe of the Superficies is Subduple to the extenſion in length, as <lb></lb>I could Geometrically demonſtrate to you.</s></p><p type="main"> <s>SAGR. </s> <s>I beſeech you, both in the behalf of my ſelf, and of <lb></lb><emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> to favour us with that Demonſtration, if ſo be you <lb></lb>think that we can comprehend it.</s></p><p type="main"> <s>SALV. </s> <s>I will ſee whether I can, thus upon the ſudden, recall <lb></lb>it to mind. </s> <s>It is already manifeſt, that that ſame firſt groſs Cylin<lb></lb>der of Silver, and the Wyer diſtended to ſo great a length are two <lb></lb>equal Cylinders, in regard that they are the ſame Silver; ſo that <pb xlink:href="069/01/046.jpg" pagenum="44"></pb>if I ſhall ſhew you what proportion the Superficies of equall Cy<lb></lb>linders have to one another, we ſhall obtain our deſire. </s> <s>I ſay there<lb></lb>fore, that</s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s><emph type="italics"></emph>The Superficies of Equal Cylinders, their Baſes being <lb></lb>ſubſtracted, are to one another in ſubduple proportion <lb></lb>of their lengths.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Take two equall Cylinders, the heights of which let be A B, <lb></lb>and C D: and let the Line E be a Mean-proportional <lb></lb>between them. </s> <s>I ſay, the Superficies of the Cylinder A B, <lb></lb>the Baſes ſubſtracted, hath the ſame proportion to the Superficies <lb></lb>of the Cylinder C D, the Baſes in like manner ſubſtracted, as the <lb></lb>Line A B hath to the Line E, which is ſubduple of the proportion <lb></lb>of A B to C D. </s> <s>Cut the part of the Cylinder A B in F, and let the <lb></lb>height A F be equal to C D: And becauſe the Baſes of equal Cy<lb></lb>linders anſwer Reciprocally to their heights, the Circle, Baſe of <lb></lb>the Cylinder C D, to the Circle, Baſe of the <lb></lb><figure id="id.069.01.046.1.jpg" xlink:href="069/01/046/1.jpg"></figure><lb></lb>Cylinder A B, ſhall be as the height B A to <lb></lb>D C: And becauſe Circles are to one ano<lb></lb>ther as the Squares of their Diameters, the <lb></lb>ſaid Squares ſhall have the ſame proportion, <lb></lb>that B A hath to C D: But as B A, is to <lb></lb>C D, ſo is the Square B A to the Square of <lb></lb>E: Therefore thoſe four Squares are Pro<lb></lb>portionals: And therefore their Sides ſhall <lb></lb>be Proportionals. </s> <s>And as the Line A B is to <lb></lb>E, ſo is the Diameter of the Circle C to the <lb></lb>Diameter of the Circle A: But as are the <lb></lb>Diameters, ſo are the Circumferences; and <lb></lb>as are the Circumferences, ſo likewiſe are the Superficies of Cylin<lb></lb>ders equal in Height. </s> <s>Therefore as the Line A B is to E, ſo is the <lb></lb>Superficies of the Cylinder C D to the Superficies of the Cylinder <lb></lb>A F. </s> <s>Becauſe therefore the height A F to the height A B, is as the <lb></lb>Superficies A F to the Superficies A B: And as is the height A B <lb></lb>to the Line E, ſo is the Superficies C D to the Superficies A F: <lb></lb>Therefore by Perturbation of Proportion as the height A F is to <lb></lb>E, ſo is the Superficies C D to the Superficies A B: And, by Con<lb></lb>verſion, as the Superficies of the Cylinder A B is to the Superficies <lb></lb>of the Cylinder C D, ſo is the Line E to the Line A F; that is, to <lb></lb>the Line C D: or as A B to E: Which is in ſubduple proportion <lb></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"></pb><p type="main"> <s>Now if we apply this, that hath been demonſtrated, to our <lb></lb>purpoſe; preſuppoſing that that ſame Cylinder of Silver, that was <lb></lb>gilded whilſt it was no more than half a yard long, and four or five <lb></lb>Inches thick, being diſgroſſed to the ſineneſs of an hair, is prolon<lb></lb>ged unto the extenſion of twenty thouſand yards (for its length <lb></lb>would be much greater) we ſhall find its Superficies augmented <lb></lb>to two hundred times its former greatneſs: and conſequently, thoſe <lb></lb>Leaves of Gold, which were laid on ten in number, being diſten<lb></lb>ded on a Superficies two hundred times bigger, aſſure us that the <lb></lb>Gold which covereth the Superficies of the ſo many yards of Wyer <lb></lb>is left of no greater thickneſs than the twentieth part of a Leaf of <lb></lb>ordinary Beaten-Gold. </s> <s>Conſider, now, how great its thinneſs is, and <lb></lb>whether it is poſſible to imagine it done without an immenſe di<lb></lb>ſtention of its parts: and whether this ſeem to you an Experi<lb></lb>ment, that tendeth likewiſe towards a compoſition of infinite In<lb></lb>diviſibles in Phyſical Matters: Howbeit there want not other more <lb></lb>ſtrong and neceſſary proofs of the ſame.</s></p><p type="main"> <s>SAGR. </s> <s>The Demonſtration ſeemeth to me ſo ingenuous, that <lb></lb>although it ſhould not be of force enough to prove that firſt intent <lb></lb>for which it was produced, (and yet, in my opinion, it plainly <lb></lb>makes it out) yet nevertheleſs that ſhort ſpace of time was well <lb></lb>ſpent which hath been employed in hearing of it.</s></p><p type="main"> <s>SALV. </s> <s>In regard I ſee, that you are ſo well pleaſed with theſe <lb></lb>Geometrical Demonſtrations, which bring with them certain pro. <lb></lb></s> <s>fit, I will give you the fellow to this, which ſatisfieth to a very cu<lb></lb>rious Queſtion. </s> <s>In the former we have that which hapneth in <lb></lb>Cylinders that are equall, but of different heights or lengths: it <lb></lb>will be convenient, that you alſo hear that which occurreth in Cy<lb></lb>linders equal in Superficies, but unequal in heights; my meaning <lb></lb>alwaies is, in thoſe Superficies only that encompaſs them about, <lb></lb>that is, not comprehending the two Baſes ſuperiour and inferiour. <lb></lb></s> <s>I ſay, therefore, that</s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s><emph type="italics"></emph>Upon Cylinders, the Superficies of which the Baſes be<lb></lb>ing ſubſtracted are equal, have the ſame proportion <lb></lb>to one another as their heights Reciprocally taken.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let the Superficies of the two Cylinders A E and C F be <lb></lb>equall; but the height of this C D greater than the height <lb></lb>of the other A B. </s> <s>I ſay, that the Cylinder A E hath the <lb></lb>ſame proportion to the Cylinder C F, that the height C D hath <lb></lb>to A B. </s> <s>Becauſe therefore the Superficies C F is equall to the <pb xlink:href="069/01/048.jpg" pagenum="46"></pb>ſuperficies A E, the Cylinder C F ſhall be leſſe than A E: For <lb></lb>if they were equal, its Superficies, by the laſt Propoſition would <lb></lb>be greater than the Superficies A E, and <lb></lb><figure id="id.069.01.048.1.jpg" xlink:href="069/01/048/1.jpg"></figure><lb></lb>much the more, if the ſaid Cylinder C F <lb></lb>were greater than A E. </s> <s>Let the Cylinder <lb></lb>I D be ſuppoſed equal to A E: There<lb></lb>fore, by the precedent Propoſition, the <lb></lb>Superficies of the Cylinder I D ſhall be <lb></lb>to the Superficies A E, as the height I F <lb></lb>to the Mean-proportional betwixt I F & <lb></lb>A B. </s> <s>But the Superficies A E being by <lb></lb>Suppoſition equal to C F and I D, ha<lb></lb>ving the ſame proportion to C F that the <lb></lb>height I F hath to C D: Therefore <lb></lb>C D is the Mean-Proportional between <lb></lb>I F and A B. Moreover, the Cylinder <lb></lb>I D being equal to the Cylinder A E, <lb></lb>they ſhall both have the ſame proporti<lb></lb>on to the Cylinder C F: But I D is to <lb></lb>C F, as the height I F is to C D: Therefore the Cylinder A E <lb></lb>ſhall have the ſame proportion to the Cylinder C F, that the line <lb></lb>I F hath to C D; that is, that C D hath to A B: Which was to be <lb></lb>demonſtrated.<lb></lb><arrow.to.target n="marg1042"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1042"></margin.target><emph type="italics"></emph>Of Corn-ſacks <lb></lb>with a Board at <lb></lb>the Bottom, made <lb></lb>of the ſame Stuffe, <lb></lb>but different in <lb></lb>height, which are <lb></lb>the more capa<lb></lb>cious.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>From hence is collected the Cauſe of an Accident, which the <lb></lb>Vulgar do not hearken to without admiration; and it is, how it <lb></lb>is poſſible that the ſame piece of ^{*}Cloth, being longer one way than <lb></lb>another, if a Sack be made thereof to hold Corn, as the uſual <lb></lb>manner is, with a Board at the bottom, will hold more, making <lb></lb>uſe of the leſſer 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></lb>and with the other encompaſſing the Board at the bottom, than if <lb></lb>it be made up the other way: As if for Example, the Cloth were <lb></lb>one way ſix foot, and the other way twelve, it will hold more, <lb></lb>when with the length of twelve one encompaſſeth the Board at the <lb></lb>bottom, the Sack being ſix foot high, than if it encompaſſed a <lb></lb>bottom of ſix foot, having twelve for its height. </s> <s>Now, by what <lb></lb>hath been demonſtrated, there is added to the Knowledge in ge<lb></lb>neral that it holds more that way than this, the Specifick, and <lb></lb>particular Knowledge of how much it holdeth more: which is, <lb></lb>That it will hold more in proportion as it is lower, and leſſer, as <lb></lb>it is higher. </s> <s>And thus in the meaſures afore taken, the Cloth be<lb></lb>ing twice as long as broad, when it is ſewed the length-ways it will <lb></lb>hold but half ſo much, as it will do the other way. </s> <s>And likewiſe <lb></lb><arrow.to.target n="marg1044"></arrow.to.target><lb></lb>having a Mat to make a ^{*} Frale or Basket twenty five foot long, <lb></lb>and ſuppoſe ſeven broad; made up the long-way it will hold but <lb></lb>onely ſeven of thoſe meaſures, whereof the other way it will hold <lb></lb>five and twenty.</s></p><pb xlink:href="069/01/049.jpg" pagenum="47"></pb><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></lb>Veſſel made of <lb></lb>Rushes or Wick<lb></lb>er.</s></p><p type="main"> <s><emph type="italics"></emph>S<emph.end type="italics"></emph.end>AGR. </s> <s>And thus to our particular content we continually diſ<lb></lb>cover new Notions of great Curioſity, and not unaccompanyed <lb></lb>with Utility. </s> <s>But in the particular glanced at but even now, I <lb></lb>really believe, that amongſt ſuch as are altogether void of the <lb></lb>knowledge of Geometry, there would not be found one in twen<lb></lb>ty, but at the firſt daſh would not be miſtaken, and wonder <lb></lb>that thoſe Bodies that are contained within equal Superficies, <lb></lb>ſhould not likewiſe be in every reſpect equal; like as they run in<lb></lb>to the ſame errour, ſpeaking of the Superficies, when for deter<lb></lb>mining, as it frequently falls out, of the ampleneſſe of ſeveral <lb></lb>Cities, they think they have obtained their deſire ſo ſoon as they <lb></lb>know the ſpace of their Circuits, not knowing that one Circuit <lb></lb>may be equal to another, and yet the place conteined by this <lb></lb>much larger than the place of that: which befalleth not onely in <lb></lb>irregular Superficies, but in the regular; amongſt which thoſe <lb></lb>of more Sides are alwayes more capacious than thoſe of fewer; <lb></lb>ſo that in fine, the Circle, as being a Poligon of infinite Sides, is <lb></lb>more capacious than all other Poligons of equal Perimeter; of <lb></lb>which I remember, that I with particular delight ſaw the Demon<lb></lb>ſtration on a time when I ſtudied the Sphere of <emph type="italics"></emph>Sacroboſco,<emph.end type="italics"></emph.end> with <lb></lb>a very learned Commentary upon the ſame.</s></p><p type="main"> <s>SALV. </s> <s>It is moſt certain; and I having likewiſe light upon <lb></lb>that very place, it gave me occaſion to inveſtigate, how it may <lb></lb>with one ſole Demonſtration be concluded, that the Circle is <lb></lb>greater than all the reſt of regular Iſoperemitral Figures, and of <lb></lb>others, thoſe of more Sides bigger than thoſe of fewer.</s></p><p type="main"> <s>SAGR. </s> <s>And I that take great pleaſure in certain ſelect and no<lb></lb>wiſe-trivial Demonſtrations, entreat you with all importunity to <lb></lb>make me a partaker therein.</s></p><p type="main"> <s>SALV. </s> <s>I ſhall diſpatch the ſame in few words, demonſtrating <lb></lb>the following Theorem, namely;</s></p><pb xlink:href="069/01/050.jpg" pagenum="48"></pb><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s><emph type="italics"></emph>The Circle is a Mean-Proportional betwixt any two <lb></lb>Regular Homogeneal Poligons, one of which is cir<lb></lb>cumſcribed about it, and the other Iſoperimetral to <lb></lb>it: Moreover, it being leſſe than all the circumſcri<lb></lb>bed, it is, on the contrary, bigger than all the Iſoperi<lb></lb>metral. </s> <s>And, again of the circumſcribed, thoſe that <lb></lb>have more angles are leſſer than thoſe that have <lb></lb>fewer; and on the other ſide of the Iſoperimetral, <lb></lb>thoſe of more angles are bigger.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Of the two like Poligons A and B, let A be circumſcribed <lb></lb>about the Circle A, and let the other B, be Iſoperime<lb></lb>tral to the ſaid Circle: I ſay, that the Circle is the Mean<lb></lb>proportional betwixt them. </s> <s>For that (having drawn the Semidi<lb></lb>ameter A C) the Circle being equal to that Right-angled Trian<lb></lb>gle, of whoſe Sides including the Right angle, the one is equal <lb></lb><figure id="id.069.01.050.1.jpg" xlink:href="069/01/050/1.jpg"></figure><lb></lb>to the Semidiameter A C, and the other to the Circumference: <lb></lb>And likewiſe the Poligon A being equal to the right angled Tri<lb></lb>angle, that about the right angle hath one of its Sides equal to <lb></lb>the ſaid right line A C, and the other to the Perimeter of the ſaid <lb></lb>Poligon: It is manifeſt, that the circumſcribed Poligon hath the <lb></lb>ſame proportion to the Circle, that its Perimeter hath to the Cir<lb></lb>cumference of the ſaid Circle; that is, to the Perimeter of the <lb></lb>Poligon B, which is ſuppoſed equal to the ſaid Circumference: <lb></lb>But the Poligon A hath a proportion to the Poligon B, double to <lb></lb>that of its Perimeter, to the Perimeter of B (they being like Fi<lb></lb>gures:) Therefore the Circle A is the Mean-proportional be<lb></lb>tween the two Poligons A and B. </s> <s>And the Poligon A being <lb></lb>bigger than the Circle A, it is manifeſt that the ſaid Circle <lb></lb>A is bigger than the Poligon B, its Iſoperimetral, and conſe<lb></lb>quently the greateſt of all Regular Poligons that are Iſoperimetral <pb xlink:href="069/01/051.jpg" pagenum="49"></pb>to it. </s> <s>As to the other particular, that is to prove, that of the <lb></lb>Poligons circumſcribed about the ſame Circle, that of fewer <lb></lb>Sides is bigger than that of more Sides; but that, on the contrary, of <lb></lb>the Iſoperimetral Poligons, that of more Sides is bigger than that <lb></lb>of fewer Sides, we will thus demonſtrate. </s> <s>In the Circle whoſe <lb></lb>Center is O, and Semidiameter O A, let there be a Tangent <lb></lb>A D, and in it let it be ſuppoſed, for example, that A D is the <lb></lb>half of the Side of the Pentagon circumſcribed, and A C the half <lb></lb>of the Side of the Heptagon, and draw the right lines O G C, <lb></lb>and O F D; and on the Center O, at the diſtance O C, draw the <lb></lb>Arch E C I: And becauſe the Triangle D O C is greater than the <lb></lb>Sector E O C, and the Sector C O I greater than the Triangle <lb></lb>C O A; the Triangle D O C ſhall have greater proportion to <lb></lb>the Triangle C O A, than the Sector E O C, to the Secant C O I, <lb></lb>that is, than the Secant F O G to the Secant G O A. And, by <lb></lb>Compoſition, Permutation of Proportion, the Triangle D O A <lb></lb>ſhall have greater proportion to the Secant F O A, than the Tri<lb></lb>angle C O A to the Secant G O A: And ten Triangles D O A <lb></lb>ſhall have greater proportion to ten Secants F O A, than four<lb></lb>teen Triangles C O A to fourteen Sectors G O A: That is the <lb></lb>circumſcribed Pentagon ſhall have greater proportion to the Cir<lb></lb>cle, than hath the Heptagon: And therefore the Pentagon ſhall <lb></lb>be greater than the Heptagon. </s> <s>Let us now ſuppoſe an Hep<lb></lb>tagon and a Pentagon Iſoperimetral to the ſame Circle. </s> <s>I ſay, that <lb></lb>the Heptagon is bigger than the Pentagon. </s> <s>For that the ſaid Cir<lb></lb>cle being the Mean proportional between the Pentagon circum<lb></lb>ſcribed and the Pentagon its Iſoperimetral, and likewiſe the Mean <lb></lb>between the Circumſcribed and Iſoperimetral Heptagon: It ha<lb></lb>ving been proved that the Circumſcribed Pentagon is greater then <lb></lb>the Circumſcribed Heptagon, the ſaid Pentagon ſhall have greater <lb></lb>proportion to the Circle, than the Heptagon: that is, the Circle <lb></lb>ſhall have greater proportion to its Iſoperimetral Pentagon, than <lb></lb>to its Iſoperimetral Heptagon: Therefore the Pentagon is leſſer <lb></lb>than the Iſoperimetral Heptagon. </s> <s>Which was to be demon<lb></lb>ſtrated</s></p><p type="main"> <s>SAGR. </s> <s>A moſt ingenious Demonſtration, and very acute. </s> <s>But <lb></lb>whither are we run to ingulph our ſelves in Geometry, when as <lb></lb>we were about to conſider the Difficulties propoſed by <emph type="italics"></emph>Simpli<lb></lb>cius,<emph.end type="italics"></emph.end> which indeed are very conſiderable, and in particular, that <lb></lb>of Condenſation, is in my opinion, very abſtruce.</s></p><p type="main"> <s>SALV. </s> <s>If Condenſation and Rarefaction are oppoſite Motions, <lb></lb>where there is ſeen an immenſe Rarefaction, one cannot deny an <lb></lb>extraordinary Condenſation: but immenſe Rarefactions, and, <lb></lb>which encreaſeth the wonder, almoſt Momentary, we ſee every <lb></lb>day: for what a boundleſſe Rarefaction is that of a little quan<pb xlink:href="069/01/052.jpg" pagenum="50"></pb><arrow.to.target n="marg1045"></arrow.to.target><lb></lb>tity of Gunpowder reſolved into a vaſt maſſe of Fire? </s> <s>And what, <lb></lb>beyond this, the (I could almoſt ſay) indeterminate Expanſion <lb></lb>of its Light? </s> <s>And if that Fire and this Light ſhould reunite toge<lb></lb>ther, which yet is no impoſſibility, in regard, that at the firſt <lb></lb>they lay in that little room, what a Condenſation would this be? <lb></lb></s> <s>If you ſtudy for them, you will find hundreds of ſuch Rarefacti<lb></lb>ons, which are much more readily obſerved, than Condenſati<lb></lb>ons: for Denſe matters are more tractable, and ſubject to our <lb></lb>Senſes. </s> <s>For we can eaſily order Wood at pleaſure, and we ſee <lb></lb>it reſolved into Fire, and into Light, but we do not in the ſame <lb></lb>manner ſee the Fire and the Light Condenſe to the making of <lb></lb>Wood: We ſee Fruits, Flowers, and many other ſolid matters <lb></lb>reſolved in a great meaſure into Odors, but we do not after the <lb></lb>ſame manner ſee the odoriferous Atomes concurre to the conſtitu<lb></lb>tion of the Oderate Solids; but where Senſible Obſervation is <lb></lb>wanting, we are to ſupply it with Reaſon, which will ſuffice to <lb></lb>make us apprehenſive, no leſſe of the Motion to the Rarefaction <lb></lb>and reſolution of Solids, than, to the Condenſation of rare and <lb></lb>moſt tenuous Subſtances. </s> <s>Moreover, we queſtion how to effect <lb></lb>the Condenſation and Rarefaction of the Bodies which may be <lb></lb>rarefied and condenſed, ſtudying in what manner it may be done <lb></lb>without introducing of a <emph type="italics"></emph>Vacuum,<emph.end type="italics"></emph.end> and Penetration of Bodies; <lb></lb>which doth not hinder, but that in Nature there may be matters <lb></lb>which admit no ſuch accidents, and conſequently do not allow <lb></lb>roome for thoſe things which you phraſe inconvenient and im<lb></lb>poſſible. </s> <s>And laſtly, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> I have on the the ſcore of ſatis<lb></lb>fying you, and thoſe Philoſophers that hold with you, taken <lb></lb>ſome pains in conſidering how Condenſation and Rarefaction <lb></lb>may be underſtood to be performed without admitting Penetra<lb></lb>tion of Bodies, and introducing the Void Spaces called Vacuities, <lb></lb>Effects which you deny and abhorre: for if you would but grant <lb></lb>them, I would no longer ſo reſolutely contradict you. </s> <s>There<lb></lb>fore either admit theſe Inconveniences, or accept of my Spe<lb></lb>culations, or elſe finde out others more conducing to the <lb></lb>purpoſe.</s></p><p type="margin"> <s><margin.target id="marg1045"></margin.target><emph type="italics"></emph>Rarefaction im<lb></lb>minſe is that of <lb></lb>a little Gunpow<lb></lb>der into a vaſt <lb></lb>maſs of Fire.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>As to the denying of Penetration, I am wholly of opi<lb></lb>nion with the Peripatetick Philoſophers; as to that of a <emph type="italics"></emph>Vacuum,<emph.end type="italics"></emph.end><lb></lb>I would ſee the Demonſtration of <emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end> thorowly examined, <lb></lb>wherewith he oppoſeth the ſame, and what you, <emph type="italics"></emph>Salviatus,<emph.end type="italics"></emph.end> will <lb></lb>anſwer to it. <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> ſhall do me the favour punctually to <lb></lb>recite the proof of the Philoſopher; and you, <emph type="italics"></emph>Salviatus,<emph.end type="italics"></emph.end> to an<lb></lb>ſwer it.</s></p><p type="main"> <s>SIMP. <emph type="italics"></emph>Ariſtotle,<emph.end type="italics"></emph.end> as neer as I can remember, breaks out againſt <lb></lb>certain of the Ancients, who introduced Vacuity, as neceſſary <lb></lb>to Motion, ſaying, that this without that could not be effected; <pb xlink:href="069/01/053.jpg" pagenum="51"></pb>to this <emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end> making oppoſition, demonſtrateth, that on the <lb></lb>contrary, the effecting of Motion (as we ſee) deſtroyeth the Poſiti<lb></lb>on of <emph type="italics"></emph>Vacuum<emph.end type="italics"></emph.end>; and his method therein is this. </s> <s>He maketh two <lb></lb>Suppoſitions, one is touching Moveables different in Gravity <lb></lb>moved in the ſame <emph type="italics"></emph>Medium:<emph.end type="italics"></emph.end> the other is concerning the ſame <lb></lb>Moveable moved in ſeveral <emph type="italics"></emph>Medium's.<emph.end type="italics"></emph.end> As to the firſt, he ſuppo<lb></lb>ſeth that Moveables different in Gravity, move in the ſame <lb></lb><emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> with unequal Velocities, which bear to each other the <lb></lb>ſame proportion as their Gravities: ſo that, for example, a Move<lb></lb>able ten times heavier than another, moveth ten times more ſwift<lb></lb>ly. </s> <s>In the other Poſition he aſſumes, that the Velocity of the <lb></lb>ſame Moveable in different <emph type="italics"></emph>Medium's<emph.end type="italics"></emph.end> are in Reciprocal to that of <lb></lb>the thickneſſe or Denſity of the ſaid <emph type="italics"></emph>Medium's<emph.end type="italics"></emph.end>: ſo that ſuppo<lb></lb>ſing <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> that the Craſſitude of the Water was ten times as great <lb></lb>as that of the Air, he will have the Velocity in the Air to be <lb></lb>ten times more than the Velocity in the Water. </s> <s>And from this ſe<lb></lb>cond Aſſumption he draweth his Demonſtration in this manner. <lb></lb></s> <s>Becauſe the tenuity of <emph type="italics"></emph>Vacuum<emph.end type="italics"></emph.end> infinitely ſurpaſſeth the corpu<lb></lb>lence, though never ſo ſubtil, of any whatever Replete <emph type="italics"></emph>Medi<lb></lb>um,<emph.end type="italics"></emph.end> every Moveable that in the Replete <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> moveth a cer<lb></lb>tain ſpace in a certain time, in a <emph type="italics"></emph>Vacuum<emph.end type="italics"></emph.end> would paſſe the ſame <lb></lb>in an inſtant: But to make a Motion in an inſtant is impoſſible: <lb></lb>Therefore to introduce Vacuity in the accompt of Motion is im<lb></lb>poſſible.</s></p><p type="main"> <s>SALV. </s> <s>The Argument one may ſee to be <emph type="italics"></emph>ad hominem,<emph.end type="italics"></emph.end> that is, <lb></lb><arrow.to.target n="marg1046"></arrow.to.target><lb></lb>againſt thoſe who would make a <emph type="italics"></emph>Vacuum<emph.end type="italics"></emph.end> neceſſary to Motion; <lb></lb>but if I ſhall admit of the Argument as concludent, granting <lb></lb>withal, that in Vacuity there would be no Motion; yet the Poſi<lb></lb>tion of Vacuity taken abſolutely, and not in relation to Motion, <lb></lb>is not thereby overthrown. </s> <s>But to tell you what thoſe Ancients, <lb></lb>peradventure, might anſwer, that ſo we may the better diſcover <lb></lb>how far the Demonſtration of <emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end> holds good, methinks that <lb></lb>one might oppoſe his Aſſumptions, denying them both. </s> <s>And as <lb></lb>to the firſt: I greatly doubt that <emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end> never experimented <lb></lb>how true it is, that two ſtones, one ten times heavier than the o<lb></lb>ther, let fall in the ſame inſtant from an height, <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> of an hun<lb></lb>dred yards, were ſo different in their Velocity, that upon the <lb></lb>arrival of the greater to the ground, the other was found not to <lb></lb>have deſcended ſo much as ten yards.</s></p><p type="margin"> <s><margin.target id="marg1046"></margin.target>Ariſtotle's <emph type="italics"></emph>Argu<lb></lb>ment againſt a<emph.end type="italics"></emph.end><lb></lb>Vacuum <emph type="italics"></emph>is<emph.end type="italics"></emph.end> ad <lb></lb>hominem.</s></p><p type="main"> <s>SIMP. Why, it may be ſeen by his own words, that he confeſ<lb></lb>ſeth he had made the Experiment, for he ſaith, [<emph type="italics"></emph>We ſee the more <lb></lb>grave<emph.end type="italics"></emph.end>] now that <emph type="italics"></emph>Seeing<emph.end type="italics"></emph.end> implieth that he had tried the Experi<lb></lb>ment.</s></p><p type="main"> <s>SAGR. </s> <s>But I, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> that have made proof thereof, do aſ<lb></lb>ſure you, that a Cannon bullet that weigheth one hundred, rwo <pb xlink:href="069/01/054.jpg" pagenum="52"></pb>hundred, and more pounds, will not one Palme anticipate the ar<lb></lb>rival of a Musket-bullet to the ground, that weigheth but half <lb></lb>a pound, falling likewiſe from an height of two hundred yards.</s></p><p type="main"> <s><emph type="italics"></emph>S<emph.end type="italics"></emph.end>ALV. </s> <s>But without any other Experiments, we may by ſhort <lb></lb>and neceſſary Demonſtrations cleerly prove, that it is not true that <lb></lb>a Moveable more grave moveth more ſwiftly than another leſſe <lb></lb>grave, confining our meaning ſtill to Moveables of the ſame Mat<lb></lb>ter; and, in ſhort, to thoſe of which <emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end> ſpeaketh. </s> <s>For tell <lb></lb>me, <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> whether you admit, that to every cadent grave <lb></lb>Body there belongeth by nature one determinate Velocity; ſo <lb></lb>as that it cannot be encreaſed or diminiſhed in it without uſing vi<lb></lb>olence to it, or impoſing ſome impediment upon it?</s></p><p type="main"> <s>SIMP. </s> <s>It cannot be doubted, but that the ſame Moveable in <lb></lb>the ſame <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> hath one eſtabliſhed and by-nature-determinate <lb></lb>Velocity, which cannot be increaſed, unleſſe with new <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end><lb></lb>conferred on it, or diminiſhed, ſave onely by ſome impediment <lb></lb>that retards it.</s></p><p type="main"> <s>SALV. </s> <s>If therefore we had two Moveables, the natural Velo<lb></lb>cities of which were unequal, it is manifeſt, that if we joyned the <lb></lb>ſlower with the ſwifter, this would be in part retarded by the <lb></lb>ſlower, and that in part accelerated by the other more ſwift. </s> <s>Do <lb></lb>not you concur with me in this opinion?</s></p><p type="main"> <s>SIMP. </s> <s>I think that it ought undoubtedly ſo to ſucceed.</s></p><p type="main"> <s>SALV. </s> <s>But if this be ſo, and, it be likewiſe true that a great <lb></lb>Stone moveth with (ſuppoſe) eight degrees of Velocity, and a leſ<lb></lb>ſer with fewer, then joyning them both together, the compound <lb></lb>of them will move with a Velocity leſſe than eight Degrees: But <lb></lb>the two Stones joyned together make one Stone greater than <lb></lb>that before, which moved with eight degrees of Velocity: There<lb></lb>fore this greater Stone moveth leſſe ſwiftly than the leſſer, which <lb></lb>is contrary to your Suppoſition. </s> <s>You ſee therefore, that from the <lb></lb>ſuppoſing that the more grave Moveable moveth more ſwiftly <lb></lb>than the leſſe grave, I prove unto you that the more grave mo<lb></lb>veth leſſe ſwiftly.</s></p><p type="main"> <s>SIMP. </s> <s>I find my ſelf at a loſſe, for the truth is, that the leſ<lb></lb>ſer Stone being joyned to the greater, weight is added unto it, and <lb></lb>weight being added to it, I cannot ſee why there ſhould not Ve<lb></lb>locity be added to it, or at leaſt why it ſhould be diminiſhed <lb></lb>in it.</s></p><p type="main"> <s>SALV. </s> <s>Here you run into another errour, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> for it <lb></lb>is not true, that that ſame leſſer Stone encreaſeth the weight of <lb></lb>the greater.</s></p><p type="main"> <s>SIMP. </s> <s>Oh wonderful! this quite ſurpaſſeth my apprehenſion.</s></p><p type="main"> <s>SALV. </s> <s>Not at all, if you will but ſtay till I have diſcovered <lb></lb>to you the Equivokes, of which you are in doubt: Therefore <pb xlink:href="069/01/055.jpg" pagenum="53"></pb>you muſt know that it is neceſſary to diſtinguiſh betwixt grave <lb></lb>Bodies ſet on Moving, and the ſame conſtituted in Reſt; a Stone <lb></lb>put into the Ballance not onely acquireth greater weight, by lay<lb></lb>ing another Stone upon it, but alſo the addition of, a Flake of <lb></lb>Hemp will make it weigh more by thoſe ſix or ten ounces that <lb></lb>the Hemp ſhall weigh; but if you ſhould freely let fall the Stone <lb></lb>tied to the Hemp from an high place, do you think that in the <lb></lb>Motion the Hemp weigheth down the Stone, ſo as to accelerate <lb></lb>its Motion; or elſe do you believe that it will retard it, ſuſtain<lb></lb>ing it in part? </s> <s>We indeed feel our ſhoulders laden, ſo long as we <lb></lb>will oppoſe the Motion that the weight would make which lyeth <lb></lb>upon our backs; but if we ſhould deſcend with the ſame Velocity <lb></lb>wherewith that ſame grave Body would naturally deſcend, in what <lb></lb>manner will you that it preſſe or bear upon us? </s> <s>Do not you ſee <lb></lb>that this would be a wounding one with a Lance that runneth <lb></lb>before you, with as much or more ſpeed than you purſue him. <lb></lb></s> <s>You may conclude therefore that in the free and natural fall, the <lb></lb>leſſer Stone doth not bear upon the greater, and conſequently doth <lb></lb>not encreaſe their weight, as it doth in Reſt.</s></p><p type="main"> <s>SIMP. </s> <s>But what if the greater was put upon the leſſer?</s></p><p type="main"> <s>SALV. </s> <s>It would encreaſe their weight, in caſe its Motion were <lb></lb>more ſwift; but it hath been already concluded, that in caſe the <lb></lb>leſſer ſhould be more ſlow it would in part retard the Velocity of <lb></lb>the greater, ſo that there Compound would move leſſe ſwiftly; <lb></lb>being greater than the other, which is contrary to your Aſſumpti<lb></lb>on: Let us conclude therefore, that great Moveables, and like<lb></lb>wiſe little, being of the ſame Specifical Gravity, move with like <lb></lb>Velocity.</s></p><p type="main"> <s>SIMP. </s> <s>Your diſcourſe really is full of ingenuity, yet methinks <lb></lb>it is hard to conceive that a drop of Bird-ſhot, ſhould move as <lb></lb>ſwiftly as a Canon-bullet.</s></p><p type="main"> <s>SALV. </s> <s>You may ſay a grain of Sand as faſt as a Mill-ſtone. <lb></lb></s> <s>I would not have you, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> to do as ſome others are wont <lb></lb>to do, and diverting the diſcourſe from the principal deſign, fa<lb></lb>ſten upon ſome one ſaying of mine that may want an hairs-breadth <lb></lb>of the truth, and under this hair hide a defect of another man as <lb></lb>big as the Cable of a Ship. <emph type="italics"></emph>Aristotle<emph.end type="italics"></emph.end> ſaith, a Ball of Iron of an <lb></lb>hundred pounds weight falling, from an height of an hundred yards, <lb></lb>commeth to the ground before that one of one pound is deſcended <lb></lb>one ſole yard: I ſay, that they arrive at the earth both in the ſame <lb></lb>time: You find, that the bigger anticipates the leſſer two Inches, <lb></lb>that is to ſay, that when the great one falls to the ground, the o<lb></lb>ther is diſtant from it two inches: you go about to hide under <lb></lb>theſe two inches the ninety nine yards of <emph type="italics"></emph>Ariſtotle,<emph.end type="italics"></emph.end> and ſpeaking <lb></lb>onely to my ſmall errour, paſſe over in ſilence the other great one. <pb xlink:href="069/01/056.jpg" pagenum="54"></pb><emph type="italics"></emph>Ariſtotlee<emph.end type="italics"></emph.end> affirmeth, that Moveables of different Gravities in the <lb></lb>ſame <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> move (as far as concerneth Gravity) with Veloci<lb></lb>ties proportionate to their Weights; and exemplifieth it by <lb></lb>Moveables, wherein one may diſcover the pure and abſolute effect <lb></lb>of Weight, omitting the other Conſiderations, as well of Figures, <lb></lb>as of the minute Motions; which things receive great alteration <lb></lb>from the <emph type="italics"></emph>Medium,<emph.end type="italics"></emph.end> which altereth the ſimple effect of the ſole <lb></lb>Gravity; wherefore we ſee Gold, that is heavier than any other <lb></lb>matter, being reduced into a very thin Leaf, to go flying to and <lb></lb>again through the Air, the like do Stones beaten to very ſmall <lb></lb>Powder. </s> <s>But if you would maintain the Univerſal Propoſition, it <lb></lb>is requiſite that you ſhew the proportion of the Velocities to be <lb></lb>obſerved in all grave Bodies, and that a Stone of twenty pounds <lb></lb>moveth ten times ſwifter than one of two: which, I tell you, is <lb></lb>falſe, and that falling from an height of fifty or an hundred yards, <lb></lb>they come to the ground in the ſame inſtant.</s></p><p type="main"> <s>SIMP. </s> <s>Perhaps in very great heights of Thouſands of yards <lb></lb>that would happen, which is not ſeen to occur in theſe leſſer <lb></lb>heights.</s></p><p type="main"> <s>SALV. </s> <s>If this was the Meaning of <emph type="italics"></emph>Ariſtotle,<emph.end type="italics"></emph.end> you have in<lb></lb>volved him in another Errour, which will be found a Lie; for <lb></lb>there being no ſuch perpendicular altitudes found on the Earth, <lb></lb>its a clear caſe, that <emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end> was not able to have made an Experi<lb></lb>ment thereof; and yet would perſwade us that he had, whilſt he <lb></lb>ſaith, that the ſaid effect is <emph type="italics"></emph>ſeen.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. <emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end> indeed makes no uſe of this Principle, but of <lb></lb>that other, which I believe is not obnoxious to theſe doubts.</s></p><p type="main"> <s>SALV. </s> <s>Why that alſo is no leſſe falſe than this; and I admire <lb></lb>that you do not of your ſelf perceive the fallacy, and diſcern, that <lb></lb>ſhould it be true, that the ſame Moveable in <emph type="italics"></emph>Medium's<emph.end type="italics"></emph.end> of dif<lb></lb>ferent Subtilty and Rarity, and, in a word, of different Ceſſion, <lb></lb>ſuch, for example, as are Water and Air, move with a greater <lb></lb>Velocity in the Air than in the Water, according to the propor<lb></lb>tion of the Airs Rarity to the Rarity of the Water, it would <lb></lb>follow that every Moveable that deſcendeth in the Air would <lb></lb>deſcend alſo in the Water: Which is ſo falſe, that very many <lb></lb>Bodies deſcend in the Air, that in the Water do not onely not <lb></lb>deſcend, but alſo riſe upwards.</s></p><p type="main"> <s>SIMP I do not underſtand the neceſſity of your Conſequence: <lb></lb>and I will ſay farther, that <emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end> ſpeaketh of thoſe Grave<lb></lb>bodies that deſcend in the one <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> and in the other, and not <lb></lb>of thoſe that deſcend in the Air and aſcend in the Water.</s></p><p type="main"> <s>SALV. </s> <s>You produce for the Philſopher ſuch Pleas as he, with<lb></lb>out all doubt, would never alledge, for that they aggravate the <lb></lb>firſt miſtake. </s> <s>Therefore tell me, if the Craſsitude of the Water, <pb xlink:href="069/01/057.jpg" pagenum="55"></pb>or whatever it be that retardeth the Motion, hath any proporti<lb></lb>on to the Craſſitude of the Air that leſſe retards it; and if it have; <lb></lb>do you aſſign it us, at pleaſure.</s></p><p type="main"> <s>SIMP. </s> <s>It hath ſuch a proportion, and we will ſuppoſe it to be <lb></lb>decuple; and that therefore the Velocity of a Grave Body, that <lb></lb>deſcends in both the Elements, ſhall be ten times ſlower in the Wa<lb></lb>ter than in the Air.</s></p><p type="main"> <s>SALV. </s> <s>I will take one of thoſe Grave-Bodies that deſcend in <lb></lb>the Air, but not in the Water; as for inſtance, a Ball of Wood, <lb></lb>and deſire that you will aſſign it what Velocity you pleaſe, whilſt it <lb></lb>deſcends through the Air.</s></p><p type="main"> <s>SIMP. </s> <s>Suppoſe we, that it move with twenty degrees of Velo<lb></lb>city.</s></p><p type="main"> <s>SALV. </s> <s>Very well: And it is manifeſt, that that Velocity to <lb></lb>ſome other leſſer, may have the ſame proportion, that the Craſſi<lb></lb>tude of the Water hath to that of the Air; and that this ſhall be <lb></lb>the Velocity of the two only degrees: ſo that exactly to an hair, <lb></lb>and in direct conformity to the Aſſumption of <emph type="italics"></emph>Ariſtotle,<emph.end type="italics"></emph.end> it ſhould <lb></lb>be concluded, That the Ball of Wood, which in the Air, ten times <lb></lb>more yielding, moveth deſcending with twenty degrees of Veloci<lb></lb>ty, in the Water ſhould deſcend with two, and not return from the <lb></lb>bottom to flote a-top, as it doth: unleſs you will ſay, that the <lb></lb>aſcending of the Wood to the top is the ſame in the Water, as its <lb></lb>ſinking to the bottom with two degrees of Velocity; which I do <lb></lb>not believe. </s> <s>But ſeeing that the Ball of Wood deſcends not to the <lb></lb>bottom, I rather think that you will grant me, that ſome other Ball, <lb></lb>of other matter different from Wood, might be found that deſcends <lb></lb>in the Water with two degrees of Velocity.</s></p><p type="main"> <s>SIMP. </s> <s>Queſtionleſſe there might; but it muſt be of a matter <lb></lb>conſiderably more grave than Wood.</s></p><p type="main"> <s>SALV. </s> <s>This is that which I deſired to know. </s> <s>But this ſecond <lb></lb>Ball, which in the Water deſcendeth with two degrees of Velocity, <lb></lb>with what Velocity will it deſcend in the Air? </s> <s>It is requiſite (if <lb></lb>you will maintain <emph type="italics"></emph>Ariſtotles<emph.end type="italics"></emph.end> Rule) that you anſwer that it will <lb></lb>move with twenty degrees: But you your ſelf have aſſigned twen<lb></lb>ty degrees of Velocity to the Ball of Wood; Therefore this, and <lb></lb>the other that is much more grave, will move thorow the Air with <lb></lb>equall Velocity. </s> <s>Now how doth the Philoſopher reconcile this <lb></lb>Concluſion with that other of his, that the Moveables of different <lb></lb>Gravity, move in the ſame <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> with different Velocities, and <lb></lb>ſo different as are their Gravities? </s> <s>But, without any deep ſtudies, <lb></lb>how comes it to paſs that you have not obſerved very frequent, <lb></lb>and very palpable Accidents, and not conſidered two Bodies, that in <lb></lb>the Water will move one an hundred times more ſwiftly than the <lb></lb>other, but that again in the Air that ſwifter one will not out-go the <pb xlink:href="069/01/058.jpg" pagenum="56"></pb>other, one ſole Centeſm? </s> <s>As for example, an Egge of Marble will <lb></lb>deſcend in the Water an hundred times faſter than one of an Hen, <lb></lb>when as in the Air, at the height of twenty Yards it will not anti<lb></lb>cipate it four Inches: and, in a word, ſuch a certain Grave Body <lb></lb>will ſink to the bottom in three hours in ten fathom Water, that <lb></lb>in the Air will paſs the ſame ſpace in one or two pulſes, and ſuch <lb></lb>another (as for inſtance a Ball of Lead) will paſs that number of <lb></lb>fathoms with eaſe in leſs than double the time. </s> <s>And here I ſee <lb></lb>plainly, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> that you find, that herein there is no place left <lb></lb>for any diſtinction, or reply. </s> <s>Conclude we therefore, that that <lb></lb>ſame Argument concludeth nothing againſt <emph type="italics"></emph>Vacuum<emph.end type="italics"></emph.end>; and if it <lb></lb>ſhould, it would only overthrow Spaces conſiderably great, which <lb></lb>neither I, nor, as I take it, thoſe <emph type="italics"></emph>Ancients<emph.end type="italics"></emph.end> did ſuppoſe to be natu<lb></lb>rally allowed, though, perhaps, with violence they may be effe<lb></lb>cted, as, me thinks, one may collect from ſeveral Experiments, which <lb></lb>it would be two tedious to go about at preſent to produce.</s></p><p type="main"> <s>SAGR. </s> <s>Seeing that <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> is ſilent, I will take leave to ſay <lb></lb>ſomething. </s> <s>In regard you have with ſufficient plainneſſe demon<lb></lb>ſtrated, that it is not true, That Moveables unequally grave move in <lb></lb>the ſame <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> with Velocities proportionate to their Gravities, <lb></lb>but with equal: deſiring to be underſtood to ſpeak of Bodies of the <lb></lb>ſame Matter, or of the ſame Specifick Gravity, but not (as I con<lb></lb>ceive) of Gravities different <emph type="italics"></emph>in Spetie,<emph.end type="italics"></emph.end> (for I do not think that <lb></lb>you intend to prove unto us, that a Ball of Cork moveth with like <lb></lb>Velocity to one of Lead;) and having moreover very manifeſtly <lb></lb>demonſtrated, that it is not true, That the ſame Moveable in <emph type="italics"></emph>Me<lb></lb>diums<emph.end type="italics"></emph.end> of different Reſiſtances retain in their Velocities and Tardi<lb></lb>ties the ſame proportion as have their Reſiſtances: to me it would <lb></lb>be a very pleaſing thing to hear, what thoſe be which are obſerved <lb></lb>as well in the one caſe as in the other.</s></p><p type="main"> <s>SALV. </s> <s>The Queſtions are ingenuous, and I have many times <lb></lb>thought of them: I will relate unto you the Contemplations made <lb></lb>upon them, and what at length I did from thence infer. </s> <s>After I <lb></lb>had aſſured my ſelf that it was not true, That the ſame Moveable <lb></lb>in <emph type="italics"></emph>Medium's<emph.end type="italics"></emph.end> of different Reſiſtance obſerveth in its Velocity the <lb></lb>proportion of the Ceſſion of thoſe <emph type="italics"></emph>Media<emph.end type="italics"></emph.end>; nor yet, again, That in <lb></lb>the ſame <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> Moveables of different Gravity retain in their <lb></lb>Velocities the proportion of thoſe Gravities (ſpeaking alwaies of <lb></lb>Gravitles different <emph type="italics"></emph>in ſpecie<emph.end type="italics"></emph.end>) I began to put both theſe Accidents <lb></lb>together, obſerving that which befell the Moveables different in <lb></lb>Gravity put into <emph type="italics"></emph>Mediums<emph.end type="italics"></emph.end> of different Reſiſtance, and I perceived <lb></lb>that the inequality of the Velocities were found to be alwaies <lb></lb>greater in the more reſiſting <emph type="italics"></emph>Medium's,<emph.end type="italics"></emph.end> than in the more yielding; <lb></lb>and that with ſuch a diverſity, that of two Moveables that, de<lb></lb>ſcending thorow the Air, differ very little in Velocity of Motion, <pb xlink:href="069/01/059.jpg" pagenum="57"></pb>one will, in the Water, move ten times faſter than the other; <lb></lb>yea: that ſuch, as in the Air do ſwiftly deſcend, in the Water not <lb></lb>only will not deſcend, but will be wholly deprived of Motion, <lb></lb>and, which is yet more, will move upwards: for one ſhall ſome<lb></lb>times find ſome kind of Wood, or ſome knot, or root of the ſame, <lb></lb>that in the Water will lye ſtill, when as in the Air it will ſwiftly <lb></lb>deſcend.</s></p><p type="main"> <s>SAGR. </s> <s>I have many times ſet my ſelf with an extream patience <lb></lb>to ſee if I could reduce a Ball of Wax, (which of it ſelf doth not <lb></lb>go to the bottom) by adding to it grains of ſand, to ſuch a degree <lb></lb>of Gravity like to the Water, as to make it ſtand ſtill in the <lb></lb>midſt of that Element; but I could never, by all the care I <lb></lb>uſed, ſucceed in my attempt; ſo that I cannot tell, whether any <lb></lb>Solid matter may be found ſo naturally alike in Gravity to Wa<lb></lb>ter, as that being put into any place of the ſame, it can reſt or lye <lb></lb>ſtill.</s></p><p type="main"> <s>SALV. </s> <s>In this, as well as in a thouſand other actions, many <lb></lb>Animals are more ingenuous than we. </s> <s>And, in this caſe, Fiſhes <lb></lb><arrow.to.target n="marg1047"></arrow.to.target><lb></lb>would have been able to have given you ſome light, being in this <lb></lb>affair ſo skilful, that at their pleaſure they ^{*} equilibrate themſelves, <lb></lb><arrow.to.target n="marg1048"></arrow.to.target><lb></lb>not only with one kind of Water, but with ſuch, as, either of their <lb></lb>own nature, or by means of ſome ſupervenient muddineſs, or for <lb></lb>their ſaltneſs (which maketh a great alteration) are very diffe<lb></lb>rent; equilibrate themſelves, I ſay, ſo exactly, that without ſtir<lb></lb>ring in the leaſt they lye ſtill in every place: and this, in my opi<lb></lb>nion, they do, by making uſe of the Inſtrument given them by Na<lb></lb>ture to that end, <emph type="italics"></emph>ſcilicet,<emph.end type="italics"></emph.end> of that Bladder which they have in their <lb></lb>Bodies, which by a very narrow neck anſwereth to their mouth; <lb></lb>and by that they either, when they would ſtand ſtill, ſend forth <lb></lb>part of the Air that is contained in the ſaid Bladders, or, ſwimming <lb></lb>to the top they draw in more, making themſelves by that art one <lb></lb>while more, another while leſs heavy than the Water, and at their <lb></lb>pleaſures equilibrating themſelves to the ſame.</s></p><p type="margin"> <s><margin.target id="marg1047"></margin.target><emph type="italics"></emph>Fiſhes equilibrate <lb></lb>themſelves admi<lb></lb>rably in the Water.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1048"></margin.target>* Or poiſe.</s></p><p type="main"> <s>SAGR I deceived ſome of my Friends with another device; <lb></lb>for I had made my boaſt unto them, that I would reduce that Ball <lb></lb>of Wax to an exact <emph type="italics"></emph>equilibrium<emph.end type="italics"></emph.end> with the Water, and having put <lb></lb>ſome ſalt Water in the bottom of the Veſſel, and a-top of that ſome <lb></lb>freſh, I ſhewed them the Ball, which in the midſt of the Water <lb></lb>ſtood ſtill, and being thruſt to the bottom, or to the top, ſtaid nei<lb></lb>ther in this nor that ſcituation, but returned to the midſt.</s></p><p type="main"> <s>SALV. </s> <s>This ſame Experiment is not void of utility; for Phyſi<lb></lb><arrow.to.target n="marg1049"></arrow.to.target><lb></lb>cians, in particular, treating of ſundry qualities of Waters, and <lb></lb>amongſt other things, principally of the more or leſs Gravity or <lb></lb>Levity of this or that: by ſuch a Ball, in ſuch manner poiſed and <lb></lb>adjuſted that it may reſt ambiguous, if I may ſo ſay, between <pb xlink:href="069/01/060.jpg" pagenum="58"></pb>aſcending and deſcending in a Water, upon the leaſt difference <lb></lb>of weight between two Waters, if that Ball ſhall deſcend in the <lb></lb>one; in the other, that is more grave, it ſhall aſcend. </s> <s>And the <lb></lb>Experiment is ſo exact, that the addition of but only two grains <lb></lb>of Salt, put into ſix pounds of Water, ſhall make that Ball to <lb></lb>aſcend from the bottom to the ſurface, which was but a little be<lb></lb><arrow.to.target n="marg1050"></arrow.to.target><lb></lb>fore deſcended thither. </s> <s>And moreover, I will tell you this in con<lb></lb>firmation of the exactneſs of this Experiment, and withall for a <lb></lb>clear proof of the Non-reſiſtance of Water to diviſion, that not <lb></lb>only the ingravitating it with the mixture of ſome matter heavier <lb></lb>than it, maketh that ſo notable difference, but the warming or <lb></lb>cooling of it a little produceth the ſame effect, and with ſo ſubtil <lb></lb>an operation, that the infuſing four diops of other Water, a lit<lb></lb>tle warmer, or a little colder, than the ſix pounds, ſhall cauſe the <lb></lb>Ball to riſe or ſink in the ſame; to ſink in it upon the infuſion of <lb></lb>the warm, and to riſe at the infuſion of the cold. </s> <s>Now ſee how <lb></lb>much thoſe Philoſophers are deceived, who would introduce in <lb></lb>Water viſcoſity, or other conjunction of parts which make it to <lb></lb>reſiſt Diviſion or Penetration.</s></p><p type="margin"> <s><margin.target id="marg1049"></margin.target><emph type="italics"></emph>A Ball of Wax <lb></lb>prepared to make <lb></lb>the Experiment of <lb></lb>the different Gra<lb></lb>vities of Waters.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1050"></margin.target><emph type="italics"></emph>Water bath no <lb></lb>Reſiſtance to Di<lb></lb>viſion.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>I have ſeen many Convincing Diſcourſes touching <lb></lb><arrow.to.target n="marg1051"></arrow.to.target><lb></lb>this Argument in a ^{*} Treatiſe of our <emph type="italics"></emph>Accademick<emph.end type="italics"></emph.end>; yet never the leſs <lb></lb>there is reſting in me a ſtrong ſcruple, which I know not how to <lb></lb>remove: For if nothing of Tenacity, or Coherence reſides amongſt <lb></lb>the parts of Water, how can it bear it ſelf up in reaſonable big <lb></lb>and high Tumours; in particular, upon the leaves of Cole-worts <lb></lb>without diſperſing or levelling?</s></p><p type="margin"> <s><margin.target id="marg1051"></margin.target>* The Tract cited <lb></lb>in this place is <lb></lb>that which we <lb></lb>diſpoſe firſt in <lb></lb>Order, in the <lb></lb>firſt part of this <lb></lb>Tome,</s></p><p type="main"> <s>SALV. </s> <s>Although it be true, that he who is Maſter of a true <lb></lb>Concluſion, may reſolve all Objections that can be brought againſt <lb></lb>it, yet will not I arrogate to my ſelf the power ſo to do; nor <lb></lb>ought my inſufficiency becloud the ſplendour of Truth. </s> <s>Firſt, <lb></lb>therefore, I confeſs that I know not how it cometh to paſs, that <lb></lb>thoſe Globes of Water ſuſtain themſelves at ſuch an height and <lb></lb>bigneſs, albeit I certainly know that it doth not proceed from any <lb></lb><arrow.to.target n="marg1052"></arrow.to.target><lb></lb>internal Tenacity that is between its parts; ſo that it remaineth <lb></lb>neœſſary, that the Cauſe of that Effect do reſide without. </s> <s>That it <lb></lb>is not Internal, beſides thoſe Experiments already ſhewn you, I can <lb></lb>prove by another moſt convincing one. </s> <s>If the parts of that Wa<lb></lb>ter, which conſerveth it ſelf in a Globe or Tumour whilſt it is en<lb></lb>compaſſed by the Air, had an internal Cauſe for ſo doing, they <lb></lb>would much better ſuſtain themſelves being environed by a <emph type="italics"></emph>Medi<lb></lb>um,<emph.end type="italics"></emph.end> in which they had leſs propenſion to deſcend, than they have <lb></lb>in the Ambient Air: But every Fluid Body more grave than the <lb></lb>Air would be ſuch a <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end>; as, for inſtance, Wine: And there<lb></lb>fore, infuſing Wine about that Globe of Water, it might raiſe it <lb></lb>ſelf on every ſide, and yet the parts of the Water, conglutinated <pb xlink:href="069/01/061.jpg" pagenum="59"></pb>by the internal Viſcoſity, never diſſolve: But it doth not happen <lb></lb>ſo; nay, no ſooner doth the circumfuſed liquor approach thereto, <lb></lb>but, without ſtaying till it riſe much about it, the little globes of <lb></lb>Water will diſſolve and become flat, reſting under the Wine, if it <lb></lb>was red. </s> <s>The Cauſe therefore of this Effect is External, and per<lb></lb>haps in the Ambient Air: and, indeed, one may obſerve a great <lb></lb>diſſention between the Air and Water; which I have obſerved <lb></lb>in another Experiment; and this it is: If I fill a ^{*} Ball of Chriſtal, <lb></lb><arrow.to.target n="marg1053"></arrow.to.target><lb></lb>that hath a mouth as narrow as the hollow of a ſtraw, with water, <lb></lb>and when it is thus full, turn it with its mouth downwards, yet will <lb></lb>not the Water, although very heavy, and prone to deſcend tho<lb></lb>row the Air, nor the Air, as much diſpoſed on the other hand, as <lb></lb>being very light, to aſcend thorow the Waters, yet will they not <lb></lb>(I ſay) agree that that ſhould deſcend, iſſuing out at the mouth, <lb></lb>and this aſcend, entering in at the ſame: but they both continue <lb></lb>averſe and contumacious. </s> <s>Again, on the contrary, if I preſent to <lb></lb>that mouth a veſſel of red Wine, which is almoſt inſenſibly leſs <lb></lb>grave than Water, we ſhall ſee it in an inſtant gently to aſcend by <lb></lb>red ſtreams thorow the Water, and the Water with like Tardity to <lb></lb>deſcend through the Wine, without ever mixing with each other, <lb></lb>till that in the end, the Ball will be full of Wine, and the Water <lb></lb>Will all ſink unto the bottom of the Veſſel underneath. </s> <s>Now <lb></lb>what are we to ſay, or what are we to infer, but a diſagreement <lb></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"></emph>Water formed into <lb></lb>great drops upon <lb></lb>the Leaves of Col<lb></lb>worts, how they <lb></lb>conſiſt.<emph.end type="italics"></emph.end></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 ſcarce refrain my laughter to ſee the great Anti<lb></lb>pathy that <emph type="italics"></emph>Salviatus<emph.end type="italics"></emph.end> hath to Antipathy, ſo that he will not ſo much <lb></lb>as name it, and yet it is ſo accommodate to reſolve the doubt.</s></p><p type="main"> <s>SALV. </s> <s>Now let this, for the ſake of <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> be the ſoluti<lb></lb>on of our ſcruple; and leaving the Digreſſion, let us return to our <lb></lb>purpoſe. </s> <s>Seeing that the difference of Velocity in Moveables of <lb></lb>divers Gravities is found to be more and more, as the <emph type="italics"></emph>Mediums<emph.end type="italics"></emph.end> are <lb></lb>more and more Reſiſting: And withall, that in a <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> of <lb></lb>Quickſilver, Gold doth not only go to the bottom more ſwiftly <lb></lb>than Lead, but it alone deſcends in it, and all other Metals and <lb></lb>Stones move upwards therein, and flote thereon; whereas between <lb></lb>Balls of Gold, Lead, Braſs, Porphiry, or other grave matters, the in<lb></lb>equality of motion in the Air ſhall be almoſt wholly inſenſible, for <lb></lb>it is certain, that a Ball of Gold in the end of the deſcent of an <lb></lb><arrow.to.target n="marg1054"></arrow.to.target><lb></lb>hundred yards ſhall not out-ſtrip one of Braſs four Inches: ſeeing <lb></lb>this, I ſay, I have thought, that if we wholly took away the <lb></lb>Reſiſtance of the <emph type="italics"></emph>Medium,<emph.end type="italics"></emph.end> all Matters would deſcend with equall <lb></lb>Velocity.</s></p><p type="margin"> <s><margin.target id="marg1054"></margin.target><emph type="italics"></emph>Reſiſtance of the<emph.end type="italics"></emph.end><lb></lb>Medium <emph type="italics"></emph>remo<lb></lb>ved, all Matters, <lb></lb>though of different <lb></lb>Gravities would <lb></lb>move with like <lb></lb>Velocity.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. </s> <s>This is a bold ſpeech, <emph type="italics"></emph>Salviatus,<emph.end type="italics"></emph.end> I ſhall never believe <lb></lb>that in <emph type="italics"></emph>Vacuity<emph.end type="italics"></emph.end> it ſelf, if ſo be one ſhould allow Motion in it, a lock <lb></lb>of Wooll would move as ſwiftly as a piece of Lead.</s></p><pb xlink:href="069/01/062.jpg" pagenum="60"></pb><p type="main"> <s>SALV. </s> <s>Fair and ſoftly, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> your ſcruple is not ſo ab<lb></lb>ſtruce, nor I ſo incautelous, that you ſhould need to think that I <lb></lb>was not adviſed of it, and that conſequently I have not found a re<lb></lb>ply to it. </s> <s>Therefore, for my explanation, and your information, <lb></lb>hearken to what I ſhall ſay. </s> <s>We are upon the examination of <lb></lb>what would befall Moveables exceeding different in weight in a <lb></lb><emph type="italics"></emph>Medium,<emph.end type="italics"></emph.end> in caſe it ſhould have no Reſiſtance, ſo that all the diffe<lb></lb>rence of Velocity that is found between the ſaid Moveables ought <lb></lb>to be referred to the ſole inequality of Weight. </s> <s>And becauſe on<lb></lb>ly a Space altogether void of Air, and of every other, though te<lb></lb>nuous and yielding Body, would be apt ſenſibly to ſhew us what <lb></lb>we ſeek, ſince we want ſuch a Space, let us ſucceſſively obſerve that <lb></lb>which happeneth in the more ſubtill and leſſe reſiſting <emph type="italics"></emph>Mediums,<emph.end type="italics"></emph.end><lb></lb>in compariſon of that which we ſee to happen in others leſſe ſubtill <lb></lb>and more reſiſting: for if we ſhould really find the Moveables <lb></lb>different in Gravity to differ leſſe and leſſe in Velocity, according <lb></lb>as the <emph type="italics"></emph>Mediums<emph.end type="italics"></emph.end> are found more and more yielding; and that, <lb></lb>finally, although extreamly unequal in weight, in a <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> more <lb></lb>tenuous than any other, though not void, the difference of Velo<lb></lb>city diſcovers it ſelf to be very ſmall, and almoſt unobſervable, I <lb></lb>conceive that we may, and that upon very probable conjecture, <lb></lb>believe, that in a <emph type="italics"></emph>Vacuum<emph.end type="italics"></emph.end> their Velocities would be exactly equal. <lb></lb></s> <s>Therefore let us conſider that which hapneth in the Air; wherein <lb></lb>to have a Figure of an uniform Superficies, and very light Matter, <lb></lb>I will that we take a blown Bladder, in which the included Air <lb></lb>will weigh little or nothing in a <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> of the Air it ſelf, becauſe <lb></lb>it can make but very ſmall Compreſſion therein, ſo that the Gravi<lb></lb>ty is only that little of the ſaid film, which would not be the thou<lb></lb>ſandth part of the weight of a lump of Lead of the bigneſs of <lb></lb>the ſaid Bladder when blown. </s> <s>Theſe, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> being let fall <lb></lb>from the height of four or ſix yards, how great a ſpace, do you <lb></lb>judge, that the Lead would anticipate the Bladder in its deſcent? <lb></lb></s> <s>Aſſure your ſelf that would not move thrice, no nor twice as faſt, <lb></lb>although even now you would have had it to have been a thou<lb></lb>ſand times more ſwift.</s></p><p type="main"> <s>SIMP. </s> <s>It is poſſible that at the beginning of the Motion, that <lb></lb>is, in the firſt five or ſix yards this might happen that you ſay; but <lb></lb>in the progreſſe, and in a long continuation I believe, that the Lead <lb></lb>would leave it behind, not only ſix, but alſo eight and ten parts of <lb></lb>twelve.</s></p><p type="main"> <s>SALV. </s> <s>And I alſo believe the ſame: and make no queſtion, <lb></lb>but that in very great diſtances the Lead will have paſſed an hun<lb></lb>dred miles of <emph type="italics"></emph>way,<emph.end type="italics"></emph.end> ere the Bladder will have paſſed ſo much as one. <lb></lb></s> <s>But this, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> which you propound, as an effect contrary to <lb></lb>my Aſſertion, is that which moſt eſpecially confirmeth it. </s> <s>It is (I <pb xlink:href="069/01/063.jpg" pagenum="61"></pb>once more tell you) my intent to declare, That the difference of <lb></lb>Gravity is in no wiſe the cauſe of the divers velocities of Movea<lb></lb>bles of different Gravity, but that the ſame dependeth on exteri<lb></lb>our accidents, & in particular, on the Reſiſtance of the <emph type="italics"></emph>Medium,<emph.end type="italics"></emph.end> ſo <lb></lb>that, this being removed, all Moveables move with the ſame de<lb></lb>grees of Velocity. </s> <s>And this I chiefly deduce from that which but <lb></lb>now you your ſelf did admit, and which is very true, namely, that <lb></lb>of two Moveables, very different in weight, the Velocities more and <lb></lb>more differ, according as the ^{*} Spaces are greater and greater that <lb></lb><arrow.to.target n="marg1055"></arrow.to.target><lb></lb>they paſſe: an Effect which would not follow, if it did depend on <lb></lb>the different Gravities: for they being alwaies the ſame, the pro<lb></lb>portion betwixt the Spaces would likewiſe alwaies continue the <lb></lb>ſame, which proportion we ſee ſtill ſucceſſively to encreaſe in the <lb></lb>continuance of the Motion; for that the heavieſt Moveable in the <lb></lb>deſcent of one yard will not anticipate the lighteſt the tenth part <lb></lb>of that Space or Way, but in the fall of twelve yards will out-go <lb></lb>it a third part, in that of an hundred will outſtrip 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 ſtep by ſtep, if the dif<lb></lb>ference of weight in Moveables of different Gravities cannot <lb></lb>cauſe the difference of proportion in their Velocities, for that the <lb></lb>Gravities do not alter; neither then can the <emph type="italics"></emph>Medium,<emph.end type="italics"></emph.end> which is <lb></lb>ſuppoſed alwaies to continue the ſame, cauſe any alteration in the <lb></lb>proportion of the Velocities.</s></p><p type="main"> <s>SALV. </s> <s>You wittily bring an inſtance againſt my Poſition, that <lb></lb><arrow.to.target n="marg1056"></arrow.to.target><lb></lb>it is very neceſſary to remove. </s> <s>I ſay therefore, that a Grave Body <lb></lb>hath, by Nature, an intrinſick Principle of moving towards the <lb></lb>Common Center of heavy things, that is to that of our Terreſtrial <lb></lb>Globe, with a Motion continually accelerated, and accelerated <lb></lb>alwaies equally, <emph type="italics"></emph>ſcilicet,<emph.end type="italics"></emph.end> that in equal times there are made equal <lb></lb>^{*} additions of new Moments, and degrees of Velocities: and this <lb></lb>ought to be underſtood to hold true at all times when all acciden<lb></lb><arrow.to.target n="marg1057"></arrow.to.target><lb></lb>tal and external impediments are removed; amongſt which there <lb></lb>is one that we cannot obviate, that is the Impediment of the <emph type="italics"></emph>Me<lb></lb>dium,<emph.end type="italics"></emph.end> which is Repleat, when as it ſhould be opened and latterally <lb></lb>moved by the falling Moveable, to which tranſverſe Motion the <lb></lb><emph type="italics"></emph>Medium,<emph.end type="italics"></emph.end> though fluid, yielding and tranquile, oppoſeth it ſelf <lb></lb>with a Reſiſtance one while leſſer, and another while greater and <lb></lb>greater, according as it is more ſlowly or haſtily to open to give <lb></lb>paſſage to the Moveable, which, becauſe, as I have ſaid, it goeth <lb></lb>of its own nature continually accelerating, it cometh of conſe<lb></lb>quence to encounter continually greater Reſiſtance in the <emph type="italics"></emph>Medi<lb></lb>um,<emph.end type="italics"></emph.end> and therefore Retardment, and diminution in the acquiſt of <lb></lb>new degrees of Velocity; ſo that in the end, the Velocity arriveth <lb></lb>to that ſwiftneſſe, and the Reſiſtance of the <emph type="italics"></emph>Medium,<emph.end type="italics"></emph.end> to that <lb></lb>ſtrength, that ballancing each other, they take away all further <pb xlink:href="069/01/064.jpg" pagenum="62"></pb>Acceleration, and reduce the Moveable to an Equable and Uni<lb></lb>form Motion, in which it afterwards continually abides. </s> <s>There is <lb></lb>therefore in the <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> augmentation of Reſiſtance, not becauſe <lb></lb>it changeth its Eſſence, but becauſe the Velocity altereth where<lb></lb>with it ought to open, and laterally move, to give paſſage to the <lb></lb>falling Body, which goeth continually accelerating. </s> <s>Now the <lb></lb>obſerving, that the Reſiſtance of the Air to the ſmall Moment or <lb></lb><emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of the Bladder is very great, and to the great weight of <lb></lb>the Lead is very ſmall, makes me hold for certain, that if one ſhould <lb></lb>wholly remove it, by adding to the Bladder great aſſiſtance, and <lb></lb>but very little to the Lead, their Velocities would equalize each <lb></lb>other. </s> <s>Taking this Principle therefore for granted, That in the <lb></lb><emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> wherein, either by reaſon of Vacuity, or otherwiſe, there <lb></lb>were no Reſiſtance that might abate the Velocity of the Motion, <lb></lb>ſo that of all Moveables the Velocities were alike, we might con<lb></lb><arrow.to.target n="marg1058"></arrow.to.target><lb></lb>gruouſly enough aſſign the proportions of the Velocities of like <lb></lb>and unlike Moveables, in the ſame and in different, Replear, and <lb></lb>therefore Reſiſting <emph type="italics"></emph>Medium's.<emph.end type="italics"></emph.end> And this we might effect by ſtudy<lb></lb>ing how much the Gravity of the <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> abateth from the Gra<lb></lb>vity of the Moveable, which Gravity is the Inſtrument wherewith <lb></lb>the Moveable makes its Way, repelling the parts of the <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end><lb></lb>on each Side: an operation that doth not occur in void <emph type="italics"></emph>Mediums<emph.end type="italics"></emph.end>; <lb></lb>and therefore there is no difference to be expected from the di<lb></lb>verſe Gravity: and becauſe it is manifeſt, that the <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> abateth <lb></lb>from the Gravity of the Body by it contained, as much as is the <lb></lb>weight of ſuch another maſs of its own Matter, if the Velocities of <lb></lb>the Moveables that in a non-reſiſting <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> would be (as hath <lb></lb>been ſuppoſed) equal, ſhould diminiſh in that proportion, we <lb></lb>ſhould have what we deſired. </s> <s>As for example; ſuppoſing that <lb></lb>Lead be ten thouſand times more grave than Air, but Ebony a <lb></lb>thouſand times only; of the Velocities of theſe two Matters, which <lb></lb>abſolutely taken, that is, all Reſiſtance being removed, would be <lb></lb>equal, the Air ſubſtracts from the ten thouſand degrees of the <lb></lb>Lead one, and from the thouſand degrees of the Ebony likewiſe <lb></lb>abateth one, or, if you will, of its ten thouſand, ten. </s> <s>If there<lb></lb>fore the Lead and the Ebony ſhall deſcend thorow the Air from <lb></lb>any height, which, the retardment of the Air removed, they would <lb></lb>have paſſed in the ſame time, the Air will abate from the ten <lb></lb>thouſand degrees of the Leads Velocity one, but from the ten <lb></lb>thouſand degrees of Ebony's Velocity it will abate ten: which is <lb></lb>as much as to ſay, that dividing that Altitude, from which thoſe <lb></lb>Moveables departed into ten thouſand parts, the Lead will arrive <lb></lb>at the Earth, the Ebony being left behind, ten, nay, nine of thoſe <lb></lb>ſame ten thouſand parts. </s> <s>And what elſe is this, but that a Ball of <lb></lb>Lead, falling from a Tower two hundred yards high, to find how <pb xlink:href="069/01/065.jpg" pagenum="63"></pb>much it will anticipate one of Ebony of leſſe than four Inches? <lb></lb></s> <s>The Ebony weigheth a thouſand times more than the Air, but that <lb></lb>Bladder ſo blown, weigheth only four times ſo much; the Air <lb></lb>therefore from the intrinſick and natural Velocity of the Ebony <lb></lb>ſubducteth one degree of a thouſand, but from that, which alſo in <lb></lb>the Bladder would abſolutely have been the ſame, the Air ſub<lb></lb>ducts one part of four: ſo that by that time the Ball of Ebony <lb></lb>falling from the Tower, ſhall come to the ground, the Bladder <lb></lb>ſhall have paſſed but three quarters of that height. </s> <s>Lead is twelve <lb></lb>times heavier than Water, but Ivory only twice as heavy; the <lb></lb>Water therefore, from their abſolute Velocities which would be <lb></lb>equal, ſhall abate in the Lead the twelfth part, but in the Ivory <lb></lb>the half: when therefore, in the Water, the Lead ſhall have de<lb></lb>ſcended eleven fathom, the Ivory ſhall have deſcended ſix. </s> <s>And, <lb></lb>arguing by this Rule, I believe, that we ſhall find the Experiment <lb></lb>much more exactly agree with this ſame Computation, than with <lb></lb>that of <emph type="italics"></emph>Ariſtotle.<emph.end type="italics"></emph.end> By the like method we might find the Veloci<lb></lb>ties of the ſame Moveable in different fluid <emph type="italics"></emph>Mediums,<emph.end type="italics"></emph.end> not compa<lb></lb>ring the different Reſiſtances of the <emph type="italics"></emph>Mediums,<emph.end type="italics"></emph.end> but conſidering the <lb></lb>exceſſes of the Gravity of the Moveable over and above the Gra<lb></lb>vities of the <emph type="italics"></emph>Mediums: v. </s> <s>gr.<emph.end type="italics"></emph.end> ^{*} Tin is a thouſand times heavier than <lb></lb><arrow.to.target n="marg1059"></arrow.to.target><lb></lb>Air, and ten times heavier than Water; therefore dividing the ab<lb></lb>ſolute Velocity of the Tin into a thouſand degrees, it ſhall move <lb></lb>in the Air, (which deducteth from it the thouſandth part,) with nine <lb></lb>hundred ninety nine, but in the Water with nine hundred only; <lb></lb>being that the Water abateth the tenth part of its Gravity, and <lb></lb>the Air the thouſandth part. </s> <s>Take a Solid ſomewhat heavier than <lb></lb>Water, as for inſtance, the Wood called Oake, a Ball of which <lb></lb>weighing, as we will ſuppoſe, a thouſand drams, a like quantity <lb></lb>of Water will weigh nine hundred and fifty, but ſo much Air will <lb></lb>weigh but two drams,: it is manifeſt, that ſuppoſing that its abſo<lb></lb>lute Velocity were of a thouſand degrees, in Air there would re<lb></lb>main nine hundred ninety eight, but in the Water only fifty; be<lb></lb>cauſe that the Water of the thouſand degrees of Gravity taketh <lb></lb>away nine hundred and fifty, and leaves fifty only; that Solid there<lb></lb>fore would move well-near twenty times as faſt in the Air as Wa<lb></lb>ter; like as the exceſſe of its Gravity above that of the Water is <lb></lb>the twentieth part of its own. </s> <s>And here I deſire that we may con<lb></lb>ſider, that no matters, having a power to move downwards in the <lb></lb>Water, but ſuch as are more grave in Species than it; and conſe<lb></lb>quently many hundreds of times, more grave than the Air, in <lb></lb>ſeeking what the proportions of their Velocities are in the Air and <lb></lb>Water, we may, without any conſiderable errour, make account <lb></lb>that the Air doth not deduct any thing of moment from the abſo<lb></lb>lute Gravity, and conſequently, from the abſolute Velocity of ſuch <pb xlink:href="069/01/066.jpg" pagenum="66"></pb>matters: ſo that having eaſily found the exceſſe of their Gravi<lb></lb>ty above the Gravity of the Water, we may ſay that their Velo<lb></lb>city in the Air, to their Velocity in the Water hath the ſame propor<lb></lb>tion, that their total Gravity hath to the exceſſe of this above <lb></lb>the Gravity of the Water. </s> <s>For example, a Ball of Ivory weigh<lb></lb>eth twenty ounces, a like quantity of Water weigheth ſeventeen <lb></lb>ounces: therefore the Velocity of the Ivory in Air, to its Velocity <lb></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"></emph>The Velocity of <lb></lb>Grave Bodies de<lb></lb>ſcending Natural<lb></lb>ly to the Center do <lb></lb>go continually en<lb></lb>creaſing till that <lb></lb>by the encreaſe of <lb></lb>the Reſiſtance of <lb></lb>the<emph.end type="italics"></emph.end> Medium <emph type="italics"></emph>it <lb></lb>becometh uniform.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1057"></margin.target>* Or aquiſts.</s></p><p type="margin"> <s><margin.target id="marg1058"></margin.target><emph type="italics"></emph>To find the Pro<lb></lb>portions of the Ve<lb></lb>locities of different <lb></lb>Moveables in the <lb></lb>ſame, and in diffe<lb></lb>rent<emph.end type="italics"></emph.end> 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 acquiſt in a buſineſſe of it ſelf cu<lb></lb>rious, and in which, but without any benefit, I have many times <lb></lb>wearied my-thoughts: nor would there any thing be wanting for <lb></lb>the putting theſe Speculations in practice, ſave onely the way <lb></lb>how one ſhould come to know of what Gravity the Air, is in com<lb></lb>pariſon to the Water, and conſequently to other heavy matters.</s></p><p type="main"> <s>SIMP. </s> <s>But in caſe one ſhould finde, that the Air inſtead of <lb></lb>Gravity had Levity, what ought one to ſay of the foregoing diſ<lb></lb>courſes, otherwiſe very ingenuous?</s></p><p type="main"> <s>SALV. </s> <s>It would be neceſſary to confeſſe that they were truly <lb></lb>Aerial, Light, and Vain. </s> <s>But will you queſtion whether the Air <lb></lb>be heavy, having the expreſſe <emph type="italics"></emph>Text<emph.end type="italics"></emph.end> of <emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end> that affirmeth it, <lb></lb>ſaying, That all the Elements have Gravity, even the Air it ſelf; <lb></lb><arrow.to.target n="marg1060"></arrow.to.target><lb></lb>a ſigne of which (ſubjoyns he) we have in that a ^{*} Bladder blown, <lb></lb>weigheth heavier than unſwell'd.</s></p><p type="margin"> <s><margin.target id="marg1060"></margin.target>* Or <emph type="italics"></emph>Boracho<emph.end type="italics"></emph.end>; a <lb></lb>bottle made of a <lb></lb>Goat skin, uſed <lb></lb>to hold wine and <lb></lb>other Liquids.</s></p><p type="main"> <s>SIMP. </s> <s>That a <emph type="italics"></emph>Boracho,<emph.end type="italics"></emph.end> or Bladder blown, weigheth more, <lb></lb>might proceed, as I could ſuppoſe, not from the Gravity that is <lb></lb>in the Air, but in the many groſſe Vapours intermixed with it in <lb></lb>theſe our lower Regions; by means whereof I might ſay, that the <lb></lb>Gravity of the Bladder, or <emph type="italics"></emph>Boracho<emph.end type="italics"></emph.end> encreaſeth.</s></p><p type="main"> <s>SALV. </s> <s>I would not have you ſay it, and much leſſe that you <lb></lb>ſhould make <emph type="italics"></emph>Aristotle<emph.end type="italics"></emph.end> ſpeak it, for he treating of the Elements, <lb></lb>and deſiring to perſwade me that the Element of Air is grave, <lb></lb>making me to ſee it by an Experement: if in comming to the proof <lb></lb>he ſhould ſay: Take a Bladder, and fill it with groſſe Vapours; <lb></lb>and obſerve that its weight will encreaſe; I would tell him that <lb></lb>it would weigh yet more if one ſhould fill it with bran; but would <lb></lb>afterwards adde; that thoſe Experiments prove, that bran, and <lb></lb>groſſe Vapours are grave: but as to the Element of Air, I ſhould <lb></lb>be left in the ſame doubt as before. </s> <s>The Experiment of <emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end><lb></lb>therefore is good, and the Propoſition true. </s> <s>But I will not ſay ſo <lb></lb>much, for a certain other reaſon taken expreſly out of a Philoſo<lb></lb>pher whoſe name I do not remember, but am ſure that I have read <lb></lb>it, who argueth the Air to be more grave than light, becauſe it <lb></lb>more eaſily carrieth grave Bodies downwards, than the light up<lb></lb>wards.</s></p><p type="main"> <s>SAGR. </s> <s>Good i-faith. </s> <s>By this reaſon then, the Air ſhall be <pb xlink:href="069/01/067.jpg" pagenum="65"></pb>much heavier than the Water, ſince, that all Bodies are carried <lb></lb>more eaſily downwards thorow the Air than thorow the Water, <lb></lb>and all light Bodies more eaſily upwards in this than in that: nay, <lb></lb>infinite matters aſcend in the Water, that in the Air deſcend. <lb></lb></s> <s>But be the Gravity of the Bladder, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> either by reaſon of <lb></lb>the groſſe Vapours, or pure Air, this nothing concerns our pur<lb></lb>poſe, for we ſeek that which happeneth to Moveables that move <lb></lb>in this our Vaporous Region. </s> <s>Therefore, returning to that which <lb></lb>more concerneth me, I would for a full and abſolute informati<lb></lb>on in the preſent buſineſſe, not onely be aſſured that the Air is <lb></lb>grave, as I hold for certain, but I would, if it be poſſible, know <lb></lb>what its Gravity is. </s> <s>Therefore, <emph type="italics"></emph>Salviatus,<emph.end type="italics"></emph.end> if you have wherewith <lb></lb>to ſatisfie me in this alſo, I entreat you to favour me with the <lb></lb>ſame.</s></p><p type="main"> <s>SALV. </s> <s>That there reſideth in the Air poſitive Gravity, and <lb></lb><arrow.to.target n="marg1061"></arrow.to.target><lb></lb>not, as ſome have thought, Levity, which haply is in no Mat<lb></lb>ter to be found, the Experiment of the Blown-Bladder, alledged <lb></lb>by <emph type="italics"></emph>Ariſtotle,<emph.end type="italics"></emph.end> affordeth us a ſufficiently-convincing Argument; for <lb></lb>if the quality of abſolute and poſitive Levity were in the Air, <lb></lb>then the Air being multiplied and compreſſed, the Levity would <lb></lb>encreaſe, and conſequently the propenſion of going upwards: <lb></lb>but Experience ſhews the contrary. </s> <s>As to the other demand, that <lb></lb><arrow.to.target n="marg1062"></arrow.to.target><lb></lb>is, of the Method how to inveſtigate its Gravity, I have tried to <lb></lb>do it in this manner: I have taken a pretty bigge Glaſſe ^{*} Bottle, <lb></lb><arrow.to.target n="marg1063"></arrow.to.target><lb></lb>with its neck bended, and a Finger-ſtall of Leather faſt about <lb></lb>it, having in the top of the ſaid Finger-ſtall inſerted and fa<lb></lb>ſtened a Valve of Leather, by which with a Siringe I have made <lb></lb>paſſe into the Bottle by force a great quantity of Air, of which, <lb></lb>becauſe it admits of great Condenſation, it may take in two or <lb></lb>three other Bottles-ful over and above that which is naturally con<lb></lb>tained therein. </s> <s>Then I have in an exact Ballance very preciſely <lb></lb>weighed that Bottle with the Air compreſſed within it, adjuſting <lb></lb>the weight with ſmall Sands. </s> <s>Afterwards, the Valve being opened, <lb></lb>and the Air let out, that was violently conteined in the Veſſel, I <lb></lb>have put it again into the Scales, and finding it notably aleviated, <lb></lb>I have by degrees taken ſo much Sand from the other Scale, keep<lb></lb>ing it by it ſelf, that the Ballance hath at laſt ſtood <emph type="italics"></emph>in Equilibrio<emph.end type="italics"></emph.end><lb></lb>with the remaining counter-poiſe, that is with the Bottle. </s> <s>And <lb></lb>here there is no queſtion, but that the weight of the reſerved Sand <lb></lb>is that of the Air that was forceably driven into the Bottle, and <lb></lb>which is at laſt gone out thence. </s> <s>But this Experiment hitherto aſ<lb></lb>ſureth me of no more but this, that the Air violently deteined in <lb></lb>the Veſſel, weigheth as much as the reſerved Sand, but how much <lb></lb>the Air reſolutely and determinately weigheth in reſpect of the <lb></lb>Water, or other grave matter, I do not as yet know, nor can <pb xlink:href="069/01/068.jpg" pagenum="66"></pb>I tell, unleſſe I meaſure the quantity of the Air compreſſed: and <lb></lb>for the diſcovering of this a Rule is neceſſary, which I have <lb></lb>found may be performed two manner of wayes, one of which <lb></lb>is to take ſuch another Bottle or Flask as the former, and in like <lb></lb>manner bended, with a Finger-ſtall of Leather, the end of which <lb></lb>may cloſely imbrace the Volve of the other, and let it be very <lb></lb>faſt tied about it. </s> <s>It's requiſite, that this ſecond Bottle be bored in <lb></lb>the bottom, ſo that as by that hole we may thruſt in a Wier, <lb></lb>wherewith we may, at pleaſure, open the ſaid Volve, to let out <lb></lb>the ſuperfluous Air of the other Veſſel, after it hath been weighed: <lb></lb>but this ſecond Bottle ought to be full of Water. </s> <s>All being pre<lb></lb>pared in the manner aforeſaid, and with the Wier opening the <lb></lb>Volve, the Air iſſuing out with impetuoſity, and paſſing into the <lb></lb>Veſſel of Water, ſhall drive it out by the hole at the Bottom: <lb></lb>and it is manifeſt, that the quantity of Water which ſhall be <lb></lb>thruſt out, is equal to the Maſſe and quantity of Air that ſhall <lb></lb>have iſſued from th'other Veſſel: that Water therefore being <lb></lb>kept, and returning to weigh the Veſſel lightned of the Air com<lb></lb>preſſed (which I ſuppoſe to have been weighed likewiſe firſt with <lb></lb>the ſaid forced Air) and the ſuperfluous ſand being laid by, as I <lb></lb>directed before; it is manifeſt, that this is the juſt weight of ſo <lb></lb>much Air in maſſe, as is the maſſe of the expulſed and reſerved <lb></lb>Water; which we are to weigh, and ſee how many times its <lb></lb>weight ſhall contain the weight of the reſerved ſand: and we may <lb></lb>without errour affirme, that the Water is ſo many times heavier <lb></lb>than Air; which ſhall not be ten times, as it ſeemeth <emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end><lb></lb>held, but very neer four hundred, as the ſaid Experiment ſheweth.</s></p><p type="margin"> <s><margin.target id="marg1061"></margin.target><emph type="italics"></emph>The Air hath Po<lb></lb>ſitive Gravity.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1062"></margin.target><emph type="italics"></emph>How that Gravity <lb></lb>may be computed.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1063"></margin.target>* <emph type="italics"></emph>Un Fiaſco,<emph.end type="italics"></emph.end> thoſe <lb></lb>long-neckt glaſſe <lb></lb>bottles in which <lb></lb>we have our <lb></lb><emph type="italics"></emph>Florence<emph.end type="italics"></emph.end> Wine <lb></lb>brought to us.</s></p><p type="main"> <s>The other way is more expeditious, and it may be done with <lb></lb>one Veſſel onely, that is with the firſt accomodated after the man<lb></lb>ner before directed, into which I will not that any other Air be <lb></lb>put, more than that which naturally is found therein; but I will, <lb></lb>that we inject Water without ſuffering any Air to come out, <lb></lb>which being forced to yield to the ſupervenient Water muſt of <lb></lb>neceſſity be compreſſed: having gotten in, therefore, as much <lb></lb>Water as is poſſible, (but yet without great violence one cannot get <lb></lb>in three quarters of what the Bottle will hold) put it into the <lb></lb>Scales, and very carefully weigh it: which done, holding the <lb></lb>Veſſel with the neck upwards, open the Volve, letting out the <lb></lb>Air, of which there will preciſely iſſue forth ſo much as there is <lb></lb>Water in the Bottle. </s> <s>The Air being gone out, put the Veſſel again <lb></lb>into the Scales, which by the departure of the Air will be found <lb></lb>lightened, and abating from the oppoſite Scale the ſuperfluous <lb></lb>weight, it ſhall give us the weight of as much Air as there is <lb></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"></pb>feſſed to be witty and very ingenuous, but whilſt, me thinks, they <lb></lb>fully ſatisfie my underſtanding, they another way occaſion in <lb></lb>me much Confuſion, for it being undoubtedly true that the Ele<lb></lb>ments in their proper Region are neither heavy nor light, I can<lb></lb>not comprehend, how and which way that portion of Air, which <lb></lb>ſeemeth to have weighed <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> four drams of ſand, ſhould af<lb></lb>terwards have that ſame Gravity in the Air, in which the ſand is <lb></lb>contained that weigheth againſt it: and therefore me thinks that <lb></lb>the Experiment ought not to be practiced in the Element of Air, <lb></lb>but in a <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> in which the Air it ſelf might exerciſe its quality <lb></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"></emph>Simplicius<emph.end type="italics"></emph.end> is very acute, <lb></lb>and therefore its neceſſary, either that it be unanſwerable, or that <lb></lb>the Solution be no leſſe acute. </s> <s>That that Air, which compreſ<lb></lb>ſed, appeared to weigh as much as that ſand, left at liberty in its <lb></lb>Element is no longer to weigh any thing as the Sand doth, is a thing <lb></lb>manifeſt: and therefore for making of ſuch an Experiment, its <lb></lb>requiſite to chooſe a place and <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> wherein the Air as well as <lb></lb>the Sand might weigh: for, as hath ſeveral times been ſaid, the <lb></lb><emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> ſubſtracts from the Weight of every Matter that is im<lb></lb>merged therein, ſo much, as ſuch another quantity of the ſaid <lb></lb><emph type="italics"></emph>Medium,<emph.end type="italics"></emph.end> as is that of the maſſe immerſed, weigheth: ſo that <lb></lb>the Air depriveth the Air of all its Gravity. </s> <s>The operation, there<lb></lb><arrow.to.target n="marg1064"></arrow.to.target><lb></lb>fore, to the end it were made exactly, ought to be tried in a <emph type="italics"></emph>Va<lb></lb>cuum,<emph.end type="italics"></emph.end> wherein every grave Body would exerciſe its Moment <lb></lb>without any diminution. </s> <s>In caſe therefore, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> that we <lb></lb>ſhould weigh a portion of Air in a <emph type="italics"></emph>Vacuum,<emph.end type="italics"></emph.end> would you then be <lb></lb>convinced and aſſured of the buſineſſe?</s></p><p type="margin"> <s><margin.target id="marg1064"></margin.target><emph type="italics"></emph>The Air compreſ<lb></lb>ſed and violently <lb></lb>pent up, weigheth in <lb></lb>a<emph.end type="italics"></emph.end> Vacuum; <emph type="italics"></emph>and <lb></lb>how its weight is to <lb></lb>be eſtimated.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. </s> <s>Verily I ſhould: but this is to defire, or enjoyn that <lb></lb>which is impoſſible.</s></p><p type="main"> <s>SALV. </s> <s>And therefore the obligation muſt needs be great that <lb></lb>you owe to me, when ever I ſhall for your ſake effect an impoſſibi<lb></lb>lity: but I will not ſell you that which I have already given you: <lb></lb>for we, in the foregoing Experiment, weigh the Air in a <emph type="italics"></emph>Vacuum,<emph.end type="italics"></emph.end><lb></lb>and not in the Air, or in any other Replete <emph type="italics"></emph>Medium.<emph.end type="italics"></emph.end> That from <lb></lb>the Maſs, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> that in the fluid <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> is immerged certain <lb></lb>Gravity is ſubſtracted by the ſaid <emph type="italics"></emph>Medium,<emph.end type="italics"></emph.end> this commeth to paſs <lb></lb>by reaſon that it reſiſteth its being opened, driven back, and in a <lb></lb>word commoved; a ſign of which is its proneneſs to return inſtant<lb></lb>ly to fill the Space up again, that the immerſed maſs occupied in it, <lb></lb>as ſoon as ever it departeth thence; for if it ſuffered not by that <lb></lb>immerſion, it would not operate againſt the ſame. </s> <s>Now tell me, <lb></lb>when you have in the Air the Bottle before filled with the ſame Air <lb></lb>naturally contained therein, what diviſion, repulſe, or, in ſhort, <lb></lb>what mutation doth the external ambient Air receive from the ſe<pb xlink:href="069/01/070.jpg" pagenum="68"></pb>cond Air that was newly infuſed with force into the Veſſel? </s> <s>Doth <lb></lb>it enlarge the Bottle, whereupon the Ambient ought the more to <lb></lb>retire it ſelf to make room for it? </s> <s>Certainly no: And therefore <lb></lb>we may ſay, that the ſecond Air is not immerſed in the Ambient, <lb></lb>not occupying any Space therein; but is as if it was in a <emph type="italics"></emph>Vacuum,<emph.end type="italics"></emph.end><lb></lb>nay more, is really conſtituted in it, and is placed in Vacuities that <lb></lb>were not repleted by the former un-condenſed Air. </s> <s>And, really, I <lb></lb>know not how to diſcern any difference between the two Conſti <lb></lb>tutions of Incloſed and <emph type="italics"></emph>Ambient,<emph.end type="italics"></emph.end> whilſt in this the <emph type="italics"></emph>Ambient<emph.end type="italics"></emph.end> doth <lb></lb>no-ways preſs the Incloſed, and in that the Incloſed doth not re<lb></lb>repulſe the <emph type="italics"></emph>Ambient<emph.end type="italics"></emph.end>: and ſuch is the placing of any matter in a <lb></lb><emph type="italics"></emph>Vacuum,<emph.end type="italics"></emph.end> and the ſecond Air compreſsed in the Flask. </s> <s>The weight <lb></lb>therefore that is found in that ſame condenſed Air, is the ſame that <lb></lb>it would have, were it freely diſtended in a <emph type="italics"></emph>Vacuum.<emph.end type="italics"></emph.end> Tis true in<lb></lb>deed, that the weight of the Sand that weigheth againſt it, as ha<lb></lb>ving been in the open Air, would in a <emph type="italics"></emph>Vacuum<emph.end type="italics"></emph.end> have been a little <lb></lb>more than juſt ſo heavy; and therefore it is neceſſary to ſay, that <lb></lb>the weighed Air is in reality ſomewhat leſſe heavy than the Sand <lb></lb>that counterpoiſeth it, that is, ſo much, by how much the like <lb></lb>quantity of Air would weigh in a <emph type="italics"></emph>Vacuum.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. </s> <s>I had thought that there was ſomething to have been <lb></lb>wiſhed for in the Experiments before produced; but now I am <lb></lb>thorowly ſatisfied.