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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>Mechanics</title> <date>1665</date> <place>London</place> <translator>Thomas Salusbury</translator> <lang>en</lang> <cvs_file>galil_mecha_070_en_1665.xml</cvs_file> <cvs_version></cvs_version> <locator>070.xml</locator> </info> <text> <front><section> <pb xlink:href="070/01/001.jpg" pagenum="271"></pb><p type="head"> <s>GALILEUS, <lb></lb>HIS <lb></lb>MECHANICKS: <lb></lb>OF THE BENEFIT DERIVED <lb></lb>FROM THE SCIENCE OF MECHANICKS, <lb></lb>AND FROM ITS INSTRUMENTS.</s></p> </section> </front> <body> <chap> <p type="main"> <s>I judged it extreamly neceſſary, before our <lb></lb>deſcending to the Speculation of Mecha<lb></lb>nick Inſtruments, to conſider how I might, <lb></lb>as it were, ſet before your eyes in a gene<lb></lb>ral Diſcourſe, the many benefits that are <lb></lb>derived from the ſaid Inſtruments: and <lb></lb>this I have thought my ſelf the more ob<lb></lb>liged to do, for that (if I am not miſtaken) <lb></lb>I have ſeen the generality of <emph type="italics"></emph>M<emph.end type="italics"></emph.end>echaniti<lb></lb>ans deceive themſelves in going about to apply Machines to many <lb></lb>operations of their own nature impoſſible; by the ſucceſſe where<lb></lb>of they have been diſappointed, and others likewiſe fruſtrate of <lb></lb>the hope which they had conceived upon the promiſe of thoſe pre<lb></lb>ſumptuous undertakers: of which miſtakes I think I have found <lb></lb>the principall cauſe to be the belief and conſtant opinion theſe <pb xlink:href="070/01/002.jpg" pagenum="272"></pb>Artificers had, and ſtill have, that they are able with a ſmall force <lb></lb>to move and raiſe great weights; (in a certain manner with their <lb></lb>Machines cozening nature, whoſe Inſtinct, yea moſt poſitive con<lb></lb>ſtitution it is, that no Reſiſtance can be overcome, but by a Force <lb></lb>more potent then it:) which conjecture how falſe it is, I hope by <lb></lb>the enſuing true and neceſſary Demonſtrations to evince.</s></p><p type="main"> <s>In the mean time, ſince I have hinted, that the benefit and help <lb></lb>derived from Machines is not, to be able with leſſe Force, by help <lb></lb>of the Machine to move thoſe weights, which, without it, could <lb></lb>not be moved by the ſame Force: it would not be beſides the <lb></lb>purpoſe to declare what the Commodities be which are derived to <lb></lb>us from ſuch like faculties, for if no profit were to be hoped for, <lb></lb>all endeavours employed in the acquiſt thereof will be but loſt <lb></lb>labour.</s></p><p type="main"> <s>Proceeding therefore according to the nature of theſe Studies, <lb></lb>let us firſt propoſe four things to be conſidered. </s> <s>Firſt, the weight <lb></lb>to be transferred from place to place; and ſecondly, the Force <lb></lb>and Power which ſhould move it; thirdly, the Diſtance between <lb></lb>the one and the other Term of the Motion; Fourthly, the Time <lb></lb>in which that mutation is to be made: which Time becometh the <lb></lb>ſame thing with the Dexterity, and Velocity of the Motion; we <lb></lb>determining that Motion to be more ſwift then another, which in <lb></lb>leſſe Time paſſeth an equal Diſtance.</s></p><p type="main"> <s>Now, any determinate Reſiſtance and limited Force whatſoever <lb></lb>being aſſigned, and any Diſtance given, there is no doubt to be <lb></lb>made, but that the given Force may carry the given Weight to the <lb></lb>determinate Diſtance; for, although the Force were extream <lb></lb>ſmall, yet, by dividing the Weight into many ſmall parts, none <lb></lb>of which remain ſuperiour to the Force, and by transferring them <lb></lb>one by one, it ſhall at laſt have carried the whole Weight to the <lb></lb>aſſigned Term: and yet one cannot at the end of the Work with <lb></lb>Reaſon ſay, that that great Weight hath been moved, and tranſ<lb></lb>ported by a Force leſſe then it ſelf, howbeit indeed it was done <lb></lb>by a Force, that many times reiterated that Motion, and that <lb></lb>Space, which ſhall have been meaſured but only once by the whole <lb></lb>Weight. </s> <s>From whence it appears, that the Velocity of the Force <lb></lb>hath been as many times Superiour to the Reſiſtance of the weight, <lb></lb>as the ſaid Weight was ſuperiour to the Force; for that in the <lb></lb>ſame Time that the moving Force hath many times meaſured the <lb></lb>intervall between the Terms of the Motion, the ſaid Moveable <lb></lb>happens to have paſt it onely once: nor therefore ought we to <lb></lb>affirm a great Reſiſtance to have been overcome by a ſmall Force, <lb></lb>contrary to the conſtitution of Nature. </s> <s>Then onely may we ſay <lb></lb>the Natural Conſtitution is overcome, when the leſſer Force tranſ<lb></lb>fers the greater Reſiſtance, with a Velocity of Motion like to that <pb xlink:href="070/01/003.jpg" pagenum="273"></pb>wherewith it ſelf doth move; which we affirm abſolutely to be <lb></lb>impoſſible to be done with any Machine imaginable. </s> <s>But becauſe <lb></lb>it may ſometimes come to paſſe, that having but little Force, it is <lb></lb>required to move a great Weight all at once, without dividing it <lb></lb>in pieces, on this occaſion it will be neceiſary to have recourſe to <lb></lb>the Machine, by means whereof the propoſed Weight may be <lb></lb>transferred to the aſſigned Space by the Force given. </s> <s>But yet <lb></lb>this doth not hinder, but that the ſame Force is to move, meaſuring <lb></lb>that ſame Space, or another equall to it, as many ſeverall times as <lb></lb>it is exceeded by the ſaid Weight. </s> <s>So that in the end of the a<lb></lb>ction we ſhall ſind that we have received from the Machine no <lb></lb>other benefit tnen only that of tranſporting the ſaid Weight with <lb></lb>the given Force to the Term given, all at once. </s> <s>Which Weight, <lb></lb>being divided into parts, would without any Machine have been <lb></lb>carried by the ſame Force, in the ſame Time, through the ſame <lb></lb>Intervall. </s> <s>And this ought to paſſe for one of the benefits taken <lb></lb>from the Mechanicks: for indeed it frequently happens, that be<lb></lb>ing ſcanted in Force but not Time, we are put upon moving great <lb></lb>Weights unitedly or in groſſe: but he that ſhould hope, and at<lb></lb>tempt to do the ſame by the help of Machines without increaſe of <lb></lb>Tardity in the Moveable, would certainly be deceived, and would <lb></lb>declare his ignorance of the uſe of Mechanick Inſtruments, and <lb></lb>the reaſon of their effects.</s></p><p type="main"> <s>Another benefit is drawn from the Inſtruments, which depend<lb></lb>eth on the place wherein the operation is to be made: for all In<lb></lb>ſtruments cannot be made uſe of in all places with equall conve<lb></lb>nience. </s> <s>And ſo we ſee (to explain our ſelves by an example) that <lb></lb>for drawing of Water out of a Well, we make uſe of onely a <lb></lb>Rope and a Bucket fitted to receive and hold Water, wherewith <lb></lb>we draw up a determinate quantity of Water, in a certain Time, <lb></lb>with our limited ſtrength: and he that ſhould think he could with <lb></lb>a Machine of whatſoever Force, with the ſame ſtrength, and in <lb></lb>the ſame Time, take up a great quantity of Water, is in a groſſe <lb></lb>Errour. </s> <s>And he ſhall find himſelf ſo much the more deceived, <lb></lb>the more he ſhall vary and multiply his Inventions: Yet never<lb></lb>theleſſe we ſee Water drawn up with other Engines, as with a Pump <lb></lb>that drinks up Water in the Hold of Ships; where you muſt note <lb></lb>that the Pump was not imployed in thoſe Offices, for that it draws <lb></lb>up more Water in the ſame Time, and with the ſame ſtrength <lb></lb>then that which a bare Bucket would do, but becauſe in that place <lb></lb>the uſe of the Bucket or any ſuch like Veſſel could not effect what <lb></lb>is deſired, namely to keep the Hold of the Ship quite dry from e<lb></lb>very little quantity of Water; which the Bucket cannot do, for <lb></lb>that it cannot dimerge and dive, where there is not a conſiderable <lb></lb>depth of Water. </s> <s>And thus we ſee the Holds of Ships by the <pb xlink:href="070/01/004.jpg" pagenum="274"></pb>ſaid Inſtrument kept dry, when Water cannot but onely oblique<lb></lb>ly be drawn up, which the ordinary uſe of the Bucket would not <lb></lb>effect, which riſeth and deſcends with its Rope perpendicu<lb></lb>larly.</s></p><p type="main"> <s>The third is a greater benefit, haply, then all the reſt that are <lb></lb>derived from Mechanick Inſtruments, and reſpects the aſſiſtance <lb></lb>which is borrowed of ſome Force exanimate, as of the ſtream of a <lb></lb>River, or elſe animate, but of leſſe expence by far, then that which <lb></lb>would be neceſſary for maintaining humane ſtrength: as when to <lb></lb>turn Mills, we make uſe of the Current of a River, or the ſtrength <lb></lb>of a Horſe, to effect that, which would require the ſtrength of five <lb></lb>or fix Men. </s> <s>And this we may alſo advantage our ſelves in raiſing <lb></lb>Water, or making other violent Motions, which muſt have been <lb></lb>done by Men, if there were no other helps; becauſe with one ſole <lb></lb>Veſſel we may take Water, and raiſe, and empty it where occaſion <lb></lb>requires; but becauſe the Horſe, or ſuch other Mover wanteth <lb></lb>Reaſon, and thoſe Inſtruments which are requiſite for holding and <lb></lb>emptying the Veſſel in due time, returning again to fill it, and one<lb></lb>ly is endued with Force, therefore it's neceſſary that the Mecha<lb></lb>nitian ſupply the naturall defect of that Mover, furniſhing it with <lb></lb>ſuch devices and inventions, that with the ſole application of it's <lb></lb>Force the defired effect may follow. </s> <s>And therein is very great <lb></lb>advantage, not becauſe that a Wheel or other Machine can enable <lb></lb>one to tranſport the ſame Weight with leſſe Force, and greater <lb></lb>Dexterity, or a greater Space than an equall Force, without thoſe <lb></lb>Inſtruments, but having Judgment and proper Organs, could have <lb></lb>done; but becauſe that the ſtream of a River coſteth little or <lb></lb>nothing, and the charge of keeping of an Horſe or other Beaſt, <lb></lb>whoſe ſtrength is greater then that of eight, or it may be more <lb></lb>Men, is far leſſe then what ſo many Men would be kept <lb></lb>for.