<lb></lb><arrow.to.target n="marg1065"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1065"></margin.target><emph type="italics"></emph>The difference, <lb></lb>though very great, <lb></lb>of the Gravity of <lb></lb>Moveables hath <lb></lb>no part in differer<lb></lb>cing their Veloci<lb></lb>ties.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>The things by me hitherto alledged, and in particular, <lb></lb>this, That the difference of Gravity, although exceeding great, <lb></lb>hath no part in diverſifying the Velocities of Moveables, ſo that, <lb></lb>notwithſtanding any thing depending on that, they would all <lb></lb>move with equal Celerity, is ſo new, and at the firſt apprehenſi<lb></lb>on ſo remote from probability, that, were there not a way to de<lb></lb>lucidate it, and make it as clear as the Sun, it would be better <lb></lb>to paſſe it over in ſilence, than to divulge it: therefore ſeeing <lb></lb>that I have let it eſcape from me, its fit that I omit neither Expe<lb></lb>riment nor Reaſon that may corroborate it.</s></p><p type="main"> <s>SAGR. </s> <s>Not onely this, but many other alſo of your Aſſerti<lb></lb>ons are ſo remote from the Opinions and Doctrines commonly <lb></lb>received, that ſending them abroad, you would ſtir up a great <lb></lb>number of Antagoniſts: in regard, that the innate Diſpoſition of <lb></lb>Men doth not ſee with good eyes, when others in their Studies <lb></lb>diſcover Truths or Fallacies, that were not diſcovered by them<lb></lb>ſelves: and with the title of Innovators of Doctrines, little plea<lb></lb>ſing to the ears of many, they ſtudy to cut thoſe knots which <lb></lb>they cannot untie, and with ſub-terranean Mines to blow up <lb></lb>thoſe Structures, which have been with the ordinary Tools by <lb></lb>patient Architects erected: but with us here, who are far from <lb></lb>any ſuch thoughts, your Experiments and Arguments are <pb xlink:href="069/01/071.jpg" pagenum="69"></pb>ſufficient to give full ſatisfaction: yet nevertheleſſe, if ſo be you <lb></lb>have other more palpable Experiments, and more convincing <lb></lb>Reaſons we would very gladly hear them.</s></p><p type="main"> <s>SALV. </s> <s>The Experiment made with two Moveables, as different <lb></lb>in weight as may be, by letting them deſcend from a place on <lb></lb>high, thereby to ſee whether their Velocity be equal, meets with <lb></lb>ſome difficulty: for if the height ſhall be great, the <emph type="italics"></emph>Medium,<emph.end type="italics"></emph.end><lb></lb>which is to be opened and laterally repelled by the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of the <lb></lb>cadent Body, ſhall be of much greater prejudice to the ſmall Mo<lb></lb>ment of the light Moveable, than to the violence of the heavy <lb></lb>one; whereupon in a long way the light one will be left behind: <lb></lb>and in a little altitude it might be doubted whether there were <lb></lb>really any difference, or if there were, whether it would be <lb></lb>ſenſible. </s> <s>Therefore I have oft been thinking to reiterate the de<lb></lb>ſcent ſo many times from ſmall heights, and to accumulate toge<lb></lb>ther ſo many of thoſe minute differences of time, as might inter<lb></lb>cede between the arrival or fall of the heavy Body to the ground, <lb></lb>and the arrival of the light one, which ſo conjoyned, would make <lb></lb>a time not onely obſervable, but obſervable with much facility <lb></lb>Moreover, that I might help my ſelf with Motions as ſlow as poſ<lb></lb>ſible may be, in which the Reſiſtance of the <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> operates <lb></lb>leſſe in altering the effect that dependeth on ſimple Gravity, I <lb></lb>have had thoughts to cauſe the Moveable to deſcend upon a de<lb></lb>clining Plane, not much raiſed above the Plane of the Horizon; <lb></lb>for upon this, no leſſe than in perpendicularity, we may diſcover <lb></lb>that which is done by Grave Bodies different in weight: and pro<lb></lb>ceeding farther, I have deſired to free my ſelf from any whatſo<lb></lb>ever impediment, that might ariſe from the Contact of the ſaid <lb></lb>Moveables upon the ſaid declining Plane: and laſtly, I have ta<lb></lb>ken two Balls, one of Lead, and one of Cork, that above an hun<lb></lb>dred times more grave than this, and have faſtened them to two <lb></lb>ſmall threads, each equally four or five yards long, tyed on <lb></lb>high: and having removed aſwel the one as the other Ball from <lb></lb>the ſtate of Perpendicularity, I have let them both go in the ſame <lb></lb>Moment, and they deſcending by the Circumferences of Circles <lb></lb>deſcribed by the equal Strings their Semidiameters, and having <lb></lb>paſſed beyond the Perpendicular, they afterwards by the ſame <lb></lb>way returned back, and reiterating theſe Vibrations, and re<lb></lb>turns of themſelves neer an hundred times, they have ſhewn ve<lb></lb>ry ſenſibly, that the grave <emph type="italics"></emph>Pendulum<emph.end type="italics"></emph.end> moveth ſo exactly under the <lb></lb>time of the light one, that it doth not in an hundred, no nor in a <lb></lb>thouſand Vibrations, anticipate the time of one ſmall moment, <lb></lb>but that they keep an equal paſſe in their Recurſions. </s> <s>They alſo <lb></lb>ſhew the Operation of the <emph type="italics"></emph>Medium,<emph.end type="italics"></emph.end> which conferring ſome im<lb></lb>pediment on the Motion, doth much more diminiſh the Vibrati<pb xlink:href="069/01/072.jpg" pagenum="70"></pb>ons of the Cork, than that of the Lead: not that it maketh them <lb></lb>more or leſſe frequent, nay, when the Arches paſſed by the Cork <lb></lb>were not of above five or ſix degrees, and thoſe of the Lead fif<lb></lb>ty, they did paſs them under the ſame times.</s></p><p type="main"> <s>SIMP. </s> <s>If this be ſo, how is it then that the Velocity of the <lb></lb>Lead is not greater than that of the Cork? </s> <s>that paſſing a jour<lb></lb>ney of ſixty degrees, in the time that this paſseth hardly ſix?</s></p><p type="main"> <s><emph type="italics"></emph>S<emph.end type="italics"></emph.end>ALV. </s> <s>But what would you ſay, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> in caſe they <lb></lb>ſhould both diſpatch their Recurſions in the ſame time, when the <lb></lb>Cork being removed thirty degrees from the Perpendicular, <lb></lb>ſhould paſs an arch of ſixty, and the Lead removed from the <lb></lb>ſame middle point onely two degrees, ſhould run an arch of four? <lb></lb></s> <s>would not then the Cork be ſo much more ſwift than the Lead? <lb></lb></s> <s>and yet Experience ſhews that ſo it happeneth: therefore obſerve, <lb></lb>The <emph type="italics"></emph>Pendulum<emph.end type="italics"></emph.end> of Lead being carried <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> fifty degrees from the <lb></lb>Perpendicular, and thence let go, ſwingeth, and paſſing beyond <lb></lb>the Perpendicular, neer fifty more degrees, deſcribeth an arch <lb></lb>of well neer an hundred degrees; and returning of its ſelf back <lb></lb>again, it deſcribeth another arch, not much leſſe than the former, <lb></lb>and continuing its Vibrations, after a great number of them, it <lb></lb>finally returneth to Reſt: Each of thoſe Vibrations are made un<lb></lb>der equal times aſwel thoſe of ninety degrees, as thoſe of fifty, <lb></lb>twenty, ten, or four; ſo that by conſequence, the Velocity of the <lb></lb>Moveable doth ſucceſſively languiſh and abate, in regard, that <lb></lb>under equal times it doth ſucceſſively paſſe arches continually <lb></lb>leſſer and leſſer. </s> <s>The like, yea the ſelf ſame effect is performed <lb></lb>by the Cork, hanging by a ſtring of the like length, ſave that <lb></lb>in a leſſe number of Vibracions it returneth to Reſt, as being leſs <lb></lb>apt, by means of its Levity, to overcome the obſtacle of the Air: <lb></lb>and yet nevertheleſs all the Vibrations, both great and ſmall, are <lb></lb>made under times equal to one another, and equal alſo to the <lb></lb>times of the times of the Vibrations of the Lead. </s> <s>Whereupon it <lb></lb>is true, that if whilſt the Lead paſſeth an arch of fifty degrees, <lb></lb>the Cork paſseth one but of ten, the Cork is then more ſlow <lb></lb>than the Lead: but it will alſo happen on the other ſide, that the <lb></lb>Cork paſseth the arch of fifty degrees, when the Lead paſseth <lb></lb>but that of ten or ſix; and ſo in ſeveral times the Lead ſhall be <lb></lb>ſwifter onewhile, and the Cork another while: but if the ſame <lb></lb>Moveables ſhall alſo under the ſame equal times, paſs arches that <lb></lb>are equal, one may then very ſafely ſay, that their Velocities are <lb></lb>equal.</s></p><p type="main"> <s>SIMP. </s> <s>This diſcourſe ſeems to me concluding, and not con<lb></lb>cluding, and I finde in my thoughts ſuch a Confuſion, ariſing <lb></lb>from the one-while ſwift, another-while ſlow, another-while ex<lb></lb>treme ſlow motion of both the one and other Moveable; as that <pb xlink:href="069/01/073.jpg" pagenum="71"></pb>it permits me not to diſcern clearly, whether it be true, That their <lb></lb>Velocities are alwaies equal.</s></p><p type="main"> <s>SAGR. </s> <s>Give me leave, I pray you, <emph type="italics"></emph>Salviatus,<emph.end type="italics"></emph.end> to interpoſe two <lb></lb>words. </s> <s>And tell me, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> whether you admit, that it may be <lb></lb>ſaid with abſolute verity that the Velocities of the Cork and of <lb></lb>the Lead are equal, in caſe, that both of them departing at the <lb></lb>ſame moment from Reſt, and moving by the ſame declivities, they <lb></lb>ſhould alwaies paſſe 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></lb>paſſeth now ſixty degrees, now fifty, now thirty, now ten, now <lb></lb>eight, four, and two; and when each of them paſſeth the Arch of <lb></lb>ſixty degrees they paſſe it in the ſame time; in the Arch of fifty the <lb></lb>ſame time is ſpent by both the one and the other Moveable; ſo in <lb></lb>the Arch of thirty, of ten, and of the reſt: and therefore it is con<lb></lb>cluded, that the Velocity of the Lead in the Arch of ſixty degrees, <lb></lb>is equal to the Velocity of the Cork in the ſame Arch of ſixty de<lb></lb>grees: and that the Velocities in the Arch of fifty, are likewiſe <lb></lb>equal to one the other, and ſo in the reſt. </s> <s>But it is not ſaid, that the <lb></lb>Velocity that is exerciſed in the Arch of ſixty is equal to the Ve<lb></lb>locity that is exerciſed in the Arch of fifty, nor this to that of the <lb></lb>Arch of thirty. </s> <s>But the Velocities are alwaies leſſer, in the leſſer <lb></lb>Arches. </s> <s>And this is collected from our ſenſibly ſeeing the ſame <lb></lb>Moveable conſume as much time in paſſing the great Arch of ſixty <lb></lb>degrees, as in paſſing the leſſer of fifty, or the leaſt of ten: and, in a <lb></lb>word, in their being all paſſed alwaies under equal times. </s> <s>It is true <lb></lb>therefore, that both the Lead and the Cork ſucceſſively retard the <lb></lb>Motion, according to the Diminution of the Arches, but yet do <lb></lb>not alter their harmony in keeping the equality of Velocity in all <lb></lb>the ſame Arches by them paſſed. </s> <s>I deſired to ſay thus much, more <lb></lb>to try whether I have rightly apprehended the Conceit of <emph type="italics"></emph>Salvia<lb></lb>tus,<emph.end type="italics"></emph.end> than out of any neceſſity that I thought <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> to ſtand in <lb></lb>of a more plain Explanation than that of <emph type="italics"></emph>Salviatus,<emph.end type="italics"></emph.end> which is, as <lb></lb>in all other things, extreamly clear, and ſuch, that, it being fre<lb></lb>quent with him to reſolve Queſtions, in appearance not only ob<lb></lb>ſcure, but repugnant to Nature, and to the Truth, with Reaſons, <lb></lb>or Obſervations, or Experiments very trite and familiar to every <lb></lb>one, it hath (as I have underſtood from divers) given occaſion to <lb></lb>one of the moſt eſteemed Profeſſors of our Age to put the leſſe <lb></lb>eſteem upon his Novelties, holding them to have as much of Sor<lb></lb>didneſſe, for that they depend on over low and popular Funda<lb></lb>mentals: as if the moſt admirable and moſt-to-be-prized Proper<lb></lb>ty of the Demonſtrative Sciences, were not to ſpring and ariſe <lb></lb>from Principles known, underſtood, and granted by every one. <lb></lb></s> <s>But let us, for all that, continue to banquet our ſelves with this diet <pb xlink:href="069/01/074.jpg" pagenum="72"></pb>that is ſo light of digeſtion; and ſuppoſing that <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> is fully <lb></lb>ſatisfied in underſtanding and admitting, That the intern Gravity <lb></lb>of different Moveables hath no ſhare in differencing their Veloci<lb></lb>ties, ſo that all of them, for ought that dependeth on that, would <lb></lb>move with the ſame Velocities; tell us, <emph type="italics"></emph>Salviatus,<emph.end type="italics"></emph.end> in what you <lb></lb>place the ſenſible and apparent inequalities of Motion; and an<lb></lb>ſwer to that Inſtance that <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> produceth, and which I like<lb></lb>wiſe confirm, I mean, of ſeeing a Cannon Bullet move more ſwift<lb></lb>ly than a drop of Bird-ſhot, for the difference of Velocity ſhall be <lb></lb>but ſmall, in reſpect of that which I object againſt you of Movea<lb></lb>bles of the ſame matter, of which ſome of the greater will deſcend <lb></lb>in a <emph type="italics"></emph>Medium,<emph.end type="italics"></emph.end> in leſſe than one beat of the Pulſe, that ſpace, that <lb></lb>others which are leſſer will not paſſe in an hour, nor in four, nor in <lb></lb>twenty; ſuch are pebbles and minute gravel-ſtones, eſpecially, <lb></lb>that ſmall ſand which muddieth the Water; in which <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end><lb></lb>they will not deſcend in many hours ſo much as two fathoms, <lb></lb>which Stones, and thoſe of no great bigneſſe, do paſſe in one beat <lb></lb>of the Pulſe.</s></p><p type="main"> <s>SALV. </s> <s>That which the <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> operates, in retarding Movea<lb></lb>bles, the more according as they are compared to one another, leſs <lb></lb>grave <emph type="italics"></emph>in ſpecie,<emph.end type="italics"></emph.end> hath been already declared, ſhewing that it pro<lb></lb>ceeds from the ſubſtraction of weight. </s> <s>But how one and the ſame <lb></lb><emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> can with ſo great difference diminiſh the Velocity in <lb></lb>Moveables that differ only in Magnitude, although they are of <lb></lb>the ſame Matter, and of the ſame Figure, requireth for its expli<lb></lb>cation a more ſubtil diſcourſe, than that which ſufficeth for under<lb></lb>ſtanding how the more dilated Figure of the Moveable, or the <lb></lb>Motion of the <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> that is made contrary to the Moveable, re</s></p><p type="main"> <s><arrow.to.target n="marg1066"></arrow.to.target><lb></lb>tardeth the Velocity of the ſaid Moveable. </s> <s>I reduce the cauſe of <lb></lb>the ſaid Problem to the Scabroſity, and Poroſity, that is common<lb></lb>ly, and, for the moſt part, neceſſarily found in the Superficies of <lb></lb>Solid Bodies, the which Scabroſities, in their Motion, go repulſing <lb></lb>and commoving the Air, or other Ambient <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end>: of which we <lb></lb>have an evident teſtimony, in that we hear the Bodies, though made <lb></lb>as round as is poſſible for them to be, to hum whilſt they paſſe ve<lb></lb>ry ſwiftly thorow the Air; and they are not only heard to hum, but <lb></lb>to whir and whiſtle, if there be but in them ſome more than ordi<lb></lb>nary cavity or prominency. </s> <s>We ſee alſo, that in turning round <lb></lb>every rotund Solid maketh a little wind: And what need more? <lb></lb></s> <s>Do we not hear a notable whirring, and in a very ſharp Accent, <lb></lb>made by a Top, while it turneth round on the ground with great <lb></lb>Celerity? </s> <s>The ſhrilneſs of which whizzing groweth flatter accor<lb></lb>ding as the Velocity of the <emph type="italics"></emph>Vertigo<emph.end type="italics"></emph.end> doth by degrees more and <lb></lb>more ſlacken: a neceſſary Argument likewiſe of the commotion <lb></lb>and percuſſion of the Air by thoſe (though very ſmall) Scabroſi<pb xlink:href="069/01/075.jpg" pagenum="73"></pb>ties of their Superficies. </s> <s>It is not to be doubted, but that theſe in the <lb></lb>deſcent of Moveables, grating upon, and repulſing the fluid Am<lb></lb>bient, procure retardment in the Velocity, and ſo much the greater, <lb></lb>by how much the Superficies ſhall be greater, as is that of leſſer <lb></lb>Solids compared to bigger.</s></p><p type="margin"> <s><margin.target id="marg1066"></margin.target><emph type="italics"></emph>The greater or leſs <lb></lb>Scabroſity and Po<lb></lb>roſity of the Super<lb></lb>ficies of Movea<lb></lb>bles, a probable <lb></lb>cauſe of their grea<lb></lb>ter or leſſer Retar<lb></lb>dation.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. Stay, I pray you, for here I begin to be at a loſſe: for <lb></lb>though I underſtand and admit, that the Confrication of the <emph type="italics"></emph>Medi<lb></lb>um<emph.end type="italics"></emph.end> with the Superficies of the Moveable retardeth the Motion, <lb></lb>and that it more retardeth it where <emph type="italics"></emph>(ceteris paribus)<emph.end type="italics"></emph.end> the Superficies <lb></lb>is greater, yet do I not comprehend upon what ground you call the <lb></lb>Superficies of leſſer Solids greater: & farthermore if, as you affirm, the <lb></lb>greater Superficies ought to cauſe greater retardment, the greater <lb></lb>Solids ought to be the ſlower, which is not ſo: but this Objection <lb></lb>may eaſily be removed, by ſaying, that although the greater hath <lb></lb>a greater Superficies, it hath alſo a greater Gravity, upon which <lb></lb>the impediment of the greater Superficies hath not ſo much more <lb></lb>prevalent influence, than the impediment of the leſſer Superficies <lb></lb>hath upon the leſſer Gravity, as that the Velocity of the greater <lb></lb>Solid ſhould become the leſſer. </s> <s>And therefore I ſee no reaſon why <lb></lb>one ſhould alter the equality of the Velocities, whilſt, that looking <lb></lb>how much the Moving Gravity diminiſheth, the faculty of the Re<lb></lb>tarding Superficies doth diminiſh at the ſame rate.</s></p><p type="main"> <s>SALV. </s> <s>I will reſolve all that which you object in one word. <lb></lb></s> <s>Therefore, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> you will without controverſie admit, that <lb></lb>when, of two equal Moveables of the ſame Matter, and alike in Fi<lb></lb>gure (which undoubtedly would move with equal ſwiftneſſe) as <lb></lb>well the Gravity, as the Superficies of one of them diminiſheth, <lb></lb>(yet ſtill retaining the ſimilitude of Figure) the Velocity like<lb></lb>wiſe, for the ſame reaſon, would not be diminiſhed in that which <lb></lb>was leſſened.</s></p><p type="main"> <s>SIMP. Really, I think, that it ought ſo to follow as you ſay, <lb></lb>granting the preſent Doctrine with a <emph type="italics"></emph>ſalvo<emph.end type="italics"></emph.end> ſtill to our Doctrine, <lb></lb>which teacheth, that the greater or leſſer Gravity hath no operati<lb></lb>on in accelerating or retarding Motion.</s></p><p type="main"> <s>SALV. </s> <s>And this I confirm; and grant you likewiſe your Po<lb></lb>ſition, from whence, in my opinion, may be inferred, That in caſe <lb></lb>the Gravity diminiſheth more than the Superficies, there may be <lb></lb>introduced in the Moveable, in that manner diminiſhed, ſome re<lb></lb>tardment of Motion, and that greater and greater, by how much in <lb></lb>proportion, the diminution of the Weight was greater than the di<lb></lb>minution of the Superficies</s></p><p type="main"> <s>SIMP. </s> <s>I make not the leaſt queſtion of it.<lb></lb><arrow.to.target n="marg1067"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1067"></margin.target><emph type="italics"></emph>Solids cannot be <lb></lb>diminiſhed at the <lb></lb>ſame rate in Super<lb></lb>ficies as in Weight, <lb></lb>retaining the ſimi<lb></lb>litude of the Fi<lb></lb>gures.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>Now know, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> that in Solids one cannot di<lb></lb>miniſh the Superficies ſo much as the Weight keeping the ſimili<lb></lb>tude of Figure. </s> <s>For it being manifeſt, that in diminiſhing of grave <pb xlink:href="069/01/076.jpg" pagenum="74"></pb>Solids, the Weight leſſeneth as much as the Bulk, when ever the <lb></lb>Bulk happens to be diminiſhed more than the Superficies, (care <lb></lb>being had to retain the ſimilitude of Figure) the Gravity likewiſe <lb></lb>would come to be more diminiſhed than the Superficies. </s> <s>But <emph type="italics"></emph>Geo<lb></lb>metry<emph.end type="italics"></emph.end> teacheth us, that there is much greater proportion between <lb></lb>the Bulk and the Bulk in like Solids, than between their Superfi<lb></lb>cies. </s> <s>Which for your better underſtanding, I ſhall explain in ſome <lb></lb>particular caſe. </s> <s>Therefore fancy to your ſelf, for example, a Dye, <lb></lb>one of the Sides of which is <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> two Inches long, ſo that one of <lb></lb>its Surfaces ſhall be four Square Inches, and all ſix, that is, all its <lb></lb>Superficies twenty four Square Inches. </s> <s>Then ſuppoſe the ſame <lb></lb>Dye at three ſawings cut into eight ſmall Dice, the Side of every <lb></lb>one of which will be one Inch, and one of its Surfaces an Inch <lb></lb>Square, and its whole Superficies ſix Square Inches, of which the <lb></lb>whole Dye contained twenty four in its Superficial content. </s> <s>Now, <lb></lb>you ſee, that the Superficial content of the little Dye is the fourth <lb></lb>part of the Superficial content of the great one, (for ſix is the <lb></lb>fourth part of twenty four) but the Solid content of the ſaid Dye <lb></lb>is only the eighth part: therefore the Bulk, and conſequently the <lb></lb>Weight, doth much more diminiſh than the Superficies. </s> <s>And if <lb></lb>you ſubdivide the little Dye into eight others, we ſhall have for <lb></lb>the whole Superficial content of one of theſe, one and an half <lb></lb>Square Inches, which is the ſixteenth part of the Superficies of the <lb></lb>firſt Dye; but its Bulk, or Maſs, is only the ſixty fourth part of that. <lb></lb></s> <s>You ſee therefore, how that in only theſe two diviſions the Bulks <lb></lb>decreaſe four times faſter than their Superficies: and if we ſhould <lb></lb>proſecute the Subdiviſion, untill that we had reduced the firſt So<lb></lb>lid into a ſmall powder, we ſhould find the Gravity of the minute <lb></lb>Atomes to be leſſened an hundred and an hundred times more <lb></lb>than their Superficies. </s> <s>And this which I have exemplified in <lb></lb>Cubes, hapneth in all like Solids, the Bulks of which are in Seſ<lb></lb>quialter proportion of their Superficies. </s> <s>You ſee, therefore, in how <lb></lb>much greater proportion the Impediment of the Contact of the <lb></lb>Superficies of the Moveable with the <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> encreaſeth in ſmall <lb></lb>Moveables, than in greater: and if we ſhould add, that the Sca<lb></lb>broſities in the very ſmall Superficies of the minute Atomes are <lb></lb>not happily leſſer than thoſe of the Superficies of greater Solids, <lb></lb>that are diligently poliſhed, obſerve how fluid, and void of all Re<lb></lb>ſiſtance being opened, the <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> is required to be, when it is to <lb></lb>give paſſage to ſo feeble a Virtue. </s> <s>And therefore take notice, <emph type="italics"></emph>Sim<lb></lb>plicius,<emph.end type="italics"></emph.end> that I did not equivocate, when even now I ſaid, That the <lb></lb>Superficies of leſſer Solids is greater, in compariſon of that of <lb></lb>bigger.</s></p><p type="main"> <s>SIMP. </s> <s>I am wholly ſatisfied: and I verily believe, that if I were <lb></lb>to begin my Studies again, I ſhould follow the Counſel of <emph type="italics"></emph>Plato,<emph.end type="italics"></emph.end><pb xlink:href="069/01/077.jpg" pagenum="75"></pb>and enter my ſelf firſt in the Mathematicks, which I ſee to proceed <lb></lb>very ſcrupulouſly, and refuſe to admit any thing for certain, ſave <lb></lb>that which they neceſſarily demonſtrate.</s></p><p type="main"> <s>SAGR. </s> <s>I have taken great delight in this Diſcourſe; but, be<lb></lb>fore we paſſe any further, I would be glad to be ſatisfied in one <lb></lb>particular, which newly came into my thoughts, when but juſt <lb></lb>now you ſaid, that Like-Solids are in Seſquialter proportion to <lb></lb>their Superficies for I have ſeen, and underſtood the Propoſition </s></p><p type="main"> <s><arrow.to.target n="marg1068"></arrow.to.target><lb></lb>with its Demonſtration, in which it is proved, That the Superficies <lb></lb>of Like-Solids are in duplicate proportion of their Sides; and ano<lb></lb>ther that proveth the ſame Solids to be in triple proportion of the <lb></lb>ſame Sides; but the proportion of Solids to their Superficies, I do <lb></lb>not remember that I ever ſo much as heard it mentioned.</s></p><p type="margin"> <s><margin.target id="marg1068"></margin.target><emph type="italics"></emph>Solids are to each <lb></lb>other in Seſquial<lb></lb>ter proportion to <lb></lb>their Superficies.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>You your ſelf have anſwered and declared the doubt. <lb></lb></s> <s>For that which is triple of a thing of which another is double, doth <lb></lb>it not come to be Seſquialter of this double? </s> <s>Yes doubtleſſe. </s> <s>Now, <lb></lb>if Superficies are in double proportion of the Lines, of which the <lb></lb>Solids are in triple proportion, may not we ſay, That the Solids are <lb></lb>in Seſquialter proportion of their Superficies?</s></p><p type="main"> <s>SAGR. </s> <s>I underſtand you very well. </s> <s>And although other par<lb></lb>ticulars, pertaining to the matter of which we have treated, do re<lb></lb>main for me to ask, yet if we ſhould thus run from one Digreſſion <lb></lb>to another, it will be late before we ſhould come to the Queſtions <lb></lb>principally intended, which concern the diverſities of the Acci<lb></lb>dents of the Reſiſtances of Solids againſt Fraction; and therefore, <lb></lb>if you ſo pleaſe, we may return to the firſt Theme, which we pro<lb></lb>poſed in the beginning.</s></p><p type="main"> <s>SALV. </s> <s>You ſay very well; but the ſo many, and ſo different <lb></lb>things that have been examined, have ſtoln ſo much of our time, <lb></lb>that there is but little of it left in this day to ſpend in our other <lb></lb>principal Argument, which is full of Geometrical Demonſtrati<lb></lb>ons that are to be conſidered with attention: ſo that I ſhould think <lb></lb>it were better to adjourn our meeting till to morrow, as well for <lb></lb>this which I have told you, as alſo becauſe I might bring with me <lb></lb>ſome Papers, on which I have, in order, ſet down the Theorems and <lb></lb>Problems, in which are propoſed and demonſtrated the different <lb></lb>Paſſions of this Subject, which, it may be, would not otherwiſe <lb></lb>with requiſite Method come into my mind.</s></p><p type="main"> <s>SAGR. </s> <s>I very gladly comply with your advice, and ſo much the <lb></lb>more willingly, in regard that, for a Concluſion of this daies Con<lb></lb>ference, I ſhall have time to hear you reſolve ſome doubts that I <lb></lb>find in my mind concerning the Point laſt handled. </s> <s>Of which one <lb></lb>is, Whether we are to hold, that the Impediment of the <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end><lb></lb>may be ſufficient to aſſign bounds to the Acceleration of Bodies of <lb></lb>very grave Matter, that are of great Bulk, and of a Spherical Figure: <pb xlink:href="069/01/078.jpg" pagenum="76"></pb>and I inſtance in the Spherical Figure, that I might take that which <lb></lb>is contained under the leaſt Superficies, and therefore leſſe ſubject <lb></lb>to Retardment. </s> <s>Another ſhall be, touching the Vibrations of Pen<lb></lb>dulums, and this hath many heads: One ſhall be, Whether all, <lb></lb>both Great, Mean, and Little, are made really and preciſely under <lb></lb>equal Times: And another, What is the proportion of the Times <lb></lb>of Moveables, ſuſpended at unequal ſtrings, of the Times of their <lb></lb>Vibrations I mean.</s></p><p type="main"> <s>SALV. </s> <s>The Queſtions are ingenious, and, like as it is incident <lb></lb>to all Truths, I ſuppoſe, that, which ever of them we handle, it will <lb></lb>draw after it ſo many other Truths, and curious Conſequences, <lb></lb>that I cannot tell whether the remainder of this day may ſuffice <lb></lb>for the diſcuſſing of them all.</s></p><p type="main"> <s>SAGR. </s> <s>If they ſhall be but as delightful as the precedent, it <lb></lb>would be more grateful for me to employ as many daies, not to ſay, <lb></lb>hours, as it is unto night, and I believe that <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> will not be <lb></lb>cloy'd with ſuch Argumentations as theſe.</s></p><p type="main"> <s>SIMP. </s> <s>No certainly: and eſpecially, when the Queſtions trea<lb></lb>ted of are Phyſical, touching which we read not the Opinions or <lb></lb>Diſcourſes of other Philoſophers.<lb></lb><arrow.to.target n="marg1069"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1069"></margin.target><emph type="italics"></emph>Any Body, of any <lb></lb>Figure, Greatneſs, <lb></lb>and Gravity, is <lb></lb>checked by the Re<lb></lb>nitence of the<emph.end type="italics"></emph.end> Me<lb></lb>dium, <emph type="italics"></emph>though ne<lb></lb>ver ſo tenuous, in <lb></lb>ſuch ſort, that the <lb></lb>Motion continuing, <lb></lb>it is reduced to <lb></lb>equability.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>I come therefore to the firſt, affirming without any <lb></lb>hæſitation, that there is not a Sphere ſo big, nor of Matter ſo grave, <lb></lb>but that the Renitence of the <emph type="italics"></emph>Medium,<emph.end type="italics"></emph.end> though very tenuous, checks <lb></lb>its Acceleration, and in the continuation of the Motion reduceth <lb></lb>it to Equability, of which we may draw a very clear Argument <lb></lb>from Experience it ſelf. </s> <s>For if any falling Moveable were able in <lb></lb>its continuation of Motion to attain any degree of Velocity, no <lb></lb>Velocity that ſhould be conferred upon it, could be ſo great but <lb></lb>that it would depoſe it, and free it ſelf of it by help of the Impe<lb></lb>diment of the <emph type="italics"></emph>Medium.<emph.end type="italics"></emph.end> And thus, a Cannon-bullet, that had de <lb></lb>ſcended through the Air, <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> four yards, and had, for example, <lb></lb>acquired ten degrees of Velocity, and that with theſe ſhould enter <lb></lb>into the Water, in caſe the Impediment of the Water were not <lb></lb>able to prohibit ſuch a certain <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> in the Ball, it would en<lb></lb>creaſe it, or at leaſt would continue it unto the bottom; which is <lb></lb>not obſerved to enſue: nay, the Water, although it were but a few <lb></lb>fathoms in depth, would impede and debilitate it in ſuch a man<lb></lb>ner, that it will make but a ſmall impreſſion in the bottom of the <lb></lb>River or Lake. </s> <s>It is therefore manifeſt, that that Velocity, of <lb></lb>which the Water had ability to deprive it in a very ſhort way, <lb></lb>would never be permitted to be acquired by it, though in a depth <lb></lb>of a thouſand Fathoms. </s> <s>And why ſhould it be permitted to gain <lb></lb>it in a thouſand, to be taken from it again in four? </s> <s>What need we <lb></lb>more? </s> <s>Do we not ſee the immenſe <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of the Ball, ſhot from <lb></lb>the Cannon it ſelf, to be in ſuch a manner flatted by the interpo<pb xlink:href="069/01/079.jpg" pagenum="77"></pb>ſition of a few Fathom of Water, that without any harm to the <lb></lb>Ship, it but very hardly reacheth to make a dent in it? </s> <s>The Air al<lb></lb>ſo, though very yielding, doth nevertheleſſe repreſſe the Velocity <lb></lb>of the falling Moveable, although it be very heavy, as we may by <lb></lb>ſuch like Experiments collect; for if from the top of a very high <lb></lb>Tower we ſhould diſcharge a Muſquet downwards, this will make <lb></lb>a leſſer impreſſion on the ground, than if we ſhould diſcharge the <lb></lb>Muſquet at the height of four or ſix yards above the Plane: an <lb></lb>evident ſign, that the <emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> wherewith the Bullet iſſueth from <lb></lb>the Gun, diſcharged on the top of the Tower, doth gradually di<lb></lb>miniſh in deſcending thorow the Air: therefore the deſcending <lb></lb>from any whatſoever great height will not ſuffice to make it ac<lb></lb>quire that <emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> of which the Reſiſtance of the Air deprived <lb></lb>it, when it had in any manner been conferred upon it. </s> <s>The batte<lb></lb>ry likewiſe that the force of a Bullet, ſhot from a Culverin, ſhall <lb></lb>make in a Wall at the diſtance of twenty Paces, would not, I be<lb></lb>lieve, be ſo great, if the Bullet was ſhot perpendicularly from any <lb></lb>immenſe Altitude. </s> <s>I believe, therefore, that there is a Bound or <lb></lb>term belonging to the Acceleration of every Natural Moveable <lb></lb>that departs from Reſt, and that the Impediment of the <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> in <lb></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></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></lb>the purpoſe: nor doth any thing remain, unleſſe the Adverſary <lb></lb>ſhould fortifie himſelf, by denying, that they will hold true in great <lb></lb>and ponderous Maſſes, and that a Cannon-bullet coming from the <lb></lb>Concave of the Moon, or from the upper Region of the Air, <lb></lb>would make a greater percuſſion than coming from the Cannon.</s></p><p type="main"> <s>SALV. </s> <s>There is no queſtion, but that many things may be <lb></lb>objected, and that they may not be all ſalved by Experiments; ne<lb></lb>vertheleſſe in this contradiction, me thinks, there is ſomething that <lb></lb>may fall under conſideration; <emph type="italics"></emph>ſcilicet,<emph.end type="italics"></emph.end> that it is very probable, <lb></lb><arrow.to.target n="marg1071"></arrow.to.target><lb></lb>that the Grave Body, falling from an Altitude, acquireth ſo much <lb></lb><emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> at its arrival to the ground, as would ſuffice to return it <lb></lb>to that height, as is plainly ſeen in a <emph type="italics"></emph>Pendulum<emph.end type="italics"></emph.end> reaſonable weighty, <lb></lb>that being removed fifty or ſixty degrees from the Perpendicular, <lb></lb>gaineth that Velocity and Virtue which exactly ſufficeth to force it <lb></lb>to the like Recurſion, that little abated, which is taken from it by <lb></lb>the Impediment of the Air. </s> <s>To conſtitute, therefore, the Cannon<lb></lb>bullet in ſuch an Altitude as may ſuffice for the acquiſt of an <emph type="italics"></emph>Impe<lb></lb>tus,<emph.end type="italics"></emph.end> as great as that which the Fire giveth it in its iſſuing from the <lb></lb>Piece, it would ſuffice to ſhoot it upwards perpendicularly with <lb></lb>the ſaid Cannon, and then obſerving, whether in its fall it maketh <lb></lb>an impreſſion equal to that of the percuſſion made near at hand in <lb></lb>its iſſuing forth; but, indeed, I believe, that it would not be any <pb xlink:href="069/01/080.jpg" pagenum="78"></pb>whit near ſo forcible. </s> <s>And therefore I hold that the Velocity, <lb></lb>which the Bullet hath near to its going out of the Piece, would <lb></lb>be one of thoſe that the Impediment of the Air would never ſuffer <lb></lb>it to acquire, whilſt it ſhould with a natural Motion deſcend, leaving <lb></lb>the ſtate of Reſt, from any great height. </s> <s>I come now to the other <lb></lb>Queſtions belonging to <emph type="italics"></emph>Pendulums,<emph.end type="italics"></emph.end> matters which to many would <lb></lb>ſeem very frivolous, and more eſpecially to thoſe Philoſophers that <lb></lb>are continually buſied in the more profound Queſtions of Natural <lb></lb>Philoſophy: yet, notwithſtanding, will not I contemn them, being <lb></lb>encouraged by the Example of <emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end> himſelf, in whom I admire <lb></lb>this above all things; that he hath not, as one may ſay, omitted any <lb></lb>matter that any waies merited conſideration, which he hath not <lb></lb>ſpoken of: and now upon the Queſtions you propounded, I think <lb></lb>I can tell you a certain conceit of mine upon ſome Problems con<lb></lb>cerning Muſick, a noble Subject, of which ſo many famous men, <lb></lb>and <emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end> himſelf, have written; and touching it, he conſide<lb></lb>reth many curious Problems: ſo that if I likewiſe ſhall from ſo fa<lb></lb>miliar and ſenſible Experiments, draw Reaſons of admirable acci<lb></lb>dents on the Argument of Sounds, I may hope that my diſcourſes <lb></lb>will be accepted by you.</s></p><p type="margin"> <s><margin.target id="marg1071"></margin.target><emph type="italics"></emph>A Grave Body, <lb></lb>falling from an <lb></lb>Altitude, acqui<lb></lb>reth ſo much<emph.end type="italics"></emph.end> Im<lb></lb>petus <emph type="italics"></emph>at its arri<lb></lb>val to the ground, <lb></lb>as in all probabili<lb></lb>ty, would ſuffice to <lb></lb>recarry it to the <lb></lb>ſame height from <lb></lb>whence it fell.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR Not only accepted, but by me, in particular, moſt paſ<lb></lb>ſionately deſired, in regard that I taking a great delight in all Mu<lb></lb>ſical Inſtruments, and being reaſonably well inſtructed concerning <lb></lb>Conſonances, have alwaies been ignorant and perplexed with <lb></lb>endeavouring to know, whence it cometh that one ſhould more <lb></lb>pleaſe and delight me than another; and that ſome not only pro<lb></lb>cure me no delight, but highly diſpleaſe me: the trite Ptoblem al<lb></lb>ſo of the two Chords ſet to an Uniſon, one of which moveth and <lb></lb>actually ſoundeth at the touching of the other, I alſo am unreſol<lb></lb>ved in: nor am I very clearly informed concerning the Forms of <lb></lb>Conſonances, and other particularities.</s></p><p type="main"> <s><emph type="italics"></emph>S<emph.end type="italics"></emph.end>ALV. </s> <s>We will ſee, if from theſe our <emph type="italics"></emph>Peudulums<emph.end type="italics"></emph.end> one may ga<lb></lb>ther any ſatisfaction in all theſe Doubts. </s> <s>And as to the firſt Que<lb></lb>ſtion, that is, Whether the ſame <emph type="italics"></emph>Pendulum<emph.end type="italics"></emph.end> doth really and punctu<lb></lb>ally perform all its Vibrations, great, leſſer, and leaſt, under Times <lb></lb>preciſely equal; I refer my ſelf to that which I have heretofore <lb></lb>learnt from our <emph type="italics"></emph>Academian,<emph.end type="italics"></emph.end> who plainly demonſtrateth, that the <lb></lb><arrow.to.target n="marg1072"></arrow.to.target><lb></lb>Moveable that ſhould deſcend along the Chords, that are Subten<lb></lb>ſes to any Arch, would neceſſarily paſſe them all in equal Times, <lb></lb>as well the Subtenſe under an hundred and eighty degrees, (that <lb></lb>is, the whole Diameter) as the Subtenſes of an hundred, ſixty, ten, <lb></lb>two, or half a degree, or of four minutes: ſtill ſuppoſing that they <lb></lb>all determine in the loweſt Point touching the Horizontal Plane. <lb></lb></s> <s>Next as to the deſcendents by the Arches of the ſame Chords eli<lb></lb>vated above the Horizon, and that are not greater than a Qua<pb xlink:href="069/01/081.jpg" pagenum="79"></pb>drant, that is, than ninety degrees, Experience likewiſe ſhews, that <lb></lb><arrow.to.target n="marg1073"></arrow.to.target><lb></lb>they paſſe all in Times equal, but yet ſhorter than the Times of <lb></lb>the paſſages by the Chords: an effect which hath ſo much of won<lb></lb>der in it, by how much at the firſt apprehenſion one would think <lb></lb>the contrary ought to follow: For the terms of the beginning, <lb></lb>and the end of the Motion being common, and the Right-Line be<lb></lb>ing the ſhorteſt, that can be comprehended between the ſaid <lb></lb>Terms, it ſeemeth reaſonable, that the Motion made by it ſhould <lb></lb>be finiſhed in the ſhorteſt Time, which yet is not ſo: but the ſhor<lb></lb>teſt Time, and conſequently, the ſwifteſt Motion, is that made by <lb></lb>the Arch of which the ſaid Right-Line is Chord. </s> <s>In the next <lb></lb><arrow.to.target n="marg1074"></arrow.to.target><lb></lb>place, as to the Times of the Vibrations of Moveables, ſuſpended <lb></lb>by ſtrings of different lengths, thoſe Times are in Subduple pro<lb></lb>portion to the lengths of the ſtrings, or, if you will, the lengths <lb></lb>are in duplicate proportion to the Times, that is, are as the Squares <lb></lb>of the Times: ſo that if, for example, the Time of a Vibration <lb></lb>of one <emph type="italics"></emph>Pendulum<emph.end type="italics"></emph.end> is double to the Time of a Vibration of another, <lb></lb>it followeth, that the length of the ſtring of that is quadruple to <lb></lb>the length of the ſtring of this. </s> <s>And in the Time of one Vibration <lb></lb>of that, another ſhall then make three Vibrations, when the ſtring <lb></lb>of that ſhall be nine times as long as the other. </s> <s>From whence doth <lb></lb>follow, that the length of the ſtrings have to each other the ſame <lb></lb>proportion, that the Squares of the Numbers of the Vibrations that <lb></lb>are made in the ſame Times have.</s></p><p type="margin"> <s><margin.target id="marg1072"></margin.target><emph type="italics"></emph>Moveables deſcen<lb></lb>ding along the <lb></lb>Chords, that are <lb></lb>Subtenſes to any <lb></lb>Arch of a Circle, <lb></lb>paſſe as well the <lb></lb>greater as the leſ<lb></lb>ſer Chords in equal <lb></lb>Times.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1073"></margin.target><emph type="italics"></emph>Moveables and<emph.end type="italics"></emph.end><lb></lb>Pendula <emph type="italics"></emph>deſcend<lb></lb>ing along the Ar<lb></lb>ches of the ſame <lb></lb>Chords, elivated as <lb></lb>far as 90 deg. </s> <s>paſs <lb></lb>the ſaid Arches in <lb></lb>Times equal, but <lb></lb>that are ſhorter <lb></lb>than the tranſiti<lb></lb>ons along the <lb></lb>Chords.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1074"></margin.target><emph type="italics"></emph>The Times of the <lb></lb>Vibrations of Mo<lb></lb>vables, hanging at <lb></lb>alonger or ſhorter <lb></lb>thread, are to one <lb></lb>another in propor<lb></lb>tion ſubduple the <lb></lb>lengths of the <lb></lb>ſtrings, at which <lb></lb>they hang.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. Then, if I have rightly underſtood you, I may eaſily <lb></lb><arrow.to.target n="marg1075"></arrow.to.target><lb></lb>know the length of a ſtring, hanging at any never-ſo-great height, <lb></lb>although the ſublime term of the ſuſpenſion were inviſible to me, <lb></lb>and I only ſaw the other lower extream. </s> <s>For if I ſhall faſten a <lb></lb>weight of ſufficient Gravity to the ſaid ſtring here below, and ſet <lb></lb>it on vibrating to and again, and a friend telling ſome of its Recur<lb></lb>ſions, and I at the ſame time tell the Recurſions of another Movea<lb></lb>ble, ſuſpended at a ſtring that is preciſely a yard long, by the <lb></lb>Numbers of the Vibrations of theſe <emph type="italics"></emph>Pendula,<emph.end type="italics"></emph.end> made in the ſame <lb></lb>Time, I will find the length of the ſtring. </s> <s>As for example, ſuppoſe <lb></lb>that in the time that my friend hath counted twenty Recurſions of <lb></lb>the long ſtring, I had told two hundred and forty of my ſtring, <lb></lb>that is one yard long: ſquaring the two numbers twenty and two <lb></lb>hundred and forty, which are 400, and 57600, I will ſay, that the <lb></lb>long ſtring containeth 57600 of thoſe Meaſures, of which my <lb></lb>ſtring containeth 400. and becauſe the ſtring is one ſole yard, I will <lb></lb>divide 57600 by 400, and the quotient will be 144, and I will af<lb></lb>firm that ſtring to be 144 yards long.</s></p><p type="margin"> <s><margin.target id="marg1075"></margin.target><emph type="italics"></emph>To find the Length <lb></lb>of any Rope, or <lb></lb>ſtring, at which a <lb></lb>Moveable hang<lb></lb>eth, by the frequen<lb></lb>cy of its Vibrations<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>Nor will you be miſtaken one Inch; and eſpecially, if <lb></lb>you take a great Number of Vibrations.</s></p><p type="main"> <s>SAGR. </s> <s>You give me frequent occaſion to admire the Riches, <pb xlink:href="069/01/082.jpg" pagenum="80"></pb>and withal the extraordinary bounty of Nature, whil'ſt by things <lb></lb>ſo common, and, I might in a certain ſence ſay, vile, you go col<lb></lb>lecting of Notions very curious, new, and oftentimes, remote <lb></lb>from all imagination. </s> <s>I have an hundred times conſidered the Vi<lb></lb>brations, in particular, of the Lamps in ſome Churches, hanging <lb></lb>by very long ropes, when they have been unawares ſtirred by <lb></lb>any one: but the moſt that I inferred from that ſame Obſervati<lb></lb>on, was the improbability of the Opinion of thoſe who hold, <lb></lb>that ſuch-like Motions are maintained and continued by the <emph type="italics"></emph>Medi<lb></lb>um,<emph.end type="italics"></emph.end> that is by the Air: for it ſhould ſeem to me, that the Air had <lb></lb>a great judgment, and withal but little buſineſſe to ſpend ſo ma<lb></lb>ny hours time in vibrating an hanging Weight with ſo much Regu<lb></lb>larity: but that I ſhould have learnt, that that ſame Moveable, <lb></lb>ſuſpended at a ſtring of an hundred yards long, being removed <lb></lb>from Perpendicularity one while ninety degrees, and another <lb></lb>while one degree onely, or half a degree, ſhould ſpend as much time <lb></lb>in paſſing this little, as in paſſing that great Arch, certainly would <lb></lb>never have come into my head, for I ſtill think, that it bordereth <lb></lb>upon Impoſsibility. </s> <s>Now I am in expectation to hear that theſe <lb></lb>petty Notions will aſsign me ſuch Reaſons of thoſe Muſical Pro<lb></lb>blems, as may, in part at leaſt, give me ſatisfaction.<lb></lb><arrow.to.target n="marg1076"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1076"></margin.target><emph type="italics"></emph>Every<emph.end type="italics"></emph.end> Pendulum <lb></lb><emph type="italics"></emph>hath the Time of <lb></lb>its Vibration ſo li<lb></lb>mited; that it is <lb></lb>not poſſible to make <lb></lb>it move under any <lb></lb>other Period.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>Above all things, you are to know, that every <emph type="italics"></emph>Pendu<lb></lb>lum<emph.end type="italics"></emph.end> hath the Time of its Vibrations ſo limited, and prefixed, that <lb></lb>it is impoſſible to make it move under any other Period, than that <lb></lb>onely one, which is natural unto it. </s> <s>Let any one take the ſtring in <lb></lb>hand, to which the Weight is faſtened, and trie all the wayes <lb></lb>he can to encreaſe or decreaſe the frequency of its Vibrations, <lb></lb>and he ſhall finde it labour in vain: but we may, on the contrary, <lb></lb>on a <emph type="italics"></emph>Pendulum,<emph.end type="italics"></emph.end> though grave and at reſt, by onely blowing up<lb></lb>on it, conferre a Motion, and a Motion conſiderably great, by <lb></lb>reiterating the blaſts, but under the Time that is properly be<lb></lb>longing to its Vibrations: for if at the firſt blaſt we ſhould have re<lb></lb>moved it from Perpendicularity half an Inch, adding a ſecond, <lb></lb>after that it being returned towards us, is ready to begin the ſe<lb></lb>cond Vibration, we ſhould conferre new Motion on it, and ſo <lb></lb>ſucceſſively with other blaſts, but given in Time, and not when <lb></lb>the <emph type="italics"></emph>Pendulum<emph.end type="italics"></emph.end> is comming towards us (for ſo we ſhould impede; <lb></lb>and not help the Motion) and ſo continuing with many Impul<lb></lb>ſes, we ſhould confer upon it ſuch an <emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> that a greater <lb></lb>force by much than that of a blaſt of our breath, will be required <lb></lb>to ſtay it.</s></p><p type="main"> <s>SAGR. </s> <s>I have, from my childhood, obſerved, that one man a<lb></lb>lone, by means of theſe Impulſes, given in Time, hath been able <lb></lb>to towl a very great Bell, and when it was to ceaſe, I have ſeen <lb></lb>four or ſix men more lay hold on the Bell-rope, and they have all <pb xlink:href="069/01/083.jpg" pagenum="81"></pb>been raiſed from the ground: ſo many together being unable to <lb></lb>arreſt that <emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> which one alone, with regular Pulls, had con<lb></lb>ferred upon the Bell.</s></p><p type="main"> <s>SALV. </s> <s>An example, that declareth my meaning with no leſſe </s></p><p type="main"> <s><arrow.to.target n="marg1077"></arrow.to.target><lb></lb>propriety than this that I have premiſed, doth ſute to render the <lb></lb>reaſon of the admirable Problem of the Chord of the Lute or Viol, <lb></lb>which moveth, and maketh not onely that really to ſound, which <lb></lb>is tuned to the Uniſon, but that alſo which is ſet to an Eighth <lb></lb>and a Fifth. </s> <s>The Chord being toucht, its Vibrations begin, and <lb></lb>continue all the Time that its Sound is heard to endure: theſe <lb></lb>Vibrations make the Air neer adjacent to vibrate and tremble, <lb></lb>whoſe tremblings and quaverings diſtend themſelves a great way, <lb></lb>and ſtrike upon all the Chords of the Inſtrument, and alſo of o<lb></lb><arrow.to.target n="marg1078"></arrow.to.target><lb></lb>thers neer unto it: the Chord that is ſet to an Uniſon, with that <lb></lb>which is toucht, being diſpoſed to make its Vibrations ^{*} in the <lb></lb>ſame Time, beginneth at the firſt impulſe to move a little, and <lb></lb><arrow.to.target n="marg1079"></arrow.to.target><lb></lb>a ſecond, a third, a twentieth, and many more, overtaking it, all <lb></lb>in juſt and Periodick Times, it receiveth at laſt, the ſame Tre<lb></lb>mulation, with that firſt touched, and one may clearly ſee it go, <lb></lb>dilating its Vibrations exactly according to the Pace of its Mo<lb></lb>ver. </s> <s>This Undulation that diſtendeth it ſelf thorow the Air, mo<lb></lb>veth, and makes to vibrate, not onely the Chords, but likewiſe <lb></lb>any other Body diſpoſed to trembling, and to vibrate in the very <lb></lb>Time of the trembling Chord: ſo that if we fix in the Sides of <lb></lb>the Inſtrument ſeveral ſmall pieces of Briſtles, or of other flexible <lb></lb>matters, you ſhall ſee upon the ſounding of the Viol, now one, <lb></lb>now another of thoſe Corpuſcles tremble, according as that <lb></lb>Chord is toucht, whoſe Vibrations return in the ſame Time: the <lb></lb>others will not move at the ſtriking of this Chord, nor will that <lb></lb>Briſtle tremble at the ſtriking of another Chord. </s> <s>If with the Bow <lb></lb>one ſmartly ſtrike the Baſe-Chord of a Viol, and ſet a drinking <lb></lb>Glaſſe, thin and ſmooth, neer unto it, if the Tone of the Chord <lb></lb>be an Uniſon to the Tone of the Glaſſe, the Glaſſe ſhall dance, <lb></lb>and ſenſibly re-ſound. </s> <s>Again, the ample dilating of the Tremor <lb></lb>or Undulation of the <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> about the Body reſounding, is ap<lb></lb>parently ſeen in making the Glaſſe to ſound, by putting a little <lb></lb>Water in it, and then chafing the brim or edge of it with the tip <lb></lb>of the finger: for the included Water is obſerved to undulate in <lb></lb>a moſt regular order: and the ſame effect will be yet more clearly <lb></lb>ſeen, by ſetting the foot of the Glaſſe in the bottom of a reaſo<lb></lb>nable large Veſſel, in which there is Water as high almoſt as to <lb></lb>the brim of the Glaſſe, for making it to ſound, as before, with <lb></lb>the Confrication of the finger, we ſhall ſee the trembling of the <lb></lb>Water to diffuſe it ſelf moſt regularly, and with great Velocity, <lb></lb>to a great diſtance round about the Glaſſe; and it hath many <pb xlink:href="069/01/084.jpg" pagenum="82"></pb>times been my fortune, in making a reaſonable big Glaſſe, almoſt <lb></lb>full of Water, to ſound as aforeſaid, to ſee the Waves in the <lb></lb>Water, at firſt formed with an exact equality; and it hapning <lb></lb>ſometimes, that the Tone of the Glaſſe riſeth an Eighth higher, at <lb></lb>the ſame inſtant, I have ſeen every one of the ſaid Waves to divide <lb></lb>themſelves in two: an accident that very clearly proveth the <lb></lb>forme of the Octave to be the double.</s></p><p type="margin"> <s><margin.target id="marg1077"></margin.target><emph type="italics"></emph>The Chord of a <lb></lb>Muſical Inſtru<lb></lb>ment touched, mo<lb></lb>veth, and maketh <lb></lb>the Chords ſet to an <lb></lb>Uniſon, Fifth and <lb></lb>Eighth, with it to <lb></lb>ſound; and why.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1078"></margin.target><emph type="italics"></emph>Sundry Problems <lb></lb>touching Muſical <lb></lb>Proportions, and <lb></lb>their Solutions.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1079"></margin.target>* Or under.</s></p><p type="main"> <s>SAGR. </s> <s>The ſame hath alſo befaln me more than once, to my <lb></lb>delight, and alſo benefit: for I ſtood a long time perplexed a<lb></lb>bout theſe Forms of Conſonants, not conceiving, that the Rea<lb></lb>ſon, commonly given thereof by the Authours that have hither<lb></lb>to written learnedly of Muſick, were ſufficiently convincing, <lb></lb>they tell us, that the Diapaſon, that is the Eighth, is contained <lb></lb>by the double, the Diapente, which we call the Fifth, by the <lb></lb>Seſquialter: for a Chord being diſtended on the ^{*} Monochord, <lb></lb><arrow.to.target n="marg1080"></arrow.to.target><lb></lb>ſtriking it all; and afterwards ſtriking but the half of it, by pla<lb></lb>cing a Bridge in the middle, one heareth an Eighth; and if the <lb></lb>Bridge be placed at a third of the whole Chord, touching the <lb></lb>whole, and then the two thirds, it ſoundeth a Fifth; whereupon <lb></lb>they infer, that the Eighth is contained between two and one, and <lb></lb>the Fifth between three and two. </s> <s>This Reaſon, I ſay, ſeemed to <lb></lb>me not neceſſarily concluding for the aſſigning juſtly the double <lb></lb>and the Seſquialter, for the natural Forms of the Diapaſon and <lb></lb>the Diapente. </s> <s>And that which moved me ſo to think, was this. <lb></lb></s> <s>There are three ways, by which we may ſharpen the Tone of a <lb></lb>Chord: one is, by making it ſhorter, the other is by diſtending; <lb></lb>or making it more tenſe; and the third is by making it thinner. </s> <s>If, <lb></lb>retaining the ſame Tention and thickneſſe, we would hear an <lb></lb>Eighth, it is neceſſary to ſhorten it to one half, which is done by <lb></lb>ſtriking it all, and then half. </s> <s>But if, retaining the ſame length <lb></lb>and thickneſſe, we would have it riſe to an Eighth, by ſcrewing <lb></lb>it higher, it will not ſuffice to ſtretch it double as much, but we <lb></lb>ſhall need the quadruple, ſo that, if before it was ſtretched by a <lb></lb>Weight of one pound, it will be needful to faſten four pound <lb></lb>to it to ſharpen it to an Eighth. </s> <s>And laſtly, if, keeping the ſame <lb></lb>length and Tention, we would have a Chord, that by being ſmal<lb></lb>ler, rendereth an Eighth, it will be neceſſary, that it retain onely <lb></lb>a fourth part of the thickneſſe of the other more Grave. </s> <s>And this <lb></lb>which I ſpeak of the Eighth, that is, that its form taken from the <lb></lb>Tention, or from the thickneſſe of the Chord, is in duplicate <lb></lb>proportion to that which it receiveth from the length, is to be <lb></lb>underſtoood of all other Muſical Intervals: for that which the <lb></lb>length giveth us in a Seſquialter proportion, <emph type="italics"></emph>i. </s> <s>e.<emph.end type="italics"></emph.end> by ſtriking it all, <lb></lb>and then the two thirds, if you would have it proceed from the <lb></lb>Tention, or from the diſgroſſing, you muſt double the Seſqui<pb xlink:href="069/01/085.jpg" pagenum="83"></pb>alter proportion, taking the double Seſquiquartan: and if the <lb></lb>Grave Chord were ſtretched by four pound weight, faſten to the <lb></lb>Acute not ſix, but nine: and, as to the thickneſſe, make the Grave <lb></lb>Chord thicker than the Acute, according to the proportion of <lb></lb>nine to four, to have the Fifth. </s> <s>Theſe being moſt exact Experi<lb></lb>ments, I thought, that I ſaw no reaſon, why theſe Sage Philoſo<lb></lb>phers ſhould eſtabliſh the form of the Eighth to be rather the dou<lb></lb>ble, than quadruple; and the Form of the Fifth to be rather the <lb></lb>Seſquialter, than the double Seſquiquartan. </s> <s>But becauſe the <lb></lb>numbring of the Vibrations of a Chord, which in giving a ſound, <lb></lb>are extreme frequent, is altogether impoſſible, I ſhould always <lb></lb>have been in doubt, whether or no it were true, that the more <lb></lb>Acute Chord of the Eighth, made in the ſame time, double the <lb></lb>number of the Vibrations of the more Grave, if the Waves, <lb></lb>which may be continued as long as you pleaſe, by making the <lb></lb>Glaſs to ſound and vibrate, had not ſenſibly ſhewn me, that in <lb></lb>the ſelf ſame moment that (ſometimes) the Sound is heard to riſe <lb></lb>to an Eighth, there are ſeen to ariſe other Waves more minute, <lb></lb>which with infinite ſmoothneſs cut in the middle each of thoſe <lb></lb>firſt.</s></p><p type="margin"> <s><margin.target id="marg1080"></margin.target>* An Inſtrument <lb></lb>of but one ſtring; <lb></lb>called by <emph type="italics"></emph>Mar<lb></lb>ſennus la Tromper<lb></lb>te Marine.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>An excellent Obſervation for diſtinguiſhing one by <lb></lb>one the Undulations ariſing from the Tremulation of the re<lb></lb>ſounding Body: which are thoſe that diffuſing themſelves tho<lb></lb>row the Air, make the titillation upon the Drum of our Ear, that <lb></lb>in our Soul becommeth a Sound: But whereas beholding and ob<lb></lb>ſerving them in the Water, endure no longer than the confrica<lb></lb>tion of the finger laſteth, and alſo in that time they are not per<lb></lb>manent, but are continually made and diſſolved, would it not <lb></lb>be an ingenious undertaking, if one could make, with much <lb></lb>exquiſiteneſſe, ſuch, as would continue a long time; I mean <lb></lb>Moneths and Years, ſo as to give a man opportunity meaſure, <lb></lb>and with eaſe to number them?</s></p><p type="main"> <s>SAGR. </s> <s>I aſſure you I ſhould highly value ſuch an Invention.</s></p><p type="main"> <s>SALV. </s> <s>The diſcovery was accidental, and the Obſervation <lb></lb>and applicative improvement of it onely were mine, and I hold <lb></lb>it to be a Circumſtance of noble Contemplation, althongh a buſi<lb></lb>neſſe in its ſelf ſufficiently homely. </s> <s>Scraping a Braſſe Plate with <lb></lb>an Iron Chizzel to fetch out ſome Spots, in moving the Chizzel to <lb></lb>and again upon it pretty quick, I heard it (once or twice amongſt <lb></lb>many gratings) to Sibilate and ſend forth a whiſtling noiſe, very <lb></lb>ſhrill and audible: and looking upon the Plate, I ſaw a long <lb></lb>row of ſmall ſtreaks, parallel to one another, and diſtant from <lb></lb>one another by moſt equal Intervals: returning to my ſcraping <lb></lb>again, I perceived by ſeveral trials, that in thoſe ſcrapings, and <lb></lb>thoſe onely that whiſtled, the Chizzel left the ſtreaks upon the <pb xlink:href="069/01/086.jpg" pagenum="84"></pb>Plate: but when the Scraping paſſed without any Sibilation, <lb></lb>there was not ſo much as the leaſt ſign of any ſuch ſtreaks. </s> <s>Re<lb></lb>peating the Experiment ſeveral times afterwards, ſcraping now <lb></lb>with greater, now with leſſe velocity, the Sibilation hapned to <lb></lb>be of a Tone ſometimes acuter, ſometimes graver; and I obſerved <lb></lb>the marks made in the more acute ſounds to be cloſer together, <lb></lb>and thoſe of the more grave farther aſunder: and ſometimes alſo, <lb></lb>according as the ſelf ſame ſcrape was made towards the end, with <lb></lb>greater velocity than at the beginning, the ſound was heard to <lb></lb>grow ſharper, and the ſtreaks were obſerved to ſtand thicker, <lb></lb>but ever with extream neatneſſe, and marked with exact equidi<lb></lb>ſtance: and farther-more, in the Sibilating ſcrapes; I felt the <lb></lb>Chizzel to ſhake or tremulate in my hand, and a certain chilneſſe <lb></lb>to run along my arm; and in ſhort, I ſaw the ſame effected upon <lb></lb>the Toole, which we uſe to obſerve in whiſpering, and after<lb></lb>wards ſpeaking aloud, for ſending forth the breath without <lb></lb>forming a ſound, we do not perceive any moving in the throat <lb></lb>and mouth, in compariſon of that which we diſcern to be in the <lb></lb>Wind-pipe and Throat of every one, in ſending forth the voice; <lb></lb>and eſpecially in grave and loud Tones. </s> <s>I have likewiſe ſome<lb></lb>times amongſt the Chords of the Viols, obſerved two that were <lb></lb>Uniſons to the Sibilations made by ſcraping after the manner I <lb></lb>told you, and that were moſt different in Tone, from which two <lb></lb>they preciſely were diſtant a perfect Fifth, and then meaſuring <lb></lb>the intervals of the ſtreaks of both the Scrapes, I ſaw the di<lb></lb>ſtance that conteined forty five ſpaces of the one, conteined <lb></lb>thirty of the other: which, indeed, is the Form attributed to the <lb></lb>Diapente. </s> <s>But here, before I proceed any farther, I will tell you, <lb></lb>that of the three manners of rendring a Sound Acute, that which <lb></lb>you refer to the ſlenderneſſe or fineneſſe of the Chord, may <lb></lb>with more truth be aſcribed to the Weight. </s> <s>For the alteration ta<lb></lb>ken from the thickneſſe, anſwereth, when the Chords are of the <lb></lb>ſame matter; and ſo a Gut-ſtring to make an Eighth, ought to be <lb></lb>four times thicker than the other Gut-ſtring; and one of Wier four <lb></lb>times thicker than another of Wier. </s> <s>But if I would make an Eighth <lb></lb>with one of Wier to one of Gut-ſtring, I am not to make it four <lb></lb>times thicker, but four times graver, ſo that, as to thickneſſe, <lb></lb>this of Wier ſhall not be four times thicker, but quadruple in <lb></lb>Gravity, for ſome times it ſhall be more ſmall than its reſpon<lb></lb>dent to the Acuter Eighth, that is of Gut-ſtring. </s> <s>Hence it com<lb></lb>meth to paſſe that, ſtringing an Inſtrument with Chords of Gold, <lb></lb>and another with Chords of Braſſe, if they ſhall be of the ſame <lb></lb>length, thickneſſe, and Tention, Gold being almoſt twice as <lb></lb>heavy, the Strings ſhall prove about a Fifth more Grave. </s> <s>And <lb></lb>here it is to be noted, that the Gravity of the Moveable more re<pb xlink:href="069/01/087.jpg" pagenum="85"></pb>ſiſteth the Velocity, than the thickneſſe doth; contrary to what <lb></lb>others at the firſt would think: for indeed, in appearance, its more <lb></lb>reaſonable, that the Velocity ſhould be retarded by the Reſiſtance <lb></lb>of the <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> againſt Opening in a Moveable thick and light, <lb></lb>than in one grave and ſlender: and yet in this caſe it happeneth <lb></lb>quite contrary. </s> <s>But purſuing our firſt Intent, I ſay, That the <lb></lb>ncereſt and immediate reaſons of the Forms of Muſical Intervals, <lb></lb>is neither the length of the Chord, nor the Tention, nor the <lb></lb>thickneſſe, but the proportion of the numbers of the Vibrations, <lb></lb>and Percuſſions of the Undulations of the Air that beat upon the <lb></lb>Drum of our Ear, which it ſelf alſo doth tremulate under the <lb></lb>ſame meaſures of Time. </s> <s>Having eſtabliſhed this Point, we may, <lb></lb>perhaps, aſſign a very apt reaſon, whence it commeth, that of <lb></lb>thoſe Sounds that are different in Tone, ſome Couples are re<lb></lb>ceived with great delight by our Sence, others with leſs, and <lb></lb>others occaſion in us a very great diſturbance; which is to ſeek a <lb></lb>reaſon of the Conſonances more or leſſe perfect, and of Diſlo<lb></lb>nances. </s> <s>The moleſtation and harſhneſſe of theſe proceeds, as I <lb></lb>believe, from the diſcordant Pulſations of two different Tones, <lb></lb>which diſproportionally ſtrike the Drum of our Ear: and the <lb></lb>Diſſonances ſhall be extreme harſh, in caſe the Times of the Vi<lb></lb>brations were incommenſurable. </s> <s>For one of which take that, <lb></lb>when of two Chords ſet to an Uniſon, one is ſounded, and ſuch <lb></lb>a part of another, as is the Side of the Square of its Diameter; <lb></lb>a Diſſonance like to the ^{*} Tritone, or Semi-diapente. </s> <s>Conſonan<lb></lb><arrow.to.target n="marg1081"></arrow.to.target><lb></lb>ces, and with pleaſure received, ſhall thoſe Couples of Sounds <lb></lb>be, that ſhall ſtrike in ſome order upon the Drum; which order <lb></lb>requireth, firſt, that the Pulſations made in the ſame Time be <lb></lb>commenſurable in number, to the end, the Cartillage of the Drum, <lb></lb>may not ſtand in the perpetual Torment of a double inflection of <lb></lb>allowing and obeying the ever diſagreeing Percuſſions. </s> <s>Therefore <lb></lb>the firſt and moſt grateful Conſonance ſhall be the Eighth, being, <lb></lb>that for every ſtroke, that the Grave-ſtring or Chord giveth upon <lb></lb>the Drum, the Acute giveth, two; ſo that both beat together <lb></lb>in every ſecond Vibration of the Acute Chord; and ſo of the <lb></lb>whole number of ſtrokes, the one half accord to ſtrike together, <lb></lb>but the ſtrokes of the Chords that are Uniſons, alwayes joyn <lb></lb>both together, and therefore they are, as if they were of the <lb></lb>ſame Chord, nor make they a Conſonance. </s> <s>The Fifth delighteth <lb></lb>likewiſe, in regard, that for every two ſtroaks of the Grave <lb></lb>Chord, the Acute giveth three: from whence it followeth, that <lb></lb>numbering the Vibrations of the Acute Chord, the third part of <lb></lb>that number will agree to beat together; that is, two Solitary ones <lb></lb>interpoſe between every couple of Conſonances; and in the Di<lb></lb>ateſſeron there interpoſe three. </s> <s>In the ſecond, that is in the <emph type="italics"></emph>Seſ-<emph.end type="italics"></emph.end><pb xlink:href="069/01/088.jpg" pagenum="86"></pb><emph type="italics"></emph>quioctave<emph.end type="italics"></emph.end> Tone for every nine Pulſations, one onely ſtrikes in Con<lb></lb>ſort with the other of the Graver Chord; all the reſt are Diſcords, <lb></lb>and received upon the Drum with regret, and are judged Diſſo<lb></lb>nances by the Ear.</s></p><p type="margin"> <s><margin.target id="marg1081"></margin.target>* Or a falſe Fifth.</s></p><p type="main"> <s>SIMP. </s> <s>I could wiſh this Diſcourſe were a little explained.</s></p><p type="main"> <s>SALV. </s> <s>Suppoſe this line A B the Space, and dilating of a Vi<lb></lb>bration of the Grave Chord; and the line C D that of the Acute <lb></lb>Chord, which with the other giveth the Eighth: and let A B be <lb></lb>divided in the midſt in E. </s> <s>It is manifeſt, that the Chords begin<lb></lb>ing to move at the terms A and C, by that time the Acute Vibra<lb></lb>tion ſhall be come to the term D, the other <lb></lb><figure id="id.069.01.088.1.jpg" xlink:href="069/01/088/1.jpg"></figure><lb></lb>ſhall be diſtended onely to the half E, which <lb></lb>not being the bound or term of the Motion, <lb></lb>it ſtrikes not: but yet a ſtroak is made in D. <lb></lb></s> <s>The Vibrations afterwards returning from D <lb></lb>to C, the other paſſeth from E to B, where<lb></lb>upon the two Percuſſions of B and C ſtrike <lb></lb>both together upon the Drum: and ſo con<lb></lb>tinuing to reiterate the like ſubſequent Vi<lb></lb>brations; one ſhall ſee, that the union of the <lb></lb>Percuſſions of the Vibrations C D with thoſe of A B, happen al<lb></lb>ternately every other time: but the Pullations of the terms A B <lb></lb>are alwayes accompanied with one of C D, and that alwayes the <lb></lb>ſame: which is manifeſt, for ſuppoſing that A and C ſtrike to<lb></lb>gether; in the time that A is paſſing to B, C goeth to D, and <lb></lb>returneth back to C: ſo that the ſtroaks at B and C are alſo <lb></lb>together. </s> <s>But now let the two Vibrations A B and C D be thoſe <lb></lb>that produce the Diapente, the times of which are in proportion <lb></lb>Seſquialter, and divide A B of the Grave Chord, in three equal <lb></lb>parts in E and O; And ſuppoſe the Vibrations to begin at the <lb></lb>ſame moment from the terms A and C: It is manifeſt, that at the <lb></lb>ſtroke that ſhall be made in D, the Vibration of A B ſhall have <lb></lb>got no farther than O, the Drum therefore receiveth the Pulſa<lb></lb>tion D onely: again in the return from D to C, the other Vibra<lb></lb>tion paſſeth from O to B, and returneth to O, making the Pul<lb></lb>ſation in B, which likewiſe is ſolitary, and in Counter-time, (an <lb></lb>accident to be conſidered:) for we having ſuppoſed the firſt <lb></lb>Pulſations to be made at the ſame moment in the terms A and C, <lb></lb>the ſecond, which was onely by the term D, was made as long after <lb></lb>as the time of the tranſition C D, that is A O, imports; but <lb></lb>that which followeth, made in B, is diſtant from the other one<lb></lb>ly ſo much as is the time O B, which is the half: afterwards con<lb></lb>tinuing the Recurſion from O to A, whilſt the other goeth from <lb></lb>C to D, the two Pulſations come to be made both at once in A <lb></lb>and D. </s> <s>There afterwards follow other Periods like to theſe, that <pb xlink:href="069/01/089.jpg" pagenum="87"></pb>is, with the interpoſition of two ſingle and ſolitary Pulſations of <lb></lb>the Acute Chord, and one of the Grave Chord, likewiſe ſolita<lb></lb>ry, is interpoſed between the two ſolitary ſtrokes of the Acute. </s> <s>So <lb></lb>that if we did but ſuppoſe the Time divided into Moments, that is, <lb></lb>into ſmall equal Particles: ſuppoſing that in the two firſt moments, <lb></lb>I paſſed from the Concordant Pulſations made in A and C to O <lb></lb>and D, and that in D, I make a Percuſſion: and that in the third <lb></lb>and fourth moment I return from D to C, ſtriking in C, and <lb></lb>that from O, I paſt to B, and returned to O, ſtriking in B; and <lb></lb>that laſtly in the fifth and ſixth moment from O and C, I paſt to <lb></lb>A and D ſtriking in both: we ſhall have the Pulſations diſtributed <lb></lb>with ſuch order upon the Drum, that ſuppoſing the Pulſations of <lb></lb>the two Chords in the ſame inſtant, it ſhall two moments after <lb></lb>receive a ſolitary Percuſſion, in the third moment anothor, ſoli<lb></lb>tary likewiſe, in the fourth another ſingle one, and two moments <lb></lb>after, that is, in the ſixth, two together; and here ends the <lb></lb>Period, and, if I may ſo ſay, Anomaly; which Period is oft-times <lb></lb>afterwards replicated.</s></p><p type="main"> <s>SAGR. </s> <s>I can hold no longer, but muſt needs expreſſe the con<lb></lb>tent I take in hearing reaſons ſo appoſitely aſſigned of effects that <lb></lb>have ſo long time held me in darkneſſe and blindneſſe. </s> <s>Now I <lb></lb>know why the Uniſon differeth not at all from a ſingle Tone: I <lb></lb>ſee why the Eighth is the principal Conſonance, but withal ſo <lb></lb>like to an Uniſon, that, as an Uniſon, it is taken and cojoyned <lb></lb>with others: it reſembleth an Uniſon, for that whereas the Pul<lb></lb>ſations of Chords ſet to an Uniſon, keep time in ſtriking, theſe <lb></lb>of the Grave Chord in an Eighth alwayes keep time with thoſe <lb></lb>of the Acute, and of theſe one interpoſeth alone, and in equal <lb></lb>diſtances, and as, one may ſay, without any variety, whereupon <lb></lb>that Conſonance is over ſweet. </s> <s>But the Fifth, with thoſe its <lb></lb>Counter-times, and with the interpoſures of two ſolitary Pulſa<lb></lb>tions of the Acute Chord, and one of the Grave Chord, <lb></lb>between the Couples of Diſcordant Pulſations, and thoſe <lb></lb>three ſolitary ones, with an interval of time, as great as the half of <lb></lb>that which interpoſeth between each Couple, and the ſolitary <lb></lb>Percuſſions of the Acute Chord, maketh ſuch a Titillation and <lb></lb>Tickling upon the Cartillage of the Drum of the Ear, that al<lb></lb>laying the Dulcity with a mixture of Acrimony, it ſeemeth at <lb></lb>one and the ſame time to kiſſe and bite.</s></p><p type="main"> <s>SALV. </s> <s>It is convenient, in regard I ſee, that you take ſuch de<lb></lb>light in theſe Novelties, that I ſhew you the way whereby the Eye <lb></lb>alſo, and not the Ear alone, may recreate it ſelf in beholding <lb></lb>the ſame ſports that the Ear feeleth. </s> <s>Suſpend Balls of Lead or o<lb></lb>ther heavy matter on three ſtrings of different lengths, but in <lb></lb>ſuch proportion, that while the longer maketh two Vibrations, <pb xlink:href="069/01/090.jpg" pagenum="88"></pb>the ſhorter may make four, and the middle one three; which <lb></lb>will happen, when the longeſt containeth ſixteen feet, or other <lb></lb>meaſures, of which the middle one containeth nine, and the <lb></lb>ſhorteſt four: and removing them all together from Perpendi<lb></lb>cularity, and then letting them go, you ſhall ſee a pleaſing In<lb></lb>termixtion of the ſaid <emph type="italics"></emph>Pendulums<emph.end type="italics"></emph.end> with various encounters, but <lb></lb>ſuch, that, at every fourth Vibration of the longeſt, all the three <lb></lb>will concurre in one and the ſame term together, and then again <lb></lb>will depart from it, reiterating anew the ſame Period: the which <lb></lb>commixture of Vibrations, is the ſame, that being made by the <lb></lb>Chords, preſents to the Ear an Eighth, with a Fifth in the midſt. <lb></lb></s> <s>And if you qualifie the length of other ſtrings in the like diſpo<lb></lb>ſure, ſo that their Vibrations anſwer to thoſe of other Muſical, <lb></lb>but Conſonant Intervals, you ſhall ſee other and other Inter<lb></lb>weavings, and alwaies ſuch, that in determinate times, and after <lb></lb>determinate numbers of Vibrations, all the ſtrings (be they three, <lb></lb>or be they four) will agree to joyn in the ſame moment, in the <lb></lb>term of their Recurſions, and from thence to begin ſuch another <lb></lb>Period: but if the Vibrations of two or more ſtrings are either <lb></lb>Incommenſurable, ſo, that they never return harmoniouſly to ter<lb></lb>minate determinate numbers of Vibrations, or though they be <lb></lb>not Incommenſurable, yet if they return not till after a long time, <lb></lb>and after a great number of Vibrations, then the ſight is con<lb></lb>founded in the diſorderly order of irregular Intermixtures, and <lb></lb>the Ear with wearineſſe and regret receiveth the intemperate Im<lb></lb>pulſes of the Airs Tremulations, that without Order or Rule, <lb></lb>ſucceſſively beat upon its Drum.</s></p><p type="main"> <s>But whither, my Maſters, have we been tranſported for ſo <lb></lb>many hours by various Problems, and unlook't for Diſcourſes? <lb></lb></s> <s>We have made it Night, and yet we have handled few or none of <lb></lb>the points propounded; nay we have ſo loſt our way, that I <lb></lb>ſcarſe remember our firſt entrance, and that ſmall Introduction, <lb></lb>which we laid down, as the Hypotheſis and beginning of the fu<lb></lb>ture Demonſtrations.</s></p><p type="main"> <s>SAGR It will be convenient, therefore, that we break up our <lb></lb>Conference for this time, giving our Minds leave to compoſe <lb></lb>themſelves in the Nights Repoſe, that we may to Morrow (if <lb></lb>you pleaſe ſo far to favour us) reaſſume the Diſcourſes deſired, <lb></lb>and chiefly intended.</s></p><p type="main"> <s>SALV. </s> <s>I ſhall not fail to be here to Morrow at the uſual <lb></lb>hour, to ſerve and enjoy you.</s></p><p type="head"> <s><emph type="italics"></emph>The End of the Firſt Dialogue.<emph.end type="italics"></emph.end></s></p></chap><chap><pb xlink:href="069/01/091.jpg" pagenum="89"></pb><p type="head"> <s>GALILEUS, <lb></lb>HIS <lb></lb>DIALOGUES <lb></lb>OF <lb></lb>MOTION.</s></p><p type="head"> <s>The Second Dialogue.</s></p><p type="head"> <s><emph type="italics"></emph>INTERLOCUTORS,<emph.end type="italics"></emph.end></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"></emph>Simplicius,<emph.end type="italics"></emph.end> and I, ſtaid expecting your com<lb></lb>ing, and we have been trying to recall to <lb></lb>memory our laſt Conſideration, which, as <lb></lb>the Principle and Suppoſition, on which <lb></lb>you ground the Concluſions that you in<lb></lb>tended to Demonſtrate to us, was that <lb></lb>Reſiſtance, that all Bodies have to <emph type="italics"></emph>Fracti<lb></lb>on,<emph.end type="italics"></emph.end> depending on that Cement, that con<lb></lb>nects and glutinates the parts, ſo, as that <lb></lb>they do not ſeparate and divide without a powerful attraction: <lb></lb>and our enquiry hath been, what might be the Cauſe of that <lb></lb>Coherence, which in ſome Solids is very vigorous; propounding <lb></lb>that of <emph type="italics"></emph>Vacuum<emph.end type="italics"></emph.end> for the principal, which afterwards occaſioned ſo <lb></lb>many Digreſſions as held us the whole day, and far from the <pb xlink:href="069/01/092.jpg" pagenum="90"></pb>matter at firſt propoſed, which was the Contemplation of the Re<lb></lb>ſiſtances of Solids to Fraction.</s></p><p type="main"> <s>SALV. </s> <s>I remember all that hath been ſaid, and returning to <lb></lb>our begun diſcourſe; What ever this Reſiſtance of Solids to brea<lb></lb>king by a violent attraction, is ſuppoſed to be, it is ſufficient, that it <lb></lb>is to be found in them: which, though it be very great againſt the <lb></lb>ſtrength of one that draweth them ſtreight out, it is obſerved to be <lb></lb>leſſe in forcing them tranſverſely, or ſidewaies: and thus we ſee, <lb></lb>for example, a rod of Steel, or Glaſſe to ſuſtain the length-waies a <lb></lb>weight of a thouſand pounds, which, faſtned at Right-Angles in<lb></lb>to a Wall, will break if you hang upon it but only fifty. </s> <s>And of <lb></lb>this ſecond Reſiſtance we are to ſpeak, enquiring, according to <lb></lb>what proportions it is found in Priſmes, and Cylinders of like and <lb></lb>unlike figure, length, and thickneſs, and, withal, of the ſame mat<lb></lb>ter. </s> <s>In which Speculation, I take for a known Principle, that which <lb></lb>in the Mechanicks is demonſtrated amongſt the Paſſions of the <lb></lb>Vectis, which we call the Leaver: namely, That in that uſe of the <lb></lb>Leaver, the Force is to the Reſiſtance in Reciprocal proportion, <lb></lb>as the Diſtances from the Fulciment to the ſaid Force and the Re<lb></lb>ſiſtance.</s></p><p type="main"> <s>SIMP. </s> <s>This <emph type="italics"></emph>Ariſtotle,<emph.end type="italics"></emph.end> in his Mechanicks, demonſtrated before <lb></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></lb>for the firmneſſe oſ Demonſtration, I think, that <emph type="italics"></emph>Archimedes<emph.end type="italics"></emph.end><lb></lb>ought to be preferred far before him, on one ſole Propoſition of <lb></lb>whom, by him demonſtrated in his Book, <emph type="italics"></emph>De Equiponderantium,<emph.end type="italics"></emph.end><lb></lb>depend the Reaſons, not only of the Leaver, but of the greater <lb></lb>part of the other Mechanick Inſtruments.</s></p><p type="main"> <s>SAGR. </s> <s>But ſince that this Principle is the foundation of all <lb></lb>that which you intend to demonſtrate to us, it would be very re<lb></lb>quiſite, that you produce us the proof of this ſame Suppoſition, <lb></lb>if it be not too long a work, giving us a full and perfect informati<lb></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></lb>you into the field of all our future Speculations, by an enterance <lb></lb>ſomewhat different from that of <emph type="italics"></emph>Archimedes<emph.end type="italics"></emph.end>; and that, ſuppo<lb></lb>ſing no more, but only that equal Weights, put into a Ballance of <lb></lb>equal Arms, make an <emph type="italics"></emph>Equilibrium,<emph.end type="italics"></emph.end> (a Principle likewiſe ſuppoſed <lb></lb>by <emph type="italics"></emph>Archimedes<emph.end type="italics"></emph.end> himſelf.) I come, in the next place, to demon<lb></lb>ſtrate to you, that not only it is as true as the other, That unequal <lb></lb>Weights make an <emph type="italics"></emph>Equilibrium<emph.end type="italics"></emph.end> in a Stiliard of Armes unequal, ac<lb></lb>cording to the proportion of thoſe Weights Reciprocally ſuſpen<lb></lb>ded, but that it is one and the ſame thing to place equal Weights <lb></lb>at equal diſtances, as to place unequal Weights at diſtances that <lb></lb>are in Reciprocal Proportion to the Weights. </s> <s>Now for a plain <pb xlink:href="069/01/093.jpg" pagenum="91"></pb>Demonſtration of what I ſay, deſcribe a Solid Priſm or Cylinder <lb></lb>A B, [<emph type="italics"></emph>as in<emph.end type="italics"></emph.end> Figure 1. <emph type="italics"></emph>at the end of this Dialogue,<emph.end type="italics"></emph.end>] ſuſpended by <lb></lb>its ends at the Line H I, and ſuſtained by two Cords, H A, and I B. <lb></lb></s> <s>It is manifeſt, that if I ſuſpend the whole by the Cord C, placed <lb></lb>in the middle of the Beam or Ballance H I, the Priſm A B will be <lb></lb>equilibrated, one half of its weight, being on one ſide, and the other <lb></lb>half on the other ſide of the Point of Suſpenſion C by the Princi<lb></lb>ple that we preſuppoſed. </s> <s>Now let the Priſm be divided into un<lb></lb>equal parts by the Line D, and let the part D A be grea<lb></lb>ter, and D B leſſer; and to the end, that ſuch diviſion being made, <lb></lb>the Parts of the Priſm may reſt in the ſame ſcituation and conſti<lb></lb>tution, in reſpect of the Line H I, let us help it with a Cord E D, <lb></lb>which, being faſtened in the Point E, ſuſtaineth the parts A D, and <lb></lb>D B: It is not to be doubted, but that there being no local muta<lb></lb>tion in the Priſm, in reſpect of the Ballance H I, it ſhall remain in <lb></lb>the ſame ſtate of Equilibration. </s> <s>But it will reſt in the ſame Con<lb></lb>ſtitution likewiſe, if the Part of the Priſm, that is now ſuſpended at <lb></lb>the two extreams, or ends with Cords A H and D E, be hanged at <lb></lb>one ſole Cord G L, placed in the midſt: and likewiſe the other <lb></lb>part D B, will not change ſtate, if ſuſpended by the middle, and <lb></lb>ſuſtained by the Cord F M. </s> <s>So that the Cords H A, E D, and I B <lb></lb>being untied, and only the two Cords G L, and F M being left, the <lb></lb><emph type="italics"></emph>Equilibrium<emph.end type="italics"></emph.end> will ſtill remain, the Suſpenſion being ſtill made at <lb></lb>the Point C. Now, here let us confider, that we have two Grave <lb></lb>Bodies A D, and D B, hanging at the terms G and F of a Beam <lb></lb>G F, in which the <emph type="italics"></emph>Equilibrium<emph.end type="italics"></emph.end> is made at the Point C: in ſuch <lb></lb>manner, that the diſtance of the ſuſpenſion of the Weight A D <lb></lb>from the Point C, is the Line C G, and the other part C F, is the <lb></lb>diſtance at which the other Weight D B hangeth. </s> <s>It remaineth, <lb></lb>therefore, only to be demonſtrated, that thoſe Diſtances have the <lb></lb>ſame proportion to one another, as the Weights themſelves have, <lb></lb>but reciprocally taken: that is, that the diſtance G C is to the di<lb></lb>ſtance C F, as the Priſm D B to the Priſm D A, which we prove <lb></lb>thus. </s> <s>The Line G E being the half of E H, and E F the half of <lb></lb>E I, all G F ſhall be equall to all H I, and therefore equal to C I: <lb></lb>and taking away the common part C F, the remainder G C ſhall <lb></lb>be equal to the remainder F I, that is, to F E: and C E taken in <lb></lb>common, the two Lines G E and C F ſhall be equal: and, there<lb></lb>fore, as G E, is to E F, ſo is F C, to C G: but as G C is to E F, ſo is <lb></lb>the double to the double; that is H E to E I; that is, the Priſm <lb></lb>A D to the Priſm D B. </s> <s>Therefore by Equality of proportion, <lb></lb>and by Converſion, as the diſtance G C is to the diſtance C F, ſo <lb></lb>is the Weight B D to the Weight D A: which is that that I was to <lb></lb>demonſtrate. </s> <s>If you underſtand this, I believe that you will not <lb></lb>ſcruple to admit, that the two Priſmes A D, and D B make an <pb xlink:href="069/01/094.jpg" pagenum="92"></pb><emph type="italics"></emph>Equilibrium<emph.end type="italics"></emph.end> in th Point C, for the half of the whole Solid A B is <lb></lb>on the right hand of the Suſpenſion C, and the other half on the <lb></lb>left; and that in this manner there are repreſented two equal <lb></lb>Weights, diſpoſed and diſtended at two equal diſtances. </s> <s>Again, <lb></lb>that the two Priſmes A D, and D B, being reduced into two Dice, <lb></lb>or two Balls, or into any two other Figures, (provided that they <lb></lb>keep the ſame Suſpenſions G and F) do continue to make their <lb></lb><emph type="italics"></emph>Equilibrium<emph.end type="italics"></emph.end> in the Point C, I believe none can deny, for that it is <lb></lb>moſt manifeſt, that Figures change not weight, where the ſame <lb></lb>quantity of matter is retained. </s> <s>From which we may gather the <lb></lb>general Concluſion, That two Weights, whatever they be, make <lb></lb>an <emph type="italics"></emph>Equilibrium<emph.end type="italics"></emph.end> at Diſtances reciprocally anſwering to their Gra<lb></lb>vities. </s> <s>This Principle, therefore, being eſtabliſhed, before we paſs <lb></lb>any farther, I am to propoſe to Conſideration, how theſe Forces, <lb></lb>Reſiſtances, Moments, Figures, may be conſidered in Abſtract, <lb></lb>and ſeparate from Matter, as alſo in Concrete and conjoyned <lb></lb>with Matter; and in this manner thoſe Accidents that agree with <lb></lb>Figures, conſidered as Immaterial, ſhall receive certain Modifica<lb></lb>tions, when we ſhall come to add Matter to them, and conſequent<lb></lb>ly Gravity. </s> <s>As for example, if we take a Leaver, as for inſtance <lb></lb>B A [<emph type="italics"></emph>as in<emph.end type="italics"></emph.end> Fig. </s> <s>2.] which, reſting upon the Fulciment E, we ap<lb></lb>ply to raiſe the heavy Stone D: It is manifeſt by the Principle de<lb></lb>monſtrated, that the Force placed at the end B, ſhall ſuffice to <lb></lb>equal the Reſiſtance of the Weight D, if ſo be, that its Moment <lb></lb>have the ſame proportion to the Moment of the ſaid D, that the <lb></lb>Diſtance A C hath to the Diſtance C B: and this is true, if we <lb></lb>confider no other Moments than thoſe of the ſimple Force in B, <lb></lb>and of the Reſiſtance in D, as if the ſaid Leaver were immaterial, <lb></lb>and void of Gravity. </s> <s>But if we bring to account the Gravity alſo <lb></lb>of the Inſtrument or Leaver it ſelf, which hapneth ſometimes to be <lb></lb>of Wood, and ſometimes of Iron; it is manifeſt, that the weight <lb></lb>of the Leaver, being added to the Force in B, it will alter the pro<lb></lb>portion, which it will be requiſite to deliver in other terms. </s> <s>And <lb></lb>therefore before we paſſe any farther, it is neceſſary, that we di<lb></lb>ſtinguiſh between theſe two waies of Conſideration, calling that a <lb></lb>taking it abſolutely, when we ſuppoſe the Inſtrument to be taken <lb></lb>in Abſtract, that is, disjunct from the Gravity of its own Matter; <lb></lb>but conjoyning the Matter, as alſo the Gravity, with ſimple and <lb></lb>abſolute Figures, we will phraſe the Figures conjoyn'd with the <lb></lb>Matter, Moment, or Force compounded.</s></p><p type="main"> <s>SAGR I muſt of neceſſity break the Reſolution I had taken, <lb></lb>not to give occaſion of digreſſing, for I ſhould not be able to ſet <lb></lb>my ſelf to hear what remaines with attention, if a certain ſcruple <lb></lb>were not removed that cometh into my head; and it is this, That <lb></lb>I gueſſe you make compariſon between the Force placed in B, and <pb xlink:href="069/01/095.jpg" pagenum="93"></pb>the total Gravity of the Stone D, of which Gravity me thinks, that <lb></lb>one, and that, very probably, the greater part, reſteth upon the <lb></lb>Plane of the Horizon: ſo that----</s></p><p type="main"> <s>SALV. </s> <s>I have rightly apprehended you, ſo that you need ſay <lb></lb>no more, but only take notice, that I named not the total Gravity <lb></lb>of the Stone, but ſpake of the Moment that it hath, and exerciſeth <lb></lb>at the Point A, the extream term of the Leaver B A, which is ever <lb></lb>leſs than the entire weight of the Stone; and is variable according <lb></lb>to the Figure of the Stone, and according as it hapneth to be more <lb></lb>or leſſe elevated.</s></p><p type="main"> <s>SAGR. </s> <s>I am ſatisfied in that particular, but I have one thing <lb></lb>more to deſire, namely, that for my perfect information, you would <lb></lb>demonſtrate to me the way, if there be one, how I may find what <lb></lb>part of the total weight that is, which cometh to be born by the <lb></lb>ſubjacent Plane, and what that which gravitates upon the Leaver <lb></lb>at the extream A.</s></p><p type="main"> <s><emph type="italics"></emph>S<emph.end type="italics"></emph.end>ALV. </s> <s>Becauſe I can give you ſatisfaction in few words, I will <lb></lb>not fail to ſerve you: therefore, deſcribing a ſlight Figure thereof, <lb></lb>be pleaſed to ſuppoſe, that the Weight, whoſe Center of Gravity is <lb></lb>A, [<emph type="italics"></emph>as in<emph.end type="italics"></emph.end> Fig. </s> <s>3.] reſteth upon the Horizon with the term B, and <lb></lb>at the other end is born up by the Leaver C G, on the Fulciment <lb></lb>N, by a Power placed in G: and that from the Center A, and term <lb></lb>C, Perpendiculars be let fall to the Horizon, A O, and C F. </s> <s>I ſay, <lb></lb>That the Moment of the whole Weight ſhall have to the Moment <lb></lb>of the whole Power in G, a proportion compounded of the Di<lb></lb>ſtance G N to the Diſtance N C, and of F B to B O. Now, as the <lb></lb>Line F B is to B O, ſo let N C be to X. </s> <s>And the whole Weight A <lb></lb>being born by the two Powers placeed in B and C, the Power B is <lb></lb>to C, as the diſtance F O to O B: and by Compoſition, the <lb></lb>two Powers B and C together, that is, the total Moment of <lb></lb>the whole Weight A, is to the Power in C, as the Line F B is <lb></lb>to the Line B O; that is, as N C to X: But the Moment of <lb></lb>the Power in C is to the Moment of the Power in G, as the Di<lb></lb>ſtance G N is to N C: Therefore, by Perturbation of proportion, <lb></lb>the whole Weight A is to the Moment of the Power in G, as G N <lb></lb>to X: But the proportion of G N to X is compounded of the pro<lb></lb>portion G N to N C, and of that of N C to X; that is, of F B to <lb></lb>B O: Therefore the Weight A is to the Power that bears it up in <lb></lb>G, in a proportion compounded of G N to N C, and of that of <lb></lb>F B to B O: which is that that was to be demonſtrated. </s> <s>Now re<lb></lb>turning to our firſt intended Argument, all things hitherto decla<lb></lb>red being underſtood, it will not be hard to know the reaſon, <lb></lb>whence it cometh to paſſe that</s></p><pb xlink:href="069/01/096.jpg" pagenum="94"></pb><p type="head"> <s>PROPOSITION I.</s></p><p type="main"> <s><emph type="italics"></emph>A Solid Priſm or Cylinder of Glaſſe, Steel, Wood, or <lb></lb>other Frangible Matter, that being ſuſpended length<lb></lb>waies, will ſuſtain a very great Weight hanged <lb></lb>Thereat, will, Sidewaies, (as we ſaid even now) be <lb></lb>broken in pieces by a far leſſer Weight, according as <lb></lb>its length ſhall exceed its thickneſs.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Wherefore let us deſcribe the Solid Priſm A B C D, <lb></lb>fixed into a Wall by the Part A B, and in the <lb></lb>other extream ſuppoſe the Force of the Weight E; <lb></lb>(alwaies underſtanding the Wall to be erect to the Horizon, <lb></lb>and the Priſm or Cylinder faſtened in the Wall at Right-An<lb></lb>gles) it is manifeſt, that being to break, it will be broken in the place <lb></lb>B, where the Mortace in the Wall ſerveth for Fulciment, and B C <lb></lb>for the part of the Leaver in which lieth the force, and the thick<lb></lb>neſſe of the Solid B A is the other part of the Leaver, in which <lb></lb>lieth the Reſiſtance, which conſiſteth in the unfaſtening, or divi<lb></lb>ding, that is to be made of the part of the Solid B D, that is with<lb></lb>out the Wall from that which is within: and by what hath been <lb></lb>declared, the Moment <lb></lb><figure id="id.069.01.096.1.jpg" xlink:href="069/01/096/1.jpg"></figure><lb></lb>of the Force placed in <lb></lb>C, is to the Moment of <lb></lb>the Reſiſtance that lieth <lb></lb>in the thickneſſe of the <lb></lb>Priſm, that is, in the <lb></lb>Connection of the Baſe <lb></lb>B A, with the parts con<lb></lb>tiguous to it, as the <lb></lb>length C B is to the half <lb></lb>of B A: And therefore <lb></lb>the abſolute Reſiſtance <lb></lb>againſt Fraction that is <lb></lb>in the Priſm B D, <lb></lb>(which abſolute Reſi<lb></lb>ſtance is that which is <lb></lb>made by drawing it <lb></lb>downwards, for at that <lb></lb>time the motion of the Mover is the ſame with that of the Body <lb></lb>Moved) againſt the fracture to be made by help of the Leaver <pb xlink:href="069/01/097.jpg" pagenum="95"></pb>B C, is as the Length B C to the half of A B in the Priſm, which <lb></lb>in the Cylinder is the Semidiameter of its Baſe. </s> <s>And this is our firſt <lb></lb>Propoſition. </s> <s>And obſerve, that what I have ſaid ought to be un<lb></lb>derſtood, when the Confideration of the proper Weight of the So<lb></lb>lid B D is removed: which Solid I have taken as weighing nothing. <lb></lb></s> <s>But in caſe we would bring its Gravity to account, conjoyning it <lb></lb>with the Weight E, we ought to add to the Weight E the half of <lb></lb>the Weight of the Solid B D: ſo that the Weight B D being <lb></lb><emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> two pounds, and the Weight of E ten pounds, we are to <lb></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"></emph>Simplicius,<emph.end type="italics"></emph.end> hanging at the term C, <lb></lb>gravitates in reſpect of B C, with all its Moment of ten pounds, <lb></lb>whereas if only B D were pendent, it would weigh with its whole <lb></lb>Moment of two pounds; but, as you ſee, that Solid is diſtributed <lb></lb>thorow all the length B C, uniformly, ſo that its parts near to the <lb></lb>extream B, gravitate leſſe than the more remote: ſo that, in a word, <lb></lb>compenſating thoſe with theſe, the weight of the whole Priſm is <lb></lb>brought to operate under the Center of its Gravity, which anſwe<lb></lb>reth to the middle of the Leaver B C: But a Weight hanging at <lb></lb>the end C, hath a Moment double to that which it would have <lb></lb>hanging at the middle: And therefore the half of the Weight of <lb></lb>the Priſm ought to be added to the Weight E, when we would uſe <lb></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 ſelf, <lb></lb>me thinks, that the Power of both the Weights B D and E, ſo placed, <lb></lb>would have the ſame Moment, as if the whole Weight of B D, and <lb></lb>the double of E were hanged in the midſt of the Leaver B C.</s></p><p type="main"> <s>SALV. </s> <s>It is exactly ſo, and you are to bear it in mind. </s> <s>Here we <lb></lb>may immediatly underſtand</s></p><p type="head"> <s>PROPOSITION II.</s></p><p type="main"> <s><emph type="italics"></emph>How, and with what proportion, a Ruler, or Priſm, <lb></lb>more broad than thick, reſiſteth Fraction, better if it <lb></lb>be forced according to its breadth, than according to <lb></lb>its thickneſſe.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>For underſtanding of which, let a Priſm be ſuppoſed A D: <lb></lb>[<emph type="italics"></emph>as in<emph.end type="italics"></emph.end> Fig. </s> <s>4.] whoſe breadth is A C, and its thickneſs much <lb></lb>leſſer C B: It is demanded, why we would attempt to break <lb></lb>it edge-waies, as in the firſt Figure it will reſiſt the great Weight <lb></lb>T, but placed flat-waies, as in the ſecond Figure, it will not reſiſt <pb xlink:href="069/01/098.jpg" pagenum="96"></pb>X, leſſer than T: Which is manifeſted, ſince we underſtand the <lb></lb>Fulciment, one while under the Line B C, and another while under <lb></lb>C A, and the Diſtances of the Forces to be alike in both Caſes, to <lb></lb>wit, the length <emph type="italics"></emph>B<emph.end type="italics"></emph.end> D. </s> <s>But in the firſt Caſe, the Diſtance of the Re<lb></lb>ſiſtance from the Fulciment, which is the half of the Line C A, is <lb></lb>greater than the Diſtance in the other Caſe, which is the half of B <lb></lb>C: Therefore the Force of the Weight T, muſt of neceſſity be grea<lb></lb>ter than X, as much as the half of the breadth C A is greater than <lb></lb>half the thichneſſe B C, the firſt ſerving for the Counter-Leaver of <lb></lb>C A, and the ſecond of C B to overcome the ſame Reſiſtance, that <lb></lb>is the quantity of the <emph type="italics"></emph>Fibres,<emph.end type="italics"></emph.end> or ſtrings of the whole Baſe A B. <lb></lb></s> <s>Conclude we therefore, that the ſaid Priſm or Ruler, which is <lb></lb>broader than it is thick, reſiſteth, bresking more the edge-waies <lb></lb>than the flat-waies, according to the Proportion of the breadth to <lb></lb>the thickneſs.</s></p><p type="main"> <s>It is requiſite that we begin in the next place</s></p><p type="head"> <s>PROPOSITION III.</s></p><p type="main"> <s><emph type="italics"></emph>To find according to what proportion the encreaſe of the <lb></lb>Moment of the proper Gravity is made in a Priſm <lb></lb>or Cylinder, in relation to the proper Reſiſtance <lb></lb>againſt Fraction, whilſt that being parallel to the <lb></lb>Horizon, it is made longer and longer: Which Mo<lb></lb>ment I find to encreaſe ſucceſsively in duplicate Pro<lb></lb>portion to that of the prolongation.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>For demonſtration whereof, deſcribe the Priſm or Cylin<lb></lb>der A D, firmly faſtned in the Wall at the end A, and let <lb></lb>it be equidiſtant from the Horizon, and let the ſame be <lb></lb>underſtood to be prolonged as far as E, adding thereto the part <lb></lb><emph type="italics"></emph>B<emph.end type="italics"></emph.end> E. </s> <s>It is manifeſt, that the prolongation of the Leaver A <emph type="italics"></emph>B<emph.end type="italics"></emph.end><lb></lb>to C encreaſeth, by it ſelf alone, that is taken abſolutely, the <lb></lb>Moment of the Force preſſing againſt the Reſiſtance of the <lb></lb>Separation and Rupture to be made in A, according to the pro<lb></lb>portion of C A to <emph type="italics"></emph>B<emph.end type="italics"></emph.end> A: but, moreover, the Weight of the Solid <lb></lb>affixed <emph type="italics"></emph>B<emph.end type="italics"></emph.end> E, encreaſeth the Moment of the preſſing Gravity of <lb></lb>the Weight of the Solid A <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> according to the Proportion of <lb></lb>the Priſm A E to the Priſm A <emph type="italics"></emph>B<emph.end type="italics"></emph.end>; which proportion is the ſame <lb></lb>as that of the length A C, to the length A <emph type="italics"></emph>B<emph.end type="italics"></emph.end>: Therefore it is clear <pb xlink:href="069/01/099.jpg" pagenum="97"></pb>that the two augmentations of the Lengths and of the Gravities <lb></lb>being put together, the Moment compounded of both is in double <lb></lb><figure id="id.069.01.099.1.jpg" xlink:href="069/01/099/1.jpg"></figure><lb></lb>proportion to ei<lb></lb>ther of them. </s> <s>We <lb></lb>conclude there<lb></lb>fore, That the Mo<lb></lb>ments of the For<lb></lb>ces of Priſmes and <lb></lb>Cylinders of equal <lb></lb>thickneſſe, but of <lb></lb>unequal length, are <lb></lb>to one another in <lb></lb>duplicate proporti<lb></lb>on to that of their <lb></lb>Lengths; that is, <lb></lb>are as the Squares of <lb></lb>their Lengths.</s></p><p type="main"> <s>We will ſhew, in <lb></lb>the ſecond place, <lb></lb>according to what proportion the Reſiſtance of Fraction in Priſmes <lb></lb>and Cylinders encreaſeth, when they continue of the ſame length, <lb></lb>and encreaſe in thickneſs. </s> <s>And here I ſay, that</s></p><p type="head"> <s>PROPOSITION IV.</s></p><p type="main"> <s><emph type="italics"></emph>In Priſmes and Cylinders of equal length, but unequal <lb></lb>thickneſs, the Reſiſtance againſt Fraction encreaſeth <lb></lb>in a proportion iriple to the Diameters of their <lb></lb>Thickneſſes, that is, of their Baſes.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let the two Cylinders be theſe A and <emph type="italics"></emph>B, [as in<emph.end type="italics"></emph.end> Fig. </s> <s>5.] <lb></lb>whoſe equal lengths are D G, and F H, the unequal <emph type="italics"></emph>B<emph.end type="italics"></emph.end>aſes <lb></lb>the Circles, whoſe Diameters are C D, and E F. </s> <s>I ſay, <lb></lb>that the Reſiſtance of the Cylinder <emph type="italics"></emph>B<emph.end type="italics"></emph.end> is to the Reſiſtance of the <lb></lb>Cylinder A againſt Fraction, in a proportion triple to that which <lb></lb>the Diameter F E hath to the Diameter D C. </s> <s>For if we conſider <lb></lb>the abſolute and ſimple Reſiſtance that reſides in the <emph type="italics"></emph>B<emph.end type="italics"></emph.end>aſes, that <lb></lb>is, in the Circles E F, and D C to breaking, offering them vio<lb></lb>lence by pulling them end-waies, without all doubt, the Reſiſtance <lb></lb>of the Cylinder <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> is ſo much greater than that of the Cylinder A, <lb></lb>by how much the Circle E F is greater than C D; for the Fibres, <lb></lb>Filaments, or tenacious parts, which hold together the Parts of the <lb></lb>Solid, are ſo many the more. <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut if we conſider, that in offering <pb xlink:href="069/01/100.jpg" pagenum="98"></pb>them violence tranſverſly we make uſe of two Leavers; of which <lb></lb>the Parts or Diſtances, at which the Forces are applied are the Lines <lb></lb>D G, and F H, the Fulciments are in the Points D and F; but the <lb></lb>other Parts or Diſtances, at which the Reſiſtances are placed, are <lb></lb>the Semidiameters of the Circles D C and E F, becauſe the Fila<lb></lb>ments diſperſed thorow the whole Superficies of the Circles are as <lb></lb>if they were all reduced into the Centers: conſidering, I ſay, thoſe <lb></lb>Leavers, we would be underſtood to intend, that the Reſiſtance in <lb></lb>the Center of the Baſe E F againſt the Force of H, is ſo much grea<lb></lb>ter than the Reſiſtance of the Baſe C D, againſt the Force placed <lb></lb>in G, (and the Forces in G and H are of equal Leavers D G, and <lb></lb>F H) as the Semidiameter F E is greater than the Semidiameter <lb></lb>D C, the Reſiſtance againſt Fraction, therefore, in the Cylinder <lb></lb>B, encreaſeth above the Reſiſtance of the Cylinder A, according <lb></lb>to both the proportions of the Circles E F and D C, and of their <lb></lb>Semidiameters, or, if you will, Diameters: <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut the proportion of <lb></lb>the Circles is double of that of the Diameters; Therefore the pro<lb></lb>portion of the Reſiſtances, which is compounded of them, is in <lb></lb>triplicate proportion of the ſaid Diameters: Which is that which <lb></lb>I was to prove. <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut becauſe alſo the Cubes are in triplicate pro<lb></lb>portion to their Sides, we may likewiſe conclude, <emph type="italics"></emph>That the Reſi<lb></lb>ſtances of Cylinders of equal Length, are to one another as the Cubes <lb></lb>of their Diameters.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>From that which we have Demonſtrated we may likewiſe con<lb></lb>clude, that</s></p><p type="head"> <s>COROLARY.</s></p><p type="main"> <s><emph type="italics"></emph>The Reſiſtances of Priſms, and Cylinders of equal length are in <lb></lb>Seſquialter proportion to that of the ſaid Cylinders.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>The which is manifeſt, becauſe the Priſms and Cylinders, <lb></lb>equal in height, are to one another, in the ſame proportion as <lb></lb>their <emph type="italics"></emph>B<emph.end type="italics"></emph.end>aſes; that is, the double of the Sides or Diameters of the <lb></lb>ſaid <emph type="italics"></emph>B<emph.end type="italics"></emph.end>aſes: <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut the Reſiſtances (as hath been demonſtrated) are <lb></lb>in triplicate proportion to the ſaid Sides or Diameters: Therefore <lb></lb>the proportion of the Reſiſtances is Seſquialter to the proportion <lb></lb>of the ſaid Solids, and, conſequently, to the Weights of the ſaid <lb></lb>Solids.</s></p><p type="main"> <s>SIMP. </s> <s>It is convenient, that, before we proceed any farther, I <lb></lb>be reſolved of a certain Doubt, and this it is, That I have not hi<lb></lb>therto heard propoſed to Conſideration another certain kind of <lb></lb>Reſiſtance, that, in my opinion, is ſucceſſively diminiſhed in So<lb></lb>lids, according as they are more and more prolonged, and not on<lb></lb>ly in uſing them ſidelongs, but alſo leng thwaies, in the ſelf ſame <pb xlink:href="069/01/101.jpg" pagenum="99"></pb>manner juſt as we ſee a very long Cord to be much leſſe apt to <lb></lb>ſuſtain a great weight, than if it were ſhort: ſo that I believe, that <lb></lb>a Ruler of Wood or Iron will ſuſtain a much greater weight, if it <lb></lb>ſhall be ſhort, than if it ſhall be very long; underſtanding it al<lb></lb>waies to be uſed lengthwaies, and not tranſverſly; and alſo <lb></lb>its own weight being accounted for, which in the longer is <lb></lb>greater.</s></p><p type="main"> <s>SALV. </s> <s>I fear, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> that in this Point you, with many <lb></lb>others, are deceived, if ſo be, that I have rightly apprehended your <lb></lb>meaning, ſo that you would ſay, that a Cord <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> forty yards <lb></lb>long cannot ſuſtain ſo much, as if uſe were made but of one or two <lb></lb>yards of the ſame Rope.</s></p><p type="main"> <s>SIMP. </s> <s>That is it, which I would have ſaid, and as yet it ſeemeth <lb></lb>a very probable Propoſition.</s></p><p type="main"> <s>SALV. </s> <s>But I hold it not only improbable, but falſe: and think <lb></lb>that I can very eaſily reclaim you from your Errour. </s> <s>Therefore <lb></lb>let us ſuppoſe this Rope A B, [<emph type="italics"></emph>as in<emph.end type="italics"></emph.end> Fig. </s> <s>6.] faſtned on high by <lb></lb>the end A, and by the other end let there hang the Weight C, <lb></lb>by the force of which, the ſaid Rope is to be broken. </s> <s>Do you <lb></lb>aſſign me the particular place, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> where the Rupture is <lb></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 cauſe why it ſhould break in D.</s></p><p type="main"> <s>SIMP. </s> <s>The reaſon thereof is, becauſe the Rope was not ſtrong <lb></lb>enough in that part, to ſuſtain <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> an hundred pounds of weight, <lb></lb>for ſo much is the Rope D B with the Stone C.</s></p><p type="main"> <s>SALV. </s> <s>Therefore when ever ſuch a Rope ſhall come to be vio<lb></lb>lently ſtretched by thoſe hundred pounds of weight, it ſhall break <lb></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 ſame Weight, not <lb></lb>at the end of the Rope <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> but near to the point D, as for inſtance, <lb></lb>in E, or elſe did tye the Rope not at the height A, but very near, <lb></lb>and almoſt at the Point D it ſelf, as in F, tell me, I ſay, whether <lb></lb>the Point D would feel the ſame weight of an hundred pounds.</s></p><p type="main"> <s>SIMP. </s> <s>It would ſo, ſtill joyning the piece of Rope E <emph type="italics"></emph>B<emph.end type="italics"></emph.end> to the <lb></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></lb>by the ſaid hundred pounds of weight, it will break by your con<lb></lb>ceſſion. </s> <s>And yet F E, is a ſmall piece of the length A <emph type="italics"></emph>B<emph.end type="italics"></emph.end>: why do <lb></lb>you ſay then, that the long Rope is weaker than the ſhort one? <lb></lb><emph type="italics"></emph>B<emph.end type="italics"></emph.end>e content, therefore, to ſuffer your ſelf to be reclaimed from an <lb></lb>Errour, in which you have had many Companions, and thoſe in <lb></lb>other things very knowing. </s> <s>And let us go on: and having demon<lb></lb>ſtrated, that Priſms and Cylinders encreaſe their Moments above <pb xlink:href="069/01/102.jpg" pagenum="100"></pb>their Reſiſtances, according to the Squares of their Lengths (alwaies <lb></lb>provided, that they retain the ſame thickneſſe) and that likewiſe, <lb></lb>theſe that are equally long, but different in thickneſſe, encreaſe <lb></lb>their Reſiſtances according to the proportion of the Cubes of the <lb></lb>Sides or Diameters of their Baſes, we may enquire what befal<lb></lb>leth to thoſe Solids, being different in length and thickneſs, in which <lb></lb>I obſerve, that</s></p><p type="head"> <s>PROPOSITION V.</s></p><p type="main"> <s><emph type="italics"></emph>Priſms and Cylinders, of different length and thickneſs, <lb></lb>have their Reſiſtances againſt Fraction, in a propor<lb></lb>tion compounded of the proportion of the Cubes of the <lb></lb>Diameters of their Baſes, and of the proportion of <lb></lb>their lengths reciprocally taken.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let theſe two A B C, and D E F, [<emph type="italics"></emph>as in<emph.end type="italics"></emph.end> Fig. </s> <s>7.] be ſuch Cy<lb></lb>linders. </s> <s>I ſay, the Reſiſtance of the Cylinder A C ſhall be to <lb></lb>the Reſiſtance of the Cylinder D F, in a proportion com<lb></lb>pounded of the proportion of the Cube of the Diameter A B, to <lb></lb>the Gube of the Diameter D E, and of the proportion of the <lb></lb>Length E F to the Length B C. </s> <s>Suppoſe E G equal to B C, and to <lb></lb>the Lines A B, and D E, let C H be a third proportional, and I, <lb></lb>a fourth; and as E F is to B C, ſo let I be to S. </s> <s>And becauſe the <lb></lb>Reſiſtance of the Cylinder A C is to the Reſiſtance of the Cylin<lb></lb>der D G, as the Cube A B to the Cube D E; that is, as the Line <lb></lb>A B to the Line I: and the Reſiſtance of the Cylinder G D is to <lb></lb>the Reſiſtance of the Cylinder D F, as the Length F E is to the <lb></lb>Length E G; that is, as the Line I is to S: Therefore by Equali<lb></lb>ty of proportion, as the Reſiſtance of the Cylinder A C is to the <lb></lb>Reſiſtance of the Cylinder D F, ſo is the Line A B to S: But the <lb></lb>Line A B is to S, in a proportion compounded of A B to I, and of <lb></lb>I to S: Therefore the Reſiſtance of the Cylinder A C is to the Re<lb></lb>ſiſtance of the Cylinder D F, in a proportion compounded of A B <lb></lb>to I, that is, as the Cube of A B to the Cube of D E, and of the <lb></lb>proportion of the Line I to S; that is, of the Length E F to the <lb></lb>Length B C: Which was to be demonſtrated.</s></p><p type="main"> <s>After the Propoſition laſt demonſtrated, we will conſider what <lb></lb>hapneth between like Cylinders and Priſms, of which we will de<lb></lb>monſtrate, how that</s></p><pb xlink:href="069/01/103.jpg" pagenum="101"></pb><p type="head"> <s>PROPOSITION VI.</s></p><p type="main"> <s><emph type="italics"></emph>Of like Cylinders and Priſms the Moments compoun<lb></lb>ded, that is to ſay, reſulting from their Gravities, <lb></lb>and from their Lengths, which are, as it were, Lea<lb></lb>vers, have to one another a proportion Seſquialter to <lb></lb>that which is between the Reſiſtances of their ſame <lb></lb>Baſes.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>For demonſtration of which let us deſcribe the two like Cy<lb></lb>linders A B, and C D, [<emph type="italics"></emph>as in<emph.end type="italics"></emph.end> Fig. </s> <s>8.] I ſay, that the Mo<lb></lb>ment of the Cylinder A B, to overcome the Reſiſtance of its <lb></lb>Baſe B, hath to the Moment of C D, to overcome the Reſiſtance <lb></lb>of its Baſe C, a proportion Seſquialter to that which the ſame Re<lb></lb>ſiſtance of the Baſe B, hath to the Reſiſtance of the Baſe D: <lb></lb>And becauſe the Moments of the Solids A B, and C D, to over<lb></lb>come the Reſiſtances of their Baſes B and D, are compounded of <lb></lb>their Gravities, and of the Forces of their Leavers, and the Force <lb></lb>of the Leaver A B is equal to the Force of the Leaver C D, and <lb></lb>that becauſe the length A B hath the ſame proportion to the Semi<lb></lb>diameter of the Baſe B, (by the ſimilitude of the Cylinders) that <lb></lb>the Length C D hath to the Semidiameter of the Baſe D; it re<lb></lb>maineth, that the total Moment of the Cylinder A B, be to the <lb></lb>total Moment of C D, as the ſole Gravity of the Cylinder A B is <lb></lb>to the ſole Gravity of the Cylinder C D; that is, as the ſaid Cy<lb></lb>linder A B is to the ſaid C D: But theſe are in triplicate propor<lb></lb>tion to the Diameters of their Baſes <emph type="italics"></emph>B<emph.end type="italics"></emph.end> and D; and the Reſiſtances <lb></lb>of the ſame <emph type="italics"></emph>B<emph.end type="italics"></emph.end>aſes, being to one another as the ſaid <emph type="italics"></emph>B<emph.end type="italics"></emph.end>aſes, they are <lb></lb>conſequently in duplicate proportion to their ſame <emph type="italics"></emph>B<emph.end type="italics"></emph.end>aſes: There<lb></lb>fore the Moments of Cylinders are in Seſquialter proportion to <lb></lb>the Reſiſtances of their <emph type="italics"></emph>B<emph.end type="italics"></emph.end>aſes.</s></p><p type="main"> <s>SIMP. </s> <s>This Propoſition, indeed, is not only new, but unexpe<lb></lb>cted to me, and at firſt ſight, very remote from the judgment that <lb></lb>I ſhould have conjecturally paſt upon it: for in regard, that theſe <lb></lb>Figures are in all other reſpects alike, I ſhould have thought that <lb></lb>their Moments likewiſe ſhould have retained the ſame proportion <lb></lb>towards their proper Reſiſtances.</s></p><p type="main"> <s>SAGR. </s> <s>This is the Demonſtration of that Propoſition, that in <lb></lb>the beginning of our Diſcourſes, I ſaid, I thought------I had ſome <lb></lb>glimps of.</s></p><p type="main"> <s>SALV. </s> <s>That which now befalleth, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> hapned for ſome <pb xlink:href="069/01/104.jpg" pagenum="102"></pb>time to my ſelf, believing, that the Reſiſtances of like Solids were <lb></lb>alike, till that a certain, and that no very fixed or accurate Obſer<lb></lb>vation ſeemed to repreſent unto me, that Solids do not contain <lb></lb>an equal tenure in their Toughneſs, but that the bigger are leſſe <lb></lb>apt to ſuffer violent accidents, as luſty men are more damnified by <lb></lb>their falls than little children; and, as in the begining we ſaid, we <lb></lb>ſee a great <emph type="italics"></emph>B<emph.end type="italics"></emph.end>eam or Column break to pieces falling from the ſame <lb></lb>height, and not a ſmall Priſin or little Cylinder of Marble. </s> <s>This <lb></lb>ſame Obſervation gave me the hint for finding of that which I am <lb></lb>now about to demonſtrate; a Quality truly admirable, for that <lb></lb>amongſt the infinite Solid-like Figures, there are not ſo much <lb></lb>as two, whoſe Moments retain the ſame proportion towards their <lb></lb>proper Reſiſtances.</s></p><p type="main"> <s>SIMP. </s> <s>Now you put me in mind of ſomething inſerted by <emph type="italics"></emph>Ari<lb></lb>ſtotle<emph.end type="italics"></emph.end> amongſt his Mechanical Queſtions, where he would give a <lb></lb>Reaſon, whence it is, that <emph type="italics"></emph>B<emph.end type="italics"></emph.end>eams the longer they are, they are by ſo <lb></lb>much the more weak, and bend more and more, although the ſhort <lb></lb>ones be the ſlendereſt, and the long ones thickeſt: and, if I well re<lb></lb>member, he reduceth the Reaſon to the ſimple Leaver.</s></p><p type="main"> <s>SALV. </s> <s>It is very true, and becauſe the Solution ſeemeth not <lb></lb>wholly to remove the cauſe of doubting <emph type="italics"></emph>Monſignore di Guevara,<emph.end type="italics"></emph.end><lb></lb>who, the truth is, with his moſt learned <emph type="italics"></emph>Commentaries<emph.end type="italics"></emph.end> hath highly <lb></lb>enobled and illuſtrated that Work, enlargeth himſelf with other <lb></lb>accute Speculations for the obviating all difficulties, yet himſelf <lb></lb>alſo remaining perplexed in this point, whether, the lengths and <lb></lb>thickneſſes of ſuch Solid Figures, encreaſing with the ſelf ſame <lb></lb>proportion, they ought to retain the ſame tenure in their Tough<lb></lb>neſſes and Reſiſtances againſt their breaking, and likewiſe againſt <lb></lb>their bending. </s> <s>After I had long conſidered thereon, I have, in <lb></lb>this manner found, that which I am about to tell you. </s> <s>And firſt <lb></lb>I will demonſtrate that</s></p><pb xlink:href="069/01/105.jpg" pagenum="103"></pb><p type="head"> <s>PROPOSITION VII.</s></p><p type="main"> <s><emph type="italics"></emph>Of like and heavy Priſms or Cylinders there is one only, <lb></lb>and no more, that is reduced (being charged with its <lb></lb>own weight) to the ultimate ſtate between breaking <lb></lb>and holding it ſelf together: ſothat every greater, as <lb></lb>being unable to reſiſt its own weight, will break, <lb></lb>and every leſſer reſiſteth ſome Force that is employed <lb></lb>againſt it to break, it.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let the heavy Priſm be A B [<emph type="italics"></emph>as in<emph.end type="italics"></emph.end> Fig 9.] reduced to the <lb></lb>utmoſt length of its Conſiſtance, ſo that being lengthned <lb></lb>never ſo little more it will break: I ſay, that this is the only <lb></lb>one amongſt all thoſe that are like unto it, (which yet are infinite) <lb></lb>that is capable of being reduced to that dubious and tickliſh ſtate; <lb></lb>ſo that every greater being oppreſſed with its own weight will <lb></lb>break, and every leſſer not, nay, will be able to reſiſt ſome additi<lb></lb>on of a new violence, over and above that of its own weight. <lb></lb></s> <s>Firſt, take the Priſm C E, like to, and greater than A B. </s> <s>I ſay, that <lb></lb>this cannot conſiſt, but will break, being overcome by its own <lb></lb>Gravity. </s> <s>Suppoſe the part C D as long as A B. </s> <s>And becauſe the <lb></lb>Reſiſtance C D is to that of A B, as the Cube of the thickneſſe of <lb></lb>C D to the Cube of the thickneſs of A B; that is, as the Priſm <lb></lb>C E to the Priſm A B (being alike:) Therefore the Weight of <lb></lb>C E is the greateſt that can be ſuſtained at the length of the Priſm <lb></lb>C D: But the Length C E is greater: Therefore the Priſm C E <lb></lb>will break. </s> <s>But let F G be leſſet: it ſhall be demonſtrated like<lb></lb>wiſe (ſuppoſing F H equal to B A) that the Reſiſtance of F G is <lb></lb>to that of A B, as the Priſm F G is to the Priſm A B, in caſe that the <lb></lb>Diſtance A B, that is F H, were equal to F G, but it is greater: <lb></lb>Therefore the Moment of the Priſm F G, placed in G, doth not <lb></lb>ſuffice to break the Priſm F G.</s></p><p type="main"> <s>SAGR. </s> <s>A moſt manifeſt and brief Demonſtration, inferring the <lb></lb>truth and neceſſity of a Propoſition that at firſt ſight ſeemeth far <lb></lb>from probability. </s> <s>It would be requiſite, therefore, to alter much <lb></lb>the proportion betwixt the Length and Thickneſſe of the greater <lb></lb>Priſm by making it thicker or ſhorter, to the end it might be re<lb></lb>duced to that nice ſtate of indifferency between holding and brea<lb></lb>king; and the Inveſtigation of that ſame State, as I think, would <lb></lb>be no leſſe ingenuous.</s></p><p type="main"> <s>SALV. Nay, rather more, as it is alſo more laborious: and I am <pb xlink:href="069/01/106.jpg" pagenum="104"></pb>ſure I have ſpent no ſmall time to find it; and I will now impart it <lb></lb>to you: Therefore</s></p><p type="head"> <s>PROP. VIII. PROBL. I.</s></p><p type="main"> <s><emph type="italics"></emph>A Cylinder or Priſm of the utmoſt length not to be bro<lb></lb>ken by its own weight, and alſo a greaver length, be<lb></lb>ing given, to find the thickneſſe of another Cylinder <lb></lb>or Priſm that under-given length is the only one, and <lb></lb>biggeſt, that can reſiſt its own weight.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let the Cylinder B C [<emph type="italics"></emph>as in<emph.end type="italics"></emph.end> Fig. </s> <s>10.] be the biggeſt that <lb></lb>can reſiſt its own weight, and let D E be a Length greater <lb></lb>than A C; it is required to find the Thickneſſe of the Cylin<lb></lb>der, that under the Length D E is the greateſt reſiſting its own <lb></lb>weight. </s> <s>Let I be a third proportional to the Lengths D E, and <lb></lb>A C; and as D E is to I, ſo let the Diameter F D be to the Dia<lb></lb>meter B A: and make the Cylinder F E. </s> <s>I ſay, that this is the big<lb></lb>geſt, and only one amongſt all that are like to it that reſiſteth its <lb></lb>own weight. </s> <s>To the Lines D C and I let M be a third propor<lb></lb>tional, and O a fourth. </s> <s>And ſuppoſe F G equal to A C. </s> <s>And be<lb></lb>cauſe the Diameter F D is to the Diameter A B, as the Line D E <lb></lb>to I, and O is a fourth proportional to D E and I, the Cube of <lb></lb>F D ſhall be to the Cube of B A as D E is to O: But as the Cube of <lb></lb>F D is to the Cube of B A, ſo is the Reſiſtance of the Cylinder <lb></lb>D G to the Reſiſtance of the Cylinder B C: Therefore the Reſi<lb></lb>ſtance of the Cylinder D G is to that of the Cylinder B C, as the <lb></lb>Line D F is to O. </s> <s>And becauſe the Moment of the Cylinder B C <lb></lb>is equal to its Reſiſtance, if we ſhew that the Moment of the Cylin<lb></lb>der F E is to the Moment of the Cylinder B C, as the Reſiſtance <lb></lb>D F to the Reſiſtance B A; that is, as the Cube of F D to the Cube <lb></lb>of B A; that is, as the Line D E to O, we ſhall have our intent: <lb></lb>that is, that the Moment of the Cylinder F E is equal to the Reſi<lb></lb>ſtance placed in F D. </s> <s>The Moment of the Cylinder F E is to the <lb></lb>Moment of the Cylinder D G, as the Square of D E is to the <lb></lb>Square of A C; that is, as the Line D E to I: But the Moment of <lb></lb>the Cylinder D G is to the Moment of the Cylinder B C, as the <lb></lb>Square D F to the Square B A; that is, as the Square of D E to the <lb></lb>Square of I; that is, as the Square of I to the Square of M; that <lb></lb>is, as I to O: Therefore, by Equality of proportion, as the Mo<lb></lb>ment of the Cylinder F E is to the Moment of the Cylinder B C, <lb></lb>ſo is the Line D E to O; that is, the Cube D F to the Cube <lb></lb>B A; that is, the Reſiſtance of the Baſe D F to the Reſiſtance <pb xlink:href="069/01/107.jpg" pagenum="105"></pb>of the Baſe B A: Which is that that was ſought.</s></p><p type="main"> <s>SAGR This, <emph type="italics"></emph>Salviatus,<emph.end type="italics"></emph.end> is a long Demonſtration, and very hard <lb></lb>to be born in mind at the firſt hearing, therefore I could wiſh, that <lb></lb>you would pleaſe to repeat it.</s></p><p type="main"> <s>SALV. </s> <s>I will do what you ſhall command; but haply it would <lb></lb>be better to produce one more conciſe and ſhort: but then it will <lb></lb>be requiſite to deſcribe a Figure ſomewhat different.</s></p><p type="main"> <s>SAGR. </s> <s>The favour will then be the greater: and beſtow upon <lb></lb>me the draught of that already explained, that I may at my leaſure <lb></lb>conſider it again.</s></p><p type="main"> <s>SALV. </s> <s>I will not fail to ſerve you. </s> <s>Now, ſuppoſe a Cylinder A, <lb></lb><arrow.to.target n="marg1082"></arrow.to.target><lb></lb>[<emph type="italics"></emph>as in<emph.end type="italics"></emph.end> Fig. </s> <s>11.] the Diameter of whoſe Baſe let be the Line D C, <lb></lb>and let this A be the greateſt that can ſuſtain it ſelf and not break, <lb></lb>than which we will find a bigger, which likewiſe ſhall be the big<lb></lb>geſt alſo, and the only one that ſuſtaineth it ſelf. </s> <s>Let us deſire one <lb></lb>like to the ſaid A, and as long as the aſſigned Line, and let this be <lb></lb><emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> E, the Diameter of whoſe Baſe let be K L; and to the two <lb></lb>Lines D C, and K L let M N be a third proportional; which let be <lb></lb>the Diameter of the Baſe of the Cylinder X, in length equal to E. <lb></lb></s> <s>I ſay, that this X is that which we ſeek. </s> <s>And becauſe the Reſi<lb></lb>ſtance D C is to the Reſiſtance K L, as the Square D C to the <lb></lb><emph type="italics"></emph>S<emph.end type="italics"></emph.end>quare K L; that is, as the Square K L to the Square M N; that <lb></lb>is, as the Cylinder E to the Cylinder X; that is, as the Moment E <lb></lb>to the Moment X: But the Reſiſtance K L is to M N, as the Cube <lb></lb>of K L is to the Cube of M N; that is, as the Cube B C to the <lb></lb>Cube K L; that is, as the Cylinder A to the Cylinder E; that is, <lb></lb>as the Moment A to the Moment E: Therefore, by Perturbation <lb></lb>of proportion, as the Reſiſtance D C is to M N, ſo is the Moment <lb></lb>A to the Moment X: Therefore the Priſm X, is in the ſame Conſti<lb></lb>tution of Moment and Reſiſtance as the Priſm A.</s></p><p type="margin"> <s><margin.target id="marg1082"></margin.target><emph type="italics"></emph>The laſt Problem <lb></lb>performed another <lb></lb>way.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>But let us make the Problem more general, and let the Propo<lb></lb>ſition be this:</s></p><p type="main"> <s><emph type="italics"></emph>The Cylinder<emph.end type="italics"></emph.end> A C <emph type="italics"></emph>being given, and its Moment to-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1083"></arrow.to.target><lb></lb><emph type="italics"></emph>wards its Reſiſtance being ſuppoſed at pleaſure, and <lb></lb>any Length<emph.end type="italics"></emph.end> D E <emph type="italics"></emph>being aſsigned, to find the Thick<lb></lb>neſſe af the Cylinder whoſe Length is<emph.end type="italics"></emph.end> D E, <emph type="italics"></emph>and whoſe <lb></lb>Moment towards its Reſiſtance retaineth the ſame <lb></lb>proportion, that the Moment of the Cylinder<emph.end type="italics"></emph.end> A C <lb></lb><emph type="italics"></emph>doth to its Reſiſtance.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/108.jpg" pagenum="106"></pb><p type="margin"> <s><margin.target id="marg1083"></margin.target><emph type="italics"></emph>The laſt Propoſi<lb></lb>tion made more ge<lb></lb>neral.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Reaſſuming the above ſaid Figure and almoſt the ſame Me<lb></lb>thod, we will ſay: Becauſe the Moment of the Cylinder <lb></lb>F E hath the ſame proportion to the Moment of the part <lb></lb>D G, that the Square E D hath to the Square F G; that is that <lb></lb>the Line D E hath to I: and becauſe the Moment of the Cylinder <lb></lb>F G is to the Moment of the Cylinder A C, as the Square F D to <lb></lb>the Square A B; that is, as the Square D E to the Square I; that <lb></lb>is, as the Square I to the Square M; that is, as the Line I to O: <lb></lb>Therefore, <emph type="italics"></emph>ex æquali,<emph.end type="italics"></emph.end> the Moment of the Cylinder F E hath the <lb></lb>ſame proportion to the Moment of the Cylinder A C, that the <lb></lb>Line D E hath to the Line O; that is, that the Cube D E hath <lb></lb>to the Cube of I; that is, that the Cube of F D hath to the <lb></lb>Cube of A B; that is, that the Reſiſtance of the Baſe F D hath to <lb></lb>the Reſiſtance of the Baſe A B: Which was to be performed.</s></p><p type="main"> <s>Now, let it be obſerved, that from the things hitherto demonſtra<lb></lb>ted, we may plainly gather, how Impoſſible it is, not only for Art, but <lb></lb><arrow.to.target n="marg1084"></arrow.to.target><lb></lb>for Nature her ſelf to encreaſe her Machines to an immenſe Vaſt<lb></lb>neſſe: ſo that it would be impoſſible by Art to build extraordina<lb></lb>ry vaſt Ships, Palaces, or Temples, whoſe ^{*} Oars, Sail-yards, Beams, <lb></lb>Iron Bolts, and, in a word, their other parts ſhould conſiſt or hold <lb></lb>together: neither again could Nature make Trees of unmeaſura<lb></lb><arrow.to.target n="marg1085"></arrow.to.target><lb></lb>ble greatneſſe, for that their Arms or Bows being oppreſſed with <lb></lb>their own weight would at laſt break: and likewiſe it would be <lb></lb>impoſſible for her to make ſtructures of Bones for men, Horſes, or <lb></lb>other Animals, that might ſubſiſt, and proportionatly perform <lb></lb>their Offices, when thoſe Animals ſhould be augmented to im<lb></lb>menſe heights, unleſſe ſhe ſhould take Matter much more hard and <lb></lb>Refiſting than that which ſhe commonly uſeth, or elſe ſhould de<lb></lb>form thoſe Bones by augmenting them beyond their due Symetry, <lb></lb>and making the Figure or ſhape of the Animal to become mon<lb></lb>ſtrouſly big: Which haply was hinted by my moſt Witty Poet, <lb></lb>where deſcribing an huge Giant, he ſaith,</s></p><p type="margin"> <s><margin.target id="marg1084"></margin.target>* Oares are uſed <lb></lb>in the Ships or <lb></lb>Gallies of the <lb></lb>Mediterrane, up<lb></lb>on which our <lb></lb>Author was a <lb></lb>Coaſter.</s></p><p type="margin"> <s><margin.target id="marg1085"></margin.target><emph type="italics"></emph>Bones of Animals <lb></lb>magnified beyond <lb></lb>their ratural ſize, <lb></lb>would not ſubſiſt, if <lb></lb>it be required to <lb></lb>retain the ſame <lb></lb>proportion of thick<lb></lb>neſs and hardneſs <lb></lb>in them that is in <lb></lb>thoſe of Natural <lb></lb>Animals.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Non ſi puo compartir quanto ſia lungo,<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Si ſmiſuratamente è tutto groſſo.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1086"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1086"></margin.target><emph type="italics"></emph>Example of the <lb></lb>Bone of an Animal <lb></lb>enlarged to thrice <lb></lb>the Natural pro<lb></lb>portion, how much <lb></lb>thicker it ought to <lb></lb>be to perform its <lb></lb>office.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>And for a ſhort example of this that I ſay, [<emph type="italics"></emph>as in<emph.end type="italics"></emph.end> Fig. </s> <s>12.] I <lb></lb>have heretofore drawn the Figure of a Bone only trebled in <lb></lb>Length, and augmented in Thickneſſe in ſuch proportion, as that <lb></lb>it may in its great Animal perform the office proportionate to that <lb></lb>of the leſſer Bone in a ſmaller Animal, and the Figures are theſe: <lb></lb>whereby you ſee what a diſproportionate Figure that of the aug<lb></lb>mented Bone becometh. </s> <s>Whence it is manifeſt, that he that would <lb></lb>in an huge Giant keep the proportions that the Members have in <pb xlink:href="069/01/109.jpg" pagenum="107"></pb>an ordinary Man, muſt either find Matter much more hard and re<lb></lb>ſiſting to make Bone of, or elſe muſt admit that its Strength is in <lb></lb>proportion much more weak than in Men of middle Stature: other<lb></lb>wiſe, encreaſing the Giant to an immeaſurable height he would be <lb></lb>oppreſſed, and fall under his own weight. </s> <s>Whereas on the con<lb></lb>trary, in diminiſhing of Bodies we do not ſee the Strength and <lb></lb>Forces to diminiſh in the ſame proportion, nay, in the leſſer the <lb></lb>Robuſtiouſneſſe encreaſeth with a great proportion. </s> <s>So that I <lb></lb>believe, that a little Dog could carry on his back two or three Dogs <lb></lb>equal to himſelf, but I do not think that an Horſe could carry ſo <lb></lb>much as one ſingle Horſe of his own ſize.</s></p><p type="main"> <s>SIMP. </s> <s>But if it be ſo, I have great reaſon to doubt the Im<lb></lb>menſe bulks that we ſee in Fiſhes, for (if I rightly underſtand <lb></lb>you) a Whale ſhall be as big as ten Elephants, and yet they ſu<lb></lb>ſtain themſelves.</s></p><p type="main"> <s>SALV. </s> <s>Your doubt, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> prompts me with another Con<lb></lb>dition which I perceived not before, which is alſo able to make <lb></lb>Giants and other very big Animals to conſiſt, and act themſelves <lb></lb>no leſſe than ſmaller, and this will happen when not only Strength <lb></lb>is added to the Bones and other Parts, whoſe office it is to ſuſtain <lb></lb>their own and the ſupervenient weight; but the ſtructure of the <lb></lb>Bones being left with the ſame proportions, the ſame Fabricks <lb></lb>would juſt in the ſame manner, yea, with much more eaſe, con<lb></lb>ſiſt, when the Gravity of the matter of thoſe Bones, or that of <lb></lb>the Fleſh, or whatever elſe is to reſt it ſelf upon the Bones is dimini<lb></lb>ſhed in that proportion: and of this ſecond Artifice, Nature hath <lb></lb>made uſe in the framing of Fiſhes, making their Bones, and Pulps, <lb></lb>not only very light, but without any Gravity.</s></p><p type="main"> <s>SIMP. </s> <s>I ſee very well, <emph type="italics"></emph>Salviatus,<emph.end type="italics"></emph.end> whither your Diſcourſe ten<lb></lb>deth: you will ſay, that becauſe the Element of Water is the Ha<lb></lb>bitation of Fiſhes, which by its Corpulence, or, as others will, by <lb></lb>its Gravity diminiſheth the weight of Bodies demerged in it, for <lb></lb>that reaſon the Matter of Fiſhes, not weighing any thing, may be <lb></lb>ſuſtained without ſurcharging their Bones: but this doth not ſuf<lb></lb>fice, for although the reſt of the ſubſtance of the Fiſh weigh not, <lb></lb>yet without doubt the matter of their Bones hath its weight: <lb></lb>and who will ſay, that the Rib of a Whale that is as big as a <lb></lb>Beam doth not weigh very much, and in Water ſinketh to the Bot<lb></lb>tom? </s> <s>Theſe therefore ſhould not be able to ſubſiſt in ſo vaſt a <lb></lb>Bulk.</s></p><p type="main"> <s>SALV. </s> <s>You argue very cunningly; and for an anſwer to your <lb></lb>Doubt, tell me, whether you have obſerved Fiſhes to ſtand im<lb></lb>moveable under water at their pleaſures, and not to deſcend to<lb></lb>wards the Bottom, or raiſe themſelves towards the top without <lb></lb>making ſome motion with their Fins?</s></p><pb xlink:href="069/01/110.jpg" pagenum="108"></pb><p type="main"> <s>SIMP. </s> <s>This is a very manifeſt Obſervation.</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"></emph>The Cauſe why <lb></lb>Fiſhes do equili<lb></lb>brate themſelves <lb></lb>in the Water.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>This power therefore that the Fiſhes have to ſtay them<lb></lb>ſelves, as if they were immoveable in the midſt of the Water, is a <lb></lb>moſt infallible argument, that the Compofition of their Corporeal <lb></lb>Maſſe equalleth the Specifick Gravity of the Water, ſo that if <lb></lb>there be found in them ſome parts that are more grave than the <lb></lb>Water, it is neceſſarily requiſite that they have others ſo much <lb></lb>leſſe grave, ſo that the <emph type="italics"></emph>Equilibrium<emph.end type="italics"></emph.end> may be ballanced. </s> <s>If therefore <lb></lb>the Bones be more grave, it is neceſſary that the Pulps, or other <lb></lb>Matters that are in them, be more light; and theſe will with their <lb></lb>lightneſſe counterpoiſe and compenſate the weight of the Bones. <lb></lb></s> <s>So that in Aquatick Animals the quite contrary hapneth to that <lb></lb>which befals the Terreſtrial, namely, that in the latter it is the of<lb></lb>fice of the Bones to ſuſtain their own weight, and the weight of <lb></lb>the Fleſh; and in the former, the <emph type="italics"></emph>Fleſh [if one may ſo call it]<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1088"></arrow.to.target><lb></lb>beareth up its own weight, and that of the Bones. </s> <s>And therefore <lb></lb>ceaſe to wonder how there may be moſt vaſt Animals in the Wa<lb></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"></emph>Aquatick Animals <lb></lb>greater than the <lb></lb>Terreſtrial, and for <lb></lb>what Reaſon.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. </s> <s>I am ſatisfied, and moreover obſerve, that theſe which <lb></lb>we call Terreſtrial Animals, ought with more reaſon to be called <lb></lb>Aerial; becauſe in the Air they really live, and by the Air they are <lb></lb>environ'd, and of the Air they breath.</s></p><p type="main"> <s>SAGR. </s> <s>The Diſcourſe of <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> pleaſeth me, as alſo his <lb></lb>Doubt and its Solution. </s> <s>And farthermore I comprehend very ea<lb></lb>ſily, that one of theſe huge Fiſhes being haul'd on ſhore, could not <lb></lb>perchance be able to ſuſtain it ſelf for any time; but that the Con<lb></lb>nections of the Bones being relaxed, its Maſſe would be cruſh'd un<lb></lb>der its own weight.</s></p><p type="main"> <s>SALV. </s> <s>For the preſent, I encline to the ſame Opinion: nor am <lb></lb>I far from thinking that the ſame would happen to that huge Ship, <lb></lb>which floating in the Sea is not diſſolved by its weight, and the bur<lb></lb>den of its Lading and Artilery, but on dry ground, and environed <lb></lb>with Air, it perhaps would fall in pieces. </s> <s>But let us purſue our bu<lb></lb>ſineſſe, and demonſtrate, that</s></p><pb xlink:href="069/01/111.jpg" pagenum="109"></pb><p type="head"> <s>PROP. IX. PROBL. II.</s></p><p type="main"> <s><emph type="italics"></emph>A Priſme or Cylinder with its weight, and the great<lb></lb>eſt Weight ſuſtained by it being given, to find the <lb></lb>greateſt Length, beyond which being prolonged. </s> <s>it <lb></lb>would break under its own Weight.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let there be given the Priſme A C (<emph type="italics"></emph>as in<emph.end type="italics"></emph.end> Fig. </s> <s>13.) with its <lb></lb>weight, and likewiſe let the Weight D be given, the great<lb></lb>eſt that can be ſuſtained by the extreme C: it is required to <lb></lb>finde the greateſt Length unto which the ſaid Priſme may be pro<lb></lb>longed, without breaking. </s> <s>As the weight of the Priſme A C is to <lb></lb>the Compound of the weights A C, with the double of the <lb></lb>Weight D, ſo let the length C A be to C A H: between which <lb></lb>let A G be a Mean-Proportional. </s> <s>I ſay that A G is the Length <lb></lb>ſought. </s> <s>For the depreſſing Moment of the Weight D in C, is <lb></lb>equal to the Moment of the double weight D, if it be placed in <lb></lb>the middle of A C, where is alſo the Center of the Moment of <lb></lb>the Priſme A C: The Moment, therefore, of the Reſiſtance of <lb></lb>the Priſme A C, which reſides in A, is equivalent to the gravi<lb></lb>tation of the double of the Weight D with the weight A C, but <lb></lb>hanged in the midſt of A C. </s> <s>And becauſe it hath been made, <lb></lb>that as the Moment of the ſaid Weights ſo ſituated, that is, of <lb></lb>the double of D, with A C, is to the Moment of A C, ſo is H A <lb></lb>to A C, between which A G is a Mean Proportional: There<lb></lb>fore the Moment of D doubled with the Moment of A C, is to <lb></lb>the Moment A C, as the Square G A to the Square A C: But the <lb></lb>preſſing Moment of the Priſme G A, is to the Moment of A C, <lb></lb>as the Square G A to the Square A C: Therefore the Length <lb></lb>A G is the greateſt that was ſought, namely, that unto which the <lb></lb>Priſme A G being prolonged, it would ſuſtain it ſelf, but beyond <lb></lb>it would break.</s></p><p type="main"> <s>Hitherto we have conſidered the Moments and Reſiſtances of <lb></lb>ſolid Priſmes and Cylinders, one end of which is ſuppoſed im<lb></lb>moveable, and to the other onely the Force of a preſſing weight <lb></lb>is applyed, conſidering it by it ſelf alone, or joyned with the <lb></lb>Gravity of the ſame Solid, or elſe the ſole Gravity of the ſaid <lb></lb>Solid. </s> <s>Now I deſire that we may ſpeak ſomething of thoſe ſame <lb></lb>Priſmes or Cylinders, in caſe they were ſuſtained at both ends, or <lb></lb>did reſt upon one ſole point taken between the ends. </s> <s>And firſt, <lb></lb>I ſay that,</s></p><pb xlink:href="069/01/112.jpg" pagenum="110"></pb><p type="head"> <s>PROPOSITION X.</s></p><p type="main"> <s><emph type="italics"></emph>The Cylinder that being charged with its own Weight <lb></lb>ſhall be reduced to its greateſt Length, beyond which <lb></lb>it would not ſuſtain it ſelf, whether it be born up in <lb></lb>the middle by one ſole Fulciment, or elſe by two at <lb></lb>the ends, may be double in length to that which <lb></lb>ſhould be faſtned in the Wall, that is ſuſtained at but <lb></lb>one end.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Which of it ſelt is very obvious; for if we ſhall ſup<lb></lb>poſe of the Cylinder which I deſcribe A B C, its <lb></lb>half A B to be the utmoſt Length that is able to be <lb></lb>ſuſtained, being faſtened at the end B, it ſhall be ſuſtained in the <lb></lb>ſame manner, if being laid upon the Fulciment G, it ſhall be <lb></lb>counterpoiſed by its other half B C. </s> <s>And likewiſe, if of the Cy<lb></lb>linder D E F, the Length ſhall be ſuch that onely one half of it <lb></lb>can be ſuſtained, being faſtened at the end D, and conſequent<lb></lb>ly the other E F, fixed at the end F; it is manifeſt, that placing <lb></lb>the Fulciments H and I under the ends D and F, every Moment <lb></lb>of Force or of Weight that is added in E, will there make the <lb></lb>Fracture.</s></p><p type="main"> <s>That which requireth a more ſubtil Speculation is, when ſub<lb></lb>ſtracting from the proper Gravity of ſuch Solids, it were pro<lb></lb>pounded to us</s></p><p type="head"> <s>PROP. XI. PROBL. III.</s></p><p type="main"> <s><emph type="italics"></emph>To find whether that Force or weight, that being ap<lb></lb>plied to the middle of a Cylinder ſuſtained at the <lb></lb>ends, would ſuffice to break it, could do the ſame, <lb></lb>applied in any other place, neerer to one end than to <lb></lb>the other.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>As for Example, whether we deſiring to break a Staffe <lb></lb>and took it with the ends in our hands, and ſetting our <lb></lb>knee, to the midſt of it, the ſame Force that ſhould ſuf<lb></lb>fice to break it in that manner, would alſo ſuffice in caſe the knee <pb xlink:href="069/01/113.jpg" pagenum="111"></pb>were ſet, not in the midſt, 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"></emph>Ariſtotle<emph.end type="italics"></emph.end> in his <lb></lb><emph type="italics"></emph>Mechanical Queſtions.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>The Queſtion of <emph type="italics"></emph>Aristotle<emph.end type="italics"></emph.end> is not preciſely the ſame, <lb></lb>for he ſeeks no more, but to render a reaſon why leſſe labour is <lb></lb>required to break the Staffe, holding the hands at the ends of it, <lb></lb>that is, far diſtant from the Knee, than if we held them neerer: <lb></lb>and he giveth a general Reaſon of the ſame, reducing the cauſe <lb></lb>of it to the Leavers, which are longer when the Arms are ex<lb></lb>tended, graſping the ends. </s> <s>Our Queſtion addeth ſomething <lb></lb>more, ſeeking whether, ſetting the Knee in the midſt, or in ano<lb></lb>ther place, but alwayes keeping the hands at the ends, the ſame <lb></lb>Force ſerveth in all ſituations.</s></p><p type="main"> <s>SAGR. </s> <s>At firſt apprehenſion it ſhould ſeem that it doth, for <lb></lb>that the two Leavers retain in a certain faſhion the ſame Moment, <lb></lb>ſeeing that as the one is ſhortned, the other is lengthened.</s></p><p type="main"> <s>SALV. </s> <s>Now you ſee, how eaſie it is to make Equivocations, <lb></lb>and with what caution and circumſpection we are to walk, leaſt <lb></lb>we run into them. </s> <s>This that you ſay, and which indeed at the <lb></lb>firſt ſight carrieth with it ſo much of probability, is in the ſtrict<lb></lb>neſſe of it ſo falſe, that whether the Knee, which is the Fulci<lb></lb>ment of the two Leavers, be placed or not placed in the midſt, <lb></lb>it maketh ſuch alteration, that of that Force which would ſuffice <lb></lb>to make the Fracture in the midſt, it being to be made in ſome <lb></lb>other place, it will not ſuffice to apply four times ſo much, nor <lb></lb>ten, nor an hundred, no nor a thouſand. </s> <s>Upon this we will <lb></lb>make ſome general Conſideration, and then we will come to the <lb></lb>Specifick Determination of the Propoſition, according to which, <lb></lb>the Forces for making of Fractures gradually vary more in one <lb></lb>point than in another.</s></p><p type="main"> <s>Let us firſt deſigne this Truncheon A B to be broken in the <lb></lb>midſt upon the Fulciment C, and neer unto that let us deſigne <lb></lb>it again, but under the Characters D E, to be broken on the <lb></lb>Fulciment F, remote from the middle. </s> <s>Firſt it is manifeſt, that <lb></lb>the Diſtances A C and C B being equal, the Force ſhall be ſha<lb></lb>red equally in the ends B and A. Again, according as the Di<lb></lb>ſtance D F groweth leſſe than the Diſtance A C, the Moment <lb></lb>of the Force placed in D groweth leſſe than the Moment in A, <lb></lb>that is placed at the Diſtance C A, and leſſeneth according to <lb></lb>the proportion of the Line D F to A C; and conſequently, it is <lb></lb>requiſite to encreaſe it to equalize or exceed the Reſiſtance of F: <lb></lb>But the Diſtance D F may diminiſh <emph type="italics"></emph>in infinitum,<emph.end type="italics"></emph.end> in relation to <lb></lb>the Diſtance A C: Therefore it is requiſite, that it be poſſible for <lb></lb>the Force to be applyed in D, to encreaſe <emph type="italics"></emph>in infinitum,<emph.end type="italics"></emph.end> that it <lb></lb>may countervail the Reſiſtance in F. But, on the contrary, ac<pb xlink:href="069/01/114.jpg" pagenum="112"></pb>cording as the Diſtance F E encreaſeth above C B, it is requiſite <lb></lb>to diminiſh the Force in E, that it may compenſate the Reſi<lb></lb>ſtance in F: But the Diſtance F E in relation to C B, cannot en<lb></lb>creaſe <emph type="italics"></emph>in infinitum,<emph.end type="italics"></emph.end> by drawing the Fulciment F towards the end <lb></lb>D, no nor yet to the double: Therefore, the Force in E, that it <lb></lb>may compenſate the Reſiſtance in F, ſhall be alwayes more than <lb></lb>half of the Force in B. </s> <s>We may comprehend, therefore, the ne<lb></lb>ceſſity of augmenting the Moments of the Collected Forces in E <lb></lb>and D infinitely to equalize or exceed the Reſiſtance placed in F, <lb></lb>according as the Fulciment F ſhall approach neerer and neerer <lb></lb>to the end D.</s></p><p type="main"> <s><emph type="italics"></emph>S<emph.end type="italics"></emph.end>AGR. </s> <s>What will <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> ſay to this? </s> <s>Muſt he not con<lb></lb>feſſe the Virtue of Geometry to be a more powerful inſtrument <lb></lb>than all others, to ſharpen the Wit, and diſpoſe it to diſcourſe <lb></lb>and ſpeculate well? </s> <s>and that <emph type="italics"></emph>Plato<emph.end type="italics"></emph.end> had great reaſon to deſire that <lb></lb>his Scholars ſhould be well grounded in the Mathematicks? </s> <s>I <lb></lb>have very well underſtood the nature of the Leaver, and how <lb></lb>that its Length encreaſing or decreaſing, the Moment of the <lb></lb>Force and of the Reſiſtance augmented or diminiſhed, and yet in <lb></lb>the determination of the preſent Problem I deceived my ſelf, and <lb></lb>that not a little, but infinitely much.</s></p><p type="main"> <s>SIMP. </s> <s>The truth is, I begin to ſee that Logick, although it <lb></lb>be a moſt appoſite Inſtrument to regulate our Diſcourſe, doth <lb></lb>not attain, as to the prompting of the Mind with Invention, <lb></lb>unto the acuteneſſe of Geometry.</s></p><p type="main"> <s>SAGR. </s> <s>In my conceit, Logick giveth us to underſtand, whe<lb></lb>ther the Diſcourfes and Demonſtrations already made and found <lb></lb>are concluding, but that it teacheth us how to finde concluding <lb></lb>Diſcourſes and Demonſtrations; the truth is, I do not believe: <lb></lb>But it will be better, that <emph type="italics"></emph>Salviatus<emph.end type="italics"></emph.end> ſhew us according to what pro<lb></lb>portion the Moments of the Forces do go increaſing, to overcome <lb></lb>the Reſiſtance of the ſame Piece of Wood, according to the ſe<lb></lb>veral places of the Fracture.</s></p><p type="main"> <s>SALV. </s> <s>The proportion that you ſeek, proceedeth after ſuch <lb></lb>a manner, that</s></p><pb xlink:href="069/01/115.jpg" pagenum="113"></pb><p type="head"> <s>PROPOSITION XII.</s></p><p type="main"> <s><emph type="italics"></emph>If in the length of a Cylinder we ſhall marke two places, <lb></lb>upon which we would make the Fracture of the ſaid <lb></lb>Cylinder, the Reſiſtances of thoſe two places have <lb></lb>the ſame proportion to each other, as have the Re<lb></lb>ctangles made by the Diſtances of thoſe places <lb></lb>reciprocally taken.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let the two Forces (<emph type="italics"></emph>as in<emph.end type="italics"></emph.end> Fig. </s> <s>16.) be A and B the leaſt, to <lb></lb>break in C, and E and F likewiſe the leaſt, to break in D. <lb></lb></s> <s>I ſay the Forces A and B have the ſame proportion to the <lb></lb>Forces E and F, that the Rectangle A D B hath to the Rectan<lb></lb>gle A C B. </s> <s>For the Forces A and B, have to the Forces E and F, a <lb></lb>proportion compounded of the Forces A and B, to the Force <lb></lb>B, of B to F, and of F to E and E: But as the Forces A and <lb></lb>B are to the Force B, ſo is the Length B A to A C; and as the <lb></lb>Force B is to F, ſo is the Line D B to B C; and as the Force F is <lb></lb>to F and E, ſo is the Line D A to A B: Therefore the Forces A <lb></lb>and B have to the Forces E and F a proportion compounded of <lb></lb>theſe three, namely, of B A to A C, of D B to B C, and of D A <lb></lb>A B. </s> <s>But of the two proportions D A to A B, and A B to A C, <lb></lb>is compounded the proportion of D A to A C: Therefore the <lb></lb>Forces A and B have to the Forces E and F, the proportion com<lb></lb>pounded of this D A to A C, and of the other D B to D C. <lb></lb></s> <s>But the Rectangle A D B hath to the Rectangle A C B, a pro<lb></lb>portion compounded of the ſame D A to A C, and of D B to <lb></lb>B C: Therefore the Forces A and B are to the Forces E and F, <lb></lb>as the Rectangle A D B is to the Rectangle A C B; which is as <lb></lb>much as to ſay, the Reſiſtance againſt Fraction in C, hath the <lb></lb>ſame proportion to the Reſiſtance againſt Fraction in D, that <lb></lb>the Rectangle A D B hath to the Rectangle A C B: Which was <lb></lb>to be demonſtrated.</s></p><p type="main"> <s>In conſequence of this Theorem we may reſolve a Problem of <lb></lb>great Curioſity; and it is this:</s></p><pb xlink:href="069/01/116.jpg" pagenum="114"></pb><p type="head"> <s>PROP. XIII. PROBL. IV.</s></p><p type="main"> <s><emph type="italics"></emph>There being given the greateſt Weight that can be ſup<lb></lb>ported at the middle of a Cylinder or Priſme, where <lb></lb>the Reſiſtance is leafl; and there being given a <lb></lb>Weight greater than that, to find in the ſaid Cylin<lb></lb>der, the point at which the given greater Weight may <lb></lb>be ſupporited as the greateſt Weight.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let the given weight greater than the greateſt weight that <lb></lb>can be ſupported at the middle of the Cylinder A B, have <lb></lb>unto the ſaid greateſt weight, the proportion of the line E <lb></lb>to F: it is required to find the point in the Cylinder at which the <lb></lb>ſaid given weight commeth to be ſupported as the biggeſt. </s> <s>Be<lb></lb>tween E and F let G be a Mean-Proportional; and as E is to G, <lb></lb>ſo let A D be to S, S ſhall be leſſer than A D. </s> <s>Let A D be the <lb></lb>Diameter of the Semicircle A H D: in which ſuppoſe A H equal <lb></lb>to S; and joyn together H and D, and take D R equal to it. <lb></lb></s> <s>I ſay that R is the point ſought, at which the given weight, <lb></lb>greater than the greateſt that can be ſupported at the middle of the <lb></lb>Cylinder D, would become as the greateſt weight. </s> <s>On the length <lb></lb><emph type="italics"></emph>B<emph.end type="italics"></emph.end>A deſcribe the Semicircle A N B, and raiſe the Perpendicular <lb></lb>RN, and conjoyn N and <lb></lb>D: And becauſe the two <lb></lb><figure id="id.069.01.116.1.jpg" xlink:href="069/01/116/1.jpg"></figure><lb></lb>Squares N R and R D are <lb></lb>equal to the Square N D; <lb></lb>that is, to the Square A D; <lb></lb>that is, to the two A H and <lb></lb>and H D; and H D is equal <lb></lb>to the Square D R: There<lb></lb>fore the Square N R, that <lb></lb>is, the Rectangle A R B <lb></lb>ſhall be equal to the Square A H; that is, to the Square S: But <lb></lb>the Square S is to the Square A D, as F to E; that is, as the <lb></lb>greateſt ſupportable Weight at D to the given greater Weight: <lb></lb>Therefore this greater ſhall be ſupported at R, as the greateſt <lb></lb>that can be there ſuſtained. </s> <s>Which is that that we ſought.</s></p><p type="main"> <s>SAGR. </s> <s>I underſtand you very well, and am conſidering that <lb></lb>the Priſme A B having alwayes more ſtrength and reſiſtance a<lb></lb>gainſt Preſſion in the parts that more and more recede from the <lb></lb>middle, whether in very great and heavy Beams one may take <pb xlink:href="069/01/117.jpg" pagenum="115"></pb>away a pretty big part towards the end with a notable alleviation <lb></lb>of the weight; which in Beams of great Rooms would be commo<lb></lb>dious, and of no ſmall proſit. </s> <s>And it would be pretty, to find what <lb></lb>Figure that Solid ought to have, that it might have equal Reſi<lb></lb>ſtance in all its parts; ſo as that it were not with more eaſe to be <lb></lb>broken by a weight that ſhould preſſe it in the midſt, than in any <lb></lb>other place.</s></p><p type="main"> <s>SALV. </s> <s>I was juſt about to tell you a thing very notable and <lb></lb>pleaſant to this purpoſe. </s> <s>I will aſſume a brief Scheme for the bet<lb></lb>ter explanation of my meaning. </s> <s>This Figure D B is a Priſm, whoſe <lb></lb>Reſiſtance againſt Fraction in the term A D by a Force preſſing <lb></lb>at the term B, is leſſe than the Reſiſtance that would be found in <lb></lb>the place C I, by how much the length C B is leſſer than B A; as <lb></lb>hath already been demon<lb></lb>ſtrated. </s> <s>Now ſuppoſe the <lb></lb><figure id="id.069.01.117.1.jpg" xlink:href="069/01/117/1.jpg"></figure><lb></lb>ſaid Priſme to be ſawed <lb></lb>Diagonally according to the <lb></lb>Line FB, ſo that the oppo<lb></lb>ſite Surfaces may be two <lb></lb>Triangles, one of which to<lb></lb>wards us is F A B. </s> <s>This So<lb></lb>lid obtains a quality contrary to the Priſme, to wit, that it leſſe re<lb></lb>ſiſteth Fraction by the Force placed in B at the term C than at A, <lb></lb>by as much the Length C <emph type="italics"></emph>B<emph.end type="italics"></emph.end> is leſſe than <emph type="italics"></emph>B<emph.end type="italics"></emph.end> A; Which we will ea <lb></lb>ſily prove: For imagining the Section C N O parallel to the other <lb></lb>A F D, the Line <emph type="italics"></emph>F<emph.end type="italics"></emph.end> A ſhall be to C N in the Triangle F A <emph type="italics"></emph>B<emph.end type="italics"></emph.end> in the <lb></lb>ſame proportion, as the Line A <emph type="italics"></emph>B<emph.end type="italics"></emph.end> is to <emph type="italics"></emph>B<emph.end type="italics"></emph.end> C: and therefore if we <lb></lb>ſuppoſe the Fulciment of the two Leavers to be in the Points A <lb></lb>and C, whoſe Diſtances are <emph type="italics"></emph>B<emph.end type="italics"></emph.end> A, A F, <emph type="italics"></emph>B<emph.end type="italics"></emph.end> C, and C N, theſe, I ſay, <lb></lb>ſhall be like: and therefore that Moment which the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce placed <lb></lb>at <emph type="italics"></emph>B<emph.end type="italics"></emph.end> hath at the Diſtance <emph type="italics"></emph>B<emph.end type="italics"></emph.end> A above the Reſiſtance placed at the <lb></lb>Diſtance A <emph type="italics"></emph>F<emph.end type="italics"></emph.end>, the ſaid <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce at <emph type="italics"></emph>B<emph.end type="italics"></emph.end> ſhall have at the Diſtance <emph type="italics"></emph>B<emph.end type="italics"></emph.end>C <lb></lb>above the ſame Reſiſtance, were it placed at the Diſtance C N: <lb></lb><emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut the Reſiſtance to be overcome at the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>ulciment C, being pla<lb></lb>ced at the Diſtance C N, from the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce in <emph type="italics"></emph>B<emph.end type="italics"></emph.end> is leſſer than the <lb></lb>Reſiſtance in A ſo much as the Rectangle C O is leſſe than the <lb></lb>Rectangle A D; that is, ſo much as the Line C N is leſs than A <emph type="italics"></emph>F<emph.end type="italics"></emph.end>; <lb></lb>that is, C <emph type="italics"></emph>B<emph.end type="italics"></emph.end> than B A: Therefore the Reſiſtance of the part O C B <lb></lb>againſt <emph type="italics"></emph>F<emph.end type="italics"></emph.end>raction in C is ſo much leſs than the Reſiſtance of the <lb></lb>whole D A O againſt <emph type="italics"></emph>F<emph.end type="italics"></emph.end>racture in O, as the Length C B is leſs than <lb></lb>A B. </s> <s>We have therefore from the Beam or Priſme D B, taken <lb></lb>away a part, that is half, cutting it Diagonally, and left the Wedge <lb></lb>or triangular Priſm <emph type="italics"></emph>F<emph.end type="italics"></emph.end> B A; and they are two Solids of contrary <lb></lb>Qualities, namely, that more reſiſts the more it is ſhortned, and this <lb></lb>in ſhortning loſeth its toughneſs as faſt. </s> <s>Now this being granted, <pb xlink:href="069/01/118.jpg" pagenum="116"></pb>it ſeemeth very reaſonable, nay, neceſſary, that one may give it <lb></lb>a cut, by which taking away that which is ſuperfluous, there remai<lb></lb>neth a Solid of ſuch a <emph type="italics"></emph>F<emph.end type="italics"></emph.end>igure, as in all its parts hath equal Reſi<lb></lb>ſtance.</s></p><p type="main"> <s>SIMP. </s> <s>It muſt needs be ſo; for where there is a tranſition from <lb></lb>the greater to the leſſer, one meeteth alſo with the equal.</s></p><p type="main"> <s>SAGR. </s> <s>But the buſineſſe is to find how we are to guide the <lb></lb>Saw for making of this Section.</s></p><p type="main"> <s>SIMP. </s> <s>This ſeemeth to me as if it were a very eaſie buſineſſe; <lb></lb>for if in ſawing the Priſm diagonally, taking away half of it, the <lb></lb>Figure that remains retaineth a contrary quality to that of the <lb></lb>whole Priſm, ſo as that in all places wherein this acquireth ſtrength, <lb></lb>that as faſt loſeth it, me thinks, that keeping the middle way, that <lb></lb>is, taking only the half of that half, which is the fourth part of the <lb></lb>whole, the remaining Figure will not gain or loſe ſtrength in any <lb></lb>of all thoſe places wherein the loſſe and the gain of the other two <lb></lb>Figures were alwaies equal.</s></p><p type="main"> <s>SALV. </s> <s>You have not hit the mark, <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end>; and as I ſhall <lb></lb>ſhew you, it will appear in reality, that that which may be cut off <lb></lb>from the Priſm, and taken away without weakening it is not its <lb></lb>fourth part, but the third. </s> <s>Now it remaineth (which is that that <lb></lb>was hinted by <emph type="italics"></emph>Sagredus<emph.end type="italics"></emph.end>)</s></p><p type="head"> <s>PROP. XIV. PROBL. V.</s></p><p type="main"> <s><emph type="italics"></emph>To find according to what Line the Section is to be <lb></lb>made; Which I will prove to be a Parabolical <lb></lb>Line.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>But firſt it is neceſſary to demonſtrate a certain Lemma, which <lb></lb>is this:</s></p><p type="head"> <s>LEMMA I.</s></p><p type="main"> <s><emph type="italics"></emph>If there ſhall be two Ballances or Leavers divided by their Fulci<lb></lb>ments in ſuch ſort that the two Distances where at the Forces <lb></lb>are to be placed, have to each other double the proportion of <lb></lb>the Diſtances at which the Reſiſtances ſball be, which Reſi<lb></lb>ſtances are to each other as their Diſtances, the ſuſtaining <lb></lb>Powers ſhall be equal.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let A B and C D be two Leavers divided upon their Fulciments <lb></lb>E and F, in ſuch ſort that the Diſtance E B hath to F D a pro<lb></lb>portion double to that which the Diſtance E A hath to F C. </s> <s>I ſay, <pb xlink:href="069/01/119.jpg" pagenum="117"></pb>the Powers that in BD ſhall ſuſtain the Reſiſtances A and C ſhall <lb></lb>be equal to each other. </s> <s>Let E G be ſuppoſed a Mean-Proporti<lb></lb>onal between E B and F D; therefore as B E is to E G, ſo ſhall <lb></lb>G E be to F D, and A E to C <emph type="italics"></emph>F<emph.end type="italics"></emph.end>; and ſo is ſuppoſed the Reſiſtance <lb></lb>of A to the Reſiſtance of C. </s> <s>And becauſe that as E G is to <emph type="italics"></emph>F<emph.end type="italics"></emph.end> D, <lb></lb>ſo is A E to C <emph type="italics"></emph>F<emph.end type="italics"></emph.end>; by Permutation as G E is to E A, ſo ſhall D <emph type="italics"></emph>F<emph.end type="italics"></emph.end><lb></lb>be to <emph type="italics"></emph>F<emph.end type="italics"></emph.end> C: And therefore (in <lb></lb>regard that the two Leavers <lb></lb><figure id="id.069.01.119.1.jpg" xlink:href="069/01/119/1.jpg"></figure><lb></lb>D C and G A are divided pro<lb></lb>portionally in the Points <emph type="italics"></emph>F<emph.end type="italics"></emph.end> and <lb></lb>E) in caſe the Power that being <lb></lb>placed at D compenſates the <lb></lb>Reſiſtance of C were at G, it <lb></lb>would countervail the ſame Reſiſtance of C placed in A: But by <lb></lb>what hath been granted, the Reſiſtance of A hath the ſame propor<lb></lb>tion to the Reſiſtance of C, that AE hath to C <emph type="italics"></emph>F<emph.end type="italics"></emph.end>; that is, B E <lb></lb>hath to E G: Therefore the Power G, or if you will D, placed at <lb></lb>B will ſuſtain the Reſiſtance placed at A: Which was to be de<lb></lb>monſtrated.</s></p><p type="main"> <s>This being underſtood: in the Surface <emph type="italics"></emph>F<emph.end type="italics"></emph.end> B of the Priſme D B, <lb></lb>let the Parabolical Line <emph type="italics"></emph>F<emph.end type="italics"></emph.end> N B be drawn, whoſe Vertex is B, ac<lb></lb>cording to which let the ſaid Priſme be ſuppoſed to be ſawed, the <lb></lb>Solid compriſed between the Baſe A D, the Rectangular Plane <lb></lb>A G, the Bight Line B G, and the Superficies D G B <emph type="italics"></emph>F<emph.end type="italics"></emph.end> being leſt <lb></lb>incurvated according to the Curvity of the Parabolical Line <lb></lb><emph type="italics"></emph>F<emph.end type="italics"></emph.end> N B. </s> <s>I ſay, that <lb></lb>that Solid is through<lb></lb><figure id="id.069.01.119.2.jpg" xlink:href="069/01/119/2.jpg"></figure><lb></lb>out of equal Reſi<lb></lb>ſtance. </s> <s>Let it be cut <lb></lb>by the Plane C O pa<lb></lb>rallel to A D; and <lb></lb>imagine two Leavers <lb></lb>divided and ſuppor<lb></lb>ted upon the Fulciments A and C; and let the Diſtances of one <lb></lb>be B A and A F, and of the other B C, and C N. </s> <s>And becauſe in <lb></lb>the Parabola <emph type="italics"></emph>F B<emph.end type="italics"></emph.end> A, A <emph type="italics"></emph>B<emph.end type="italics"></emph.end> is to <emph type="italics"></emph>B<emph.end type="italics"></emph.end> C, as the Square of <emph type="italics"></emph>F<emph.end type="italics"></emph.end> A to the <lb></lb>Square of C N, it is manifeſt, that the Diſtance <emph type="italics"></emph>B<emph.end type="italics"></emph.end> A of one Leaver, <lb></lb>hath to the Diſtance <emph type="italics"></emph>B<emph.end type="italics"></emph.end> C of the other a proportion double to that <lb></lb>which the other Diſtance A <emph type="italics"></emph>F<emph.end type="italics"></emph.end> hath to the other C N, And be<lb></lb>cauſe the Reſiſtance that is to be equal by help of the Leaver <lb></lb><emph type="italics"></emph>B<emph.end type="italics"></emph.end> A hath the ſame proportion to the Reſiſtance that is to be <lb></lb>equal by help of the Leaver <emph type="italics"></emph>B<emph.end type="italics"></emph.end> C, that the Rectangle D A hath to <lb></lb>the Rectangle O C; which is the ſame that the Line A <emph type="italics"></emph>F<emph.end type="italics"></emph.end> hath to <lb></lb>N C, which are the other two Diſtances of the Leavers; it is ma<lb></lb>nifeſt by the fore going Lemma, that the ſame Force that being <pb xlink:href="069/01/120.jpg" pagenum="118"></pb>applyed to the Line <emph type="italics"></emph>B<emph.end type="italics"></emph.end> G will equal the Reſiſtance D A, will like<lb></lb>wiſe equal the Reſiſtance C O. </s> <s>And the ſame may be demonſtra<lb></lb>ted, if one cut the Solid in any other place: therefore that Parabo<lb></lb>lical Solid is throughout of equal Reſiſtance. </s> <s>In the next place, <lb></lb>that cutting the Priſme according to the Parabolical Line F N B, <lb></lb>the third part of it is taken away, appeareth, For that the Semi<lb></lb>Parabola F N <emph type="italics"></emph>B<emph.end type="italics"></emph.end> A and the Rectangle F <emph type="italics"></emph>B<emph.end type="italics"></emph.end> are Baſes of two Solids <lb></lb>contained between two parallel Planes, that is, between the Rect<lb></lb>angles F B and D G, whereby they retain the ſame Proportion, as <lb></lb>thoſe their Baſes: But the Rectangle F <emph type="italics"></emph>B<emph.end type="italics"></emph.end> is Seſquialter to the Se<lb></lb>miparabola F N <emph type="italics"></emph>B<emph.end type="italics"></emph.end> A: Therefore cutting the Priſine according to <lb></lb>the Parabolick Line, we take away the third part of it. </s> <s>Hence we <lb></lb>ſee, that <emph type="italics"></emph>B<emph.end type="italics"></emph.end>eams may be made with the diminution of their Weight <lb></lb>more than thirty three in the hundred, without diminiſhing their <lb></lb>Strength in the leaſt; which in great Ships, in particular, for bea<lb></lb>ring the Decks may be of no ſmall benefit; for that in ſuch kind <lb></lb>of Fabricks Lightneſſe is of infinite importance.</s></p><p type="main"> <s>SAGR. </s> <s>The Commodities are ſo many, that it would be tedi<lb></lb>ous, if not impoſſible, to mention them all. <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut I, laying aſide <lb></lb>theſe, would more gladly underſtand that the alleviation is made <lb></lb>according to the aſſigned proportions. </s> <s>That the Section, according <lb></lb>to the Diagonal Line, cuts away half of the weight I very well <lb></lb>know: but that the other Section according to the Parabolical Line <lb></lb>takes away the third part of the Priſme I can believe upon the <lb></lb>word of <emph type="italics"></emph>Salviatus,<emph.end type="italics"></emph.end> who evermore ſpeaks the truth, but in this <lb></lb>Caſe Science would better pleaſe me than Faith.</s></p><p type="main"> <s>SALV. </s> <s>I ſee then that you would have the Demonſtration, <lb></lb>whether or no it be true, that the exceſſe of the Priſme over and <lb></lb>above this, which for this time we will call a Parabolical Solid, is <lb></lb>the third part of the whole Priſme. </s> <s>I am certain that I have for<lb></lb>merly demoſtrated it; I will try now whether I can put the <lb></lb>Demonſtration together again: to which purpoſe I do remember <lb></lb>that I made uſe of a Certain Lemma of <emph type="italics"></emph>Archimedes,<emph.end type="italics"></emph.end> inſerted by <lb></lb>him in his <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ook <emph type="italics"></emph>de Spiralibus,<emph.end type="italics"></emph.end> and it is this:</s></p><p type="head"> <s>LEMMA II.</s></p><p type="main"> <s><emph type="italics"></emph>If any number of Lines at pleaſure ſhall exceed one another equal<lb></lb>ly, and the exceſſes be equal to the leaſt of them, and there be as <lb></lb>many more, each of them equal to the greateſt; the Squares of all <lb></lb>theſe ſhall be leſſe than the triple of the Squares of thoſe that <lb></lb>exceed one another: but they ſhall be more than triple to thoſe <lb></lb>others that remain, the Square of the greateſt being ſub<lb></lb>ſtracted.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/121.jpg" pagenum="119"></pb><p type="main"> <s>This being granted: Let the Parabolick Line A <emph type="italics"></emph>B<emph.end type="italics"></emph.end> be inſcribed <lb></lb>in this Rectangle A C <emph type="italics"></emph>B<emph.end type="italics"></emph.end> P: we are to prove the Mixt Triangle <lb></lb><emph type="italics"></emph>B<emph.end type="italics"></emph.end> A P, whoſe ſides are <emph type="italics"></emph>B<emph.end type="italics"></emph.end> P and P A, and <emph type="italics"></emph>B<emph.end type="italics"></emph.end>aſe the Parabolical Line <lb></lb><emph type="italics"></emph>B<emph.end type="italics"></emph.end> A, to be the third part of the whole Rectangle C P. </s> <s>For if it be <lb></lb>not ſo, it will be either more than the third part, or leſſe. </s> <s>Let it be <lb></lb>ſuppoſed that it may be <lb></lb>leſſe, and to that which is <lb></lb><figure id="id.069.01.121.1.jpg" xlink:href="069/01/121/1.jpg"></figure><lb></lb>wanting ſuppoſe the Space <lb></lb>X to be equal. </s> <s>Then di<lb></lb>viding the Rectangle con<lb></lb>tinually into equal parts <lb></lb>with Lines parallel to the <lb></lb>Sides <emph type="italics"></emph>B<emph.end type="italics"></emph.end> P and C A, we <lb></lb>ſhall in the end arrive at <lb></lb>ſuch parts, as that one of them ſhall be leſſe than the Space X. <lb></lb></s> <s>Now let one of them be the Rectangle O <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> and by the Points <lb></lb>where the other Parallels interſect the Parabolick Line, let the Pa<lb></lb>rallels to A P paſſe: and here I will ſuppoſe a Figure to be cir<lb></lb>cumſcribed about our Mixt-Triangle, compoſed of Rectangles, <lb></lb>which are <emph type="italics"></emph>B<emph.end type="italics"></emph.end> O, I N, H M, F L, E K, G A; which Figure ſhall alſo <lb></lb>yet be leſs than the third part of the Rectangle C P, in regard that <lb></lb>the exceſſe of that Figure over and above the Mixed Triangle is <lb></lb>much leſſe than the Rectangle <emph type="italics"></emph>B<emph.end type="italics"></emph.end> O, which yet again is leſſe than <lb></lb>the Space X.</s></p><p type="main"> <s>SAGR. Softly, I pray you, for I do not ſee how the exceſſe of <lb></lb>this circumſcribed Figure above the Mixt Triangle is conſiderably <lb></lb>leſſer than the Rectangle <emph type="italics"></emph>B<emph.end type="italics"></emph.end> O.</s></p><p type="main"> <s>SALV. </s> <s>Is not the Rectangle <emph type="italics"></emph>B<emph.end type="italics"></emph.end> O equal to all theſe ſmall Rect<lb></lb>angles by which our Parabolical Line paſſeth; I mean theſe, <emph type="italics"></emph>B<emph.end type="italics"></emph.end> I, <lb></lb>I H, H F, F E, E G, and G A, of which one part only lyeth with<lb></lb>out the Mixt Triangle? </s> <s>And the Rectangle <emph type="italics"></emph>B<emph.end type="italics"></emph.end> O, is it not alſo ſup<lb></lb>poſed to be leſſe than the Space X? </s> <s>Therefore if the Triangle to<lb></lb>gether with X did, as the Adverſary ſuppoſeth, equalize the third <lb></lb>part of the Rectangle C P the circumſcribed Figure that adjoyns <lb></lb>to the Triangle ſo much leſſe than the Space X, will remain even <lb></lb>yet leſſe than the third part of the ſaid Rectangle C P. <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut this <lb></lb>cannot be, for it is more than a third part, therefore it is not true <lb></lb>that our Mixt Triangle is leſſe than one third of the Rectangle.</s></p><p type="main"> <s>SAGR. </s> <s>I underſtand the Solution of my Doubt. <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut it is <lb></lb>requiſite now to prove unto us, that the Circumſcribed Figure is <lb></lb>more than a third part of the Rectangle C P; which, I believe, will <lb></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></lb><arrow.to.target n="marg1089"></arrow.to.target><lb></lb>D E hath the ſame proportion to the Square of Z G, that the Line <pb xlink:href="069/01/122.jpg" pagenum="120"></pb>D A hath to A Z; which is the ſame that the Rectangle K E hath to <lb></lb>the Rectangle A G, their heights A K and K L being equal. </s> <s>There<lb></lb>fore the proportion that the Square E D hath to the Square Z G; <lb></lb>that is, the Square L A hath to the Square A K, the Rectangle K E <lb></lb>hath likewiſe to the Rectangle K Z. </s> <s>And in the ſelf-ſame manner <lb></lb>we might prove that the other Rectangles L F, M H, N I, O B are <lb></lb>to one another as the Squares of the Lines M A, N A, O A, P A. <lb></lb></s> <s>Conſider we in the next place, how the Circumſcribed Figure is <lb></lb>compounded of certain Spaces that are to one another as the <lb></lb>Squares of the Lines that exceed with Exceſſes equal to the leaſt, <lb></lb>and how the Rectangle C P is compounded of ſo many other Spa<lb></lb>ces each of them equal to the Greateſt, which are all the Rectan<lb></lb>gles equal to O B. Therefore, by the Lemma of <emph type="italics"></emph>Archimedes,<emph.end type="italics"></emph.end> the <lb></lb>Circumſcribed Figure is more than the third part of the Rectangle <lb></lb>C P: But it was alſo leſſe, which is impoſſible: Therefore the <lb></lb>Mixt-Triangle is not leſſe than one third of the Rectangle C P. <lb></lb></s> <s>I ſay likewiſe, that it is not more: For if it be more than one <lb></lb>third of the Rectangle C P, ſuppoſe the Space X equal to the ex<lb></lb>ceſſe of the Triangle above the third part of the ſaid Rectangle <lb></lb>C P, and the diviſion and ſubdiviſion of the Rectangle into Rect<lb></lb>angolets, but alwaies equal, being made, we ſhall meet with ſuch as <lb></lb>that one of them is leſſer than the Space X; which let be done: <lb></lb>and let the Rectangle <emph type="italics"></emph>B<emph.end type="italics"></emph.end> O be leſſer than X; and, having deſcribed <lb></lb>the Figure as before, we ſhall have inſcribed in the Mixt-Triangle <lb></lb>a Figure compounded of the Rectangles V O, T N, S M, N L, Q K, <lb></lb>which yet ſhall not be leſs <lb></lb><figure id="id.069.01.122.1.jpg" xlink:href="069/01/122/1.jpg"></figure><lb></lb>than the third part of the <lb></lb>great Rectangle C P, for <lb></lb>the Mixt Triangle doth <lb></lb>much leſſe exceed the In<lb></lb>ſcribed Figure than it doth <lb></lb>exceed the third part of <lb></lb>the Rectangle C P; Be<lb></lb>cauſe the exceſſe of the <lb></lb>Triangle above the third part of the Rectangle C P is equal to <lb></lb>the Space X which is greater than the Rectangle <emph type="italics"></emph>B<emph.end type="italics"></emph.end> O, and this al<lb></lb>ſo is conſiderably greater than the exceſſe of the Triangle above <lb></lb>the Inſcribed Figure: For to the Rectangle <emph type="italics"></emph>B<emph.end type="italics"></emph.end> O, all the Rectan<lb></lb>golets A G, G E, E <emph type="italics"></emph>F,<emph.end type="italics"></emph.end> F H, H I, I <emph type="italics"></emph>B<emph.end type="italics"></emph.end> are equal, of which the Ex<lb></lb>ceſſes of the Triangle above the Inſcribed <emph type="italics"></emph>F<emph.end type="italics"></emph.end>igure are leſſe than <lb></lb>half: And therefore the Triangle exceeding the third part of the <lb></lb>Rectangle C P, by much more (exceeding it by the Space X) <lb></lb>than it exceedeth its inſcribed <emph type="italics"></emph>F<emph.end type="italics"></emph.end>igure, that ſame <emph type="italics"></emph>F<emph.end type="italics"></emph.end>igure ſhall alſo <lb></lb>be greater than the third part of the Rectangle C P: <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut it is leſſer, <lb></lb>by the Lemma preſuppoſed: <emph type="italics"></emph>F<emph.end type="italics"></emph.end>or that the Rectangle C P, as being <pb xlink:href="069/01/123.jpg" pagenum="127"></pb>the Aggregate of all the biggeſt Rectangles, hath the ſame pro<lb></lb>portion to the Rectangles compounding the Inſcribed <emph type="italics"></emph>F<emph.end type="italics"></emph.end>igure, that <lb></lb>the Aggregate of of all the Squares of the Lines equal to the big<lb></lb>geſt, hath to the Squares of the Lines that exceed equally, ſubſtra<lb></lb>cting the Square of the biggeſt: And therefore (as it hapneth in <lb></lb>Squares) the whole Aggregate of the biggeſt (that is the Rectan<lb></lb>gle C P) is more than triple the Aggregate of the exceeding <lb></lb>ones, the biggeſt deducted, that compound the Inſcribed <emph type="italics"></emph>F<emph.end type="italics"></emph.end>i<lb></lb>gure. </s> <s>Therefore the Mixt-Triangle is neither greater nor leſſer <lb></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"></emph>The Quadrature of <lb></lb>the Parabola ſhewn <lb></lb>by one ſingle De<lb></lb>monſtration.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>A pretty and ingenuous Demonſtration: and ſo much <lb></lb>the more, in that it giveth us the Quadrature of the Parabola, ſhew<lb></lb>ing it to be <emph type="italics"></emph>Seſquitertial<emph.end type="italics"></emph.end> of the Triangle inſcribed in the ſame; <lb></lb>proving that which <emph type="italics"></emph>Archimedes<emph.end type="italics"></emph.end> demonſtrateth by two very diffe<lb></lb>rent, but both very admirable, methods of a great number of Pro<lb></lb>poſitions. </s> <s>As hath likewiſe been demonſtrated lately by <emph type="italics"></emph>Lucas <lb></lb>Valerius,<emph.end type="italics"></emph.end> another ſecond <emph type="italics"></emph>Archimedes<emph.end type="italics"></emph.end> of our Age, which Demon<lb></lb>ſtration is ſet down in the Book that he writ of the Center of the <lb></lb>Gravity of Solids.</s></p><p type="main"> <s><emph type="italics"></emph>S<emph.end type="italics"></emph.end>ALV. </s> <s>A Treatiſe which verily is not to come behind any one <lb></lb>that hath been written by the moſt <emph type="italics"></emph>F<emph.end type="italics"></emph.end>amous Geometricians of the <lb></lb>preſent and all paſt Ages: which when it was read by our <emph type="italics"></emph>Acade<lb></lb>mick,<emph.end type="italics"></emph.end> it made him deſiſt from proſecuting his Diſcoveries that he <lb></lb>was then proceeding to write on the ſame Subject: in regard he <lb></lb>ſaw the whole buſineſs ſo happily found and demonſtrated by the <lb></lb>ſaid <emph type="italics"></emph>Valerius.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>I was informed of all theſe things by our <emph type="italics"></emph>Academick<emph.end type="italics"></emph.end>; <lb></lb>and have beſought him withall that he would one day let me ſee <lb></lb>his Demonſtrations that he had ſound at the time when he met <lb></lb>with the <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ook of <emph type="italics"></emph>Valerius:<emph.end type="italics"></emph.end> but I never was ſo happy as to ſee them.</s></p><p type="main"> <s>SALV. </s> <s>I have a Copy of them, and will impart them to you, <lb></lb>for you will be much pleaſed to ſee the variety of Methods, which <lb></lb>theſe two Authors take to inveſtigate the ſame Concluſions, and <lb></lb>their Demonſtrations: wherein alſo ſome of the Concluſions have <lb></lb>different Explanations, howbeit in effect equally true.</s></p><p type="main"> <s>SAGR. </s> <s>I ſhall be very glad to ſee them, therefore when you re<lb></lb>turn to our wonted Conferences you may do me the favour to <lb></lb>bring them with you. <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut in the mean time, this ſame of the Re<lb></lb>fiſtance of the Solid taken from the Priſme by a Parabolick Secti<lb></lb>on, being an Operation no leſſe ingenuous than beneficial in many <lb></lb>Mechanical Works, it would be good that Artificers had ſome ea<lb></lb>ſie and expedite Rule how they may draw the ſaid Parabolick <lb></lb>Line upon the Plane of the Priſme.</s></p><p type="main"> <s>SALV. </s> <s>There are ſeveral waies to draw thoſe Lines, but two <lb></lb><arrow.to.target n="marg1090"></arrow.to.target><lb></lb>that are more expedite than all the reſt, I will deſcribe unto you. <pb xlink:href="069/01/124.jpg" pagenum="122"></pb>One of which is truly admirable, ſince that thereby, in leſſe time <lb></lb>than another can with Compaſſes ſlightly draw upon a paper <lb></lb>four or ſix Circles of different ſizes, I can deſign thirty or forty <lb></lb>Parabolick Lines no leſſe exact, ſmall, and ſmooth than the Cir<lb></lb>cumferences of thoſe Circles. </s> <s>I have a <emph type="italics"></emph>B<emph.end type="italics"></emph.end>all of <emph type="italics"></emph>B<emph.end type="italics"></emph.end>raſſe exquiſitely <lb></lb>round, no bigger than a Nut, this thrown upon a Steel Mirrour <lb></lb>held, not erect to the Horizon, but ſomewhat inclined, ſo that the <lb></lb><emph type="italics"></emph>B<emph.end type="italics"></emph.end>all in its motion may run along preſſing lightly upon it, leaveth <lb></lb>a Parabolical Line finely and ſmoothly deſcribed, and wider or <lb></lb>narrower according as the Projection ſhall be more or leſs elevated. <lb></lb></s> <s>Whereby alſo we have a clear and ſenſible Experiment that the <lb></lb>Motion of Projects is made by Parabolick Lines: an Effect obſer<lb></lb>ved by none before our <emph type="italics"></emph>Academick,<emph.end type="italics"></emph.end> who alſo layeth down the <lb></lb>Demonſtration of it in his <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ook of Motion, which we will joynt<lb></lb>ly peruſe at our next meeting. </s> <s>Now the <emph type="italics"></emph>B<emph.end type="italics"></emph.end>all, that it may deſcribe <lb></lb>by its motion thoſe Parabola's, muſt be rouled a little in the hands <lb></lb>that it may be warmed, and ſomewhat moyſtned, for by this <lb></lb>means it will leave its track more apparent upon the Mirrour. </s> <s>The <lb></lb>other way to draw the Line that we deſire upon the Priſme is after <lb></lb>this manner. </s> <s>Let two Nailes be faſtned on high in a Wall, at an <lb></lb>equal diſtance from the Horizon, and remote from one another <lb></lb>twice the breadth of the Rectangle upon which we would trace the <lb></lb>Semiparabola, and to theſe two Nails tye a ſmall thread of ſuch a <lb></lb>length that its doubling may reach as far as the length of the <lb></lb>Priſme; this ſtring will hang in a Parabolick <emph type="italics"></emph>F<emph.end type="italics"></emph.end>igure: ſo that tra<lb></lb>cing out upon the Wall the way that the ſaid String maketh on it, <lb></lb>we ſhall have a whole Parabola deſcribed: which a Perpendicular <lb></lb>that hangeth in the midſt between theſe two Nailes will divide <lb></lb>into two equal parts. </s> <s>And for the transferring or ſetting off of <lb></lb>that Line afterwards upon the oppoſite Surfaces of the Priſme it is <lb></lb>not difficult at all, ſo that every indifferent Artiſt will know how <lb></lb>to do it. </s> <s>The ſame Line might be drawn upon the ſaid Sur<lb></lb>face of the Priſme by help of the Geometrical Lines delineated up<lb></lb>on the <emph type="italics"></emph>Compaſſe<emph.end type="italics"></emph.end> of our <emph type="italics"></emph>Friend,<emph.end type="italics"></emph.end> without any more ado.