</s></p><p type="main"> <s>Theſe then are the benefits that may be derived from Mecha<lb></lb>nick Inſtruments, and not thoſe which ignorant Engineers dream <lb></lb>of, to their own diſgrace, and the abuſe of ſo many Princes, <lb></lb>whilſt they undertake impoſſible enterprizes; of which, both <lb></lb>by the little which hath been hinted, and by the much which <lb></lb>ſhall be demonſtrated in the Progreſſe of this Treatiſe, we ſhall <lb></lb>come to aſſure our ſelves, if we attentively heed that which ſhall <lb></lb>be ſpoken.</s></p><pb xlink:href="070/01/005.jpg" pagenum="275"></pb><p type="head"> <s>DEFINITIONS.</s></p><p type="main"> <s>That which in all Demonſtrative Sciences is neceſſary to be <lb></lb>obſerved, we ought alſo to follow in this Diſcourſe, that is; <lb></lb>to propound the Definitions of the proper Terms of this <lb></lb>Art, and the primary Suppoſitions, from which, as from ſeeds full <lb></lb>of fecundity, may of conſequence ſpring and reſult the cauſes, <lb></lb>and true Demonſtrations, of the Nature of all the Mechanick <lb></lb>Engines which are uſed, for the moſt part about the Motions of <lb></lb>Grave Matters, therefore we will determine, firſt, what is <emph type="italics"></emph>GRA<lb></lb>VITIE.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>We call <emph type="italics"></emph>GRAVITIE<emph.end type="italics"></emph.end> then, That propenſion of moving <lb></lb>naturally downwards, which is found in ſolid Bodies, cauſed by <lb></lb>the greater or leſſe quantity of matter, whereof they are conſti<lb></lb>tuted.</s></p><p type="main"> <s><emph type="italics"></emph>MOMENT<emph.end type="italics"></emph.end> is the propenſion of deſcending, cauſed not ſo <lb></lb>much by the Gravity of the moveable, as by the diſpoſure which <lb></lb>divers Grave Bodies have in relation to one another; by means of <lb></lb>whichMoment, we oft ſee a Body leſs Grave counterpoiſe another <lb></lb>of greater Gravity: as in the Stiliard, a great Weight is raiſed by <lb></lb>a very ſmall counterpoiſe, not through exceſs of Gravity, but <lb></lb>through the remoteneſſe from the point whereby the Beam is up<lb></lb>held, which conjoyned to the Gravity of the leſſer weight adds <lb></lb>thereunto Moment, and <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of deſcending, wherewith the <lb></lb>Moment of the other greater Gravity may be exceeded. <emph type="italics"></emph>MO<lb></lb>MENT<emph.end type="italics"></emph.end> then is that IMPETUS of deſcending, compounded <lb></lb>of Gravity, Poſition, and the like, whereby that propenfion may <lb></lb>be occaſioned</s></p><p type="main"> <s>The <emph type="italics"></emph>CENTER<emph.end type="italics"></emph.end> of <emph type="italics"></emph>GRAVITY<emph.end type="italics"></emph.end> we define to be that point <lb></lb>in every Grave Body, about which conſiſt parts of equall Moment: <lb></lb>ſo that, imagining ſome Grave Body to be ſuſpended and ſuſtain<lb></lb>ed by the ſaid point, the parts on the right hand will Equilibrate <lb></lb>thoſe on the left, the Anteriour, the Poſteriour, and thoſe above <lb></lb>thoſe below; ſo that be it in any whatſoever fite, and poſition, <lb></lb>provided it be ſuſpended by the ſaid <emph type="italics"></emph>CENTER,<emph.end type="italics"></emph.end> it ſhall ſtand <lb></lb>ſtill: and this is that point which would gladly unite with the <lb></lb>univerſall Center of Grave Bodies, namely withthat of the Earth, <lb></lb>if it might thorow ſome free <emph type="italics"></emph>Medium<emph.end type="italics"></emph.end> deſcend thither. </s> <s>From <lb></lb>whence we take theſe Suppoſitions.</s></p><pb xlink:href="070/01/006.jpg" pagenum="276"></pb><p type="head"> <s>SUPPOSITIONS.</s></p><p type="main"> <s>Any Grave Body, (as to what belongeth to it's proper ver<lb></lb>tue) moveth downwards, ſo that the Center of it's Gravity <lb></lb>never ſtrayeth out of that Right Line which is produced <lb></lb>from the ſaid Center placed in the firſt Term of the Motion unto <lb></lb>the univerſal Center of Grave Bodies. </s> <s>Which is a Suppoſition <lb></lb>very manifeſt, becauſe that ſingle Center being obliged to endea<lb></lb>vour to unite with the common Center, it's neceſſary, unleſſe ſome <lb></lb>impediment intervene, that it go ſeeking it by the ſhorteſt Line, <lb></lb>which is the Right alone: And from hence may we ſecondarily <lb></lb>ſuppoſe</s></p><p type="main"> <s>Every Grave Body putteth the greateſt ſtreſſe, and weigheth <lb></lb>moſt on the Center of it's Gravity, and to it, as to its proper ſeat, <lb></lb>all <emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> all Ponderoſity, and, in ſome, all Moment hath re<lb></lb>courſe.</s></p><p type="main"> <s>We laſtly ſuppoſe the Center of the Gravity of two Bodies e<lb></lb>qually Grave to be in the midſt of that Right Line which conjoyns <lb></lb>the ſaid two Centers; or that two equall weights, ſuſpended in <lb></lb>equall diſtence, ſhall have the point of <emph type="italics"></emph>Equilibrium<emph.end type="italics"></emph.end> in the common <lb></lb>Center, or meeting of thoſe equal Diſtances. </s> <s>As for Example, <lb></lb>the Diſtance C E being equall to the Diſtance E D, and there be<lb></lb>ing by them two equall weights ſuſpended, A and B, we ſuppoſe <lb></lb>the point of <emph type="italics"></emph>Equilibrium<emph.end type="italics"></emph.end> to be in the point E, there being no <lb></lb>greater reaſon for inclining to <lb></lb>one, then to the other part. </s> <s>But <lb></lb><figure id="id.070.01.006.1.jpg" xlink:href="070/01/006/1.jpg"></figure><lb></lb>here is to be noted, that the Di<lb></lb>ſtances ought to be meaſured <lb></lb>with Perpendicular Lines, which <lb></lb>from the point of Suſpenſion E, <lb></lb>fall on the Right Lines, that from <lb></lb>the Center of the Gravity of the <lb></lb>Weights A and B, are drawn to <lb></lb>the common Center of things <lb></lb>Grave; and therefore if the Diſtance E D were tranſported into <lb></lb>E F, the weight B would not counterpoiſe the weight A, becauſe <lb></lb>drawing from the Centers of Gravity two Right Lines to the Cen<lb></lb>ter of the Earth, we ſhall ſee that which cometh from the Center <lb></lb>of the Weight I, to be nearer to the Center E, then the other <lb></lb>produced from the Center of the weight A. </s> <s>Therefore our ſaying <lb></lb>that equal Weights are ſuſpended by [or at] equal Diſtances, is <lb></lb>to be underſtood to be meant when as the Right Lines that go from <lb></lb>their Centers & to ſeek out the common Center of Gravity, ſhall be <lb></lb>equidiſta nt from that Right Line, which is produced from the ſaid <pb xlink:href="070/01/007.jpg" pagenum="277"></pb>Term of thoſe Diſtances, that is from the point of Suſpenſion, to <lb></lb>the ſame Center of the Earrh.</s></p><p type="main"> <s>Theſe things determined and ſuppoſed, we come to the explica<lb></lb>tion of a Principle, the moſt common and materiall of the greater <lb></lb>part of Mechanick Inſtruments: demonſtrating, that unequall <lb></lb>Weights weigh equally when ſuſpended by [or at] unequal Diſtan<lb></lb>ces, which have contrary proportion to that which thoſe weights <lb></lb>are found to have, See the Demonſtration in the beginning of the <lb></lb>ſecond Dialogue of Local-Motions.</s></p><p type="head"> <s><emph type="italics"></emph>Some Adveriiſements about what hath been ſaid.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Now being that Weights unequall come to acquire equall <lb></lb>Moment, by being alternately ſuſpended at Diſtances that <lb></lb>have the ſame proportion with them; I think it not fit to <lb></lb>over paſſe with ſilence another congruicy and probability, which <lb></lb>may confirm the ſame truth; for let the Ballance A B, be conſide<lb></lb>red, as it is divided into unequal parts in the point C, and let the <lb></lb>Weights be of the ſame propor<lb></lb><figure id="id.070.01.007.1.jpg" xlink:href="070/01/007/1.jpg"></figure><lb></lb>tion that is between the Diſtan<lb></lb>ces B C, and C A, alternately <lb></lb>ſuſpended by the points A, and <lb></lb>B: It is already manifeſt, that <lb></lb>the one will counterpoiſe the <lb></lb>other, and conſequently, that <lb></lb>were there added to one of them <lb></lb>a very ſmall Moment of Gravity, it would preponderate, raiſing <lb></lb>the other, ſo that an inſenſible Weight put to the Grave B, the <lb></lb>Ballance would move and deſcend from the point B towards E, <lb></lb>and the other extream A would aſcend into D, and in regard that <lb></lb>to weigh down B, every ſmall Gravity is ſufficient, therefore not <lb></lb>keeping any accompt of this inſenſible Moment, we will put no <lb></lb>difference between one Weights <emph type="italics"></emph>ſuſtaining,<emph.end type="italics"></emph.end> and one Weights <lb></lb><emph type="italics"></emph>moving<emph.end type="italics"></emph.end> another. </s> <s>Now, let us conſider the Motion which the <lb></lb>Weight B makes, deſcending into E, and that which the other <lb></lb>A