</s></p><p type="margin"> <s><margin.target id="marg1090"></margin.target><emph type="italics"></emph>Several waies to <lb></lb>deſcribe a Para<lb></lb>bola.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>We have hitherto demonſtrated ſo many Concluſions touching <lb></lb>the Contemplation of theſe Reſiſtances of Solids againſt Fraction <lb></lb>by having firſt opened the way unto the Science with ſuppoſing the <lb></lb>direct Reſiſtance for known, that we may in purſuance of them <lb></lb>proceed forwards to the finding of other, and other Concluſions, <lb></lb>with their Demonſtrations of thoſe which in Nature are infinite. <lb></lb></s> <s>Only at preſent, for a final concluſion of this daies Conferences, <lb></lb>I will add the Speculation of the Reſiſtances of the Hollow Solids <lb></lb>which Art, and chiefly Nature, uſeth in an hundred Operations, <lb></lb>when without encreaſing the weight ſhe greatly augmenteth the <lb></lb>ſtrength: as is ſeen in the Bones of Birds, and in many Canes that <pb xlink:href="069/01/125.jpg" pagenum="123"></pb>are light and of great Reſiſtance againſt bending and breaking. <lb></lb></s> <s>For if a Wheat Straw that ſupports an Ear that is heavier than the <lb></lb>whole Stalk, were made of the ſame quantity of matter but were <lb></lb>maſſie or ſolid, it would be much leſſe repugnant to Fraction or <lb></lb>Flection. </s> <s>And with the ſame Reaſon Art hath obſerved, and Ex<lb></lb>perience confirmed, that an hollow Cane, or a Trunk of Wood <lb></lb>or Metal, is much more firm and tough than if being of the ſame <lb></lb>weight and length it were ſolid, which conſequently would be <lb></lb>more flender, and therefore Art hath contrived to make Lances hol<lb></lb>low within when they are deſired to be ſtrong and light. </s> <s>We will <lb></lb>ſhew therefore, that</s></p><p type="head"> <s>PROPOSITION XV.</s></p><p type="main"> <s><emph type="italics"></emph>The Reſiſtances of two Cylinders, equall, and equally <lb></lb>long, one of which is Hollow, and the other Maſsie, <lb></lb>have to each other the ſame proportion, as their Dia<lb></lb>meters.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let the Cane or Hollow Cylinder be A E, [<emph type="italics"></emph>as in<emph.end type="italics"></emph.end> Fig. </s> <s>17.] <lb></lb>and the Cylinder I N Maſſie, and equall in weight and length. <lb></lb></s> <s>I ſay, the Reſiſtance of the Cane A E hath the ſame propor<lb></lb>tion to the Reſiſtance of the ſolid Cylinder, as the Diameter <lb></lb>A B hath to the Diameter I L. </s> <s>Which is very manifeſt; For the <lb></lb>Cane and the Cylinder I N being equal, and of equal lengths, the <lb></lb>Circle I L that is Baſe of the Cylinder ſhall be equal to the Ring <lb></lb>A B that is Baſe of the Cane A E, (I call the Superficies that re<lb></lb>maineth when a leſſer Circle is taken out of a greater that is Con<lb></lb>centrick with it a Ring:) and therefore their Abſolute Reſiſtan<lb></lb>ces ſhall be equal: but becauſe in breaking croſſe-waies we make <lb></lb>uſe in the Cylinder I N of the length L N for a Leaver, and of the <lb></lb>point L for a Fulciment, and of the Semidiameter or Diameter L I <lb></lb>for a Counter-Leaver; and in the Cane the part of the Leaver, <lb></lb>that is the Line B E is equal to L N; but the Counter-Leaver at <lb></lb>the Fulciment B is the Diameter or Semidiameter A B: It is mani<lb></lb>feſt therefore that the Reſiſtance of the Cane exceedeth that of <lb></lb>the Solid Cylinder as much as the Diameter A B exceeds the Dia<lb></lb>meter I L; Which is that that we ſought. </s> <s>Toughneſs therefore is ac<lb></lb>quired in the hollow Cane above the Toughneſs of the ſolid Cylin<lb></lb>der according to the proportion of the Diameters: provided al<lb></lb>waies that they be both of the ſame matter, weight, and length.</s></p><p type="main"> <s>It would be well, that in conſequence of this we try to inveſtigate <lb></lb>that which hapneth in other Caſes indifferently between all Canes <lb></lb>and ſolid Cylinders of equal length, although unequal in quantity <lb></lb>of weight, and more or leſs evacuated. </s> <s>And firſt we will demon<lb></lb>ſtrate, that</s></p><pb xlink:href="069/01/126.jpg"></pb><figure id="id.069.01.126.1.jpg" xlink:href="069/01/126/1.jpg"></figure><p type="caption"> <s><emph type="italics"></emph>Place this at the end of the ſecond Dialogue pag:<emph.end type="italics"></emph.end> 124,</s></p><pb xlink:href="069/01/127.jpg" pagenum="124"></pb><p type="head"> <s>PROP. XVI. PROBL. VI.</s></p><p type="main"> <s><emph type="italics"></emph>A Trunk or Hollow Cane being given, a Solid Cylinder <lb></lb>may be found equal to it.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>This Operation is very eaſie. </s> <s>For let the Line A B, be the Dia<lb></lb>meter of the Cane, and C D the Diameter of the Hollow or <lb></lb>Cavity. </s> <s>Let the Line A E be ſet off upon the greater Circle <lb></lb>equal to the Diameter C D, and conjoyn E B. </s> <s>And becauſe in <lb></lb><figure id="id.069.01.127.1.jpg" xlink:href="069/01/127/1.jpg"></figure><lb></lb>the Semicircle A E B the Angle E is Right<lb></lb>Angle, the Circle whoſe Diameter is A B <lb></lb>ſhall be equall to the two Circles of the Di<lb></lb>ameters A E and E B: But A E is the Dia<lb></lb>meter of the Hollow of the Cane: Therefore <lb></lb>the Circle whoſe Diameter is E B, ſhall be <lb></lb>equal to the Ring A C B D: And therefore <lb></lb>the ſolid Cylinder, the Circle of whoſe Baſe <lb></lb>hath the Diameter E B ſhall be equal to the <lb></lb>Cane, they being of the ſame length. </s> <s>This demonſtrated, we may <lb></lb>preſently be able</s></p><p type="head"> <s>PROP. XVII. PROBL. VII.</s></p><p type="main"> <s><emph type="italics"></emph>To find what proportion is betwixt the Reſiſtances of <lb></lb>any whatſoever Cane and Cylinder, their lengths be<lb></lb>ing equal.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>LET the Cane A B E, and the Cylinder R S M, be of equal <lb></lb>length: it is required to find what proportion the Reſiſtances <lb></lb>have to each other. </s> <s>By the precedent let the Cylinder I L N <lb></lb>be found equal to the Cane, and of the ſame length; and to the <lb></lb>Lines I L and R S (Diameters of the Baſes of the Cylinders I N and <lb></lb><figure id="id.069.01.127.2.jpg" xlink:href="069/01/127/2.jpg"></figure><lb></lb>R M) let the Line V be a fourth <lb></lb>Proportional. </s> <s>I ſay, the Reſiſtance <lb></lb>of the Cane A E is to the Reſi<lb></lb>ſtance of the Cylinder R M, as the <lb></lb>Line A B is to V. </s> <s>For the Cane <lb></lb>A E being equal to, and of the <lb></lb>ſame length with the Cylinder <lb></lb>I N, the Reſiſtance of the Cane <lb></lb>ſhall be to the Reſiſtance of the <lb></lb>Cylinder, as the Line A B is to I L: <lb></lb>But the Reſiſtance of the Cylinder I N is to the Reſiſtance of the <lb></lb>Cylinder R M, as the Cube I L is to the Cube R S; that is, as the <lb></lb>Line I L to V: Therefore, <emph type="italics"></emph>ex æquali,<emph.end type="italics"></emph.end> the Reſiſtance of the Cane <lb></lb>A E hath the ſame proportion to the Reſiſtance of the Cylinder <lb></lb>R M, that the Line A B hath to V: Which is that that was ſought.</s></p><p type="head"> <s><emph type="italics"></emph>The End of the Second Dialogue.<emph.end type="italics"></emph.end></s></p></chap><chap><pb xlink:href="069/01/128.jpg" pagenum="125"></pb><p type="head"> <s>GALILEUS, <lb></lb>HIS <lb></lb>DIALOGUES <lb></lb>OF <lb></lb>MOTION.</s></p><p type="head"> <s>The Third Dialogue.</s></p><p type="head"> <s><emph type="italics"></emph>INTERLOCUTORS,<emph.end type="italics"></emph.end></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"></emph>We promote a very new Science, but of a very <lb></lb>old Subject. </s> <s>There is nothing in Nature more <lb></lb>antient than<emph.end type="italics"></emph.end> MOTION, <emph type="italics"></emph>of which <lb></lb>many and great Volumns have been written <lb></lb>by Philoſophers: But yet there are ſundry <lb></lb>Symptomes and Properties in it worthy of <lb></lb>our Notice, which I find not to have been hi<lb></lb>therto obſerved, much leſſe demonſtrated by <lb></lb>any. </s> <s>Some ſlight particulars have been no<lb></lb>ted: as that the Natural Motion of Grave Bodies continually accelle-<emph.end type="italics"></emph.end><pb xlink:href="069/01/129.jpg" pagenum="126"></pb><emph type="italics"></emph>rateth, as they deſcend towards their Center: but it hath not been as yet <lb></lb>declared in what proportion that Acceleration is made. </s> <s>For no man, <lb></lb>that I know, hath ever demonſtrated, That there is the ſame proportion <lb></lb>between the Spaces, thorow which a thing moveth in equal Times, as <lb></lb>there is between the Odde Numbers which follow in order after a Vnite. <lb></lb></s> <s>It hath been obſerved that Projects [or things thrown or darted with vi<lb></lb>olence] make a Line that is ſomewhat curved; but that this line is a Pa<lb></lb>rabola, none have hinted: Yet theſe, and ſundry other things, no <lb></lb>leſſe worthy of our knowledg, will I here demonſtrate: And which <lb></lb>is more, I will open a way to a moſt ample and excellent Science, <lb></lb>of which theſe our Labours ſhall be the Elements: into which more <lb></lb>ſubtil and piercing Wits than mine will be better able to dive.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>We divide this Treatiſe into three parts. </s> <s>In the firſt part we conſider <lb></lb>ſuch things as reſpect Equable or Vniforme Motion. </s> <s>In the ſecond we <lb></lb>write of Motion naturally accelerate. </s> <s>In the third we treat of Violent <lb></lb>Motion, or<emph.end type="italics"></emph.end> De Projectis.</s></p><p type="head"> <s>OF EQVABLE MOTION.</s></p><p type="main"> <s><emph type="italics"></emph>Concerning Equable or Vniform Motion we have need of onely one <lb></lb>Definition, which I thus deliver.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>DEFINITION.</s></p><p type="main"> <s>By an Equable or Uniform Motion, I underſtand that by which a <lb></lb>Moveable in all equal Times paſſeth thorow equal Spaces.</s></p><p type="head"> <s>ADVERTISEMENT.</s></p><p type="main"> <s><emph type="italics"></emph>I thought good to add to the old Definition (which ſimply termeth <lb></lb>that an Equable Motion, whereby equal Spaces are paſt in equal <lb></lb>Times) this Particle<emph.end type="italics"></emph.end> All, <emph type="italics"></emph>that is, any whatſoever Times that are equal: <lb></lb>for it may happen, that a Moveable may paſſe thorow equal Spaces in cer<lb></lb>tain equal Times, though the Spaces be not equal which it hath gone in <lb></lb>leſſer, though equal parts of the ſame Time. </s> <s>From this our Definition <lb></lb>follow theſe four Axiomes:<emph.end type="italics"></emph.end> ſcilicet,</s></p><p type="head"> <s>AXIOMEL</s></p><p type="main"> <s>In the ſame Equable Motion that Space is greater which is paſſed <lb></lb>in a longer Time, and that leſſer which is paſt in a ſhorter.</s></p><pb xlink:href="069/01/130.jpg" pagenum="127"></pb><p type="head"> <s>AXIOME II.</s></p><p type="main"> <s>In the ſame Equable Motion, the greater the Space is that hath <lb></lb>been gone thorow, the longer was the Time in which the Move<lb></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 paſſeth in any Time, is great<lb></lb>er than the Space which a leſſer Velocity paſſeth in the ſame <lb></lb>Time.</s></p><p type="head"> <s>AXIOME IV.</s></p><p type="main"> <s>The Velocity which paſſeth a greater Space, is greater than the <lb></lb>Velocity which paſſeth a leſſer Space in the ſame 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></lb>and with the ſame Velocity paſſe two ſeveral <lb></lb>Spaces, the Times of the Motion ſhall be to <lb></lb>one another as the ſaid Spaces.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Moveable by an Equable Motion with the ſame Velocity paß <lb></lb>the two Spaces A B and B C: and let D E be the Time of the Moti<lb></lb>on thorow A B; and let the Time of the Motion thorow B C be E F <lb></lb>I ſay that the Time D E to the Time E F, is as the Space A B to the <lb></lb>Space B C. </s> <s>Protract the Spaces and Times on both ſides, towards <lb></lb>G H and I K, and in A G take any number of Spaces equal to A B,<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.130.1.jpg" xlink:href="069/01/130/1.jpg"></figure><lb></lb><emph type="italics"></emph>and in D I the like number of Times equal to D E. Again, in C H take <lb></lb>any number of Spaces equal to B C, and in F K take the ſame number <lb></lb>of Times equal to the Time E F. </s> <s>This done, the Space B G will con<lb></lb>tain juſt as many Spaces equal to B A, as the Time E I containeth <lb></lb>Times equal to E D, equimultiplied according to what ever Rate; And <lb></lb>likewiſe the Space B H will contain as many Spaces equal to B C, as<emph.end type="italics"></emph.end><pb xlink:href="069/01/131.jpg" pagenum="128"></pb><emph type="italics"></emph>the Time K E containeth Times equal to F E, at what ever rate equi<lb></lb>multiplied. </s> <s>And foraſmuch as D E is the Time of the Motion thorow <lb></lb>A B, the whole Time E I, ſhall be the Time of the whole Space of the <lb></lb>Motion thorow B G, by reaſon that the Motion is Equable, and that the <lb></lb>number of the Times in E I equal to D E, is the ſame with the number <lb></lb>of Spaces in B G, equal to B A: For the ſame reaſon E K is the Time <lb></lb>of the Motion thorow H B. </s> <s>Now in regard the Motion is Equable, if the <lb></lb>Space G B were equal to H B, the Time I E would be equal to E K: <lb></lb>and if G B be greater than B H, I E ſhall likewiſe be greater than E K: <lb></lb>and if leſſer, leſſer. </s> <s>They are therefore four Magnitudes; A B the firſt, <lb></lb>B C the ſecond, D E the third, and E F the Fourth; and the firſt <lb></lb>and third, to wit, the Space A B, and Time D E, there were taken the <lb></lb>Time I E, and the Space G B equimultiple, according to any multi<lb></lb>plication; and it hath been demonſtrated that theſe do at once either <lb></lb>equal, or fall ſhort of, or elſe exceed the Time E K, and Space B H, <lb></lb>which are equimultiple of the ſecond and fourth: Therefore the firſt <lb></lb>bath to the ſecond, to wit the Space A B to the Space B C, the ſame <lb></lb>proportion that the third hath to the fourth, to wit, the Time D E to <lb></lb>the Time E F. </s> <s>Which was to be demonſtrated.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>THEOR. II. PROP. II.</s></p><p type="main"> <s>If a Moveable in equal Times paſſe thorow two <lb></lb>Spaces, the ſaid Spaces will be to each other, <lb></lb>as the Velocities. </s> <s>And if the Spaces are to each <lb></lb>other as the Velocities, the Times will be <lb></lb>equal.</s></p><p type="main"> <s><emph type="italics"></emph>Let us ſuppoſe A B and B C in the former Figure, to be two <lb></lb>Spaces paſt, by the Moveable in equal times; the Space A B with <lb></lb>the Velocity D E, and the Space B C with the Velocity E F. </s> <s>I <lb></lb>ſay, that the Space A B is to the Space B C, as the Velocity D E is to <lb></lb>the Velocity E F: and thus I prove it. </s> <s>Take as before, of the Spaces <lb></lb>and Velocities equi-multiples, accordieg to any what ever Rate, ſci<lb></lb>licet G B and I E, of A B and D E, and likewiſe H B and K E, of <lb></lb>B C and E F: It may be concluded as above, that G B and I E are <lb></lb>both at once either equal to, or fall ſhort of, or elſe exceed the equi-mul<lb></lb>tiples of D H and E K. </s> <s>Therefore the Propoſition is proved.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/132.jpg" pagenum="129"></pb><p type="head"> <s>THEOR. III. PROP. III.</s></p><p type="main"> <s>The Times in which the ſame Space is paſt tho<lb></lb>row by unequal Velocities, have the ſame pro<lb></lb>portion to each other as their Velocities contra<lb></lb>rily taken.</s></p><p type="main"> <s><emph type="italics"></emph>Let the two unequal Velocities be A the greater, and B the leſſe: <lb></lb>and according to both theſe let a Motion be made thorow the ſame <lb></lb>Space C D. </s> <s>I ſay the Time in which the Velocity A paſſeth the <lb></lb>Space C D, ſhall be to the Time in which the Velocity B paſſeth the <lb></lb>ſaid Space, as the Velocity B to the Velocity A. </s> <s>As A is to B, ſo let <lb></lb>C D be to C E: Then, by the <lb></lb>former Propoſition, the Time in<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.132.1.jpg" xlink:href="069/01/132/1.jpg"></figure><lb></lb><emph type="italics"></emph>which the Velocity A paſſeth <lb></lb>C D, ſhall be the ſame with <lb></lb>the Time in which B paſſeth <lb></lb>C E. </s> <s>But the Time in which <lb></lb>the Velocity B paſſeth C E, is <lb></lb>to the Time in which it paſſeth C D, as C E is to C D: Therefore <lb></lb>the Time in which the Velocity A paſſeth C D, is to the Time in which <lb></lb>the Velocity B paſſeth the ſame C D, as C E is to C D; that is, the Ve<lb></lb>locity B is to the Velocity A: Which was to be proved.<emph.end type="italics"></emph.end></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></lb>tion, but with unequal Velocities, the Spaces <lb></lb>which they paſſe in unequal Times, are to each <lb></lb>other in a proportion compounded of the pro<lb></lb>portion of the Velocities, and of the propor<lb></lb>tion of the Times.</s></p><p type="main"> <s><emph type="italics"></emph>Let the two Moveables moving with an Equable Motion, be E and <lb></lb>F: And let the proportion of the Velocity of the Moveable E be <lb></lb>to the Velocity of the Moveable F, as A is to B: And let the Time <lb></lb>in which E is moved, be unto the Time in which F is moved, as C is <lb></lb>to D. </s> <s>I ſay the Space paſſed by E, with the Velocity A in the Time C, is to <lb></lb>the Space paſſed by F, with the Velocity B in the Time D, in a proportion <lb></lb>compounded of the proportion of the Velocity A to the Velocity B, and of<emph.end type="italics"></emph.end><pb xlink:href="069/01/133.jpg" pagenum="130"></pb><emph type="italics"></emph>the proportion of the Time C to the Time D. </s> <s>Let the Space paſſed by the <lb></lb>Moveable E, with the Velocity A in the Time C, be G: And as the <lb></lb>Velocity A is to the Velocity B, <lb></lb><figure id="id.069.01.133.1.jpg" xlink:href="069/01/133/1.jpg"></figure><lb></lb>ſo let G be to I: And as the <lb></lb>Time C is to the Time D, ſo <lb></lb>let I be to L: It is manifeſt, <lb></lb>that I is the Space paſſed by F <lb></lb>in the ſame Time in which E <lb></lb>paſſeth thorow G; ſeeing that <lb></lb>the Spaces G and I are as the <lb></lb>Velocities A and B; and ſeeing that as the Time C is to the Time D, ſo <lb></lb>is I unto L; and ſince that I is the Space paſſed by the Moveable F in the <lb></lb>Time C: Therefore L ſhall be the Space that F paſſeth in the Time D, <lb></lb>with the Velocity B: But the proportion of G to L, is compounded of the <lb></lb>proportions of G to I, and of I to L; that is, of the proportions of the <lb></lb>Velocity A to the Velocity B, and of the Time C to the Time D: <lb></lb>Therefore the Propoſition is demonſtrated.<emph.end type="italics"></emph.end></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></lb>but with unequal Velocities, and if the Spaces <lb></lb>paſſed be alſo unequal, the Times ſhall be to <lb></lb>each other in a proportion compounded of the <lb></lb>proportion of the Spaces, and of the proporti<lb></lb>on of the Velocities contrarily taken.</s></p><p type="main"> <s><emph type="italics"></emph>Let A and B be the two Moveables, and let the Velocity of A be to <lb></lb>the Velocity of B, as V to T, and let the Spaces paſſed, be as S to <lb></lb>R. </s> <s>I ſay the proportion of the Time in which A is moved to the <lb></lb>Time in which B is moved, ſhall be compounded of the proportions of the <lb></lb>Velocity T to the Velocity V, and of the Space S to the Space R. </s> <s>Let C be <lb></lb>the Time of the Motion A;<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.133.2.jpg" xlink:href="069/01/133/2.jpg"></figure><lb></lb><emph type="italics"></emph>and as the Velocity T is to <lb></lb>the Velocity V, ſo let the <lb></lb>Time C be to the Time E: <lb></lb>And for aſmuch as C is the <lb></lb>Time in which A with <lb></lb>the Velocity V paſſeth the <lb></lb>Space S; and that the <lb></lb>Time C is to the Time E, as the Velocity T of the Moveable B is to the <lb></lb>Velocity V, E ſhall be the Time in which the Moveable B would paſſe<emph.end type="italics"></emph.end><pb xlink:href="069/01/134.jpg" pagenum="131"></pb><emph type="italics"></emph>the ſame Space S. </s> <s>Again as the Space S is to the Space R, ſo let the <lb></lb>Time E be to the Time G: Therefore G is the Time in which B would <lb></lb>paſſe the Space R. </s> <s>And becauſe the proportion of C to G is compounded <lb></lb>of the proportions of C to E, and of E to G; And ſince the proportion <lb></lb>of C to E is the ſame with that of the Velocities of the Moveables A and <lb></lb>B contrarily taken; that is, with that of T and V; And the proportion <lb></lb>of E to G is the ſame with the proportion of the Spaces S and R: There<lb></lb>fore the Propoſition is demonſtrated.<emph.end type="italics"></emph.end></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></lb>the proportion of their Velocities ſhall be com<lb></lb>pounded of the proportion of the Spaces paſ<lb></lb>ſed, and of the proportion of the Times con<lb></lb>trarily taken.</s></p><p type="main"> <s><emph type="italics"></emph>Let A and B be the two Moveables moving with an Equable <lb></lb>Motion; and let the Spaces by them paſſed, be as V to T; and <lb></lb>let the Times be as S to R. </s> <s>I ſay that the proportion of the Ve<lb></lb>locity of the Moveable A, to that of the Velocity of B, ſhall be <lb></lb>compounded of the proportions of the Space V to the Space T, and <lb></lb>of the Time R to the Time S. </s> <s>Let C be the Velocity with which the <lb></lb>Moveable A paſſeth the Space V in the Time S: And let the Velocity C <lb></lb>be to the Velo-<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.134.1.jpg" xlink:href="069/01/134/1.jpg"></figure><lb></lb><emph type="italics"></emph>city E, as the <lb></lb>Space V is to <lb></lb>the Space T; <lb></lb>And E ſhall <lb></lb>be the Veloci<lb></lb>ty with which <lb></lb>the Moveable <lb></lb>B paſſeth the Space T in the Time S: Again, let the Velocity E be to the <lb></lb>other Velocity G, as the Time R is to the Time S; And G ſhall be the <lb></lb>Velocity with which the Moveable B paſſeth the Space T in the Time R. <lb></lb></s> <s>We have therefore the Velocity C, wherewith the Moveable A paſſeth <lb></lb>the Space V in the Time S; and the Velocity G, wherewith the Move<lb></lb>able B paſſeth the Space T in the Time R: And the proportion of C to <lb></lb>G is compounded of the proportions of C to E and of E to G: But the <lb></lb>proportion of C to E, is ſuppoſed the ſame with that of the Space V to <lb></lb>the Space T; and the proportion of E to G, is the ſame with that of R <lb></lb>to S: Therefore the Propoſition is manifest.<emph.end type="italics"></emph.end><pb xlink:href="069/01/135.jpg" pagenum="132"></pb><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></lb>cademick, <emph type="italics"></emph>i. </s> <s>e. <lb></lb></s> <s>Galileus.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>This that we have read, is what our ^{*} <emph type="italics"></emph>Author<emph.end type="italics"></emph.end> hath written <lb></lb>of the Equable Motion. </s> <s>We will paſs therefore to a more ſubtil and <lb></lb>new Contemplation touching the Motion Naturally Accelerate: <lb></lb>and behold here the Title and Introduction.</s></p><p type="head"> <s>OF MOTION <lb></lb>NATVRALLY ACCELERATE.</s></p><p type="main"> <s><emph type="italics"></emph>In the former Book we have conſidered the Accidents which ac<lb></lb>company Equable Motion; we are now to treat of another kind of <lb></lb>Motion which we call Accelerate. </s> <s>And firſt it will be expedient to <lb></lb>find out and explain a Definition beſt agreeing to that which Nature <lb></lb>makes uſe of. </s> <s>For though it be not nconvenient to feign a Motion at plea<lb></lb>ſure, and then to conſider the Accidents that attend it (as thoſe have <lb></lb>done, who having framed in their imagination Helixes and Conchoi<lb></lb>des, which are Lines ariſing from certain Motions, although not uſed <lb></lb>by Nature, and upon that Suppoſition have laudably demonſtrated the <lb></lb>Symptomes thereof) yet in regard that Nature maketh uſe of a certain <lb></lb>kind of Acceleration in the deſcent of Grave Bodies, we are reſolved to <lb></lb>ſearch out and contemplate the paſſions thereof, and ſee whether the <lb></lb>Definition that we are about to produce of this our Accelerate Motion, <lb></lb>doth aptly and congruouſly ſute with the Eſſence of Motion Naturally <lb></lb>Accelerate. </s> <s>After many long and laborious Studies we have found out <lb></lb>a Definition which ſeemeth to expreſſe the true nature of this Accelerate <lb></lb>Motion, in regard that all the Natural Experiments that fall under <lb></lb>the Obſervation of our Senſes, do agree with thoſe its properties that <lb></lb>we intend anon to demonſtrate. </s> <s>In this Diſquiſition we have been aſſi<lb></lb>ſted, and as it were led by the hand by that obſervation of the uſual <lb></lb>Method and common procedure of Nature her ſelf in her other Operati<lb></lb>ons, wherein ſhe conſtantly makes uſe of the Firſt, Simpleſt, and Ea<lb></lb>ſieſt Means that are: for I believe that no man can think that Swim<lb></lb>ming or flying can be performed in a more ſimple or eaſie way, than that <lb></lb>which Fiſhes and Birds do uſe out of a Natural Inſtinct. </s> <s>Why there<lb></lb>fore ſhall not I be perſwaded, that, when I ſee a Stone to acquire conti<lb></lb>nually new additions of Velocity in its deſcending from its Reſt out of ſome <lb></lb>high place, this encreaſe made in the ſimpleſt eaſieſt and moſt obvious <lb></lb>manner that we can imagine? </s> <s>Now if we ſeriouſly examine all the ways <lb></lb>that can be deviſed, we ſhall find no encreaſes, no acquiſitions <lb></lb>leſſe intricate or more intelligible than that which ever encreaſeth or <lb></lb>makes its additions after the ſame manner. </s> <s>This appeareth by the great<emph.end type="italics"></emph.end><lb></lb>Affinity <emph type="italics"></emph>that is between Time and Motion. </s> <s>For as the Equability or <lb></lb>Vniformity of Motion is defined and expreſſed by the Equability of the<emph.end type="italics"></emph.end><pb xlink:href="069/01/136.jpg" pagenum="133"></pb><emph type="italics"></emph>Times and Spaces, (for we call that Motion or Lation Equable, by which <lb></lb>equal Spaces are paſt in equal Times) ſo by the ſame Equability of the <lb></lb>parts of Time, we may perceive, that the encreaſe of Celerity in the Natu<lb></lb>ral Motion of Grave Bodies, is made after a Simple and plain manner; <lb></lb>conceiving in our Mind that their Motion is continually accelerated uni<lb></lb>formly and at the ſame Rate, whilſt equal additions of Celerity are <lb></lb>conferred upon them in all equal Times. </s> <s>So that taking any equal par<lb></lb>ticles of Time beginning from the firſt Inſtant in which the Moveable <lb></lb>departeth from Reſt, and entereth upon its Deſcent, the Degree of <lb></lb>Velocity acquired in the firſt and ſecond Particles of Time, is double the <lb></lb>degree of Velocity that the Moveable acquired in the firſt Particle: and <lb></lb>the degree of Velocity that it acquireth in three Particles, is triple, and <lb></lb>that in four quadruple to the ſame Degree of the firſt Time: As, for <lb></lb>our better underſtanding, if a Moveable ſhould continue its Motion <lb></lb>according to the degree or moment of Velocity acquired in the firſt Parti<lb></lb>cle of Time, and ſhould extend its courſe equably with that ſame De<lb></lb>gree; this Motion would be twice as ſlow as that which it would obtain <lb></lb>according to the degree of Velocity acquired in two Particles of Time: <lb></lb>So that it will not be improper if we underſtand the Intention of the Ve<lb></lb>locity, to proceed according to the Extenſion of the Time. </s> <s>From whence <lb></lb>we may frame this Definition of the Motion of which we are about to <lb></lb>treat.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>DEFINITION.</s></p><p type="main"> <s>Motion Accelerate in an Equable or Vniform <lb></lb>Proportion, I call that which departing from <lb></lb>Reſt, ſuperaddeth equal moments of Velocity <lb></lb>in equal Times.</s></p><p type="main"> <s>SAGR. </s> <s>Though it were Irrational for me to oppoſe this or any <lb></lb>other Definition aſſigned by any whatſoever Author, they being all <lb></lb>Arbitrary, yet I may very well, without any offence, queſtion whe<lb></lb>ther this Definition, which is underſtood and admitted in Abſtract, <lb></lb>doth ſute, agree, and hold true in that ſort of Accelerate Motion, <lb></lb>which Grave Bodies deſcending naturally do exerciſe. </s> <s>And becauſe <lb></lb>the Authour ſeemeth to promiſe us, that the Natural Motion of <lb></lb>Grave Bodies is ſuch as he hath defined it, I could wiſh that ſome <lb></lb>Scruples were removed that trouble my mind; that ſo I might apply <lb></lb>my ſelf afterwards with greater attention to the Proportions and <lb></lb>Demonſtrations which are expected.</s></p><p type="main"> <s>SALV. </s> <s>I like well, that you and <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> do propound <lb></lb>Doubts as they come in the way: which I do imagine will be the <pb xlink:href="069/01/137.jpg" pagenum="134"></pb>ſame that I my ſelf did meet with when I firſt read this Treatiſe, <lb></lb>and that, either were reſolved by conferring with the Author, or <lb></lb>removed by my own conſidering of them.</s></p><p type="main"> <s>SAGR. </s> <s>Whilſt I am fancying to my ſelf a Grave Deſcending <lb></lb>Moveable to depart from Reſt, that is from the privation of all <lb></lb>Velocity, and to enter into Motion, and in that to go encrea<lb></lb>ſing, according to the proportion after which the Time encreaſeth <lb></lb>from the firſt inſtant of the Motion; and to have <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> in eight <lb></lb>Pulſations, acquired eight degrees of Velocity, of which in the <lb></lb>fourth Pulſation it had gained four, in the ſecond two, in the <lb></lb>firſt one, Time being ſubdiviſible <emph type="italics"></emph>in infinitum,<emph.end type="italics"></emph.end> it followeth, that <lb></lb>the Antecedent Velocity alwayes diminiſhing at that Rate, there <lb></lb>will bt no degree of Velocity ſo ſmall, or, if you will, of Tardity <lb></lb>ſo great, in which the ſaid Moveable is not found to be conſti<lb></lb>tuted, after its departure from infinite Tardity, that is, from <lb></lb>Reſt. </s> <s>So that if that degree of Velocity which it had at four Pul<lb></lb>ſations of Time, was ſuch, that maintaining it Equable, it would <lb></lb>have run two Miles in an hour, and with the degree of Velocity <lb></lb>that it had in the ſecond Pulſation, it would have gone one mile <lb></lb>an hour, it muſt be granted, that in the Inſtants of Time neeter <lb></lb>and neerer to its firſt Inſtant of moving from Reſt, it is ſo ſlow, <lb></lb>as that (continuing to move with that Tardity) it would not have <lb></lb>paſſed a Mile in an hour, nor in a day, nor in a year, nor in a <lb></lb>thouſand; nay, nor have gone one ſole foot in a greater time: <lb></lb>An accident to which me thinks the Imagination but very unea<lb></lb>ſily accords, ſeeing that Senſe ſheweth us, that a Grave Falling <lb></lb>Body commeth down ſuddenly, and with great Velocity.</s></p><p type="main"> <s>SALV. </s> <s>This is one of thoſe Doubts that alſo fell in my way <lb></lb>upon my firſt thinking on this affair, but not long after I remo<lb></lb>ved it: and that removal was the effect of the ſelf ſame Expe<lb></lb>riment which at preſent ſtarts it to you. </s> <s>You ſay, that in your <lb></lb>opinion, Experience ſheweth that the Moveable hath no ſooner <lb></lb>departed from Reſt, but it entereth into a very notable Velocity: <lb></lb>and I ſay, that this very Experiment proves it to us, that the firſt <lb></lb>Impetus's of the Cadent Body, although it be very heavy, are <lb></lb>moſt ſlack and ſlow. </s> <s>Lay a Grave Body upon ſome yielding mat<lb></lb>ter, and let it continue upon it till it hath preſſed into it as far as <lb></lb>it can with its ſimple Gravity; it is manifeſt, that raiſing it a yard <lb></lb>or two, and then letting it fall upon the ſame matter, it ſhall <lb></lb>with its percuſſion make a new preſſure, and greater than that <lb></lb>made at firſt by its meer weight: and the effect ſhall be cauſed <lb></lb>by the falling Moveable conjoyned with the Velocity acquired in <lb></lb>the Fall: which impreſſion ſhall be greater and greater, accord<lb></lb>ing as the Percuſſion ſhall come from a greater height; that is, <lb></lb>according as the Velocity of the Percutient ſhall be greater. </s> <s>We <pb xlink:href="069/01/138.jpg" pagenum="135"></pb>may therefore without miſtake conjecture the quantity of the Ve<lb></lb>locity of a falling heavy Body; by the quality and quantity of <lb></lb>the Percuſſion. </s> <s>But tell me Sirs, that Beetle which being let fall <lb></lb>upon a Stake from an height of four yards, driveth it into the <lb></lb>ground, <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> four inches, comming from an height of two yards, <lb></lb>ſhall drive it much leſſe, and leſſe from an height of one, and <lb></lb>leſſe from a foot; and laſtly lifting it up an inch, what will it do <lb></lb>more than if without any blow it were laid upon it? </s> <s>Certainly <lb></lb>but very little, and the operation would be wholly impercep<lb></lb>tible, if it were raiſed the thickneſſe of a leaf. </s> <s>And becauſe the <lb></lb>effect of the Percuſſion is regulated by the Velocity of the Percu<lb></lb>tient, who will queſtion but that the Motion is very ſlow, and <lb></lb>the Velocity extreme ſmall, where its operation is impercep<lb></lb>tible? </s> <s>See now of what power Truth is, ſince the ſame Experi<lb></lb>ment that ſeemed at the firſt bluſh to hold forth one thing, be<lb></lb>ing better conſidered, aſcertains us of the contrary. </s> <s>But without <lb></lb>having recourſe to that Experiment (which without doubt is moſt <lb></lb>perſwaſive) me-thinks that it is not hard to penetrate ſuch a <lb></lb>Truth as this by meer Diſcourſe. </s> <s>We have an heavy ſtone ſu<lb></lb>ſtained in the Air at Reſt: let it be diſengaged from its uphol<lb></lb>der, and ſet at liberty; and, as being more grave than the Air, it <lb></lb>goeth deſcending downwards, and that not with a Motion Equa<lb></lb>ble, but ſlow in the beginning, and continually afterwards ac<lb></lb>celerate: and ſeeing that the Velocity is Augmentable and Di<lb></lb>miniſhable <emph type="italics"></emph>in infinitum,<emph.end type="italics"></emph.end> what Reaſon ſhall perſwade me, that that <lb></lb>Moveable departing from an infinite Tardity (for ſuch is Reſt) <lb></lb>entereth immediately into ten degrees of Velocity, rather than in <lb></lb>one of four, or in this more than in one of two, of one, of half <lb></lb>one, or of the hundredth part of one; and to be ſhort, in all <lb></lb>the infinite leſſer? </s> <s>Pray you hear me. </s> <s>I do not think that you <lb></lb>would ſcruple to grant me, that the acquiſt of the Degrees of Ve<lb></lb>locity of the falling Stone may be made with the ſame Order as <lb></lb>is the Diminution and loſſe of the ſame degrees, when with an <lb></lb>impellent Virtue it is driven upwards to the ſame height: But if <lb></lb>that be ſo, I do not ſee how it can be ſuppoſed that in the diminu<lb></lb>tion of the Velocity of the aſcendent Stone, ſpending it all, it <lb></lb>can come to the ſtate of Reſt before it hath paſſed thorow all the <lb></lb>degrees of Tardity.</s></p><p type="main"> <s>SIMP. </s> <s>But if the greater and greater degrees of Tardity are <lb></lb>infinite, it ſhall never ſpend them all; ſo that the aſcendent <lb></lb>Grave will never attain to Reſt, but will move <emph type="italics"></emph>ad infinitum,<emph.end type="italics"></emph.end> ſtill <lb></lb>retarding: a thing which we ſee not to happen.</s></p><p type="main"> <s>SALV. </s> <s>This would happen, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> in caſe the Moveable <lb></lb>ſhould ſtay for ſome time in each degree: but it paſſeth thorow <lb></lb>them, without ſtaying longer than an inſtant in any of them. <pb xlink:href="069/01/139.jpg" pagenum="136"></pb>And becauſe in every quantitative Time, though never ſo ſmall, <lb></lb>there are infinite Inſtants, therefore they are ſufficient to anſwer <lb></lb>to the infinite degrees of Velocity diminiſhed. </s> <s>And that the <lb></lb>aſcendent Grave Body perſiſts not for any quantitative Time in <lb></lb>one and the ſame degree of Velocity, may thus be made out: <lb></lb>Becauſe, a certain quantitative Time being aſſigned it in the firſt <lb></lb>inſtant of that Time, and likewiſe in the laſt, the Moveable <lb></lb>ſhould be found to have one and the ſame degree of Velocity, it <lb></lb>might by this ſecond degree be likewiſe driven upwards ſuch an<lb></lb>other Space, like as from the firſt it was tranſported to the ſe<lb></lb>cond; and by the ſame reaſon it would paſſe from the ſecond to <lb></lb>the third, and, in ſhort, would continue its Motion Uniform <emph type="italics"></emph>ad <lb></lb>infinitum.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>From this Diſcourſe, as I conceive, one might derive a <lb></lb>very appoſite Reaſon of the Queſtion controverted amongſt Philo. <lb></lb></s> <s>ſophers, Touching what ſhould be the Cauſe of the acceleration <lb></lb>of the Natural Motion of Grave Moveables. </s> <s>For when I confider <lb></lb>in the Grave Body driven upwards, its continual Diminution of <lb></lb>that Virtue impreſſed upon it by the Projicient, which ſo long as <lb></lb>it was ſuperiour to that other contrary one of Gravity, forced it <lb></lb>upwards, this and that being come to an <emph type="italics"></emph>Equilibrium,<emph.end type="italics"></emph.end> the Move<lb></lb>able ceaſeth to riſe any higher, and paſſeth thorow the ſtate of <lb></lb>Reſt, in which the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> impreſſed is not annihilated, but one<lb></lb>ly that exceſſe is ſpent, which it before had above the Gravity of <lb></lb>the Moveable, whereby prevailing over the ſame, it did drive <lb></lb>it upwards. </s> <s>And the Diminution of this forrein <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> continu<lb></lb>ing, and conſequently the advantage beginning to be on the part <lb></lb>of the Gravity, the Deſcent alſo beginneth but ſlow, in regard <lb></lb>of the oppoſition of the Virtue impreſſed, a conſiderable part of <lb></lb>which ſtill remaineth in the Moveable: but becauſe it doth go <lb></lb>continually diminiſhing, and is ſtill with a greater and greater <lb></lb>proportion overcome by the Gravity, hence ariſeth the continual <lb></lb>Acceleration of the Motion.</s></p><p type="main"> <s>SIMP. </s> <s>The conceit is witty, but more ſubtil than ſolid: for in <lb></lb>caſe it were concludent, it ſalveth onely thoſe Natural Motions <lb></lb>to which a Violent Motion preceded, in which part of the extern <lb></lb>Virtue ſtill remains in force: but where there is no ſuch remaining <lb></lb>impulſe, as where the Moveable departeth from a long Quieſ<lb></lb>cence, the ſtrength of your whole Diſcourſe vaniſheth.</s></p><p type="main"> <s>SAGR. </s> <s>I believe that you are in an Errour, and that this Di<lb></lb>ſtinction of Caſes which you make, is needleſſe, or, to ſay bet<lb></lb>ter, <emph type="italics"></emph>Null.<emph.end type="italics"></emph.end> Therefore tell me, whether may there be impreſſed <lb></lb>on the Project by the Projicient ſometimes much, and ſometimes <lb></lb>little Vertue; ſo as that it may be ſtricken upwards an hundred <lb></lb>yards, and alſo twenty, or four, or one?</s></p><pb xlink:href="069/01/140.jpg" pagenum="137"></pb><p type="main"> <s>SIMP. </s> <s>No doubt but there may.</s></p><p type="main"> <s>SAGR. </s> <s>And no leſſe poſſible is it, that the ſaid Virtue impreſſed <lb></lb>ſhall ſo little ſeperate the Reſiſtance of the Gravity, as not to <lb></lb>raiſe the Project above an inch: and finally the Virtue of the <lb></lb>Projicient may be onely ſo much, as juſt to equalize and com<lb></lb>penſate the Reſiſtance of the Gravity, ſo as that the Moveable <lb></lb>is not driven upwards, but onely ſuſtained. </s> <s>So that when you <lb></lb>hold a Stone in your hand, what elſe do you, but impreſſe on it <lb></lb>ſo much Virtue impelling upwards, as is the faculty of its Gra<lb></lb>vity drawing downwards? </s> <s>And this your Virtue, do you not <lb></lb>continue to keep it impreſſed on the Stone all the time that you <lb></lb>hold it in your hand? </s> <s>What ſay you, is it diminiſhed by your <lb></lb>long holding it? </s> <s>And this ſuſtention which impedeth the Stones <lb></lb>deſcent, what doth it import, whether it be made by your hand, <lb></lb>or by a Table, or by a Rope, that ſuſpends it? </s> <s>Doubtleſſe no <lb></lb>thing at all. </s> <s>Conclude with your ſelf therefore, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> that <lb></lb>the precedence of a long, a ſhort, or a Momentary Reſt to the <lb></lb>Fall of the Stone, makes no alteration at all, ſo that the Stone <lb></lb>ſhould not alwaies depart affected with ſo much Virtue contrary <lb></lb>to Gravity, as did exactly ſuffice to have kept it in Reſt.</s></p><p type="main"> <s>SALV. </s> <s>I do not think it a ſeaſonable time at preſent to enter <lb></lb>upon the Diſquiſition of the Cauſe of the Acceleration of Natu<lb></lb>ral Motion: touching which ſundry Philoſophers have produced <lb></lb>ſundry opinions: ſome reducing it to the approximation unto <lb></lb>the Center others to the leſſe parts of the <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> ſucceſſively re<lb></lb>maining to be perforated; others to a certain Extruſion of the <lb></lb>Ambient <emph type="italics"></emph>Medium,<emph.end type="italics"></emph.end> which in reuniting upon the back of the <lb></lb>Moveable, goeth driving and continually thruſting it; which <lb></lb>Fancies, and others of the like nature, it would be neceſſary to <lb></lb>examine, and with ſmall benefit to anſwer. </s> <s>It ſerveth our Au<lb></lb>thours turn at the preſent, that we underſtand that he will de<lb></lb>clare and demonſtrate to us ſome Paſſions of an Accelerate Mo<lb></lb>tion (be the Cauſe of its Acceleration what it will) ſo as that the <lb></lb>Moments of its Velocity do go encreaſing, after its departure from <lb></lb>Reſt with that moſt ſimple proportion wherewith the Continua<lb></lb>tion of the Time doth encreaſe: which is as much as to ſay, that <lb></lb>in equal Times there are made equal additaments of Velocity. <lb></lb></s> <s>And if it ſhall be found, that the Accidents that ſhall hereafter <lb></lb>be demonſtrated, do hold true in the Motion of Naturally De<lb></lb>ſcendent and Accelerate Grave Moveables, we may account, <lb></lb>that the aſſumed Definition taketh in that Motion of Grave Bo<lb></lb>dies, and that it is true, that their Acceleration doth encreaſe ac<lb></lb>cording as the Time and Duration of the Motion encreaſeth.</s></p><p type="main"> <s>SAGR. </s> <s>By what as yet is ſet before my Intellectuals, it appears <lb></lb>to me that one might with (haply) more plainneſſe define, and yet <pb xlink:href="069/01/141.jpg" pagenum="138"></pb>never alter the Conceit; ſaying that, A Motion uniformly accele<lb></lb>rate is that in which the Velocity goeth encreaſing according as <lb></lb>the Space encreaſeth that is paſſed thorow: So that, for example, <lb></lb>the degree of Velocity acquired by the Moveable in a deſcent of <lb></lb>four yards ſhould be double to that that it would have after it had <lb></lb>deſcended a Space of two, and this double to that acquired in the <lb></lb>Space of the firſt Yard. </s> <s>For I do not think that it can be doubted, <lb></lb>but that that Grave Moveable which falleth from an height of ſix <lb></lb>yards hath, and percuſſeth with an <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> double to that which <lb></lb>it had when it had deſcended three yards, and triple to that which <lb></lb>it had at two, and ſextuple to that had in the Space of one.</s></p><p type="main"> <s>SALV. </s> <s>I comfort my ſelf in that I have had ſuch a Companion <lb></lb>in my Errour: and I will tell you farther, that your Diſcourſe hath <lb></lb>ſo much of likelihood and probability in it, that our Author himſelf <lb></lb>did not deny unto me, when I propoſed it to him, that he likewiſe <lb></lb>had been for ſome time in the ſame miſtake. <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut that which I af<lb></lb>terwards extreamly wondred at, was to ſee in four plain words, <lb></lb>diſcovered, not only the falfity, but impoſſibility of two Propoſi<lb></lb>tions that carry with them ſo much of ſeeming truth, that having <lb></lb>propounded them to many, I never met with any one but did freely <lb></lb>admit them to be ſo.</s></p><p type="main"> <s>SIMP. </s> <s>Certainly I ſhould be of the number, and that the De<lb></lb>ſcendent Grave Moveable <emph type="italics"></emph>vires acquir at eundo,<emph.end type="italics"></emph.end> encreaſing its Ve<lb></lb>locity at the rate of the Space, and that the Moment of the ſame <lb></lb>Percutient is double, coming from a double height, ſeem to me Pro<lb></lb>poſitions to be granted without any hæſitation or controverſie.</s></p><p type="main"> <s>SALV. </s> <s>And yet they are as falſe and impoſſible, as that Moti<lb></lb>on is made in an inſtant. </s> <s>And hear a clear proof of the ſame. </s> <s>In <lb></lb>caſe the Velocities have the ſame proportion as the Spaces paſſed, <lb></lb>or to be paſſed, thoſe Spaces ſhall be paſſed in equal Times: if <lb></lb>therefore the Velocities with which the falling Moveable paſſeth <lb></lb>the Space of four yards, were double to the Velocities with which it <lb></lb>paſſeth the two firſt yards (like as the Space is double to the Space) <lb></lb>then the Times of thoſe Tranſitions are equal: but the ſame Move<lb></lb>able's paſſing the four yards, and the two in one and the ſame Time, <lb></lb>hath place only in Inſtantaneous Motion. <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut we ſee, that the <lb></lb>falling grave <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ody maketh its Motion in Time, and paſſeth the two <lb></lb>yards in a leſſer than it doth the four. </s> <s>Therefore it is falſe that its <lb></lb>Velocity encreaſeth as its Space. </s> <s>The other Propoſition is demon<lb></lb>ſtrated to be falſe with the ſame perſpicuity. </s> <s>For that which per<lb></lb>cuſſeth being the ſame, the difference and Moment of the Percuſſton <lb></lb>cannot be determined but by the difference of Velocity; If there<lb></lb>fore the percutient, coming from a double height, make a Percuſſi<lb></lb>on with a double Moment, it is neceſſary that it ſtrike with a dou<lb></lb>ble Velocity: <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut the double Velocity paſſeth the double Space in <pb xlink:href="069/01/142.jpg" pagenum="139"></pb>the ſame Time; and we ſee the Time of the Deſcent from the grea<lb></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></lb>wherewith you manifeſt abſtruce Concluſions: this extream eaſi<lb></lb>neſs rendreth them of leſſe value than they were whilſt they lay hid <lb></lb>under contrary appearances. </s> <s>I believe that the Generality of men <lb></lb>little preſſe thoſe Notions which are eaſily obtained, in compari<lb></lb>ſon of thoſe about which men make ſo long and inexplicable alter<lb></lb>cations.</s></p><p type="main"> <s>SALV. </s> <s>To thoſe which with great brevity and clarity ſhew the <lb></lb>fallacies of Propoſitions that have been commonly received for <lb></lb>true by the generality of people, it would be a very tolerable in<lb></lb>jury to return them only ſlighting inſtead of thanks: but there is <lb></lb>much diſpleaſure and moleſtation in another certain affection <lb></lb>ſometimes found in ſome men, that pretending in the ſame Studies <lb></lb>at leaſt Parity with any whomſoever, do ſee that they have let <lb></lb>paſs ſuch and ſuch for true Concluſions, which afterwards by <lb></lb>another, with a ſhort and eaſie diſquiſition, have been detected and <lb></lb>convicted for falſe. </s> <s>I will not call that affection Envy, that is ac<lb></lb>cuſtomed to convert in time to hatred and deſpite againſt the diſ<lb></lb>coverers of ſuch Fallacies, but I will call it an itch, and a deſire to <lb></lb>be able rather to maintain their inveterate Errours, than to per<lb></lb>mit the reception of new-diſcovered Truths. </s> <s>Which humour ſome<lb></lb>times induceth them to write in contradiction of thoſe truths <lb></lb>which are but too perfectly known unto themſelves only to keep <lb></lb>the Reputation of others low in the opinion of the numerous and <lb></lb>ill-informed Vulgar. </s> <s>Of ſuch falſe Concluſions received for true, <lb></lb>and very eaſie to be confuted, I have heard no ſmall number from <lb></lb>our <emph type="italics"></emph>Academick,<emph.end type="italics"></emph.end> of ſome of which I have kept account.</s></p><p type="main"> <s>SAGR. </s> <s>And you muſt not deprive us of them; but in due time <lb></lb>impart them to us, when a particular Meeting ſhall be appointed <lb></lb>for them. </s> <s>For the preſent, continuing the diſcourſe we are about, <lb></lb>I think that by this time we have eſtabliſhed the Definition of Mo<lb></lb>tion uniformly Accelerate, treated of in the enſuing diſcourſes, <lb></lb>and it is this;</s></p><p type="main"> <s><emph type="italics"></emph>A Motion Equable, or Vniformly Accelerate, we call that which <lb></lb>departing from Reſt ſuperadds equal Moments of Velocity in <lb></lb>equal Times.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>That Definition being confirmed, the Author asketh <lb></lb>and ſuppoſeth but one only Principle to be true, namely:</s></p><pb xlink:href="069/01/143.jpg" pagenum="140"></pb><p type="head"> <s>SVPPOSITION.</s></p><p type="main"> <s><emph type="italics"></emph>I ſuppoſe that the degrees of Velocity acquired by the <lb></lb>ſame Moveable upon Planes of different inclinations <lb></lb>are equal then, when the Elevations of the ſaid <lb></lb>Planes are equal.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>By the Elevation of an inclined Plane he meaneth the Per<lb></lb>pendicular, which from the higher term of the ſaid Plane <lb></lb>falleth upon the Horizontal Line produced along by the <lb></lb>lower term of the ſaid Plane inclined: as for better underſtanding; <lb></lb>the Line A B being parallel to the Horizon, upon which let the two <lb></lb><figure id="id.069.01.143.1.jpg" xlink:href="069/01/143/1.jpg"></figure><lb></lb>Planes C A, and C D be inclined: <lb></lb>the Perpendicular C B falling up<lb></lb>on the Horizontal Line B A the <lb></lb>Author calleth the Elevation <lb></lb>of the Planes C A and C D; <lb></lb>and ſuppoſeth that the degrees of <lb></lb>Velocity of the ſame Moveable <lb></lb>deſcending along the inclined Planes C A and C D, acqui<lb></lb>red in the Terms A and D are equal, for that their Elevation is <lb></lb>the ſame C B. </s> <s>And ſo great alſo ought the degree of Velocity be <lb></lb>underſtood to be which the ſame Moveable falling from the Point <lb></lb>C would acquire in the term B.</s></p><p type="main"> <s>SAGR. </s> <s>The truth is, this Suppoſition hath in it ſo much of pro<lb></lb>bability, that it deſerveth to be granted without diſpute, alwaies <lb></lb>preſuppoſing that all accidental and extern Impediments are re<lb></lb>moved, and that the Planes be very Solid and Terſe, and the Move<lb></lb>able in Figure moſt perfectly Rotund, ſo that neither the Plane, <lb></lb>nor the Moveable have any unevenneſs. </s> <s>All Contraſts and Im<lb></lb>pediments, I ſay, being removed, the light of Nature dictates to <lb></lb>me without any difficulty, that a Ball heavy and perfectly round <lb></lb>deſcending by the Lines C A, C D, and C B would come to the <lb></lb>terms A D, and B with equal <emph type="italics"></emph>Impetus's.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>You argue very probably; but over and above the pro<lb></lb>bability, I will by an Experiment ſo increaſe the likelihood, as that <lb></lb>it wants but little of being equal to a very neceſſary Demonſtrati<lb></lb>on. </s> <s>Imagine this leafe of Paper to be a Wall erect at Right-angles <lb></lb>to the Horizon, and at a Nail, faſtned in the ſame, hang a Ball or <lb></lb>Plummet of Lead, weighing an ounce or two, ſuſpended by the <lb></lb>ſmall thread A B, two or three yards long, perpendicular to the <lb></lb>Horizon: and on the Wall draw an Horizontal Line D C, cutting <pb xlink:href="069/01/144.jpg" pagenum="141"></pb>the Perpendicular A B at Right angles, which A B muſt hang two <lb></lb>Inches, or thereabouts, from the Wall: Then transferring the <lb></lb>ſtring A B with the Ball into C, let go the ſaid Ball; which you will <lb></lb><figure id="id.069.01.144.1.jpg" xlink:href="069/01/144/1.jpg"></figure><lb></lb>ſee firſt to deſcend <lb></lb>deſcribing C B D, and <lb></lb>to paſs ſo far beyond <lb></lb>the Term B, that run<lb></lb>ning along the Arch <lb></lb>B D it will riſe almoſt <lb></lb>as high as the deſigned <lb></lb>Parallel C D, wanting <lb></lb>but a very ſmall mat<lb></lb>ter of reaching to it, <lb></lb>the preciſe arrival thi<lb></lb>ther being denied it by <lb></lb>the Impediment of the Air, and of the Thread. </s> <s>From which we <lb></lb>may truly conclude, that the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> acquired in the point B by <lb></lb>the Ball in its deſcent along the Arch C B, was ſo much as ſufficed <lb></lb>to carry it upwards along ſuch another Arch B D unto the ſame <lb></lb>height: having made, and often reiterated this Experiment, let <lb></lb>us drive into the Wall, along which the Perpendicular A B paſſeth, <lb></lb>another Nail, as in E or in F, which is to ſtand out five or ſix In<lb></lb>ches; and this to the end that the thread A B, returning as before <lb></lb>to carry back the Ball C along the Arch C B, when it is come to <lb></lb>B, the Thread ſtopping at the Nail E may be conſtrained to move <lb></lb>along the Circumference B G, deſcribed about the Center E: by <lb></lb>which we ſhall ſee what that ſame <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> is able to do, which be<lb></lb>fore, being conceived in the ſame term B, carried the ſame Move<lb></lb>able along the Arch B D unto the height of the Horizontal Line <lb></lb>C D. Now, Sirs, you ſhall with delight ſee the Ball carried unto <lb></lb>the Horizontal Line in the Point G; and the ſame will happen if <lb></lb>the ſtop be placed lower, as in F, where the Ball would deſcribe <lb></lb>the Arch B I, evermore terminating its aſcent exactly in the Line <lb></lb>C D: and in caſe the Check were ſo low that the overplus of the <lb></lb>thread beneath it cannot reach to the height of C D, (which would <lb></lb>happen if it were nearer to the point B than to the interſection of <lb></lb>A B with the Horizontal Line C D) then the thread would <lb></lb>whirle and twine about the Nail. </s> <s>This experiment leaveth no <lb></lb>place for our doubting of the truth of the Suppoſition: for the <lb></lb>two Arches C B and D B being equall, and ſcituate alike, the <lb></lb>acquiſt of Moment made along the Deſcent in the Arch C B, is <lb></lb>the ſame with that made along the Deſcent in the Arch D B. </s> <s>But <lb></lb>the Moment acquired in <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> along the Arch C <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> is able to carry the <lb></lb>ſame Moveable upwards along the Arch <emph type="italics"></emph>B<emph.end type="italics"></emph.end> D: Therefore the Mo<lb></lb>ment acquired in the Deſcent D <emph type="italics"></emph>B<emph.end type="italics"></emph.end> is equall to that which driveth <pb xlink:href="069/01/145.jpg" pagenum="142"></pb>the ſame Moveable along the ſame Arch from <emph type="italics"></emph>B<emph.end type="italics"></emph.end> to D: So that ge<lb></lb>nerally every Moment acquired along the Deſcent of an Arch is <lb></lb>equall to that which hath power to make the ſame Moveable re<lb></lb>aſcend along the ſame Arch: <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut all the Moments that make the <lb></lb>Moveable aſcend along all the Arches <emph type="italics"></emph>B<emph.end type="italics"></emph.end> D, <emph type="italics"></emph>B<emph.end type="italics"></emph.end> G, <emph type="italics"></emph>B<emph.end type="italics"></emph.end> I are equal, <lb></lb>ſince they are made by one and the ſame Moment acquired along <lb></lb>the Deſcent C <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> as Experience ſhews: Therefore all the Moments <lb></lb>that are acquired by the Deſcents along the Arches D <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> G <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> and <lb></lb>I <emph type="italics"></emph>B<emph.end type="italics"></emph.end> are equal.</s></p><p type="main"> <s>SAGR. </s> <s>Your Diſcourſe is in my Judgment very Rational, and <lb></lb>the Experiment ſo appoſite and pertinent to verifie the <emph type="italics"></emph>Poſtulatum,<emph.end type="italics"></emph.end><lb></lb>that it very well deſerveth to be admitted as if it were Demon<lb></lb>ſtrated.</s></p><p type="main"> <s>SALV. </s> <s>I will not conſent, <emph type="italics"></emph>Sagredus,<emph.end type="italics"></emph.end> that we take more to our <lb></lb>ſelves than we ought; and the rather for that we are chiefly to <lb></lb>make uſe of this Aſſumption in Motions made upon ſtreight and <lb></lb>not curved Superficies; in which the Acceleration proceedeth with <lb></lb>degrees very different from thoſe wherewith we ſuppoſe it to pro<lb></lb>ceed in ſtreight Planes. </s> <s>Inſomuch, that although the Experiment <lb></lb>alledged ſhews us, that the deſcent along the Arch C <emph type="italics"></emph>B<emph.end type="italics"></emph.end> conferreth <lb></lb>on the Moveable ſuch a Moment, as that it is able to re-carry it <lb></lb>to the ſame height along any other Arch <emph type="italics"></emph>B<emph.end type="italics"></emph.end> C, <emph type="italics"></emph>B<emph.end type="italics"></emph.end> G, and <emph type="italics"></emph>B<emph.end type="italics"></emph.end> I, yet <lb></lb>we cannot with the like evidence ſhew, that the ſame would hap<lb></lb>pen in caſe a moſt exact <emph type="italics"></emph>B<emph.end type="italics"></emph.end>all were to deſcend by ſtreight Planes in<lb></lb>clined according to the inclinations of the Chords of theſe ſame <lb></lb>Arches: yea, it is credible, that Angles being formed by the ſaid <lb></lb>Right Planes in the term <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> the <emph type="italics"></emph>B<emph.end type="italics"></emph.end>all deſcended along the Declivi<lb></lb>ty according to the Chord C <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> finding a ſtop in the Planes aſcend<lb></lb>ing according to the Chords <emph type="italics"></emph>B<emph.end type="italics"></emph.end> D, <emph type="italics"></emph>B<emph.end type="italics"></emph.end> G, and <emph type="italics"></emph>B<emph.end type="italics"></emph.end> I, in juſtling againſt <lb></lb>them, would loſe of its <emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> and could not be able in riſing to <lb></lb>attain the height of the Line C D. <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut the Obſtacle being remo<lb></lb>ved, which prejudiceth the Experiment, I do believe, that the un<lb></lb>derſtanding may conceive, that the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> (which in effect de<lb></lb>riveth vigour from the quantity of the Deſcent) would be able to <lb></lb>remount the Moveable to the ſame height. </s> <s>Let us therefore take <lb></lb>this at preſent for a <emph type="italics"></emph>Poſtulatum<emph.end type="italics"></emph.end> or Petition, the abſolute truth of <lb></lb>which will come to be eſtabliſhed hereafter by ſeeing other Con<lb></lb>cluſions raiſed upon this Hypotheſis to anſwer, and exactly jump <lb></lb>with the Experiment. </s> <s>The Author having ſuppoſed this only Prin<lb></lb>ciple, he paſſeth to the Propoſitions, demonſtratively proving them; <lb></lb>of which the firſt is this;</s></p><pb xlink:href="069/01/146.jpg" pagenum="143"></pb><p type="head"> <s>THEOR. I. PROP. I.</s></p><p type="main"> <s>The time in which a Space is paſſed by a Movea<lb></lb>ble with a Motion Vniformly Accelerate, out of <lb></lb>Reſt, is equal to the Time in which the ſame <lb></lb>Space would be paſt by the ſame Moveable <lb></lb>with an Equable Motion, the degree of whoſe <lb></lb>Velocity is ſubduple to the greateſt and ulti <lb></lb>mate degree of the Velocity of the former Vni<lb></lb>formly Accelerate Motion.</s></p><p type="main"> <s><emph type="italics"></emph>Let us by the extenſion A B repreſent the Time, in which the <lb></lb>Space<emph.end type="italics"></emph.end> C D <emph type="italics"></emph>is paſſed by a Moveable with a Motion Vniformly <lb></lb>Accelerate, out of Reſt in C: and let the greateſt and laſt de-<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.146.1.jpg" xlink:href="069/01/146/1.jpg"></figure><lb></lb><emph type="italics"></emph>gree of Velocity acquired in the Inſtants of the Time<emph.end type="italics"></emph.end><lb></lb>A B <emph type="italics"></emph>be repreſented by<emph.end type="italics"></emph.end> E B; <emph type="italics"></emph>and conſtitute at plea<lb></lb>ſure upon<emph.end type="italics"></emph.end> A B <emph type="italics"></emph>any number of parts, and thorow the <lb></lb>points of diviſion draw as many Lines, continued <lb></lb>out unto the Line<emph.end type="italics"></emph.end> A E, <emph type="italics"></emph>and equidiſtant to<emph.end type="italics"></emph.end> B E, <lb></lb><emph type="italics"></emph>which will repreſent the encreaſe of the degrees of <lb></lb>Velocity after the firſt Inſtant A. </s> <s>Then divide<emph.end type="italics"></emph.end> B E <lb></lb><emph type="italics"></emph>into two equall parts in<emph.end type="italics"></emph.end> F, <emph type="italics"></emph>and draw<emph.end type="italics"></emph.end> F G <emph type="italics"></emph>and<emph.end type="italics"></emph.end> A G <lb></lb><emph type="italics"></emph>parallel to B A and<emph.end type="italics"></emph.end> B F<emph type="italics"></emph>: The Parallelogram<emph.end type="italics"></emph.end> A G <lb></lb>F B <emph type="italics"></emph>ſhall be equall to the Triangle<emph.end type="italics"></emph.end> A E B, <emph type="italics"></emph>its Side<emph.end type="italics"></emph.end><lb></lb>G F <emph type="italics"></emph>dividing<emph.end type="italics"></emph.end> A E <emph type="italics"></emph>into two equall parts in I: For <lb></lb>if the Parallels of the Triangle<emph.end type="italics"></emph.end> A E <emph type="italics"></emph>B be continued <lb></lb>out unto<emph.end type="italics"></emph.end> I G F, <emph type="italics"></emph>we ſhall have the Aggregate of all <lb></lb>the Parallels contained in the Quadrilatural Figure <lb></lb>equal to the Aggregate of all the Parallels compre<lb></lb>hended in the Triangle<emph.end type="italics"></emph.end> A E <emph type="italics"></emph>B; For thoſe in the Triangle<emph.end type="italics"></emph.end> I E F <emph type="italics"></emph>are equal <lb></lb>to thoſe contained in the Triangle<emph.end type="italics"></emph.end> G I A, <emph type="italics"></emph>and thoſe that are in the<emph.end type="italics"></emph.end> Tra<lb></lb>pezium <emph type="italics"></emph>are in common. </s> <s>Now ſince all and ſingular the Inſtants of Time <lb></lb>do anſwer to all and ſingular the Points of the Line A B; and ſince the <lb></lb>Parallels contained in the Triangle<emph.end type="italics"></emph.end> A E <emph type="italics"></emph>B do repreſent the degrees of Ac<lb></lb>celeration or encreaſing Velocity, and the Parallels contained in the Pa<lb></lb>rallelogram do likewiſe repreſent as many degrees of Equable Motion or <lb></lb>unencreaſing Velocity: It appeareth, that as many Moments of Velocity <lb></lb>paſſed in the Accelerate Motion according to the encreaſing Parallels of the <lb></lb>Triangle A E B, as in the Equable Motion according to the Parallels of <lb></lb>the Parallelogram G B: Becauſe what is wanting in the firſt half of the<emph.end type="italics"></emph.end><pb xlink:href="069/01/147.jpg" pagenum="144"></pb><emph type="italics"></emph>Accelerate Motion of the Velocity of the Equable Motion (which defi<lb></lb>cient Moments are repreſented by the Parallels of the Triangle A<emph.end type="italics"></emph.end> G I) <lb></lb><emph type="italics"></emph>is made up by the moments repreſented by the Parallels of the Triangle<emph.end type="italics"></emph.end><lb></lb>I E F. <emph type="italics"></emph>It is manifeſt, therefore, that thoſe Spaces are equal which are <lb></lb>in the ſame Time by two Moveables, one whereof is moved with a Mo<lb></lb>tion uniformly Accelerated from Reſt, the other with a Motion Equable <lb></lb>according to the Moment ſubduple of that of the greateſt Velocity of the <lb></lb>Accelerated Motion: Which was to be demonſtrated.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>THEOR. II. PROP. II.</s></p><p type="main"> <s>If a <emph type="italics"></emph>M<emph.end type="italics"></emph.end>oveable deſcend out of Reſt with a <emph type="italics"></emph>M<emph.end type="italics"></emph.end>oti<lb></lb>on uniformly Accelerate, the Spaces which it <lb></lb>paſſeth in any whatſoever Times are to each <lb></lb>other in a proportion Duplicate of the ſame <lb></lb>Times; that is, they are as the Squares of <lb></lb>them.</s></p><p type="main"> <s><emph type="italics"></emph>Let<emph.end type="italics"></emph.end> A B <emph type="italics"></emph>repreſent a length of Time beginning at the firſt Inſtant A; <lb></lb>and let<emph.end type="italics"></emph.end> A D <emph type="italics"></emph>and<emph.end type="italics"></emph.end> A E <emph type="italics"></emph>repreſent any two parts of the ſaid Time; <lb></lb>and let<emph.end type="italics"></emph.end> H I <emph type="italics"></emph>be a Line in which the Moveable out of H, (as the firſt <lb></lb>beginning of the Motion) deſcendeth uniformly accelerating; and let the<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.147.1.jpg" xlink:href="069/01/147/1.jpg"></figure><lb></lb><emph type="italics"></emph>Space<emph.end type="italics"></emph.end> H L <emph type="italics"></emph>be paſſed in the firſt Time<emph.end type="italics"></emph.end> A D; <emph type="italics"></emph>and let<emph.end type="italics"></emph.end> H M <lb></lb><emph type="italics"></emph>be the Space that it ſhall deſcend in the Time<emph.end type="italics"></emph.end> A E. <emph type="italics"></emph>I ſay, <lb></lb>the Space<emph.end type="italics"></emph.end> M H <emph type="italics"></emph>is to the Space<emph.end type="italics"></emph.end> H L <emph type="italics"></emph>in duplicate propor<lb></lb>tion of that which the Time<emph.end type="italics"></emph.end> E A <emph type="italics"></emph>hath to the Time<emph.end type="italics"></emph.end> A D<emph type="italics"></emph>: <lb></lb>Or, if you will, that the Spaces<emph.end type="italics"></emph.end> M H <emph type="italics"></emph>and<emph.end type="italics"></emph.end> H L <emph type="italics"></emph>are to one <lb></lb>another in the ſame proportion as the Squares<emph.end type="italics"></emph.end> E A <emph type="italics"></emph>and<emph.end type="italics"></emph.end><lb></lb>A D. <emph type="italics"></emph>Draw the Line<emph.end type="italics"></emph.end> A C <emph type="italics"></emph>at any Angle with<emph.end type="italics"></emph.end> A B, <emph type="italics"></emph>and <lb></lb>from the points D and E draw the Parallels<emph.end type="italics"></emph.end> D O <emph type="italics"></emph>and<emph.end type="italics"></emph.end><lb></lb>P E<emph type="italics"></emph>: of which<emph.end type="italics"></emph.end> D O <emph type="italics"></emph>will repreſent the greateſt degree <lb></lb>of Velocity acquired in the Inſtant D of the Time<emph.end type="italics"></emph.end> A D; <lb></lb><emph type="italics"></emph>and<emph.end type="italics"></emph.end> P <emph type="italics"></emph>the greateſt degree of Velocity acquired in the In<lb></lb>ſtant E of the Time<emph.end type="italics"></emph.end> A E. <emph type="italics"></emph>And becauſe we have de<lb></lb>monſtrated in the laſt Propoſition concerning Spaces, that <lb></lb>thoſe are equal to one another, of which two Moveables <lb></lb>have paſt in the ſame Time, the one by a Moveable out <lb></lb>of Reſt with a Motion uniformly Accelerate, and the <lb></lb>other by the ſame Moveable with an Equable Motion, <lb></lb>whoſe Velocity is ſubduple to the greateſt acquired by the <lb></lb>Accelerate Motion: Therefore<emph.end type="italics"></emph.end> M H <emph type="italics"></emph>and<emph.end type="italics"></emph.end> H L <emph type="italics"></emph>are the Spaces that two <lb></lb>Lquable Motions, whoſe Velocities ſhould be as the half of<emph.end type="italics"></emph.end> P E, <emph type="italics"></emph>and<emph.end type="italics"></emph.end><pb xlink:href="069/01/148.jpg" pagenum="145"></pb><emph type="italics"></emph>half of<emph.end type="italics"></emph.end> O D, <emph type="italics"></emph>would paſſe in the Times<emph.end type="italics"></emph.end> E A <emph type="italics"></emph>and<emph.end type="italics"></emph.end> D A. <emph type="italics"></emph>If it be proved <lb></lb>therefore that theſe Spaces<emph.end type="italics"></emph.end> M H <emph type="italics"></emph>and<emph.end type="italics"></emph.end> L H <emph type="italics"></emph>are in duplicate proportion to <lb></lb>the Times<emph.end type="italics"></emph.end> E A <emph type="italics"></emph>and<emph.end type="italics"></emph.end> D A; <emph type="italics"></emph>We ſhall have done that which was intended. <lb></lb></s> <s>But in the fourth Propoſition of the Firſt Book we have demonſtrated: <lb></lb>That the Spaces paſt by two Moveables with an Equable Motion are <lb></lb>to each other in a proportion compounded of the proportion of the Velo<lb></lb>cities and of the proportion of the Times: But in this caſe the propor<lb></lb>tion of the Velocities and the proportion of the Times is the ſame<emph.end type="italics"></emph.end> (<emph type="italics"></emph>for <lb></lb>as the half of<emph.end type="italics"></emph.end> P E <emph type="italics"></emph>is to the half of<emph.end type="italics"></emph.end> O D, <emph type="italics"></emph>or the whole<emph.end type="italics"></emph.end> P E <emph type="italics"></emph>to the whole<emph.end type="italics"></emph.end><lb></lb>O D, <emph type="italics"></emph>ſo is<emph.end type="italics"></emph.end> A E <emph type="italics"></emph>to<emph.end type="italics"></emph.end> A D<emph type="italics"></emph>: Therefore the proportion of the Spaces paſ<lb></lb>ſed is double to the proportion of the Times. </s> <s>Which was to be demon<lb></lb>ſtrated.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Hence likewiſe it is manifeſt, that the proportion of the ſame Spaces <lb></lb>is double to the proportions of the greateſt degrees of Velocity: that is, <lb></lb>of the Lines<emph.end type="italics"></emph.end> P E <emph type="italics"></emph>and<emph.end type="italics"></emph.end> O D<emph type="italics"></emph>: becauſe<emph.end type="italics"></emph.end> P E <emph type="italics"></emph>is to<emph.end type="italics"></emph.end> O D, <emph type="italics"></emph>as<emph.end type="italics"></emph.end> E A <emph type="italics"></emph>to<emph.end type="italics"></emph.end> D A.</s></p><p type="head"> <s>COROLARY I.</s></p><p type="main"> <s><emph type="italics"></emph>Hence it is manifeſt, that if there were many equal Times taken in or<lb></lb>der from the firſt Inſtant or beginniug of the Motion, as ſuppoſe<emph.end type="italics"></emph.end><lb></lb>A D, D E, E F, F G, <emph type="italics"></emph>in which the Spaces<emph.end type="italics"></emph.end> H L, L M, M N, N I <lb></lb><emph type="italics"></emph>are paſſed, thoſe Spaces ſhall be to one another as the odd numbers <lb></lb>from an Vnite:<emph.end type="italics"></emph.end> ſcilicet, <emph type="italics"></emph>as 1, 3, 5, 7. For this is the Rate or pro<lb></lb>portion of the exceſſes of the Squares of Lines that equally exceed <lb></lb>one another, and the exceſſe of which is equal to the least of them, <lb></lb>or, if you will, of Squares that follow one another, beginning<emph.end type="italics"></emph.end> ab <lb></lb>Unitate. <emph type="italics"></emph>Whilſt therefore the degree of Velocity is encreaſed ac<lb></lb>cording to the ſimple Series of Numbers in equal Times, the Spaces <lb></lb>paſt in thoſe Times make their encreaſe according to the Series of <lb></lb>odd Numbers from an Vnite.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>Be pleaſed to ſtay your Reading, whilſt I do paraphraſe <lb></lb>touching a certain Conjecture that came into my mind <lb></lb>but even now; for the explanation of which, unto your under<lb></lb>ſtanding and my own, I will deſcribe a ſhort Scheme: in which I <lb></lb>fanſie by the Line A I the continuation of the Time after the firſt <lb></lb>Inſtant, applying the Right Line A F unto A according to any <lb></lb>Angle: and joyning together the Terms I F, I divide the Time A I <lb></lb>in half at C, and then draw C B parallel to I F. <emph type="italics"></emph>A<emph.end type="italics"></emph.end>nd then conſide<lb></lb>ring B C, as the greateſt degree of Velocity which beginning from <lb></lb>Reſt in the firſt Inſtant of the Time <emph type="italics"></emph>A<emph.end type="italics"></emph.end> goeth augmenting accord<lb></lb>ing to the encreaſe of the Parallels to B C, drawn in the Triangle <lb></lb><emph type="italics"></emph>A<emph.end type="italics"></emph.end> B C, (which is all one as to encreaſe according to the encreaſe <lb></lb>of the Time) I admit without diſpute, upon what hath been ſaid <lb></lb>already, That the Space paſt by the falling Moveable with the <pb xlink:href="069/01/149.jpg" pagenum="146"></pb>Velocity encreaſed in the manner aforeſaid would be equal to the <lb></lb>Space that the ſaid Moveable would paſſe, in caſe it were in the <lb></lb>ſame Time <emph type="italics"></emph>A<emph.end type="italics"></emph.end> C, moved with an Uniform Motion, whoſe degree of <lb></lb>Velocity ſhould be equal to E C, the half of B C. </s> <s>I now proceed <lb></lb>farther, and imagine the Moveable; having deſcended with an <lb></lb><emph type="italics"></emph>A<emph.end type="italics"></emph.end>ccelerate Motion, to have in the Inſtant <lb></lb>C the degree of Velocity B C: It is ma<lb></lb><figure id="id.069.01.149.1.jpg" xlink:href="069/01/149/1.jpg"></figure><lb></lb>nifeſt, that if it did continue to move <lb></lb>with the ſame degree of Velocity B C, <lb></lb>without farther <emph type="italics"></emph>A<emph.end type="italics"></emph.end>cceleration, it would <lb></lb>paſſe in the following Time C I, a Space <lb></lb>double to that which it paſſed in the equal <lb></lb>Time <emph type="italics"></emph>A<emph.end type="italics"></emph.end> C, with the degree of Uniform <lb></lb>Velocity E C, the half of the Degree B C. <lb></lb></s> <s>But becauſe the Moveable deſcendeth <lb></lb>with a Velocity encreaſed alwaies Uni<lb></lb>formly in all equal Times; it will add to <lb></lb>the degree C B in the following Time <lb></lb>C I, thoſe Tame Moments of Velocity <lb></lb>that encreaſe according to the Parallels of <lb></lb>the Triangle B F G, equal to the Triangle <lb></lb><emph type="italics"></emph>A<emph.end type="italics"></emph.end> B C. </s> <s>So that adding to the degree of <lb></lb>Velocity G I, the half of the degree F G, the greateſt of thoſe ac<lb></lb>quired in the <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ccelerate Motion, and regulated by the Parallels of <lb></lb>the Triangle B F G, we ſhall have the degree of Velocity I N, with <lb></lb>which, with an Uniform Motion, it would have moved in the <lb></lb>Time C I: Which degree I N, being triple the degree E C, pro<lb></lb>veth that the Space paſſed in the ſecond Time C I ought to be tri<lb></lb>ple to that of the firſt Time C <emph type="italics"></emph>A. A<emph.end type="italics"></emph.end>nd if we ſhould ſuppoſe to be <lb></lb>added to <emph type="italics"></emph>A<emph.end type="italics"></emph.end> I another equal part of Time I O, and the Triangle to <lb></lb>be enlarged unto <emph type="italics"></emph>A<emph.end type="italics"></emph.end> P O; it is manifeſt, that if the Motion ſhould <lb></lb>continue for all the Time I O with the degree of Velocity I F, <lb></lb>acquired in the <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ccelerate Motion in the Time <emph type="italics"></emph>A<emph.end type="italics"></emph.end> I, that degree <lb></lb>I F being Quadruple to E C, the Space paſſed would be Quadruple <lb></lb>to that paſſed in the equal firſt Time <emph type="italics"></emph>A<emph.end type="italics"></emph.end> C: But continuing the <lb></lb>encreaſe of the Uniform <emph type="italics"></emph>A<emph.end type="italics"></emph.end>cceleration in the Triangle F P Q like <lb></lb>to that of the Triangle <emph type="italics"></emph>A<emph.end type="italics"></emph.end> B C, which being reduced to equable <lb></lb>Motion addeth the degree equal to E C, Q R being added, equal <lb></lb>to E C, we ſhall have the whole Equable Velocity exerciſed in the <lb></lb>Time I O, quintuple to the Equable Velocity of the firſt Time <emph type="italics"></emph>A<emph.end type="italics"></emph.end> C, <lb></lb>and therefore the Space paſſed quintuple to that paſt in the firſt <lb></lb>Time <emph type="italics"></emph>A<emph.end type="italics"></emph.end> C. </s> <s>We ſee therefore, even by this familiar computation, <lb></lb>That the Spaces paſſed in equal Times by a Moveable which <lb></lb>departing from Reſt goeth acquiring Velocity, according to the <lb></lb>encreaſe of the Time, are to one another as the odd Numbers <emph type="italics"></emph>ab<emph.end type="italics"></emph.end><pb xlink:href="069/01/150.jpg" pagenum="147"></pb><emph type="italics"></emph>unitate 1, 3, 5: A<emph.end type="italics"></emph.end>nd that the Spaces paſſed being conjunctly taken, <lb></lb>that paſſed in the double Time is quadruple to that paſſed in the <lb></lb>ſubduple, that paſſed in the triple Time is nonuple; and, in a word, <lb></lb>that the Spaces paſſed are in duplicate proportion to their Times; <lb></lb>that is, as the Squares of the ſaid Times.</s></p><p type="main"> <s>SIMP. </s> <s>I muſt confeſſe that I have taken more pleaſure in this <lb></lb>plain and clear diſcourſe of <emph type="italics"></emph>Sagredus,<emph.end type="italics"></emph.end> than in the to-me-more <lb></lb>obſcure Demonſtration of the <emph type="italics"></emph>A<emph.end type="italics"></emph.end>uthor: ſo that I am very well <lb></lb>ſatisfied, that the buſineſſe is to ſucceed as hath been ſaid, the <lb></lb>Definition of Uniformly <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ccelerate Motion being ſuppoſed, and <lb></lb>granted. </s> <s>But whether this be the <emph type="italics"></emph>A<emph.end type="italics"></emph.end>cceleration of which Nature <lb></lb>maketh uſe in the Motion of its deſcending Grave Bodies, I yet <lb></lb>make a queſtion: and therefore for information of me, and of <lb></lb>others like unto me, me thinks it would be ſeaſonable in this place <lb></lb>to produce ſome Experiment amongſt thoſe which were ſaid to be <lb></lb>many, which in ſundry Caſes agree with the Concluſions demon<lb></lb>ſtrated.</s></p><p type="main"> <s>SALV. You, like a true <emph type="italics"></emph>A<emph.end type="italics"></emph.end>rtiſt, make a very reaſonable demand, <lb></lb>and ſo it is uſual and convenient to do in Sciences that apply <lb></lb>Mathematical Demonſtrations to Phyſical Concluſions, as we ſee <lb></lb>in the Profeſſors of Perſpection, <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ſtronomy, Mechanicks, Muſick, <lb></lb>and others, who with Senſible Experiments confirm thoſe their <lb></lb>Principles that are as the foundations of all the following Structure: <lb></lb>and therefore I deſire that it may not be thought ſuperfluous, that <lb></lb>we diſcourſe with ſome prolixity upon this firſt and grand funda<lb></lb>mental on which we lay the weight of the Immenſe Machine of <lb></lb>infinite Concluſions, of which we have but a very ſmall part ſet <lb></lb>down in this Book by our <emph type="italics"></emph>A<emph.end type="italics"></emph.end>uthor, who hath done enough to open <lb></lb>the way and door that hath been hitherto ſhut unto all Specula<lb></lb>tive Wits. </s> <s>Touching Experiments, therefore, the <emph type="italics"></emph>A<emph.end type="italics"></emph.end>uthor hath <lb></lb>not omitted to make ſeveral; and to aſſure us, that the <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ccelerati<lb></lb>on of natural-deſcending Graves hapneth in the aforeſaid propor<lb></lb>tion, I have many times in his company ſet my ſelf to make a triall <lb></lb>thereof in the following Method.</s></p><p type="main"> <s>In a priſme or Piece of Wood, about twelve yards long, and <lb></lb>half a yard broad one way, and three Inches the other, we made, <lb></lb>upon the narrow Side or edge a Groove of little more than an Inch <lb></lb>wide; we ſhot it with the Grooving Plane very ſtraight, and to <lb></lb>make it very ſmooth and ſleek, we glued upon it a piece of Vellum, <lb></lb>poliſhed and ſmoothed as exactly as can be poſſible: and in it we <lb></lb>have let a brazen Ball, very hard, round, and ſmooth, deſcend. <lb></lb></s> <s>Having placed the ſaid Priſme Pendent, raiſing one of its ends <lb></lb>above the Horizontal Plane a yard or two at pleaſure, we have let <lb></lb>the Ball (as I ſaid) deſcend along the Grove, obſerving, in the <lb></lb>manner that I ſhall tell you preſently, the Time which it ſpent in <pb xlink:href="069/01/151.jpg" pagenum="148"></pb>runing it all; repeating the ſame obſervation again and again to <lb></lb>aſſure our ſelves of the Time, in which we never found any diffe<lb></lb>rence, no not ſo much as the tenth part of one beat of the Pulſe. <lb></lb></s> <s>Having done, and preciſely ordered this buſineſſe, we made the <lb></lb>ſame Ball to deſcend only the fourth part of the length of that <lb></lb>Grove: and having meaſured the time of its deſcent, we alwaies <lb></lb>found it to be punctually half the other. </s> <s>And then making trial of <lb></lb>other parts, examining one while the Time of the whole Length <lb></lb>with the Time of half the Length, or with that of 2/3, or of 3/4, or, in <lb></lb>brief, with any whatever other Diviſion, by Experiments repeated <lb></lb>near an hundred Times, we alwaies found the Spaces to be to one <lb></lb>another as the Squares of the Times. </s> <s>And this in all Inclinations <lb></lb>of the Plane, that is, of the Grove in which the Ball was made to <lb></lb>deſcend. </s> <s>In which we obſerved moreover, that the Times of the <lb></lb>Deſcents along ſundry Inclinations did retain the ſame proportion <lb></lb>to one another, exactly, which anon you will ſee aſſigned to them, <lb></lb>and demonſtrated by the Author. </s> <s>And as to the meaſuring of the <lb></lb>Time; we had a good big Bucket full of Water hanged on high, <lb></lb>which by a very ſmall hole, pierced in the bottom, ſpirted, or, as <lb></lb>we ſay, ſpin'd forth a ſmall thread of Water, which we received <lb></lb>with a ſmall cup all the while that the Ball was deſcending in the <lb></lb>Grove, and in its parts; and then weighing from time to time the <lb></lb>ſmall parcels of Water, in that manner gathered, in an exact pair <lb></lb>of ſcales, the differences and proportions of their Weights gave <lb></lb>juſtly the differences and proportions of the Times; and this with <lb></lb>ſuch exactneſſe, that, as I ſaid before, the trials being many <lb></lb>and many times repeated, they never differed any conſiderable <lb></lb>matter.</s></p><p type="main"> <s>SIMP. </s> <s>I ſhould have received great ſatisfaction by being preſent <lb></lb>at thoſe Experiments: but being confident of your diligence in <lb></lb>making them, and veracity in relating them, I content my ſelf, and <lb></lb>admit them for true and certain.</s></p><p type="main"> <s>SALV. </s> <s>We may, then, reaſſume our Reading, and go on.</s></p><p type="head"> <s>COROLLARY II.</s></p><p type="main"> <s>It is collected in the ſecond place, that if any two Spaces are ta<lb></lb>ken from the beginning of the Motion, paſſed in any Times, <lb></lb>thoſe Times ſhall be unto each other as one of them is to a <lb></lb>Space that is the Mean proportional between them.</s></p><p type="main"> <s><emph type="italics"></emph>For taking two Spaces S T, and S V from the beginning of the Mo<lb></lb>tion S, to which S X is a Mean-proportional, the Time of the deſcent <lb></lb>along S T, ſhall be to the Time of the deſcent along S V, as S T to S X; <lb></lb>or, if you will, the Time along S V ſhall be to the Time along S T,<emph.end type="italics"></emph.end><pb xlink:href="069/01/152.jpg" pagenum="149"></pb><figure id="id.069.01.152.1.jpg" xlink:href="069/01/152/1.jpg"></figure><lb></lb><emph type="italics"></emph>as VS is to SX. </s> <s>For it is demonſtrated, that Spaces <lb></lb>paſſed are in duplicate proportion to the Times, or, (which <lb></lb>is the ſame) are as the Squares of the Times: But the pro<lb></lb>portion of the Space VS to the Space ST is double to the <lb></lb>proportion of V S to SX, or is the ſame that V S, and S X <lb></lb>ſquared have to one another: Therefore, the proportion of <lb></lb>the Times of the Motion by V S, and ST, is as the Spaces or <lb></lb>Lines V S to S X.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>SCHOLIUM.</s></p><p type="main"> <s><emph type="italics"></emph>That which is demonſtrated in Motions that are made Perpendicu<lb></lb>larly, may be underſtood alſo to hold true in the Motions made along <lb></lb>Planes of any whatever Inclination; for it is ſuppoſed, that in them <lb></lb>the degree of Acceleration encreaſeth in the ſame proportion; that <lb></lb>is, according to the encreaſe of the Time; or, if you will, according <lb></lb>to the ſimple and primary Series of Numbers.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>Here I deſire <emph type="italics"></emph>Sagredus,<emph.end type="italics"></emph.end> that I alſo may be allowed, al<lb></lb>beit perhaps with too much tediouſneſſe in the opinion of <emph type="italics"></emph>Simplici<lb></lb>us,<emph.end type="italics"></emph.end> to defer for a little time the preſent Reading, untill I may have <lb></lb>explained what from that which hath been already ſaid and de<lb></lb>monſtrated, and alſo from the knowledge of certain Mechanical <lb></lb>Concluſions heretofore learnt of our <emph type="italics"></emph>Academick,<emph.end type="italics"></emph.end> I now remember <lb></lb>to adjoyn for the greater confirmation of the truth of the Princi<lb></lb>ple, which hath been examined by us even now with probable <lb></lb>Reaſons and Experiments: and, which is of more importance, for <lb></lb>the Geometrical proof of it, let me firſt demonſtrate one ſole Ele<lb></lb>mental <emph type="italics"></emph>Lemma<emph.end type="italics"></emph.end> in the Contemplation of <emph type="italics"></emph>Impetus's.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>If our advantage ſhall be ſuch as you promiſeus, there <lb></lb>is no time that I would not moſt willingly ſpend in diſcourſing <lb></lb>about the confirmation and thorow eſtabliſhing theſe Sciences of <lb></lb>Motion: and as to my own particular, I not only grant you liber<lb></lb>ty to ſatisfie your ſelf in this particular, but moreover entreat you <lb></lb>to gratifie, as ſoon as you can, the Curioſity which you have begot <lb></lb>in me touching the ſame: and I believe that <emph type="italics"></emph>Simplicius<emph.end type="italics"></emph.end> alſo is of the <lb></lb>ſame mind.</s></p><p type="main"> <s>SIMP. </s> <s>I cannot deny what you ſay.</s></p><p type="main"> <s><emph type="italics"></emph>S<emph.end type="italics"></emph.end>ALV. </s> <s>Seeing then that I have your permiſſion, I will in the <lb></lb>firſt place conſider, as an Effect well known, That</s></p><pb xlink:href="069/01/153.jpg" pagenum="150"></pb><p type="head"> <s>LEMMA.</s></p><p type="main"> <s><emph type="italics"></emph>That the Moments or Velocities of the ſame Moveable are different <lb></lb>upon different Inclinations of Planes, and the greateſt is by the <lb></lb>Line elevated perpendicularly above the Horizon, and by the <lb></lb>others inclined, the ſaid Velocity diminiſheth according as they <lb></lb>more and more depart from Perpendicularity, that is, as they in<lb></lb>cline more obliquely: ſo that the Impetus, Talent, Energy, or, we <lb></lb>may ſay, Moment of deſcending is diminiſhed in the Moveable by <lb></lb>the ſubjected Plane, upon which the ſaid Moveable lyeth and <lb></lb>deſcendeth.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>And the better to expreſs my ſelf, let the Line A B be perpen<lb></lb>dicularly erected upon the Horizon A C: then ſuppoſe the <lb></lb>ſame to be declined in ſundry Inclinations towards the Horizon, as <lb></lb>in A D, A E, A F, <emph type="italics"></emph>&c.