makes in aſcending into D, we ſhall without doubt find the <lb></lb>Space B E to be ſo much greater than the Space A D, as the Di<lb></lb>ſtance B C is greater than C A, forming in the Center C two an<lb></lb>gles D C A, and E C B, equall as being at the Cock, and conſe<lb></lb>quently two Circumferences A D and B E alike; and to have the <lb></lb>ſame proportion to one another, as have the Semidiameters B C, <lb></lb>and C A, by which they are deſcribed: ſo that then the Velocity <lb></lb>of the Motion of the deſcending Grave B cometh to be ſo much <lb></lb>Superiour to the Velocity of the other aſcending Moveable A, as <lb></lb>the Gravity of this exceeds the Gravity of that; and it not being <pb xlink:href="070/01/008.jpg" pagenum="278"></pb>poſſible that the Weight A ſhould be raiſed to D, although ſlow<lb></lb>ly, unleſſe the other Weight B do move to E ſwiftly, it will not <lb></lb>be ſtrange, or inconſiſtent with the Order of Nature, that the <lb></lb>Velocity of the Motion of the Grave B, do compenſate the greater <lb></lb>Reſiſtance of the Weight A, ſo long as it moveth ſlowly to D, <lb></lb>and the other deſcendeth ſwiftly to E, and ſo on the contrary, <lb></lb>the Weight A being placed in the point D, and the other B in <lb></lb>the point E, it will not be unreaſonable that that falling leaſurely <lb></lb>to A, ſhould be able to raiſe the other haſtily to B, recovering by <lb></lb>its Gravity what it had loſt by it's Tardity of Motion. </s> <s>And by <lb></lb>this Diſcourſe we may come to know how the Velocity of the <lb></lb>Motion is able to encreaſe Moment in the Moveable, according to <lb></lb>that ſame proportion by which the ſaid Velocity of the Motion is <lb></lb>augmented.</s></p><p type="main"> <s>There is alſo another thing, before we proceed any farther, to <lb></lb>be confidered; and this is touching the Diſtances, whereat, or <lb></lb>wherein Weights do hang: for it much imports how we are to <lb></lb>underſtand Diſtances equall, and unequall; and, in ſum, in what <lb></lb>manner they ought to be mea<lb></lb><figure id="id.070.01.008.1.jpg" xlink:href="070/01/008/1.jpg"></figure><lb></lb>ſured: for that A B being the <lb></lb>Right Line, and two equall <lb></lb>Weights being ſuſpended at <lb></lb>the very ends thereof, the point <lb></lb>C being taken in the midſt of <lb></lb>the ſaid Line, there ſhall be an <lb></lb><emph type="italics"></emph>Equilibrium<emph.end type="italics"></emph.end> upon the ſame: <lb></lb>And the reaſon is for that the <lb></lb>Diſtance C B is equal to C A. <lb></lb></s> <s>But if elevating the Line C B, moving it about the point C, it <lb></lb>ſhall be transferred into CD, ſo that the Ballance ſtand according <lb></lb>to the two Lines A C, and C D, the two equall Weights hanging <lb></lb>at the Terms A and D, ſhall no longer weigh equally on that <lb></lb>point C, becauſe the diſtance of the Weight placed in D, is made <lb></lb>leſſe then it was when it hanged in B. </s> <s>For if we confider the Lines, <lb></lb>along [or by] which the ſaid Graves make their Impulſe, and <lb></lb>would deſcend, in caſe they were freely moved, there is no doubt <lb></lb>but that they would make or deſcribe the Lines A G, D F, B H: <lb></lb>Therefore the Weight hanging on the point D, maketh it's Moment <lb></lb>and <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> according to the Line D F: but when it hanged in <lb></lb>B, it made <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> in the Line B H: and becauſe the Line D F is <lb></lb>nearer to the Fulciment C, then is the Line B H Therefore we <lb></lb>are to underſtand that the Weights hanging on the points A and D, <lb></lb>are not equi-diſtant from the point C, as they be when they are <lb></lb>conſtituted according to their Right Line A C B: And laſtly, <lb></lb>we are to take notice, that the Diſtance is to be meaſured by <pb xlink:href="070/01/009.jpg" pagenum="279"></pb>Lines, which fall at Right Angles on thoſe whereon the Weights <lb></lb>hang, and would move, if ſo be they were permitted to deſcend <lb></lb>freely.</s></p><p type="head"> <s>Of the BALLANCE and LEAVER.</s></p><p type="main"> <s>Having underſtood by certain Demonſtration, one of the <lb></lb>firſt Principles, from which, as from a plentiſul Fountain, <lb></lb>many of the Mechanical Inſtruments are derived, we may <lb></lb>take occaſion without any difficulty to come to the knowledge of <lb></lb>the nature of them: and firſt ſpeaking of the Stiliard, an Inſtru<lb></lb>ment of moſt ordinary uſe, with which divers Merchandizes are <lb></lb>weighed, ſuſtaining them, though very heavy, with a very ſmall <lb></lb>counterpoiſe, which is com<lb></lb>monly called the Roman or <lb></lb><figure id="id.070.01.009.1.jpg" xlink:href="070/01/009/1.jpg"></figure><lb></lb>Plummet, we ſhall prove that <lb></lb>there is no more to be done in <lb></lb>ſuch an operation, but to re<lb></lb>duce into act and practice <lb></lb>what hath been above contemplated. </s> <s>For if we propoſe the Bal<lb></lb>lance A B, whoſe Fulciment or Lanquet is in the point C, by <lb></lb>which, at the ſmall Diſtance C A, hangeth the heavy Weight D, <lb></lb>and if along the other greater C B, (which we call the Needle of <lb></lb>the Stiliard) we ſhould ſuppoſe the Roman F, though of but little <lb></lb>weight in compariſon of the Grave Body D to be ſlipped to and <lb></lb>fro, it ſhall be pofſible to place it ſo remotely from the Lanquet C, <lb></lb>that the ſame proportion may be found between the two Weights <lb></lb>D and F, as is between the Diſtances F C, and C A: and then ſhall <lb></lb>an <emph type="italics"></emph>Equilibrium<emph.end type="italics"></emph.end> ſucceed; unequall Weights hanging at Diſtances <lb></lb>alternately proportional to them.</s></p><p type="main"> <s>Nor is this Inſtrument different from that other called <emph type="italics"></emph>Vectis,<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1107"></arrow.to.target><lb></lb>and vulgarly the ^{*} Leaver, wherewith great Weights are moved <lb></lb>by ſmall Force; the application of which is according to the Fi<lb></lb>gure prefixed; wherein the Leaver <lb></lb>is repreſented by the Bar of wood <lb></lb>or other ſolid matter, <emph type="italics"></emph>B<emph.end type="italics"></emph.end> C D, let <lb></lb><figure id="id.070.01.009.2.jpg" xlink:href="070/01/009/2.jpg"></figure><lb></lb>the heavy Weight to be raiſed be <lb></lb>A, and let the ſteadfaſt ſupport <lb></lb>or Fulciment on which the Leaver <lb></lb>reſts and moves be ſuppoſed to be <lb></lb>E, and putting one end of the <lb></lb>Leaver under the Weight A, as <lb></lb>may be ſeen in the point C, en<lb></lb>creaſing the Weight or Force at the other end D, it will be able <lb></lb>to lift up the Weight A, though not much, whenever the Force in <pb xlink:href="070/01/010.jpg" pagenum="280"></pb>D hath the ſame proportion to the Reſiſtance made by the Weight <lb></lb>A, in the point C: as the Diſtance <emph type="italics"></emph>B<emph.end type="italics"></emph.end> C hath to the Diſtance C D, <lb></lb>whereby it's clear, that the nearer the Fulciment E ſhall approach <lb></lb>to the Term B, encreaſing the proportion of the Diſtance D C to <lb></lb>the Diſtance C <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> the more may one diminiſh the Force in D which <lb></lb>is to raiſe the Weight A. </s> <s>And here it is to be noted, which I ſhall <lb></lb>alſo in its place remember you of, that the benefit drawn from all <lb></lb>Mechanical Inſtruments, is not that which the vulgar Mechanitians <lb></lb>do perſwade us, to wit, ſuch, that there by Nature is overcome, and <lb></lb>in a certain manner deluded, a ſmall Force over-powring a very <lb></lb>great Reſiſtance with help of the Leaver; for we ſhall demonſtrate, <lb></lb>that without the help of the length of the Leaver, the ſame Force, <lb></lb>in the ſame Time, ſhall work the ſame effect. </s> <s>For taking the ſame <lb></lb>Leaver B C D, whoſe reſt or Fulci<lb></lb>ment is in C, let the Diſtance C D <lb></lb><figure id="id.070.01.010.1.jpg" xlink:href="070/01/010/1.jpg"></figure><lb></lb>be ſuppoſed, for example, to be <lb></lb>in quintuple proportion to the <lb></lb>Diſtance C <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> & the ſaid Leaver to <lb></lb>be moved till it come to I C G: In <lb></lb>the Time that the Force ſhall have <lb></lb>paſſed the Space D I, the Weight <lb></lb>ſhall have been moved from B <lb></lb>to G: and becauſe the Diſtance <lb></lb>D C, was ſuppoſed quintuple to the other C B, it is manifeſt from <lb></lb>the things demonſtrated, that the Weight placed in B may be five <lb></lb>times greater then the moving Force ſuppoſed to be in D: but now, <lb></lb><arrow.to.target n="marg1108"></arrow.to.target><lb></lb>if on the contrary, we take notice of the ^{*} Way paſſed by <lb></lb>the Force from D unto I, whilſt the Weight is moved from B unto <lb></lb>G, we ſhall find likewiſe the Way D I, to be quintuple to the Space <lb></lb>B G. </s> <s>Moreover if we take the Diſtance C L, equal to the Diſtance <lb></lb>C B, and place the ſame Force that was in D, in the point L, and <lb></lb>in the point B the fifth part onely of the Weight that was put there <lb></lb>at firſt, there is no queſtion, but that the Force in L being now <lb></lb>equal to this Weight in B, and the Diſtances L C and C B being <lb></lb>equall, the ſaid Force ſhall be able, being moved along the Space LM <lb></lb>to transfer the Weight equall to it ſelf, thorow the other equall <lb></lb>Space B G: which five times reiterating this ſame action, ſhall tranſ<lb></lb>port all the parts of the ſaid Weight to the ſame Term G: But <lb></lb>the repeating of the Space L M, is certainly nothing more nor leſſe <lb></lb>then the onely once meaſuring the Space D I, quintuple to the <lb></lb>ſaid L M. </s> <s>Therefore the transferring of the Weight from B to G, <lb></lb>requireth no leſſe Force, nor leſſe Time, nor a ſhorter Way if it <lb></lb>wee placed in D, than it would need if the ſame were applied <lb></lb>in L: And, in ſhort, the benefit that is derived from the length of <lb></lb>the Leaver C D, is no other, ſave the enabling us to move that <pb xlink:href="070/01/011.jpg" pagenum="281"></pb>Body all at once, which would not have been moved by the ſame <lb></lb>Force, in the ſame Time, with an equall Motion, ſave onely in <lb></lb>pieces, without the help of the Leaver.</s></p><p type="margin"> <s><margin.target id="marg1107"></margin.target>If of Iron, it is <lb></lb>called a Crow, <lb></lb>if of wood, a Bar <lb></lb>or Hand-ſpike.</s></p><p type="margin"> <s><margin.target id="marg1108"></margin.target>Or Space.</s></p><p type="head"> <s><emph type="italics"></emph>Of the<emph.end type="italics"></emph.end> CAPSTEN <emph type="italics"></emph>and of the<emph.end type="italics"></emph.end> CRANE.</s></p><p type="main"> <s>The Inſtruments which we are now about to declare, have <lb></lb>immediate dependence upon the Leaver, nay, are no other <lb></lb>but a perpetual Vectis or Leaver. </s> <s>For if we ſhall ſuppoſe the <lb></lb>Leaver B A C to be ſuſtained in <lb></lb>the point A, and the Weight G to <lb></lb><figure id="id.070.01.011.1.jpg" xlink:href="070/01/011/1.jpg"></figure><lb></lb>hang at the point B, the Force be<lb></lb>ing placed in C; It is manifeſt, <lb></lb>that transferring the Leaver unto <lb></lb>the points D A E, the Weight G <lb></lb>doth alter according to the Di<lb></lb>ſtance B D, but cannot much far<lb></lb>ther continue to raiſe it, ſo that <lb></lb>if it were required to elevate it yet <lb></lb>higher, it would be neceſſary to <lb></lb>ſtay it by ſome other Fulciment <lb></lb>in this Poſition, and to remit or return the Leaver to its former Po<lb></lb>ſition B A C, and ſuſpending the Weight anew thereat, to raiſe it <lb></lb>once again to the like height B D; and in this manner repeating <lb></lb>the work, many times one ſhall come with an interrupted Motion <lb></lb>to effect the drawing up of the Weight, which for many reſpects <lb></lb>will not prove very beneficial: whereupon this difficulty hath bin <lb></lb>thought on, and remedied, by finding out a way how to unite to<lb></lb>gether almoſt infinite Leavers, perpetuating the operation without <lb></lb>any interruption; and this hath been done by framing a Wheel <lb></lb>about the Center A, according to the Semidiameter A C, and an <lb></lb>Axis or Nave, about the ſame Center, of which let the Line A B <lb></lb>be the Semidiameter; and all this of very tough wood, or of other <lb></lb>ſtrong and ſolid matter, afterwards ſuſtaining the whole Machine <lb></lb>upon a Gudgeon or Pin of Iron planted in the point A, which <lb></lb>paſſeth quite thorow, where it is held faſt by two fixed Fulciments, <lb></lb>and the Rope D B G, at which the weight G hangeth, being be-laid <lb></lb>or wound about the Axis or Barrell, and applying another Rope <lb></lb>about the greater Wheel, at which let the other Grave I be hang<lb></lb>ed: It is manifeſt, that the length C A having to the other A B <lb></lb>the ſelf-ſame proportion that the Weight G hath to the Weight I, <lb></lb>it may ſuſtain the Grave G, and with any little Moment more ſhall <lb></lb>move it: and becauſe the Axis turning round together with the <lb></lb>Wheel, the Ropes that ſuſtain the Weights are alwaies pendent and <lb></lb>contingent with the extream Circumferences of that Wheel and <pb xlink:href="070/01/012.jpg" pagenum="282"></pb>Axis, ſo that they ſhall conſtantly maintain alike Site and Poſition <lb></lb>in reſpect of the Diſtances B A and A C, the Motion ſhall be <lb></lb>perpetuated, the Weight I deſcending, and forcing the other G <lb></lb>to aſcend. </s> <s>Where we are to obſerve the neceſſity of be-laying <lb></lb>or winding the Rope about the Wheel, that ſo the Weight I may <lb></lb>hang according to the Line that is tangent to the ſaid Wheel: for <lb></lb>if one ſhould ſuſpend the ſaid Weight, ſo as that it did hang by the <lb></lb>point F, cutting the ſaid Wheel, as is ſeen along the Line F N M, <lb></lb>the Motion would ceaſe, the Moment of the Weight M being di<lb></lb>miniſhed; which would weigh no more then if it did hang by the <lb></lb>point N: becauſe the Diſtance of its Suſpenſion from the Center <lb></lb>A, cometh to be determined by the Line A N, which falleth per<lb></lb>pendicularly upon the Rope F M, and is no longer terminated by <lb></lb>the Semidiameter of the Wheel A F, which falleth at unequall <lb></lb>Angles upon the ſaid Line F M. </s> <s>A violence therefore being offered <lb></lb>in the Circumference of the Wheel by a Grave and Exanimate <lb></lb>Body that hath no other <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> then that of Deſcending, it is <lb></lb>neceſſary that it be ſuſtained by a Line that is contingent with <lb></lb>the Wheel, and not by one that cutteth it. </s> <s>But if in the ſame <lb></lb>Circumference an Animate Force were employed, that had a Mo<lb></lb>ment or Faculty of making an <emph type="italics"></emph>Impulſe<emph.end type="italics"></emph.end> on all ſides, the work might <lb></lb>be effected in any whatever place of the ſaid Circumference. </s> <s>And <lb></lb>thus being placed in F, it would draw up the Weight by turning <lb></lb>the Wheel about, pulling not according to the Line F M down<lb></lb>wards, but ſide-waies according to the Contingent Line F L, which <lb></lb>maketh a Right Angle, with that which is drawn from the Center <lb></lb>A unto the point of Contact F: ſo, that if in this manner one do <lb></lb>meaſure the Diſtance from the Center A to the Force placed in <lb></lb>F, according to the Line A F perpendicular to F L, along which <lb></lb>the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> is made, a man ſhall not in any part have altered the <lb></lb>uſe of the ordinary Leaver. </s> <s>And we muſt note, that the ſame <lb></lb>would be poſſible to be done likewiſe with an Exanimate Force, <lb></lb>in caſe that a way were found out to cauſe that its Moment might <lb></lb>make Impulſe in the point F, drawing according to the Contingent <lb></lb>Line F L: which would be done by adjoyning beneath the Line F L <lb></lb>a turning Pulley, making the Rope wound about the Wheel to <lb></lb>paſſe along upon it, as it is ſeen to do by the Line F L X, ſuſpending <lb></lb>at the end thereof the Weight X equall to the other I, which ex<lb></lb>erciſing its Force according to the Line F L, ſhall alwaies keep a <lb></lb>Diſtance from the Center A equall unto the Semidiameter of the <lb></lb>Wheel. </s> <s>And from what hath been declared we will gather for a <lb></lb>Concluſion, That in this Inſtrument the Force hath alwaies the <lb></lb>ſame proportion to the Weight, as the Semidiameter of the Axis <lb></lb>or Barrell hath to the Semidiameter of the Wheel.</s></p><pb xlink:href="070/01/013.jpg" pagenum="283"></pb><p type="main"> <s>From the Inſtrument laſt deſcribed, the other Inſtrument which <lb></lb>we call the Crane is not much different, as to form, nay, differeth <lb></lb>nothing, ſave in the way of applying or employing it: For that the <lb></lb>Capſten moveth and is conſtituted perpendicular to the Horizon, <lb></lb>and the Crane worketh with its Moment parallel to the ſame Ho<lb></lb><figure id="id.070.01.013.1.jpg" xlink:href="070/01/013/1.jpg"></figure><lb></lb>rizon. </s> <s>For if upon the Circle D A E we ſuppoſe an Axis to be <lb></lb>placed Column-wiſe, turning about the Center B, and about which <lb></lb>the Rope D H, faſtened to the Weight that is to be drawn, is be<lb></lb>laid, and if the Bar F E B D be let into the ſaid Axis [<emph type="italics"></emph>by the Mor<lb></lb>tace B<emph.end type="italics"></emph.end>] and the Force of a Man, of an Horſe, or of ſome other <lb></lb>Animal apt to draw, be applyed at its end F, which moving round, <lb></lb>paſſeth along the Circumference F G C, the Crane ſhall be framed <lb></lb>and finiſhed, ſo that by carrying round the Bar F B D, the Barrell <lb></lb>or Axis E A D ſhall turn about, and the Rope which is twined a<lb></lb>bout it, ſhall conſtrain the Weight H to go forward: And becauſe <lb></lb>the point of the Fulciment about which the Motion is made, is the <lb></lb>point B, and the Moment keeps at a Diſtance from it according to <lb></lb>the Line B F, and the Reſiſtor at the Diſtance B D, the Leaver <lb></lb>F B D is formed, by vertue of which the Force acquireth Moment <lb></lb>equall to the Reſiſtance, if ſo be, that it be in proportion to it, as <lb></lb>the Line B D is to B F, that is, as the Semidiameter of the Axis to <lb></lb>the Semidiameter of the Circle, along whoſe Circumference the <lb></lb>Force moveth. </s> <s>And both in this, and in the other Inſtrument we <lb></lb>are to obſerve that which hath been frequently mentioned, that is, <lb></lb>That the benefit which is derived from theſe Machines, is not that <lb></lb>which the generality of the Vulgar promiſe themſelves from the <lb></lb>Mechanicks; namely, that being too hard for Nature, its poſſible <pb xlink:href="070/01/014.jpg" pagenum="284"></pb>with a Machine to overcome a Reſiſtance, though great, with a <lb></lb>ſmall Force, in regard, that we ſhall manifeſtly prove that the ſame <lb></lb>Force placed in F, might in the ſame Time conveigh the ſame <lb></lb>Weight, with the ſame Motion, unto the ſame Diſtance, without <lb></lb>any Machine at all: For ſuppoſing, for example, that the Reſiſtance <lb></lb>of the Grave H