<emph.end type="italics"></emph.end> I ſay, that the greateſt and total <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end><lb></lb>of the Grave Body in deſcending is along the Perpendicular B A, <lb></lb>and leſs than that along D A, <lb></lb><figure id="id.069.01.153.1.jpg" xlink:href="069/01/153/1.jpg"></figure><lb></lb>and yet leſs along E A; and <lb></lb>ſucceſſively diminiſhing along <lb></lb>the more inclined F <emph type="italics"></emph>A,<emph.end type="italics"></emph.end> and fi<lb></lb>nally is wholly extinct in the <lb></lb>Horizontal C <emph type="italics"></emph>A,<emph.end type="italics"></emph.end> where the <lb></lb>Moveable is indifferent either <lb></lb>to Motion or Reſt, and hath not <lb></lb>of it ſelf any Inclination to <lb></lb>move one way or other, nor yet <lb></lb>any Reſiſtance to its being mo<lb></lb>ved: for as it is impoſſi<lb></lb>ble that a Grave Body, or a <lb></lb>Compound thereof ſhould move naturally upwards, receding from <lb></lb>the Common Center, towards which all Grave Matters conſpire <lb></lb>to go, ſo it is impoſſible that it do ſpontaneouſly move, unleſs <lb></lb>with that Motion its particular Center of Gravity do acquire Proxi<lb></lb>mity to the ſaid Common Center: ſo that upon the Horizontal <lb></lb>which here is underſtood to be a Superficies equidiſtant from the <lb></lb>ſaid Center, and therefore altogether void of Inclination, the <emph type="italics"></emph>Im<lb></lb>petus<emph.end type="italics"></emph.end> or Moment of that ſame Moveable ſhall be nothing at all. <lb></lb></s> <s>Having underſtood this mutation of <emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> I am to explain that <lb></lb>which, in an old Treatiſe of the Mechanicks, written heretofore <lb></lb>in <emph type="italics"></emph>Padona<emph.end type="italics"></emph.end> by our <emph type="italics"></emph>Academick,<emph.end type="italics"></emph.end> only for the uſe of his Scholars, was <lb></lb>diffuſely and demonſtratively proved, upon the occaſion of con<lb></lb>ſidering the Original and Nature of the admirable Inſtrument cal<lb></lb>led the Screw, and it is, With what proportion that mutation of <pb xlink:href="069/01/154.jpg" pagenum="151"></pb><emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> is made along ſeveral Inclinations or Declivities of <lb></lb>Planes.</s></p><p type="main"> <s>As, for example, in the inclined Plane A F, drawing its Eleva<lb></lb>tion above the Horizontal, that is, the Line F C, along the which <lb></lb>the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of a Grave Body, and the Moment of Deſcent is the <lb></lb>greateſt; it is ſought what proportion this Moment hath to the <lb></lb>Moment of the ſame Moveable along the Declivity F A: Which <lb></lb>Proportion, I ſay, is Reciprocal to the ſaid Lengths. </s> <s>And this is <lb></lb>the <emph type="italics"></emph>Lemma<emph.end type="italics"></emph.end> that was to go before the Theorem, which I hope to be <lb></lb>able anon to Demonſtrate. </s> <s>Hence it is manifeſt, That the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end><lb></lb>of Deſcent of a Grave Body is as much as the Reſiſtance or leaſt <lb></lb>force that ſufficeth to arreſt and ſtay it. </s> <s>For this Force or Reſi<lb></lb>ſtance, and its meaſure, I will make uſe of the Gravity of another <lb></lb>Moveable. </s> <s>Let us now upon the Plane F A put the Moveable G <lb></lb>tyed to a thread which ſliding over F hath faſtned at its other end <lb></lb>the Weight H: and let us conſider that the Space of the Deſcent <lb></lb>or Aſcent of the Weight H along the Perpendicular, is alwaies <lb></lb>equal to the whole Aſcent or Deſcent of the other Moveable G <lb></lb>along the ^{*} Declivity A F, but yet not to the Aſcent or Deſcent </s></p><p type="main"> <s><arrow.to.target n="marg1092"></arrow.to.target><lb></lb>along the Perpendicular, in which only the ſaid Moveable G (like <lb></lb>as every other Moveable) exerciſeth its Reſiſtance. </s> <s>Which is <lb></lb>manifeſt: for conſidering in the Triangle AFC the Motion of <lb></lb>the Moveable G, as for example, upwards from A to F, to be com<lb></lb>poſed of the tranſverſe Horizontal Line A C, and of the Perpendi<lb></lb>cular C F: <emph type="italics"></emph>A<emph.end type="italics"></emph.end>nd in regard, that as to the Horizontal Plane along <lb></lb>which the Moveable, as hath been ſaid, hath no Reſiſtance to mo<lb></lb>ving (it not making by that Motion any loſs, nor yet acquiſt in <lb></lb>regard of its particular diſtance from the Common Center of Grave <lb></lb>Matters, which in the Horizon continueth ſtill the ſame) it remai<lb></lb>neth that the Reſiſtance be only in reſpect of the <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ſcent that it is to <lb></lb>make along the Perpendicular C F. </s> <s>Whilſt therefore the Grave <lb></lb>Moveable G, moving from <emph type="italics"></emph>A<emph.end type="italics"></emph.end> to F, hath only the Perpendicular <lb></lb>Space C F to reſiſt in its <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ſcent, and whilſt the other Grave Move<lb></lb>able H deſcendeth along the Perpendicular of neceſſity as far as <lb></lb>the whole Space F <emph type="italics"></emph>A,<emph.end type="italics"></emph.end> and that the ſaid proportion of <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ſcent and <lb></lb>Deſcent maintains it ſelf alwaies the ſame, be the Motion of the <lb></lb>ſaid Moveables little or much (by reaſon they are tyed toge<lb></lb>ther) we may confidently affirm, that in caſe there were an <emph type="italics"></emph>Equi<lb></lb>librium,<emph.end type="italics"></emph.end> that is Reſt, to enſue betwixt the ſaid Moveables, the Mo<lb></lb>ments, the Velocities, or their Propenſions to Motion, that is the <lb></lb>Spaces which they would paſs in the ſame Time ſhould anſwer re<lb></lb>ciprocally to their Gravities, according to that which is demonſtra<lb></lb>ted in all caſes of Mechanick Motions: ſo that it ſhall ſuffice to <lb></lb>impede the deſcent of G, if H be but ſo much leſs grave than it, as <lb></lb>in proportion the Space C F is leſſer than the Space F <emph type="italics"></emph>A.<emph.end type="italics"></emph.end> Therefore <pb xlink:href="069/01/155.jpg" pagenum="152"></pb>ſuppoſe that the Moveable G is to the Moveable H, as F <emph type="italics"></emph>A<emph.end type="italics"></emph.end> is to <lb></lb>F C; and then the <emph type="italics"></emph>Equilibrium<emph.end type="italics"></emph.end> ſhall follow, that is, the Moveables <lb></lb>H and G ſhall have equal Moments, and the Motion of the ſaid <lb></lb>Moveables ſhall ceaſe. <emph type="italics"></emph>A<emph.end type="italics"></emph.end>nd becauſe we ſee that the <emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end><lb></lb>Energy, Moment, or Propenſion of a Moveable to Motion is the <lb></lb>ſame as is the Force or ſmalleſt Reſiſtance that ſufficeth to ſtop it; <lb></lb>and becauſe it hath been concluded, that the Grave Body H is ſuf. <lb></lb></s> <s>ficient to arreſt the Motion of <lb></lb><figure id="id.069.01.155.1.jpg" xlink:href="069/01/155/1.jpg"></figure><lb></lb>the Grave Body G: Therefore <lb></lb>the leſſer Weight H, which in <lb></lb>the Perpendicular F C imploy<lb></lb>eth its total Moment, ſhall be <lb></lb>the preciſe meaſure of the par<lb></lb>tial Moment that the greater <lb></lb>Weight G exerciſeth along the <lb></lb>inclined Plane F <emph type="italics"></emph>A<emph.end type="italics"></emph.end>: But the <lb></lb>meaſure of the total Moment of <lb></lb>the ſaid Grave Body G, is the <lb></lb>ſelf ſame, (ſince that to impede <lb></lb>the Perpendicular Deſcent of a <lb></lb>Grave Body there is required the oppoſition of ſuch another Grave <lb></lb>Body, which likewiſe is at liberty to move Perpendicularly:) <lb></lb>Therefore the partial <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> or Moment of G along the inclined <lb></lb>Plane F A ſhall be to the grand and total <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of the ſame G <lb></lb>along the Perpendicular F C, as the Weight H to the Weight G: <lb></lb>that is, by Conſtruction, as the ſaid Perpendicular F C, the Eleva<lb></lb>tion of the inclined Plane, is to the ſame inclined Plane F A: <lb></lb>Which is that that by the <emph type="italics"></emph>Lemma<emph.end type="italics"></emph.end> was propoſed to be demon<lb></lb>ſtrated, and which by our Author, as we ſhall ſee, is ſuppoſed as <lb></lb>known in the ſecond part of the Sixth Propoſition of the preſent <lb></lb>Treatiſe.</s></p><p type="margin"> <s><margin.target id="marg1092"></margin.target>* Or inclined <lb></lb>Plane.</s></p><p type="main"> <s>SAGR. </s> <s>From this that you have already concluded I conceive <lb></lb>one may eaſily deduce, arguing <emph type="italics"></emph>ex æquali<emph.end type="italics"></emph.end> by perturbed Proportion, <lb></lb>that the Moments of the ſame Moveable, along Planes variouſly <lb></lb>inclined (as F A and F I) that have the ſame Elevation, are to each <lb></lb>other in Reciprocal proportion to the ſame Planes.</s></p><p type="main"> <s>SALV. <emph type="italics"></emph>A<emph.end type="italics"></emph.end> moſt certain Concluſion. </s> <s>This being agreed on, we <lb></lb>will paſs in the next place to demonſtrate the <emph type="italics"></emph>Theoreme,<emph.end type="italics"></emph.end> namely, <lb></lb>that</s></p><pb xlink:href="069/01/156.jpg" pagenum="153"></pb><p type="head"> <s>THEOREM.</s></p><p type="main"> <s><emph type="italics"></emph>The degrees of Velocity of a Moveable deſcending with a Natural <lb></lb>Motion from the ſame height along Planes in any manner inclined <lb></lb>at the arrival to the Horizon are alwaies equal, Impediments be<lb></lb>ing removed.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Here we are in the firſt place to advertiſe you, that it having <lb></lb>been proved, that in any Inclination of the Plane the Move<lb></lb>able from its receſſion from Quieſſence goeth encreaſing its Ve<lb></lb>locity, or quantity of its <emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> with the proportion of the <lb></lb>Time (according to the Definition which the Author giveth of <lb></lb>Motion naturally Accelerate) whereupon, as he hath by the pre<lb></lb>cedent Propoſition demonſtrated, the Spaces paſſed are in dupli<lb></lb>cate proportion to the Times, and, conſequently, to the degrees <lb></lb>of Velocity: look what the <emph type="italics"></emph>Impetus's<emph.end type="italics"></emph.end> were in that which was firſt <lb></lb>moved, ſuch proportionally ſhall be the degrees of Velocity gai<lb></lb>ned in the ſame Time; ſeeing that both theſe and thoſe encreaſe <lb></lb>with the ſame proportion in the ſame Time.</s></p><p type="main"> <s>Now let the inclined Plane be A B, its elevation above the Ho <lb></lb>rizon the Perpendicular A C, and the Horizontal Plane C B: and <lb></lb>becauſe, as was even now concluded, the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of a Moveable <lb></lb>along the Perpendicular A C is to the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of the ſame along <lb></lb>the inclined Plane A B, as A B is to A C, let there be taken in the <lb></lb>inclined Plane A B, A D a third proportional to A B and A C: <lb></lb>The <emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> therefore, along A C is to the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> along A B, <lb></lb>that is along A D, as A C is to <lb></lb><figure id="id.069.01.156.1.jpg" xlink:href="069/01/156/1.jpg"></figure><lb></lb>A D: And therefore the Move<lb></lb>able in the ſame Time that it <lb></lb>would paſs the Perpendicular <lb></lb>Space AC, ſhall likewiſe paſs the <lb></lb>Space A D, in the inclined Plane <lb></lb>A B, (the Moments being as <lb></lb>the Spaces:) And the degree of Velocity in C ſhall have the ſame <lb></lb>proportion to the degree of Velocity in D, as A C hath to A D: <lb></lb>But the degree of Velocity in B is to the ſame degree in D, as the <lb></lb>Time along A B is to the Time along AD, by the definition of <lb></lb>Accelerate Motion; And the Time along AB is to the Time along <lb></lb>A D, as the ſame A C, the Mean Proportional between B A and <lb></lb>A D, is to A D, by the laſt Corollary of the ſecond Propoſition: <lb></lb>Therefore the degrees of Velocity in B and in C have to the de<lb></lb>gree in D, the ſame Proportion as A C hath to A D; and therefore <lb></lb>are equal: Which is the <emph type="italics"></emph>Theorem<emph.end type="italics"></emph.end> intended to be demonſtrated.</s></p><p type="main"> <s>By this we may more concludingly prove the enſuing third <pb xlink:href="069/01/157.jpg" pagenum="154"></pb>Propoſition of the Author, in which he makes uſe of this Princi<lb></lb>ple; and it is, That the Time along the inclined Plane, hath to the <lb></lb>Time along the Perpendicular, the ſame proportion as the ſaid In<lb></lb>clined Plane and Perpendicular. </s> <s>For if we put the caſe that BA <lb></lb>be the Time along A B, the Time along A D ſhall be the Mean <lb></lb>between them, that is A C, by the ſecond Corollary of the ſecond <lb></lb>Propoſition: But if C A be the Time along A D, it ſhall likewiſe <lb></lb>be the Time along <emph type="italics"></emph>A<emph.end type="italics"></emph.end> C, by reaſon that <emph type="italics"></emph>A<emph.end type="italics"></emph.end> D and <emph type="italics"></emph>A<emph.end type="italics"></emph.end> C are paſt in <lb></lb>equal Times: And therefore in caſe B <emph type="italics"></emph>A<emph.end type="italics"></emph.end> be the Time along A B, <lb></lb><emph type="italics"></emph>A<emph.end type="italics"></emph.end> C ſhall be the Time along <emph type="italics"></emph>A<emph.end type="italics"></emph.end> C: Therefore, as <emph type="italics"></emph>A<emph.end type="italics"></emph.end> B is to A C, ſo <lb></lb>is the Time along <emph type="italics"></emph>A<emph.end type="italics"></emph.end> B to the Time along <emph type="italics"></emph>A<emph.end type="italics"></emph.end> C.</s></p><p type="main"> <s>By the ſame diſcourſe one ſhall prove, that the Time along <emph type="italics"></emph>A<emph.end type="italics"></emph.end> C <lb></lb>is to the Time along the inclined Plane <emph type="italics"></emph>A<emph.end type="italics"></emph.end> E, as <emph type="italics"></emph>A<emph.end type="italics"></emph.end> C is to <emph type="italics"></emph>A<emph.end type="italics"></emph.end> E: <lb></lb>Therefore, <emph type="italics"></emph>ex æquali,<emph.end type="italics"></emph.end> the Time along the inclined Plane <emph type="italics"></emph>A B<emph.end type="italics"></emph.end> is, <lb></lb>Directly, to the Time along the inclined Plane <emph type="italics"></emph>A<emph.end type="italics"></emph.end> E as <emph type="italics"></emph>A B<emph.end type="italics"></emph.end> to <lb></lb><emph type="italics"></emph>A E, &c.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>One might alſo by the ſame application of the <emph type="italics"></emph>Theorem,<emph.end type="italics"></emph.end> as <emph type="italics"></emph>Sa<lb></lb>gredus<emph.end type="italics"></emph.end> ſhall very evidently ſee anon, immediately demonſtrate the <lb></lb>ſixth Propoſition of the <emph type="italics"></emph>A<emph.end type="italics"></emph.end>uthor: <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut let this Digreſſion ſuffice <lb></lb>for the preſent, which he perhaps thinketh too tedious, though in<lb></lb>deed it is of ſome importance in theſe matters of Motion.</s></p><p type="main"> <s>SAGR. </s> <s>You may ſay extreamly delightful, and moſt neceſſary <lb></lb>to the perfect underſtanding 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 Reſt do move along <lb></lb>an Inclined Plane, and alſo along the Perpendi<lb></lb>cular whoſe heights are the ſame, the Times of <lb></lb>their Motions ſhall be to one another as the <lb></lb>Lengths of the ſaid Plane and Perpendicular.</s></p><p type="main"> <s><emph type="italics"></emph>Let the inclined Plane be A C, and the Perpendicular A B, <lb></lb>whoſe heights are the ſame above the Horizon C B, to wit, <lb></lb>the ſelf ſame Line B A. </s> <s>I ſay, that the Time of the Deſcent <lb></lb>of the ſame Moveable upon the Plane A C, hath the ſame Proporti<lb></lb>on to the Time of the Deſcent along the Perpendicular A B, as the <lb></lb>Length of the Plane A C hath to the Length of the ſaid Perpendi<lb></lb>cular. </s> <s>For let any number of Lines D G, E I, F L, be drawn, Paral<lb></lb>lel to the Horizon C B: It is manifeſt from the Aſſumption fore<lb></lb>going, that the degrees of Velocity of the Moveable, departing from <lb></lb>A the beginning of Motion, acquired in the Points G and D are<emph.end type="italics"></emph.end><pb xlink:href="069/01/158.jpg" pagenum="155"></pb><emph type="italics"></emph>equal, their exceſſe or elevation above the Horizon being equal; <lb></lb>and ſo the degrees in the Points I and E; as alſo the degrees in L <lb></lb>and F. </s> <s>And if not only theſe Parallels, but many more were ſup<lb></lb>poſed to be drawn from all the points imagined to be in the Line <lb></lb>A B, untill they meet the Line A C, the Mo-<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.158.1.jpg" xlink:href="069/01/158/1.jpg"></figure><lb></lb><emph type="italics"></emph>ments, or degrees of the Velocities along the <lb></lb>extreams [or ends] of every one of thoſe <lb></lb>Parallels, ſhall be alwaies equal to one ano<lb></lb>ther: Therefore the two Spaces A C and A B <lb></lb>are paſt with the ſame degree of Velocity: <lb></lb>But it hath been demonſtrated, that if two <lb></lb>Spaces be paſſed by a Moveable with one <lb></lb>and the ſame degree of Velocity, the Times <lb></lb>of the Motions have the ſame proportion as <lb></lb>thoſe Spaces: Therefore the Time of the Motion along A C is to the <lb></lb>Time along A B, as the Length of the Plane A C to the length of the <lb></lb>Perdendicular A B. </s> <s>Which was to be demonſtrated.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>It ſeemeth to me, that the ſame might very clearly and <lb></lb>conciſely be concluded, it having firſt been proved that the ſum of <lb></lb>the Accelerate Motion of the Tranſitions along A C and A B, is <lb></lb>as much as the Equable Motion, whoſe degree of Velocity is ſub<lb></lb>duple to the greateſt degree C B: Therefore the two Spaces AC <lb></lb>and A B being paſſed with the ſame Equable Motion, it hath been <lb></lb>ſhewn, by the Firſt Propoſition of the firſt, that the Times of the <lb></lb>Tranſitions ſhall be as the ſaid Spaces.</s></p><p type="head"> <s>COROLLARY.</s></p><p type="main"> <s>Hence is collected, that the Times of the Deſcents along Planes <lb></lb>of different Inclination, but of the ſame Elevation, are to <lb></lb>one another according to their Lengths.</s></p><p type="main"> <s><emph type="italics"></emph>For if we ſuppoſe another Plane A M, coming from A, and ter<lb></lb>minated by the ſame Horizontal C B; it ſhall in like manner be <lb></lb>demonſtrated, that the Time of the Deſcent along A M, is to the <lb></lb>Time along A B, as the Line A M to A B: But as the Time A B is <lb></lb>to the Time along A C, ſo is the Line A B to A C: Therefore,<emph.end type="italics"></emph.end> ex <lb></lb>æquali, <emph type="italics"></emph>as A M is to A C, ſo is the Time along A M to the Time <lb></lb>along A C.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/159.jpg" pagenum="156"></pb><p type="head"> <s>THEOR. IV. PROP. IV.</s></p><p type="main"> <s>The Times of the Motions along equal Planes, <lb></lb>but unequally inclined, are to each other in <lb></lb>ſubduple proportion of the Elevations of thoſe <lb></lb>Planes Reciprocally taken.</s></p><p type="main"> <s><emph type="italics"></emph>Let there proceed from the term B two equal Planes, but une<lb></lb>qually inclined, B A and B C, and let A E and C D be Hori<lb></lb>zontal Lines, drawn as far as the Perpendicular B D: Let the <lb></lb>Elevation of the Plane B A be B E; and let the Elevation of the <lb></lb>Plane B C be B D: And let B I be a Mean Proportional between the <lb></lb>Elevations D B and B E: It is manifeſt<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.159.1.jpg" xlink:href="069/01/159/1.jpg"></figure><lb></lb><emph type="italics"></emph>that the proportion of D B to B I, is ſub<lb></lb>duple the proportion of D B to B E. </s> <s>Now <lb></lb>I ſay, that the proportion of the Times <lb></lb>of the Deſcents or Motions along the <lb></lb>Planes B A and B C, are the ſame with <lb></lb>the proportion of D B to B I Reciprocal<lb></lb>ly taken: So that to the Time B A the <lb></lb>Elevation of the other Plane B C, that is <lb></lb>B D be Homologal; and to the Time along <lb></lb>B C, B I be Homologal: Therefore it is <lb></lb>to be demonſtrated, That the Time along B A is to the Time along <lb></lb>B C, as D B is to B I. </s> <s>Let I S be drawn equidiſtant from D C. </s> <s>And <lb></lb>becauſe it hath been demonſtrated that the Time of the Deſcent <lb></lb>along B A, is to the Time of the Deſcent along the Perpendicular <lb></lb>B E, as the ſaid B A is to B E; and the Time along B E is to the <lb></lb>Time along B D, as B E is to B I; and the Time along B D is to the <lb></lb>Time along B C, as B D to B C, or as B I to B S: Therefore,<emph.end type="italics"></emph.end> ex æqua<lb></lb>li, <emph type="italics"></emph>the Time along B A ſhall be to the Time along B C as B A to B S, <lb></lb>or as C B to BS: But C B is to B S, as D B to B I: Therefore the <lb></lb>Propoſition is manifeſt:<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/160.jpg" pagenum="157"></pb><p type="head"> <s>THEOR. V. PROP. V.</s></p><p type="main"> <s>The proportion of the Times of the Deſcents <lb></lb>along Planes that have different Inclinations <lb></lb>and Lengths, and the Elivations unequal, is <lb></lb>compounded of the proportion of the Lengths <lb></lb>of thoſe Planes, and of the ſubduple proporti<lb></lb>on of their Elevations Reciprocally taken.</s></p><p type="main"> <s><emph type="italics"></emph>Let A B and A C be Planes inclined after different manners, <lb></lb>whoſe Lengths are unequal, as alſo their Elevations. </s> <s>I ſay, <lb></lb>the proportion of the Time of the Deſcent along A C to the <lb></lb>Time along A B, is compounded of the proportion of the ſaid A C <lb></lb>to A B, and of the ſubduple proportion of their Elevation Recipro<lb></lb>cally taken. </s> <s>For let the Perpendicular A D be drawn, with which <lb></lb>let the Horizontal Lines B G and C D interſect, and let A L be a <lb></lb>Mean-proportional between C A and A E; and from the point L let <lb></lb>a Parallel be drawn to the Horizon interſecting<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.160.1.jpg" xlink:href="069/01/160/1.jpg"></figure><lb></lb><emph type="italics"></emph>the Plane A C in F; and A F ſhall be a Mean <lb></lb>proportional between C A and A E. </s> <s>And becauſe <lb></lb>the Time along A C is to the Time along A E, as <lb></lb>the Line F A to A E; and the Time along A E is <lb></lb>to the Time along A B, as the ſaid A E to the ſaid <lb></lb>A B: It is manifeſt that the Time along A C is to <lb></lb>the Time along A B, as A F to A B. </s> <s>It remaineth, <lb></lb>therefore, to be demonſtrated, that the proportion <lb></lb>of A F to A B is compounded of the proportion of <lb></lb>C A to A B, and of the proportion of G A to A L; <lb></lb>which is the ſubduple proportion of the Elevati<lb></lb>ons D A and A G Reciprocally taken. </s> <s>But that is manifeſt, C A <lb></lb>being put between F A and A B: For the proportion of F A to A C <lb></lb>is the ſame as that of L A to A D, or of G A to A L; which is ſub<lb></lb>duple of the proportion of the Elevations G A and A D; and the <lb></lb>proportion of C A to A B is the proportion of the Lengths: Therefore <lb></lb>the Propoſition is manifeſt.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/161.jpg" pagenum="158"></pb><p type="head"> <s>THEOR. VI. PROP. VI.</s></p><p type="main"> <s>If from the higheſt or loweſt part of a Circle, <lb></lb>erect upon the Horizon, certain Planes be <lb></lb>drawn inclined towards the Circumference, <lb></lb>the Times of the Deſcents along the ſame <lb></lb>ſhall be equal.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Circle be erect upon the Horizon G H, whoſe Diameter <lb></lb>recited upon the loweſt point, that is upon the contact with the <lb></lb>Horizon, let be F A, and from the higheſt point A let certain <lb></lb>Planes A B and A C incline towards the Circumference: I ſay that the <lb></lb>Times of the Deſcents along the ſame are equal. </s> <s>Let B D and C E be <lb></lb>two Perpendiculars let fall unto the Diameter; and let A I be a Mean<lb></lb>Proportional between the Altitudes<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.161.1.jpg" xlink:href="069/01/161/1.jpg"></figure><lb></lb><emph type="italics"></emph>of the Planes E A and A D. </s> <s>And <lb></lb>becauſe the Rectangles F A E and <lb></lb>F A D are equal to the Squares of <lb></lb>A C and A B; And alſo becauſe <lb></lb>that as the Rectangle F A E, is to <lb></lb>the Rectangle F A D, ſo is E A to <lb></lb>A D. </s> <s>Therefore as the Square of <lb></lb>C A is to the Square of B A, <lb></lb>ſo is the Line E A to the Line <lb></lb>A D. </s> <s>But as the Line E A is to <lb></lb>D A, ſo is the Square of I A to the Square of A D: Therefore <lb></lb>the Squares of the Lines C A and A B are to each other as the Squares <lb></lb>of the Lines I A and A D: And therefore as the Line C A is to A B, <lb></lb>ſo is I A to A D: But in the precedent Propoſition it hath been demon<lb></lb>ſtrated that the proportion of the Time of the Deſcent along A C to the <lb></lb>Time of the Deſcent by A B, is compounded of the proportions of C A <lb></lb>to A B, and of D A to A I, which is the ſame with the proportion of <lb></lb>B A to A C: Therefore the proportion of the Time of the Deſcent along <lb></lb>A C, to the Time of the Deſcent along A B, is compounded of the pro<lb></lb>portions of C A to A B, and of B A to A C: Therefore the proporti<lb></lb>on of thoſe Times is a proportion of equality: Therefore the Propoſition <lb></lb>is evident.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>The ſame is another way demonſtrated from the Mechanicks: Name<lb></lb>ly that in the enſuing Figure the Moveable paſſeth in equal Times along <lb></lb>C A and D A. </s> <s>For let B A be equal to the ſaid D A, ond let fall the <lb></lb>Perpendiculars B E and D F: It is manifeſt by the Elements of the<emph.end type="italics"></emph.end><pb xlink:href="069/01/162.jpg" pagenum="159"></pb><emph type="italics"></emph>Mechanicks: That the Moment of the Weight elevated upon the Plane <lb></lb>according to the Line A B C, is <lb></lb>to its total Moment, as B E to B A;<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.162.1.jpg" xlink:href="069/01/162/1.jpg"></figure><lb></lb><emph type="italics"></emph>And that the Moment of the ſame <lb></lb>Weight upon the Elevation A D, <lb></lb>is to its total Moment, as D F to <lb></lb>D A or B A: Therefore the Mo<lb></lb>ment of the ſaid Weight upon the <lb></lb>Plane inclined according to D A, <lb></lb>is to the Moment upon the Plane <lb></lb>inclined according to A B C, as <lb></lb>the Line D F to the Line B E: <lb></lb>Therefore the Spaces which the <lb></lb>ſaid Weight ſhall paſſe in equal <lb></lb>Times along the Inclined Planes C A and D A, ſhall be to each other as <lb></lb>the Line B E to D F; by the ſecond Propoſition of the Firſt Book: <lb></lb>But as B E is to D F, ſo A C is demonſtrated to be to D A: <lb></lb>Therefore the ſame Moveable will in equal Times paſſe the Lines <lb></lb>C A and D A.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>And that C A is to D A as B E is to D F, is thus demonſtrated.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Draw a Line from C to D; and by D and B draw the Lines <lb></lb>D G L, (cutting C A in the point I) and B H, Parallels to A F: <lb></lb>And the Angle A D I ſhall be equal to the Angle D C A, for that <lb></lb>the parts L A and A D of the Circumference ſubtending them, are <lb></lb>equal, and the Angle D A C common to them both: Therefore of <lb></lb>the equiangled Triangles C A D and D A I, the ſides about the <lb></lb>equal Angles ſhall be proportional: And as C A is to A D, ſo is <lb></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></lb>D F: Which was to be proved.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Or elſe the ſame ſhall be demonſtrated more ſpeedily thus.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Vnto the Horizon A B, let a Circle be erect, whoſe Diameter is <lb></lb>perpendicular to the Horizon: and <lb></lb>from the higheſt Term D let a Plane<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.162.2.jpg" xlink:href="069/01/162/2.jpg"></figure><lb></lb><emph type="italics"></emph>at pleaſure D F, be inclined to the <lb></lb>Circumference. </s> <s>I ſay that the De<lb></lb>ſcent along the Plane D F, and the <lb></lb>Fall along the Diameter B C, will <lb></lb>be paſſed by the ſame Moveable in <lb></lb>equal Times. </s> <s>For let F G be drawn <lb></lb>parallel to the Horizon A B, which <lb></lb>ſhall be perpendicular to the Diameter <lb></lb>D C, and let a Line conjoyn F and <lb></lb>C: and becauſe the Time of the Fall <lb></lb>along D C, is to the Time of the Fall along D G, as the Mean <lb></lb>Proportional between C D and D G, is to the ſaid D G; and the<emph.end type="italics"></emph.end><pb xlink:href="069/01/163.jpg" pagenum="160"></pb><emph type="italics"></emph>Mean between C D and D G being D F, (for that the Angle D F C <lb></lb>in the Semicircle, is a Right Angle, and F G perpendicular to D C:) <lb></lb>Therefore the Time of the Fall along D C is to the Time of the Fall <lb></lb>along D G, as the Line F D to D G: But it hath been demonſtrated <lb></lb>that the Time of the Deſcent along D F, is to the Time of the Fall <lb></lb>along D G, as the ſame Line D F is to D G: The Times, therefore, <lb></lb>of the Deſcent along D F and Fall along D C, are to the Time of the <lb></lb>Fall along the ſaid D G in the ſame proportion: Therefore they are <lb></lb>equal. </s> <s>It will likewiſe be demonſtrated, if from the loweſt Term C, <lb></lb>one ſhould raiſe the Chord C E, and draw E H parallel to the Hori<lb></lb>zon, and conjoyn E and D, that the Time of the Deſcent along E C <lb></lb>equals the Time of the Fall along the Diameter D C.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>COROLLARY I.</s></p><p type="main"> <s>Hence is collected that the Times of the Deſcents along all the <lb></lb>Chords drawn from the Terms C or D are equal to one <lb></lb>another.</s></p><p type="head"> <s>COROLLARY II.</s></p><p type="main"> <s>It is alſo collected that if the Perpendicular and inclined Plane <lb></lb>deſcend from the ſame point along which the Deſcents are <lb></lb>made in equal Times, they are in a Semicircle whoſe Dia<lb></lb>meter is the ſaid 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></lb>Planes, are then equal, where the Elevations of equal parts of <lb></lb>thoſe Planes ſhall be to one another as their Longitudes.</s></p><p type="main"> <s><emph type="italics"></emph>For it hath been ſhewn that the Times C A and D A in the laſt Fi<lb></lb>gure ſave one are equal, the Elevation of the part A B being equal <lb></lb>to A D, that is, that B E ſhall be to the Elevation D F, as C A <lb></lb>to D A.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>Pray you Sir be pleaſed to ſtay your Reading of what <lb></lb>followeth until that I have ſatisfied my ſelf in a Contemplation <lb></lb>that juſt now cometh into my mind, which if it be not a deluſi<lb></lb>on, is not far from being a pleaſing divertiſement: as are all ſuch <lb></lb>that proceed from Nature or neceſſity.</s></p><p type="main"> <s>It is manifeſt, that if from a point aſſigned in an Horizontal <lb></lb>Plane, one ſhall produce along the ſame Plane infinite right Lines <lb></lb>every way, upon each of which a point is underſtood to move with <lb></lb>an Equable Motion, all beginning to move in the ſame inſtant <pb xlink:href="069/01/164.jpg" pagenum="161"></pb>of Time from the aſſigned point, and the Velocities of them all <lb></lb>being equal, there ſhall conſequently be deſcribed by thoſe move<lb></lb>able points Circumferences of Circles alwayes bigger and bigger, <lb></lb>all concentrick about the firſt point aſſigned: juſt in the ſame <lb></lb>manner as we ſee it done in the Undulations of ſtanding Water, <lb></lb>when a ſtone is dropt into it; the percuſſion of which ſerveth to <lb></lb>give the beginning to the Motion on every ſide, and remaineth <lb></lb>as the Center of all the Circles that happen to be deſigned ſucceſ<lb></lb>ſively bigger and bigger by the ſaid Undulations. </s> <s>But if we ima<lb></lb>gine a Plane erect unto the Horizon, and a point be noted in the <lb></lb>ſame on high, from which infinite Lines are drawn inclined, ac<lb></lb>cording to all inclinations, along which we fancy grave Movea<lb></lb>bles to deſcend, each with a Motion naturally Accelerate <lb></lb>with thoſe Velocities that agree with the ſeveral Inclinations; <lb></lb>ſuppoſing that thoſe deſcending Moveables were continually viſi<lb></lb>ble, in what kind of Lines ſhould we ſee them continually diſpoſed? <lb></lb></s> <s>Hence my wonder ariſeth, ſince that the precedent Demonſtrati<lb></lb>ons aſſure me, that they ſhall all be alwayes ſeen in one and the <lb></lb>ſame Circumference of Circles ſucceſſively encreaſing, according <lb></lb>as the Moveables in deſcending go more and more ſucceſſively re<lb></lb>ceding from the higheſt point in which their Fall began: And the <lb></lb>better to declare my ſelf, let the chiefeſt point A be marked, from <lb></lb>which Lines deſcend according to any Inclinations A F, A H, and <lb></lb>the Perpendicular A B, in which taking the points C and D, de<lb></lb>ſcribe Circles about them that paſs by <lb></lb><figure id="id.069.01.164.1.jpg" xlink:href="069/01/164/1.jpg"></figure><lb></lb>the point A, interſecting the inclined <lb></lb>Lines in the points F, H, B, and E, G, <lb></lb>I. </s> <s>It is manifeſt, by the fore-going <lb></lb>Demonſtrations, that Moveables de<lb></lb>ſcendent along thoſe Lines departing <lb></lb>at the ſame Time from the term A, <lb></lb>one ſhall be in E, the other ſhall be in <lb></lb>G, and the other in I; and ſo con<lb></lb>tinuing to deſcend they ſhall arrive <lb></lb>in the ſame moment of Time at F, H, <lb></lb>and B: and theſe and infinite others continuing to move along the <lb></lb>infinite differing Inclinations, they ſhall alwayes ſucceſſively arrive <lb></lb>at the ſelf-ſame Circumferences made bigger & bigger <emph type="italics"></emph>in infinitum.<emph.end type="italics"></emph.end><lb></lb>From the two Species, therefore, of Motion of which Nature makes <lb></lb>uſe, ariſeth, with admirable harmonious variety, the generation of in<lb></lb>ſinite Circles. </s> <s>She placeth the one as in her Seat, and original be<lb></lb>ginning, in the Center of infinite concentrick Circles; the other <lb></lb>is conſtituted in the ſublime or higheſt Contact of infinite Circum<lb></lb>ferences of Circles, all excentrick to one another: Thoſe proceed <lb></lb>from Motions all equal and Equable; Theſe from Motions all al<pb xlink:href="069/01/165.jpg" pagenum="162"></pb>wayes Inequable to themſelves, and all unequal to one another, <lb></lb>that deſcend along the infinite different Inclinations. </s> <s>But we fur<lb></lb>ther adde, that if from the two points aſſigned for the Emanations, <lb></lb>we ſhall ſuppoſe Lines to proceed, not onely along two Superfi<lb></lb>cies Horizontal and Upright [or erect] but along all every ways <lb></lb>like as from thoſe, beginning at one ſole point, we paſſed to the <lb></lb>production of Circles from the leaſt to the greateſt, ſo beginning <lb></lb>from one ſole point we ſhall ſucceſſively produce inſinite Spheres, <lb></lb>or we may ſay one Sphere, that ſhall <emph type="italics"></emph>gradatim<emph.end type="italics"></emph.end> increaſe to infinite <lb></lb>bigneſſes: And this in two faſhions; that is, either with placing <lb></lb>the original in the Center, or elſe in the Circumference of thoſe <lb></lb>Spheres.</s></p><p type="main"> <s>SALV. </s> <s>The Contemplation is really ingenuous, and adequate <lb></lb>to the Wit of <emph type="italics"></emph>Sagredus.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. </s> <s>Though I am at leaſt capable of the Speculation, ac<lb></lb>cording to the two manners of the production of Circles and <lb></lb>Spheres, with the two different Natural Motions, howbeit I do <lb></lb>not perfectly underſtand the production depending on the Acce<lb></lb>lerate Motion and its Demonſtration, yet notwithſtanding that <lb></lb>licence of aſſigning for the place of that Emanation as well the <lb></lb>loweſt Center, as the higheſt Spherical Superficies, maketh me to <lb></lb>think that its poſſible that ſome great Miſtery may be contained <lb></lb>in theſe true and admirable Concluſions: ſome Miſtery I ſay <lb></lb>touching the Creation of the Univerſe, which is held to be of <lb></lb>Spherical form, and concerning the Reſidence of the Firſt <lb></lb>Cauſe.</s></p><p type="main"> <s>SALV. </s> <s>I am not unwilling to think the ſame: but ſuch pro<lb></lb>found Speculations are to be expected from Sharper Wits than <lb></lb>ours. </s> <s>And it ſhould ſuffice us, that if we be but thoſe leſſe noble <lb></lb>Workmen that diſcover and draw forth of the Quarry the <lb></lb>Marbles, in which the Induſtrious Statuaries afterwards make <lb></lb>wonderful Images appear, that lay hid under rude and miſhaped <lb></lb>Cruſts. </s> <s>Now, if you pleaſe, 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 ſhall have a pro<lb></lb>portion double to that of their Lengths, the <lb></lb>Motions in them from Reſt ſhall be finiſhed in <lb></lb>equal Times.</s></p><p type="main"> <s><emph type="italics"></emph>Let A E and A B be two unequal Planes, and unequally inclined, <lb></lb>and let their Elevations be F A and D A, and let F A have the <lb></lb>ſame proportion to D A, as A E hath to A B. </s> <s>I ſay that the Times <lb></lb>of the Motions along the Planes A E and A B, out of Reſt in A are<emph.end type="italics"></emph.end><pb xlink:href="069/01/166.jpg" pagenum="163"></pb><emph type="italics"></emph>equal. </s> <s>Draw Horizontal Parallels to the Line of Elevation E F and <lb></lb>B D, which cutteth A E in G. </s> <s>And be-<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.166.1.jpg" xlink:href="069/01/166/1.jpg"></figure><lb></lb><emph type="italics"></emph>cauſe the proportion of F A to A D, is <lb></lb>double the proportion of E A to A B; and <lb></lb>as F A to A D, ſo is E A to A G: There<lb></lb>fore the proportion of E A to A G, is dou<lb></lb>ble the proportion of E A to A B: There<lb></lb>fore A B is a Mean-Proportional between <lb></lb>E A and A G: And becauſe the Time of the <lb></lb>Deſcent along A B, is to the Time of the De<lb></lb>ſcent along A G, as A B to A G; and the <lb></lb>Time of the Deſcent along AG, is to the Time of the Deſcent along A E, as <lb></lb>A G is to the Mean-proportional between A G and A E, which is A B: <lb></lb>Therefore<emph.end type="italics"></emph.end> ex equali, <emph type="italics"></emph>the Time along A B is to the Time along A E, as A B <lb></lb>unto it ſelf: Therefore the Times are equal: Which was to be demonſtrated.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>THEOR. VIII. PROP. VIII.</s></p><p type="main"> <s>In Planes cut by the ſame Circle, erect to the <lb></lb>Horizon, in thoſe which meet with the end of <lb></lb>the erect Diameter, whether upper or lower, <lb></lb>the Times of the Motions are equal to the <lb></lb>Time of the Fall along the Diameter: and in <lb></lb>thoſe which fall ſhort of the Diameter, the <lb></lb>Times are ſhorter; and in thoſe which inter<lb></lb>ſect the Diameter, they are longer.</s></p><p type="main"> <s><emph type="italics"></emph>Let A B be the Perpendicular Diameter of the Circle erect to the <lb></lb>Horizon. </s> <s>That the Times of the Motions along the Planes pro<lb></lb>duced out of the Terms A and B unto the Circumference are equal, <lb></lb>hath already been demonſtrated: That the Time of the Deſcent along <lb></lb>the Plane D F, not reaching to the<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.166.2.jpg" xlink:href="069/01/166/2.jpg"></figure><lb></lb><emph type="italics"></emph>Diameter is ſborter, is demonſtrated <lb></lb>by drawing the Plane D B, which <lb></lb>ſhall be both longer and leſſe decli<lb></lb>ning than D F. </s> <s>Therefore the Time <lb></lb>along D F is ſhorter than the Time <lb></lb>along D B, that is, along A B. </s> <s>And <lb></lb>that the Time of the Deſcent along <lb></lb>the Plane that interſecteth the Dia<lb></lb>meter, as C O is longer, doth in the <lb></lb>ſame manner appear, for that it is <lb></lb>longer and leſſe declining than C B: Therefore the Propoſition is de<lb></lb>monſtrated.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/167.jpg" pagenum="164"></pb><p type="head"> <s>THEOR. IX. PROP. IX.</s></p><p type="main"> <s>If two Planes be inclined at pleaſure from a point <lb></lb>in a Line parallel to the Horizon, and be inter<lb></lb>ſected by a Line which may make Angles Al<lb></lb>ternately equal to the Angles contained be<lb></lb>tween the ſaid Planes and Horizontal Parallel, <lb></lb>the Motion along the parts cut off by the ſaid <lb></lb>Line, ſhall be performed in equal Times.</s></p><p type="main"> <s><emph type="italics"></emph>From off the point C of the Horizontal Line X, let any two Planes <lb></lb>be inclined at pleaſure C D and C E, and in any point of the <lb></lb>Line C D make the Angle C D F equal to the Angle X C E: <lb></lb>and let the Line D F cut the Plane C E in F, in ſuch a manner that <lb></lb>the Angles C D F and C F D may be equal to the Angles X C E, L C D <lb></lb>Alternately taken. </s> <s>I ſay, that<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.167.1.jpg" xlink:href="069/01/167/1.jpg"></figure><lb></lb><emph type="italics"></emph>the Times of the Deſcents along <lb></lb>C D and C F are equal. </s> <s>And <lb></lb>that (the Angle C D F being <lb></lb>ſuppoſed equal to the Angle <lb></lb>X C E) the Angle C F D is <lb></lb>equal to the Angle D C L, is <lb></lb>manifeſt. </s> <s>For the Common An<lb></lb>gle D C F being taken from the <lb></lb>three Angles of the Triangle <lb></lb>C D F equal to two Right An<lb></lb>gles, to which are equal all the Angles made with to the Line L X <lb></lb>at the point C, there remains in the Triangle two Angles C D F and <lb></lb>C F D, equal to the two Angles X C E and L C D: But it was ſup<lb></lb>poſed that C D F is equal to the Angle X C E: Therefore the remaining <lb></lb>Angle C F D is equal to the remaining angle D C L. </s> <s>Let the Plane <lb></lb>C E be ſuppoſed equal to the Plane C D, and from the points D and <lb></lb>E raiſe the Perpendiculars D A and E B, unto the Horizontal Paral<lb></lb>lel X L; and from C unto D F let fall the Perpendicular C G. </s> <s>And <lb></lb>becauſe the Angle C D G is equal to the Angle E C B; and becauſe <lb></lb>D G C and C B E are Right Angles; The Triangles C D G and <lb></lb>C B E ſhall be equiangled: And as D C is to C G, ſo let C E be <lb></lb>to E B: But D C is equal to C E: Therefore C G ſhall be equal to <lb></lb>E B. </s> <s>And inregard that of the Triangles D A C and C G F, the An<lb></lb>gles C and A are equal to the Angles F and G: Therefore as C D is to <lb></lb>D A, ſo ſhall F C be to C G; and Alternately, as D C is to C F, ſo<emph.end type="italics"></emph.end><pb xlink:href="069/01/168.jpg" pagenum="165"></pb><emph type="italics"></emph>is D A to C G, or B E. </s> <s>The proportion therefore of the Elevations <lb></lb>of the Planes equal to C D and C E, is the ſame with the proportion <lb></lb>of the Longitudes D C and C E: Therefore, by the firſt Corollary of <lb></lb>the precedent Sixth Propoſition, the Times of the Dcſcent along the <lb></lb>ſame ſhall be equal: Which mas to be proved.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Take the ſame another way: Draw F S perpendicular to the <lb></lb>Horizontal Parallel A S. </s> <s>Becauſe the Triangle C S F is like to <lb></lb>the Triangle D G C, it ſhall be, that as S F is to F C, ſo is G C <lb></lb>to C D. </s> <s>And becauſe the Triangle C F G is like to the Triangle <lb></lb>D C A, it ſhall be, that as F C is to C G, ſo is C D to D A: <lb></lb>Therefore,<emph.end type="italics"></emph.end> ex æquali, <emph type="italics"></emph>as <lb></lb>S F is to C G, ſo is C G to <lb></lb>D A: Thorefore C G is a<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.168.1.jpg" xlink:href="069/01/168/1.jpg"></figure><lb></lb><emph type="italics"></emph>Mean-proportional between <lb></lb>S F and D A: And as DA <lb></lb>is to S F, ſo is the Square <lb></lb>D A unto the Square C G <lb></lb>Again, the Triangle A C D <lb></lb>being like to the Triangle <lb></lb>C G F, it ſhall be, that as <lb></lb>D A is to D C, ſo is G C <lb></lb>to C F: and, Alternately, <lb></lb>as D A is to G C, ſo is D C to C F; and as the Square of D A <lb></lb>is to the Square of C G, ſo is the Square of D C to the Square of <lb></lb>C F. </s> <s>But it hath been proved that the Square D A is to the <lb></lb>Square C G as the Line D A is to the Line F S: Therefore, as the <lb></lb>Square D C is to the Square C F, ſo is the Line D E to F S: There<lb></lb>fore, by the ſeventh fore-going, in regard that the Elevations D A <lb></lb>and F S, of the Planes C D, and C F are in double proportion to <lb></lb>their Planes; the Times of the Motions along the ſame ſhall be <lb></lb>equal.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>THEOR. X. PROP. X.</s></p><p type="main"> <s>The Times of the Motions along ſeveral Inclina<lb></lb>tions of Planes whoſe Elevations are equal, <lb></lb>are unto one another as the Lengths of thoſe <lb></lb>Planes, whether the Motions be made from <lb></lb>Reſt, or there hath proceeded a Motion from <lb></lb>the ſame height.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Motions be made along A B C, and along A B D, until <lb></lb>they come to the Horizon D C, in ſuch ſort as that the Motion <lb></lb>along A B precedeth the Motions along B D and B C. </s> <s>I ſay, <lb></lb>that the Time of the Motion along B D, is to the Time along B C, as<emph.end type="italics"></emph.end><pb xlink:href="069/01/169.jpg" pagenum="166"></pb><emph type="italics"></emph>the Length B D is to B C. </s> <s>Let A F be drawn parallel to the Ho<lb></lb>rizon, to which continue out D B, meeting it in F; and let F E be <lb></lb>a Mean-proportional between D F and F B; and draw E O parallel <lb></lb>to D C, and A O ſhall be a Mean-proportional between C A and <lb></lb>A B: But if we ſuppoſe the Time <lb></lb>along A B, to be as A B, the Time a-<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.169.1.jpg" xlink:href="069/01/169/1.jpg"></figure><lb></lb><emph type="italics"></emph>long F B ſhall be as F B. </s> <s>And the <lb></lb>Time along all A C, ſhall be as the <lb></lb>Mean-proportional A O; and along <lb></lb>all F D ſhall be F E: Wherefore the <lb></lb>Time along the remainder B C ſhall <lb></lb>be B O; and along the remainder <lb></lb>B D ſhall be B E. </s> <s>But as B E is to <lb></lb>B O, ſo is B D to B C: Therefore <lb></lb>the Times along B D and B C, after the Deſcent along A B and <lb></lb>F B, or which is the ſame, along the Common part A B, ſhall be to <lb></lb>one another as the Lengths B D and B C: But that the Time along <lb></lb>B D, is to the Time along B C, out of Reſt in B, as the Length <lb></lb>B D to B C, hath already been demonſtrated. </s> <s>Therefore the Times <lb></lb>of the Motions along different Planes whoſe Elevations are equal, are <lb></lb>to one another as the Lengths of the ſaid Planes, whether the Motion <lb></lb>be made along the ſame out of Reſt, or whether another Motion of <lb></lb>the ſame Altitude do precede thoſe Motions: Which was to be de<lb></lb>monſtrated.<emph.end type="italics"></emph.end></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></lb>Reſt, be divided at pleaſure, the Time of <lb></lb>the Motion along the firſt part, is to the Time <lb></lb>of the Motion along the ſecond, as the ſaid <lb></lb>firſt part is to the exceſſe whereby the ſame <lb></lb>part ſhall be exceeded by the Mean-Propor<lb></lb>tional between the whole Plane and the ſame <lb></lb>firſt part.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Motion be along the whole Plane A B, ex quiete in A, <lb></lb>which let be divided at pleaſure in C; and let A F be a Mean <lb></lb>proportional between the whole B A and the firſt part A C; <lb></lb>C F ſhall be the exceſſe of the Mean proportional F A above the part <lb></lb>A C. </s> <s>I ſay the Time of the Motion along A C is to the Time of the <lb></lb>following Motion along C B, as A C to C F. </s> <s>Which is manifeſt;<emph.end type="italics"></emph.end><pb xlink:href="069/01/170.jpg" pagenum="167"></pb><emph type="italics"></emph>For the Time along A C is to the Time along all <lb></lb>A B, as A C to the Mean-proportional A F: There-<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.170.1.jpg" xlink:href="069/01/170/1.jpg"></figure><lb></lb><emph type="italics"></emph>fore, by Diviſion, the Time along A C, ſhall be to <lb></lb>the Time along the remainder C B as A C to C F: <lb></lb>If therefore the Time along A C be ſuppoſed to be <lb></lb>the ſaid A C, the Time along C B ſhall be C F: <lb></lb>Which was the Propoſition.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>But if the Motion be not made along the continu<lb></lb>ate Plane A C B, but by the inflected Plane A C D <lb></lb>until it come to the Horizon B D, to which from F a Parallel is <lb></lb>drawn F E. </s> <s>It ſhall in like manner be<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.170.2.jpg" xlink:href="069/01/170/2.jpg"></figure><lb></lb><emph type="italics"></emph>demonſtrated, that the Time along <lb></lb>A C is to the Time along the reflected <lb></lb>Plane C D, as A C is to C E. </s> <s>For <lb></lb>the Time along A C is to the Time a<lb></lb>long C B, as A C is to C F: But the <lb></lb>Time along C B, after A C hath been <lb></lb>demonſtrated to be to the Time along <lb></lb>C D, after the ſaid Deſoent along <lb></lb>A C, as C B is to C D; that is, as <lb></lb>C F to C E: Therefore,<emph.end type="italics"></emph.end> ex æquali, <emph type="italics"></emph>the Time along A C ſhall be to <lb></lb>the Time along C D, as the Line A C to C E.<emph.end type="italics"></emph.end></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></lb>ſure, be cut between the ſame Horizontal <lb></lb>Lines, and Mean-Proportionals between <lb></lb>them and the parts of them contained betwixt <lb></lb>the common Section and upper Horizontal <lb></lb>Line be given; the Time of the Motion a<lb></lb>long the Perpendicular ſhall have the ſame <lb></lb>proportion to the Time of the Motion along <lb></lb>the upper part of the Perpendicular, and af<lb></lb>terwards along the lower part of the interſe<lb></lb>cted Plane, as the Length of the whole Per<lb></lb>pendicular hath to the Line compounded of <lb></lb>the Mean-Proportional given upon the Per<lb></lb>pendicular, and of the exceſſe by which the <lb></lb>whole Plane exceeds its Mean-Proporttonal.</s></p><pb xlink:href="069/01/171.jpg" pagenum="168"></pb><p type="main"> <s><emph type="italics"></emph>Let the Horizontal Lines be A F the upper, and C D the low<lb></lb>er; between which let the Perpendicular A C, and inclined <lb></lb>Plane D F, be cut in B; and let A R be a Mean-Proportional <lb></lb>between the whole Perpendicular C A, and the upper part A B; and <lb></lb>let F S be a Mean-proportional between the whole Inclined Plane D F, <lb></lb>and the upper part B F. </s> <s>I ſay, that the Time of the Fall along the <lb></lb>whole Perpendicular A C hath the ſame proportion to the Time along <lb></lb>its upper part A B, with the lower of the Plane, that is, with B D, <lb></lb>as A C hath to the Mean-proporti<lb></lb>onal of the Perpendicular, that is<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.171.1.jpg" xlink:href="069/01/171/1.jpg"></figure><lb></lb><emph type="italics"></emph>A R, with S D, which is the ex<lb></lb>ceſſe of the whole Plane D F above <lb></lb>its Mean-proportional F S. </s> <s>Let a <lb></lb>Line be drawn from R to S, which <lb></lb>ſhall be parallel to the two Horizon<lb></lb>tal Lines. </s> <s>And becauſe the Time of <lb></lb>the Fall along all A C, is to the <lb></lb>Time along the part A B, as C A is <lb></lb>to the Mean proportional A R, if we ſuppoſe A C to be the Time of <lb></lb>the Fall along A C, A R ſhall be the Time of the Fall along A B, <lb></lb>and R C that along the remainder B C. </s> <s>For if the Time along A C <lb></lb>be ſuppoſed, as was done, to be A C it ſelf the Time along F D ſhall <lb></lb>be F D; and in like manner D S may be concluded to be the Time a<lb></lb>long B D, after F B, or after A B. </s> <s>The Time therefore along the <lb></lb>whole A C, is A R, with R C; And the Time along the inflected <lb></lb>Plane A B D, ſhall be A R, with S D: Which was to be proved.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>The ſame happeneth, if inſtead of the Perpendicular, another <lb></lb>Plane were taken, as ſuppoſe N O; and the Demonstration is the <lb></lb>ſame.<emph.end type="italics"></emph.end></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></lb>unto it, along which, when it hath the ſame <lb></lb>Elevation with the ſaid Perpendicular, it may <lb></lb>make a Motion after its Fall along the Per<lb></lb>pendicular in the ſame Time, as along the <lb></lb>ſame Perpendicular <emph type="italics"></emph>ex quiete.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Let the Perpendicular given be A B, to which extended to C, <lb></lb>let the part B C be equal; and draw the Horizontal Lines <lb></lb>C E and A G. </s> <s>It is required from B to inflect a Plane reach<lb></lb>ing to the Horizon C E, along which a Motion, after the Fall out<emph.end type="italics"></emph.end><pb xlink:href="069/01/172.jpg" pagenum="169"></pb><emph type="italics"></emph>of A, ſhall be made in the ſame Time, as along A B from Reſt in A. </s> <s>Let <lb></lb>C D be equal to C B, and drawing B D, let B E be applied equal to both <lb></lb>B D and D C. </s> <s>I ſay B E is the Plane required. </s> <s>Continue out E B to <lb></lb>meet the Horizontal Line A G in G;<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.172.1.jpg" xlink:href="069/01/172/1.jpg"></figure><lb></lb><emph type="italics"></emph>and let G F be a Mean-Proportional be<lb></lb>tween the ſaid E G and G B. </s> <s>E F ſhall <lb></lb>be to F B, as E G is to G F; and the <lb></lb>Square E F ſhall be to the Square F B, as <lb></lb>the Square E G is to the Square G F; <lb></lb>that is as the Line E G to G B: But <lb></lb>E G is double to G B: Therefore the <lb></lb>Square of E F is double to the Square of F B: But alſo the Square of <lb></lb>D B is double to the Square of B C: Therefore, as the Line E F is to <lb></lb>F B, ſo is D B to B C: And by Compoſition and Permutation, as E B is <lb></lb>to the two D B and B C, ſo is B F to B C: But B E is equal to the two <lb></lb>D B and B C: Therefore B F is equal to the ſaid B C, or B A. </s> <s>If there<lb></lb>fore A B be underſtood to be the Time of the Fall along A B, G B ſhall <lb></lb>be the Time along G B, and G F the Time along the whole G E: There<lb></lb>fore B F ſhall be the Time along the remainder B E, after the Fall from <lb></lb>G, or from A, which was the Propoſition.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROBL. II. PROP. XIV.</s></p><p type="main"> <s>A <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicular and a <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane inclined to it being <lb></lb>given, to find a part in the upper <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicu<lb></lb>lar which ſhall be paſt <emph type="italics"></emph>ex quiete<emph.end type="italics"></emph.end> in a Time <lb></lb>equal to that in which the inclined <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane is <lb></lb>paſt after the Fall along the part found in the <lb></lb>Perpendicular.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Perpendicular be D B, and the Plane inclined to it A C. </s> <s>It is <lb></lb>required in the Perpendicular A D to find a part which ſhall be <lb></lb>paſt<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in a Time equal to that in which the Plane A C is <lb></lb>paſt after the Fall along the ſaid part. </s> <s>Draw the Horizontal Line C B; <lb></lb>and as B A more twice A C is to A C, ſo let E A be to A R; And from <lb></lb>R let fall the Perpendicular R X unto D B. </s> <s>I ſay X is the point requi<lb></lb>red. </s> <s>And becauſe as B A more twice A C is to A C, ſo is C A to A E, <lb></lb>by Diviſion it ſhall be that as B A more A C is to A C, ſo is C E to E A: <lb></lb>And becauſe as B A is to A C, ſo is E A to A R, by Compoſition it ſhall <lb></lb>be that as B A more A C is to A C, ſo is E R to R A: But as B A more <lb></lb>A C is to A C, ſo is C E to E A: Therefore, as C E is to E A, ſo is E R, <lb></lb>to R A, and both the Antecedents to both the Conſequents, that is, C R<emph.end type="italics"></emph.end><pb xlink:href="069/01/173.jpg" pagenum="170"></pb><emph type="italics"></emph>to R E: Therefore C R, R E, and R A are Proportionals. </s> <s>Farther<lb></lb>more, becauſe as B A is to A C, ſo E A is ſuppoſed to be to A R, and,<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.173.1.jpg" xlink:href="069/01/173/1.jpg"></figure><lb></lb><emph type="italics"></emph>in regard of the likeneſſe of the Triangles, <lb></lb>as B A is to A C, ſo is X A to A R: There<lb></lb>fore, as E A is to A R, ſo is X A to A R: <lb></lb>Therefore E A and X A are equal. </s> <s>Now if <lb></lb>we underſtand the Time along R A to be as <lb></lb>R A, the Time along R C ſhall be R E, the <lb></lb>Mean-Proportional between C R and R A: <lb></lb>And A E ſhall be the Time along A C after <lb></lb>R A or after X A: But the Time along X A <lb></lb>is X A, ſo long as R A is the Time along R <lb></lb>A: But it hath been proved that X A and <lb></lb>A E are equal: Therefore the Propoſition is proved.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROBL. III. <emph type="italics"></emph>P<emph.end type="italics"></emph.end>RO<emph type="italics"></emph>P.<emph.end type="italics"></emph.end> XV.</s></p><p type="main"> <s>A <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicular and a <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane inflected to it being <lb></lb>given, to find a part in the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicular ex<lb></lb>tended downwards which ſhall be paſſed in the <lb></lb>ſame. </s> <s>Time as the inflected <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane after the Fall <lb></lb>along the given Perpendicular.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Perpendicular be A B, and the Plane Inſlected to it B C. </s> <s>It <lb></lb>is required in the Perpendicular extended downwards to find a <lb></lb>part which from the Fall out of A ſhall be paſt in the ſame Time as <lb></lb>B C is paſſed from the ſame Fall out of A. </s> <s>Draw the Horizontal Line <lb></lb>A D, with which let C B meet extended to D; and let D E be a Mean<lb></lb>proportional between C D and D B;<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.173.2.jpg" xlink:href="069/01/173/2.jpg"></figure><lb></lb><emph type="italics"></emph>and let B F be equal to B E; and let <lb></lb>A G be a third Proportional to B A and <lb></lb>A F. </s> <s>I ſay, B G is the Space that after <lb></lb>the Fall A B ſhall be paſt in the ſame <lb></lb>Time, as the Plane B C ſhall be paſt af<lb></lb>ter the ſame Fall. </s> <s>For if we ſuppoſe <lb></lb>the Time along A B to be as A B, the <lb></lb>Time along D B ſhall be as D B: And <lb></lb>becauſe D E is the Mean-proportional <lb></lb>between B D and D C, the ſame D E <lb></lb>ſhall be the Time along the whole D C, and B E the Time along the Part <lb></lb>or Remainder B C<emph.end type="italics"></emph.end> ex quiete, <emph type="italics"></emph>in D, or<emph.end type="italics"></emph.end> ^{*} ex caſu <emph type="italics"></emph>A B: And it may in<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1093"></arrow.to.target><lb></lb><emph type="italics"></emph>like manner be proved, that B F is the Time along B G, after the ſame <lb></lb>Fall: But B F is equal to B E: Which was the Propoſition to be proved.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/174.jpg" pagenum="171"></pb><p type="margin"> <s><margin.target id="marg1093"></margin.target>* From or after <lb></lb>the Fall A B.</s></p><p type="head"> <s>THEOR. XIII. <emph type="italics"></emph>P<emph.end type="italics"></emph.end>RO<emph type="italics"></emph>P.<emph.end type="italics"></emph.end> XVI.</s></p><p type="main"> <s>If the parts of an inclined <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane and <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicu<lb></lb>lar, the Times of whoſe Motions <emph type="italics"></emph>ex quiete<emph.end type="italics"></emph.end> are <lb></lb>equal, be joyned together at the ſame point, a <lb></lb>Moveable coming out of any ſublimer Height <lb></lb>ſhall ſooner paſſe the ſaid part of the inclined <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane, than that part of the Perpendicular.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Perpendicular be E B, and the Inclined Plane C E, joyned <lb></lb>at the ſame Point E, the Times of whoſe Motions from off Reſt in <lb></lb>E are equal, and in the Perpendicular continued out, let a ſublime <lb></lb>point A be taken at pleaſure, out of which the Moveables may be let <lb></lb>fall. </s> <s>I ſay, that the Inclined Plane E C ſhall be paſſed in a leſſe Time <lb></lb>than the Perpendicular E B, after the Fall A E. </s> <s>Draw a Line from C <lb></lb>to B, and having drawn the Horizontal Line A D continue out C E till <lb></lb>it meet the ſame in D; and let D F be a Mean-Proportional between <lb></lb>C D and D E; and let A G be a<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.174.1.jpg" xlink:href="069/01/174/1.jpg"></figure><lb></lb><emph type="italics"></emph>Mean-Proportional between B A and <lb></lb>A E; and draw F G and D G. </s> <s>And <lb></lb>becauſe the Time of the Motion along <lb></lb>E C and E B out of Reſt in E are <lb></lb>equal, the Angle C ſhall be a Right <lb></lb>Angle, by the ſecond Corollary of the <lb></lb>Sixth Propoſition; and A is a Right <lb></lb>Angle, and the Vertical Angles <lb></lb>at E are equal: Therefore the Tri<lb></lb>angles A E D and C E B are equian<lb></lb>gled, and the Sides about equal An<lb></lb>gles are Proportionals: Therefore as <lb></lb>B E is to E C, ſo is D E to E A. <lb></lb></s> <s>Therefore the Rectangle B E A is <lb></lb>equal to the Rectangle C E D: And <lb></lb>becauſe the Rectangle C D E ex<lb></lb>ceedeth the Rectangle C E D, by the Square E D, and the Rectangle <lb></lb>B A E doth exceed the Rectangle B E A, by the Square E A: The <lb></lb>exceſſe of the Rectangle C D E above the Rectangle B A E, that is of <lb></lb>the Square F D above the Square A G ſhall be the ſame as the exceſſe <lb></lb>of the Square D E above the Square A E; which exceſs is the <lb></lb>Square D A: Therefore the Square F D is equal to the two Squares <lb></lb>G A and A D, to which the Square G D is alſo equal: Therefore the<emph.end type="italics"></emph.end><pb xlink:href="069/01/175.jpg" pagenum="172"></pb><emph type="italics"></emph>Line D F is equal to D G, and the Angle D G F is equal to the An<lb></lb>gle D F G, and the Angle E G F is leſſc than the Angle E F G, and <lb></lb>the oppoſite Side E F leſſe than the Side E G. </s> <s>Now if we ſuppoſe <lb></lb>the Time of the Fall along A E to be as A E, the Time by D E ſhall <lb></lb>be as D E; and A G being a Mean-Proportional between B A and A E, <lb></lb>A G ſhall be the Time along the whole A B, and the part E G ſhall be <lb></lb>the Time along the Part E B<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in A. </s> <s>And it may in like man<lb></lb>ner be proved that E F is the Time along E C after the Deſcent D E, or <lb></lb>after the Fall A E: But E F is proved to be leſſer than E G: Therefore <lb></lb>the Propoſition is proved.<emph.end type="italics"></emph.end></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 paſ<lb></lb>ſed along the Perpendicular after the Fall from above in the <lb></lb>ſame Time in which the Inclined Plane is paſt, is leſſe than <lb></lb>that which is paſt in the ſame Time as in the Inclined, no fall <lb></lb>from above preceding, yet greater than the ſaid Inclined <lb></lb>Plane.</s></p><p type="main"> <s><emph type="italics"></emph>For it having been proved, but now, that of the Moveables coming <lb></lb>from the ſublime Term A the Time of the Converſion along E C is <lb></lb>ſhorter than the Time of the Progreſſion along E B; It is manifeſt that <lb></lb>the Space that is paſt along E B in a Time equal to the Time along E C <lb></lb>is leſs than the whole Space E B. </s> <s>And that the ſame Space along the <lb></lb>Perpendicular is greater than E C is mani<lb></lb>feſted by reaſſuming the Figure of the pre-<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.175.1.jpg" xlink:href="069/01/175/1.jpg"></figure><lb></lb><emph type="italics"></emph>cedent Propoſition, in which the part of the <lb></lb>Perpendicular B G hath been demonſtrated <lb></lb>to be paſſed in the ſame Time as B C after <lb></lb>the Fall A B: But that B G is greater than <lb></lb>B C is thus collected. </s> <s>Becauſe B E and F B <lb></lb>are equal, and B A leſſer than B D, F B, <lb></lb>hath greater proportion to B A, than E B <lb></lb>hath to B D: And, by Compoſition, F A <lb></lb>hath greater proportion to A B, than E D <lb></lb>to D B: But as F A is to A B, ſo is G F <lb></lb>to F B, (for A F is the Mean-Proportional <lb></lb>between B A and A G:) And in like man<lb></lb>ner, as E D is to B D, ſo is C E to E B: Therefore G B hath greater <lb></lb>proportion to B F, than C B hath to B E: Therefore G B is greater <lb></lb>than B C.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/176.jpg" pagenum="173"></pb><p type="head"> <s>PROBL. IV. <emph type="italics"></emph>P<emph.end type="italics"></emph.end>RO<emph type="italics"></emph>P.<emph.end type="italics"></emph.end> XVII.</s></p><p type="main"> <s>A <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicular and <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane Inflected to it being <lb></lb>given, to aſſign a part in the given <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane, in <lb></lb>which after the Fall along the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicular <lb></lb>the Motion may be made in a Time equal to <lb></lb>that in which the Moveable <emph type="italics"></emph>ex quiete<emph.end type="italics"></emph.end> paſſeth <lb></lb>the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicular given.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Perpendicular be A B, and a Plane Inflected to it B E: It is <lb></lb>required in B E to aſſign a Space along which the Moveable af<lb></lb>ter the Fall along A B may move in a Time equal to that in which <lb></lb>the ſaid Perpendicular A B is paſſed<emph.end type="italics"></emph.end> ex quiete. <emph type="italics"></emph>Let the Line A D be <lb></lb>parallel to the Horizon, with which let the Plane prolonged meet in D; <lb></lb>and ſuppoſe F B equal to B A; and as B D <lb></lb>is to D F, ſo let F D be to D E. </s> <s>I ſay, that<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.176.1.jpg" xlink:href="069/01/176/1.jpg"></figure><lb></lb><emph type="italics"></emph>the Time along B E after the Fall along A B <lb></lb>equalleth the Time along A B, out of Reſt <lb></lb>in A. </s> <s>For if we ſuppoſe A B to be the Time <lb></lb>along A B, D B ſhall be the Time along <lb></lb>D B. </s> <s>And becauſe, as B D is to D F, ſo is <lb></lb>F D to D E, D F ſhall be the Time along <lb></lb>the whole Plane D E, and B F along the <lb></lb>part B E out of D: But the Time along <lb></lb>B E after D B, is the ſame as after A B: Therefore the Time along B E <lb></lb>after A B ſhall be B F, that is, equal to the Time<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in A: <lb></lb>Which was the Propoſition.<emph.end type="italics"></emph.end></s></p><p type="head"> <s><emph type="italics"></emph>P<emph.end type="italics"></emph.end>ROBL. V. PROP. XVIII.</s></p><p type="main"> <s>Any Space in the Perpendicular being given from <lb></lb>the aſſigned beginning of Motion that is <lb></lb>paſſed in a Time given, and any other leſſer <lb></lb>Time being alſo given, to find another Space in <lb></lb>the ſaid Perpendicular that may be paſſed in <lb></lb>the given leſſer Time.</s></p><pb xlink:href="069/01/177.jpg" pagenum="174"></pb><p type="main"> <s><emph type="italics"></emph>Let the Perpendicular be A D, in which let the Space aſſigned be <lb></lb>A B, whoſe Time from the beginning A let be A B: and let the <lb></lb>Horizon be C B E, and let a Time be given leſs than A B, to <lb></lb>which let B C be noted equal in the Horizon: It is required in the <lb></lb>ſaid Perpendicular to find a Space equal to the ſame A B that ſhall be <lb></lb>paſſed in the Time B C. </s> <s>Draw a Line from A to<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.177.1.jpg" xlink:href="069/01/177/1.jpg"></figure><lb></lb><emph type="italics"></emph>C. </s> <s>And becauſe B C is leſſe than B A, the Angle <lb></lb>B A C ſhall be leſſe than the Angle B C A. </s> <s>Let <lb></lb>C A E be made equal to it, and the Line A E meet <lb></lb>with the Horizon in the Point E, to which ſup<lb></lb>poſe E D a Perpendicular, cutting the Perpendi<lb></lb>cular in D, and let D F be cut equal to B A. </s> <s>I <lb></lb>ſay, that the ſaid F D is a part of the Perpendi<lb></lb>cular along which the Lation from the beginning <lb></lb>of Motion in A, the Time B C given will be ſpent. <lb></lb></s> <s>For if in the Right-angled Triangle A E D, a <lb></lb>Perpendicular to the oppoſite Side A D, be drawn <lb></lb>E B, A E ſhall be a Mean-Proportional betwixt <lb></lb>D A and A B, and B E a Mean-Proportional betwixt D B and B A, <lb></lb>or betwixt F A and A B (for F A is equal to D B.) And in regard <lb></lb>A B hath been ſuppoſed to be the Time along A B, A E, or E C ſhall be <lb></lb>the Time along the whole A D, and E B the Time along A F: There<lb></lb>fore the part B C ſhall be the Time along the part F D: Which was <lb></lb>intended.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROBL. VI. PROP. XIX.</s></p><p type="main"> <s>Any Space in the Perpendicular paſſed from the <lb></lb>beginning of the Motion being given, and the <lb></lb>Time of the Fall being aſſigned, to find the <lb></lb>Time in which another Space. </s> <s>equal to the gi<lb></lb>ven one, and taken in any part of the ſaid Per<lb></lb>pendicular, ſhall be afterwards paſt by the <lb></lb>ſame Moveable.</s></p><p type="main"> <s><emph type="italics"></emph>In the Perpendicular A B let A C be any Space taken from the be<lb></lb>ginning of the Motion in A, to which let D B be another equal Space <lb></lb>taken any where at pleaſure, and let the Time of the Motion along <lb></lb>A C be given, and let it be A C. </s> <s>It is required to ſind the Time of the<emph.end type="italics"></emph.end><pb xlink:href="069/01/178.jpg" pagenum="175"></pb><emph type="italics"></emph>Motion along D B after the Fall from A. </s> <s>About the whole A B de<lb></lb>ſcribe a Semicircle A E B, and from<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.178.1.jpg" xlink:href="069/01/178/1.jpg"></figure><lb></lb><emph type="italics"></emph>C let fall C E, a Perpendicular to A <lb></lb>B, and draw a Line from A to E; <lb></lb>which ſhall be greater than E C. <lb></lb></s> <s>Let E F be out equall to E C: I ſay, <lb></lb>that the remainder F A is the Time <lb></lb>of the Motion along D B. </s> <s>For be<lb></lb>cauſe A E is a Mean-proportional be<lb></lb>twixt B A and and A C, and A C <lb></lb>is the Time of the Fall along A C; <lb></lb>A E ſhall be the Time along the <lb></lb>Whole A B. </s> <s>And becauſe C E is a <lb></lb>Mean-proportional betwixt D A and <lb></lb>A C, (for D A is equal to B C) <lb></lb>C E, that is E F ſhall be the Time <lb></lb>along A D: Therefore the Remainder A F ſhall be the Time along the <lb></lb>Remainder B B: Which is the Propoſition.<emph.end type="italics"></emph.end></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"></emph>ex quiete<emph.end type="italics"></emph.end> be <lb></lb>as the ſaid Spaec, the Time thereof after another Space is ad<lb></lb>ded ſhall be the exceſſe of the Mean-proportional betwixt <lb></lb>the Addition and Space taken together, and the ſaid Space <lb></lb>above the Mean-proportional betwixt the firſt Space and the <lb></lb>Addition.</s></p><p type="main"> <s><emph type="italics"></emph>As for example, it being ſuppoſed that the Time along<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.178.2.jpg" xlink:href="069/01/178/2.jpg"></figure><lb></lb><emph type="italics"></emph>A B, out of Reſt in A, be A B; A S being another Space <lb></lb>added, The Time along A B after S A ſhall be the exceſſe of <lb></lb>the Mean-proportional betwixt S B and B A above the <lb></lb>Mean-proportional betwixt B A and A S.<emph.end type="italics"></emph.end></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></lb>of the Motion being given, to find another <lb></lb>part towards the end that ſhall be paſt in the <lb></lb>ſame Time as the firſt part given.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Space be C B, and let the part in it given after the begin<lb></lb>ing of the Motion in C be C D. </s> <s>It is required to find another <lb></lb>part towards the end B, which ſhall be paſt in the ſame Time as<emph.end type="italics"></emph.end><pb xlink:href="069/01/179.jpg" pagenum="176"></pb><emph type="italics"></emph>the given part C D. </s> <s>Take a Mean-proportional betwixt B C and C D, <lb></lb>to which ſuppoſe B A equal; and let C E be a third proportional be-<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.179.1.jpg" xlink:href="069/01/179/1.jpg"></figure><lb></lb><emph type="italics"></emph>tween B C and C A. </s> <s>I ſay, that E B is the Space that after <lb></lb>the Fall out of C ſhall be past in the ſame Time as the ſaid <lb></lb>C D is paſſed. </s> <s>For if we ſuppoſe the Time along C B <lb></lb>to be as C B; B A (that is the Mean-proportional betwixt <lb></lb>B C and C D) ſhall be the Time along C D. </s> <s>And becauſe <lb></lb>C A is the Mean proportional betwixt B C and C E, C A <lb></lb>ſhall be the Time along C E: But the whole B C is the <lb></lb>Time along the Whole C B: Therefore the part B A ſhall be <lb></lb>the Time along the part E B, after the Fall out of C: But <lb></lb>the ſaid B A was the Time along C D: Therefore C D and <lb></lb>E B ſhall be paſt in equal Times out of Reſt in C: Which <lb></lb>was to be done.<emph.end type="italics"></emph.end></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"></emph>ex quie<lb></lb>te,<emph.end type="italics"></emph.end> in which from the begining of the Motion <lb></lb>a part is taken at pleaſure, paſſed in any Time, <lb></lb>after which an Inflex Motion followeth along <lb></lb>any Plane however Inclined, the Space which <lb></lb>along that Plane is paſſed in a Time equal to <lb></lb>the Time of the Fall already made along the <lb></lb>Perpendicular ſhall be to the Space then paſ<lb></lb>ſed along the Perpendicular more than double, <lb></lb>and leſſe than triple.</s></p><p type="main"> <s><emph type="italics"></emph>From the Horizon A E let fall a Perpendicular A B, along which <lb></lb>from the begining A let a Fall be made, of which let a part A C <lb></lb>be taken at pleaſure; then out of C let any Plane G be inclined at <lb></lb>pleaſure: along which after the Fall along A C let the Motion be con<lb></lb>tinued. </s> <s>I ſay, the Space paſſed by that Motion along C G in a Time <lb></lb>equall to the Time of the Fall along A C, is more than double, and leſs <lb></lb>than triple that ſame Space A C. </s> <s>For ſuppoſe C F equal to A C, and <lb></lb>extending out the Plane G C as far as the Horizon in E, and as C E <lb></lb>is to E F, ſo let F E be to E G. </s> <s>If therefore we ſuppoſe the Time of<emph.end type="italics"></emph.end><pb xlink:href="069/01/180.jpg" pagenum="177"></pb><emph type="italics"></emph>the Fall along A C to be as the Line A C; C E ſhall be the Time along <lb></lb>E C, and C F or C A the Time of the Motion along C G. </s> <s>Therefore <lb></lb>it is to be proved that the<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.180.1.jpg" xlink:href="069/01/180/1.jpg"></figure><lb></lb><emph type="italics"></emph>Space C G is more than <lb></lb>double, and leſſe than <lb></lb>triple the ſaid C A. </s> <s>For <lb></lb>in regard that as C E is <lb></lb>to E F, ſo is F E to E G; <lb></lb>therefore alſo ſo is C F to <lb></lb>F G. </s> <s>But E C is leſſe <lb></lb>than E F: Therefore C F <lb></lb>ſhall be leſſe than F G, and <lb></lb>G C more than double to <lb></lb>F C or A C. </s> <s>And moreover, in regard that F E is leſſe than double to <lb></lb>E C, (for E C is greater than C A or C F) G F ſhall alſo be leſſe <lb></lb>than double to F C, and G C leſſe than triple to C F or C A: Which <lb></lb>was to be demonſtrated.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>And the ſame may be more generally propounded: for that which <lb></lb>hapneth in the Perpendicular and Inclined Plane, holdeth true alſo if <lb></lb>after the Motion a Plane ſomewhat inclined it be inflected along a more <lb></lb>inclining Plane, as is ſeen in the other Figure: And the Demonſtration <lb></lb>is the ſame.<emph.end type="italics"></emph.end></s></p><p type="head"> <s><emph type="italics"></emph>P<emph.end type="italics"></emph.end>ROBL. VIII. <emph type="italics"></emph>P<emph.end type="italics"></emph.end>RO<emph type="italics"></emph>P.<emph.end type="italics"></emph.end> XXII.</s></p><p type="main"> <s>Two unequall Times being given, and a Space <lb></lb>that is paſt <emph type="italics"></emph>ex quiete<emph.end type="italics"></emph.end> along the Perpendicular <lb></lb>in the ſhorteſt of thoſe given Times, to inflect <lb></lb>a Plane from the higheſt point of the Perpen<lb></lb>dicular unto the Horizon, along which the <lb></lb>Moveable may deſcend in a Time equal to the <lb></lb>longeſt of thoſe Times given.</s></p><p type="main"> <s><emph type="italics"></emph>Let the unsqual Times be A the greater, and B the leſſer; and let <lb></lb>the Space that is paſt<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>along the Perpendicular in the <lb></lb>Time B, be C D. </s> <s>It is required from the Term C to inflect<emph.end type="italics"></emph.end> [or <lb></lb><figure id="id.069.01.180.2.jpg" xlink:href="069/01/180/2.jpg"></figure><lb></lb>bend] <emph type="italics"></emph>a Plane untill it reach the Horizon that may be paſſed in the<emph.end type="italics"></emph.end><pb xlink:href="069/01/181.jpg" pagenum="178"></pb><emph type="italics"></emph>Time A. </s> <s>As B is to A, ſo let C D be to another Line, to which let C X <lb></lb>be equal that deſcendeth from C unto the Horizon: It is manifeſt that <lb></lb>the Plane C X is that along which the Moveable deſcendeth in the Gi<lb></lb>ven Time A. </s> <s>For it hath been demonſtrated, that the Time along the <lb></lb>inclined Plane hath the ſame proportion to the Time along its ^{*} Eleva-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1094"></arrow.to.target><lb></lb><emph type="italics"></emph>tion, as the Length of the Plane hath to the Length of its Elevation,. <lb></lb>The Time, therefore, along C X is to the Time along C D, as C X is to <lb></lb>C D, that is, as the Time A is to the Time B: But the Time B is that <lb></lb>in which the Perpendicular is paſt<emph.end type="italics"></emph.end> ex quiete: <emph type="italics"></emph>Therefore the Time A is <lb></lb>that in which the Plane C X is paſſed.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1094"></margin.target>* Or Perpendi<lb></lb>cular.</s></p><p type="head"> <s><emph type="italics"></emph>P<emph.end type="italics"></emph.end>ROBL. IX. PROP. XXIII.</s></p><p type="main"> <s>A Space paſt <emph type="italics"></emph>ex quiete<emph.end type="italics"></emph.end> along the Perpendicular in <lb></lb>any Time being given, to inflect a Plane from <lb></lb>the loweſt term of that Space, along which, <lb></lb>after the Fall along the Perpendicular, a Space <lb></lb>equal to any Space given may be paſſed in the <lb></lb>ſame Time: which nevertheleſſe is more than <lb></lb>double, and leſſe than triple the Space paſſed <lb></lb>along the Perpendicular.</s></p><p type="main"> <s><emph type="italics"></emph>Along the Perpendicular A S, in the Time A C, let the Space <lb></lb>A C be paſt<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in A; to which let I R be more than <lb></lb>double, and leſſe than triple. </s> <s>It is required from the Terme C <lb></lb>to inflect a Plane, along which a Moveable after the Fall along A C <lb></lb>may in the ſame Time A C paſſe a Space equal to the ſaid I R. </s> <s>Let <lb></lb>R N, and N M be equal to A C: And look what proportion the part <lb></lb>I M hath to M N, the ſame ſhall the Line A C have to another, equal <lb></lb>to which draw C E from C to<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.181.1.jpg" xlink:href="069/01/181/1.jpg"></figure><lb></lb><emph type="italics"></emph>the Horizon A E, which con<lb></lb>tinue out towards O, and take <lb></lb>C F, F G, and G O, equal to <lb></lb>the ſaid R N, N M, and M I. <lb></lb></s> <s>I ſay, that the Time along the <lb></lb>inflected Plane C O, after the <lb></lb>Fall A G, is equal to the Time <lb></lb>A C out of Reſt in A. </s> <s>For in <lb></lb>regard that as O G is to G F, <lb></lb>ſo is F C to C E by Compoſition it ſhall be that as O F is to F G or F C, <lb></lb>ſo is F E to E C; and as one of the Antecedents is to one of the Con<lb></lb>ſequents, ſo are all to all; that is, the whole O E is to E F as F E to <lb></lb>E C: Therefore O E, E F, and E C are Continual Proportionals:<emph.end type="italics"></emph.end><pb xlink:href="069/01/182.jpg" pagenum="179"></pb><emph type="italics"></emph>And ſince it was ſuppoſed that the Time along A C is as A C, C E ſhall <lb></lb>be the Time along E C; and E F the Time along the whole E O; and <lb></lb>the part C F that along the part C O: But C F is equal to the ſaid C A: <lb></lb>Therefore that is done which was required: For the Time C A is the <lb></lb>Time of the Fall along A C<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in A; and C F (which is equal <lb></lb>to C A) is the Time along C O, after the Deſcent along E C, or after <lb></lb>the Fall along A C: Which was the Propoſition.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>And here it is to be noted, that the ſame may happen if the preceding <lb></lb>Motion be not made along the Perpendicular, but along an Inclined Plane: <lb></lb>As in the following Figure, in which let the preceding Lation be made <lb></lb>along the inclined Plane A S beneath the Horizon A E: And the Demon<lb></lb>ſtration is the very ſame.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>SCHOLIUM.</s></p><p type="main"> <s>If one obſerve well, it ſhall be manifeſt, that the leſſe the given <lb></lb>Line I R wanteth of being triple to the ſaid A C, the nearer <lb></lb>ſhall the Inflected Plane, along which the ſecond Motion is <lb></lb>to be made, which ſuppoſe to be C O, come to the Perpen<lb></lb>dicular, along which in a Time equal to A C a Space ſhall <lb></lb>be paſſed triple to A C.</s></p><p type="main"> <s><emph type="italics"></emph>For in caſe I R were very near the triple of A C, I M ſhould be well<lb></lb>near equal to M N: And if, as I M is to M N by Conſtruction, ſo <lb></lb>A C is to C E, then it is evident that the ſaid C E will be found but <lb></lb>little bigger than C A, and, which followeth of conſequence, the point E <lb></lb>ſhall be found very near the point A, and C O to containe a very acute<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.182.1.jpg" xlink:href="069/01/182/1.jpg"></figure><lb></lb><emph type="italics"></emph>Angle with C S, and <lb></lb>almoſt to concur both in <lb></lb>one Line. </s> <s>And on the <lb></lb>contrary, if the ſaid I R <lb></lb>were but a very little <lb></lb>more than double the <lb></lb>ſaid A C, I M ſhould <lb></lb>be a very ſhort Line. <lb></lb></s> <s>Hence it may happen <lb></lb>alſo that A C may come <lb></lb>to be very ſhort in reſpect of C E which ſhall be very long, and ſhall ap<lb></lb>proach very near the Horizontal Parallel drawn from C. </s> <s>And from <lb></lb>hence we may collect, that if in the preſent Figure after the Deſcent along <lb></lb>the inclined Plane A C, a Reflexion be made along the Horizontal Line, <lb></lb>as<emph.end type="italics"></emph.end> v. </s> <s>gr. <emph type="italics"></emph>C T, the Space along which the Moveable afterwards moved <lb></lb>in a Time equal to the Time of the Deſcent along A C would be exactly <lb></lb>double to the Space A C. </s> <s>And it appears that the like Diſcourſe may be <lb></lb>here applied: For it is apparent by what hath been ſaid, that ſince O E<emph.end type="italics"></emph.end><pb xlink:href="069/01/183.jpg" pagenum="180"></pb><emph type="italics"></emph>is to E F, as F E is to E C, that F C determineth the Time along C O: <lb></lb>And if a part of the Horizontal Line T C double to C A be divided in <lb></lb>two equal parts in V, the extenſion towards X ſhall be prolonged<emph.end type="italics"></emph.end> in in<lb></lb>finitum, <emph type="italics"></emph>whilſt it ſeeks to meet with the prolonged Line A E: And the <lb></lb>proportion of the Infinite Line T X to the Infinite Line V X, ſhall be <lb></lb>no other than the proportion of the Infinite Line V X to the Infinite <lb></lb>Line X C.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>We may conclude the ſelf-ſame thing another way by reaſſuming the <lb></lb>ſame Reaſoning that we uſed in the Demonſtration of the firſt Propoſi<lb></lb>tion. </s> <s>For reſuming the Triangle A B C, repreſenting to us by its Pa<lb></lb>rallels to the Baſe B C the Degrees of Velocity continually encreaſed ac<lb></lb>cording to the encreaſes of the Time; from which, ſince they are infi<lb></lb>nite, like as the Points are infinite in the Line A C, and the Inſtants <lb></lb>in any Time, ſhall reſult the Superficies of that ſame Triangle, if we <lb></lb>underſtand the Motion to continue for ſuch another Time, but no far<lb></lb>ther with an Accelerate, but with an Equable Motion, according to the <lb></lb>greateſt degree of Velocity acquired, which degree is repreſented <lb></lb>by the Line B C. </s> <s>Of ſuch degrees ſhall be made up an Aggregate like to <lb></lb>a Parallelogram A D B C, which is the double of<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.183.1.jpg" xlink:href="069/01/183/1.jpg"></figure><lb></lb><emph type="italics"></emph>the Triangle A B C. </s> <s>Wherefore the Space which <lb></lb>with degrees like to thoſe ſhall be paſſed in the ſame <lb></lb>Time, ſhall be double to the Space paſt with the de<lb></lb>grees of Velocity repreſented by the Triangle A B C: <lb></lb>But along the Horizontal Plane the Motion is Equa<lb></lb>ble, for that there is no cauſe of Acceleration, or Re<lb></lb>tardation: Therefore it may be concluded that the <lb></lb>Space C D, paſſed in a Time equall to the Time A C is double to the <lb></lb>Space A C: For this Motion is made<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>Accelerate according <lb></lb>to the Parallels of the Triangle; and that according to the Parallels <lb></lb>of the Parallelogram, which, becauſe they are infinite, are donble to <lb></lb>the infinite Parallels of the Triangle.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Moreover it may farther be obſerved, that what ever degree of <lb></lb>ſwiftneſs is to be found in the Moveable, is indelibly impreſſed upon it <lb></lb>of its own nature, all external cauſes of Acceleration or Retardation <lb></lb>being removed; which hapneth only in Horizontal Planes: for in de<lb></lb>clining Planes there is cauſe of greater Acceleration, and in the riſing <lb></lb>Planes of greater Retardation. </s> <s>From whence in like manner it fol<lb></lb>loweth that the Motion along the Horizontal Plane is alſo Perpetual: <lb></lb>for if it be Equable, it can neither be weakned nor retarded, nor much <lb></lb>leſſe deſtroyed. </s> <s>Farthermore, the degree of Celerity acquired by the <lb></lb>Moveable in a Natural Deſcent, being of its own Nature Indelible and <lb></lb>Penpetual, it is worthy conſideration, that if after the Deſcent along a <lb></lb>declining Plane a Reflexion be made along another Plane that is riſing, <lb></lb>in this latter there is cauſe of Retardation, for in theſe kind of Planes<emph.end type="italics"></emph.end><pb xlink:href="069/01/184.jpg" pagenum="181"></pb><emph type="italics"></emph>the ſaid Moveable doth naturally deſcend; whereupon there reſults a <lb></lb>mixture of certain contrary Affections, to wit, that degree of Celerity <lb></lb>acquired in the precedent Deſcent, which would of it ſelf carry the Move<lb></lb>able uniformly<emph.end type="italics"></emph.end> in infinitum, <emph type="italics"></emph>and of Natural Propenſion to the Motion of <lb></lb>Deſcent according to that ſame proportion of Acceleration wherewith it <lb></lb>alwaies moveth. </s> <s>So that it will be but reaſonable, if, enquiring what <lb></lb>accidents happen when the Moveable after the Deſcent along any incli<lb></lb>ned Plane is Reflected along ſome riſing Plane, we take that greateſt de<lb></lb>gree acquired in the Deſcent to keep it ſelf perpetually the ſame in the <lb></lb>Aſcending Plane; But that there is ſuperadded to it in the Aſcent the <lb></lb>Natural Inclination downwards, that is the Motion from Reſt Accelerate <lb></lb>according to the received proportion: And leſt this ſhould, perchance, be <lb></lb>ſomewhat intricate to be underſtood, it ſhall be more clearly explained by a <lb></lb>Scheme.