be ten times greater than the Force placed in F, it <lb></lb><figure id="id.070.01.014.1.jpg" xlink:href="070/01/014/1.jpg"></figure><lb></lb>will be requiſite for the mo<lb></lb>ving of the ſaid Reſiſtance, <lb></lb>that the Line F B be decuple <lb></lb>to B D; and conſequently, <lb></lb>that the Circumference of the <lb></lb>Circle F G C be alſo decuple <lb></lb>to the Circumference E A D: <lb></lb>and becauſe when the Force <lb></lb>ſhall be moved once along the <lb></lb>whole Circumference of the <lb></lb>Circle F G C, the Barrel EAD, <lb></lb>about which the Rope is be-laid which draweth the Weight, ſhall <lb></lb>likewiſe have given one onely turn; it is manifeſt, that the Weight <lb></lb>H ſhall not have been moved more than the tenth part of that way <lb></lb>which the Mover ſhall have gone. </s> <s>If therefore the Force that is to <lb></lb>move a Reſiſtance that is greater than it ſelf, for ſuch an aſſigned <lb></lb>Space by help of this Machine, muſt of neceſſity move ten times as <lb></lb>far, there is no doubt, but that dividing that Weight into ten parts, <lb></lb>each of them ſhall be equall to the Force, and conſequently, might <lb></lb>have been tranſported one at a Time, as great a Space as that <lb></lb>which it ſelf did move, ſo that making ten journeys, each equal to <lb></lb>the Circumference E A D, it ſhall not have gone any farther than <lb></lb>if it did move but once alone about the Circumference F G C; <lb></lb>and ſhall have conveighed the ſame Weight H to the ſame Di<lb></lb>ſtance. </s> <s>The benefit therefore that is to be derived from theſe <lb></lb>Machines is, that they carry all the Weight together, but not with <lb></lb>leſſe Labour, or with greater Expedition, or a greater Way than <lb></lb>the ſame Force might have done conveying it by parcels.</s></p><p type="head"> <s>Of PULLIES.</s></p><p type="main"> <s>The Inſtruments, whoſe Natures are reducible unto the Bal<lb></lb>lance, as to their Principle and Foundation, and others little <lb></lb>differing from them, have been already deſcribed; now for <lb></lb>the underſtanding of that which we have to ſay touching Pullies, <lb></lb>it is requiſite, that we conſider in the firſt place another way to uſe <lb></lb>the Leaver, which will conduce much towards the inveſtigation of <lb></lb>the Force of Pullies, and towards the underſtanding of other Me<lb></lb>chanical Effects. </s> <s>The uſe of the Leaver above declared ſuppoſed <pb xlink:href="070/01/015.jpg" pagenum="285"></pb>the Weight to be at one extream, and the Force at the other, and <lb></lb>the Fulciment placed in ſome point between the extreams: but we <lb></lb>may make uſe of the Leaver another way, yet, placing, as we ſee, <lb></lb>the Fulciment in the extream A, the Force in the other extream C, <lb></lb>and ſuppoſing the Weight D to hang by ſome point in the midſt, <lb></lb><figure id="id.070.01.015.1.jpg" xlink:href="070/01/015/1.jpg"></figure><lb></lb>as here we ſee by the point B, in <lb></lb>this example it's manifeſt, that if <lb></lb>the Weight did hang at a point <lb></lb>Equi-diſtant from the two ex<lb></lb>treams A and C, as at the point F, <lb></lb>the labour of ſuſtaining it would <lb></lb>be equally divided betwixt the <lb></lb>two points A and C, ſo that half <lb></lb>the Weight would be felt by the <lb></lb>Force C, the other half being ſu<lb></lb>ſtained by the Fulciment A: but if the Grave Body ſhall be hanged <lb></lb>at another place, as at B, we ſhall ſhew that the Force in C is ſuffi<lb></lb>cient to ſuſtain the Weight in B, as it hath the ſame proportion <lb></lb>to it, that the Diſtance, A B hath to the Diſtance A C. </s> <s>For De<lb></lb>monſtration of which, let us imagine the Line B A to be continued <lb></lb>right out unto G, and let the Diſtance B A be equall to A G, and <lb></lb>let the Weight hanging at G, be ſuppoſed equall to D: It is ma<lb></lb>nifeſt, that by reaſon of the equality of the Weights D and E, and <lb></lb>of the Diſtances G A and A B, the Moment of the Weight E <lb></lb>ſhall equalize the Moment of the Weight D, and is ſufficient to <lb></lb>ſuſtain it: Therefore whatever Force ſhall have Moment equall to <lb></lb>that of the Weight E, and that ſhall be able to ſuſtain it, ſhall be <lb></lb>ſufficient likewiſe to ſuſtain the Weight D: But for ſuſtaining the <lb></lb>Weight E, let there be placed in the point C ſuch a Force, whoſe <lb></lb>Moment hath that proportion to the Weight E, that the Diſtance <lb></lb>G A hath to the Diſtance A C, it ſhall be ſufficient to ſuſtain it: <lb></lb>Therefore the ſame Force ſhall likewiſe be able to ſuſtain the <lb></lb>Weight D, whoſe Moment is equall to the of E: But look what <lb></lb>Proportion the Line G A hath to the Line A C; and A B alſo hath <lb></lb>the ſame to the ſaid A C, G A having been ſuppoſed equall to A B: <lb></lb>And becauſe the Weights E and D are equall, each of them ſhall <lb></lb>have the ſame proportion to the Force placed in C: Therefore the <lb></lb>Force in C is concluded to equall the Moment of the Weight D, <lb></lb>as often as it hath unto it the ſame proportion that the Diſtance B A <lb></lb>hath to the Diſtance C A. </s> <s>And by moving the Weight, with the <lb></lb>Leaver uſed in this manner, it is gathered in this alſo, as well as in <lb></lb>the other Inſtruments, that what is gained in Force is loſt in Velo<lb></lb>city: for the Force C raiſing the Leaver, and transferring it to A I, <lb></lb>the Weight is moved the Space B H, which is as much leſſer than <lb></lb>the Space C I paſſed by the Force, as the Diſtance A B is leſſer <pb xlink:href="070/01/016.jpg" pagenum="286"></pb>than the Diſtance A C; that is, as the Force is leſſe than the <lb></lb>Weight.</s></p><p type="main"> <s>Theſe Principles being declared, we will paſſe to the Contem<lb></lb>plation of Pullies, the compoſition and ſtructure of which, together <lb></lb>with their uſe, ſhall be deſcribed by us. </s> <s>And firſt let us ſuppoſe the <lb></lb><arrow.to.target n="marg1109"></arrow.to.target><lb></lb>^{*} Little Pulley A B C, made of Mettall or hard Wood, voluble a<lb></lb>bout it's Axis which paſſeth thorow it's Center D, and about this <lb></lb><figure id="id.070.01.016.1.jpg" xlink:href="070/01/016/1.jpg"></figure><lb></lb>Pulley let the Rope E A B C be put, <lb></lb>at one end of whichlet the Weight E <lb></lb>hang, and at the other let us ſuppoſe <lb></lb>the Force F. </s> <s>I ſay, that the Weight <lb></lb>being ſuſtained by a Force equall to <lb></lb>it ſelf in the upper Nut or Pulley <lb></lb>A B C, bringeth ſome benefit, as the <lb></lb>moving or ſuſtaining of the ſaid <lb></lb>Weight with the Force placed in F: <lb></lb>For if we ſhall underſtand, that from <lb></lb>the Center D, which is the place of the Fulciment, two Lines be <lb></lb>drawn out as far as the Circumference of the Pulley in the points <lb></lb>A and C, in which the pendent Cords touch the Circumference, we <lb></lb>ſhall have a Ballance of equal Arms which determine the Diſtance <lb></lb>of the two Suſpenſions from the Center and Fulciment D: Where<lb></lb>upon it is manifeſt, that the Weight hanging at A cannot be ſuſtain<lb></lb>ed by a leſſer Weight hanging at G, but by one equal to it; ſuch <lb></lb>is the nature of equal Weights hanging at equal Diſtances. </s> <s>And <lb></lb>although in moving downwards, the Force F cometh to turn about <lb></lb>the Pulley A B C, yet there followeth no alteration of the Alti<lb></lb>tude or Reſpect, that the Weight and Force have unto the two <lb></lb>Diſtances A D and D C, nay, the Pulley encompaſſed becometh a <lb></lb>Ballance equal to A C, but perpetuall. </s> <s>Whence we may learn, <lb></lb>how childiſhly <emph type="italics"></emph>Ariſtotle<emph.end type="italics"></emph.end> deceiveth himſelf, who holds, that by making <lb></lb>the ſmall Pulley A B C bigger, one might draw up the Weight with <lb></lb>a leſſer Force; he conſidering that upon the enlargement of the <lb></lb>ſaid Pulley, the Diſtance D C encreaſed, but not conſidering that <lb></lb>there was as great an encreaſe of the other Diſtance of the Weight, <lb></lb>that is, the other Semidiameter D A. </s> <s>The benefit therefore that may <lb></lb>be drawn from the Inſtrument above ſaid, is nothing at all as to the <lb></lb>diminution of the labour: and if any one ſhould ask how it hap<lb></lb>pens, that on many occaſions of raiſing Weights, this means is made <lb></lb>uſe of to help the Axis, as we ſee, for example, in drawing up the <lb></lb>Water of Wells; it is anſwered, that that is done, becauſe that <lb></lb>by this means the manner of employing the Force is found more <lb></lb>commodious: for being to pull downwards, the proper Gravity of <lb></lb>our Arms and other parts help us, whereas if we were to draw <lb></lb>the fame Weight upwards with a meer Rope, by the ſole ſtrength <pb xlink:href="070/01/017.jpg" pagenum="287"></pb>of the Members and Muſcles, and as we uſe to ſay, by Force of <lb></lb>Armes, beſides the extern Weight, we are to lift up the Weight of <lb></lb>our own Armes, in which greater pains is required. </s> <s>Conclude we, <lb></lb>therefore, that this upper Pulley doth not bring any Facility to the <lb></lb>Force ſimply conſidered, but onely to the manner of applying it: <lb></lb>but if we ſhall make uſe of the like Machine <lb></lb><figure id="id.070.01.017.1.jpg" xlink:href="070/01/017/1.jpg"></figure><lb></lb>in another manner, as we are now about to <lb></lb>declare; we may raiſe the Weight with di<lb></lb>minution of Forces: For let the Pulley <lb></lb>B D C be voluble about the Center E placed <lb></lb>in it's Frame B L C, at which hang the <lb></lb>Grave G; and let the Rope A B D C F <lb></lb>paſſe about the Pulley; of which let the end <lb></lb>A be faſtned to ſome fixed ſtay, and in the <lb></lb>other F let the Force be placed; which <lb></lb>moving to wards H ſhall raiſe the Machine <lb></lb>B L C, and conſequently the Weight G: <lb></lb>and in this operation I ſay, that the Force in <lb></lb>F is the half of the Weight ſuſtained by it. <lb></lb></s> <s>For the ſaid Weight being kept to Rights by the two ^{*} Ropes A B <lb></lb><arrow.to.target n="marg1110"></arrow.to.target><lb></lb>and F C, it is manifeſt, that the Labour is equally ſhared betwixt <lb></lb>the Force F and the Fulciment A: and more ſubtilly examining the <lb></lb>nature of this Inſtrument, if we but continue forth the Diameter <lb></lb>B E C, we ſhall ſee a Leaver to be made, at the midſt of which, that <lb></lb>is at the point E, the Grave doth hang, and the Fulciment cometh <lb></lb>to be at the end B, and the Force in the Term C: whereupon, by <lb></lb>what hath been above demonſtrated, the Force ſhall have the ſame <lb></lb>proportion to the Weight, that the Diſtance E B hath to the Di<lb></lb>ſtance; Therefore it ſhall be the half of the ſaid Weight: And <lb></lb>becauſe the Force riſing towards A, the Pulley turneth round, <lb></lb>therefore that Reſpect or Conſtitution which the Fulciment B and <lb></lb>Center E, on which the Weight and Term C, in which the Force <lb></lb>is employed do depend, ſhall not change all the while; but yet in <lb></lb>the Circuinduction the Terms B and C happen to vary in number, <lb></lb>but not in vertue, others and others continually ſucceeding in their <lb></lb>place, whereby the Leaver B C cometh to be perpetuated. </s> <s>And <lb></lb>here (as hath been done in the other Inſtruments, and ſhall be in <lb></lb>thoſe that follow) we will not paſſe without conſidering how that <lb></lb>the journey that the Force maketh, is double to the Moment of the <lb></lb>Weight. </s> <s>For in caſe the Weight ſhall be moved ſo far, till that <lb></lb>the Line B C come to arrive with it's points B and C, at the points <lb></lb>A and F, it is neceſſary that the two equal Ropes be diſtended in <lb></lb>one ſole Line F H, and conſequently, when the Weight ſhall have <lb></lb>aſcended along the Intervall B A, the Force ſhall have been moved <lb></lb>twice as far, that is, from <emph type="italics"></emph>F<emph.end type="italics"></emph.end> unto H. </s> <s>Then conſidering that the <pb xlink:href="070/01/018.jpg" pagenum="288"></pb>Force in <emph type="italics"></emph>F,<emph.end type="italics"></emph.end> that it may raiſe the Weight, muſt move upwards, which <lb></lb>to exanimate Movers, as being for the moſt part Grave Bodies, is al<lb></lb><figure id="id.070.01.018.1.jpg" xlink:href="070/01/018/1.jpg"></figure><lb></lb>together impoſſible, or at leaſt more laborious, <lb></lb>than the making of the ſame <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce down<lb></lb>wards: Therefore to help this inconvenience, <lb></lb>a Remedy hath been found by adjoyning an<lb></lb>other Nut or Pulley above, as in the adjacent <lb></lb><emph type="italics"></emph>F<emph.end type="italics"></emph.end>igure is ſeen, where the Rope C E <emph type="italics"></emph>F<emph.end type="italics"></emph.end> hath <lb></lb>been made to paſs about the upper Pulley <emph type="italics"></emph>F<emph.end type="italics"></emph.end> G <lb></lb>upheld by the Hook L, ſo that the Rope paſſing <lb></lb>to H, and thither transferring the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce E, it <lb></lb>ſhall be able to move the Weight X by pulling <lb></lb>downwards, but not that it may be leſſer than <lb></lb>it was in E: <emph type="italics"></emph>F<emph.end type="italics"></emph.end>or the Motions of the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce <lb></lb><emph type="italics"></emph>F<emph.end type="italics"></emph.end> H, hanging at the equal Diſtances <emph type="italics"></emph>F<emph.end type="italics"></emph.end> D and <lb></lb>D G of the upper Pulley, do alwaies continue <lb></lb>equal; nor doth that upper Pulley (as hath <lb></lb>been ſhewn above) come to produce any di<lb></lb>minution in the Labour. </s> <s>Moreover it having been neceſſary by <lb></lb>the addition of the upper Pulley to introduce the Appendix B, by <lb></lb>which it is ſuſtained, it will prove of ſome benefit to us to raiſe <lb></lb>the other A, to which one end of the Rope was faſtned, transferring <lb></lb>it to a Ring annexed to the lower part of the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>rame of the upper <lb></lb>Pulley, as we ſee it done in M. </s> <s>Now finally, this Machine com<lb></lb>pounded of upper and lower Pullies, is that which the Greeks call <lb></lb><arrow.to.target n="marg1111"></arrow.to.target><lb></lb><foreign lang="grc">Τποχίλιον.</foreign></s></p><p type="margin"> <s><margin.target id="marg1109"></margin.target>*Called by ſome <lb></lb>a Nut.</s></p><p type="margin"> <s><margin.target id="marg1110"></margin.target>* Or two ends of <lb></lb>the ſame Rope.</s></p><p type="margin"> <s><margin.target id="marg1111"></margin.target>In Latine <emph type="italics"></emph>Tro<lb></lb>chlea.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>We have hitherto explained, how by help of Pullies one may <lb></lb>double the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce, it remaineth that with the greateſt brevity poſ<lb></lb>ſible, we ſhew the way how to encreaſe it according to any Multi<lb></lb>plicity. </s> <s>And firſt we will ſpeak of the Multiplicity according to <lb></lb>the even numbers, and then the odde: To ſhew how we may mul<lb></lb>tiply the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce in a quadruple Proportion, we will propound the <lb></lb>following Speculation as the Soul of all that followeth.</s></p><p type="main"> <s>Take two Leavers, A B, C D, with the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>ulciments in the ex<lb></lb><figure id="id.070.01.018.2.jpg" xlink:href="070/01/018/2.jpg"></figure><lb></lb>treams A and C; and at the middles <lb></lb>of each of them let the Grave G hang, <lb></lb>ſuſtained by two <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orces of equal Mo<lb></lb>ment placed in B and D. </s> <s>I ſay, that <lb></lb>the Moment of each of them will <lb></lb>equal the Moment of the fourth part <lb></lb>of the Weight G. <emph type="italics"></emph>F<emph.end type="italics"></emph.end>or the two <emph type="italics"></emph>F<emph.end type="italics"></emph.end>or<lb></lb>ces B and D bearing equally, it is <lb></lb>manifeſt, that the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce D hath not <lb></lb>contraſted with more then one half of the Weight G: But if the <lb></lb><emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce D do by benefit of the Leaver D C ſuſtain the half of the <pb xlink:href="070/01/019.jpg" pagenum="289"></pb>Weight G hanging at <emph type="italics"></emph>F,<emph.end type="italics"></emph.end> it hath been already demonſtrated, that <lb></lb>the ſaid <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce D hath to the Weight ſo by it ſuſtained, that ſame <lb></lb>proportion which the Diſtance <emph type="italics"></emph>F<emph.end type="italics"></emph.end> C hath to the Diſtance C D: <lb></lb>Which is ſubduple proportion: Therefore the Moment D is ſub<lb></lb>duple to the Moment of half of the Weight G ſuſtained by it: <lb></lb>Wherefore it followeth, that it is the fourth part of the Moment <lb></lb>of the whole Weight. </s> <s>And in the ſame manner the ſame thing is <lb></lb>demonſtrated, of the Moment <emph type="italics"></emph>B<emph.end type="italics"></emph.end>; and it is but reaſonable, that the <lb></lb>Weight G being ſuſtained by the four points, A, <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> C, D, each of <lb></lb>them ſhould feel an equall part of the Labour.</s></p><p type="main"> <s>Let us come now to apply this Conſideration to Pullies, and let <lb></lb>the Weight X be ſuppoſed to hang at the two Pullies A B and D E <lb></lb>entwining about them, and about the uppermoſt Pulley G H, the <lb></lb>Rope, as we ſee, I D E H G A B, ſuſtaining the whole Machine in <lb></lb>the point K. </s> <s>Now I ſay, that placing the Force in L, it ſhall be able <lb></lb>to ſuſtain the Weight X, if ſo be, it be equal to the fourth part of <lb></lb>it. </s> <s>For if we do imagine the two Diameters D E and A B, and the <lb></lb>Weights hanging at the middle points F and C, we ſhall have two <lb></lb>Leavers like to thoſe before deſcribed, the Fulciments of which an<lb></lb>ſwer to the points D and A. </s> <s>Whereupon the Force placed in B, <lb></lb><figure id="id.070.01.019.1.jpg" xlink:href="070/01/019/1.jpg"></figure><lb></lb>or if you will, in L, ſhall be able to ſu<lb></lb>ſtain the Weight X, being the fourth <lb></lb>part of it: And if we adde another Pul<lb></lb>ley above the other two, making the <lb></lb>Rope or Cord to paſs along L M N, trans<lb></lb>ferring the Force L into N, it ſhall be <lb></lb>able to bear the ſame Weight gravitating <lb></lb>downwards, the upper Pulley neither aug<lb></lb>menting or diminiſhing the Force, as hath <lb></lb>been declared. </s> <s>And we will likewiſe <lb></lb>note, that to make the: Weight aſcend the <lb></lb><arrow.to.target n="marg1112"></arrow.to.target><lb></lb>four Ropes B L, E H, D I, and A G <lb></lb>ought to paſs, whereupon the Mover will <lb></lb>be to begin, as much as thoſe Ropes are <lb></lb>long; and yet nevertheleſs the Weight <lb></lb>ſhall move but only as much as the length <lb></lb>of one of them: So that we may ſay by <lb></lb>way of advertiſement, and for confirma<lb></lb>tion of what hatn been many times ſpo<lb></lb>ken, namely, that look with what proportion the Labour of the <lb></lb><arrow.to.target n="marg1113"></arrow.to.target><lb></lb>Mover is diminiſhed, the length of the Way, on the contrary, is <lb></lb>encreaſed with the ſame proportion</s></p><p type="margin"> <s><margin.target id="marg1112"></margin.target>* Or four parts <lb></lb>of the ſame Rope</s></p><p type="margin"> <s><margin.target id="marg1113"></margin.target>* The word <emph type="italics"></emph>Gy<lb></lb>rilla<emph.end type="italics"></emph.end> ſignifieth a <lb></lb>Shiver, Rundle, <lb></lb>or ſmall Wheel <lb></lb>of a Pulley, tran<lb></lb>ſlated by we <lb></lb>ſometimes Pul<lb></lb>ley, ſometimes <lb></lb>Nut or Girill.</s></p><p type="main"> <s>But if we would encreaſe the Force in ſexcuple proportion, it <lb></lb>will be requiſite that we adjoyn another ^{*} ſmall Pulley or Gyrill <lb></lb>to the inferiour Pulley which that you may the better underſtand <pb xlink:href="070/01/020.jpg" pagenum="290"></pb>we will ſet before you the preſent Contemplation. </s> <s>Suppoſe, there<lb></lb>fore, that A B, C D, and E F are three Leavers; and that on the <lb></lb>middle points of them G, H, and I the Weight K doth hang in <lb></lb>common, ſo that every one of them ſhall ſuſtain the third part of <lb></lb><figure id="id.070.01.020.1.jpg" xlink:href="070/01/020/1.jpg"></figure><lb></lb>it: And becauſe the Power in <lb></lb>B, ſuſtaining with the Leaver <lb></lb>B A thependent Weight in G, <lb></lb>hapneth to be the half of the <lb></lb>ſaid Weight, and it hath been <lb></lb>already ſaid, that it ſuſtaineth <lb></lb>the third part of the Weight <lb></lb>K: Therefore the Moment of <lb></lb>the Force B is equal to half of <lb></lb>the third part of the Weight K; that is, to the ſixth part of it: <lb></lb>And the ſame ſhall be demonſtrated of the other Forces D and F: <lb></lb>From whence we may eaſily gather, that putting three Gyrils or <lb></lb>Rundles into the inferiour Pulley, and two or three into the upper<lb></lb><figure id="id.070.01.020.2.jpg" xlink:href="070/01/020/2.jpg"></figure><lb></lb>moſt, we may multiply the Force accor<lb></lb><arrow.to.target n="marg1114"></arrow.to.target><lb></lb>ding to our ^{*} <emph type="italics"></emph>Senarius.<emph.end type="italics"></emph.end> And if we would <lb></lb>encreaſe it according to any other even <lb></lb>Number, the Gyrils of the Pulley below <lb></lb>muſt be multiplyed according to the half <lb></lb>of that Number, according to which the <lb></lb>Force is to be multiplyed, circumpoſing <lb></lb>the Rope about the Pulleys, ſo as that one <lb></lb>of the ends be faſtned to the upper Pul<lb></lb>ley, and let the Force be in the other; as <lb></lb>in this Figure adjoyning may manifeſtly <lb></lb>be gathered.</s></p><p type="margin"> <s><margin.target id="marg1114"></margin.target>* Or in Sexcuple <lb></lb>proportion.</s></p><p type="main"> <s>Now paſſing to the Declaration of the <lb></lb>manner how to multiply the Force ac<lb></lb>cording to the odd Numbers, and begin<lb></lb><figure id="id.070.01.020.3.jpg" xlink:href="070/01/020/3.jpg"></figure><lb></lb>ning at the triple proportion: firſt, let us <lb></lb>propoſe the preſent Contemplation, as <lb></lb>that, on the underſtanding of which the <lb></lb>knowledge of all the Work in hand <lb></lb>doth depend. </s> <s>Let therefore the Leaver <lb></lb>be A B, its Fulciment A, and from the <lb></lb>middle of it, that is, at the point C let <lb></lb>the Grave D be hanged; and let it be ſu<lb></lb>ſtained by two equal Forces; and let one of them be applied to the <lb></lb>point C, and the other to the term B. </s> <s>I ſay, that each of thoſe Powers <lb></lb>have Moment equal to the third part of the Weight D. </s> <s>For the <lb></lb>Force in C ſuſtaineth a Weight equal to it ſelf, being placed in the <lb></lb>ſame Line in which the Weight D doth hang & Gravitate: But the <pb xlink:href="070/01/021.jpg" pagenum="291"></pb>Force in B ſuſtaineth a part of the Weight D double to it ſelf, its <lb></lb>Diſtance from the Fulciment A, that is, the Line B A being dou<lb></lb>ble to the Diſtance A C at which the Grave hangeth: But becauſe <lb></lb>the two Forces in B and C are ſuppoſed to be equal to each other: <lb></lb>Therefore the part of the Weight D, which is ſuſtained by the <lb></lb>Force in B, is double to the part ſuſtained by the Force in C. </s> <s>If <lb></lb>therefore, of the Grave D two parts be made, the one double to <lb></lb>the remainder, the greater is ſuſtained by the Force in B, and the <lb></lb>leſſer by the Force in C: But this leſſer is the third part of the <lb></lb>Weight D: Therefore the Moment of the Force in C is equal to <lb></lb>the Moment of the third part of the Weight D; to which, of <lb></lb>conſequence, the Force B ſhall be equal, we having ſuppoſed it <lb></lb>equal to the other Force C: Wherefore our intention is manifell, <lb></lb>which we were to demonſtrate, how that each of the two Powers <lb></lb>C and B is equal to the third part of the Weight D. </s> <s>Which be<lb></lb>ing demonſtrated, we will paſs forwards to the Pulleys, and will <lb></lb>deſcribe the inferiour Gyrils of A C B, voluble about the Center <lb></lb>G, and the Weight H hanging thereat, we will draw the other up<lb></lb>per one E F, winding about them both the Rope D F E A C B I, <lb></lb>of which let the end D be faſtned to the inferiour Pulley, and to <lb></lb><figure id="id.070.01.021.1.jpg" xlink:href="070/01/021/1.jpg"></figure><lb></lb>the other I let the Force be applyed: <lb></lb>Which, I ſay, ſuſtaining or moving the <lb></lb>Weight H, ſhall feele no more than the <lb></lb>third part of the Gravity of the ſame. </s> <s>For <lb></lb>conſidering the contrivance of this Ma<lb></lb>chine, we ſhall find that the Diameter A B <lb></lb>ſupplieth the place of a Leaver, in whoſe <lb></lb>term B the Force I is applied, and in the <lb></lb>other A the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>uiciment is placed, at the mid<lb></lb>dle G the Grave H is hanged, and another <lb></lb><emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce D applied at the ſame place: ſo that <lb></lb><arrow.to.target n="marg1115"></arrow.to.target><lb></lb>the Weight is faſtned to the ^{*} three Ropes <lb></lb>I B, <emph type="italics"></emph>F<emph.end type="italics"></emph.end> D, and E A, which with equal Labour <lb></lb>ſuſtain the Weight. </s> <s>Now, by what hath <lb></lb>already been contemplated, the two <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orces <lb></lb>D and B being applied, one, to the midſt of the Leaver A B, and <lb></lb>the other to the extream term B, it is manifeſt, that each of them <lb></lb>holdeth no more but the third part of the Weight H: Therefore <lb></lb>the Power I, having a Moment equal to the third part of the <lb></lb>Weight H, ſhall be able to ſuſtain and move it: but yet the Way <lb></lb>of the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce in I ſhall be triple to the Way that the Weight ſhall <lb></lb>paſs; the ſaid Force being to diſtend it ſelf according to the <lb></lb>Length of the three Ropes I B, <emph type="italics"></emph>F<emph.end type="italics"></emph.end> D, and E A, of which one alone <lb></lb>meaſureth the Way of the Weight H.</s></p><pb xlink:href="070/01/022.jpg" pagenum="292"></pb><p type="margin"> <s><margin.target id="marg1115"></margin.target>* Or three parts <lb></lb>of one Rope.</s></p><p type="head"> <s><emph type="italics"></emph>Of the<emph.end type="italics"></emph.end> SCREW.</s></p><p type="main"> <s>Amongſt the reſt of Mechanick Inſtruments for ſundry uſes <lb></lb>found out by the Wit of Man, the Screw doth, in my opi<lb></lb>nion, both for Invention and for Utility, hold the firſt <lb></lb>place, as that which is appoſitely accommodated, and ſo contrived <lb></lb>not only to move, but alſo to ſtay and preſs with very great Force, <lb></lb>that taking up but little room, it worketh thoſe effects which other <lb></lb>Inſtruments cannot, unleſs they were reduced to a great Machine. <lb></lb></s> <s>The Screw therefore being of moſt ingenious and commodious <lb></lb>contrivance, we ought deſervedly to be at ſome pains in explaining, <lb></lb>with all the plainneſs that is poſſible, the Original and Nature of <lb></lb>it. </s> <s>The which that we may do, we will begin at a Speculation, <lb></lb>which, though at firſt bluſh it may appear ſomewhat remote from <lb></lb>the conſideration of this Inſtrument, yet is the <emph type="italics"></emph>Baſis<emph.end type="italics"></emph.end> and Founda<lb></lb>tion thereof.