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Let the Deſcent therefore be ſuppoſed to be made along the Declining <lb></lb>Plane A B, from which let the Reflex Motion be continued along another <lb></lb>Riſing Plane B C: And in the firſt place let the Planes be equal, and <lb></lb>elevated at equal Angles to the Horizon G H. </s> <s>Now it is manifeſt, that <lb></lb>the Moveable<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in A deſcending along A B acquireth degrees of <lb></lb>Velocity according to the increaſe of its Time, and that the degree in B <lb></lb>is the greateſt of thoſe acquired and by Nature immutably impreſſed, I <lb></lb>mean the Cauſes of new Acceleration or Retardation being removed: <lb></lb>of Acceleration, I ſay, if it ſhould paſſe any farther along the extended <lb></lb>Plane; and of Retardation, whilſt the Reflection is making along the <lb></lb>Acclivity B C: But along the Horizontal Plane G H the Equable Mo<lb></lb>tion according to the de-<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.184.1.jpg" xlink:href="069/01/184/1.jpg"></figure><lb></lb><emph type="italics"></emph>gree of Velocity acquired <lb></lb>from A unto B would ex<lb></lb>tend<emph.end type="italics"></emph.end> in infinitum. <emph type="italics"></emph>And <lb></lb>ſuch a Velocity would <lb></lb>that be which in a Time <lb></lb>equal to the Time of the <lb></lb>Deſcent along A B would paſſe a Space in double the Horizon to the ſaid <lb></lb>A B. </s> <s>Now let us ſuppoſe the ſame Moveable to be Equably moved with <lb></lb>the ſame degree of Swiftneſſe along the Plane B C, in ſuch ſort that alſo <lb></lb>in this Time equal to the Time of the Deſcent along A B a Space may be <lb></lb>paſſed a long B C extended double to the ſaid A B. </s> <s>And let us under<lb></lb>ſtand that as ſoon as it beginneth to aſcend there naturally befalleth the <lb></lb>ſame that hapneth to it from A along the Plane A B, to wit, a certain <lb></lb>Deſcent<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>according to thoſe degrees of Acceleration, by vertue <lb></lb>of which, as it befalleth in A B, it may deſcend as much in the ſame <lb></lb>Time along the Reflected Plane as it doth along A B: It is manifeſt, that <lb></lb>by this ſame Mixture of the Equable Motion of Aſcent, and the Acce<lb></lb>lerate of Deſcent the Moveable may be carried up to the Term C along <lb></lb>the Plane B C according to thoſe degrees of Velocity, which ſhall be<emph.end type="italics"></emph.end><pb xlink:href="069/01/185.jpg" pagenum="182"></pb><emph type="italics"></emph>equal. </s> <s>And that two points at pleaſure D and E being taken, equally <lb></lb>remote from the Angle B, the Tranſition along D B is made in a Time <lb></lb>equal to the Time of the Reflection along B E, we may collect from hence: <lb></lb>Draw D F, which ſhall be Parallel to B C; for it is manifeſt that the <lb></lb>Deſcent along A D is reflected along D F: And if after D the Move<lb></lb>able paſſe along the Horizontal Plane D E, the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>in E ſhall be <lb></lb>the ſame as the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>in D: Therefore it will aſcend from E to C: <lb></lb>And therefore the degree of Velocity in D is equal to the degree in E. <lb></lb></s> <s>From theſe things, therefore, we may rationally affirm, that, if a de<lb></lb>ſcent be made along any inclined Plane, after which a Reflection may <lb></lb>follow along an elevated Plane, the Moveable may by the conceived<emph.end type="italics"></emph.end><lb></lb>Impetus <emph type="italics"></emph>aſcend untill it attain the ſame beight, or Elevation from the <lb></lb>Horizon. </s> <s>As if a Deſcent be made along A B, the Moveable would <lb></lb>paſſe along the Reflected Plane B C, untill it arrive at the Horizon <lb></lb>A C D; and that not only when the Inclinations of the Planes are <lb></lb>equal, but alſo when they are unequal, as is the Plane B D: For it was <lb></lb>first ſuppoſed, that the degrees of Velocity are equal, which are acqui<lb></lb>red upon Planes unequally inclined, ſo long as the Elevation of thoſe <lb></lb>Planes above the Horizon was the ſame: But, if there being the ſame <lb></lb>Inclination of the Planes E B and B D, the Deſcent along E B ſufficeth <lb></lb>to drive the Moveable along the Plane BD as far as D, ſeeing this Impulſe<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.185.1.jpg" xlink:href="069/01/185/1.jpg"></figure><lb></lb><emph type="italics"></emph>is made by the<emph.end type="italics"></emph.end> Impe<lb></lb>tus <emph type="italics"></emph>of Velocity in the <lb></lb>point B; and if the<emph.end type="italics"></emph.end><lb></lb>Impetus <emph type="italics"></emph>be the ſame <lb></lb>in B, whether the <lb></lb>Moveable deſcend a<lb></lb>long A B, or along E B: It is manifeſt, that the Moveable ſhall be in <lb></lb>the ſame manner driven along B D, after the Deſcent along A B, and <lb></lb>after that along E B: But it will happen that the Time of the Aſcent <lb></lb>along B D ſhall be longer than along B C, like as the Deſcent along <lb></lb>E B is made in a longer time than along A B: But the Proportion of <lb></lb>thoſe Times was before demonſtrated to be the ſame as the Lengths of <lb></lb>thoſe Planes. </s> <s>Now it follows, that we ſeek the proportion of the Spaces <lb></lb>paſt in equal Times along Planes, whoſe Inclinations are different, but <lb></lb>their Elevations the ſame; that is, which are comprehended between <lb></lb>the ſame Horizontal Parallels. </s> <s>And this hapneth according to the fol<lb></lb>lowing Propoſition.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/186.jpg" pagenum="183"></pb><p type="head"> <s>THEOR. XV. PROP. XXIV.</s></p><p type="main"> <s>There being given between the ſame Horizontal <lb></lb>Parallels a Perpendicular and a <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane eleva<lb></lb>ted from its loweſt term, the Space that a <lb></lb>Moveable after the Fall along the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendi<lb></lb>cular paſſeth along the Elevated <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane in a <lb></lb>Time equal to the Time of the Fall, is greater <lb></lb>than that <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicular, but leſſe than double <lb></lb>the ſame.</s></p><p type="main"> <s><emph type="italics"></emph>Between the ſame Horizontal Parallels B C and H G let there <lb></lb>be the Perpendicular A E; and let the Elevated Plane be E B, <lb></lb>along which after the Fall along the Perpendicular A E out of <lb></lb>the Term E let a Reflexion be made towards B. </s> <s>I ſay, that the Space, <lb></lb>along which the Moveable aſcendeth in a Time equal to the Time of the <lb></lb>Deſcent A E, is greater than A E, but leſſe than double the ſame A E. <lb></lb></s> <s>Let E D be equal to A E, and as E B is to B D, ſo let D B be to B F. </s> <s>It <lb></lb>ſhall be proved, firſt that the point F is the Term at which the Moveable <lb></lb>with a Reflex Motion along E B arriveth in a Time equal to the Time <lb></lb>A E: And then, that E F is greater than E A, but leſſe than double the <lb></lb>ſame. </s> <s>If we ſuppoſe the Time of the Deſcent along A E to be as A E, <lb></lb>the Time of the Deſcent along B E, or Aſcent along E B ſhall be as the <lb></lb>ſame Line B E: And D B being a Mean-Proportional betwixt E B <lb></lb>and B F, and B E being the Time of Deſcent along the whole B E, B D <lb></lb>ſhall be the Time of the Deſcent along B F, and the Remaining part <lb></lb>D E the Time of the<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.186.1.jpg" xlink:href="069/01/186/1.jpg"></figure><lb></lb><emph type="italics"></emph>Deſcent along the Re<lb></lb>maining part F E: But <lb></lb>the Time along F E<emph.end type="italics"></emph.end> ex <lb></lb>quiete <emph type="italics"></emph>in B, and the <lb></lb>Time of the Aſcent a<lb></lb>long E F is the ſame, ſince that the Degree of Velocity in E was acqui<lb></lb>red along the Deſcent B E, or A E: Therefore the ſame Time D E ſhall <lb></lb>be that in which the Moveable after the Fall out of A along A E, <lb></lb>with a Reflex Motion along E B ſhall reach to the Mark F: But it hath <lb></lb>been ſuppoſed that E D is equal to the ſaid A E: Which was firſt to be <lb></lb>proved. </s> <s>And becauſe that as the whole E B is to the whole B D, ſo is the <lb></lb>part taken away D B to the part taken away B F, therefore, as the whole <lb></lb>E B is to the whole B D, ſo ſhall the Remainder E D be to D F: <lb></lb>But E B is greater than B D: Therefore E D is greater than D F, and <lb></lb>E F leſſe than double to D E or A E: Which was to be proved.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/187.jpg" pagenum="184"></pb><p type="main"> <s><emph type="italics"></emph>And the ſame alſo hapneth if the precedent Motion be not made <lb></lb>along the Perpendicular, but along an Inclined Plane; and the Demon<lb></lb>ſtration is the ſame, provided that the Reflex Plane be leſſe riſing, that is, <lb></lb>longer than the declining Plane.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>THEOR. XVI. <emph type="italics"></emph>P<emph.end type="italics"></emph.end>RO<emph type="italics"></emph>P.<emph.end type="italics"></emph.end> XXV.</s></p><p type="main"> <s>If after the Deſcent along any Inclined Plane a <lb></lb>Motion follow along the Plane of the Hori<lb></lb>zon, the Time of the Deſcent along the Incli<lb></lb>ned Plane ſhall be to the Time of the Motion <lb></lb>along any Horizontal Line; as the double <lb></lb>Length of the Inclined Plane is to the Line ta<lb></lb>ken in the Horizon.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Horizontal Line be C B, the inclined Plane A B, and after <lb></lb>the Deſcent along A B let a Motion follow along the Horizon, in <lb></lb>which take any Space B D. </s> <s>I ſay, that the Time of the Deſcent <lb></lb>along A B to the Time of the Motion along B D is as the double of A B <lb></lb>to B D. </s> <s>For B C being ſuppoſed <lb></lb>the double of A B, it is manifeſt by<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.187.1.jpg" xlink:href="069/01/187/1.jpg"></figure><lb></lb><emph type="italics"></emph>what hath already been demonſtra<lb></lb>ted that the Time of the Deſcent <lb></lb>along A B is equal to the Time of <lb></lb>the Motion along B C: But the <lb></lb>Time of the Motion along B C is to <lb></lb>the Time of the Motion along B D, as the Line C B is to the Line B D: <lb></lb>Therefore the Time of the Motion along A B is the Time along B D, as <lb></lb>the Double of A B is to B D: Which was to be proved.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROBL X. PROP. XXVI.</s></p><p type="main"> <s>A Perpendicular between two Horizontal <emph type="italics"></emph>P<emph.end type="italics"></emph.end>aral<lb></lb>lel Lines, as alſo a Space greater than the ſaid <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicular, but leſſe than double the ſame, <lb></lb>being given, to raiſe a <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane between the ſaid <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>arallels from the loweſt Term of the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>er<lb></lb>pendicular, along which the Moveable may <lb></lb>with a Reflex Motion after the Fall along the <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicular paſſe a Space equal to the Space <lb></lb>given, and in a Time equal to the Time of the <lb></lb>Fall along the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicular.</s></p><pb xlink:href="069/01/188.jpg" pagenum="185"></pb><p type="main"> <s><emph type="italics"></emph>Let A B be a Perpendicular between the Horizontal Parallels A O <lb></lb>and B C; and let F E be greater than B A, but leſſe than double <lb></lb>the ſame. </s> <s>It is required between the ſaid Parallels from the point <lb></lb>B to raiſe a Plane, along which the Moveable after the Fall from A to <lb></lb>B may with a Reflex Motion in a Time equal to the Time of the Fall <lb></lb>along A B paſſe a Space aſcending equal to the ſaid E F. </s> <s>Suppoſe E D <lb></lb>equall to A B, the Remaining Part D F ſhall be leſſe, for that the whole <lb></lb>E F is leſſe than double to A B: Let D I be equal to D F, and as E I is <lb></lb>to I D, ſo let D F be to another Space F X, and out of B let the Right-<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.188.1.jpg" xlink:href="069/01/188/1.jpg"></figure><lb></lb><emph type="italics"></emph>Line B O be reflected, equal to E X. </s> <s>I ſay, that the Plane along B O <lb></lb>is that along which after the Fall A B a Moveable in a Time equal <lb></lb>to the Time of the Fall along A B paſſeth aſcending a Space equal to <lb></lb>the given Space E F. </s> <s>Suppoſe B R and R S equal to the ſaid E D and <lb></lb>D F. </s> <s>And becauſe that as E I is to I D, ſo is D F to F X; therefore, <lb></lb>by Compoſition, as E D is to D I, ſo ſhall D X be to X F; that is, as <lb></lb>E D is to D F, ſo ſhall D X be to X F, and E X to X D; that is, as <lb></lb>B O is to O R, ſo ſhall R O be to O S: And if we ſuppoſe the Time <lb></lb>along A B to be A B, the Time along O B ſhall be the ſame O B, and <lb></lb>R O the Time along O S, and the Remaining Part B R the Time along <lb></lb>the Remaining Part S B, deſcending from O to B: But the Time of <lb></lb>the Deſcent along S B from Rest in O, is equal to the Time of the <lb></lb>Aſcent from B to S after the Fall A B: Therefore B O is the Plane ele<lb></lb>vated from B, along which after the Fall along A B the Space B S <lb></lb>equal to the given Space E F is paſſed in the Time B R or B A: Which <lb></lb>was required to be done.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/189.jpg" pagenum="186"></pb><p type="head"> <s>THEOR. XVII. PROP. XXVII.</s></p><p type="main"> <s>If a Moveable deſcend along unequal <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lanes, <lb></lb>whoſe Elevation is the ſame, the Space that <lb></lb>ſhall be paſt along the lower part of the longeſt <lb></lb>in a Time equal to that in which the whole <lb></lb>ſhorter <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane is paſſed, is equal to the Space <lb></lb>that is compounded of the ſaid ſhorter <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane <lb></lb>and of the part to which that ſhorter <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane <lb></lb>hath the ſame <emph type="italics"></emph>P<emph.end type="italics"></emph.end>roportion that the longer <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane hath to the Exceſſe by which the longeſt <lb></lb>exceedeth the ſhorteſt.</s></p><p type="main"> <s><emph type="italics"></emph>Let A C be the longer Plane, and A B the ſhorter, whoſe Elevation <lb></lb>A D is the ſame; and in the lower part of A C take the Space <lb></lb>C E, equal to the ſaid A B; and as C A is to A E, (that is to <lb></lb>the exceſſe of the Plane C A above A B) ſo let C E be to E F. </s> <s>I ſay, <lb></lb>that the Space F C is that which is paſt after the Deſcent out of A in <lb></lb>a Time equal to the Time of<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.189.1.jpg" xlink:href="069/01/189/1.jpg"></figure><lb></lb><emph type="italics"></emph>the Deſcent along A B. </s> <s>For <lb></lb>the whole C A, being to the <lb></lb>whole A E, as the part taken <lb></lb>away C E is to the part taken <lb></lb>away E F, therefore the re<lb></lb>maining part E A ſhall be to <lb></lb>the remaining part A F, as the <lb></lb>whole C A is to the whole A E: Therefore the three Spaces C A, <lb></lb>A E, and A F are three Continual proportionals. </s> <s>And if the Time <lb></lb>along A B be ſuppoſed to be as A B, the Time along A C ſhall be as <lb></lb>A C, and the Time along A F ſhall be as A E, and along the remain<lb></lb>ing part F C ſhall be as E C: But E C is equal to the ſaid A B: There<lb></lb>fore the Propoſition is manifeſt.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>THEOR. XVIII. PROP. XXVIII.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Horizontal Line A G be Tangent to a Circle, and from the <lb></lb>point of Contact let A B be the Diameter, and A E B two Chords <lb></lb>at pleaſure: We are to aſſign the proportion of the Time of the <lb></lb>Fall along A B to the Time of the Deſcent along both the Chords <lb></lb>A E B. </s> <s>Let B E be continued out till it meet the Tangent in G, and<emph.end type="italics"></emph.end><pb xlink:href="069/01/190.jpg" pagenum="187"></pb><emph type="italics"></emph>let the Angle B A E be cut in two equal parts, and draw A F. </s> <s>I ſay, <lb></lb>that the Time along A B is to the Time along A E B, as A E is to A E F. <lb></lb></s> <s>For in regard the Angle F A B is equal to the Angle F A E, and the An<lb></lb>gle E A G to the Angle A B F, the whole Angle G A F ſhall be equal to <lb></lb>the two Angles F A B, and A B F; <lb></lb>to which alſo the Angle G F A<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.190.1.jpg" xlink:href="069/01/190/1.jpg"></figure><lb></lb><emph type="italics"></emph>is equal: Therefore the Line G F <lb></lb>is equal to G A. </s> <s>And becauſe the <lb></lb>Rectangle B G E is equal to the <lb></lb>Square of G A, it ſhall likewiſe <lb></lb>be equal to the Square of G F, and <lb></lb>the three Lines B G, G F, and <lb></lb>G E ſhall be proportionals. </s> <s>And <lb></lb>if we ſuppoſe A E to be the Time <lb></lb>along A E, G E ſhall be the Time <lb></lb>along G E, and G F the Time along the whole G B, and E F the Time <lb></lb>along E B, after the Deſcent out of G, or out of A, along A E: The Time, <lb></lb>therefore, along A E, or along A B ſhall be to the Time along A E B, as <lb></lb>A E is to A E F: Which was to be determined.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>More briefly thus. </s> <s>Let G F be cut equal to G A: It is manifeſt <lb></lb>that G F is the Mean-proportional between B G, and G E. </s> <s>The reſt as <lb></lb>before.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROBL. <emph type="italics"></emph>XI. P<emph.end type="italics"></emph.end>RO<emph type="italics"></emph>P. XXIX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Any Horizontal Space being given upon the <lb></lb>end of which a Perpendicular is erected, <lb></lb>in which a part is taken equal to half of the <lb></lb>Space given in the Horizontal a Moveable fal<lb></lb>ling from that height, and turned along the <lb></lb>Horizon, ſhall paſſe the Horizontal Space to<lb></lb>gether with the Perpendicular in a ſhorter <lb></lb>Time than any other Space of the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendi<lb></lb>cular with the ſame Horizontal Space.</s></p><p type="main"> <s><emph type="italics"></emph>Let there be an Horizontal Space in which let any Space be given <lb></lb>B C, and on B let there be a Perpendicular erected, in which let <lb></lb>B A be the half of the foreſaid B C. </s> <s>I ſay, that the Time in which <lb></lb>a Moveable let fall out of A paſſeth both the Spaces A B and B C is the <lb></lb>ſhorteſt of all Times in which the ſaid Space B C with a part of the <lb></lb>Perpendicular, whether greater or leſſer than the part A B, ſhall be paſ<lb></lb>ſed. </s> <s>Let a greater be taken, as in the ſirſt Figure, or leſſer, as in the<emph.end type="italics"></emph.end><pb xlink:href="069/01/191.jpg" pagenum="188"></pb><emph type="italics"></emph>ſecond, which let be E B. </s> <s>It is to be proved that the Time in which the <lb></lb>Spaces E B and B C are paſſed is longer than the Time in which A B <lb></lb>and B C are paſſed. </s> <s>Let the Time along A B be as A B; the ſame ſhall <lb></lb>be the Time of the Motion along the Horizontal Space B G; becauſe <lb></lb>B C is double to A B, and the Time along both the Spaces A B C ſhall <lb></lb>be double of O B A. </s> <s>Let B O<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.191.1.jpg" xlink:href="069/01/191/1.jpg"></figure><lb></lb><emph type="italics"></emph>be a Mean-proportional between <lb></lb>E B and B A. </s> <s>B O ſhall be the <lb></lb>Time of the Fall along E B. <lb></lb>Again, let the Horizontal Space <lb></lb>B D be double to the ſaid B E: <lb></lb>It is manifeſt that the Time of it <lb></lb>after the Fall E B is the ſame <lb></lb>B O. </s> <s>As D B is to B C, or as <lb></lb>E B is to B A, ſo let O B be to <lb></lb>B N: and in regard the Motion <lb></lb>along the Horizontal Plane is Equable, and O B being the Time along <lb></lb>B D after the Fall out of E, therefore N B ſhall be the Time along B C <lb></lb>after the Fall from the ſame Altitude E. </s> <s>Hence it is manifeſt, that O B, <lb></lb>together with B N is the Time along E B C; and becauſe the double of <lb></lb>B A is the Time along A B C; it remains to be proved, that O B, to<lb></lb>gether with B N is more than double B A. </s> <s>Now becauſe O B is a Mean <lb></lb>between E B and B A, the proportion of E B to B A is double the pro<lb></lb>portion of O B to B A: and, in regard that E B is to B A, as O B is to <lb></lb>B N, the proportion of O B to B N ſhall alſo be double the proportion of <lb></lb>O B to B A: But that proportion of O B to B N is compounded of the <lb></lb>proportions of O B to B A, and of A B to B N: therefore the proportion <lb></lb>of A B to B N is the ſame with that of O B to B A. </s> <s>Therefore B O, <lb></lb>B A, and B N are three continual Proportionals, and O B, together with <lb></lb>B N, are greater than double B A: Whereupon the Propoſition is ma<lb></lb>nifeſt.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/192.jpg" pagenum="189"></pb><p type="head"> <s>THEOR. <emph type="italics"></emph>XIX.<emph.end type="italics"></emph.end> PROP. <emph type="italics"></emph>XXX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>If a Perpendicular be let fall from any point of the <lb></lb>Horizontal Line, and out of another point in <lb></lb>the ſame Horizontal Line a Plane be drawn <lb></lb>forth untill it meet the Perpendicular, along <lb></lb>which a Moveable deſcendeth in the ſhorteſt <lb></lb>time unto the ſaid Perpendicular, this Plane <lb></lb>ſhall be that which cutteth off a part equall to <lb></lb>the diſtance of the aſſigned point from the end <lb></lb>of the Perpendicular.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Perpendicular B D be let fall from the point B of the Ho<lb></lb>rizontal Line A C, in which let there be any point C, and in the <lb></lb>Perpendicular let the Diſtance B E be ſuppoſed equal to the Di<lb></lb>ſtance B C, and draw C E. </s> <s>I ſay, that of all Planes inclined out of <lb></lb>the point C till they meet the Perpendicular C E is that, along which <lb></lb>in the ſhorteſt of all Times the Deſcent<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.192.1.jpg" xlink:href="069/01/192/1.jpg"></figure><lb></lb><emph type="italics"></emph>is made unto the Perpendicular. </s> <s>For <lb></lb>let the Planes C F and C G be inclined <lb></lb>above and below, and draw I K a Tan<lb></lb>gent unto the Semidiameter B C of the <lb></lb>deſcribed Circle in C, which ſhall be <lb></lb>equidiſtant from the Perpendicular; <lb></lb>and unto the ſaid C F let E K be Paral<lb></lb>lel cutting the Circumference of the Cir<lb></lb>cle in L: It is manifeſt that the Time of <lb></lb>the Deſcent along L E is equal to the <lb></lb>Time of the Deſcent along C E: But <lb></lb>the Time along K E is longer than along <lb></lb>L E: Therefore the Time along K E is <lb></lb>longer than that along C E: But the <lb></lb>Time along K E is equal to the Time a<lb></lb>long C F, they being equal, and drawn <lb></lb>according to the ſame Inclination: Likewiſe ſince C G, and I E are <lb></lb>equal, and inclined according to the ſame Inclination, the Times of the <lb></lb>Motions along them ſhall be equal: But H E being ſhorter than I E, the <lb></lb>Time along it is alſo ſhorter than I E: Therefore the Time alſo along <lb></lb>C E, (which is equal to the Time along H E) ſhall be ſhorter than the <lb></lb>Time along I E: The Propoſition, therefore, is manifeſt.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/193.jpg" pagenum="190"></pb><p type="head"> <s>THEOR. <emph type="italics"></emph>XX.<emph.end type="italics"></emph.end> PROP. <emph type="italics"></emph>XXXI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>If a Right-Line ſhall be in any manner inclined <lb></lb>upon the Horizontal Line, the Plane produced <lb></lb>from a given point in the Horizon untill it <lb></lb>meet with the Inclined Plane, along which <lb></lb>the Deſcent is made in the ſhorteſt of all <lb></lb>Times, is that which ſhall divide the Angle <lb></lb>contained between the two <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendiculars <lb></lb>drawn from the given <emph type="italics"></emph>P<emph.end type="italics"></emph.end>oint, the one unto the <lb></lb>Horizontal Line, the other to the Inclined <lb></lb>Line, into two equal parts.</s></p><p type="main"> <s><emph type="italics"></emph>Let C D be a Line inclined in any manner upon the Hori<lb></lb>zontal Line A B, and let any point A be given in the Hori<lb></lb>zon, and from it let A C be drawn Perpendicular to A B, <lb></lb>and A E Perpendicular to C D, and let the Line F A divide the <lb></lb>Angle C A E into two equal parts. </s> <s>I ſay, that of all Planes incli<lb></lb>ned out of any point of the Line C D to the point A that ſame pro<lb></lb>duced along F A is it along<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.193.1.jpg" xlink:href="069/01/193/1.jpg"></figure><lb></lb><emph type="italics"></emph>which the Deſcent is made in <lb></lb>the ſhorteſt of all Times. </s> <s>Let <lb></lb>F G be drawn Parallel to AE; <lb></lb>the alternate Angles G F A <lb></lb>and F A E ſhall be equal: But <lb></lb>E A F is equal to that other <lb></lb>F A G: Therefore of the Tri<lb></lb>angle the Sides F G and G A <lb></lb>ſhall be equal. </s> <s>If therefore <lb></lb>about the Center G, at the di<lb></lb>ſtance G A, a Circle be deſcri<lb></lb>bed it ſhall paſſe by F, and ſhall <lb></lb>touch the Horizontal, and the Inclined Lines in the points A and F: <lb></lb>For the Angle G F C is a Right Angle, and likewiſe G F is equidiſtant <lb></lb>to A E: Whence it is manifeſt that all Lines produced from the point <lb></lb>A unto the inclined Plane do extend beyond the Circumference, and, <lb></lb>which followeth of conſequence, that the Motions along the ſame do <lb></lb>take up more Time than along F A. </s> <s>Which was to be demonſtrated.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/194.jpg" pagenum="191"></pb><p type="head"> <s>LEMMA.</s></p><p type="main"> <s>If two Circles touch one another within, the innermoſt of which <lb></lb>toucheth ſome Right Line, and the exteriour one cutteth it, <lb></lb>three Lines produced from the Contact of the Circles unto <lb></lb>three points of the Tangent Right-Line, that is, to the Con<lb></lb>tact of the interiour Circle, and to the Sections of the exte<lb></lb>riour ſhall contain equall Angles in the Contact of the <lb></lb>Circles.</s></p><p type="main"> <s><emph type="italics"></emph>Let two Circles touch one another in the point A, of which let the <lb></lb>Centers be B, that of the leſſer, and C that of the greater; and let <lb></lb>the interiour Circle touch any Line F G in the point H, and let the grea<lb></lb>ter cut it in the points F and G, and connect the three Lines A F, A H, <lb></lb>and A G. </s> <s>I ſay, that the Angles by<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.194.1.jpg" xlink:href="069/01/194/1.jpg"></figure><lb></lb><emph type="italics"></emph>them contained F A H and G A H are <lb></lb>equal. </s> <s>Produce A H untill it meeteth <lb></lb>the Circumference in I, and from the <lb></lb>Centers draw B H and C I, and thorow <lb></lb>the ſaid Centers let B C be drawn, <lb></lb>which continued forth ſhall meet with <lb></lb>the Contact A, and with the Circum<lb></lb>ferences of the Circles in O and N. <lb></lb></s> <s>And becauſe the Angles I C N and <lb></lb>H O B are equal, for as much as either <lb></lb>of them is double to the Angle I A N, <lb></lb>the Lines B H and C I ſhall be Parallels: And becauſe B H drawn <lb></lb>from the Center to the Contact is Perpendicular to F G; C I ſhall alſo be <lb></lb>Perpendicular to the ſame, and the Arch F I equal to the Arch I G, and, <lb></lb>which followeth of conſequence, the Angle F A I to the Angle I A G: <lb></lb>Which was to be demonſtrated.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/195.jpg" pagenum="192"></pb><p type="head"> <s>THEOR. <emph type="italics"></emph>XXI.<emph.end type="italics"></emph.end> PROP. <emph type="italics"></emph>XXXII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>If two points be taken in the Horizon, and any <lb></lb>Line ſhould be inclined from one of them to<lb></lb>wards the other, out of which a Right-Line is <lb></lb>drawn unto the Inclined Line, cutting off a <lb></lb>part thereof equal to that which is included <lb></lb>between the points of the Horizon, the De<lb></lb>ſcent along this laſt drawn ſhall be ſooner per<lb></lb>formed, than along any other Right Lines pro<lb></lb>duced from the ſame point unto the ſaid Incli<lb></lb>ned Line. </s> <s>And along other Lines which are <lb></lb>on each hand of this by equal Angles a De<lb></lb>ſcent ſhall be made in equal Times.</s></p><p type="main"> <s><emph type="italics"></emph>In the Horizon let there be two points A and B, and from B incline <lb></lb>the Right Line B C, in which from the Term B take B D equal to <lb></lb>the ſaid B A, and draw a Line from A to D. </s> <s>I ſay, that the De<lb></lb>ſcent along A D is more ſwiftly made, than along any other whatſoever <lb></lb>drawn from the point A unto the inclined Line B C. </s> <s>For out of the <lb></lb>points A and D unto B A and<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.195.1.jpg" xlink:href="069/01/195/1.jpg"></figure><lb></lb><emph type="italics"></emph>B D draw the Perpendiculars <lb></lb>A E and D E, interſecting one <lb></lb>another in E: and foraſmuch as <lb></lb>in the equicrural Triangle A B D <lb></lb>the Angles B A D and B D A <lb></lb>are equal, the remainders to the <lb></lb>Right-Angles D A E and E D A <lb></lb>ſhall be equal. </s> <s>Therefore a Circle <lb></lb>deſcribed about the Center E at <lb></lb>the diſtance A E ſhall alſo paſſe <lb></lb>by D; and the Lines B A and <lb></lb>B D will touch it in the points A <lb></lb>and D. </s> <s>And ſince A is the end of the Perpendicular A E, the Deſcent <lb></lb>along A D ſhall be ſooner performed, than along any other produced from <lb></lb>the ſame Term A unto the Line B C beyond the Circumference of the <lb></lb>Circle: Which was firſt to be proved.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>But if in the Perpendicular A E being prolonged any Center be taken as <lb></lb>F, and at the diſtance F A the Circle A G C be deſcribed cutting the <lb></lb>Tangent Line in the points G and C; drawing A G and A C they ſhall <lb></lb>make equal Angles with the middle Line A D by what hath been afore<emph.end type="italics"></emph.end><pb xlink:href="069/01/196.jpg" pagenum="193"></pb><emph type="italics"></emph>demonſtrated, and the Motions thorow them ſhall be performed in equal <lb></lb>Times ſeeing that they terminate in A unto the Circumference of the <lb></lb>Circle A G O from the higheſt point of it A.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROBL. XII. PROP. <emph type="italics"></emph>XXXIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>A Perpendicular and Plane inclined to it being <lb></lb>given, whoſe height is one and the ſame, as al<lb></lb>ſo the higheſt term, to find a point in the Per<lb></lb>pendicular above the common term, out of <lb></lb>which if a Moveable be demitted that ſhall <lb></lb>afterwards turn along the inclined Plane, the <lb></lb>ſaid Plane may be paſt in the ſame Time in <lb></lb>which the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicular <emph type="italics"></emph>ex quiete<emph.end type="italics"></emph.end> would be <lb></lb>paſſed.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Perpendicular and inclined Plane, whoſe Altitude is the <lb></lb>ſame, be A B and A C. </s> <s>It is required in the Perpendicular B A, <lb></lb>continued out from the point A to find a Point out of which a <lb></lb>Moveable deſcending may paſſe the Space A C in the ſame Time in <lb></lb>which it will paſſe the ſaid Perpendicular A B out of Reſt in A. </s> <s>Draw <lb></lb>D C E at Right-Angles to A C, and let C D be cut equal to A B, and <lb></lb>draw a Line from A to D: The Angle A D C ſhall be greater than the <lb></lb>Angles C A D: (for C A is greater than A B or C D:) Let the <lb></lb>Angle D A E be equal to the Angle A D E; and to A E let E F an in<lb></lb>clined Plane be Perpen-<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.196.1.jpg" xlink:href="069/01/196/1.jpg"></figure><lb></lb><emph type="italics"></emph>dicular, and let both be<lb></lb>ing prolonged meet in F, <lb></lb>and unto both A I and <lb></lb>A G ſuppoſe C F to be <lb></lb>equal, and by G draw <lb></lb>G H equidiſtant to the <lb></lb>Horizon. </s> <s>I ſay, that H <lb></lb>is the point which is <lb></lb>ſought. </s> <s>For ſuppoſing the <lb></lb>Time of the Fall along <lb></lb>the Perpendicular A B <lb></lb>to be A B, the Time along <lb></lb>A C ex quiete in A ſhall be the ſame A C. </s> <s>And becauſe in the Right<lb></lb>angled Triangle A E F, from the Right Angle E unto the Baſe A F, <lb></lb>E C is a Perpendicular, A E ſhall be a Mean-Proportional betwixt F A <lb></lb>and A C, and C E a Mean betwixt A C and C F, that is, betwixt C A <lb></lb>and A I: and foraſmuch as the Time of A C out of A is A C, A E<emph.end type="italics"></emph.end><pb xlink:href="069/01/197.jpg" pagenum="194"></pb><emph type="italics"></emph>ſhall be the Time of the whole A F, and E C the Time of A I: And be<lb></lb>cauſe in the Equicrural Triangle A E D the Side A E is equal to the <lb></lb>Side E D, E D ſhall be the Time along A F, and E C is the Time along <lb></lb>A I: Therefore C D, that is A B ſhall be the Time along A F<emph.end type="italics"></emph.end> ex qui<lb></lb>ete <emph type="italics"></emph>in A; which is the ſame as if we ſaid, that A B is the Time along <lb></lb>A G out of G, or out of H: Which was to be done.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROBL. <emph type="italics"></emph>XIII. P<emph.end type="italics"></emph.end>RO<emph type="italics"></emph>P. XXXIV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>An inclined <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane and Perpendicular whoſe ſub<lb></lb>lime term is the ſame being given, to find a <lb></lb>more ſublime point in the Perpendicular pro<lb></lb>longed out of which a Moveable falling, and <lb></lb>being turned along the inclined <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane, may <lb></lb>paſſe them both in the ſame Time, as it doth <lb></lb>the ſole inclined <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lane <emph type="italics"></emph>ex quiete<emph.end type="italics"></emph.end> in its ſuperi<lb></lb>our Term.</s></p><p type="main"> <s><emph type="italics"></emph>Let the inclined Plane and Perpendicular be A B and A C, whoſe <lb></lb>Term A is the ſame. </s> <s>It is required in the Perpendicular prolonged <lb></lb>from A to find a ſublime point, out of which the Moveable deſcen<lb></lb>ding, and being turned along the Plane A B, may paſſe the aſſigned part <lb></lb>of the Perpendicular and the Plane A B in the ſame Time, as it would the <lb></lb>ſole Plane A B out of Reſt in A.<emph.end type="italics"></emph.end></s></p><figure id="id.069.01.197.1.jpg" xlink:href="069/01/197/1.jpg"></figure><p type="main"> <s><emph type="italics"></emph>Let the Ho<lb></lb>rizontal Line <lb></lb>be B C, and <lb></lb>let A N be <lb></lb>cut equal to <lb></lb>A C; and as <lb></lb>A B is to B N, <lb></lb>ſo let A L be <lb></lb>to L C: and <lb></lb>unto A L let <lb></lb>A I be equal, <lb></lb>and unto A C <lb></lb>and B I let C <lb></lb>E be a third <lb></lb>proportional, <lb></lb>marked in the <lb></lb>Perpendicular A C produced. </s> <s>I ſay, that C E is the Space acquired; <lb></lb>ſo that the Perpendicular being extended above A, and the part A X <lb></lb>equal to C E being taken, a Moveable out of X will paſſe both the<emph.end type="italics"></emph.end><pb xlink:href="069/01/198.jpg" pagenum="195"></pb><emph type="italics"></emph>Spaces X A B in the ſame Time as it would the ſole Space A B out of A. <lb></lb></s> <s>Draw the Horizontal Line X R Parallel to B C, with which let B A <lb></lb>being prolonged meet in R, and then A B being continued out unto D <lb></lb>draw E D Parallel to C B, and upon A D deſcribe a Semicircle, and <lb></lb>from B, and Perpendicular to D A, erect B F till it meet with the Cir<lb></lb>cumference. </s> <s>It is manifeſt that F B is a Mean-proportional betwixt <lb></lb>A B and B D, and that the Line drawn from F to A is a Mean-propor<lb></lb>tional betwixt D A and A B. </s> <s>Suppoſe B S equal to B I, and F H equal <lb></lb>to F B: And becauſe, as A B is to B D, ſo is A C to C E, and becauſe <lb></lb>B F is a Mean-proportional betwixt A B and B D, and becauſe B I is a <lb></lb>Mean-proportional betwixt A C and C E; therefore as B A is to A C, <lb></lb>ſo is F B to B S. </s> <s>And becauſe as B A is to A C, or A N, ſo is F B to <lb></lb>B S, therefore, by Converſion of the proportion, B F is to F S, as A B is <lb></lb>to B N, that is, A L to L C; therefore the Rectangle under F B and <lb></lb>C L, is equal to the Rectangle under A L, and S F: But this Rectangle <lb></lb>A L, and S F, is the exceſſe of the Rectangle under A L and F B, or A I <lb></lb>and B F, over and above the Triangle A I and B S, or A I B; and the <lb></lb>Rectangle F B and L C is the exceſſe of the Rectangle A C and B F <lb></lb>over and above the Rectangle A L and B F: But the Rectangle A C and <lb></lb>B F is equal to the Rectangle A B I; (for as B A is to A C, ſo is F B to <lb></lb>B I:) The exceſſe, therefore, of the Rectangle A B I above the Rectan<lb></lb>gle A I and B F, or A I and F H, is equal to the exceſſe of the Rectangle <lb></lb>A I and F H above the Rectangle A I B: Therefore twice the Rectan<lb></lb>gle A I and F H is equal to the two Rectangles A B I and A I B; that <lb></lb>is twice A I B with the Square of B I. </s> <s>Let the Square A I be common <lb></lb>to both, and twice the Rectangle A I B with the two Squares A I, and <lb></lb>I B, (that is, the Square A B) ſhall be equal to twice the Rectangle <lb></lb>A I and F H, with the Square A I: Again, taking in commonly the <lb></lb>Square B F; the two Squares A B and B F, that is the ſole Square A F <lb></lb>ſhall be equal to twice the Rectangle A I and F H, with the two Squares <lb></lb>A I and F B, that is A I and F H: But the ſame Square A F is equal <lb></lb>to twice the Rectangle A H F, with the two Squares A H and H F: <lb></lb>Therefore twice the Rectangle A I and F H, with the Squares A I and <lb></lb>F H, are equal to twice the Rectangle A H F, with the Squares A H <lb></lb>and H F: And, the Common Square H F being taken away, twice the <lb></lb>Rectangle A I and F H, with the Square A I, ſhall be equal to twice the <lb></lb>Rectangle A H F, with the Square A H. </s> <s>And becauſe that in all the <lb></lb>Rectangles F H is the Common Side, the Line A H ſhall be equal to A I: <lb></lb>For if it ſhould be greater or leſſer, then the Rectangles F H A and the <lb></lb>Square H A would alſo be greater or leſſer than the Rectangles F H and <lb></lb>I A, and the Square I A: Contrary to what hath been demonſtrated.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Now if we ſuppoſe the Time of the Deſcent along A B to be as A B, <lb></lb>the Time along A C ſhall be as A C, and I B the Mean-proportional be<lb></lb>twixt A C and C E ſhall be the Time along C E, or along X A from <lb></lb>Reſt in X: And becauſe betwixt D A and A B, or R B and B A the<emph.end type="italics"></emph.end><pb xlink:href="069/01/199.jpg" pagenum="196"></pb><emph type="italics"></emph>Mean-proportional is A F, and between A B and B D, that is, R A and <lb></lb>A B the Mean is B F, to which F H is equal; Therefore,<emph.end type="italics"></emph.end> exprædemon<lb></lb>ſtratis, <emph type="italics"></emph>the exceſſe A H ſhall be the Time along A B<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in R, or <lb></lb>after the Fall out of X; ſince the Time along the ſaid A B<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in <lb></lb>A, ſhall be A B. </s> <s>Therefore the Time along X A is I B; and along A B <lb></lb>after R A, or after X A, is A I: Therefore the Time along X A B ſhall <lb></lb>be as A B, namely the ſelf-ſame with the Time along the ſole A B<emph.end type="italics"></emph.end> ex qui<lb></lb>ete <emph type="italics"></emph>in A. </s> <s>Which was the Propoſition.<emph.end type="italics"></emph.end></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"></emph>P<emph.end type="italics"></emph.end>erpendicular be<lb></lb>ing aſſigned, to take part in the Inflected Line, <lb></lb>along which alone <emph type="italics"></emph>ex quiete<emph.end type="italics"></emph.end> a Motion may be <lb></lb>made in the ſame Time, as it would be along <lb></lb>the ſame together with the Perpendicular.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Perpendicular be A B, and a Line inflected to it B C. </s> <s>It is <lb></lb>required in B C to take a part, along which alone out of Reſt a <lb></lb>Motion may be made in the ſame Time as it would along the ſame <lb></lb>together with the Perpendicular A B. </s> <s>Draw the Horizon A D, with <lb></lb>which let the Inclined Line C B prolonged meet in E; and ſuppoſe B F <lb></lb>equal to B A, and on the Center E at the diſtance E F deſcribe the Circle <lb></lb>F I G; and continue out F E unto the Circumference in G; and as G B <lb></lb>is to B F, ſo let B H be to H F; and let H I touch the Circle in I. </s> <s>Then <lb></lb>out of B erect B K<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.199.1.jpg" xlink:href="069/01/199/1.jpg"></figure><lb></lb><emph type="italics"></emph>Perpendicular to <lb></lb>F C, with which <lb></lb>let the Line E I L <lb></lb>meet in L; and laſt <lb></lb>of all let fall L M <lb></lb>Perpendicular to E <lb></lb>L, meeting B C in <lb></lb>M. </s> <s>I ſay, that along <lb></lb>the Line B M from <lb></lb>Rest in B a Motion <lb></lb>may be made in the <lb></lb>ſame Time, as it <lb></lb>would be<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in A along both A B and B M. </s> <s>Let E N be made <lb></lb>equal to E L. </s> <s>And becauſe as G B is to B F, ſo is B H to H F; there<lb></lb>fore, by Permutation as G B is to B H, ſo will B F be to F H; and, by <lb></lb>Diviſion, G H ſhall be to H B, as B H is to H F: Wherefore the Rect<lb></lb>angle G H F ſhall be equal to the Square H B: But the ſaid Rectangle <lb></lb>is alſo equal to the Square H I: Therefore B H is equal to the ſame H I.<emph.end type="italics"></emph.end><pb xlink:href="069/01/200.jpg" pagenum="197"></pb><emph type="italics"></emph>And becauſe in the Quadrilateral Figure I L B H the Sides H B and <lb></lb>H I are equal, and the Angles B and I Right Angles, the Side B L ſhall <lb></lb>likewiſe be equal to the Side L I: But E I is equal to E F: Therefore the <lb></lb>whole Line L E, or N E is equal to the two Lines L B and E F: Let <lb></lb>the Common Line E F be taken away, and the remainder F N ſhall be <lb></lb>equal to L B: And F B was ſuppoſed equal to B A: Therefore L B ſhall <lb></lb>be equal to the two Lines A B and B N. Again, if we ſuppoſe the <lb></lb>Time along A B to be the ſaid A B, the Time along E B ſhall be equal to <lb></lb>E B; and the Time along the whole E M ſhall be E N, namely, the <lb></lb>Mean-proportional betwixt M E and E B: I berefore the Time of the <lb></lb>Deſcent of the remaining part B M after E B, or after A B, ſhall be the <lb></lb>ſaid B N: But it hath been ſuppoſed, that the Time along A B is A B: <lb></lb>Therefore the Time of the Fall along both A B and B M is A B N: <lb></lb>And becauſe the Time along E B<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in E is E B, the Time along <lb></lb>B M<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in B ſhall be the Mean-proportional between B E and <lb></lb>B M; and this is B L: The Time, therefore, along both A B M<emph.end type="italics"></emph.end> ex quiete <lb></lb><emph type="italics"></emph>in A is A B N: And the Time along B M only<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in B is B L: <lb></lb>But it was proved that B L is equal to the two A B and B N: Therefore <lb></lb>the Propoſition is manifeſt.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Otherwiſe with more expedition.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Let B C be the Inclined Plane, and B A the Perpendicular. </s> <s>Continue <lb></lb>out C B to E, and unto E C erect a Perpendicular at B, which being <lb></lb>prolonged ſuppoſe B H equal to the exceſſe of B E above B A; and to the <lb></lb>Angle B H E let the Angle H E L be equal; and let E L continued out <lb></lb>meet with B K in L; and from L erect the Perpendicular L M unto E L <lb></lb>meeting B C in M. </s> <s>I ſay, that<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.200.1.jpg" xlink:href="069/01/200/1.jpg"></figure><lb></lb><emph type="italics"></emph>B M is the Space acquired in <lb></lb>the Plane B C. </s> <s>For becauſe <lb></lb>the Angle M L E is a Right<lb></lb>Angle, therefore B L ſhall be <lb></lb>a Mean-proportional betwixt <lb></lb>M B and B E; and L E a <lb></lb>Mean proportional betwixt M <lb></lb>E and E B; to which E L let <lb></lb>E N be cut equal: And the <lb></lb>three Lines N E, E L, and <lb></lb>L H ſhall be equal; and H B ſhall be the exceſſe of N E above B L: But <lb></lb>the ſaid H B is alſo the exceſſe of N E above N B and B A: Therefore <lb></lb>the two Lines N B and B A are equal to B L. </s> <s>And if we ſuppoſe E B <lb></lb>to be the Time along E B, B L ſhall be the Time along B M<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in <lb></lb>B; and B N ſhall be the Time of the ſame B M after E B or after A B; <lb></lb>and A B ſhall be the Time along A B: Therefore the Times along A B M, <lb></lb>namely, A B N, are equal to the Times along the ſole Line B M<emph.end type="italics"></emph.end> ex quiete <lb></lb><emph type="italics"></emph>in B: Which was intended.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/201.jpg" pagenum="198"></pb><p type="head"> <s>LEMMAI.</s></p><p type="main"> <s><emph type="italics"></emph>Let D C be Perpendicular to the Diameter B A; and from the Term <lb></lb>B continue forth B E D at pleaſure, and draw a Line from F to B. </s> <s>I <lb></lb>ſay, that F B is a Mean-proportional be-<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.201.1.jpg" xlink:href="069/01/201/1.jpg"></figure><lb></lb><emph type="italics"></emph>twixt D B and B E. </s> <s>Draw a Line from E <lb></lb>to F, and by B draw the Tangent B G; <lb></lb>which ſhall be Parallel to the former C D: <lb></lb>Wherefore the Angle D B G ſhall be equal <lb></lb>to the Angle F D B, like as the ſame G B D <lb></lb>is equal alſo to the Angle E F B in the al<lb></lb>tern Portion or Segment: Therefore the <lb></lb>Triangles F B D and F E B are alike: And, <lb></lb>as B D is to B F, ſo is F B to B E.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>LEMMA II.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Line A C be greater than D F; and let A B have greater <lb></lb>proportion to B C, than D E hath to E F. </s> <s>I ſay, that A B is greater <lb></lb>than D E. </s> <s>For becauſe A B hath to B C<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.201.2.jpg" xlink:href="069/01/201/2.jpg"></figure><lb></lb><emph type="italics"></emph>greater proportion than D E hath to D F, <lb></lb>therefore look what proportion A B hath to <lb></lb>B C, the ſame ſhall D E have to a Line leſ<lb></lb>ſer than E F; let it have it to E G: And <lb></lb>becauſe A B to B C, is as D E, to E G, there<lb></lb>fore, by Compoſition, and by converting the Proportion, as C A is to A B, <lb></lb>ſo is G D to D E: But C A is greater than G D: Therefore B A ſhall <lb></lb>be greater than D E.<emph.end type="italics"></emph.end></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"></figure><p type="main"> <s><emph type="italics"></emph>Let A C I B be the Quadrant of a Circle: <lb></lb>and to A C let B E be drawn from B Pa<lb></lb>rallel: And out of any Center taken in the <lb></lb>ſame deſcribe the Circle B O E S, touching <lb></lb>A B in B, and cutting the Circumference of <lb></lb>the Quadrant in I; and draw a Line from <lb></lb>C to B, and another from C to I continued <lb></lb>out to S. </s> <s>I ſay, that the Line C I is alwaies <lb></lb>leſſe than C O. </s> <s>Draw a Line from A to I; <lb></lb>which toucheth the Circle B O E. </s> <s>And if <lb></lb>D I be drawn it ſhall be equal to D B: And <lb></lb>becauſé D B toucheth the Quadrant, the ſaid <lb></lb>D I ſhall likewiſe touch it; and ſhall be Per-<emph.end type="italics"></emph.end><pb xlink:href="069/01/202.jpg" pagenum="199"></pb><emph type="italics"></emph>pendicular to the Diameter A I: Wherefore alſo A I toucheth the Cir<lb></lb>cle B O E in I. And, becauſe the Angle A I C is greater than the An<lb></lb>gle A B C, as inſiſting on a larger Periphery: Therefore the Angle <lb></lb>S I N ſhall be alſo greater than the ſame A B C: Therefore the Portion <lb></lb>I E S is greater than the Portion B O; and the Line C S, nearer to the <lb></lb>Center, greater than C B: Therefore alſo C O is greater than C I; <lb></lb>for that S C is to C B, as O C is to C I.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>And the ſame alſo would happen to be greater, if (as in the other <lb></lb>Figure) the Quadrant B I C were<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.202.1.jpg" xlink:href="069/01/202/1.jpg"></figure><lb></lb><emph type="italics"></emph>leſſer: For the Perpendicular D B <lb></lb>will cut the Circle C I B: Wherefore <lb></lb>D I alſo is equal to the ſaid D B; and <lb></lb>the Angle D I A ſhall be Obtuſe, and <lb></lb>therefore A I N will alſo cut B I N: <lb></lb>And becauſe the Angle A B C is leſſe <lb></lb>than the Angle A I C, which is equal <lb></lb>to S I N; and this now is leſſe than that <lb></lb>which would be made at the Contact in <lb></lb>I by the Line S I: Therefore the Porti<lb></lb>on S E I is much greater than the Por<lb></lb>tion B O: Wherefore,<emph.end type="italics"></emph.end> &c. <emph type="italics"></emph>Which was <lb></lb>to be demonſtrated.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>THEOR. <emph type="italics"></emph>XXII.<emph.end type="italics"></emph.end> PROP. <emph type="italics"></emph>XXXVI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>If from the loweſt point of a Circle erect unto <lb></lb>the Horizon a Plane ſhould be elevated ſub<lb></lb>tending a Circumference not greater than a <lb></lb>Quadrant, from whoſe Terms two other <lb></lb>Planes are Inflected to any point of the Cir<lb></lb>cumference, the Deſcent along both the Infle<lb></lb>cted Planes would be performed in a ſhorter <lb></lb>Time than along the former elevated Plane <lb></lb>alone, or than along but one of the other two, <lb></lb>namely, along the lower.</s></p><p type="main"> <s><emph type="italics"></emph>Let C B D be the Circumference not greater than a Quadrant of a <lb></lb>Circle erect unto the Horizon on the lower point C, in which let <lb></lb>C D be an elevated Plane; and let two Planes be inflected from the <lb></lb>Terms D and C to any point in the Circumference taken at pleaſure, <lb></lb>as B. </s> <s>I ſay, that the Time of the Deſcent along both thoſe Planes D B C <lb></lb>is ſhorter than the Time of the Deſcent along the ſole Plane D C, or <lb></lb>along the other only B C<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in B. </s> <s>Let the Horizontal Line M D A<emph.end type="italics"></emph.end><pb xlink:href="069/01/203.jpg" pagenum="200"></pb><emph type="italics"></emph>be drawn by D, with which let C B prolonged meet in A; and let fall <lb></lb>the Perpendiculars D N and M C to M D, and B N to B D; and about <lb></lb>the Right-angled Triangle D B N deſcribe the Semicircle D F B N, <lb></lb>cutting D C in F; and let D O be a Mean-proportional betwixt C D <lb></lb>and D F; and A V a Mean-proportional betwixt C A and A B: And <lb></lb>let P S be the time in which the whole D C, or B C, ſhall be paſſed; <lb></lb>(for it is manifeſt that they ſhall be both paſt in the ſame Time;) And <lb></lb>look what proportion C D hath to D O, the ſame ſhall the Time S P <lb></lb>have to the Time P R: the Time P R ſhall be that in which a Movea<lb></lb>ble out of D will paſſe D F; and R S that in which it ſhall paſſe the re<lb></lb>mainder F C. </s> <s>And becauſe P S is alſo the Time in which the Movea<lb></lb>ble out of B ſhall paſſe B C; if it be ſuppoſed that as B C is to C D, ſo is <lb></lb>S P to P T, P T ſhall be the Time of the Deſcent out of A to C: by <lb></lb>reaſon D C is a Mean-proportional betwixt A C and C B, by what was <lb></lb>before demonſtrated: Laſt of all, as C A is to A V, ſo let T P be to<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.203.1.jpg" xlink:href="069/01/203/1.jpg"></figure><lb></lb><emph type="italics"></emph>P G: P G ſhall be the Time, <lb></lb>in which thé Moveable out <lb></lb>of A deſcendeth to B. </s> <s>And <lb></lb>becauſe of the Circle D F N <lb></lb>the Diameter erect to the <lb></lb>Horizon is D N, the Lines <lb></lb>D F and D B ſhall be paſ<lb></lb>ſed in equal Times. </s> <s>So that <lb></lb>if it ſhould be demonſtra<lb></lb>ted that the Moveable would <lb></lb>ſooner paſſe B C after the <lb></lb>Deſcent D B, than F C after the Lation D F; we ſhould have our in<lb></lb>tent. </s> <s>But the Moveable will with the ſame Celerity of Time paſſe B C <lb></lb>coming out of D along D B, as if it came out of A along A B: for that <lb></lb>in both the Deſcents D B and A B it acquireth equal Moments of Velo<lb></lb>city: Therefore it ſhall reſt to be demonſtrated that the Time is ſhorter <lb></lb>in which B C is paſſed after A B, than that in which F C is paſt after <lb></lb>D F. </s> <s>But it hath been demonſtrated, that the Time in which B C is <lb></lb>paſſed after A B is G T; and the Time of F C after D F is R S. </s> <s>It is <lb></lb>to be proved therefore, that R S is greater than G T: Which is thus <lb></lb>done. </s> <s>Becauſe as S P is to P R, ſo is C D to D O, therefore, by Conver<lb></lb>ſion of proportion, and by Inverſion, as R S is to S P, ſo is O C to C D: <lb></lb>and as S P is to P T, ſo is D C to C A: And, becauſe as T P is to PG, <lb></lb>ſo is C A to A V: Therefore alſo, by Converſion of the proportion, as <lb></lb>P T is to T G, ſo is A C to C V: therefore, ex equali, as R S is to G T, <lb></lb>ſo is O C to C V. </s> <s>But O C is greater than C V, as ſhall anon be de<lb></lb>monſtrated: Therefore the Time R S is greater than the Time G T: <lb></lb>Which it was required to demonſtrate. </s> <s>And becauſe C F is greater than <lb></lb>C B, and F D leſſe than B A, therefore C D ſhall have greater propor<lb></lb>tion to D F than C A to A B: And as C D is to D F, ſo is the Square<emph.end type="italics"></emph.end><pb xlink:href="069/01/204.jpg" pagenum="201"></pb><emph type="italics"></emph>C O to the Square O F; foraſmuch as C D, D O, and O F are Propor<lb></lb>tionals: And as C A is to A B, ſo is the Square C V to the Square <lb></lb>V B: Therefore C O hath greater proportion to O F, than C V to V B: <lb></lb>Therefore, by the foregoing Lemma, C O is greater than C V. </s> <s>It is <lb></lb>manifeſt moreover, that the Time along D C is to the Time along <lb></lb>D B C, as D O C is to D O together with C V.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>SCHOLIUM.</s></p><p type="main"> <s>From theſe things that have been demonſtrated may evidently <lb></lb>be gathered, that the ſwifteſt of all Motions betwixt Term <lb></lb>and Term is not made along the ſhorteſt Line, that is by the <lb></lb>Right, but along a portion of a Circle.</s></p><p type="main"> <s><emph type="italics"></emph>For in the Quadrat B A E C, whoſe Side B C is erect to the Hori<lb></lb>zon, let the Arch A C be divided into any number of equal parts, <lb></lb>A D, D E, E F, F G, G C; and let Right-lines be drawn from C to <lb></lb>the Points A, D, E, F, G, H; and alſo by Lines joyn A D, D E, E F, <lb></lb>F G. and G C. </s> <s>It is manifest, that the Motion along the two Lines <lb></lb>A D C is ſooner performed than along the<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.204.1.jpg" xlink:href="069/01/204/1.jpg"></figure><lb></lb><emph type="italics"></emph>ſole Line A C, or D C out of Reſt in D: <lb></lb>But out of Reſt in A, D C is ſooner paſt <lb></lb>than the two A D C: But along the two <lb></lb>D E C out of Reſt in A the Deſcent is <lb></lb>likewiſe ſooner made than along the ſole <lb></lb>C D: Therefore the Deſcent along the <lb></lb>three Lines A D E C ſhall be performed <lb></lb>ſooner than along the two A D C. </s> <s>And <lb></lb>in like manner the Deſcent along A D E <lb></lb>preceding, the Motion is more ſpeedily con<lb></lb>ſummated along the two EFC than along the ſole FC: Therfore along the <lb></lb>four A D E F C the Motion is quicklier accompliſhed than along the <lb></lb>three A D E C: And ſo, in the laſt place, along the two F G C after the <lb></lb>precedent Deſcent along A D E F the Motion will be ſooner conſumma<lb></lb>ted than along the ſole F C: Therefore along the five A D E F G C <lb></lb>the Deſcent ſhall be effected in a yet ſhorter Time than along the four <lb></lb>A D E F C: Whereupon the nearer by inſcribed Poligons we approach <lb></lb>the Circumference, the ſooner will the Motion be performed between the <lb></lb>two aſſigned points A C.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>And that which is explained in a Quadrant, holdeth true likewiſe <lb></lb>in a Circumference leſſe than the Quadrant: and the Ratiocination is <lb></lb>the ſame.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/205.jpg" pagenum="202"></pb><p type="head"> <s>PROBL.XV. PROP. XXXVII.</s></p><p type="main"> <s>A Perpendicular and Inclined Plane of the ſame <lb></lb>Elevation being given, to find a part in the In<lb></lb>clined Plane that is equal to the Perpendicu<lb></lb>lar, and paſſed in the ſame Time as the ſaid <lb></lb>Perpendicular.</s></p><p type="main"> <s><emph type="italics"></emph>LET A B be the Perpendicular, and A C the Inclined Plane. </s> <s>It is <lb></lb>required in the Inclined to find a part equal to the Perpendicular <lb></lb>A B, that after Reſt in A may be paſſed in a Time equal to the <lb></lb>Time in which the Perpendicular is paſſed. </s> <s>Let A D be equal to A B, <lb></lb>and cut the Remainder B C in two equal parts in I; and as A C is to<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.205.1.jpg" xlink:href="069/01/205/1.jpg"></figure><lb></lb><emph type="italics"></emph>C I, ſo let C I be to another Line <lb></lb>A E; to which let D G be equal: It <lb></lb>is manifeſt that E G is equal to A D <lb></lb>and to A B. </s> <s>I ſay moreover, that <lb></lb>this ſame E G is the ſame that is <lb></lb>paſſed by the Moveable coming out <lb></lb>of Reſt in A in a Time equal to the <lb></lb>Time in which the Moveable fall eth along A B. </s> <s>For becauſe that as <lb></lb>A C is to C I, ſo is C I to A E, or I D to D G; Therefore by Converſion <lb></lb>of the proportion, as C A is to A I, ſo is D I to I G. </s> <s>And becauſe as the <lb></lb>whole C A is to the whole A I, ſo is the part taken away C I to the part <lb></lb>I G; therefore the Remaining part I A ſhall be to the Remainder A G, <lb></lb>as the whole C A is to the whole A I: Therefore A I is a Mean-propor<lb></lb>tional betwixt C A and A G; and C I a Mean-proportional betwixt <lb></lb>C A and A E: If therefore we ſuppoſe the Time along A B to be as A B; <lb></lb>A C ſhall be the Time along A C, and C I or I D the Time along A E: <lb></lb>And becauſe A I is a Mean-proportional betwixt C A and A G; and <lb></lb>C A is the Time along the whole A C: Therefore A I ſhall be the Time <lb></lb>along. </s> <s>A G; and the Remainder I C that along the Remainder G C: But <lb></lb>D I was the Time along A E: Therefore D I and I C are the Times <lb></lb>along both the Spaces A E and C G: Therefore the Remainder D A ſhall <lb></lb>be the Time along E G, to wit, equal to the Time along A B. </s> <s>Which was <lb></lb>to be done.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>COROLLARIE.</s></p><p type="main"> <s>Hence it is manifeſt, that the Space required is an intermedial be<lb></lb>tween the upper and lower parts that are paſt in equal <lb></lb>Times.</s></p><pb xlink:href="069/01/206.jpg" pagenum="203"></pb><p type="head"> <s><emph type="italics"></emph>P<emph.end type="italics"></emph.end>ROBL. XVI. <emph type="italics"></emph>P<emph.end type="italics"></emph.end>RO<emph type="italics"></emph>P.<emph.end type="italics"></emph.end> XXXVIII.</s></p><p type="main"> <s>Two Horizontal Planes cut by the Perpendicular <lb></lb>being given, to find a ſublime point in the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>er<lb></lb>pendicular, out of which Moveables falling <lb></lb>and being reflected along the Horizontal <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>lanes may in Times equal to the Times of <lb></lb>the Deſcents along the ſaid Horizontal <emph type="italics"></emph>P<emph.end type="italics"></emph.end>lanes, <lb></lb>namely, along the upper and along the lower, <lb></lb>paſſe Spaces that have to each other any given <lb></lb>proportion of the leſſer to the greater.</s></p><p type="main"> <s><emph type="italics"></emph>LET the Planes C D and B E be interſected by the Perpendicular <lb></lb>A C B, and let the given proportion of the leſſe to the greater be <lb></lb>N to F G. </s> <s>It is required in the Perpendicular A B to find a point <lb></lb>on high, out of which a Moveable falling, and reflected along C D may <lb></lb>in a Time equal to the Time of its Fall, paſſe a Space, that ſhall have <lb></lb>unto the Space paſſed by the other Moveable coming out of the ſame ſub<lb></lb>lime point in a Time equal to the Time of its Fall with a Reflex Motion <lb></lb>along the Plane B E the ſame proportion as the given Line N batb to<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.206.1.jpg" xlink:href="069/01/206/1.jpg"></figure><lb></lb><emph type="italics"></emph>F G. </s> <s>Let G H be <lb></lb>made equal to the <lb></lb>ſaid N; and as F H <lb></lb>is to H G, ſo let <lb></lb>B C be to C L. </s> <s>I ſay, <lb></lb>L is the ſublime <lb></lb>point required. </s> <s>For <lb></lb>taking C M double <lb></lb>to C L, draw L M <lb></lb>meeting the Plane <lb></lb>B E in O; B O <lb></lb>ſhall be double to <lb></lb>B L: And becauſe, <lb></lb>as F H is to H G, ſo is B C to C L; therefore, by Compoſition and In<lb></lb>verſion, as H G, that is, N is to G F, ſo is C L to L B, that is, C M to <lb></lb>B O: But becauſe C M is double to L C; let the Space C M be that <lb></lb>which by the Moveable coming from L after the Fall L C is paſſed along <lb></lb>the Plane C D; and by the ſame reaſon B O is that which is paſſed after <lb></lb>the Fall L B in a Time equal to the Time of the Fall along L B; foraſ<lb></lb>much as B O is double to B L: Therefore the Propoſition is manifeſt.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/207.jpg" pagenum="204"></pb><p type="main"> <s>SAGR. </s> <s>Really me thinks that we may juſtly grant our <emph type="italics"></emph>Acade<lb></lb>mian<emph.end type="italics"></emph.end> what he without arrogance aſſumed to himſelf in the begining <lb></lb>of this his Treatiſe of ſhewing us a <emph type="italics"></emph>New Science<emph.end type="italics"></emph.end> about <emph type="italics"></emph>a very old <lb></lb>Subject.<emph.end type="italics"></emph.end> And to ſee with what Facility and Perſpicuity he deduceth <lb></lb>from one ſole Principle the Demonſtrations of ſo many Propoſiti<lb></lb>ons, maketh me not a little to wonder how this buſineſs eſcaped <lb></lb>unhandled by <emph type="italics"></emph>Archimedes, Apollonius, Euclid,<emph.end type="italics"></emph.end> and ſo many other <lb></lb><emph type="italics"></emph>I<emph.end type="italics"></emph.end>lluſtrious Mathematicians and Phyloſophers: eſpecially ſince <lb></lb>there are found many great Volumns of <emph type="italics"></emph>Motion.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>There is extant a ſmall Fragment of <emph type="italics"></emph>Euclid<emph.end type="italics"></emph.end> touching <lb></lb><emph type="italics"></emph>Motion,<emph.end type="italics"></emph.end> but there are no marks to be ſeen therein of any ſteps that he <lb></lb>took towards the diſcovery of the Proportion of <emph type="italics"></emph>Acceleration,<emph.end type="italics"></emph.end> and <lb></lb>of its Varieties along different <emph type="italics"></emph>I<emph.end type="italics"></emph.end>nclinations. </s> <s>So that indeed one <lb></lb>may ſay, that never till now was the door opened to a new Con<lb></lb>templation fraught with infinite and admirable Concluſions, which <lb></lb>in times to come may buſie other Wits.</s></p><p type="main"> <s>SAGR. <emph type="italics"></emph>I<emph.end type="italics"></emph.end> verily believe, that as thoſe few Paſſions (<emph type="italics"></emph>I<emph.end type="italics"></emph.end> will ſay <lb></lb>for example) of the Circle demonſtrated by <emph type="italics"></emph>Euclid<emph.end type="italics"></emph.end> in the third of <lb></lb>his <emph type="italics"></emph>Elements<emph.end type="italics"></emph.end> are an introduction to innumerable others more ab<lb></lb>ſtruce, ſo thoſe produced and demonſtrated in this ſhort Tractate, <lb></lb>when they ſhall come to the hands of other Speculative Wits, ſhall <lb></lb>be a manuduction unto infinite others mote admirable: and it is to <lb></lb>be believed that thus it will happen by reaſon of the Nobility of <lb></lb>the Argument above all others Phyſical.</s></p><p type="main"> <s>This daies Conference hath been very long and laborious; in <lb></lb>which <emph type="italics"></emph>I<emph.end type="italics"></emph.end> have taſted more of the ſimple Propoſitions than of their <lb></lb>Demonſtrations; many of which, <emph type="italics"></emph>I<emph.end type="italics"></emph.end> believe, will coſt me more than <lb></lb>an hour a piece well to comprehend them: a task that <emph type="italics"></emph>I<emph.end type="italics"></emph.end> reſerve to <lb></lb>my ſelf to perform at leaſure, you leaving the Book in my hands ſo <lb></lb>ſoon as we ſhall have heard this part that remains about the Moti<lb></lb>on of Projects: which ſhall, if you ſo pleaſe, be to morrow.</s></p><p type="main"> <s>SALV. <emph type="italics"></emph>I<emph.end type="italics"></emph.end> ſhall not fail to be with you.</s></p><p type="head"> <s><emph type="italics"></emph>The End of the Third Dialogue.<emph.end type="italics"></emph.end></s></p></chap><chap><pb xlink:href="069/01/208.jpg" pagenum="205"></pb><p type="head"> <s>GALILEUS, <lb></lb>HIS <lb></lb>DIALOGUES <lb></lb>OF <lb></lb>MOTION.</s></p><p type="head"> <s>The Fourth Dialogue.</s></p><p type="head"> <s><emph type="italics"></emph>INTERLOCUTORS,<emph.end type="italics"></emph.end></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"></emph>Simplicius<emph.end type="italics"></emph.end> likewiſe cometh in the nick of time, therefore <lb></lb>without interpoſing any <emph type="italics"></emph>Reſt<emph.end type="italics"></emph.end> let us proceed to <emph type="italics"></emph>Motion<emph.end type="italics"></emph.end>; <lb></lb>and ſee here the <emph type="italics"></emph>Text<emph.end type="italics"></emph.end> of our <emph type="italics"></emph>Author.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>OF THE MOTION OF <lb></lb>PROJECTS.</s></p><p type="main"> <s><emph type="italics"></emph>What accidents belong to<emph.end type="italics"></emph.end> Equable Motion, <emph type="italics"></emph>as alſo to the<emph.end type="italics"></emph.end> Na<lb></lb>turally Accelerate <emph type="italics"></emph>along all whatever Inclinations of Planes, <lb></lb>we have conſidered above. </s> <s>In this Contemplation which we are now <lb></lb>entering upon, I will attempt to declare, and with ſolid Demonſtrations<emph.end type="italics"></emph.end><pb xlink:href="069/01/209.jpg" pagenum="206"></pb><emph type="italics"></emph>to eſtabliſh ſome of the principal Symptomes, and thoſe worthy of know<lb></lb>ledge, which befall a Moveable whilſt it is moved with a Motion com<lb></lb>pounded of a twofold Lation, to wit, of the Equable and Naturally<lb></lb>Accelerate: and this is that Motion, which we call the Motion of Pro<lb></lb>jects: whoſe Generation I constitute to be in this manner.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>I fancy in my mind a certain Moveable projected or thrown along <lb></lb>an Horizontal Plane, all impediment ſecluded: Now it is manifeſt by <lb></lb>what we have elſewhere ſpoken at large, that that Motion will be Equa<lb></lb>ble and Perpetual along the ſaid Plane, if the Plane be extended<emph.end type="italics"></emph.end> in in<lb></lb>finitum<emph type="italics"></emph>: but if we ſuppoſe it terminate, and placed on high, the Move<lb></lb>able, which I conceive to be endued with Gravity, being come to the end <lb></lb>of the Plane, proceeding forward, it addeth to the Equable and Indeli<lb></lb>ble firſt Lation that propenſion downwards which it receiveth from its <lb></lb>Gravity, and from thence a certain Motion doth reſult compounded of <lb></lb>the Equable Horizontal, and of the Deſcending naturally. </s> <s>Accellerate <lb></lb>Lations: which I call<emph.end type="italics"></emph.end> Projection. <emph type="italics"></emph>Some of whoſe Accidents we will de<lb></lb>monſtrate; the firſt of which ſhall be this.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>THEOR.I. PROP.I.</s></p><p type="main"> <s><emph type="italics"></emph>A Project, when it is moved with a Motion compounded <lb></lb>of the Horizontal Equable, and of the Naturally<lb></lb>Accelerate downwards, ſhall deſcribe a Semipara<lb></lb>bolical Line in its Lation.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>It is requiſite, <emph type="italics"></emph>Salviatus,<emph.end type="italics"></emph.end> in favour of my ſelf, and, as I <lb></lb>believe, alſo of <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> here to make a pauſe; for I <lb></lb>am not ſo far gone in Geometry as to have ſtudied <emph type="italics"></emph>Apol<lb></lb>lonius,<emph.end type="italics"></emph.end> ſave only ſo far as to know that he treateth of theſe Para<lb></lb>bola's, and of the other Conick Sections, without the knowledge <lb></lb>of which, and of their Paſſions, I do not think that one can under<lb></lb>ſtand the Demonſtrations of other Propoſitions depending on <lb></lb>them. </s> <s>And becauſe already in the very firſt Propoſition it is pro<lb></lb>poſed by the Author to prove the Line deſcribed by the Project to <lb></lb>be Parabolical, I imagine to my ſelf, that being to treat of none <lb></lb>but ſuch Lines, it is abſolutely neceſſary to have a perfect know<lb></lb>ledge, if not of all the Paſſions of thoſe Figures that are demon<lb></lb>ſtrated by <emph type="italics"></emph>Apollonius,<emph.end type="italics"></emph.end> at leaſt of thoſe that are neceſſary for the Sci<lb></lb>ence in hand.</s></p><p type="main"> <s>SALV. </s> <s>You undervalue your ſelf very much, to make ſtrange <lb></lb>of thoſe Notions, which but even now you admitted as very well <lb></lb>underſtood: I told you heretofore, that in the Treatiſe of Reſi<lb></lb>ſtances we had need of the knowledge of certain Propoſitions of <pb xlink:href="069/01/210.jpg" pagenum="207"></pb><emph type="italics"></emph>Apollonius,<emph.end type="italics"></emph.end> at which you made no ſeruple.</s></p><p type="main"> <s>SAGR. </s> <s>It may be either that I knew them by chance, or that I <lb></lb>might for once gueſſe at, and take for granted ſo much as ſerved my <lb></lb>turn in that Tractate: but here where I imagine that we are to <lb></lb>hear all the Demonſtrations that concern thoſe Lines, it is not con<lb></lb>venient, as we ſay, to ſwallow things whole, loſing our time and <lb></lb>pains.</s></p><p type="main"> <s>SIMP. </s> <s>But as to what concerns me, although <emph type="italics"></emph>Sagredus<emph.end type="italics"></emph.end> were, <lb></lb>as I believe he is, well provided for his occaſions, the very firſt <lb></lb>Terms already are new to me: for though our Philoſophers have <lb></lb>handled this Argument of the Motion of Projects, I do not remem<lb></lb>ber that they have confined themſelves to deſine what the Lines <lb></lb>are which they deſcribe, ſave only in general that they are alwaies <lb></lb>Curved Lines, except it be in Projections Perpendicularly upwards. <lb></lb></s> <s>Therefore in caſe that little Geometry that I have learnt from <emph type="italics"></emph>Eu<lb></lb>clid<emph.end type="italics"></emph.end> ſince the Time that we have had other Conferences, be not ſuf<lb></lb>ficient to render me capable of the Notions requiſite for the under<lb></lb>ſtanding of the following Demonſtrations, I muſt content my ſelf <lb></lb>with bare Propoſitions believed, but not underſtood.</s></p><p type="main"> <s>SALV. </s> <s>But I will have you to know them by help of the Au<lb></lb>thor of this Book himſelf, who when he heretofore granted me a <lb></lb>ſight of this his Work, becauſe I alſo at that time was not perfect <lb></lb>in the Books of <emph type="italics"></emph>Apollonius,<emph.end type="italics"></emph.end> took the pains to demonſtrate to me <lb></lb>two moſt principal Paſſions of the Parabola without any other Pre<lb></lb>cognition, of which two, and no more, we ſhall ſtand in need in <lb></lb>the preſent Treatiſe; which are both likewiſe proved by <emph type="italics"></emph>Apollonius,<emph.end type="italics"></emph.end><lb></lb>but after many others, which it would take up a long time to look <lb></lb>over, and I am deſirous that we may much ſhorten the Journey, ta<lb></lb>king the firſt immediately from the pure and ſimple generation of <lb></lb>the ſaid Parabola, and from this alſo immediately ſhall be deduced <lb></lb>the Demonſtration of the ſecond. </s> <s>Coming therefore to the firſt;</s></p><p type="main"> <s>Deſcribe the Right Cone, whoſe Baſe let be the Circle I B K C, <lb></lb>and Vertex the point L, in which, cut by a Plane parallel to the <lb></lb><figure id="id.069.01.210.1.jpg" xlink:href="069/01/210/1.jpg"></figure><lb></lb>Side L K, ariſeth the Section B A C <lb></lb>called a Parabola; and let its Baſe <lb></lb>B C cut the Diameter I K of the <lb></lb>Circle I B K C at Right-Angles; <lb></lb>and let the Axis of the Parabola <lb></lb>A D be Parallel to the ſide L K; <lb></lb>and taking any point F in the Line <lb></lb>B F A, draw the Right-Line F E <lb></lb>parallel to B D. </s> <s>I ſay, that the Square <lb></lb>of B D hath to the Square of F E <lb></lb>the ſame proportion that the Axis <lb></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"></pb>be ſuppoſed to paſſe by the Point E, which ſhall make in the Cone <lb></lb>a Circular Section, whoſe Diameter is G E H. </s> <s>And becauſe upon <lb></lb>the Diameter I K of the Circle I B K, B D is a Perpendicular, the <lb></lb>Square of B D ſhall be equal to the Rectangle made by the parts <lb></lb>I D and D K: And likewiſe in the upper Circle which is underſtood <lb></lb>to paſſe by the points G F H, the Square of the Line F E is equal <lb></lb>to the Rectangle of the parts G E H: Therefore the Square of B D <lb></lb>hath the ſame proportion to the Square of F E, that the Rectangle <lb></lb>I D K hath to the Rectangle G E H. </s> <s>And becauſe the Line E D is <lb></lb>Parallel to H K, E H ſhall be equal to D K, which alſo are Parallels: <lb></lb>And therefore the Rectangle I D K ſhall have the ſame proportion <lb></lb>to the Rectangle G E H, as I D hath to G E; that is, that D A hath <lb></lb>to A E: Therefore the Rectangle I D K to the Rectangle G E H, <lb></lb>that is, the Square B D to the Square F E, hath the ſame proportion <lb></lb>that the Axis D A hath to the part A E: Which was to be de<lb></lb>monſtrated.</s></p><p type="main"> <s>The other Propoſition, likewiſe neceſſary to the preſent Tract, <lb></lb>we will thus make out. </s> <s>Let us deſcribe the Parabola, of which let the <lb></lb>Axis C A be prolonged out unto D; and taking any point B, let the <lb></lb>Line B C be ſuppoſed to be continued out by the ſame Parallel un<lb></lb><figure id="id.069.01.211.1.jpg" xlink:href="069/01/211/1.jpg"></figure><lb></lb>to the Baſe of the ſaid Parabola; <lb></lb>and let D A be ſuppoſed equal <lb></lb>to the part of the Axis C A. </s> <s>I ſay, <lb></lb>that the Right-Line drawn by <lb></lb>the points D and B, falleth not <lb></lb>within the Parabola, but without, <lb></lb>ſo as that it only toucheth the <lb></lb>ſame in the ſaid point B: For, if <lb></lb>it be poſſible for it to fall within, <lb></lb>it cutteth it above, or being pro<lb></lb>longed, it cutteth it below. </s> <s>And <lb></lb>in that Line let any point G be <lb></lb>taken, by which paſſeth the Right <lb></lb>Line F G E. </s> <s>And becauſe the <lb></lb>Square F E is greater than the <lb></lb>Square G E, the ſaid Square F E <lb></lb>ſhall have greater proportion to <lb></lb>the Square B C, than the ſaid Square G E hath to the ſaid B C. </s> <s>And <lb></lb>becauſe, by the precedent, the Square F E is to the Square B C as <lb></lb>E A is to A C; therefore E A hath greater proportion to A C, than <lb></lb>the Square G E hath to the Square B C; that is, than the Square <lb></lb>E D hath to the Square D C: (becauſe in the Triangle D G E as <lb></lb>G E is to the Parallel B C, ſo is E <emph type="italics"></emph>D<emph.end type="italics"></emph.end> to <emph type="italics"></emph>D<emph.end type="italics"></emph.end> C:) But the Line E A to <lb></lb>A C, that is, to A <emph type="italics"></emph>D<emph.end type="italics"></emph.end> hath the ſame proportion that four Rectangles <lb></lb>E A <emph type="italics"></emph>D<emph.end type="italics"></emph.end> hath to four Squares of A <emph type="italics"></emph>D,<emph.end type="italics"></emph.end> that is, to the Square C <emph type="italics"></emph>D,<emph.end type="italics"></emph.end><pb xlink:href="069/01/212.jpg" pagenum="209"></pb>(which is equal to four Squares of A D:) Therefore four Rectan<lb></lb>gles E A D ſhall have greater proportion to the Square C D, than <lb></lb>the Square E D hath to the Square D C: Therefore four Rectan<lb></lb>gles E A D ſhall be greater than the Square E D: which is falſe, <lb></lb>for they are leſſe; becauſe the parts E A and A D of the Line E D <lb></lb>are not equal: Therefore the Line D B toucheth the Parabola in B, <lb></lb>and doth not cut it: Which was to be demonſtrated.</s></p><p type="main"> <s>SIMP. </s> <s>You proceed in your Demonſtrations too ſublimely, <lb></lb>and ſtill, as far as I can perceive, ſuppoſe that the Propoſitions of <lb></lb><emph type="italics"></emph>Euclid<emph.end type="italics"></emph.end> are as familiar and ready with me, as the firſt Axioms them<lb></lb>ſelves, which is not ſo. </s> <s>And the impoſing upon me, juſt now, that <lb></lb>four Rectangles E A <emph type="italics"></emph>D<emph.end type="italics"></emph.end> are leſs than the Square <emph type="italics"></emph>D<emph.end type="italics"></emph.end> E becauſe the <lb></lb>parts E A and A <emph type="italics"></emph>D<emph.end type="italics"></emph.end> of the Line E <emph type="italics"></emph>D<emph.end type="italics"></emph.end> are not equal, doth not ſatisſie <lb></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></lb>ſuppoſe that the Reader hath ready by heart the Elements of <lb></lb><emph type="italics"></emph>Euclid<emph.end type="italics"></emph.end>: And here to ſupply your want, it ſhall ſuſfice to remember <lb></lb>you of a Propoſition in the ſecond Book, in which it is demonſtrated <lb></lb>that when a Line is cut into equal parts, and into unequal, the <lb></lb>Rectangle of the unequal parts is leſs than the Rectangle of the <lb></lb>equal, (that is, than the Square of the half) by ſo much as is the <lb></lb>Square of the Line comprized between the Sections. </s> <s>Whence it is <lb></lb>manifeſt, that the Square of the whole, which continueth four <lb></lb>Squares of the Half, is greater than four Rectangles of the unequal <lb></lb>parts. </s> <s>Now it is neceſſary that we bear in mind theſe two Propoſi<lb></lb>tions which have been demonſtrated, taken from the Conick Ele<lb></lb>ments, for the better underſtanding the things that follow in the <lb></lb>preſent Treatiſe: for of theſe two, and no more, the Author <lb></lb>makes uſe. </s> <s>Now we may reaſſume the Text to ſee in what manner <lb></lb>he doth demonſtrate his firſt Propoſition, in which he intendeth to <lb></lb>prove unto us, That the Line deſcribed by the Grave Moveable, <lb></lb>when it deſcends with a Motion compounded of the Equable <lb></lb>Horizontal, and of the Natural <emph type="italics"></emph>D<emph.end type="italics"></emph.end>eſcending is a Semiparabola.</s></p><p type="main"> <s><emph type="italics"></emph>Suppoſe the Horizontal Line or Plane A B placed on high; upon<emph.end type="italics"></emph.end><lb></lb>[or along] <emph type="italics"></emph>which let the Moveable paſſe with an Equable Motion out <lb></lb>of A unto B: and the ſupport of the Plane failing in B let there be <lb></lb>derived upon the Moveable from its own Gravity a Motion naturally <lb></lb>downwards according to the Perpendicular B N. </s> <s>Let the Line B E be <lb></lb>ſuppoſed applyed unto the Plane A B right out, as if it were the Efflux <lb></lb>or meaſure of the Time, on which at pleaſure note any equal parts of <lb></lb>Time, B C, C D, D E: And out of the points B C D E ſuppoſe Per<lb></lb>pendicular Lines to be let fall equidiſtant or parallel to B N: In the firſt <lb></lb>of which take any part C I, whoſe quadruple take in the following one <lb></lb>D F, nonuple E H, and ſo in the reſt that follow according to the propor-<emph.end type="italics"></emph.end><pb xlink:href="069/01/213.jpg" pagenum="210"></pb><emph type="italics"></emph>tion of the Squares of C B, D B, E B, or, if you will, in the doubled <lb></lb>proportion of the Lines. </s> <s>And if unto the Moveable moved beyond B <lb></lb>towards C with the Equable Lation we ſuppoſe the Perpendicular <lb></lb>Deſcent to be ſuperadded according to the quantity C I, in the Time <lb></lb>B C it ſhall be found conſtituted in the Term I. </s> <s>And proceeding farther,<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.213.1.jpg" xlink:href="069/01/213/1.jpg"></figure><lb></lb><emph type="italics"></emph>in the Time D B, namely, <lb></lb>in the double of B C, the <lb></lb>Space of the Deſcent down<lb></lb>wards ſhall be quadruple to <lb></lb>the firſt Space C I: For <lb></lb>it hath beendemonſtrated in <lb></lb>the firſt Trastate, that the <lb></lb>Spaces paſſed by GraveBo<lb></lb>dies with a Motion Natu<lb></lb>rally Accelerate are in du<lb></lb>plicate proportion of their Times. </s> <s>And it likewiſe followeth, that the <lb></lb>Space E H paſſed in the Time B E, ſhall be as G. </s> <s>So that it is manifeſtly <lb></lb>proved, that the Spaces E H, D F, C I, are to one another as the Squares <lb></lb>of the Lines E B, D B, C B. </s> <s>Now from the points I, F, and H draw <lb></lb>the Right Lines I O, F G, H L, Parallel to the ſaid E B; and each of <lb></lb>the Lines H L, F G, and I O ſhall be equal to each of the other Lines <lb></lb>E B, D B, and C B; as alſo each of thoſe B O, B G, and B L, ſhall be <lb></lb>equal to each of thoſe C I, D F, and E H: And the Square H L ſhall <lb></lb>be to the Square F G, as the Line L B to B G: And the Square F G <lb></lb>ſhall be to the Square I O, as G B to B O: Therefore the Points I, F, <lb></lb>and H are in one and the ſame Parabolical Line. </s> <s>And in like manner <lb></lb>it ſhall be demonſtrated, any equalparticles of Time of whatſoever Mag<lb></lb>nitude being taken, that the place of the Moveable whoſe Motion is <lb></lb>compounded of the like Lations, is in the ſame Times to be found in the <lb></lb>ſame Parabolick Line: Therefore the Propoſition is manifeſt.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>This Concluſion is gathered from the Converſion of the <lb></lb>firſt of thoſe two Propoſitions that went before, for the Parabola <lb></lb>being, for example, deſcribed by the points B H, if either of the <lb></lb>two F or I were not in the deſcribed Parabolick Line, it would be <lb></lb>within, or without; and by conſequence the Line F G would be <lb></lb>either greater or leſſer than that which ſhould determine in the Pa<lb></lb>rabolick Line; Wherefore the Square of HL would have, not to <lb></lb>the Square of F G, but to another greater or leſſer, the ſame pro<lb></lb>portion that the Line L B hath to BG, but it hath the ſame propor<lb></lb>tion to the Square of F G: Therefore the point F is in the Parabo<lb></lb>lick Line: And ſo all the reſt, <emph type="italics"></emph>&c.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>It cannot be denied but that the Diſcourſe is new, in<lb></lb>genious and concludent, arguing <emph type="italics"></emph>ex ſuppoſitione,<emph.end type="italics"></emph.end> that is, ſuppoſing <lb></lb>that the Tranſverſe Motion doth continue alwaies Equable, and <pb xlink:href="069/01/214.jpg" pagenum="211"></pb>that the Natural <emph type="italics"></emph>Dcorſum<emph.end type="italics"></emph.end> do likewiſe keep its tenour of continu<lb></lb>ally Accelerating according to a proportion double to the Times; <lb></lb>and that thoſe Motions and their Velocities in mingling be not al<lb></lb>tered, diſturbed, and impeded, ſo that finally the Line of the Pro<lb></lb>ject do not in the continuation of the Motion degenerate into an<lb></lb>other kind; a thing which ſeemeth to me to be impoſſible. </s> <s>For, in <lb></lb>regard that the Axis of our Parabola, according to which we ſup<lb></lb>poſe the Natural Motion of Graves to be made, being Perpendicu<lb></lb>lar to the Horizon, doth terminate in the Center of the Earth; and <lb></lb>in regard that the Parabolical Line doth ſucceſſively enlarge from <lb></lb>its Axis, no Project would ever come to terminate in the Center, or <lb></lb>if it ſhould come thitherwards, as it ſeemeth neceſſary that it muſt, <lb></lb>the Line of the Project ſhould deſcribe another moſt different from <lb></lb>that of the Parabola.</s></p><p type="main"> <s>SIMP. </s> <s>I add to theſe difficulties ſeveral others; one of which is <lb></lb>that we ſuppoſe, that the Horizontal Plane which hath neither accli<lb></lb>vity or declivity is a Right Line; as if that ſuch a Line were in all <lb></lb>its parts equidiſtant from the Center, which is not true: for depart<lb></lb>ing from its middle it goeth towards the extreams, alwaies more and <lb></lb>more receding from the Center, and therefore alwaies aſcending: <lb></lb>which of conſequence rendereth it Impoſſible that its Motion <lb></lb>ſhould be perpetual, or that it ſhould for any time continue Equa<lb></lb>ble, and neceſſitates it to grow continually more and more weak. <lb></lb></s> <s>Moreover, it is, in my Opinion, impoſſible to avoid the Impedi<lb></lb>ment of the <emph type="italics"></emph>Medium,<emph.end type="italics"></emph.end> but that it will take away the Equability of <lb></lb>the Tranſverſe Motion, and the Rule of the Acceleration in falling <lb></lb>Grave Bodies. </s> <s>By all which difficulties it is rendred very improba<lb></lb>ble that the things demonſtrated with ſuch inconſtant Suppoſi<lb></lb>tions ſhould afterwards hold true in the practical Experiments.</s></p><p type="main"> <s>SALV. </s> <s>All the Objections and Difficulties alledged are ſo <lb></lb>well grounded, that I eſteem it impoſſible to remove them; and <lb></lb>for my own part I admit them all, as alſo I believe the Author <lb></lb>himſelf would do. </s> <s>And I grant that the Concluſions thus demon<lb></lb>ſtrated in Abſtract, do alter and prove falſe, and that ſo egregiouſ<lb></lb>ly, in Concrete, that neither is the Tranſverſe Motion Equable, <lb></lb>nor is the Acceleration of the Natural in the proportion ſuppoſe, <lb></lb>nor is the Line of the Project Parabolical, <emph type="italics"></emph>&c. </s> <s>B<emph.end type="italics"></emph.end>ut yet on the <lb></lb>contrary, I deſire that you would not ſcruple to grant to this our <lb></lb>Author that which other famous Men have ſuppoſed, although <lb></lb>falſe. </s> <s>And the ſingle Authority of <emph type="italics"></emph>Archimedes<emph.end type="italics"></emph.end> may ſatisfie every <lb></lb>one: who in his Mechanicks, and in the firſt Quadrature of the <lb></lb>Parabola, taketh it as a true Principle, that the <emph type="italics"></emph>B<emph.end type="italics"></emph.end>eam of the <emph type="italics"></emph>B<emph.end type="italics"></emph.end>allance <lb></lb>or Stilliard is a Right Line in all its points equidiſtant from the <lb></lb>Common Center of Grave <emph type="italics"></emph>B<emph.end type="italics"></emph.end>odies, and that the Scale-ropes, to <lb></lb>which the Weights are hanged, are parallel to one another. </s> <s>Which <pb xlink:href="069/01/215.jpg" pagenum="212"></pb>Liberty of his hath been excuſed by ſome, for that in our practices <lb></lb>the Inſtruments we uſe, and the Diſtances which we take are ſo <lb></lb>ſmall in compariſon of our great remoteneſs from the Center of <lb></lb>the Terreſtrial Globe, that we may very well take a Minute of a <lb></lb>degree of the great Circle as if it were a Right Line, and two Per<lb></lb>pendiculars that ſhould hang at its extreams as if they were Paral<lb></lb>lels. </s> <s>For if we were in practical Operations to keep account of <lb></lb>ſuch like Minutes, we ſhould begin to reprove the Architects, who <lb></lb>with the Plumb Line ſuppoſe that they raiſe very high Towers <lb></lb>between Lines equidiſtant. </s> <s>And I here add, that we may ſay that <lb></lb><emph type="italics"></emph>Archimedes,<emph.end type="italics"></emph.end> and others ſuppoſe in their Contemplations that they <lb></lb>were conſtituted remote at an infinite diſtance from the Center; <lb></lb>in which caſe their Aſſumptions were not falſe: And that therefore <lb></lb>they did conclude by Abſolute Demonſtration. </s> <s>Again, if we will <lb></lb>practice the demonſtrated Concluſions in terminate Diſtances, by <lb></lb>ſuppoſing an immenſe Diſtance, we ought to defalk from the <lb></lb>truth demonſtrated that which our Diſtance from the Center doth <lb></lb>import, not being really infinite, but yet ſuch as that it may be <lb></lb>termed Immenſe in compariſon of the Artifices that we make uſe <lb></lb>of, the greateſt of which will be the Ranges of Projects, and amongſt <lb></lb>theſe that only of Canon ſhot; which though it be great, yet ſhall <lb></lb>it not exceed four of thoſe Miles of which we are remote from the <lb></lb>Center well-nigh ſo many thouſands: and theſe coming to deter<lb></lb>mine in the Surface of the Terreſtrial Globe may very well only in<lb></lb>ſenſibly alter that Parabolick Figure, which we grant would be <lb></lb>extreamly transformed in going to determine in the Center. </s> <s>In <lb></lb>the next place as to the perturbation proceeding from the Impedi<lb></lb>ment of the <emph type="italics"></emph>Medium,<emph.end type="italics"></emph.end> this is more conſiderable, and, by reaſon of <lb></lb>its ſo great multiplicity of Varieties, incapable of being brought <lb></lb>under any certain Rules, and reduced to a Science: for if we <lb></lb>ſhould propoſe to conſideration no more but the Impediment which <lb></lb>the Air procureth to the Motions conſidered by us, this alone ſhall <lb></lb>be found to diſturb all, and that infinite waies, according as we <lb></lb>infinite waies vary the Figures, Gravities, and Velocities of the <lb></lb>Moveables. </s> <s>For as to the Velocity, according as this ſhall be grea<lb></lb>ter, the greater ſhall the oppoſition be that the Air makes againſt <lb></lb>them, which ſhall yet more impede the ſaid Moveable according as <lb></lb>they are leſs Grave: ſo that although the deſcending Grave Body <lb></lb>ought to go Accelerating in a duplicate proportion to the Duration <lb></lb>of its Motion, yet nevertheleſs, albeit the Moveable were very <lb></lb>Grave, in coming from very great heights, the Impediment of the <lb></lb>Air ſhall be ſo great, as that it will take from it all power of far<lb></lb>ther encreaſing its Velocity, and will reduce it to an Uniform and <lb></lb>Equable Motion: And this Adequation ſhall be ſo much the ſooner <lb></lb>obtained, and in ſo much leſſer heights, by how much the Moveable <pb xlink:href="069/01/216.jpg" pagenum="213"></pb>ſhall be leſs Grave. </s> <s>That Motion alſo which along the Horizontal <lb></lb>Plane, all other Obſtacles being removed, ought to be Equable <lb></lb>and perpetual, ſhall come to be altered, and in the end arreſted by <lb></lb>the Impediment of the Air: and here likewiſe ſo much the ſooner, <lb></lb>by how much the Moveable ſhall be Lighter. </s> <s>Of which Accidents <lb></lb>of Gravity, of Velocity, and alſo of Figure, as being varied ſeve<lb></lb>ral waies, there can no fixed Science be given. </s> <s>And therefore that <lb></lb>we may be able Scientifically to treat of this Matter it is requiſite <lb></lb>that we abſtract from them; and, having found and demonſtrated <lb></lb>the Concluſions abſtracted from the Impediments, that we make <lb></lb>uſe of them in practice with thoſe Limitations that Experience ſhall <lb></lb>from time to time ſhew us. </s> <s>And yet nevertheleſs the benefit ſhall <lb></lb>not be ſmall, becauſe ſuch Matters, and their Figures ſhall be made <lb></lb>choice of as are leſs ſubject to the Impediments of the <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end>; <lb></lb>ſuch are the very Grave, the Rotund: and the Spaces, and the <lb></lb>Velocities for the moſt part will not be ſo great, but that their ex<lb></lb>orbitances may with eaſie ^{*} Allowance be reduced to a certainty. <lb></lb><arrow.to.target n="marg1095"></arrow.to.target><lb></lb>Yea more, in Projects practicable by us, that are of Grave Matters, <lb></lb>and of Round Figure, and alſo that are of Matters leſſe Grave, <lb></lb>and of Cylindrical Figure, as Arrows, ſhot from Slings or Bows, <lb></lb>the variation of their Motion from the exact Parabolical Figure <lb></lb>ſhall be altogether inſenſible. </s> <s>Nay, (and I will aſſume to my ſelf <lb></lb>a little more freedom) that in ^{*} Inſtruments that are practicable by <lb></lb><arrow.to.target n="marg1096"></arrow.to.target><lb></lb>us, their ſmalneſs rendreth the extern and accidental Impediments, <lb></lb>of which that of the <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> is moſt conſiderable, to be but of <lb></lb>very ſmall note, I am able by two experiments to make manifeſt. <lb></lb></s> <s>I will conſider the Motions made thorow the Air, for ſuch are thoſe <lb></lb>chiefly of which we ſpeak: againſt which the ſaid Air in two man<lb></lb>ners exerciſeth its power. </s> <s>The one is by more impeding the Movea<lb></lb>bles leſs Grave, than thoſe very Grave. </s> <s>The other is in more oppo<lb></lb>ſing the greater than the leſs Velocity of the ſame Moveable. </s> <s>As <lb></lb>to the firſt; Experience ſhewing us that two Balls of equal <lb></lb>bigneſs, but in weight one ten or twelve times more Grave than the <lb></lb>other, as, for example, one of Lead and another of Oak would <lb></lb>be, deſcending from an height of 150, or 200 Yards, arrive to the <lb></lb>Earth with Velocity very little different, it aſſureth us that the Im<lb></lb>pediment or Retardment of the Air in both is very ſmall: for if <lb></lb>the <emph type="italics"></emph>B<emph.end type="italics"></emph.end>all of Lead departing from on high in the ſame Moment with <lb></lb>that of Wood, were but little retarded, and this much, the Lead at <lb></lb>its coming to the ground ſhould leave the Wood a very conſidera<lb></lb>ble Space behind, ſince it is ten times more Grave; which never<lb></lb>theleſs doth not happen: nay, its Anticipation ſhall not be ſo <lb></lb>much as the hundredth part of the whole height. </s> <s>And between a <lb></lb><emph type="italics"></emph>B<emph.end type="italics"></emph.end>all of Lead, and another of Stone which weighs a third part, or <lb></lb>half ſo much as it, the difference of the Times of their coming to <pb xlink:href="069/01/217.jpg" pagenum="214"></pb>the ground would be hardly obſervable. </s> <s>Now becauſe the <emph type="italics"></emph>Impe<lb></lb>tus<emph.end type="italics"></emph.end> that a <emph type="italics"></emph>B<emph.end type="italics"></emph.end>all of Lead acquireth in falling from an height of 200 <lb></lb>Yards (which is ſo much that continuing it in an Equable Moti<lb></lb>on it would in a like Time run 400 Yards) is very conſiderable in <lb></lb>compariſon of the Velocity that we confer with <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ows or other Ma<lb></lb>chines, upon our Projects (excepting the <emph type="italics"></emph>Impetus's<emph.end type="italics"></emph.end> that depend <lb></lb>on the Fire) we may without any notable Errour conclude and <lb></lb>account the Propoſitions to be abſolutely true that are demonſtra<lb></lb>ted without any regard had to the alteration of the <emph type="italics"></emph>Medium.<emph.end type="italics"></emph.end> In <lb></lb>the next place as touching the other part, that is to ſhew, that the <lb></lb>Impediment that the ſaid Moveable receiveth from the Air whilſt <lb></lb>it moveth with great Velocity is not much greater than that which <lb></lb>oppoſeth it in moving ſlowly, the enſuing Experiment giveth us <lb></lb>full aſſurance of it. </s> <s>Suſpend by two threads both of the ſame <lb></lb>length, <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> four or five Yards, two equal <emph type="italics"></emph>B<emph.end type="italics"></emph.end>alls of Lead: and <lb></lb>having faſtned the ſaid threads on high, let both the <emph type="italics"></emph>B<emph.end type="italics"></emph.end>alls be re<lb></lb>moved from the ſtate of Perpendicularity; but let the one be re<lb></lb>moved 80. or more degrees, and the other not above 4 or 5: ſo <lb></lb>that one of them being left at liberty deſcendeth, and paſſing be<lb></lb>yond the Perpendicular, deſcribeth very great Arches of 160, 150, <lb></lb>140, <emph type="italics"></emph>&c.<emph.end type="italics"></emph.end> degrees, diminiſhing them by little and little: but the <lb></lb>other ſwinging freely paſſeth little Arches of 10, 8, 6, <emph type="italics"></emph>&c.<emph.end type="italics"></emph.end> this <lb></lb>alſo diminiſhing them in like manner by little and little. </s> <s>Here I <lb></lb>ſay, in the firſt place, that the firſt <emph type="italics"></emph>B<emph.end type="italics"></emph.end>all ſhall paſs its 180, 160, <emph type="italics"></emph>&c.<emph.end type="italics"></emph.end><lb></lb>degrees in as much Time as the other doth its 10, 8, <emph type="italics"></emph>&c.<emph.end type="italics"></emph.end> From <lb></lb>whence it is manifeſt, that the Velocity of the firſt <emph type="italics"></emph>B<emph.end type="italics"></emph.end>all ſhall be 16 <lb></lb>and 18 times greater than the Velocity of the ſecond: ſo that in <lb></lb>caſe the greater Velocity were to be more impeded by the Air than <lb></lb><arrow.to.target n="marg1097"></arrow.to.target><lb></lb>the leſſer, the Vibrations ſhould be more ^{*} rare in the greateſt <lb></lb>Arches of 180, or 160 degrees, <emph type="italics"></emph>&c.<emph.end type="italics"></emph.end> than in the leaſt of 10, 8, 4, <lb></lb>and alſo of 2, and of 1; but this is contradicted by Experience: <lb></lb>for if two Aſſiſtants ſhall ſet themſelves to count the Vibrations, <lb></lb>one the greateſt, the other the leaſt, they will find that they ſhall <lb></lb>number not only tens, but hundreds alſo, without diſagreeing one <lb></lb>ſingle Vibration, yea, or one ſole point. </s> <s>And this obſervati<lb></lb>on joyntly aſſureth us of the two Propoſitions, namely, that the <lb></lb>greateſt and leaſt Vibrations are all made one after another under <lb></lb>equal Times, and that the Impediment and Retardment of the Air <lb></lb>operates no more in the ſwifteſt Motion, than in the ſloweſt: <lb></lb>contrary to that which before it ſeemed that we our ſelves alſo <lb></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 ſewer.</s></p><p type="main"> <s>SAGR. Rather, becauſe it cannot be denied but that the Air <lb></lb>impedeth both thoſe and theſe, ſince they both continually grow <lb></lb>more languid, and at laſt ceaſe, it is requiſite to ſay that thoſe Re<lb></lb>tardations are made with the ſame proportion in the one and in the <pb xlink:href="069/01/218.jpg" pagenum="215"></pb>other Operation. </s> <s>And then, the being to make greater Reſiſtance <lb></lb>at one time than at another, from what other doth it proceed, but <lb></lb>only from its being aſſailed at one time with a greater <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> and <lb></lb>Velocity, and at another time with leſſer? </s> <s>And if this be ſo then the <lb></lb>ſame quantity of the Velocity of the Moveable is at once the Cauſe <lb></lb>and the Mealure of the quantity of the Reſiſtance. </s> <s>Therefore all <lb></lb>Motions, whether they be ſlow or ſwift, are retarded and impe<lb></lb>ded in the ſame proportion: a Notion in my judgment not con<lb></lb>temptible.</s></p><p type="main"> <s>SALV. </s> <s>We may alſo in this ſecond caſe conclude, That the <lb></lb>Fallacies in the Concluſions, which are demonſtrated, abſtracting <lb></lb>from the extern Accidents, are in our Inſtruments of very ſmall <lb></lb>conſideration, in reſpect of the Motions of great Velocities of <lb></lb>which for the moſt part we ſpeak, and of the Diſtances which are <lb></lb>but very ſmall in relation to the Semidiameter and great Circles of <lb></lb>the Terreſtrial Globe.</s></p><p type="main"> <s>SIMP. </s> <s>I would gladly hear the reaſon why you ſequeſtrate <lb></lb>the Projects from the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of the Fire, that is, as I conceive from <lb></lb>the force of the Powder, from the other Projects made by Slings, <lb></lb>Bows, or Croſs-bows, touching their not being in the ſame manner <lb></lb>ſubject to the Acceleration and Impediment of the Air.</s></p><p type="main"> <s>SALV. </s> <s>I am induced thereto by the exceſſive, and, as I may ſay, <lb></lb>Supernatural Fury or Impetuouſneſs with which thoſe Projects are <lb></lb>driven out: For indeed I think that the Velocity with which a <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ul<lb></lb>let is ſhot out of a Musket or Piece of Ordinance may without any <lb></lb>Hyperbole be called Supernatural. </s> <s>For one of thoſe <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ullets de<lb></lb>ſcending naturally thorow the Air from ſome immenſe height, its <lb></lb>Velocity, by reaſon of the Reſiſtance of the Air will not go in<lb></lb>creaſing perpetually: but that which in Cadent <emph type="italics"></emph>B<emph.end type="italics"></emph.end>odies of ſmall <lb></lb>Gravity is ſeen to happen in no very great ^{*} Space, I mean their <lb></lb><arrow.to.target n="marg1098"></arrow.to.target><lb></lb>being reduced in the end to an Equable Motion, ſhall alſo happen <lb></lb>after a Deſcent of thouſands of yards, in a <emph type="italics"></emph>B<emph.end type="italics"></emph.end>all of Iron or Lead: <lb></lb>and this determinate and ultimate Velocity may be ſaid to be the <lb></lb>greateſt that ſuch a <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ody can obtain or acquire thorow the Air: <lb></lb>which Velocity I account to be much leſſer than that which cometh <lb></lb>to be impreſſed on the ſame <emph type="italics"></emph>B<emph.end type="italics"></emph.end>all by the fired Powder. </s> <s>And of this <lb></lb>a very appoſite Experiment may advertiſe us. </s> <s>At an height of an <lb></lb>hundred or more yards let off a Musket charged with a Leaden <lb></lb><emph type="italics"></emph>B<emph.end type="italics"></emph.end>ullet perpendicularly downwards upon a Pavement of Stone; and <lb></lb>with the ſame Musket ſhoot againſt ſuch another Stone at the Di<lb></lb>ſtance of a yard or two, and then ſee which of the two <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ullets is <lb></lb>more flatted: for if that coming from on high be leſs ^{*} dented than <lb></lb><arrow.to.target n="marg1099"></arrow.to.target><lb></lb>the other, it ſhall be a ſign that the Air hath impeded it, and dimi<lb></lb>niſhed the Velocity conferred upon it by the Fire in the beginning <lb></lb>of the Motion: and that, conſequently, ſo great a Velocity the Air <pb xlink:href="069/01/219.jpg" pagenum="216"></pb>would not ſuffer it to gain coming from never ſo great an height: <lb></lb>for in caſe the Velocity impreſſed upon it by the Fire ſhould not <lb></lb>exceed that which it might acquire of its ſelf deſcending naturally, <lb></lb>the battery downwards ought rather to be more valid than leſs. <lb></lb></s> <s>I have not made ſuch an Experiment, but incline to think that a <lb></lb>Musket or Cannon Bullet falling from never ſo great an height, <lb></lb>will not make that percuſſion which it maketh in a Wall at a Di<lb></lb>ſtance of a few yards, that is of ſo few that the ſhort perforation, <lb></lb>or, if you will, Sciſſure to be made in the Air ſufficeth not to ob<lb></lb>viate the exceſs of the ſupernatural impetuoſity impreſſed on it by <lb></lb>the Fire. </s> <s>This exceſſive <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of ſuch like forced ſhots may <lb></lb>cauſe ſome deformity in the Line of the Projection; making <lb></lb>the beginning of the Parabola leſs inclined or curved than the end. <lb></lb><emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut this can be but of little or no prejudice to our Author in <lb></lb>practical Operations: amongſt the which the principal is the com<lb></lb>poſition of a Table for the Ranges, or Flights, which containeth <lb></lb>the diſtances of the Falls of <emph type="italics"></emph>B<emph.end type="italics"></emph.end>alls ſhot according to all Elevations. <lb></lb></s> <s>And becauſe theſe kinds of Projections are made with Mortar<lb></lb>Pieces, and with no great charge; in theſe the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> not being <lb></lb>ſupernatural, the Ranges deſcribe 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"></emph>B<emph.end type="italics"></emph.end>ut for the preſent let us proceed forwards in the Treatiſe, <lb></lb>where the Author deſireth to lead us to the Contemplation and <lb></lb>Inveſtigation of the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of the Moveable whilſt it moveth <lb></lb>with a Motion compounded of two. </s> <s>And firſt of that compoun<lb></lb>ded of two Equable Motions; the one Horizontal, and the other <lb></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></lb>ble Motion, that is, Horizontal and Perpen<lb></lb>dicular, the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> or Moment of the Lation <lb></lb>compounded of both the Motions ſhall be <emph type="italics"></emph>po<lb></lb>tentia<emph.end type="italics"></emph.end> equal to both the Moments of the firſt <lb></lb>Motions.</s></p><p type="main"> <s><emph type="italics"></emph>For let any Moveable be moved Equably with a double Lation, <lb></lb>and let the Mutations of the Perpendicular anſwer to the Space <lb></lb>A B, and let B C anſwer to the Horizontal Lation paſſed in <lb></lb>the ſame Time. </s> <s>Foraſmuch therefore as the Spa-<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.219.1.jpg" xlink:href="069/01/219/1.jpg"></figure><lb></lb><emph type="italics"></emph>ces A B, and B C are paſſed by the Equable Mo<lb></lb>tion in the ſame Time, their Moments ſhall be to <lb></lb>cach other as the ſaid A B and B C. </s> <s>But the <lb></lb>Moveable which is moved according to theſe two Mutations ſhall de-<emph.end type="italics"></emph.end><pb xlink:href="069/01/220.jpg" pagenum="217"></pb><emph type="italics"></emph>ſcribe the Diagonal A C, and its Moment ſhall be as A C. </s> <s>But A C is<emph.end type="italics"></emph.end><lb></lb>potentia <emph type="italics"></emph>equal to the ſaid A B and B C: therefore the Moment com<lb></lb>pounded of both the Moments A B and B C, is<emph.end type="italics"></emph.end> potentia <emph type="italics"></emph>equal to them <lb></lb>both taken together: Which was to be demonſtrated.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SIMP. </s> <s>It is neceſſary that you eaſe me of one Scruple that <lb></lb>cometh into my mind, it ſeemeth to me that this which is now con<lb></lb>cluded oppugneth another Propoſition of the former Tractate: in <lb></lb>which it is affirmed, That the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of the Moveable coming <lb></lb>from A into B is equal to that coming from A into C; and now it is <lb></lb>concluded, that the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> in C is greater than that in B.</s></p><p type="main"> <s>SALV. </s> <s>The Propoſitions, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> are both true, but very <lb></lb>different from one another. </s> <s>Here the Author ſpeaks of one ſole <lb></lb>Moveable moved with one ſole Motion, but compounded of two, <lb></lb>both Equable; and there he ſpeaks of two Moveables moved <lb></lb>with Motions Naturally Accelerated, one along the Perpendicular <lb></lb>A B, and the other along the Inclined Plane A C: and moreover, <lb></lb>the Times there are not ſuppoſed equal, but the Time along <lb></lb>the Inclined Plane A C is greater than the Time along the Perpen<lb></lb>dicular A B: but in the Motion ſpoken of at preſent, the Motions <lb></lb>along A B, B C and A C are underſtood to be Equable, and made <lb></lb>in the ſame Time.</s></p><p type="main"> <s>SIMP. </s> <s>Excuſe me, and go on, for I am ſatisfied.</s></p><p type="main"> <s>SALV. </s> <s>The Author proceeds to ſhew us that which hapneth <lb></lb>concerning the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of a Moveable moved in like manner with <lb></lb>one Motion compounded of two, that is to ſay, the one Horizon<lb></lb>tal and Equable, and the other Perpendicular but Naturally-Acce<lb></lb>lerate, of which in fine the Motion of the Project is compounded, <lb></lb>and by which the Parabolick Line is deſcribed; in each point of <lb></lb>which the Author endeavours to determine what the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of the <lb></lb>Project is; for underſtanding of which he ſheweth us the manner, <lb></lb>or, if you will, Method of regulating and meaſuring that ſame <emph type="italics"></emph>Im<lb></lb>petus<emph.end type="italics"></emph.end> upon the ſaid Line, along which the Motion of the Grave <lb></lb>Moveable deſcending with a Natural-Accelerate Motion departing <lb></lb>from Reſt is made, ſaying:</s></p><p type="head"> <s>THEOR. III. PROP. III.</s></p><p type="main"> <s><emph type="italics"></emph>Let a Motion be made along the Line A B out of Reſt in A, and <lb></lb>take in ſome point C; and ſuppoſe the ſaid A C to be the Time or <lb></lb>Meaſure of the Time of the ſaid Fall along the Space A C, as alſo <lb></lb>the Meaſure of the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>or Moment in the Point C acquired by <lb></lb>the Deſcent along A C. </s> <s>Now let there be taken in the ſaid Line <lb></lb>A B any other Point, as ſuppoſe B, in which we are to determine of the<emph.end type="italics"></emph.end><lb></lb>Impetus <emph type="italics"></emph>acquired by the Moveable along the Fall A B, in proportion to<emph.end type="italics"></emph.end><pb xlink:href="069/01/221.jpg" pagenum="218"></pb><emph type="italics"></emph>the<emph.end type="italics"></emph.end> Impetus, <emph type="italics"></emph>which it obtaineth in C, whoſe Meaſure is ſuppoſed to be <lb></lb>A C, Let A S be a Mean-proportional betwixt B A and A C. </s> <s>We will <lb></lb>demonſtrate that the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>in B is to the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>in C, as S A is to <lb></lb>A C. </s> <s>Let the Horizontal Line C D be double to the ſaid A C; and B E <lb></lb>double to B A. </s> <s>It appeareth by what hath been demonſtrated, That the <lb></lb>Cadent along A C being turned along the Horizon C D, and according <lb></lb>to the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>acquired in C, with an Equable Motion, ſhall paſs the <lb></lb>Space C D in a Time equal to that <lb></lb>in which the ſaid A C is paſſed<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.221.1.jpg" xlink:href="069/01/221/1.jpg"></figure><lb></lb><emph type="italics"></emph>with an Accelerate Motion; and <lb></lb>likewiſe that B E is paſſed in the <lb></lb>ſame time as A B: But the Time of <lb></lb>the Deſcent along A B is A S: There<lb></lb>fore the Horizontal Line B E is <lb></lb>paſſed in A S. </s> <s>As the Time S A is <lb></lb>to the Time A C, ſo let E B be to <lb></lb>B L. </s> <s>And becauſe the Motion by <lb></lb>B E is Equable, the Space B L ſhall be paſſed in the Time A C ac<lb></lb>cording to the Moment of Celerity in B: But in the ſame Time A C <lb></lb>the Space C D is paſſed, according to the Moment of Velocity in C: <lb></lb>the Moments of Velocity therefore are to one another as the Spaces <lb></lb>which according to the ſame Moments are paſſed in the ſame Time: <lb></lb>Therefore the Moment of Velocity in C is to the Moment of Celerity in <lb></lb>B, as D C is to B L. </s> <s>And becauſe as D C is to B E, ſo are their halfs, <lb></lb>to wit, C A to A B: but as E B is to B L, ſo is B A to A S: Therefore,<emph.end type="italics"></emph.end><lb></lb>exæquali, <emph type="italics"></emph>as D C is to B L, ſo is C A to A S: that is, as the Moment <lb></lb>of Velocity in C is to the Moment of Velocity in B, ſo is C A to A S; that <lb></lb>is, the Time along C A to the Time along A B. </s> <s>I he manner of Meaſu<lb></lb>ring the<emph.end type="italics"></emph.end> Impetus, <emph type="italics"></emph>or the Moment of Velocity upon a Line along which it <lb></lb>makes a Motion of Deſcent is therefore manifeſt; which<emph.end type="italics"></emph.end> Impetus <lb></lb><emph type="italics"></emph>is indeed ſuppoſed to encreaſe according to the Proportion of the <lb></lb>Time.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>But this, before we proceed any farther, is to be premoniſhed, that in <lb></lb>regard we are to ſpeak for the future of the Motion compounded of the <lb></lb>Equable Horizontal, and of the Naturally Accelerate downwards, (for <lb></lb>from this Mixtion reſults, and by it is deſigned the Line of the Project, <lb></lb>that is a Parabola;) it is neceſſary that we define ſome common meaſure <lb></lb>according to which we may meaſure the Velocity,<emph.end type="italics"></emph.end> Impetus, <emph type="italics"></emph>or Moment <lb></lb>of both the Motions. </s> <s>And ſeeing that of the Equable Motion the de<lb></lb>grees of Velocity are innumerable, of which you may not take any <lb></lb>promiſcuouſly, but one certain one which may be be compared and con<lb></lb>joyned with the Degree of Velocity naturally Accelerate. </s> <s>I can think of <lb></lb>no more eaſie way for the electing and determining of that, than by aſ<lb></lb>ſuming another of the ſame kind. </s> <s>And that I may the better expreſs <lb></lb>my meaning; Let A C be Perpendicular to the Horizon C B; and A C<emph.end type="italics"></emph.end><pb xlink:href="069/01/222.jpg" pagenum="219"></pb><emph type="italics"></emph>to be the Altitude, and C B the Amplitude of the Semiparabola A B; <lb></lb>which is deſcribed by the Compoſition of two Lations; of which one is <lb></lb>that of the Moveable deſcending along A C with a Motion Naturally <lb></lb>Acceler ate<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in A; the other is the Equable Tranſverſal Moti<lb></lb>on according to the Horizontal Line A D. The<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>acquired in C <lb></lb>along the Deſcent A C is determined by the quantity of the ſaid height <lb></lb>A C; for the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>of a Moveable<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.222.1.jpg" xlink:href="069/01/222/1.jpg"></figure><lb></lb><emph type="italics"></emph>falling from the ſame height is alwaies <lb></lb>one and the ſame: but in the Horizontal <lb></lb>Line one may aſſign not one, but innume<lb></lb>rable Degrees of Velocities of Equable <lb></lb>Motions: out of which multitude that I <lb></lb>may ſingle out, and as it were point with <lb></lb>the finger to that which I make choice of, <lb></lb>I extend or prolong the Altitude C A<emph.end type="italics"></emph.end> in <lb></lb>ſublimi, <emph type="italics"></emph>in which, as was done before, I <lb></lb>will pitch upon A E; from which if I <lb></lb>conceive in my mind a Moveable to fall<emph.end type="italics"></emph.end><lb></lb>ex quiete <emph type="italics"></emph>in E, it appeareth that its<emph.end type="italics"></emph.end> Im<lb></lb>petus <emph type="italics"></emph>acquired in the Time A, is one with which I conceive the ſame <lb></lb>Moveable being turned along A D to be moved; and its degree of <lb></lb>Vclocity to be that, which in the Time of the Deſcent along E A paſſeth <lb></lb>a Space in the Horizon double to the ſaid E A. </s> <s>This Præmonition I <lb></lb>judged neceſſary.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>It is moreover to be advertized that the Amplitude of the Semi<lb></lb>parabola A B ſhall be called by me the Horizontal Line<emph.end type="italics"></emph.end> [or Plane] <lb></lb><emph type="italics"></emph>C B.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>The Altitude, to with A C, the Axis of the ſaid Parabola.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>And the Line E A, by whoſe Deſcent the Horizontal<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>is de<lb></lb>termined, I call the Sublimity, or height.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Theſe things being declared and defined, I proceed to Demonſtra<lb></lb>tion.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. Stay, I pray you, for here me thinks it is convenient to <lb></lb>adorn this Opinion of our Author with the conformity of it to <lb></lb>the Conceit of <emph type="italics"></emph>Plato<emph.end type="italics"></emph.end> about the determining the different Veloci<lb></lb>ties of the Equable Motions of the Revolutions of the Cœleſtial <lb></lb>Bodies; who, having perhaps had a conjecture that no Moveable <lb></lb>could paſſe from Reſt into any determinate degree of Velocity in <lb></lb>which it ought afterwards to be perpetuated, unleſs by paſſing <lb></lb>thorow all the other leſſer degrees of Velocity, or, if you will, <lb></lb>greater degrees of Tardity, which interpoſe between the aſſigned <lb></lb>degree, and the higheſt degree of Tardity, that is of Reſt, ſaid that <lb></lb>God after he had created the Moveable Cœleſtial <emph type="italics"></emph>B<emph.end type="italics"></emph.end>odies that he <lb></lb>might aſſign them thoſe Velocities wherewith they were afterwards <pb xlink:href="069/01/223.jpg" pagenum="220"></pb>to be perpetually moved with an Equable Circular Motion, made <lb></lb>them, they departing from Reſt, to move along determinate Spaces <lb></lb>with that Natural Motion in a Right Line, according to which we <lb></lb>ſenſibly ſee our Moveables to move from the ſtate of Reſt ſucceſ<lb></lb>ſively Accelerating. </s> <s>And he addeth, that having made them to <lb></lb>acquire that degree in which it pleaſed him that they ſhould after<lb></lb>wards be perpetually conſerved, he converted their Right or direct <lb></lb>Motion into Circular; which only is apt to conſerve it ſelf Equa<lb></lb>ble, alwaies revolving without receding from, or approaching to <lb></lb>any prefixed term by them deſired. </s> <s>The Conceit is truly worthy <lb></lb>of <emph type="italics"></emph>Plato<emph.end type="italics"></emph.end>; and is the more to be eſteemed in that the grounds there<lb></lb>of paſſed over in ſilence by him, and diſcovered by our Author by <lb></lb>taking off the Mask or Poetick Repreſentation, do ſhew it to be <lb></lb>in its native aſpect a true Hiſtory. </s> <s>And I think it very credible that <lb></lb>we having by the Doctrine of Aſtronomy ſufficiently competent <lb></lb>Knowledge of the Magnitudes of the Orbes of the Planets, and of <lb></lb>their Diſtances from the Center about which they move, as alſo <lb></lb>of their Velocities, our Author (to whom <emph type="italics"></emph>Plato's<emph.end type="italics"></emph.end> Conjecture was <lb></lb>not unknown) may ſometime for his curioſity have had ſome <lb></lb>thought of attempting to inveſtigate whether one might aſſign a <lb></lb>determinate Sublimity from which the <emph type="italics"></emph>B<emph.end type="italics"></emph.end>odies of the Planets depar<lb></lb>ting, as from a ſtate of Reſt, and moved for certain Spaces with a <lb></lb>Right and Naturally Accelerate Motion, afterwards converting <lb></lb>the Acquired Velocity into Equable Motions, they might be found <lb></lb>to correſpond with the greatneſs of their Orbes, and with the Times <lb></lb>of their Revolutions.</s></p><p type="main"> <s><emph type="italics"></emph>S<emph.end type="italics"></emph.end>ALV. </s> <s>I think I do remember that he hath heretofore told me, <lb></lb>that he had once made the Computation, and alſo that he found <lb></lb>it exactly to anſwer the Obſervations; but that he had no mind to <lb></lb>ſpeak of them, doubting leſt the two many Novelties by him diſ<lb></lb>covered, which had provoked the diſpleaſure of many againſt him, <lb></lb>might blow up new ſparks. <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut if any one ſhall have the like de<lb></lb>ſire he may of himſelf by the Doctrine of the preſent Tract give <lb></lb>himſelf content. <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut let us purſue our buſineſs, which is to <lb></lb>ſhew;</s></p><p type="head"> <s>PROBL. I. PROP. IV.</s></p><p type="main"> <s>How in a Parabola given, deſcribed by the Pro<lb></lb>ject, the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of each ſeveral point may be <lb></lb>determined.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Semiparabola be B E C, whoſe Amplitude is C D and Al<lb></lb>titude D B, with which continued out on high the Tangent of the <lb></lb>Parabola C A meeteth in A; and along the<emph.end type="italics"></emph.end> Vertex <emph type="italics"></emph>B let B I be<emph.end type="italics"></emph.end><pb xlink:href="069/01/224.jpg" pagenum="221"></pb><emph type="italics"></emph>an Horizontal Line, and parallel to C D. </s> <s>And if the Amplitude C D <lb></lb>be equal to the whole Altitude D A, B I ſhall be equal to B A and B D. <lb></lb></s> <s>And if the Time of the Fall along A B, and the Moment of Velocity <lb></lb>acquired in B along the Deſcent A B<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in A be ſuppoſed to be <lb></lb>meaſured by the ſaid A B, then D C (that is twice B I) ſhall be the <lb></lb>Space which ſhall be paſſed by the<emph.end type="italics"></emph.end> Impetus A <emph type="italics"></emph>B turned along the Hori<lb></lb>zontal Line in the ſame Time: But in the ſame Time falling along B D <lb></lb>out of Reſt in B, it ſhall paſs the Altitude B D: Therefore the Movea<lb></lb>ble falling out of Reſt in A along A B, <lb></lb>being converted with the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>A B<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.224.1.jpg" xlink:href="069/01/224/1.jpg"></figure><lb></lb><emph type="italics"></emph>along the Horizontal Parallel ſhall <lb></lb>paſs a Space equal to D C. </s> <s>And the <lb></lb>Fall along B D ſupervening, it paſſeth <lb></lb>the Altitude B D, and deſcribes the <lb></lb>Parabola B C; whoſe<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>in the <lb></lb>Term C is compounded of the Equable <lb></lb>Tranſverſal whoſe Moment is as A B, <lb></lb>and of another Moment acquired in the <lb></lb>Fall B D in the Term D or C; which <lb></lb>Moments are Equal. </s> <s>If therefore we <lb></lb>ſuppoſe A B to be the Meaſure of one of them, as ſuppoſe of the Equa<lb></lb>ble Tranſverſal; and B I, which is equal to B D, to be the Meaſure of <lb></lb>the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>acquired in D or C; then the Subtenſe I A ſhall be the <lb></lb>quantity of the Moment compound of them both: Therefore it ſhall be <lb></lb>the quantity or Meaſure of the whole Moment which the Project deſcend<lb></lb>ing along the Parabola B C ſhall acquire of<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>in C. </s> <s>This pre<lb></lb>miſed, take in the Parabola any point E, in which we are to determine <lb></lb>of the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>of the Project. </s> <s>Draw the Horizontal Parallel E F, <lb></lb>and let B G be a Mean-proportional between B D and B F. </s> <s>And foraſ<lb></lb>much as A B or B D is ſuppoſed to be the Meaſure of the Time, and of <lb></lb>the Moment of the Velocity in the Fall B D<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in B: B G ſhall <lb></lb>be the Time, or the Meaſure of the Time, and of the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>in F, coming <lb></lb>out of B. </s> <s>If therefore B O be ſuppoſed equal to B G, the Diagonal <lb></lb>drawn from A to O ſhall be the quantity of the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>in E; for <lb></lb>A B hath been ſuppoſed the determinator of the Time, and of the<emph.end type="italics"></emph.end> Impe<lb></lb>tus <emph type="italics"></emph>in B, which turned along the Horizontal Parallel doth alwaies <lb></lb>continue the ſame: And B O determineth the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>in F or in E <lb></lb>along the Deſcent<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in B in the Altitude B F: But theſe two <lb></lb>A B and B O are<emph.end type="italics"></emph.end> potentia <emph type="italics"></emph>equal to the Power A O. </s> <s>Therefore that is <lb></lb>manifeſt which was ſought.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>The Contemplation of the Compoſition of theſe diffe<lb></lb>rent <emph type="italics"></emph>Impetus's,<emph.end type="italics"></emph.end> and of the quantity of that <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> which reſults <lb></lb>from this mixture, is ſo new to me, that it leaveth my mind in no <lb></lb>ſmall confuſion. </s> <s>I do not ſpeak of the mixtion of two Motions <pb xlink:href="069/01/225.jpg" pagenum="222"></pb>Equable, though unequal to one another, made the one along the <lb></lb>Horizontal Line, and the other along the Perpendicular, for I very <lb></lb>well comprehend that there is made a Motion of theſe two <emph type="italics"></emph>poten<lb></lb>tia<emph.end type="italics"></emph.end> equal to both the Compounding Motions, but my confuſion <lb></lb>ariſeth upon the mixing of the Equable-Horizontal and Perpendi<lb></lb>cular-Naturally-Accelerate Motion. </s> <s>Therefore I could wiſh we <lb></lb>might toge ther a little better conſider this buſineſs.</s></p><p type="main"> <s>SIMP. </s> <s>And I ſtand the more in need thereof in that I am not <lb></lb>yet ſo well ſatisfied in Mind as I ſhould be, in the Propoſitions that <lb></lb>are the firſt foundations of the others that follow upon them. </s> <s>I <lb></lb>will add, that alſo in the Mixtion of the two Motions Equable <lb></lb>Horizontal, and Perpendicular, I would better underſtand that <lb></lb><emph type="italics"></emph>Potentia<emph.end type="italics"></emph.end> of their Compound. </s> <s>Now, <emph type="italics"></emph>Salviatus,<emph.end type="italics"></emph.end> you ſee what we <lb></lb>want and deſire.</s></p><p type="main"> <s>SALV. </s> <s>Your deſire is very reaſonable: and I will eſſay whe<lb></lb>ther my having had a longer time to think thereon may facilitate <lb></lb>your ſatisfaction. </s> <s>But you muſt bear with and excuſe me if in diſ<lb></lb>courſing I ſhall repeat a great part of the things hitherto delivered <lb></lb>by our Author.</s></p><p type="main"> <s>It is not poſſible for us to ſpeak poſitively touching Motions and <lb></lb>their Velocities or <emph type="italics"></emph>Impetus's,<emph.end type="italics"></emph.end> be they Equable, or be they Naturally <lb></lb>Accelerate, unleſs we firſt agree upon the Meaſure that we are to <lb></lb>uſe in the commenſuration of thoſe Velocities, as alſo of the Time. <lb></lb></s> <s>As to the Meaſure of the Time, we have already that which is <lb></lb>commonly received by all of Hours, Prime-Minutes, and Se<lb></lb>conds, <emph type="italics"></emph>&c.<emph.end type="italics"></emph.end> and as for the meaſuring of Time we have that com<lb></lb>mon Meaſure received by all, ſo it is requiſite to aſſign another <lb></lb>Meaſure for the Velocities that is commonly underſtood and re<lb></lb>ceived by every one; that is, which every where is the ſame. </s> <s>The <lb></lb>Author, as hath been declared, adjudged the Velocity of Naturally <lb></lb>deſcending Grave-Bodies to be fit for this purpoſe; the encreaſing <lb></lb>Velocities of which are the ſame in all parts of the World. </s> <s>So that <lb></lb>that ſame degree of Velocity which (for example) a Ball of Lead of <lb></lb>a pound acquireth in having, departing from Reſt, deſcended Per<lb></lb>pendicularly as much as the height of a Pike, is alwaies, and in all <lb></lb>places the ſame, and therefore moſt commodious for explicating <lb></lb>the quantity of the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> that is derived from the Natural De<lb></lb>ſcent. </s> <s>Now it remains to find a way to determine likewiſe the <lb></lb>Quantity of the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> in an Equable Motion in ſuch a manner, <lb></lb>that all thoſe which diſcourſe about it may form the ſame conceit <lb></lb>of its greatneſs and Velocity; ſo that one may not imagine it more <lb></lb>ſwift, and another leſs; whereupon afterwards in conjoyning and <lb></lb>mingling this Equable Motion imagined by them with the eſtabli<lb></lb>ſhed Accelerate Motion ſeveral men may form ſeveral Conceits of <lb></lb>ſeveral greatneſſes of <emph type="italics"></emph>Impetus's.<emph.end type="italics"></emph.end> To determine and repreſent this <pb xlink:href="069/01/226.jpg" pagenum="223"></pb><emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> and particular Velocity our Author hath not found any <lb></lb>way more commodious, than the making uſe of the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> which <lb></lb>the Moveable from time to time acquires in the Naturally-Accele<lb></lb>rate Motion, any acquired Moment of which being reduced into <lb></lb>an Equable Motion retaineth its Velocity preciſely limited, and <lb></lb>ſuch, that in ſuch another Time as that wherein it did Deſcend, it <lb></lb>paſſeth double the Space of the Height from whence it fell. </s> <s>But <lb></lb>becauſe this is the principal point in the buſineſs that we are upon, <lb></lb>it is good to make it to be perfectly underſtood by ſome particular <lb></lb>Example. </s> <s>Reaſſuming therefore the Velocity and <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> acqui<lb></lb>red by the Cadent Moveable, as we ſaid before, from the height <lb></lb>of a Pike, of which Velocity we will make uſe for a Meaſure of <lb></lb>other Velocities and <emph type="italics"></emph>Impetuſſes<emph.end type="italics"></emph.end> upon other occaſions, and ſuppo<lb></lb>ſing, for example, that the Time of that Fall be four ſecond Mi<lb></lb>nutes of an hour, to find by this ſame Meaſure how great the <emph type="italics"></emph>Im<lb></lb>petus<emph.end type="italics"></emph.end> of the Moveable would be falling from any other height <lb></lb>greater, or leſſer, we ought not from the proportion that this other <lb></lb>height hath to the height of a Pike to argue and conclude the quan<lb></lb>tity of the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> acquired in this ſecond height, thinking, for <lb></lb>example, that the Moveable falling from quadruple the height <lb></lb>hath acquired quadruple Velocity, for that it is falſe: for that the <lb></lb>Velocity of the Naturally-Accelerate Motion doth not increaſe or <lb></lb>decreaſe according to the proportion of the Spaces, but according <lb></lb>to that of the Times, than which that of the Spaces is greater in a <lb></lb>duplicate proportion, as was heretofore demonſtrated. </s> <s>Therefore <lb></lb>when in a Right Line we have aſſigned a part for the Meaſure of <lb></lb>the Velocity, and alſo of the Time, and of the Space in that Time <lb></lb>paſſed (for that for brevity ſake all theſe three Magnitudes are <lb></lb>often repreſented by one ſole Line,) to find the quantity of the <lb></lb>Time, and the degree of Velocity that the ſame Moveable would <lb></lb>have acquired in another Diſtance we ſhall obtain the ſame, not <lb></lb>immediataly by this ſecond Diſtance, but by the Line which ſhall <lb></lb>be a Mean-proportional betwixt the two Diſtances. </s> <s>But I will <lb></lb>better declare my ſelf by an Example. </s> <s>In the Line A C Perpendi<lb></lb>cular to the Horizon let the part A B be underſtood to <lb></lb>be a Space paſſed by a Moveable naturally deſcending <lb></lb><figure id="id.069.01.226.1.jpg" xlink:href="069/01/226/1.jpg"></figure><lb></lb>with an Accelerate Motion: the Time of which paſ<lb></lb>ſage, in regard I may repreſent it by any Line, I will, for <lb></lb>brevity, imagine it to be as much as the ſame Line A B <lb></lb>and likewiſe for a Meaſure of the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> and Velocity <lb></lb>acquired by that Motion, I again take the ſame Line <lb></lb>A B; ſo that of all the Spaces that are in the progreſs of <lb></lb>the Diſcourſe to be conſidered the part A B may be the <lb></lb>Meaſure. </s> <s>Having all our pleaſure eſtabliſhed under one <lb></lb>ſole Magnitude A B theſe three Meaſures of different kinds of <pb xlink:href="069/01/227.jpg" pagenum="224"></pb>Quantities, that is to ſay, of Spaces, of Times, and of <emph type="italics"></emph>Impetus's,<emph.end type="italics"></emph.end> let <lb></lb>it be required to determine in the aſſigned Space, and at the height <lb></lb>A C, how much the Time of the Fall of the Moveable from A to <lb></lb>C is to be, and what the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> is that ſhall be found to have been <lb></lb>acquired in the ſaid Term C, in relation to the Time and to the <lb></lb><emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> meaſured by A B. </s> <s>Both theſe queſtions ſhall be reſolved <lb></lb>taking A D the Mean-proportional betwixt the two Lines A C <lb></lb>and A B; affirming the Time of the Fall along the whole Space <lb></lb>A C to be as the Time A D is in relation to A B, aſſigned in the <lb></lb>beginning for the Quantity of the Time in the Fall A B. </s> <s>And like<lb></lb>wiſe we will ſay that the <emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> or degree of Velocity that the <lb></lb>Cadent Moveable ſhall obtain in the Term C, in relation to the <lb></lb><emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> that it had in B, is as the ſame Line A D is in relation to <lb></lb>A B, being that the Velocity encreaſeth with the ſame proportion <lb></lb>as the Time doth: Which Concluſion although it was aſſumed as <lb></lb>a <emph type="italics"></emph>Poſtulatum,<emph.end type="italics"></emph.end> yet the Author was pleaſed to explain the Applicati<lb></lb>on thereof above in the third Propoſition.</s></p><p type="main"> <s>This point being well underſtood and proved, we come to the <lb></lb>Conſideration of the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> derived from two compound Moti<lb></lb>ons: whereof let one be compounded of the Horizontal and alwaies <lb></lb>Equable, and of the Perpendicular unto the Horizon, and it alſo <lb></lb>Equable: but let the other be compounded of the Horizontal like<lb></lb>wiſe alwaies Equable, and of the Perpendicular Naturally-Accele<lb></lb>rate. </s> <s>If both ſhall be Equable, it hath been ſeen already that the <lb></lb><emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> emerging from the compoſition of both is <emph type="italics"></emph>potentia<emph.end type="italics"></emph.end> equal to <lb></lb>both, as for more plainneſs we will thus Exemplifie. </s> <s>Let the Move<lb></lb>able deſcending along the Perpendicular A B be ſuppoſed to have, <lb></lb>for example, three degrees of Equable <emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> but being tranſ<lb></lb>ported along A B towards C, let the ſaid Velocity and <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> be <lb></lb>ſuppoſed four degrees, ſo that in the ſame Time that falling it would <lb></lb>paſs along the Perpendicular, <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> three yards, <lb></lb><figure id="id.069.01.227.1.jpg" xlink:href="069/01/227/1.jpg"></figure><lb></lb>it would in the Horizontal paſs four, but in <lb></lb>that compounded of both the Velocities it <lb></lb>cometh in the ſame Timefrom the point A un<lb></lb>to the Term C, deſcending all the way along the Diagonal Line <lb></lb>A C, which is not ſeven yards long, as that ſhould be which is com<lb></lb>pounded of the two Lines A B, 3, and B C, 4, but is 5; which 5 is <lb></lb><emph type="italics"></emph>potentia<emph.end type="italics"></emph.end> equal to the two others, 3 and 4: For having found the <lb></lb>Squares of 3 and 4, which are 9 and 16, and joyning theſe together, <lb></lb>they make 25 for the Square of A C, which is equal to the two <lb></lb>Squares of A B and B C: whereupon A C ſhall be as much as is the <lb></lb>Side, or, if you will, Root of the Square 25, which is 5. For a conſtant <lb></lb>and certain Rule therefore, when it is required to aſſign the <lb></lb>Quantity of the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> reſulting from two <emph type="italics"></emph>Impetus's<emph.end type="italics"></emph.end> given, the <lb></lb>one Horizontal, and the other Perpendicular, and both Equable, <pb xlink:href="069/01/228.jpg" pagenum="225"></pb>they are each of them to be ſquared, and their Squares being put <lb></lb>together the Root of the Aggregate is to be extracted, which ſhall <lb></lb>give us the quantity of the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> compounded of them both. <lb></lb></s> <s>And thus in the foregoing example, that Moveable that by vertue <lb></lb>of the Perpendicular Motion would have percuſſed upon the Hori<lb></lb>zon with three degrees of Force, and with only the Horizontal Mo<lb></lb>tion would have percuſſed in C with four degrees, percuſſing with <lb></lb>both the <emph type="italics"></emph>Impetus's<emph.end type="italics"></emph.end> conjoyned, the blow ſhall be like to that of the <lb></lb>Percutient moved with five degrees of Velocity and Force. </s> <s>And <lb></lb>this ſame Percuſſion would be of the ſame Impetuoſity in all the <lb></lb>points of the Diagonal A C, for that the compounded <emph type="italics"></emph>Impetus's<emph.end type="italics"></emph.end><lb></lb>are alwaies the ſame, never encreaſing or diminiſhing.</s></p><p type="main"> <s>Let us now ſee what befalls in compounding the Equable Hori<lb></lb>zontal Motion with another Perpendicular to the Horizon which <lb></lb>beginning from Reſt goeth Naturally Accelerating. </s> <s>It is already <lb></lb>manifeſt, that the Diagonal, which is the Line of the Motion com<lb></lb>pounded of theſe two, is not a Right Line, but Semiparabolical, <lb></lb>as hath been demonſtrated; ^{*} in which the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> doth go con<lb></lb><arrow.to.target n="marg1100"></arrow.to.target><lb></lb>tinually encreaſing by means of the continual encreaſe of the Ve<lb></lb>locity of the Perpendicular Motion: Wherefore, to determine what <lb></lb>the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> is in an aſſigned point of that Parabolical Diagonal, it <lb></lb>is requiſite firſt to aſſign the Quantity of the Uniform Horizontal <lb></lb><emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> and then to find what is the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of the falling Movea<lb></lb>ble in the point aſſigned: the which cannot be determined without <lb></lb>the conſideration of the Time ſpent from the beginning of the <lb></lb>Compoſition of the two Motions: which Conſideration of the <lb></lb>Time is not required in the Compoſition of Equable Motions, the <lb></lb>Velocities and <emph type="italics"></emph>Impetus's<emph.end type="italics"></emph.end> of which are alwaies the ſame: but here <lb></lb>where there is inſerted into the mixture a Motion which beginning <lb></lb>from extream Tardity goeth encreaſing in Velocity according to <lb></lb>the continuation of the Time, it is neceſſary that the quantity of <lb></lb>the Time do ſhew us the quantity of the degree of Velocity in the <lb></lb>aſſigned point: for, as to the reſt, the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> compounded of theſe <lb></lb>two (as in Uniform Motions) is <emph type="italics"></emph>potentia<emph.end type="italics"></emph.end> equal to both the others <lb></lb>compounding. </s> <s>But here again I will better explain my meaning by <lb></lb>an example. </s> <s>In A C the Perpendicular to the Horizon let any part <lb></lb>be taken A B; the which I will ſuppoſe to ſtand for the Meaſure <lb></lb>of the Space of the Natural Motion made along the ſaid Perpen<lb></lb>dicular, and likewiſe let it be the Meaſure of the Time, and alſo of <lb></lb>the degree of Velocity, or, if you will, of the <emph type="italics"></emph>Impetus's.<emph.end type="italics"></emph.end> It is ma<lb></lb>nifeſt in the firſt place, that if the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of the Moveable in B <lb></lb><emph type="italics"></emph>ex quiete<emph.end type="italics"></emph.end> in A ſhall be turned along B D parallel to the Horizon in <lb></lb>an Equable Motion, the quantity of its Velocity ſhall be ſuch that <lb></lb>in the Time A B it ſhall paſs a Space double to the Space A B, which <lb></lb>let be the Line B D. </s> <s>Then let B C be ſuppoſed equal to B A, and <pb xlink:href="069/01/229.jpg" pagenum="226"></pb>let C E be drawn parallel and equal to B D, and thus by the Points <lb></lb>B and E we ſhall deſcribe the Parabolick Line B E I. </s> <s>And becauſe <lb></lb>that in the Time A B with the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> A B the Horizontal Line B D <lb></lb>or C E is paſſed, double to A B, and in ſuch another Time the Per<lb></lb>pendicular B C is paſſed with an acquiſt of <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> in C equal to <lb></lb>the ſaid Horizontal Line; therefore the Moveable in ſuch another <lb></lb>Time as A B ſhall be found to have paſſed from B to E along the <lb></lb>Parabola B E with an <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> compounded of two, each equal to <lb></lb>the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> A B. </s> <s>And becauſe one of them is Horizontal, and the <lb></lb>other Perpendicular, the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> compound of them ſhall be equal <lb></lb>in Power to them both, that is <lb></lb><figure id="id.069.01.229.1.jpg" xlink:href="069/01/229/1.jpg"></figure><lb></lb>double to one of them. </s> <s>So that <lb></lb>ſuppoſing B F equal to B A, and <lb></lb>drawing the Diagonal A F, the <lb></lb><emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> or the Percuſſion in E <lb></lb>ſhall be greater than the Percuſ<lb></lb>ſion in B of the Moveable fal<lb></lb>ling from the Height A, or than <lb></lb>the Percuſſion of the Horizon<lb></lb>tal <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> along B D, according <lb></lb>to the proportion of A F to <lb></lb>A B. </s> <s>But in caſe, ſtill retaining <lb></lb>B A for the Meaſure of the <lb></lb>Space of the Fall from Reſt in <lb></lb>A unto B, and for the Meaſure of the Time and of the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of <lb></lb>the falling Moveable acquired in B, the Altitude B O ſhould not be <lb></lb>equal to, but greater than A B, taking B G to be a Mean-propor<lb></lb>tional betwixt the ſaid A B and B O, the ſaid B G would be the <lb></lb>Meaſure of the Time and of the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> in O, acquired in O by the <lb></lb>Fall from the height B O; and the Space along the Horizontal <lb></lb>Line, which being paſſed with the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> A B in the Time A B <lb></lb>would be double to A B, ſhall, in the whole duration of the Time <lb></lb>B G, be ſo much the greater, by how much in proportion B G is <lb></lb>greater than B A. </s> <s>Suppoſing therefore L B equal to B G, and draw<lb></lb>ing the Diagonal A L, it ſhall give us the quantity compounded of <lb></lb>the two <emph type="italics"></emph>Impetus's<emph.end type="italics"></emph.end> Horizontal and Perpendicular, by which the <lb></lb>Parabola is deſcribed; and of which the Horizontal and Equable is <lb></lb>that acquired in B by the fall of A B, and the other is that acquired <lb></lb>in O, or, if you will, in I by the Deſcent B O, whoſe Time, as alſo <lb></lb>the quantity of its Moment was B G. </s> <s>And in this Method we ſhall <lb></lb>inveſtigate the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> in the extream term of the Parabola, in caſe <lb></lb>its Altitude were leſſer than the Sublimity A B, taking the Mean<lb></lb>proportional betwixt them both: which being ſet off upon the Ho<lb></lb>rizontal Line in the place of B F, and the Diagonal drawn, as A F, <lb></lb>we ſhall hereby have the quantity of the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> in the extream <lb></lb>term of the Parabola.</s></p><pb xlink:href="069/01/230.jpg" pagenum="227"></pb><p type="margin"> <s><margin.target id="marg1100"></margin.target>* Or along <lb></lb>which.</s></p><p type="main"> <s>And to what hath hitherto been propoſed touching <emph type="italics"></emph>Impetus's,<emph.end type="italics"></emph.end><lb></lb>Blows, or if you pleaſe, Percuſſions of ſuch like Projects, it is ne<lb></lb>ceſſary to add another very neceſſary Conſideration; and this it is: <lb></lb>That it doth not ſuffice to have regard to the Velocity only of the <lb></lb>Project for the determining rightly of the Force and Violence of the <lb></lb>Percuſſion, but it is requiſite likewiſe to examine apart the State <lb></lb>and Condition of that which receiveth the Percuſſion, in the effica<lb></lb>cy of which it hath for many reſpects a great ſhare and intereſt. <lb></lb></s> <s>And firſt there is no man but knows that the thing ſmitten doth ſo <lb></lb>much ſuffer violence from the Velocity of the Percutient by how <lb></lb>much it oppoſeth it, and either totally or partially checketh its <lb></lb>Motion: For if the Blow ſhall light upon ſuch an one as yieldeth to <lb></lb>the Velocity of the Percutient without any Reſiſtance, that Blow <lb></lb>ſhall be nullified: And he that runneth to hit his Enemy with his <lb></lb>Launce, if at the overtaking of him it ſhall fall out that he moveth, <lb></lb>giving back with the like Velocity, he ſhall make no thruſt, and the <lb></lb>Action ſhall be a meer touch without doing any harm.</s></p><p type="main"> <s>But if the Percuſſion ſhall happen to be received upon an Object <lb></lb>which doth not wholly yield to the Percutient, but only partially, <lb></lb>the Percuſſion ſhall do hurt, though not with its whole <emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> but <lb></lb>only with the exceſs of the Velocity of the ſaid Percutient above <lb></lb>the Velocity of the recoile and receſſion of the Object percuſſed: <lb></lb>ſo that, if <emph type="italics"></emph>v. </s> <s>g.<emph.end type="italics"></emph.end> the Percutient ſhall come with 10 degrees of Velo<lb></lb>city upon the Percuſſed Body, which giving back in part retireth <lb></lb>with 4 degrees, the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> and Percuſſion ſhall be as if it were of <lb></lb>6 degrees. </s> <s>And laſtly, the Percuſſion ſhall be entire and perfect on <lb></lb>the part of the Percutient when the thing percuſſed yieldeth not, <lb></lb>but wholly oppoſeth and ſtoppeth the whole Motion of the Percu<lb></lb>tient; if haply there can be ſuch a caſe. </s> <s>And I ſay on the part of <lb></lb>the Percutient, for when the Body percuſſed moveth with a contra<lb></lb>ry Motion towards the Percutient, the Blow and Shock ſhall be <lb></lb>ſo much the more Impetuous by how much the two Velocities uni<lb></lb>ted are greater than the ſole Velocity of the Percutient. </s> <s>More<lb></lb>over, you are likewiſe to take notice, that the more or leſs yielding <lb></lb>may proceed not only from the quality of the Matter more or leſs <lb></lb>hard, as if it be of Iron, of Lead, or of Wooll, <emph type="italics"></emph>&c.<emph.end type="italics"></emph.end> but alſo from <lb></lb>the Poſition of the Body that receiveth the Percuſſion. </s> <s>Which Po<lb></lb>ſition if it ſhall be ſuch as that the Motion of the Percutient hap<lb></lb>neth to hit it at Right-Angles, the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of the Percuſſion ſhall <lb></lb>be the greateſt: but if the Motion ſhall proceed obliquely, and, as <lb></lb>we ſay, aſlant, the Percuſſion ſhall be weaker; and that more, and <lb></lb>more according to its greater and greater Obliquity: for an Ob<lb></lb>ject in that manner ſcituate, albeit of very ſolid matter, doth not <lb></lb>damp or arreſt the whole <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> and Motion of the Percutient, <lb></lb>which ſlanting paſſeth farther, continuing at leaſt in ſome part to <pb xlink:href="069/01/231.jpg" pagenum="228"></pb>move along the Surface of the oppoſed Body Reſiſting. </s> <s>When <lb></lb>therefore we have even now determined of the greatneſs of the <lb></lb><emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of the Project in the end of the Parabolicall Line, it ought <lb></lb>to be underſtood to be meant of the Percuſſion received upon a <lb></lb>Line at Right Angles with the ſame Parabolick Line, or with the <lb></lb>Line that is Tangent to the Parabola in the foreſaid point: for <lb></lb>although that ſame Motion be compounded of an Horizontal and <lb></lb>a Perpendicular Motion, the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> is not at the greateſt either <lb></lb>upon the Horizontal Plane, or upon that erect to the Horizon, be<lb></lb>ing received upon them both obliquely.</s></p><p type="main"> <s>SAGR. </s> <s>Your ſpeaking of theſe Blows, and theſe Percuſſions <lb></lb>hath brought into my mind a Problem, or, if you will, Queſtion <lb></lb>in the Mechanicks, the ſolution whereof I could never find in any <lb></lb>Author, nor any thing that doth diminiſh my admiration, or ſo <lb></lb>much as in the leaſt afford my judgment ſatisfaction. </s> <s>And my <lb></lb>doubt and wonder lyeth in my not being able to comprehend <lb></lb>whence that Immenſe Force and Violence ſhould proceed, and on <lb></lb>what Principle it ſhould depend, which we ſee to conſiſt in Per<lb></lb>cuſſion, in that with the ſimple ſtroke of an Hammer, that doth <lb></lb>not weigh above eight or ten pounds, we ſee ſuch Reſiſtances to be <lb></lb>overcome as would not yield to the weight of a Grave Body that <lb></lb>without Percuſſion hath an <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> only by preſſing and bearing <lb></lb>upon it, albeit the weight of this be many hundreds of pounds <lb></lb>more. </s> <s>I would likewiſe find out a way to meaſure the Force of this <lb></lb>Percuſſion, which I do not think to be infinite, but rather hold <lb></lb>that it hath its Term in which it may be compared, and in the end <lb></lb>Regulated with other Forces of preſſing Gravities, either of Lea<lb></lb>vers, or of Screws, or of other Mechanick Inſtruments, of whoſe <lb></lb>multiplication of Force I am thorowly ſatisfied.</s></p><p type="main"> <s>SALV. </s> <s>You are not alone in the admirableneſs of the effect, <lb></lb>and the obſcurity of the cauſe of ſo ſtupendious an Accident. </s> <s>I <lb></lb>ruminated a long time upon it in vain, my ſtupifaction ſtill encrea<lb></lb>ſing; till in the end meeting with our <emph type="italics"></emph>Academian,<emph.end type="italics"></emph.end> I received from <lb></lb>him a double ſatisfaction: firſt in hearing that he alſo had been a <lb></lb>long time at the ſame loſs; and next in underſtanding that after he <lb></lb>had at times ſpent many thouſands of hours in ſtudying and con<lb></lb>templating thereon, he had light upon certain Notions far from <lb></lb>our firſt conceptions, and therefore new, and for their Novelty to <lb></lb>be admired. </s> <s>And becauſe that I already ſee that your Curioſity <lb></lb>would gladly hear thoſe Conceits which are Remote from common <lb></lb>Conjecture, I ſhall not ſtay for your entreaty, but I give you my <lb></lb>word that ſo ſoon as we ſhall have finiſhed the Reading of this <lb></lb>Treatiſe of Projects, I will ſet before you all thoſe Fancies, or, I <lb></lb>might ſay, Extravagancies that are yet left in my memory of the <lb></lb>Diſcourſes of the Academick. </s> <s>In the mean time let us proſecute <lb></lb>the Propoſitions of our Author.</s></p><pb xlink:href="069/01/232.jpg" pagenum="229"></pb><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></lb>a ſublime point out of which the Moveable <lb></lb>falling ſhall deſcribe the ſaid Parabola.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Parabola be A B, its Amplitude H B, and its prolonged <lb></lb>Axis H E; in which a Sublimity is to be found, out of which the <lb></lb>Moveable falling, and converting the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>conceived in A <lb></lb>along the Horizontal Line, deſcribeth the Parabola A B. </s> <s>Draw the <lb></lb>Horizontal Line A G, which ſhall be Parallel to B H, and ſuppoſing A F <lb></lb>equal to A H draw the Right Line F B, which toucheth the Parabola in <lb></lb>B, and cutteth the Horizontal Line A G in G; and unto F A and A G <lb></lb>let A E be a third Proportional. </s> <s>I ſay, that E is the ſublime Point re<lb></lb>quired, out of which the Moveable falling<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in E, and the<emph.end type="italics"></emph.end> Im<lb></lb>petus <emph type="italics"></emph>conceived in A being converted along the Horizontal Line over<lb></lb>taking the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>of the Deſcent<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.232.1.jpg" xlink:href="069/01/232/1.jpg"></figure><lb></lb><emph type="italics"></emph>in H<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in A, deſcribeth the <lb></lb>Parabola A B. </s> <s>For if we ſuppoſe <lb></lb>E A to be the Meaſure of the Time <lb></lb>of the Fall from E to A, and of <lb></lb>the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>acquired in A, A G <lb></lb>(that is a Mean-proportional be<lb></lb>tween E A and A F) ſhall be the <lb></lb>Time and the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>coming <lb></lb>from F to A, or from A to H. </s> <s>And <lb></lb>becauſe the Moveable coming out of <lb></lb>E in the Time E A with the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>acquired in A paſſeth in the Ho<lb></lb>rizontal Lation with an Equable Motion the double of E A; There<lb></lb>fore likewiſe moving with the ſame<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>it ſhall in the Time A G <lb></lb>paſs the double of G A, to wit, the Mean-proportional B H (for the <lb></lb>Spaces paſſed with the ſame Equable Motion are to one another as the <lb></lb>Times of the ſaid Motions:) And along the Perpendicular A H ſhall <lb></lb>be paſſed with a Motion<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in the ſame Time G A: Therefore <lb></lb>the Amplitude H B, and Altitude A H are paſſed by the Moveable in the <lb></lb>ſame Time: Therefore the Parabola A B ſhall be deſcribed by the <lb></lb>Deſcent of the Project coming from the Sublimity E: Which was re<lb></lb>quired.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>COROLLARY.</s></p><p type="main"> <s>Hence it appeareth that the half of the <emph type="italics"></emph>B<emph.end type="italics"></emph.end>aſe or Amplitude of the <lb></lb>Semiparabola (which is the fourth part of the Amplitude of <lb></lb>the whole Parabola) is a Mean-proportional betwixt its Al<lb></lb>titude and the Sublimity out of which the Moveable falling <lb></lb>deſcribeth it.</s></p><pb xlink:href="069/01/233.jpg" pagenum="230"></pb><p type="head"> <s>PROBL. III. <emph type="italics"></emph>P<emph.end type="italics"></emph.end>RO<emph type="italics"></emph>P.<emph.end type="italics"></emph.end> VI.</s></p><p type="main"> <s>The Sublimity and Altitude of a Semiparabola <lb></lb>being given to find its Amplitude.</s></p><p type="main"> <s><emph type="italics"></emph>Let A C be perpendicular to the<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.233.1.jpg" xlink:href="069/01/233/1.jpg"></figure><lb></lb><emph type="italics"></emph>Horizontal Line D C, in <lb></lb>which let the Altitude C B and <lb></lb>the Sublimity B A be given: It is <lb></lb>required in the Horizontal Line <lb></lb>D C to find the Amplitude of the <lb></lb>Semiparabola that is deſcribed out of <lb></lb>the Sublimity B A with the Alti<lb></lb>tude B C. </s> <s>Take a Mean proportional <lb></lb>between C B and B A, to which let <lb></lb>C D be double, I ſay, that C D is <lb></lb>the Amplitude required. </s> <s>The which <lb></lb>is manifeſt by the precedent Propoſition.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>THEOR. IV. PROP. VII.</s></p><p type="main"> <s>In Projects which deſcribe Semiparabola's of the <lb></lb>ſame Amplitude, there is leſs <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> required <lb></lb>in that which deſcribeth that whoſe Ampli<lb></lb>tude is double to its Altitude, than in any <lb></lb>other.</s></p><p type="main"> <s><emph type="italics"></emph>For let the Semiparabola be B D, whoſe Amplitude C D is dou<lb></lb>ble to its Altitude C B; and in its Axis extended on high let B A <lb></lb>be ſuppoſed equal to the Altitude B C; and draw a Line from <lb></lb>A to D which toucheth the Semiparabola in D, and ſhall cut the Hori<lb></lb>zontal Line B E in E; and B E ſhall be equal to B C or to B A: It is <lb></lb>manifeſt that it is deſcribed by the Project whoſe Equable Horizontal<emph.end type="italics"></emph.end><lb></lb>Impetus <emph type="italics"></emph>is ſuch as is that gained in B of a thing falling from Reſt in A, <lb></lb>and the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>of the Natural Motion downwards, ſuch as is that of <lb></lb>a thing coming to C<emph.end type="italics"></emph.end> ex quiete <emph type="italics"></emph>in B. </s> <s>Whence it is manifeſt, that the<emph.end type="italics"></emph.end><lb></lb>Impetus <emph type="italics"></emph>compounded of them, and that ſtriketh in the Term D is as the <lb></lb>Diagonal A E, that is<emph.end type="italics"></emph.end> potentia <emph type="italics"></emph>equal to them both. </s> <s>Now let there be <lb></lb>another Semiparabola G D, whoſe Amplitude is the ſame C D, and the <lb></lb>Altitude C G leſs, or greater than the Altitude B C, and let H D touch <lb></lb>the ſame, cutting the Horizontal Line drawn by G in the point K; and <lb></lb>as H G is to G K, ſo let K G be to G L: by what hath been demonſtrated <lb></lb>G L ſhall be the Altitude from which the Project falling deſcribeth the<emph.end type="italics"></emph.end><pb xlink:href="069/01/234.jpg" pagenum="231"></pb><emph type="italics"></emph>Parabola G D. </s> <s>Let G M be a Mean-proportional betwixt A B and <lb></lb>G L; G M ſhall be the Time, and the Moment or<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>in G of the <lb></lb>Project falling from L, (for it hath been ſuppoſed that A B is the Mea<lb></lb>ſure of the Time and<emph.end type="italics"></emph.end> Impetus.) <emph type="italics"></emph>Again, let G N be a Mean-propor<lb></lb>tional betwixt B C and C G: this G N ſhall be the Meaſure of the <lb></lb>Time and the<emph.end type="italics"></emph.end><lb></lb>Impetus <emph type="italics"></emph>of the<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.234.1.jpg" xlink:href="069/01/234/1.jpg"></figure><lb></lb><emph type="italics"></emph>Project falling <lb></lb>from G to C. <lb></lb></s> <s>If therefore a <lb></lb>Line be drawn <lb></lb>from M to N <lb></lb>it ſhall be the <lb></lb>the Meaſure of <lb></lb>the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>of <lb></lb>the Project a<lb></lb>long the Para<lb></lb>bola B D, ſcri<lb></lb>king in the <lb></lb>term D. Which<emph.end type="italics"></emph.end><lb></lb>Impetus, <emph type="italics"></emph>I ſay, <lb></lb>is greater than the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>of the Project along the Parabola B D, <lb></lb>whoſe quantity was A E. </s> <s>For becauſe G N is ſuppoſed the Mean-pro<lb></lb>portional betwixt B C and C G, and B C is equal to B E, that is to H G; <lb></lb>(for they are each of them ſubduple to D C:) Therefore as C G is to <lb></lb>G N, ſo ſhall N G be to G K: and, as C G or H G is to G K, ſo ſhall the <lb></lb>Square N G be to the Square of G K: But as H G is to G K, ſo was <lb></lb>K G ſuppoſed to be to G L: Therefore as N G is to the Square G K, ſo <lb></lb>is K G to G L: But as K G is to G L, ſo is the Square K G unto the <lb></lb>Square G M, (for G M is the Mean between K G and G L:) Therefore <lb></lb>the three Squares N G, K G, and G M are continual proportionals: And <lb></lb>the two extream ones N G and G M taken together, that is the Square <lb></lb>M N is greater than double the Square K G, to which the Square A E <lb></lb>is double: Therefore the Square M N is greater than the Square A E: <lb></lb>and the Line M N greater than the Line A E: Which was to be de<lb></lb>monſtrated.<emph.end type="italics"></emph.end></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></lb>along the Semiparabola D B, leſs <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> is required than <lb></lb>along any other according to the greater or leſſer Elevation <lb></lb>of the Semiparabola B D, which is according to the Tan<lb></lb>gent A D, containing half a Right-Angle upon the Hori<lb></lb>zon.</s></p><pb xlink:href="069/01/235.jpg" pagenum="232"></pb><p type="head"> <s>COROLLARRY II.</s></p><p type="main"> <s>And that being ſo, it followeth, that if Projections be made with <lb></lb>the ſame <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> out of the Term D, according to ſeveral <lb></lb>Elevations, that ſhall be the greateſt Projection or Amplitude <lb></lb>of the Semiparabola or whole Parabola which followeth at <lb></lb>the Elevation of a ^{*} Semi-Right-Angle; and the reſt, made <lb></lb><arrow.to.target n="marg1101"></arrow.to.target><lb></lb>according to greater or leſſer Angles, ſhall be greater or <lb></lb>leſſer.</s></p><p type="margin"> <s><margin.target id="marg1101"></margin.target>* Or, at the Ele<lb></lb>vation of 45 de<lb></lb>grees.</s></p><p type="main"> <s>SAGR. </s> <s>The ſtrength of Neceſſary Demonſtrations are full of <lb></lb>pleaſure and wonder; and ſuch are only the Mathematical. </s> <s>I un<lb></lb>derſtood before upon truſt from the Relations of ſundry Gunners, <lb></lb>that of all the Ranges of a Cannon, or of a Mortar-piece, the grea<lb></lb>teſt, <emph type="italics"></emph>ſcilicet<emph.end type="italics"></emph.end> that which carryeth the Ball fartheſt was that made at <lb></lb>the Elevation of a Semi-Right-Angle, which they call, of the Sixth <lb></lb>point of the Square: but the knowledge of the Cauſe whence it <lb></lb>hapneth infinitely ſurpaſſeth the bare Notion that I received upon <lb></lb>their atteſtation, and alſo from many repeated Experiments.</s></p><p type="main"> <s>SALV. </s> <s>You ſay very right: and the knowledge of one ſingle <lb></lb>Effect acquired by its Cauſes openeth the Intellect to underſtand <lb></lb>and aſcertain our ſelves of other effects, without need of repairing <lb></lb>unto Experiments, juſt as it hapneth in the preſent Caſe; in which <lb></lb>having found by demonſtrative Diſcourſe the certainty of this, <lb></lb>That the greateſt of all Ranges is that of the Elevation of a Semi<lb></lb>Right-Angle, the Author demonſtrates unto us that which poſſibly <lb></lb>hath not been obſerved by Experience: and that is, that of the <lb></lb>other Ranges thoſe are equal to one another whoſe Elevations ex<lb></lb>ceed or fall ſhort by equal Angles of the Semi-right: ſo that the <lb></lb>Balls ſhot from the Horizon, one according to the Elevation of ſe<lb></lb>ven Points, and the other of 5, ſhall light upon the Horizon at <lb></lb>equal Diſtances: and ſo the Ranges of 8 and of 4 points, of 9 and <lb></lb>of 3, <emph type="italics"></emph>&c.<emph.end type="italics"></emph.end> ſhall be equal. </s> <s>Now hear the Demonſtration of it.</s></p><p type="head"> <s>THEOR. V. PROP. VIII.</s></p><p type="main"> <s>The Amplitudes of Parabola's deſcribed by Pro<lb></lb>jects expulſed with the ſame <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> according <lb></lb>to the Elevations by Angles equidiſtant above, <lb></lb><arrow.to.target n="marg1102"></arrow.to.target><lb></lb>and beneath from the ^{*} Semi-right, are equal to <lb></lb>each other.</s></p><pb xlink:href="069/01/236.jpg" pagenum="233"></pb><p type="margin"> <s><margin.target id="marg1102"></margin.target>* Or Angle of <lb></lb>45.</s></p><p type="main"> <s><emph type="italics"></emph>Of the Triangle M C B, about the Right-Angle C, let the Ho<lb></lb>rizontal Line B C and the Perpendicular C M be equal; for <lb></lb>ſo the Angle M B C ſhall be Semi-right; and prolonging C M <lb></lb>to D, let there be conſtituted in B two equal Angles above and below <lb></lb>the Diagonal M B,<emph.end type="italics"></emph.end> viz. <emph type="italics"></emph>M B E, and M B D. </s> <s>It is to be demonſtrated <lb></lb>that the Amplitudes of the Parabola's deſcribed by the Projects be<lb></lb>ing emitted<emph.end type="italics"></emph.end> [or ſhot off] <emph type="italics"></emph>with the ſame<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>out of the Term B, <lb></lb>according to the Elevations of the Angles E B C and D B C, are equal. <lb></lb></s> <s>For in regard that the extern Angle B M C, is equal to the two intern <lb></lb>M D B and M B D, the Angle M B C ſhall alſo be equal to them. </s> <s>And if <lb></lb>we ſuppoſe M B E inſtead of the Angle M B D, <lb></lb>the ſaid Angle M B C ſhall be equal to the two<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.236.1.jpg" xlink:href="069/01/236/1.jpg"></figure><lb></lb><emph type="italics"></emph>Angles M B E and B D C: And taking away <lb></lb>the common Angle M B E, the remaining An<lb></lb>gle B D C ſhall be equal to the remaining An<lb></lb>gle E B C: Therefore the Triangles D C B <lb></lb>and B C E are alike. </s> <s>Let the Right Lines <lb></lb>D C and E C be divided in the midſt in H and <lb></lb>F; and draw H I and F G parallel to the Ho<lb></lb>rizontal Line C B; and as D H is to H I, ſo <lb></lb>let I H be to H L: the Triangle I H L ſhall be <lb></lb>like to the Triangle I H D, like to which alſo is E G F. </s> <s>And ſeeing <lb></lb>that I H and G F are equal (to wit, halves of the ſame B C:) There<lb></lb>fore F E, that is F C, ſhall be equal to H L: And, adding the common <lb></lb>Line F H, C H ſhall be equal to F L. </s> <s>If therefore we underſtand the Se<lb></lb>miparabola to be deſcribed along by H and B, whoſe Altitude ſhall be <lb></lb>H C, and Sublimity H L, its Amplitude ſhall be C B, which is double <lb></lb>to HI, that is, the Mean betwixt D H, or C H, and HL: And D B <lb></lb>ſhall be a Tangent to it, the Lines C H and H D being equal. </s> <s>And if, <lb></lb>again, we conceive the Parabola to be deſcribed along by F and B from <lb></lb>the Sublimity FL, with the Altitude F C, betwixt which the Mean<lb></lb>proportional is F G, whoſe double is the Horizontal Line C B: C B, as <lb></lb>before, ſhall be its Amplitude; and E B a Tangent to it, ſince E F and <lb></lb>F C are equal: But the Angles D B C and E B C<emph.end type="italics"></emph.end> (ſcilicet, <emph type="italics"></emph>their Eleva<lb></lb>tions) ſhall be equidiſtant from the Semi-Right Angle: Therefore the <lb></lb>Propoſition is demonſtrated.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>THEOR. VI. <emph type="italics"></emph>P<emph.end type="italics"></emph.end>RO<emph type="italics"></emph>P.<emph.end type="italics"></emph.end> IX.</s></p><p type="main"> <s>The Amplitudes of Parabola's, whoſe Altitudes <lb></lb>and Sublimities anſwer to each other <emph type="italics"></emph>è contra<lb></lb>rio,<emph.end type="italics"></emph.end> are equall.</s></p><pb xlink:href="069/01/237.jpg" pagenum="234"></pb><p type="main"> <s><emph type="italics"></emph>Let the Altitude G F of the Parabola F H have the ſame proporti<lb></lb>on to the Altitude C B of the Parabola B D, as the Sublimity B A <lb></lb>hath to the Sublimity F E. </s> <s>I ſay, that the Amplitude H G is equal <lb></lb>to the Amplitude D C. </s> <s>For ſince the firſt G F hath the ſame propor<lb></lb>tion to the ſecond C B, as the third B A hath to the fourth F E; There<lb></lb>fore, the Rectangle<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.237.1.jpg" xlink:href="069/01/237/1.jpg"></figure><lb></lb><emph type="italics"></emph>G F E of the firſt and <lb></lb>fourth, ſhall be equal to <lb></lb>the Rectangle C B A <lb></lb>of the ſecond and <lb></lb>third: Therefore the <lb></lb>Squares that are equal <lb></lb>to theſe Rectangles ſhall <lb></lb>be equal to one another: <lb></lb>But the Square of half of G H is equal to the Rectangle G F E; and <lb></lb>the Square of half of C D is equal to the Rectangle C B A: There<lb></lb>fore theſe Squares, and their Sides, and the doubles of their Sides ſhall <lb></lb>be equal: But theſe are the Amplitudes G H and C D: Therefore the <lb></lb>Propoſition is manifeſt.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>LEMMA <emph type="italics"></emph>pro ſequenti.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>If a Right Line be cut according to any proportion, the Squares <lb></lb>of the Mean-proportionals between the whole and the two <lb></lb>parts are equal to the Square of the whole.</s></p><p type="main"> <s><emph type="italics"></emph>Let A B be cut according to any proportion in C. </s> <s>I ſay, that the <lb></lb>Squares of the Mean-proportional Lines between the whole A B and <lb></lb>the parts A C and C B, being taken together are equal to the Square of <lb></lb>the whole A B. </s> <s>And this appeareth, a Semi-<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.237.2.jpg" xlink:href="069/01/237/2.jpg"></figure><lb></lb><emph type="italics"></emph>circle being deſcribed upon the whole Line <lb></lb>B A, and from C a Perpendicular being ere<lb></lb>cted C D, and Lines being drawn from D to <lb></lb>A, and from D to B. </s> <s>For D A is the Mean<lb></lb>proportional betwixt A B and A C; and D B is the Mean-proporti<lb></lb>onal between A B and B C: And the Squares of the Lines D A and <lb></lb>D B taken together are equal to the Square of the whole Line A B, <lb></lb>the Angle A D B in the Semicircle being a Right-Angle: Therefore <lb></lb>the Propoſition is manifest.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/238.jpg" pagenum="235"></pb><p type="head"> <s>THEOR. VII. PROP. X.</s></p><p type="main"> <s>The <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> or Moment of any Semiparabola is <lb></lb>equal to the Moment of any Moveable falling <lb></lb>naturally along the Perpendicular to the Ho<lb></lb>rizon that is equal to the Line compounded of <lb></lb>the Sublimity and of the Altitude of the Se<lb></lb>miparabola.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Semiparabola be A B, its Sublimity D A, and Altitude <lb></lb>A C, of which the Perpendicular D C is compounded. </s> <s>I ſay, that <lb></lb>the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>of the Semiparabola in B is equal to the Moment of <lb></lb>the Moveable Naturally falling from D to C. </s> <s>Suppoſe D C it ſelf to be <lb></lb>the Meaſure of the Time and of the<emph.end type="italics"></emph.end> Impetus; <emph type="italics"></emph>and take a Mean-pro<lb></lb>portional betwixt C D and D A, to which let<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.238.1.jpg" xlink:href="069/01/238/1.jpg"></figure><lb></lb><emph type="italics"></emph>C F be equal; and withal let C E be a Mean<lb></lb>proportional between D C and C A: Now C F <lb></lb>ſhall be the Meaſure of the Time and of the Mo<lb></lb>ment of the Moveable ſalling along D A out of <lb></lb>Reſt in D; and C E ſhall be the Time and Mo<lb></lb>ment of the Moveable falling along A C, out of <lb></lb>Reſt in A, and the Moment of the Diagonal E F <lb></lb>ſhall be that compounded of both the others,<emph.end type="italics"></emph.end> ſcil. <lb></lb><emph type="italics"></emph>that of the Semiparabola in B. </s> <s>And becauſe <lb></lb>D C is cut according to any proportion in A, and becauſe C F and C E <lb></lb>are Mean-Proportionals between C D and the parts D A and A C; the <lb></lb>Squares of them taken together ſhall be equal to the Square of the <lb></lb>whole; by the Lemma aforegoing: But the Squares of them are alſo <lb></lb>equal to the Square of E F: Therefore D F is equal alſo to the Line D C: <lb></lb>Whence it is manifeſt that the Moments along D C, and along the Se<lb></lb>miparabola A B, are equal in C and B: Which was required.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>COROLLARY.</s></p><p type="main"> <s>Hence it is manifeſt, that of all Parabola's whoſe Altitudes and <lb></lb>Sublimities being joyned together are equal, the <emph type="italics"></emph>Impetus's<emph.end type="italics"></emph.end> are <lb></lb>alſo equal.</s></p><pb xlink:href="069/01/239.jpg" pagenum="236"></pb><p type="head"> <s>PROBL. IV. PROP. XI.</s></p><p type="main"> <s>The <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> and Amplitude of a Semiparabola be<lb></lb>ing given, to find its Altitude, and conſequently <lb></lb>its Sublimity.</s></p><p type="main"> <s><emph type="italics"></emph>Let the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>given be defined by the Perpendicular to the Ho<lb></lb>rizon A B; and let the Amplitude along the Horizontal Line be <lb></lb>B C. </s> <s>It is required to find the Altitude and Sublimity of the <lb></lb>Parabola whoſe<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>is A B, and Amplitude B C. </s> <s>It is manifeſt, <lb></lb>from what hath been already demonſtrated, that half the Amplitude B C <lb></lb>will be a Mean-proportional betwixt the Altitude and the Sublimity of <lb></lb>the ſaid Semiparabola, whoſe<emph.end type="italics"></emph.end> Impetus, <emph type="italics"></emph>by the precedent Propoſition, is <lb></lb>the ſame with the<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>of the Moveable falling from Reſt in A along <lb></lb>the whole Perpendicular A B: Wherefore B A is ſo to be cut that the <lb></lb>Rectangle contained by its parts may be equal to the Square of half of <lb></lb>B C, which let be B D. </s> <s>Hence it appeareth <lb></lb>to be neceſſary that D B do not exceed the<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.239.1.jpg" xlink:href="069/01/239/1.jpg"></figure><lb></lb><emph type="italics"></emph>half of B A; for of Rectangles contained by <lb></lb>the parts the greateſt is when the whole <lb></lb>Line is cut into two equal parts. </s> <s>Therefore <lb></lb>let B A be divided into two equal parts in E. <lb></lb></s> <s>And if B D be equal to B E the work is <lb></lb>done; and the Altitude of the Semipara<lb></lb>bola ſhall be B E, and its Sublimity E A: <lb></lb>(and ſee here by the way that the Amplitude <lb></lb>of the Parabola of a Semi-right Elevation, <lb></lb>as was demonſtrated above, is the greateſt of <lb></lb>all thoſe deſcribed with the ſame<emph.end type="italics"></emph.end> Impetus.) <lb></lb><emph type="italics"></emph>But let B D be leſs than the half of B A, <lb></lb>which is ſo to be cut that the Rectangle under the parts may be equal to <lb></lb>the Square B D. </s> <s>Upon E A deſcribe a Semicircle, upon which out of A <lb></lb>ſet off A F equal to B D, and draw a Line from F to E, to which cut <lb></lb>a part equal E G. </s> <s>Now the Rectangle B G A, together with the Square <lb></lb>E G, ſhall be equal to the Square E A; to which the two Squares A F <lb></lb>and F E are alſo equal: Therefore the equal Squares G E and F E be<lb></lb>ing ſubſtracted, there remaineth the Rectangle B G A equal to the <lb></lb>Square A F,<emph.end type="italics"></emph.end> ſcilicet, <emph type="italics"></emph>to B D; and the Line B D is a Mean-proportional <lb></lb>betwixt B G and G A. </s> <s>Whence it appeareth, that of the Semipa<lb></lb>rabola whoſe Amplitude is B C, and<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>A B, the Altitude is <lb></lb>B G, and the Sublimity G A. </s> <s>And if we ſet off B I below equal to G A, <lb></lb>this ſhall be the Altitude, and I A the Sublimity of the Semiparabola <lb></lb>I C. </s> <s>From what hath been already demonſtrated we are able,<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/240.jpg" pagenum="237"></pb><p type="head"> <s>PROBL. V. PROP. XII.</s></p><p type="main"> <s>To collect by Calculation of the Amplitudes of all <lb></lb>Semiparabola's that are deſcribed by Projects <lb></lb>expulſed with the ſame <emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> and to make <lb></lb>Tables thereof.</s></p><p type="main"> <s><emph type="italics"></emph>It is obvious, from the things demonſtrated, that Parabola's are de<lb></lb>ſcribed by Projects of the ſame<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>then, when their Subli<lb></lb>mities together with their Altitudes do make up equal Perpendicu<lb></lb>lars upon the Horizon. </s> <s>Theſe Perpendiculars therefore are to be com<lb></lb>prehended between the ſame Horizontal Parallels. </s> <s>Therefore let the <lb></lb>Horizontal Line C B be ſuppoſed equal to the Perpendicular B A, and <lb></lb>draw the Diagonal from A to C. </s> <s>The Angle A C B ſhall be Semi<lb></lb>right, or 45 Degrees. </s> <s>And the Perpendicular B A being divided into <lb></lb>two equal parts in D, the Semiparabola D C ſhall be that which is de<lb></lb>ſcribed from the Sublimity A D together with the Altitude D B: and <lb></lb>its<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>in C ſhall be as great as that of the Moveable coming out of <lb></lb>Reſt in A along the Perpendicular A B is in B. </s> <s>And if A G be drawn <lb></lb>parallel to B C, the united Altitudes and Sublimities of all other re<lb></lb>maining Semiparabola's whoſe future<emph.end type="italics"></emph.end> Impetus's <emph type="italics"></emph>are the ſame with thoſe <lb></lb>now mentioned muſt be bounded by the Space between the Parallels<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.240.1.jpg" xlink:href="069/01/240/1.jpg"></figure><lb></lb><emph type="italics"></emph>A G and B C. Farthermore, it having <lb></lb>been but now demonſtrated, that the Am<lb></lb>plitudes of the Semiparabola's whoſe <lb></lb>Tangents are equidiſtant either above or <lb></lb>below from the Semi right Elevation are <lb></lb>equal, the Calculations that we frame <lb></lb>for the greater Elevations will likewiſe <lb></lb>ſerve for the leſſer. </s> <s>We chooſe moreover <lb></lb>a number of ten thouſand parts for the <lb></lb>greateſt Amplitude of the Projection of <lb></lb>the Semiparabola made at the Elevation <lb></lb>of 45 degrees: ſo much therefore the Line <lb></lb>B A, and the Amplitude of the Semipa<lb></lb>rabola B C, are to be ſuppoſed. </s> <s>And we <lb></lb>make choice of the number 10000, becauſe we in our Calculation uſe <lb></lb>the Table of Tangents, in which this number agreeth with the Tangent <lb></lb>of 45 degrees. </s> <s>Now, to come to the buſineſs, let C E be drawn, contain<lb></lb>ing the Angle E C B greater (Acute nevertheleſs,) than the Angle <lb></lb>A C B; and let the Semiparabola be deſcribed which is touched by the <lb></lb>Line E C, and whoſe Sublimity united with its Altitude is equal to <lb></lb>B A. </s> <s>In the Table of Tangents take the ſaid B E for the Tangent at the<emph.end type="italics"></emph.end><pb xlink:href="069/01/241.jpg" pagenum="238"></pb><emph type="italics"></emph>given Angle B C E, which divide into two equal parts at F. </s> <s>Then <lb></lb>find a third Proportional to B F and B C, (or to the half of B C,) <lb></lb>which ſhall of neceſſity be greater than F A; therefore let it be F O: <lb></lb>Of the Semiparabola, therefore, inſcribed in the Triangle E C B, ac<lb></lb>cording to the Tangent C E, whoſe Amplitude is C B, the Altitude B F, <lb></lb>and the Sublimity F O is found: But the whole Line B O riſeth above <lb></lb>the Parallels A G and C B, whereas our work was to bound it between <lb></lb>them: For ſo both it and the Semiparabola D C ſhall be deſcribed by <lb></lb>the Projects out of C expelled with the ſame<emph.end type="italics"></emph.end> Impetus. <emph type="italics"></emph>Therefore we <lb></lb>are to ſeek another like to this, (for innumerable greater and ſmaller, <lb></lb>like to one another, may be deſcribed within the Angle B C E) to whoſe <lb></lb>united Sublimity and Altitude B A ſhall be equal. </s> <s>Therefore as O B is <lb></lb>to B A, ſo let the Amplitude B C be to C R: and C R ſhall be found,<emph.end type="italics"></emph.end><lb></lb>ſcilicet <emph type="italics"></emph>the Amplitude of the Semiparabola according to the Elevation <lb></lb>of the Angle B C E, whoſe conjoyned Sublimity and Altitude is equal <lb></lb>to the Space contained between the Parallels G A and C B: Which <lb></lb>was required. </s> <s>The work, therefore, ſhall be after this manner.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Take the Tangent of the given Angle B C E, to the half of which <lb></lb>add the third Proportional of it, and half of B C, which let be F O: <lb></lb>Then as O B is to B A, ſo let B C be to another, which let be C R, to wit, <lb></lb>the Amplitude ſought. </s> <s>Let us give an Example.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Let the Angle E C B be 50 degrees, its Tangent ſhall be 11918, <lb></lb>whoſe half, to wit, B F, is 5959, and the half of B C is 5000, the third <lb></lb>proportional of theſe halves is 4195, which added to the ſaid B F <lb></lb>maketh 10154: for the ſaid B O. Again, as O B is to B A, that is as <lb></lb>10154 is to 10000, ſo is B E, that is 10000 (for each of them is the <lb></lb>Tangent of 45 degrees) to another: and that ſhall give us the required <lb></lb>Altitude R C 9848, of ſuch as B C (the greateſt Amplitude) is <lb></lb>10000. To theſe the Amplitudes of the whole Parabola's are double,<emph.end type="italics"></emph.end><lb></lb>ſcilicet <emph type="italics"></emph>19696 and 20000. And ſo much likewiſe is the Amplitude of <lb></lb>the Parabola according to the Elevation of 40 degrees, ſince it is equal<lb></lb>ly diſtant from 45 degrees.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>For the perfect underſtanding of this Demonſtration I <lb></lb>muſt be informed how true it is, that the Third Proportional to <lb></lb>B F and B I, is (as the Author ſaith) neceſſarily greater than <lb></lb>F A.</s></p><p type="main"> <s>SALV. </s> <s>That inference, as I conceive, may be deduced thus. <lb></lb></s> <s>The Square of the Mean of three proportional Lines is equal to <lb></lb>the Rectangle of the other two: whence the Square of B I, or of <lb></lb>B D equal to it, ought to be equal to the Rectangle of the firſt F B <lb></lb>multiplied into the third to be found: which third is of neceſſity to <lb></lb>be greater than F A, becauſe the Rectangle of B F multiplied into <lb></lb>F A is leſs than the Square B D: and the Defect is as much as the <lb></lb>Square of D F, as <emph type="italics"></emph>Euclid<emph.end type="italics"></emph.end> demonſtrates in a Propoſition of his <pb xlink:href="069/01/242.jpg" pagenum="239"></pb>Second Book. </s> <s>You muſt alſo know, that the point F which divi<lb></lb>deth the Tangent E B in the middle, will many other times fall <lb></lb>above the point A, and once alſo in the ſaid A: In which caſes it is <lb></lb>evident of it ſelf, that the third proportional to the half of the Tan<lb></lb>gent, and to B I (which giveth the Sublimity) is all above A. </s> <s>But <lb></lb>the Author hath taken a Caſe in which it was not manifeſt that the <lb></lb>ſaid third Proportional is alwaies greater than F A: and which <lb></lb>therefore being ſet off above the point F paſſeth beyond the Paral<lb></lb>lel A G. </s> <s>Now let us proceed.</s></p><p type="main"> <s><emph type="italics"></emph>It will not be unprofitable if by help of this Table we compoſe ano<lb></lb>ther, ſhewing the Altitudes of the ſame Semiparabola's of Projects of <lb></lb>the ſame<emph.end type="italics"></emph.end> Impetus. <emph type="italics"></emph>And the Conſtruction of it is in this manner.<emph.end type="italics"></emph.end></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></lb>the following Table ſet down, keeping the <lb></lb>common <emph type="italics"></emph>Impeius<emph.end type="italics"></emph.end> with which every one of <lb></lb>them is deſcribed, to compute the Altitudes of <lb></lb>each ſeveral Semiparabola.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Amplitude given be B C, and of the<emph.end type="italics"></emph.end> Impetus, <emph type="italics"></emph>which is <lb></lb>ſuppoſed to be alwaies the ſame, let the Meaſure be O B, to wit, <lb></lb>the Aggregate of the Altitude and Sublimity. </s> <s>The ſaid Altitude <lb></lb>is required to be found and diſtinguiſhed. </s> <s>Which ſhall then be done when <lb></lb>B O is ſo divided as that the Rectangle contained under its parts is <lb></lb>equal to the Square of half the Amplitude B C. </s> <s>Let that ſame divi<lb></lb>ſion fall in F; and let both O B and B C be cut in the midſt at D and I.<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.242.1.jpg" xlink:href="069/01/242/1.jpg"></figure><lb></lb><emph type="italics"></emph>The Square I B, therefore, is equal to the <lb></lb>Rectangle B F O: And the Square D O is <lb></lb>equal to the ſame Rectangle together with the <lb></lb>Square F D. </s> <s>If therefore from the Square <lb></lb>D O we deduct the Square B I, which is equal <lb></lb>to the Rectangle B F O, there ſhall remain <lb></lb>the Square F D; to whoſe Side D F, B D be<lb></lb>ing added it ſhall give the deſired Altitude <lb></lb>Altitude B F. </s> <s>And it is thus compounded<emph.end type="italics"></emph.end><lb></lb>ex datis. <emph type="italics"></emph>From half of the Square B O known <lb></lb>ſubſtract the Square B I alſo known, of the remainder take the Square <lb></lb>Root, to which add D B known; and you ſhall have the Altitude ſought <lb></lb>B F. </s> <s>For example. </s> <s>The Altitude of the Parabola deſcribed at the <lb></lb>Elevation of 55 degrees is to be found. </s> <s>The Amplitude, by the follow<lb></lb>ing Table is 9396, its half is 4698, the Square of that is 22071204,<emph.end type="italics"></emph.end><pb xlink:href="069/01/243.jpg" pagenum="240"></pb><emph type="italics"></emph>this ſubſtracted from the Square of the half B O, which is alwaies <lb></lb>the ſame, to wit, 2500000, the remainder is 2928796, whoſe Square <lb></lb>Root is 1710 very near, this added to the half of B O, to wit, 5000, <lb></lb>gives 67101, and ſo much is the Altitude B F. </s> <s>It will not be unprofi<lb></lb>table, to give the Third Table, containing the Altitudes and Sublimi<lb></lb>ties of Semiparabola's, whoſe Amplitude ſhall be alwaies the ſame.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>This I would very gladly ſee ſince by it I may come to <lb></lb>know the Difference of the <emph type="italics"></emph>Impetus's,<emph.end type="italics"></emph.end> and of the Forces that are <lb></lb>required for carrying the Project to the ſame Diſtance with Ranges <lb></lb>which are called at Random: which Difference I believe is very <lb></lb>great according to the different Elevations [<emph type="italics"></emph>or Mountures:<emph.end type="italics"></emph.end>] ſo that <lb></lb>if, for example, one would at the Elevation of 3 or 4 degrees, or of <lb></lb>87 or 88 make the Ball to fall where it did, being ſhot at the Ele<lb></lb>vation of <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end> 45. (where, as hath been ſhewn, the leaſt <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> is <lb></lb>required) I believe that it would require a very much greater <lb></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></lb>full execution in all the Elevations it is requiſite to make great Pro<lb></lb>greſſions towards an infinite <emph type="italics"></emph>Impetus.<emph.end type="italics"></emph.end> Now let us ſee the Conſtru<lb></lb>ction of the Table.<pb xlink:href="069/01/244.jpg" pagenum="241"></pb><arrow.to.target n="table75"></arrow.to.target><lb></lb><arrow.to.target n="table76"></arrow.to.target><lb></lb><arrow.to.target n="table77"></arrow.to.target></s></p><pb xlink:href="069/01/245.jpg" pagenum="242"></pb><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, deſcribed with the ſame <emph type="italics"></emph>Impetus.<emph.end type="italics"></emph.end></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, whoſe <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> is the ſame.</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, whoſe Am-plitudes are the ſame, that is to ſay, 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></lb>rabola's whoſe Amplitudes ſhall be equal for <lb></lb>each degree of Elevation.</s></p><p type="main"> <s><emph type="italics"></emph>This we ſhall eaſily do. </s> <s>For ſuppoſing the Amplitude of the Semi<lb></lb>par abola to be of 10000 parts, the half of the Tangent of each <lb></lb>degree of Elevation ſhews the Altitude. </s> <s>As for example, of the <lb></lb>Semiparabola whoſe Elevation is 30 degrees, and Amplitude, as is <lb></lb>ſuppoſed, 10000 parts, the Altitude ſhall be 2887, for ſo much, very <lb></lb>near, is the half of the Tangent. </s> <s>And having found the Altitude the <lb></lb>Sublimity is to be known in this manner. </s> <s>For aſmuch as it hath been <lb></lb>demonſtrated that the half of the Amplitude of a Semiparabola is the <lb></lb>Mean proportional betwixt the Altitude and Sublimity, and the Alti<lb></lb>tude being already found, and the half of the Amplitude being alwaies <lb></lb>the ſame, to wit, 5000 parts, if we ſhall divide the Square thereof by <lb></lb>the Altitude found, the deſired Sublimity ſhall come forth. </s> <s>As in the <lb></lb>Example: The Altitude found was 2887; The Square of the 5000 <lb></lb>parts is 25000000; which being divided by 2887, giveth 8659, ve<lb></lb>ry near, for the Sublimity ſought.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SALV. </s> <s>Now here we ſee, in the ſirſt place, that the Conje<lb></lb>cture is very true which was mentioned afore, that in different <lb></lb>Elevations the farther one goeth from the middlemoſt, whether it <lb></lb>be in the Higher, or in the Lower, ſo much greater <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> and Vio<lb></lb>lence is required to carry the Project to the ſame Diſtance. </s> <s>For the <lb></lb><emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> lying in the mixture of the two Motions, Equable, Hori<lb></lb>zontal, and Perpendicular Naturally-Accelerate, of which <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end><lb></lb>the Aggregate of the Altitude and Sublimity is the Meaſure, we do <lb></lb>ſee in the propounded Table that that ſame Aggregate is leaſt in <lb></lb>the Elevation of <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end> 45, in which the Altitude and Sublimity are <lb></lb>equal, <emph type="italics"></emph>ſcilicet<emph.end type="italics"></emph.end> each 5000, and their Aggregate 10000. But if we <lb></lb>ſhould look on any greater Elevation, as, for example, of <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end> 50, we <lb></lb>ſhould ſind the Altitude to be 5959, and the Sublimity 4196, which <lb></lb>added together make 10155. And ſo much alſo we ſhould find the <lb></lb><emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end> 40 to be, this and that Elevation being equally re<lb></lb>mote from the middlemoſt. </s> <s>Where we are to note, in the ſecond <lb></lb>place, that it is true, That equal <emph type="italics"></emph>Impetus's<emph.end type="italics"></emph.end> are ſought by two, and <lb></lb>two in the Elevations equidiſtant from the middlemoſt, with this <lb></lb>pretty variation over and above that the Altitudes and the Subli<lb></lb>mities of the ^{*} ſuperiour Elevations anſwer alternally to the Sub<lb></lb><arrow.to.target n="marg1103"></arrow.to.target><lb></lb>limities and Altitudes of the Inferiour: ſo that whereas in the <pb xlink:href="069/01/246.jpg" pagenum="243"></pb>example propoſed, in the Elevation of <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end> 50. the Altitude is 5959 <lb></lb>and the Sublimity 4196, in the Elevation of <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end> 40. it falls out on <lb></lb>the contrary that the Altitude is 4196, and the Sublimity 5959: <lb></lb>And the ſame happens in all others without any difference; ſave <lb></lb>only that for the avoyding of tediouſneſs in Calculations we have <lb></lb>kept no account of ſome fractions, which in ſo great ſums are of no <lb></lb>value, but may without any prejudice be omitted.</s></p><p type="margin"> <s><margin.target id="marg1103"></margin.target>* <emph type="italics"></emph>i.e.<emph.end type="italics"></emph.end> Thoſe above <lb></lb>45 deg.</s></p><p type="main"> <s>SAGR. </s> <s>I am obſerving that of the two <emph type="italics"></emph>Impetus's<emph.end type="italics"></emph.end> Horizontal and <lb></lb>Perpendicular in Projections, the more Sublime they are, they need <lb></lb>ſo much the leſs of the Horizontal, and the more of the Perpendi<lb></lb>cular. </s> <s>Moreover in thoſe of ſmall Elevation, great muſt be the <lb></lb>Force of the Horizontal <emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> which is to carry the Project in a <lb></lb>little Altitude. </s> <s>But although I comprehend very well that in the <lb></lb>Total Elevation of <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end> 90, all the force in the world ſufficeth not <lb></lb>to drive the Project one ſingle Inch from the Perpendicular, but <lb></lb>that it muſt of neceſſity fall in the ſame place whence it was expel<lb></lb>led; yet dare I not with the like certainty affirm that likewiſe in the <lb></lb>nullity of Elevation, that is in the Horizontal Line, the Project <lb></lb>cannot by any Force leſs than infinite, be driven to any di<lb></lb>ſtance: So, as that, for example, a Culverin it ſelf ſhould not be <lb></lb>able to carry a Ball of Iron Horizontally, or, as they ſay, at Point <lb></lb>blank, that is at no point, which is when it hath no Elevation. </s> <s>I <lb></lb>ſay, in this caſe I ſtand in ſome doubt; and that I do not reſolute<lb></lb>ly deny the thing, the reaſon depends on another Accident which <lb></lb>ſeems no leſs ſtrange, and yet I have a very neceſſary Demonſtrati<lb></lb>on for it. </s> <s>And the Accident is this, the Impoſſibility of diſtending <lb></lb>a Rope, ſo, as that it may be ſtretched right out, and parallel to the <lb></lb>Horizon, but that it alwaies ſwayes and bendeth, nor is there any <lb></lb>Force that can ſtretch it otherwiſe.</s></p><p type="main"> <s>SALV. </s> <s>So then, <emph type="italics"></emph>Sagredus,<emph.end type="italics"></emph.end> your wonder ceaſeth in this caſe of <lb></lb>the Rope becauſe you have the Demonſtration of it. </s> <s>But if we <lb></lb>ſhall well conſider the matter, it may be we ſhall find ſome corre<lb></lb>ſpondence between the Accident of the Project and this of the <lb></lb>Rope. </s> <s>The Curvity of the Line of the Horizontal Projection ſeem<lb></lb>eth to be derived from two Forces, of which one, (which is that of <lb></lb>the Projicient) driveth it Horizontally, and the other, (which <lb></lb>is the Gravity of the Project) draweth it downwards Perpendicu<lb></lb>larly. </s> <s>Now ſo in the ſtretching of the Rope, there are the Forces <lb></lb>of thoſe that pull it Horizontally, and there is alſo the weight of <lb></lb>the Rope it ſelf, which naturally inclineth it downwards. </s> <s>Theſe <lb></lb>two effects are very much alike in the generation of them. </s> <s>And if <lb></lb>you allow the weight of the Rope ſo much ſtrength and power as to <lb></lb>be able to oppoſe and overcome any whatever Immenſe Force, that <lb></lb>would diſtend it right out, why will you deny the like to the weight <lb></lb>of the Bullet? </s> <s>But beſides, I ſhall tell you, and at once procure your <pb xlink:href="069/01/247.jpg" pagenum="244"></pb>wonder, and delight, that the Rope thus tentered, and ſtretcht little <lb></lb>or much, doth ſhape it ſelf into Lines that come very near to Para<lb></lb>bolical, and the reſemblance is ſo great, that if you draw a Para<lb></lb>bolical Line upon a plain Superficies that is erect unto the Horizon, <lb></lb>and holding it reverſed, that is with the Vertex downwards and <lb></lb>with the Baſe Parallel to the Horizon, you cauſe a Chain to be held <lb></lb>pendent, and ſuſtained at the extreams of the Baſe of the Deſcribed <lb></lb>Parabola, you ſhall ſee the ſaid Chain, as you ſlaken it more or leſs, <lb></lb>to incurvate and apply it ſelf to the ſame Parabola, and this ſame <lb></lb>Application ſhall be ſo much the more exact, when the deſcribed <lb></lb>Parabola is leſs curved, that is more diſtended: So that in Parabola's <lb></lb>deſcribed with Elevations under <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end> 45, the Chain anſwereth the <lb></lb>Parabola almoſt to an hair.</s></p><p type="main"> <s>SAGR. </s> <s>It ſeems then that with ſuch a Chain wrought into ſmall <lb></lb>Links one might in an inſtant trace out many Parabolick Lines up<lb></lb>on a plain Superficies.</s></p><p type="main"> <s>SALV. </s> <s>One might, and that alſo with no ſmall commodity, as I <lb></lb>ſhall tell you anon.</s></p><p type="main"> <s>SIMP. </s> <s>But before you paſs any farther, I alſo would gladly be <lb></lb>aſcertained at leaſt in that Propoſition of which you ſay there is a <lb></lb>very neceſſary Demonſtration, I mean that of the Impoſſibility of <lb></lb>diſtending a Rope, by any whatever immenſe Force, right out and <lb></lb>equidiſtant from the Horizon.</s></p><p type="main"> <s>SAGR. </s> <s>I will ſee if I remember the Demonſtration, for under<lb></lb>ſtanding of which it is neceſſary, <emph type="italics"></emph>Simplicius,<emph.end type="italics"></emph.end> that you ſuppoſe for <lb></lb>true, that which in all Mechanick Inſtruments is confirmed, not on<lb></lb>ly by Experience, but alſo by Demonſtration: and this it is, That <lb></lb>the Velocity of the Mover, though its Force be very ſmall, may <lb></lb>overcome the Reſiſtance, though very great, of a Reſiſter, which <lb></lb>muſt be moved ſlowly when ever the Velocity of the Mover hath <lb></lb>greater proportion to the Tardity of the Reſiſter, than the Reſi<lb></lb>ſtance of that which is to be moved hath to the Force of the Mo<lb></lb>ver.</s></p><p type="main"> <s>SIMP. </s> <s>This I know very well, and it is demonſtrated by <emph type="italics"></emph>Ari<lb></lb>ſtotle<emph.end type="italics"></emph.end> in his Mechanical Queſtions, and is manifeſtly ſeen in the Lea<lb></lb>ver and in the Stiliard, in which the Roman which weigheth not <lb></lb>above 4 pounds, will lift up a weight of 400 in caſe the diſtance of <lb></lb>the ſaid Roman from the Center on which the Beam turneth be <lb></lb>more than an hundred times greater than the diſtance of that point <lb></lb>at which the great weight hangeth from the ſame Center: and this <lb></lb>cometh to paſs becauſe in the deſcent which the Roman maketh <lb></lb>paſſeth a Space above an hundred times greater than the Space <lb></lb>which the great weight mounteth in the ſame Time: Which is all <lb></lb>one as to ſay, that the little Roman moveth with a Velocity above <lb></lb>an hundred times greater than the Velocity of the great Weight.</s></p><pb xlink:href="069/01/248.jpg" pagenum="245"></pb><p type="main"> <s>SAGR. </s> <s>You argue very well, and make no ſeruple at all of <lb></lb>granting, that be the Force of the Mover never ſo ſmall it ſhall ſu<lb></lb>perate any what ever great Reſiſtance at all times when that ſhall <lb></lb>more exceed in Velocity than this doth in Force and Gravity. <lb></lb></s> <s>Now come we to the caſe of the Rope. </s> <s>And drawing a ſmall <lb></lb>Scheme be pleaſed to underſtand for once that this Line A B, reſt<lb></lb>ing upon the two fixed and ſtanding points A and B, to have hang<lb></lb>ing at its ends, as you ſee, two immenſe Weights C and D, which <lb></lb>drawing it with great Force make it to ſtand directly diſtended, it <lb></lb>being a ſimple Line without any gravity. </s> <s>And here I proceed, and <lb></lb>tell you, that if at the midſt of that which is the point E, you ſhould <lb></lb>hang any never ſo little a Weight, as is this H, the Line A B would <lb></lb>yield, and inclining towards the point F, and by conſequence <lb></lb>lengthening, will conſtrain the two great Weights C and D to <lb></lb>aſcend upwards: which I demonſtrate to you in this manner: <lb></lb>About the two points A and B as Centers I deſcribe two Quadrants <lb></lb>E F G, and E L M, and in regard that the two Semidiameters AI <lb></lb>and B L are equal to the two Semidiameters A E and E B, the exceſ<lb></lb>ſes F I and F L ſhall be the quantity of the prolongations of the <lb></lb>parts A F and F B, above A E and E B; and of conſequence ſhall <lb></lb><figure id="id.069.01.248.1.jpg" xlink:href="069/01/248/1.jpg"></figure><lb></lb>determine the Aſcents <lb></lb>of the Weights C and <lb></lb>D, in caſe that the <lb></lb>Weight H had had a <lb></lb>power to deſcend to F: <lb></lb>which might then be <lb></lb>in caſe the Line E F, <lb></lb>which is the quantity <lb></lb>of the Deſcent of the <lb></lb>ſaid Weight H, had <lb></lb>greater proportion to <lb></lb>the Line F I which de<lb></lb>termineth the Aſcent of <lb></lb>the two Weights C & <lb></lb>D, than the pondero<lb></lb>ſity of both thoſe Weights hath to the ponderoſity of the Weight <lb></lb>H. </s> <s>But this will neceſſarily happen, be the ponderoſity of the <lb></lb>Weights C and D never ſo great, and that of H never ſo ſmall; for <lb></lb>the exceſs of the Weights C and D above the Weight His not ſo <lb></lb>great, but that the exceſs of the Tangent E F above the part of the <lb></lb>Secant F I may bear a greater proportion. </s> <s>Which we will prove <lb></lb>thus: Let there be a Circle whoſe Diameter is G A I; and look <lb></lb>what proportion the ponderoſity of the Weights C and D have to <lb></lb>the ponderoſity of H, let the Line B O have the ſame proportion to <lb></lb>another, which let be C, than which let D be leſſer: So that B O <pb xlink:href="069/01/249.jpg" pagenum="246"></pb>ſhall have greater proportion to D, than to C. </s> <s>Unto O B and D <lb></lb>take a third proportional B E; and as O E is to E B, ſo let the Dia<lb></lb>meter G I (prolonging it) be to I F: and from the Term F <lb></lb>draw the Tangent F N. </s> <s>And becauſe it hath been preſuppoſed, <lb></lb>that as O E is to E B, ſo is G I to I F: therefore, by Compoſition, as <lb></lb>O B is to B E, ſo is G F to F I: But betwixt O B and B E the Mean<lb></lb>proportional is D; and betwixt G F and F I the Mean-proporti<lb></lb>onal is N F: Therefore N F hath the ſame proportion to F I that <lb></lb>O B hath to D: which proportion is greater than that of the <lb></lb>Weights C and D to the Weight H. Therefore, the Deſcent or <lb></lb>Velocity of the Weight H having greater proportion to the Aſcent <lb></lb>or Velocity of the Weights C and D, than the ponderoſity of the <lb></lb>ſaid Weights C and D hath to the ponderoſity of the Weight H: <lb></lb>It is manifeſt, that the Weight H ſhall deſcend, that is, that the <lb></lb>Line A B ſhall depart from Horizontal Rectitude. </s> <s>And that which <lb></lb>befalleth the right Line A B deprived of Gravity in caſe any ſmall <lb></lb>Weight H cometh to be hanged at the ſame in E, happens alſo to <lb></lb>the ſaid Rope A B, ſuppoſed to be of ponderous Matter, without <lb></lb>the addition of any other Grave Body; for that the Weight of <lb></lb>the Matter it ſelf compounding the ſaid Rope AB is ſuſpended <lb></lb>thereat.</s></p><p type="main"> <s>SIMP. </s> <s>You have fully ſatisfied me; therefore <emph type="italics"></emph>Salviatus<emph.end type="italics"></emph.end> may ac<lb></lb>cording to his promiſe declare unto us, what the Commodity is that <lb></lb>may be drawn from ſuch like Chains, and after that relate unto us <lb></lb>thoſe Speculations which have been made by our <emph type="italics"></emph>Accademian<emph.end type="italics"></emph.end><lb></lb>touching the Force of Percuſſion.</s></p><p type="main"> <s><emph type="italics"></emph>S<emph.end type="italics"></emph.end>ALV. </s> <s>We are for this day ſufficiently employed in the Con<lb></lb>templations already delivered, and the Time, which is pretty late, <lb></lb>would not be enough to carry us through the matters you mention; <lb></lb>therefore we ſhall defer our Conference till ſome more convenient <lb></lb>time.</s></p><p type="main"> <s>SAGR. </s> <s>I concur with you in opinion, for that by ſundry diſ<lb></lb>courſes that I have had with the Friends of our <emph type="italics"></emph>Academick<emph.end type="italics"></emph.end> I have <lb></lb>learnt that this Argument of the Force of Percuſſion is very ob<lb></lb>ſcure, nor hath hitherto any one that hath treated thereof penetra<lb></lb>ted its intricacies, full of darkneſs, and altogether remote from <lb></lb>mans firſt imaginations: and amongſt the Concluſions that I have <lb></lb>heard of, one runs in my mind that is very extravagant and odde, <lb></lb>namely, That the Force of Percuſſion is Interminate, if not Infi<lb></lb>nite. </s> <s>We will therefore attend the leaſure of <emph type="italics"></emph>Salviatus.<emph.end type="italics"></emph.end> But for <lb></lb>the preſent, tell me what things are thoſe which are written at the <lb></lb>end of the Treatiſe of Projects?</s></p><p type="main"> <s>SALV. </s> <s>Theſe are certain Propoſitions touching the Center of <lb></lb>Gravity of Solids, which our <emph type="italics"></emph>Academick<emph.end type="italics"></emph.end> found out in his youth, <lb></lb><arrow.to.target n="marg1104"></arrow.to.target><lb></lb>conceiving that what ^{*} <emph type="italics"></emph>Frederico Comandino<emph.end type="italics"></emph.end> had writ touching the <pb xlink:href="069/01/250.jpg" pagenum="247"></pb>ſame was not altogether without Imperſection. </s> <s>He therefore <lb></lb>thought that with theſe Propoſitions, which here you ſee written, <lb></lb>he might ſupply that which is wanting in the Book of <emph type="italics"></emph>Comandine<emph.end type="italics"></emph.end>; <lb></lb>and he applyed himſelf to the ſame at the Inſtance of the moſt <lb></lb>Illuſtrious Lord Marqueſs <emph type="italics"></emph>Guid' Vbaldo dal Monte,<emph.end type="italics"></emph.end> the moſt ex<lb></lb>cellent Mathematician of his Time, as his ſeveral Printed Works <lb></lb>do ſpeak him; and gave a Copy thereof to that Noble Lord with <lb></lb>thoughts to have purſued the ſame Argument in other Solids not <lb></lb>mentioned by <emph type="italics"></emph>Comandine:<emph.end type="italics"></emph.end> But he chanced after ſome Time to <lb></lb>meet with the ^{*} Book of <emph type="italics"></emph>Signore Luca Valerio,<emph.end type="italics"></emph.end> a moſt famous <lb></lb><arrow.to.target n="marg1105"></arrow.to.target><lb></lb>Geometrician, and ſaw that he reſolveth all theſe matters with<lb></lb>out omiſſion of any thing, he proceeded no farther, although his <lb></lb>Agreſſions were by methods very different from theſe of <emph type="italics"></emph>Signore <lb></lb>Valerio.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1104"></margin.target>* <emph type="italics"></emph>Fredericus Co<lb></lb>mandinus.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1105"></margin.target>* <emph type="italics"></emph>De.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>SAGR. </s> <s>It would be a favour, therefore, if, for this time, which <lb></lb>interpoſeth between this and our next Meeting, you would pleaſe <lb></lb>to leave the Book in my hands: for I ſhall all the while be read<lb></lb>ing and ſtudying the Propoſitions that are conſequently therein <lb></lb>writ.</s></p><p type="main"> <s>SALV. </s> <s>I ſhall very willingly obey your Command; and hope <lb></lb>that you will take pleaſure in theſe Propoſitions.</s></p></chap><chap><pb xlink:href="069/01/251.jpg" pagenum="248"></pb><p type="head"> <s>AN <lb></lb>APPENDIX, <lb></lb>In which is contained certain <lb></lb>THE OREMS and their DEMONSTRATIONS: <lb></lb>Formerly written by the ſame Author, touching the <lb></lb><emph type="italics"></emph>CENTER<emph.end type="italics"></emph.end> of <emph type="italics"></emph>GRAVITY,<emph.end type="italics"></emph.end> of <lb></lb>SOLIDS.</s></p><p type="head"> <s>POSTVLATVM.</s></p><p type="main"> <s><emph type="italics"></emph>We preſuppoſe equall Weights to be alike diſpo<lb></lb>ſed in ſever all Ballances, if the Center of Gra<lb></lb>vity of ſome of thoſe Compounds ſhall divide the Ballance <lb></lb>according to ſome proportion, and the Ballance ſhall <lb></lb>alſo divide their Center of Gravity according to the <lb></lb>ſame proportion.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>LEMMA.</s></p><p type="main"> <s><emph type="italics"></emph>Let the line A B be cut in two equall parts in C, <lb></lb>whoſe half A C let be divided in E, ſo that as B E is to <lb></lb>E A, ſo may A E be to E C. </s> <s>I ſay that B E is double<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.251.1.jpg" xlink:href="069/01/251/1.jpg"></figure><lb></lb><emph type="italics"></emph>to E A. </s> <s>For as B E is to E <lb></lb>A, ſo is E A to E C: there<lb></lb>fore by Compoſition and by Permutation of Proportion, as <lb></lb>B A is to A C, ſo is A E to E C: But as A E is to E C, <lb></lb>that is, B A to A C, ſo is B E to E A: Wherefore B <lb></lb>E is double to E A.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>This ſuppoſed, we will Demonſtrate, That,<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/252.jpg" pagenum="249"></pb><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>If certain Magnitudes at any Rate equally exceed<lb></lb>ing one another, and whoſe exceſs is equal to <lb></lb>the leaſt of them, be ſo diſpoſed in the Balance, <lb></lb>as that they hang at equal diſtances, to divide <lb></lb>the Center of Gravity of the whole Balance <lb></lb>ſo, that the part towards the leſſer Magnitudes <lb></lb>be double to the remainder.</s></p><p type="main"> <s><emph type="italics"></emph>In the ^{*} Ballance A B, therefore, let there be ſuſpended at equal di-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1106"></arrow.to.target><lb></lb><emph type="italics"></emph>ſtances any number of Magnitudes, as hath been ſaid, F, G, H, K, <lb></lb>N; of which let the leaſt be N, and let the points of the Suſpenſions <lb></lb>be A, C, D, E, B, and let the Center of Gravity of all the Magnitudes <lb></lb>ſo diſpoſed be X. </s> <s>It is to be proved that the part of the Ballance B X <lb></lb>towards the leſſer Magnitudes is double to the remaining part X A.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1106"></margin.target>* Or Beam.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Ballance be divided in two equal parts in D, for it muſt ei<lb></lb>ther fall in ſome point of the Suſpenſions, or elſe in the middle point be<lb></lb>tween two of the points of the Suſpenſions: and let the remaining di<lb></lb>ſtances of the Suſpenſions which fall between A and D, be all divided <lb></lb>into halves by the Points M and I; and let all the Magnitudes be divi-<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.252.1.jpg" xlink:href="069/01/252/1.jpg"></figure><lb></lb><emph type="italics"></emph>ded into parts equal to <lb></lb>N: Now the parts of F <lb></lb>ſhall be ſo many in num<lb></lb>ber, as thoſe Magnitudes <lb></lb>be which are ſuſpended <lb></lb>at the Ballance, and the <lb></lb>parts of G one fewer, <lb></lb>and ſo of the reſt. </s> <s>Let <lb></lb>the parts of F therefore be N, O, R, S, T, and let thoſe of G be N, O, <lb></lb>R, S, thoſe of H alſo N, O, R, then let thoſe of K be N, O: and all the <lb></lb>Magnitudes in which are N ſhall be equal to F; and all the Magnitudes <lb></lb>in which are O ſhall be equal to G; and all the Magnitudes in which <lb></lb>are R ſhall be equal to H; and thoſe in which S ſhall be equal to K; and <lb></lb>the Magnitude T is equal to N. </s> <s>Becauſe therefore all the Magnitudes <lb></lb>in which are N are equal to one another, they ſhall equiponderate in <lb></lb>the point D, which divideth the Ballance into two equal parts; and for <lb></lb>the ſame cauſe all the Magnitudes in which are O do equiponderate in <lb></lb>I; and thoſe in which are R in C; and in which are S in M do equi<lb></lb>ponderate; and T is ſuſpended in A. </s> <s>Therefore in the Ballance A D at <lb></lb>the equal diſtances D, I, C, M, A, there are Magnitudes ſuſpended ex<lb></lb>ceeding one another equally, and whoſe exceſs is equal to the leaſt: and <lb></lb>the greateſt, which is compounded of all the N N hangeth at D, the<emph.end type="italics"></emph.end><pb xlink:href="069/01/253.jpg" pagenum="250"></pb><emph type="italics"></emph>leaſt which is T hangeth at A; and the reſt are ordinately diſpoſed. <lb></lb></s> <s>And again there is another Ballance A B in which other Magnitudes <lb></lb>equal in number and Magnitude to the former are diſpoſed in the ſame <lb></lb>order. </s> <s>Wherefore the Ballances A B and A D are divided by the Cen<lb></lb>ter of all the Magnitudes according to the ſame proportion: But the <lb></lb>Center of Gravity of the aforeſaid Magnitudes is X: Wherefore X <lb></lb>divideth the Ballances B A and A D according to the ſame proportion; <lb></lb>ſo that as B X is to X A, ſo is X A to X D: Wherefore B X is double <lb></lb>to X A, by the Lemma aforegoing: Which was to be proved.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>If in a Parabolical Conoid Figure be deſcribed, <lb></lb>and another circumſcribed by Cylinders of <lb></lb>equal Altitude; and the Axis of the ſaid Co<lb></lb>noid be divided in ſuch proportion that the <lb></lb>part towards the Vertex be double to that to<lb></lb>wards the Baſe; the Center of Gravity of the <lb></lb>inſcribed Figure of the Baſe portion ſhall be <lb></lb>neareſt to the ſaid point of diviſion; and the <lb></lb>Center of Gravity of the circumſcribed from <lb></lb>the Baſe of the Conoid ſhall be more remote: <lb></lb>and the diſtance of either of thoſe Centers <lb></lb>from that ſame point ſhall be equal to the Line <lb></lb>that is the ſixth part of the Altitude of one of <lb></lb>the Cylinders of which the Figures are com<lb></lb>poſed.</s></p><p type="main"> <s><emph type="italics"></emph>Take therefore a Parabolical Conoid, and the Figures that have <lb></lb>been mentioned: let one of them be inſcribed, the other circum<lb></lb>ſcribed; and let the Axis of the Conoid, which let be A E, be di<lb></lb>vided in N, in ſuch proportion as that A N be double to N E. </s> <s>It is to <lb></lb>be proved that the Center of Gravity of the inſcribed Figure is in the <lb></lb>Line N E, but the Center of the circumſcribed in the Line A N. </s> <s>Let <lb></lb>the Plane of the Figures ſo diſpoſed be cut through the Axis, and let <lb></lb>the Section be that of the Parabola B A C: and let the Section of the <lb></lb>cutting Plane, and of the Baſe of the Conoid be the Line B C; and <lb></lb>let the Sections of the Cylinders be the Rectangular Figures; as ap<lb></lb>peareth in the deſcription. </s> <s>Firſt, therefore, the Cylinder of the inſcri<lb></lb>bed whoſe Axis is D E, hath the ſame proportion to the Cylinder whoſe <lb></lb>Axis is D Y, as the Quadrate I D hath to the Quadrate S Y; that is, <lb></lb>as D A hath to A Y: and the Cylinder whoſe Axis is D Y is<emph.end type="italics"></emph.end> potentia <pb xlink:href="069/01/254.jpg" pagenum="251"></pb><emph type="italics"></emph>to the Cylinder Y Z as S Y to R Z, that is, as Y A to A Z: and, by the <lb></lb>ſame reaſon, the Cylinder whoſe Axis is Z Y is to that whoſe Axis is <lb></lb>Z V, as Z A is to A V. </s> <s>The ſaid Cylinders, therefore, are to one ano<lb></lb>ther as the Lines D A, A Y; Z A, A V: But theſe are equally exceed<lb></lb>ing to one another, and the exceſs is equal to the leaſt, ſo that A Z is <lb></lb>double to A V; and A Y is triple the<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.254.1.jpg" xlink:href="069/01/254/1.jpg"></figure><lb></lb><emph type="italics"></emph>ſame; and D A Quadruple. </s> <s>Thoſe <lb></lb>Cylinders, therefore, are certain Mag<lb></lb>nitudes in order equally exceeding one <lb></lb>another, whoſe exceſs is equal to the <lb></lb>leaſt of them, and is the Line X M, <lb></lb>in which they are ſuſpended at equal <lb></lb>diſtances (for that each of the Cy<lb></lb>linders hath its Center of Gravity in <lb></lb>the miaſt of the Axis.) Wherefore, <lb></lb>by what hath been above demonſtra<lb></lb>ted, the Center of Gravity of the Mag<lb></lb>nitude compounded of them all divi<lb></lb>deth the Line X M ſo, that the part <lb></lb>towards X is double to the reſt. </s> <s>Divide it, therefore, and, let X<emph.end type="italics"></emph.end> <foreign lang="grc">α</foreign> <emph type="italics"></emph>be <lb></lb>double<emph.end type="italics"></emph.end> <foreign lang="grc">α</foreign> <emph type="italics"></emph>M: therefore is<emph.end type="italics"></emph.end> <foreign lang="grc">α</foreign> <emph type="italics"></emph>the Center of Gravity of the inſcribed Fi<lb></lb>gure. </s> <s>Divide A V in two equal parts in<emph.end type="italics"></emph.end> <foreign lang="grc">ε</foreign>: <foreign lang="grc">ε</foreign> <emph type="italics"></emph>X ſhall be double to <lb></lb>M E: But X<emph.end type="italics"></emph.end> <foreign lang="grc">α</foreign> <emph type="italics"></emph>is double to<emph.end type="italics"></emph.end> <foreign lang="grc">α</foreign> <emph type="italics"></emph>M: Wherefore<emph.end type="italics"></emph.end> <foreign lang="grc">ε</foreign> <emph type="italics"></emph>E ſhall be triple E<emph.end type="italics"></emph.end> <foreign lang="grc">α.</foreign> <emph type="italics"></emph>But<emph.end type="italics"></emph.end><lb></lb><foreign lang="grc">α</foreign> <emph type="italics"></emph>E is triple E N: It is manifeſt, therefore, that E N is greater than <lb></lb>E X; and for that cauſe<emph.end type="italics"></emph.end> <foreign lang="grc">α,</foreign> <emph type="italics"></emph>which is the Center of Gravity of the in<lb></lb>ſcribed Figure, cometh nearer to the Baſe of the Conoid than N. </s> <s>And <lb></lb>becauſe that as A E is to E N, ſo is the part taken away<emph.end type="italics"></emph.end> <foreign lang="grc">ε</foreign> <emph type="italics"></emph>E to the part <lb></lb>taken away E<emph.end type="italics"></emph.end> <foreign lang="grc">α</foreign>: <emph type="italics"></emph>and the remaining part ſhall be to the remaming part, <lb></lb>that is, A<emph.end type="italics"></emph.end> <foreign lang="grc">ε</foreign> <emph type="italics"></emph>to N<emph.end type="italics"></emph.end> <foreign lang="grc">α,</foreign> <emph type="italics"></emph>as A E to E N. Therefore<emph.end type="italics"></emph.end> <foreign lang="grc">α</foreign> <emph type="italics"></emph>N is the third part of <lb></lb>A<emph.end type="italics"></emph.end> <foreign lang="grc">ε,</foreign> <emph type="italics"></emph>and the ſixt part of A V. </s> <s>And in the ſame manner the Cylinders of <lb></lb>the circumſcribed Figure may be demonſtrated to be equally exceeding <lb></lb>one another, and the exceſs to me equal to the least; and that they have <lb></lb>their Centers of Gravity at equal diſtances in the Line<emph.end type="italics"></emph.end> <foreign lang="grc">ε</foreign> <emph type="italics"></emph>M. </s> <s>If therefore<emph.end type="italics"></emph.end><lb></lb><foreign lang="grc">ε</foreign> <emph type="italics"></emph>M be divided in<emph.end type="italics"></emph.end> <foreign lang="grc">π,</foreign> <emph type="italics"></emph>ſo as that<emph.end type="italics"></emph.end> <foreign lang="grc">ε π</foreign> <emph type="italics"></emph>be double to the remaining part<emph.end type="italics"></emph.end> <foreign lang="grc">π</foreign> <emph type="italics"></emph>M;<emph.end type="italics"></emph.end><lb></lb><foreign lang="grc">π</foreign> <emph type="italics"></emph>ſhall be the Center of Gravity of the whole circumſcribed Magnitude. <lb></lb></s> <s>And ſince<emph.end type="italics"></emph.end> <foreign lang="grc">ε π</foreign> <emph type="italics"></emph>is double to<emph.end type="italics"></emph.end> <foreign lang="grc">π</foreign> <emph type="italics"></emph>M; and A<emph.end type="italics"></emph.end> <foreign lang="grc">ε</foreign> <emph type="italics"></emph>leſs than double EM: (for <lb></lb>that they are equal:) the whole A E ſhall be leſs than triple E<emph.end type="italics"></emph.end> <foreign lang="grc">π</foreign><emph type="italics"></emph>: Where<lb></lb>fore E<emph.end type="italics"></emph.end> <foreign lang="grc">π</foreign> <emph type="italics"></emph>ſhall be greater than E N. And, ſince<emph.end type="italics"></emph.end> <foreign lang="grc">ε</foreign> <emph type="italics"></emph>M is triple to M<emph.end type="italics"></emph.end> <foreign lang="grc">π,</foreign><lb></lb><emph type="italics"></emph>and M E with twice<emph.end type="italics"></emph.end> <foreign lang="grc">ε</foreign> <emph type="italics"></emph>A is likewiſe triple to M E: the whole A E with <lb></lb>A<emph.end type="italics"></emph.end> <foreign lang="grc">ε</foreign> <emph type="italics"></emph>ſhall be triple to E<emph.end type="italics"></emph.end> <foreign lang="grc">π</foreign><emph type="italics"></emph>: But A E is triple to E N: Wherefore the <lb></lb>remaining part A<emph.end type="italics"></emph.end> <foreign lang="grc">ε</foreign> <emph type="italics"></emph>ſhall be triple to the remaining part<emph.end type="italics"></emph.end> <foreign lang="grc">π</foreign> <emph type="italics"></emph>N. </s> <s>Therefore <lb></lb>N<emph.end type="italics"></emph.end> <foreign lang="grc">π</foreign> <emph type="italics"></emph>is the ſixth part of A V. </s> <s>And theſe are the things that were to be <lb></lb>demonſtrated.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/255.jpg" pagenum="252"></pb><p type="head"> <s>COROLLARY.</s></p><p type="main"> <s>Hence it is manifeſt, that a Conoid may be inſcribed in a Para<lb></lb>bolical Figure, and another circumſcribed, ſo, as that the <lb></lb>Centers of their Gravities may be diſtant from the point N <lb></lb>leſs than any Line given.</s></p><p type="main"> <s><emph type="italics"></emph>For if we aſſume a Line ſexcuple of the propoſed Line, and make the <lb></lb>Axis of the Cylinders, of which the Figures are compounded given <lb></lb>leſſer than this aſſumed Line, there ſhall fall Lines between the Centers <lb></lb>of Gravities of theſe Figures and the mark N that are leſs than the <lb></lb>Line propoſed.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>The former Propoſition another way.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Axis of the Conoid (which let be C D) be divided in <lb></lb>O, ſo, as that C O be double to O D. </s> <s>It is to be proved that the <lb></lb>Center of Gravity of the inſcribed Figure is in the Line O D; <lb></lb>and the Center of the circumſcribed in C O. </s> <s>Let the Plane of the Fi<lb></lb>gures be cut through the Axis and C, as hath been ſaid. </s> <s>Becauſe there<lb></lb>fore the Cylinders S N, T M, V I,<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.255.1.jpg" xlink:href="069/01/255/1.jpg"></figure><lb></lb><emph type="italics"></emph>X E are to one another as the Squares <lb></lb>of the Lines S D, T N, V M, X I; <lb></lb>and theſe are to one another as the <lb></lb>Lines N C, C M, C I, C E: but <lb></lb>theſe do exceed one another equally; <lb></lb>and the exceſs is equal to the leaſt, to <lb></lb>wit, C E: And the Cylinder T M is <lb></lb>equal to the Cylinder Q N; and the <lb></lb>Cylinder V I equal to P N; and X E <lb></lb>is equal to L N: Therefore the Cylin<lb></lb>ders S N, Q N, P N, and L N do <lb></lb>equally exceed one another, and the <lb></lb>exceſs is equal to the leaſt of them, <lb></lb>namely, to the Cylinder L N. </s> <s>But <lb></lb>the exceſs of the Cylinder S N, above <lb></lb>the Cylinder Q N is a Ring whoſe <lb></lb>height is Q T; that is, N D; and <lb></lb>its breadth S <expan abbr="q.">que</expan> And the exceſs of the Cylinder Q N above P N, is a <lb></lb>Ring, whoſe breadth is Q P. </s> <s>And the exceſs of the Cylinder P N above <lb></lb>L N is a Ring, whoſe breadth is P L. </s> <s>Wherefore the ſaid Rings S Q, <lb></lb>Q P, P L, are equal to another, and to the Cylinder L N. </s> <s>Therefore the <lb></lb>Ring S T equalleth the Cylinder X E: the Ring Q V, which is double <lb></lb>to S T, equalleth the Cylinder V I; which likewiſe is double to the<emph.end type="italics"></emph.end><pb xlink:href="069/01/256.jpg" pagenum="253"></pb><emph type="italics"></emph>Cylinder X E: and for the ſame cauſe the Ring P X is equal to the <lb></lb>Cylinder T M; and the Cylinder L E ſhall be equal to the Cylinder S N. <lb></lb></s> <s>In the Beam or Ballance, therefore, K F connecting the middle points of <lb></lb>the Right-lines E I and D N, and cut into equal parts in the points H <lb></lb>and G, are certain Magnitudes ſuſpended, to wit the Cylinders S N, <lb></lb>T M, V I, X E; and the Center of Gravity of the firſt Cylinder is K; <lb></lb>and of the ſecond H; of the third G; of the fourth F. </s> <s>And we have <lb></lb>another Ballance M K, which is the half of the ſaid F K, and a like <lb></lb>number of points diſtributed into equal parts, to wit, M H, H N, N K, <lb></lb>and on it other Magnitudes, equal in number and bigneſs to thoſe which <lb></lb>are on the Beam F K, and having the Centers of Gravity in the points <lb></lb>M, H, N, and K, and diſpoſed in the ſame order. </s> <s>For the Cylinder L E <lb></lb>hath its Center of Gravity in M; and is equal to the Cylinder S N that <lb></lb>hath its Center in K: And the Ring P X hath the Center H; and is <lb></lb>equal to the Cylinder T M, whoſe Center is H: And the Ring Q V ha<lb></lb>ving the Center N is equal to the V I whoſe Center is G: And laſtly, <lb></lb>the Ring S T having the Center K, is equal to the Cylinder X E whoſe <lb></lb>Center is F. </s> <s>Therefore the Center of Gravity of the ſaid Magnitudes <lb></lb>divideth the Beam in the ſame proportion: But the Center of them is <lb></lb>one, and therefore ſome point common to both the Beams or Ballance, <lb></lb>which let be Y. </s> <s>Therefore F Y and Y K ſhall be as K Y and Y M. </s> <s>F Y <lb></lb>therefore is double to Y K: and C E being divided into two equal parts <lb></lb>in Z, Z F, ſhall be double to K D: and for that cauſe Z D triple to D Y: <lb></lb>But to the Right Line D O C D is triple: Therefore the Right Line <lb></lb>D O is greater than D Y: And for the like cauſe Y the Center of the <lb></lb>inſcribed Figure approacheth nearer the Baſe than the point O. </s> <s>And <lb></lb>becauſe as C D is to D O, ſo is the part taken away Z D to the part ta<lb></lb>ken away D Y; the remaining part C Z ſhall be to the remaining part <lb></lb>Y O, as C D is to D O; that is Y O ſhall be the third part of C Z; <lb></lb>that is, the ſixth part of C E. </s> <s>Again we will, by the ſame reaſon, de<lb></lb>monſtrate the Cylinders of the circumſcribed Figure to exceed one ano<lb></lb>ther equally, and that the exceſs is equal to the leaſt, and that their <lb></lb>Centers of Gravity are conſtituted in equal diſtances upon the Beam <lb></lb>K Z: and likewiſe that the Rings equal to thoſe ſame Cylinders are in <lb></lb>like manner diſpoſed on another Beam K G, the half of the ſaid K Z, <lb></lb>and that therefore the Center of Gravity of the circumſcribed Figure, <lb></lb>which let be R, ſo divideth the Beam, as that Z R is to R K, as K R is to <lb></lb>R G. </s> <s>Therefore Z R ſhall be double to R K: But C Z is equal to the <lb></lb>Right Line K D, and not double to it. </s> <s>The whole C D ſhall be leſſer <lb></lb>than triple to D R: Wherefore the Right Line D R is greater than D O; <lb></lb>that is to ſay, the Center of the circumſcribed Figure recedeth from the <lb></lb>Baſe more than the point O. </s> <s>And becauſe Z K is triple to K R; and <lb></lb>K D with twice Z C is triple to K D; the whole C D with C Z ſhall be <lb></lb>triple to D R: But C D is triple to D O: Wherefore the remaining <lb></lb>part C Z ſhall be triple to the remaining part R O; that is, O R<emph.end type="italics"></emph.end><pb xlink:href="069/01/257.jpg" pagenum="254"></pb><emph type="italics"></emph>is the ſixth part of E C: Which was the Propoſition.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>This being pre-demonſtrated, we will prove that<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>The Center of Gravity of the Parabolick <lb></lb>Conoid doth ſo divide the Axis, as that the <lb></lb>part towards the Vertex is double to the re<lb></lb>maining part towards the Baſe.</s></p><p type="main"> <s><emph type="italics"></emph>Let there be a Parabolick Conoid whoſe Axis let be A B divided in <lb></lb>N ſo as that A N be double to N B. </s> <s>It is to be proved that the Cen<lb></lb>ter of Gravity of the Conoid is the point N. </s> <s>For if it be not N, it <lb></lb>ſhall be either above or below it. </s> <s>Firſt let it be below; and let it be X: <lb></lb>And ſet off upon ſome place by it ſelf the Line L O equal to N X; and let <lb></lb>L O be divided at pleaſure in S: and look what proportion B X and <lb></lb>O S both together have to O S, and the ſame ſhall the Conoid have to <lb></lb>the Solid R. </s> <s>And in the Conoid let Figures be deſcribed by Cylinders <lb></lb>having equal Altitudes, ſo, as that that which lyeth between the Center <lb></lb>of Gravity and the point N be leſs than L S: and let the exceſs of the <lb></lb>Conoid above it be leſs than the Solid R: and that this may be done is <lb></lb>clear. </s> <s>Take therefore the inſcribed, whoſe Center of Gravity let be I: <lb></lb>now I X ſhall be greater than S O: And becauſe that as X B with S O <lb></lb>is to S O, ſo is the Conoid to the Solid R: (and R is greater than the <lb></lb>exceſs by which the Conoid exceeds the inſcribed Figure:) the proporti<lb></lb>on of the Conoid to the ſaid exceſs ſhall be greater than both B X and <lb></lb>O S unto S O: And, by Diviſion, the inſcribed Figure ſhall have grea<lb></lb>ter proportion to the ſaid exceſs than B X to S O: But B X hath to <lb></lb>X I a proportion yet leſs than to S O: Therefore the inſcribed Figure <lb></lb>ſhall have much greater proportion to the reſt of the proportions than <lb></lb>B X to X I: Therefore what proportion the inſcribed Figure hath to <lb></lb>thereſt of the portions, the ſame ſhall a certain other Line have to X I: <lb></lb>which ſhall neceſſarily be greater than B X: Let it, therefore, be M X. <lb></lb></s> <s>We have therefore the Center of Gravity of the Conoid X: But the <lb></lb>Center of Gravity of the Figure inſcribed in it is I: of the reſt of the <lb></lb>portions by which the Conoid exceeds the inſcribed Figure the Center of <lb></lb>Gravity ſhall be in the Line X M, and in it that point in which it ſhall <lb></lb>be ſo terminated, that look what proportion the inſcribed Figure hath <lb></lb>to the exceſs by which the Conoid exceeds it, the ſame it ſhall have to <lb></lb>X I: But it hath been proved, that this proportion is that which M X <lb></lb>hath to X I: Therefore M ſhall be the Center of Gravity of thoſe pro<lb></lb>portions by which the Conoid exceeds the inſcribed Figure: Which <lb></lb>certainly cannot be. </s> <s>For if along by M a Plane be drawn equidiſtant to <lb></lb>the Baſe of the Conoid, all thoſe proportions ſhall be towards one and<emph.end type="italics"></emph.end><pb xlink:href="069/01/258.jpg" pagenum="255"></pb><emph type="italics"></emph>the ſame part, and not by it divided. </s> <s>Therefore the Center of Gravity <lb></lb>of the ſaid Conoid is not below the point N: Neither is it above. </s> <s>For, <lb></lb>if it may, let it be H: and again, as before, ſet the Line L O by it ſelf <lb></lb>equalto the ſaid H N, and divided at pleaſure in S: and the ſame pro<lb></lb>portion that B N and S O both together have to S L, let the Conoid <lb></lb>have to R: and about the Conoid let a Figure be circumſcribed conſi<lb></lb>ſting of Cylinders, as hath been ſaid: by which let it be exceeded a leſs <lb></lb>quantity than that of the Solid R: and let the Line betwixt the Center <lb></lb>of Gravity of the circumſcribed Figure and the point N be leſſer than <lb></lb>S O: the remainder V H ſhall be greater than S L. </s> <s>And becauſe that as <lb></lb>both B N and O S is to SL, ſo is the<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.258.1.jpg" xlink:href="069/01/258/1.jpg"></figure><lb></lb><emph type="italics"></emph>Conoid to R: (and R is greater <lb></lb>than the exceſs by which the circum<lb></lb>ſcribed Figure exceeds the Conoid:) <lb></lb>Therefore B N and S O hath leſs pro<lb></lb>portion to S L than the Conoid to the <lb></lb>ſaid exceſs. </s> <s>And B V is leſſer than <lb></lb>both B N and S O; and V H is grea<lb></lb>ter than S L: much greater proporti<lb></lb>on, therefore, hath the Conoid to the <lb></lb>ſaid proportions, than B V hath to <lb></lb>V H. </s> <s>Therefore whatever proporti<lb></lb>on the Conoid hath to the ſaid pro<lb></lb>portions, the ſame ſhall a Line greater <lb></lb>than B V have to V H. </s> <s>Let the ſame be M V: And becauſe the Center <lb></lb>of Gravity of the circumſcribed Figure is V, and the Center of the <lb></lb>Conoid is H. and ſince that as the Conoid to the reſt of the proportions, <lb></lb>ſois M V to V H, M ſhall be the Center of Gravity of the remaining <lb></lb>proportions: which likewiſe is impoſſible: Therefore the Center of <lb></lb>Gravity of the Conoid is not above the point N: But it hath been de<lb></lb>monſtrated that neither is it beneath: It remains, therefore, that it ne<lb></lb>ceſſarily be in the point N it ſelf. </s> <s>And the ſame might be demonſtrated <lb></lb>of Conoidal Plane cut upon an Axis not erect. </s> <s>The ſame in other terms, <lb></lb>as appears by what followeth:<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>The Center of Gravity of the Parabolick Co<lb></lb>noid falleth betwixt the Center of the cir<lb></lb>cumſcribed Figure and the Center of the in<lb></lb>ſcribed.</s></p><pb xlink:href="069/01/259.jpg" pagenum="256"></pb><p type="main"> <s><emph type="italics"></emph>Let there be a Conoid whoſe Axis is A B, and the Center of the <lb></lb>circumſcribed Figure C, and the Center of the inſcribed O. </s> <s>I ſay <lb></lb>the Center of the Conoid is betwixt the points C and O. </s> <s>For if <lb></lb>not, it ſhall be either above them, or below them, or in one of them. </s> <s>Let <lb></lb>it be below, as in R. </s> <s>And becauſe R is the Center of Gravity of the <lb></lb>whole Conoid; and the Center of Gravity of the inſcribed Figure is O: <lb></lb>Therefore of the remaining proportions by which the Conoid exceeds <lb></lb>the inſcribed Figure the Center of Gravity ſhall be in the Line O R ex<lb></lb>tended towards R, and in that point in which it is ſo determined, that, <lb></lb>what proportion the ſaid proportions have to the inſcribed Figure, the <lb></lb>ſame ſhall O R have to the Line falling betwixt R and that falling point. <lb></lb></s> <s>Let this proportion be that of O R to R X. </s> <s>Therefore X falleth either <lb></lb>without the Conoid or within, or in its<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.259.1.jpg" xlink:href="069/01/259/1.jpg"></figure><lb></lb><emph type="italics"></emph>Baſe. </s> <s>That it falleth without, or in its <lb></lb>Baſe it is already manifeſt to be an abſur<lb></lb>dity. </s> <s>Let it fall within: and becauſe X R <lb></lb>is to R O, as the inſcribed Figure is to <lb></lb>the exceſs by which the Conoid exceeds <lb></lb>it; the ſame proportion that B R hath to <lb></lb>R O, the ſame let the inſcribed Figure <lb></lb>have to the Solid K: Which neceſſarily <lb></lb>ſhall be leſſer than the ſaid exceſs. </s> <s>And let <lb></lb>another Figure be inſcribed which may be <lb></lb>exceeded by the Conoid a leſs quantity <lb></lb>than is K, whoſe Center of Gravity falleth betwixt O and C. </s> <s>Let it <lb></lb>be V. And, becauſe the firſt Figure is to K as B R to R O, and the ſe<lb></lb>cond Figure, whoſe Center V is greater than the firſt, and exceeded <lb></lb>by the Conoid a leſs quantity than is K; what proportion the ſecond <lb></lb>Figure hath to the exceſs by which the Conoid exceeds it, the ſame <lb></lb>ſhall a Line greater than B R have to R V. </s> <s>But R is the Center of Gra<lb></lb>vity of the Conoid; and the Center of the ſecond inſcribed Figure V: <lb></lb>The Center therefore of the remaining proportions ſhall be without <lb></lb>the Conoid beneath B: Which is impoſſible. </s> <s>And by the ſame means <lb></lb>we might demonſtrate the Center of Gravity of the ſaid Conoid not to <lb></lb>be in the Line C A. </s> <s>And that it is none of the points betwixt C and <lb></lb>O is manifeſt. </s> <s>For ſay, that there other Figures deſcribed, greater <lb></lb>ſomething than the inſcribed Figure whoſe Center is O, and leſs than <lb></lb>that circumſcribed Figure whoſe Center is C, the Center of the Conoid <lb></lb>would fall without the Center of theſe Figures: Which but now was <lb></lb>concluded to be impoſſible: It reſts therefore that it be betwixt the Cen<lb></lb>ter of the circumſcribed and inſcribed Figure. </s> <s>And if ſo, it ſhall ne<lb></lb>ceſſarily be in that point which divideth the Axis, ſo as that the part <lb></lb>towards the Vertex is double to the remainder; ſince N may circum<lb></lb>ſcribe and inſcribe Figures, ſo, that thoſe Lines which fall between<emph.end type="italics"></emph.end><pb xlink:href="069/01/260.jpg" pagenum="257"></pb><emph type="italics"></emph>their Centers and the ſaid points, may be leſſer than any other Lines. <lb></lb></s> <s>To expreſs the ſame in other terms, we have reduced it to an impoſſibi<lb></lb>lity, that the Center of the Conoid ſhould not fall betwixt the Centers of <lb></lb>the inſcribed and circumſcribed Figures.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>Suppoſing three proportional Lines, and that <lb></lb>what proportion the leaſt hath to the exceſs <lb></lb>by which the greateſt exceeds the leaſt, the <lb></lb>ſame ſhould a Line given have to two thirds of <lb></lb>the exceſs by which the greateſt exceeds the <lb></lb>middlemoſt: and moreover, that what pro<lb></lb>portion that compounded of the greateſt, and <lb></lb>of double the middlemoſt, hath unto that com<lb></lb>pounded of the triple of the greateſt and mid<lb></lb>dlemoſt, the ſame hath another Line given, to <lb></lb>the exceſs by which the greateſt exceeds the <lb></lb>middle one; both the given Lines taken toge<lb></lb>ther ſhall be a third part of the greateſt of the <lb></lb>proportional Lines.</s></p><p type="main"> <s><emph type="italics"></emph>Let A B, B C, and B F, be three proportional Lines; and what <lb></lb>proportion B F hath to F A, the ſame let M S have to two thirds <lb></lb>of C A. </s> <s>And what proportion that compounded of A B and the <lb></lb>double of B C hath to that compounded of the triple of both A B and <lb></lb>B C, the ſame let another, to wit S N, have to A C. </s> <s>Becauſe therefore <lb></lb>that A B, B C, and C F,<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.260.1.jpg" xlink:href="069/01/260/1.jpg"></figure><lb></lb><emph type="italics"></emph>are proportionals, A G <lb></lb>and C F ſhall, for the ſame <lb></lb>reaſon, be likewiſe ſo. <lb></lb></s> <s>Therefore, as A B is to <lb></lb>B C, ſo is A C to C F: <lb></lb>and as the triple of A B is to the triple of B C, ſo is A C to C F: <lb></lb>Therefore, what proportion the triple of A B with the triple of B C <lb></lb>hath to the triple of C B, the ſame ſhall A C have to a Line leſs than <lb></lb>C F. </s> <s>Let it be C O. </s> <s>Wherefore by Compoſition and by Converſion of <lb></lb>proportion, O A ſhall have to A C, the ſame proportion, as triple A B <lb></lb>with Sextuple B C, hath to triple A B with triple B C. </s> <s>But A C hath <lb></lb>to S N the ſame proportion, that triple A B with triple B C hath to A B <lb></lb>with double B C: Therefore,<emph.end type="italics"></emph.end> ex equali, <emph type="italics"></emph>O A to NS ſhall have the <lb></lb>ſame proportion, as triple A B with Sexcuple B C hath to A B with<emph.end type="italics"></emph.end><pb xlink:href="069/01/261.jpg" pagenum="258"></pb><emph type="italics"></emph>double B C: But triple A B with ſexcuple B C, are triple to A B with <lb></lb>double B C. </s> <s>Therefore A O is triple to S N.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Again, becauſe O C is to C A as triple C B is to triple A B with tri<lb></lb>ple C B: and becauſe as C A is to A F, ſo is triple A B to triple B C: <lb></lb>Therefore,<emph.end type="italics"></emph.end> ex equali, <emph type="italics"></emph>by perturbed proportion, as O C is to C F, ſo ſhall <lb></lb>triple A B be to triple A B with treble B C: And, by Converſion of <lb></lb>proportion, as O F is to F C, ſo is triple B C to triple A B with triple <lb></lb>B C: And as C F is to F B, ſo is A C to C B, and triple A C to triple <lb></lb>C B: Therefore,<emph.end type="italics"></emph.end> ex equali, <emph type="italics"></emph>by Perturbation of proportion, as O F is <lb></lb>to F B, ſo is triple A C to the triple of both A B and A C together. <lb></lb></s> <s>And becauſe F C and C A are in the ſame proportion as C B and B A; <lb></lb>it ſhall be that as F C is to C A, ſo ſhall B C be to B A. And, by Com<lb></lb>poſition, as F A is to A C, ſo are both B A and B C to B A: and ſo the <lb></lb>triple to the triple: Therefore as F A is to A C, ſo the compound of tri<lb></lb>ple B A and triple B C is to triple A B. Wherefore, as F A is to two <lb></lb>thirds of A C, ſo is the compound of triple B A and triple B C to two <lb></lb>thirds of triple B A; that is, to double B A: But as F A is to two thirds <lb></lb>of A C, ſo is F B to M S: Therefore, as F B is to M S, ſo is the compound <lb></lb>of triple B A and triple B C to double B A: But as O B is to F B, ſo <lb></lb>was Sexcuple A B to triple of both A B and B C: Therefore,<emph.end type="italics"></emph.end> ex equa<lb></lb>li, <emph type="italics"></emph>O B ſhall have to M S the ſame proportion as Sexcuple A B hath to <lb></lb>double B A. </s> <s>Wherefore M S ſhall be the third part of O B: And it <lb></lb>hath been demonſtrated, that S N is the third part of A O: It is mani<lb></lb>feſt therefore, that MN is a third part likewiſe of A B: And this is <lb></lb>that which was to be demonſtrated.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>Of any <emph type="italics"></emph>Fruſtum<emph.end type="italics"></emph.end> or Segment cut off from a Para<lb></lb>bolick Conoid the Center of Gravity is in the <lb></lb>Right Line that is Axis of the <emph type="italics"></emph>Fruſtum<emph.end type="italics"></emph.end>; which <lb></lb>being divided into three equal parts the Cen<lb></lb>ter of Gravity is in the middlemoſt and ſo di<lb></lb>vides it, as that the part towards the leſſer Baſe <lb></lb>hath to the part towards the greater Baſe, the <lb></lb>ſame proportion that the greater Baſe hath to <lb></lb>the leſſer.</s></p><p type="main"> <s><emph type="italics"></emph>From the Conoid whoſe Axis is R B let there be cut off the Solid <lb></lb>whoſe Axis is B E; and let the cutting Plane be equidiſtaut to <lb></lb>the Baſe: and let it be cut in another Plane along the Axis erect <lb></lb>upon the Baſe, and let it be the Section of the Parabola V R C: R B <lb></lb>ſhall be the Diameter of the proportion, or the equidiſtant Diameter<emph.end type="italics"></emph.end><pb xlink:href="069/01/262.jpg" pagenum="259"></pb><emph type="italics"></emph>L M, V C: they ſhall be ordinately applyed. </s> <s>Divide therefore E B in<lb></lb>to three equal parts, of which let the middlemoſt be Q Y: and divide <lb></lb>this ſo in the point I that Q I may have the ſame proportion to I Y, as <lb></lb>the Baſe whoſe Diameter is V C hath to the Baſe whoſe Diameter is <lb></lb>L M; that is, that the Square V C hath to Square L M. </s> <s>It is to be de<lb></lb>monſtrated that I is the Center of Gravity of the Fruſtrum L M C. <lb></lb></s> <s>Draw the Line N S, by the by, equall to B R: and let S X be equal to <lb></lb>E R: and unto N S and S X aſſume a third proportional S G: and as <lb></lb>N G is to G S, ſo let B Q be to I O. </s> <s>And it nothing matters whether <lb></lb>the point O fall above or below L M. </s> <s>And becauſe in the Section V R C <lb></lb>the Lines L M and V C are ordinately<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.262.1.jpg" xlink:href="069/01/262/1.jpg"></figure><lb></lb><emph type="italics"></emph>applyed, it ſhall be that as the Square <lb></lb>V C is to the Square L M, ſo is the Line <lb></lb>B R to R E: And as the Square V C is <lb></lb>to the Square L M, ſo is Q I to I Y: and <lb></lb>as B R is to R E, ſo is N S to S X: There<lb></lb>fore Q I is to I Y, as R S is to S X. </s> <s>Where<lb></lb>fore as G Y is to Y I, ſo ſhall both N S and <lb></lb>S X be to S X: and as E B is to Y I, ſo <lb></lb>ſhall the compound of triple N S and tri<lb></lb>ple S X be to S X: But as E B is to B Y, <lb></lb>ſo is the compound of triple N S and S X <lb></lb>both together to the compound of N S and S X: Therefore, as E B is to <lb></lb>B I, ſo is the compound of triple N S and triple S X to the compound of <lb></lb>N S and double S X. </s> <s>Therefore N S, S X, and S G are three proporti<lb></lb>onal Lines: And as S G is to G N, ſo is the aſſumed O I to two thirds <lb></lb>of E B; that is, to N X: And as the compound of N S and double <lb></lb>S X is to the compound of triple N S and triple S X, ſo is another aſſu<lb></lb>med Line I B to B E; that is, to N X. </s> <s>By what therefore hath been <lb></lb>above demonſtrated, thoſe Lines taken together are a third part of N S; <lb></lb>that is, of R B: Therefore R B is triple to B O: Wherefore O ſhall <lb></lb>be the Center of Gravity of the Conoid v R C. </s> <s>And let it be the Cen<lb></lb>ter of Gravity of the<emph.end type="italics"></emph.end> Fruſtrum <emph type="italics"></emph>L R M of the Conoid: Therefore the <lb></lb>Center of Gravity of V L M C is in the Line O B, and in that point <lb></lb>which ſo terminates it, that as V L M C of the<emph.end type="italics"></emph.end> Fruſtrum <emph type="italics"></emph>is to the <lb></lb>proportion L R M, ſo is the Line A O to that which intervenes betwixt <lb></lb>O and the ſaid point. </s> <s>And becauſe R O is two thirds of R B; and <lb></lb>R A two thirds of R E; the remaining part A O ſhall be two thirds <lb></lb>of the remaining part E B. </s> <s>And becauſe that as the<emph.end type="italics"></emph.end> Fruſtum <emph type="italics"></emph>V L M C <lb></lb>is to the proportion L R M, ſo is N G to G S: and as N G to G S, ſo is <lb></lb>two thirds of E B to O I: and two thirds of E B is equal to the Line <lb></lb>A O: it ſhall be that as the<emph.end type="italics"></emph.end> Fruſtum <emph type="italics"></emph>V L M O is to the proportion <lb></lb>L R M, ſo is A O to O I. </s> <s>It is manifeſt therefore that of the<emph.end type="italics"></emph.end> Fruſtum <lb></lb><emph type="italics"></emph>V L M C the Center of Gravity is the point I, and ſo divideth the Axis, <lb></lb>[as?] that the part towards the leſſer Baſe is to the part towards the grea-<emph.end type="italics"></emph.end><pb xlink:href="069/01/263.jpg" pagenum="260"></pb><emph type="italics"></emph>ter, as the double of the greater Baſe together with the Leſſer is to the <lb></lb>double of the leſſer together with the greater. </s> <s>Which is the Propoſition <lb></lb>more elegantly expreſſed.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>If any number of Magnitudes ſo diſpoſed to one <lb></lb>another, as that the ſecond addeth unto the firſt <lb></lb>the double of the firſt, the third addeth unto <lb></lb>the ſecond the triple of the firſt, the fourth <lb></lb>addeth unto the third the quadruple of the <lb></lb>firſt, and ſo every one of the following ones <lb></lb>addeth unto the next unto it the magnitude of <lb></lb>the firſt multiplyed according to the number <lb></lb>which it ſhall hold in order; if, I ſay, theſe <lb></lb>Magnitudes be ſuſpended ordinarily on the <lb></lb>Ballance at equal diſtances; the Center of the <lb></lb><emph type="italics"></emph>Equilibrium<emph.end type="italics"></emph.end> of all the compounding Magni<lb></lb>tudes ſhall ſo divide the Beam, as that the part <lb></lb>towards the leſſer Magnitudes is triple to the <lb></lb>remainder.</s></p><p type="main"> <s><emph type="italics"></emph>Let the Beam be L T, and let ſuch Magnitudes as were ſpoken of <lb></lb>hang upon it; and let them be A, F, G, H, K; of which A is in <lb></lb>the firſt place ſuſpended at T. </s> <s>I ſay, that the Center of the<emph.end type="italics"></emph.end> Equi<lb></lb>librium <emph type="italics"></emph>ſo cuts the Beam T L as that the part towards T is triple to the <lb></lb>reſt. </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"></emph.end><lb></lb><figure id="id.069.01.263.1.jpg" xlink:href="069/01/263/1.jpg"></figure><lb></lb><emph type="italics"></emph>and L P to L O: I P, <lb></lb>P N, N O, and O L <lb></lb>ſhall be equal. </s> <s>And <lb></lb>in F let a Magnitude <lb></lb>be placed double to A; <lb></lb>in G another trebble to <lb></lb>the ſame; in H ano<lb></lb>ther Quadruple; and <lb></lb>ſo of the reſt: and let <lb></lb>thoſe Magnitudes be <lb></lb>taken in which there <lb></lb>is A; and let the ſame <lb></lb>be done in the Magni<lb></lb>tudes F, G, H, K. </s> <s>And <lb></lb>becauſe in F the remaining Magnitude, to wit B, is equal to A; take it<emph.end type="italics"></emph.end><pb xlink:href="069/01/264.jpg" pagenum="261"></pb><emph type="italics"></emph>double in G, triple in H, &c. </s> <s>and let thoſe Magnitudes be taken in <lb></lb>which there is B: and in the ſame manner let thoſe be taken in which is <lb></lb>C, D, and E: now all thoſe in which there is A ſhall be equal to K: and <lb></lb>the compound of all the B B ſhall equal H; and the compound of C C <lb></lb>ſhall equal G; and the compound of all the D D ſhall equal F; and <lb></lb>E ſhall equal A. </s> <s>And becauſe T I is double to I L, I ſhall be the point <lb></lb>of the<emph.end type="italics"></emph.end> Equilibrium <emph type="italics"></emph>of the Magnitudes compoſed of all the A A: and <lb></lb>likewiſe ſince S P is double to P L, P ſhall be the point of the<emph.end type="italics"></emph.end> Equilibri<lb></lb>um <emph type="italics"></emph>of the compost of B B: and for the ſame cauſe N ſhall be the point <lb></lb>of the<emph.end type="italics"></emph.end> Equilibrium <emph type="italics"></emph>of the compoſt of C C: and O of the compound <lb></lb>of D D: and L that of E. </s> <s>Therefore T L is a Beam on which at <lb></lb>equal diſtances certain Magnitudes K, H, G, F, A do hang. </s> <s>And again <lb></lb>L I is another Ballance, on which, at diſtances in like manner equal, do <lb></lb>hang ſuch a number of Magnitudes, and in the ſame order equal to the <lb></lb>former. </s> <s>For the compound of all the A A, which hang on I, is equal to <lb></lb>K hanging at L; and the compoſt of all B B, which is ſuſpended at P, is <lb></lb>equal to H hanging at P; and likewiſe the compound of C C, which <lb></lb>hangeth at N do equal G; and the compoſt of D, which hang on O, <lb></lb>are equal to F; and E, hanging on L, is equal to A. </s> <s>Wherefore the <lb></lb>Ballances are divided in the ſame proportion by the Center of the com<lb></lb>pounds of the Magnitudes And the Center of the compound of, the ſaid <lb></lb>Magnitudes is one. </s> <s>Therefore the common point of the Right Line T L, <lb></lb>and of the Right Line L I ſhall be the Center, which let be X. </s> <s>Therefore <lb></lb>as T X is to X L, ſo ſhall L X be to X I; and the whole T L to the whole <lb></lb>L I. </s> <s>But T L is triple to L I: Wherefore T X ſhall alſo be triple to X L.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>If any number of Magnitudes be ſo taken, that the <lb></lb>ſecond addeth unto the firſt the triple of the <lb></lb>firſt, and the third addeth unto the ſecond the <lb></lb>quintuple of the firſt, and the fourth addeth <lb></lb>unto the third the ſeptuple of the firſt, and ſo <lb></lb>the reſt, every one encreaſing above the next to <lb></lb>it, and proceedeth ſtill to a new multiplex of <lb></lb>the firſt Magnitude according to the conſe<lb></lb>quent odd numbers, like as the Squares of <lb></lb>Lines equally exceeding one another do pro<lb></lb>ceed, whereof the exceſs is equal to the leaſt, <lb></lb>and if they be ſuſpended on a Ballance at equal <lb></lb>Diſtances, the Center of <emph type="italics"></emph>Equilibrium<emph.end type="italics"></emph.end> of all the <lb></lb>compound Magnitudes ſo divideth the Beam <pb xlink:href="069/01/265.jpg" pagenum="262"></pb>that the part towards the leſſer Magnitudes is <lb></lb>more than triple the remaining part; and alſo <lb></lb>one may take a diſtance that is to the ſame leſs <lb></lb>than triple.</s></p><p type="main"> <s><emph type="italics"></emph>In the Ballance B E let there be Magnitudes, ſuch as were ſpoken off, <lb></lb>from which let there be other Magnitudes taken away that were to <lb></lb>one another as they were diſpoſed in the precedent, and let it be of <lb></lb>the compound of all <lb></lb>the A A: the reſt<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.265.1.jpg" xlink:href="069/01/265/1.jpg"></figure><lb></lb><emph type="italics"></emph>in which are C <lb></lb>ſhall be diſtributed <lb></lb>in the ſame order, <lb></lb>but the greateſt de<lb></lb>ficient. </s> <s>Let E D be <lb></lb>triple to D B; and <lb></lb>G F triple to F B. <lb></lb></s> <s>D ſhall be the Center <lb></lb>of the<emph.end type="italics"></emph.end> Equilibrium <lb></lb><emph type="italics"></emph>of the compound con<lb></lb>ſiſting of all the A A; <lb></lb>and F that of the <lb></lb>compound of all the <lb></lb>C C. </s> <s>Wherefore the <lb></lb>Center of the com<lb></lb>pound of both A A <lb></lb>and C C falleth be<lb></lb>tween D and F. </s> <s>Let <lb></lb>it be O. </s> <s>It is there<lb></lb>fore manifeſt that <lb></lb>E O is more than triple to O B; but G O leſs thantriple to the <lb></lb>ſame O B: Which was to be demonſtrated.<emph.end type="italics"></emph.end></s></p><pb xlink:href="069/01/266.jpg" pagenum="263"></pb><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>If to any Cone or portion of a Cone a Eigure con<lb></lb>ſiſting of Cylinders of equal heights be inſcri<lb></lb>bed and another circumſcribed; and if its Axis <lb></lb>be ſo divided as that the part which lyeth be<lb></lb>twixt the point of diviſion and the Vertex be <lb></lb>triple to the reſt; the Center of Gravity of <lb></lb>the inſcribed Figure ſhall be nearer to the Baſe <lb></lb>of the Cone than that point of diviſion: and <lb></lb>the Center of Gravity of the circumſcribed <lb></lb>ſhall be nearer to the Vertex than that ſame <lb></lb>point.</s></p><p type="main"> <s><emph type="italics"></emph>Take therefore a Cone, whoſe Axis is N M. </s> <s>Let it be divided <lb></lb>in S ſo, as that N S be triple to the remainder S M. </s> <s>I ſay, that <lb></lb>the Center of Gravity of any Figure inſcribed, as was ſaid, in <lb></lb>a Cone doth conſiſt in the Axis N M, and approacheth nearer to the Baſe <lb></lb>of the Cone than the point S: and that the Center of Gravity of the <lb></lb>Circumſcribed is likewiſe in the Axis N M, and nearer to the Vertex <lb></lb>than is S. </s> <s>Let a Figure therefore be ſuppoſed to be inſcribed by the Cy<lb></lb>linders whoſe Axis M C, C B, B E, E A are equal. </s> <s>Firſt therefore <lb></lb>the Cylinder whoſe Axis is M C hath<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.266.1.jpg" xlink:href="069/01/266/1.jpg"></figure><lb></lb><emph type="italics"></emph>to the Cylinder whoſe Axis is C B the <lb></lb>ſame proportion as its Baſe hath to <lb></lb>the Baſe of the other (for their Alti<lb></lb>tudes are equal.) But this propor<lb></lb>tion is the ſame with that which the <lb></lb>Square C N hath to the Square N B. <lb></lb></s> <s>And ſo we might prove, that the Cy<lb></lb>linder whoſe Axis is C B hath to the <lb></lb>Cylinder whoſe Axis is B E the ſame <lb></lb>proportion, as the Square B N hath to <lb></lb>the Square N E: and the Cylinder <lb></lb>whoſe Axis is B E hath to the Cylin<lb></lb>der whoſe Axis is E A the ſame pro<lb></lb>portion that the Square E N hath to <lb></lb>the Square N A. </s> <s>But the Lines N C, <lb></lb>N B, E N, and N A equally exceed one <lb></lb>another, and their exceſs equalleth the <lb></lb>leaſt, that is N A. </s> <s>Therefore they are certain Magnitudes, to wit, in<lb></lb>ſcribed Cylinders having conſequently to one another the ſame proporti<lb></lb>on as the Squares of Lines that equally exceed one another, and the ex-<emph.end type="italics"></emph.end><pb xlink:href="069/01/267.jpg" pagenum="264"></pb><emph type="italics"></emph>ceſs of which is equal to the leaſt: and they are ſo diſpoſed on the Beam <lb></lb>T I that their ſeveral Centers of Gravity conſiſt in it, and that at equal <lb></lb>diſtances. </s> <s>Therefore by the things above demonſtrated it appeareth that <lb></lb>the Center of Gravity of all ſo compoſed Magnitudes do ſo divide the <lb></lb>Balance T I, that the part to wards T is more than triple to the remain<lb></lb>der. </s> <s>Let this Center be O. </s> <s>T O therefore is more than triple to O I. <lb></lb></s> <s>But T N is triple to I M. </s> <s>Therefore the whole M O will be leſs than a <lb></lb>fourth part of the whole M N, whoſe fourth part was ſuppoſed to be <lb></lb>M S. </s> <s>It is manifeſt, therefore, that the point O doth nearer approach <lb></lb>the Baſe of the Cone than S. </s> <s>And let the circumſcribed Figure be com<lb></lb>poſed of the Cylinders whoſe Axis M C, C B, B E, E A and A N are <lb></lb>equal to each other, and, like as in thoſe inſcribed, let them be to one <lb></lb>another as the Squares of the Lines M N, N C, B N, N E, A N, <lb></lb>which equally exceed one another, and the exceſs is equal to the leaſt <lb></lb>A N. Wherefore, by the premiſes, the Center of Gravity of all the Cy<lb></lb>linders ſo diſpoſed, which let be V, doth ſo divide the Beam R I, that the <lb></lb>part towards R, to wit R V, is more than triple to the remaining part <lb></lb>V I: but T V ſhall be leſs than triple to the ſame. </s> <s>But N T is triple to <lb></lb>all I M: Therefore all V M is more than the fourth part of all M N, <lb></lb>whoſe fourth part was ſuppoſed to be M S. </s> <s>Therefore the point V is <lb></lb>nearer to the Vertex than the Point S. </s> <s>Which was to be demonſtra<lb></lb>ted.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>About a given Cone a Figure may be circumſcri<lb></lb>bed and another inſcribed conſiſting of Cylin<lb></lb>ders of equal height, ſo, as that the Line which <lb></lb>lyeth betwixt the Center of Gravity of the <lb></lb>circumſcribed, and the Center of Gravity of <lb></lb>the inſcribed, may be leſſer than any Line <lb></lb>given.</s></p><p type="main"> <s><emph type="italics"></emph>Let a Cone be given, whoſe Axis is A B; and let the Right Line <lb></lb>given be K. </s> <s>I ſay; Let there be placed by the Cylinder L <lb></lb>equal to that inſcribed in the Cone, having for its Altitude half <lb></lb>of the Axis A B: and let A B be divided in C, ſo as that A C be tri<lb></lb>ple to C B: And as A C is to K, ſo let the Cylinder L be to the Solid X. <lb></lb></s> <s>And about the Cone let there be a Figure circumſcribed of Cylin<lb></lb>ders that have equal Altitude, and let another be inſcribed, ſo as that <lb></lb>the circumſcribed exceed the inſcribed a leſs quantity than the Solid X. <lb></lb></s> <s>And let the Center of Gravity of the circumſcribed be E; which falls <lb></lb>above C: and let the Center of the inſcribed be S, falling beneath C.<emph.end type="italics"></emph.end><pb xlink:href="069/01/268.jpg" pagenum="265"></pb><emph type="italics"></emph>I ſay now, that the Line E S is leſſer than K. </s> <s>For if not, then let C A <lb></lb>be ſuppoſed equal to E O. </s> <s>Becauſe therefore O E hath to K the ſame <lb></lb>proportion that L hath to X; and the inſcribed Figure is not leſs than <lb></lb>the Cylinder L; and the exceſs with which the ſaid Figure is exceeded <lb></lb>by the circumſcribed is leſs than the Solid X: therefore the inſcribed <lb></lb>Figure ſhall have to the ſaid exceſs<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.268.1.jpg" xlink:href="069/01/268/1.jpg"></figure><lb></lb><emph type="italics"></emph>greater proportion than O E hath to <lb></lb>K: But the proportion of O E to K is <lb></lb>not leſs than that which O E hath to <lb></lb>E S with E S. </s> <s>Let it not be leſs than <lb></lb>K. </s> <s>Therefore the inſcribed Figure <lb></lb>hath to the exceſs of the circumſcri<lb></lb>bed Figure above it greater propor<lb></lb>tion than O E hath to E S. </s> <s>Therefore <lb></lb>as the inſcribed is to the ſaid exceſs, <lb></lb>ſo ſhall it be to the Line E S. </s> <s>Let E R <lb></lb>be a Line greater than E O; and the <lb></lb>Center of Gravity of the inſcribed <lb></lb>Figure is S; and the Center of the cir<lb></lb>cumſcribed is E. </s> <s>It is manifeſt there<lb></lb>fore, that the Center of Gravity of <lb></lb>the remaining proportions by which <lb></lb>the circumſcribed exceedeth the in <lb></lb>ſcribed is in the Line R E, and in that point by which it is ſo termina<lb></lb>ted, that as the inſcribed Figure is to the ſaid proportions, ſo is the Line <lb></lb>included betwixt E and that point to the Line E S. </s> <s>And this propor<lb></lb>tion hath R E to E S. </s> <s>Therefore the Center of Gravity of the remain<lb></lb>ing proportions with which the circumſcribed Figure exceeds the in<lb></lb>ſcribed ſhall be R, which is impoſſible. </s> <s>For the Plane drawn thorow <lb></lb>R equidiſtant to the Baſe of the Cone doth not cut thoſe proportions. </s> <s>It <lb></lb>is therefore falſe that the Line E S is not leſſer than K. </s> <s>It ſhall therefore <lb></lb>be leſs. </s> <s>The ſame alſo may be done in a manner not unlike this in Pyra<lb></lb>mides, as ne could demonſtrate.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>COROLLARY.</s></p><p type="main"> <s>Hence it is manifeſt, that a given Cone may circumſcribe one <lb></lb>Figure and inſcribe another conſiſting of Cylinders of equal <lb></lb>Altitudes ſo, as that the Lines which are intercepted betwixt <lb></lb>their Centers of Gravity and the point which ſo divides the <lb></lb>Axis of the Cone, as that the part towards the Vertex is tri<lb></lb>ple to the leſt, are leſs than any given Line.</s></p><p type="main"> <s><emph type="italics"></emph>For, ſince it hath been demonſtrated, that the ſaid point dividing the <lb></lb>Axis, as was ſaid, is alwaies found betwixt the Centers of Gravity<emph.end type="italics"></emph.end><pb xlink:href="069/01/269.jpg" pagenum="266"></pb><emph type="italics"></emph>of the Circumſcribed and inſcribed Figures: and that it's poſſible, that <lb></lb>there be a Line in the middle betwixt thoſe Centers that is leſs than any <lb></lb>Line aſſigned; it followeth that the ſame given Line be much leſs that <lb></lb>lyeth betwixt one of the ſaid Centers and the ſaid point that divides <lb></lb>the Axis.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>The Center of Gravity divideth the Axis of any <lb></lb>Cone or Pyramid ſo, that the part next the <lb></lb>Vertex is triple to the remainder.</s></p><p type="main"> <s><emph type="italics"></emph>Let there be a Cone whoſe Axis is A B. </s> <s>And in C let it be divided, <lb></lb>ſo that A C be triple to the remaining part C B. </s> <s>It is to be proved, <lb></lb>that C is the Center of Gravity of the Cone. </s> <s>For if it be not, the <lb></lb>Cone's Center ſhall be either above or below the point C. </s> <s>Let it be firſt <lb></lb>beneath, and let it be E. </s> <s>And draw the Line L P, by it ſelf, equal to <lb></lb>C E; which divided at pleaſure in N. </s> <s>And as both B E and P N to<lb></lb>gether are to P N, ſo let the Cone be to the Solid X: and inſcribe in the <lb></lb>Cone a Solid Figure of Cylinders that have equal Baſes, whoſe Center <lb></lb>of Gravity is leſs diſtant from the point C than is the Line L N, and <lb></lb>the exceſs of the Cone above it leſs than the Solid X. </s> <s>And that this <lb></lb>may be done is manifeſt from what hath been already demonſtrated. <lb></lb></s> <s>Now let the inſcribed Figure be ſuch as<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.269.1.jpg" xlink:href="069/01/269/1.jpg"></figure><lb></lb><emph type="italics"></emph>was required, whoſe Center of Gravity <lb></lb>let be I. </s> <s>The Line I E therefore ſhall be <lb></lb>greater than N P together with L P. </s> <s>Let <lb></lb>C E and I C leſs L N be equal: And be<lb></lb>cauſe both together B E and N P is to N P <lb></lb>as the Cone to X: and the exceſs by which <lb></lb>the Cone exceeds the inſcribed Figure is <lb></lb>leſs than the Solid X: Therefore the Cone <lb></lb>ſhall have greater proportion to the ſaid <lb></lb>X S than both B E and N P to N P: and, by <lb></lb>Diviſion, the inſcribed Figure ſhall have <lb></lb>greater proportion to the exceſs by which <lb></lb>the Cone exceeds it, than B E to N P: But B E hath leſs proportion to <lb></lb>E I than to N P with I E. </s> <s>Let N P be greater. </s> <s>Then the inſcribed Fi<lb></lb>gure hath to the exceſs of the Cone above it much greater proportion <lb></lb>than B E to E I. </s> <s>Therefore as the inſcribed Figure is to the ſaid exceſs, <lb></lb>ſo ſhall a Line bigger than B E be to E I. </s> <s>Let that Line be M E. Becauſe, <lb></lb>therefore, M E is to E I as the inſcribed Figure is to the exceſs of the <lb></lb>Cone above the ſaid Figure, and D is the Center of Gravity of the <lb></lb>Cone, and I the Center of Gravity of the inſcribed Figure: Therefore<emph.end type="italics"></emph.end><pb xlink:href="069/01/270.jpg" pagenum="267"></pb><emph type="italics"></emph>M ſhall be the Center of Gravity of the remaining proportions by which <lb></lb>the Cone exceeds the inſcribed Figure. </s> <s>Which is impoſſible. </s> <s>Therefore <lb></lb>the Center of Gravity of the Cone is not below the point C. </s> <s>Nor is it <lb></lb>above it. </s> <s>For if it may be, let it be R. </s> <s>And again aſſume L P cut at <lb></lb>pleaſure in N: And as both B C and N P together are to N L, ſo let the <lb></lb>Cone be to X. </s> <s>And let a Figure be, in like manner, circumſcribed about <lb></lb>the Cone, which exceeds the ſaid Cone a leſs quantity than the Solid X. <lb></lb></s> <s>And let the Line which intercepts bet wixt its Center of Gravity and C, <lb></lb>be leſſer than N P. </s> <s>Now take the circumſcribed Figure, whoſe Center <lb></lb>let be O; the remainder O R ſhall be greater than the ſaid N L. </s> <s>And <lb></lb>becauſe, as both together B C and P N is to N L, ſo is the Cone to X: <lb></lb>And the exceſs by which the circumſcribed exceeds the Cone is leſſer <lb></lb>than X: And B O is leſſer than B C and P N together: And O R grea<lb></lb>ter than L N: The Cone therefore ſhall have much greater proportion to <lb></lb>the remaining proportions by which it was exceeded by the circumſcribed <lb></lb>Figure, than B O to O R. </s> <s>Let it be as M O is to O R. </s> <s>M O ſhall <lb></lb>be greater than B C; and M ſhall be the Center of Gravity of the pro<lb></lb>portions by which the Cone is exceeded by the circumſcribed Figure. <lb></lb></s> <s>Which is inconvenient. </s> <s>Therefore the Center of Gravity of the Cone is <lb></lb>not above the point C. </s> <s>But neither is it below it; as hath been proved. <lb></lb></s> <s>Therefore it ſhall be C it ſelf. </s> <s>And ſo in like manner may it be demon<lb></lb>ſtrated in any Pyramid.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>If there were four Lines continual proportionals; <lb></lb>and as the leaſt of them were to the exceſs by <lb></lb>which the greateſt exceeds the leaſt, ſo a Line <lb></lb>taken at pleaſure ſhould be to 3/4 the exceſs by <lb></lb>which the greateſt exceeds the ſecond; and as <lb></lb>the Line equal to theſe (<emph type="italics"></emph>viz.<emph.end type="italics"></emph.end> to the greateſt, <lb></lb>double of the ſecond, and triple of the third) <lb></lb>is to the Line equal to the quadruple of the <lb></lb>fourth, the quadruple of the ſecond, and the <lb></lb>quadruple of the third, ſo ſhould another Line <lb></lb>taken be to the exceſs of the greateſt above the <lb></lb>ſecond: theſe two Lines taken together ſhall <lb></lb>be a fourth part of the greateſt of the propor<lb></lb>tionals.</s></p><pb xlink:href="069/01/271.jpg" pagenum="268"></pb><p type="main"> <s><emph type="italics"></emph>For let A B, B C, B D, and B E be four proportional Lines. </s> <s>And <lb></lb>as B E is to E A, ſo let F G be to 3/4 of A C. </s> <s>And as the Line equal <lb></lb>to A B and to double B C and to triple B D is to the Line equal <lb></lb>to the quadruples of A B, B C, and B D, ſo let H G be to A C. </s> <s>It is <lb></lb>to be proved, that H F is a fourth part of A B. </s> <s>Foraſmuch therefore <lb></lb>as A B, B C, B D, and B E<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.271.1.jpg" xlink:href="069/01/271/1.jpg"></figure><lb></lb><emph type="italics"></emph>are proportionals, A C, <lb></lb>C D, and D E ſhall be in <lb></lb>the ſame proportion: And <lb></lb>as the quadruple of the ſaid <lb></lb>A B, B C, and B D is to <lb></lb>A B with the double of B C and triple of B D, ſo is the quadruple of <lb></lb>A C, C D, and D E; that is, the quadruple of A E; to A C with the <lb></lb>double of C D, and triple of D E. </s> <s>And ſo is A C to H G. </s> <s>Therefore <lb></lb>as the triple of A E is to A C, with the double of C D and triple of <lb></lb>D E, ſo is 3/4 of A C to H G. </s> <s>And as the triple of A E is to the triple of <lb></lb>E B, ſo is 3/4 A C to G F: Therefore, by the Converſe of the twenty <lb></lb>fourth of the fifth, As triple A E is to A C with double C D and tri<lb></lb>ple D B, ſo is 3/4 of A C to H F: And as the quadruple of A E is to A C <lb></lb>with the double of C D and triple of D B; that is, to A B with C B and <lb></lb>B D, ſo is A C to H F. And, by Permutation, as the quadruple of A E <lb></lb>is to A C, ſo is A B with C B and B D to H F. </s> <s>And as A C is to A E, ſo <lb></lb>is A B to A B with C B and B D. Therefore,<emph.end type="italics"></emph.end> ex æquali, <emph type="italics"></emph>by Perturbed <lb></lb>proportion, as quadruple A E is to A E, ſo is A B to H F. </s> <s>Wherefore it <lb></lb>is manifeſt that H F is the fourth part of A B.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROPOSITION.</s></p><p type="main"> <s>The Center of Gravity of the <emph type="italics"></emph>Fruſtum<emph.end type="italics"></emph.end> of any Py<lb></lb>ramid or Cone, cut equidiſtant to the Plane <lb></lb>of the Baſe, is in the Axis, and doth ſo divide <lb></lb>the ſame, that the part towards the leſſer Baſe <lb></lb>is to the remainder, as the triple of the greater <lb></lb>Baſe, with the double of the mean Space be<lb></lb>twixt the greater and leſſer Baſe, together <lb></lb>with the leſſer Baſe is to the triple of the leſſer <lb></lb>Baſe, together with the ſame double of the <lb></lb>mean Space, as alſo of the greater Baſe.</s></p><pb xlink:href="069/01/272.jpg" pagenum="269"></pb><p type="main"> <s><emph type="italics"></emph>From a Cone or Pyramid whoſe Axis is A D, and equidiſtant to <lb></lb>the Plane of the Baſe, let a<emph.end type="italics"></emph.end> Fruſtum <emph type="italics"></emph>be cut whoſe Axis is V D. <lb></lb></s> <s>And as the triple of the greateſt Baſe with the double of the <lb></lb>mean and leaſt is to the triple of the leaſt and double of the mean and <lb></lb>greateſt, ſo is \ O to O D. </s> <s>It is to be proved that the Center of Gra<lb></lb>vity of the<emph.end type="italics"></emph.end> Fruſtum <emph type="italics"></emph>is in O. </s> <s>Let V M be the fourth part of V D. <lb></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></lb>and unto H X K let X L be a third proportional, and X S a fourth. <lb></lb></s> <s>And as H S is to S X, ſo let M D be to the Line taken from O towards <lb></lb>A: which let be O N. </s> <s>And becauſe the greater Baſe is in proportion <lb></lb>to that which is mean betwixt the <lb></lb>greater and leſſer as D A to A V; that<emph.end type="italics"></emph.end><lb></lb><figure id="id.069.01.272.1.jpg" xlink:href="069/01/272/1.jpg"></figure><lb></lb><emph type="italics"></emph>is, as H X, to X K, but the ſaid <lb></lb>mean is to the leaſt as K X to X L; <lb></lb>the greater, mean, and leſſer Baſes <lb></lb>ſhall be in the ſame proportion as <lb></lb>H X, X K, and X L. </s> <s>Wherefore as <lb></lb>triple the greater Baſe, with double <lb></lb>the mean and leſſer, is to triple the <lb></lb>leaſt with double the mean and grea<lb></lb>teſt; that is, as V O is to O D; ſo is <lb></lb>triple H X with double X K and X L <lb></lb>to triple X L, with double X K and <lb></lb>X H: And by Compoſition and Converting the proportion, O D ſhall <lb></lb>be to V D, as H X, with double X K and triple X L, to quadruple H X, <lb></lb>X K, and X L. </s> <s>There are, therefore, four proportional Lines, H X, <lb></lb>X K, X L, and X S: And as X S is to S H, ſo is the Line taken N O <lb></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></lb>with double X K and triple X L is to quadruple H X, X K and X L; <lb></lb>ſo is another Line taken O D to D V; that is, to H K. Therefore, by <lb></lb>the things demonſtrated, D N ſhall be the fourth part of H X; that <lb></lb>is, of A D. </s> <s>Wherefore the point N ſhall be the Center of Gravity <lb></lb>of the Cone or Pyramid whoſe Axis is A D. </s> <s>Let the Center of Gra<lb></lb>vity of the Pyramid or Cone whoſe Axis is A V be I. </s> <s>It is therefore <lb></lb>manifeſt that the Center of Gravity of the<emph.end type="italics"></emph.end> Fruſtum <emph type="italics"></emph>is in the Line <lb></lb>I N inclining towards the part N, and in that point of it which with <lb></lb>the point N include a Line to which I M hath the ſame proportion that <lb></lb>the<emph.end type="italics"></emph.end> Fruſtum <emph type="italics"></emph>cut hath to the Pyramid or Cone whoſe Axis is A V. <lb></lb></s> <s>It remaineth therefore to prove that I N hath the ſame proportion <lb></lb>to N O, that the<emph.end type="italics"></emph.end> Fruſtum <emph type="italics"></emph>hath to the Cone whoſe Axis is A V. </s> <s>But <lb></lb>as the Cone whoſe Axis is D A is to the Cone whoſe Axis is A V, ſo <lb></lb>is the Cube D A to the Cube D V; that is, the Cube H X to the <lb></lb>Cube X K: But this is the ſame proportion that H X hath to X S. <lb></lb>Wherefore, by Diviſion, as H S is to S X, ſo ſhall the<emph.end type="italics"></emph.end> Fruſtum <emph type="italics"></emph>whoſe<emph.end type="italics"></emph.end><pb xlink:href="069/01/273.jpg" pagenum="270"></pb><emph type="italics"></emph>Axis is D V be to the Cone or Pyramid whoſe Axis is V A. </s> <s>And as <lb></lb>H S is to S X, ſo alſo is M D to O N. </s> <s>Wherefore the<emph.end type="italics"></emph.end> Fruſtum <emph type="italics"></emph>is to the <lb></lb>Pyramid whoſe Axis is A V, as M D to N O. </s> <s>And becauſe A N <lb></lb>is 3/4 of A D; and A I is 3/4 of A V; the remainder I N ſhall be 3/4 of the <lb></lb>remainder V D. </s> <s>Wherefore I N ſhall be equal to M D. <lb></lb></s> <s>And it hath been demonſtrated that M D is to N O, <lb></lb>as the<emph.end type="italics"></emph.end> Fruſtum <emph type="italics"></emph>to the Cone A V. </s> <s>It is mani<lb></lb>feſt, therefore, that I N hath likewiſe <lb></lb>the ſame proportion to N O: <lb></lb>Wherefore the Propo<lb></lb>ſition is manifeſt.<emph.end type="italics"></emph.end></s></p><p type="head"> <s><emph type="italics"></emph>FINIS.<emph.end type="italics"></emph.end><lb></lb></s></p> </chap> </body> <back></back> </text></archimedes>