</s></p><p type="main"> <s>No doubt, but that Natures operation in the Motions of Grave <lb></lb>Bodies is ſuch, that any whatever Body that hath a Gravity in it <lb></lb>hath a propenſion of moving, being at liberty, towards the Cen<lb></lb><arrow.to.target n="marg1116"></arrow.to.target><lb></lb>ter, and that not only ^{*} by the Right Line perpendicularly, but al<lb></lb>ſo (when it cannot do otherwiſe) by any other Line, which ha<lb></lb>ving ſome inclination towards the Center goeth more and more <lb></lb>abaſing. </s> <s>And thus we ſee the Water not only to fall downwards <lb></lb>along the Perpendicular from ſome eminent place, but alſo to run <lb></lb>about the Surface of the Earth along Lines though very little en<lb></lb>clined; as we ſee in the Courſe of Rivers, the Waters of which, if ſo <lb></lb>be that the Bed have any the leaſt declivity, go freely declining <lb></lb>downwards. </s> <s>Which very effect, like as it is diſcerned in all Fluid <lb></lb>Bodies, would appear alſo in hard Bodies, if ſo be, that their Fi<lb></lb>gure and other Accidental and Extern Impediments did not hinder <lb></lb>it. </s> <s>So that we, having a Superficies very well ſmoothed and poli<lb></lb>ſhed, as for inſtance, that of a Looking-glaſs, and a Ball exactly <lb></lb>rotund and ſleek, either of Marble, or of Glaſs, or of any other <lb></lb>Matter apt to be poliſhed, this being placed upon that Superficies <lb></lb>ſhall trundle along, in caſe that this have any, though very ſmall, <lb></lb>inclination; and ſhall lie ſtill only upon that Superficies which is <lb></lb>exactly levelled and parallel to the Plane of the Horizon: as is <lb></lb>that, for example, of a Lake or ſtanding Water being frozen, up<lb></lb>on which the ſaid Spherical Body would ſtand ſtill, but in a con<lb></lb>dition of being moved by every ſmall Force. </s> <s>For we having ſup<lb></lb>poſed that if that Plane did incline but an hairs breadth only, the <lb></lb>ſaid Ball would move along it ſpontaneouſly towards the part de<lb></lb>clining, and on the oppoſite would have a Reſiſtance, nay, would <lb></lb>not be able without ſome Violence to move towards the part <pb xlink:href="070/01/023.jpg" pagenum="293"></pb>riſing or aſcending: it of neceſſity remaineth manifeſt, that in the <lb></lb>Superficies which is exactly equilibrated, the ſaid Ball remaineth in<lb></lb>different and dubious between Motion and Reſt, ſo that every ſmall <lb></lb>Force is ſufficient to move it, as on the contrary, every ſmall Reſi<lb></lb>ſtance, and no greater than that of the meer Air that environs it, is <lb></lb>able to hold it ſtill.</s></p><p type="margin"> <s><margin.target id="marg1116"></margin.target>* Or along.</s></p><p type="main"> <s>From whence we may take this Concluſion for indubitable, That <lb></lb>Crave Bodies, all Extern and Adventitious Impediments being re<lb></lb>moved, may be moved along the Plane of the Horizon by any ne<lb></lb>ver ſo ſmall Force: but when the ſame Grave is to be thrown along <lb></lb>an Aſcending Plane, then, it beginning to ſtrive againſt that aſcent, <lb></lb>having an inclination to the contrary Motion, there ſhall be requi<lb></lb>red greater Violence, and ſtill greater the more Elevation that ſame <lb></lb>Plane ſhall have. </s> <s>As for example, the Moveable G, being poſited <lb></lb>upon the Line A B parallel to the Horizon, it ſhall, as hath been <lb></lb>ſaid, be indifferent on it either to Motion or Reſt, ſo that it may <lb></lb>be moved by a very ſmall Force: But if we ſhall have the Planes <lb></lb>Elevated, they ſhall not be driven along without Violence; which <lb></lb><figure id="id.070.01.023.1.jpg" xlink:href="070/01/023/1.jpg"></figure><lb></lb>Violence will be required to be <lb></lb>greater to move it along the Line <lb></lb>A D, than along A C; and ſtill <lb></lb>greater along A E than along A D: <lb></lb>The which hapneth, becauſe it hath <lb></lb>greater <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of going down<lb></lb>wards along A E than along A D, <lb></lb>and along A D than along A C. </s> <s>So <lb></lb>that we may likewiſe conclude <lb></lb>Grave Bodies to have greater Reſiſtance upon Planes differently <lb></lb>Elevared, to their being moved along the ſame, according as one <lb></lb>ſhall be more or leſs elevated than the other; and, in fine, that the <lb></lb>greateſt Reſiſtance of the ſame Grave to its being raiſed is in the <lb></lb>Perpendicular A F. </s> <s>But it will be neceſſary to declare exactly what <lb></lb>proportion the Force muſt have to the Weight, that it may be able <lb></lb>to carry it along ſeveral elevated Planes, before we proceed any <lb></lb>farther, to the end that we may perfectly underſtand all that which <lb></lb>remains to be ſpoken.</s></p><p type="main"> <s>Letting, therefore, Perpendiculars fall from the points C, D, <lb></lb>and E unto the Horizontal Line A B, which let be C H, D I, and <lb></lb>E K: it ſhall be demonſtrated that the ſame Weight ſhall be mo<lb></lb>ved along the Plane A C with leſſer Force than along the Perpendi<lb></lb>cular A F, (where it is raiſed by a Force equal to it ſelf) accor<lb></lb>ding to the proportion by which the Perpendicular C H is leſs than <lb></lb>A C: and that along the Plane A D, the Force hath the ſame pro<lb></lb>portion to the Weight, that the Perpendicular I D hath to D A: <lb></lb>and, laſtly, that in the Plane A E the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce to the Weight obſer<lb></lb>veth the proportion of E K and E A.</s></p><pb xlink:href="070/01/024.jpg" pagenum="294"></pb><p type="main"> <s>The preſent Speculation hath been attempted by <emph type="italics"></emph>Pappus Alex<lb></lb>andrinus<emph.end type="italics"></emph.end> in <emph type="italics"></emph>Lib.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>de Collection. </s> <s>Mathemat.<emph.end type="italics"></emph.end> but, if I be in the <lb></lb>right, he hath not hit the mark, and was overſeen in the Aſſumpti<lb></lb>on that he maketh, where he ſuppoſeth that the Weight ought to <lb></lb>be moved along the Horizontal Line by a <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce given; which is <lb></lb>falſe: there needing no ſenſible <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce (removing the Accidental <lb></lb>Impediments, which in the Theory are not regarded) to move the <lb></lb>given Weight along the Horizon, ſo that he goeth about in vain <lb></lb>afterwards to ſeek with what <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce it is to be moved along the <lb></lb>elevated Plane. </s> <s>It will be therefore better, the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce that moveth <lb></lb>the Weight upwards perpendicularly, (which equalizeth the Gra<lb></lb>vity of that Weight which is to be moved) being given, to <lb></lb>ſeek the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce that moveth it along the Elevated Plane: Which <lb></lb>we will endeavour to do in a Method different from that of <lb></lb><emph type="italics"></emph>Pappus.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let us therefore ſuppoſe the Circle A I C, and in it the Diame<lb></lb>ter A B C, and the Center B, and two Weights of equal Moment <lb></lb>in the extreams B and C; ſo that the Line A C being a Leaver, <lb></lb>or Ballance moveable about the Center B, the Weight C ſhall <lb></lb>come to be ſuſtained by the Weight A. </s> <s>But if we ſhall imagine <lb></lb>the Arm of the Ballance B C to be inclined downwards according <lb></lb>to the Line B F, but yet in ſuch a manner that the two Lines <emph type="italics"></emph>A B<emph.end type="italics"></emph.end><lb></lb>and <emph type="italics"></emph>B F<emph.end type="italics"></emph.end> do continue ſolidly conjoyned in the point <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> in this caſe <lb></lb>the Moment of the Weight C ſhall not be equal to the Moment <lb></lb><figure id="id.070.01.024.1.jpg" xlink:href="070/01/024/1.jpg"></figure><lb></lb>of the Weight <emph type="italics"></emph>A,<emph.end type="italics"></emph.end> for that the Di<lb></lb>ſtance of the point <emph type="italics"></emph>F<emph.end type="italics"></emph.end> from the Line <lb></lb>of Direction, which goeth accord<lb></lb>ing to B I, from the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>ulciment B un<lb></lb>to the Center of the Earth, is dimi<lb></lb>niſhed: But if from the point <emph type="italics"></emph>F<emph.end type="italics"></emph.end> we <lb></lb>erect a Perpendicular unto B C, as is <lb></lb><emph type="italics"></emph>F<emph.end type="italics"></emph.end> K, the Moment of the Weight in <lb></lb><emph type="italics"></emph>F<emph.end type="italics"></emph.end> ſhall be as if it did hang by the <lb></lb>Line K <emph type="italics"></emph>F,<emph.end type="italics"></emph.end> and look how much the <lb></lb>Diſtance K B is diminiſhed by the <lb></lb>Diſtance B <emph type="italics"></emph>A,<emph.end type="italics"></emph.end> ſo much is the Moment of the Weight <emph type="italics"></emph>F<emph.end type="italics"></emph.end> diminiſhed <lb></lb>by the Moment of the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>eight <emph type="italics"></emph>A. A<emph.end type="italics"></emph.end>nd in this faſhion inclining <lb></lb>the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>eight more, as for inſtance, according to B L, its Moment ſhall <lb></lb>ſtill diminiſh and ſhall be as if it did hang at the Diſtance <emph type="italics"></emph>B<emph.end type="italics"></emph.end> M, ac<lb></lb>cording to the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine M <emph type="italics"></emph>L,<emph.end type="italics"></emph.end> in which point <emph type="italics"></emph>L<emph.end type="italics"></emph.end> it ſhall be ſuſtained by <lb></lb>a <emph type="italics"></emph>W<emph.end type="italics"></emph.end>eight placed in <emph type="italics"></emph>A,<emph.end type="italics"></emph.end> ſo much leſs than it ſelf, by how much the <lb></lb>Diſtance B <emph type="italics"></emph>A<emph.end type="italics"></emph.end> is greater than the Diſtance <emph type="italics"></emph>B<emph.end type="italics"></emph.end> M. </s> <s>See therefore that <lb></lb>the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>eight placed in the extream of the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>eaver B C, in inclining <lb></lb>downwards along the Circumference C <emph type="italics"></emph>F L<emph.end type="italics"></emph.end> I, cometh to diminiſh <lb></lb>its Moment and <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of going downwards from time to time, <pb xlink:href="070/01/025.jpg" pagenum="295"></pb>more and leſs, as it is more or leſs ſuſtained by the Lines B F and <lb></lb>B L: But the conſidering that this Grave deſcending, and ſuſtained <lb></lb>by the Semidiameters B F and B L is one while leſs, and another <lb></lb>while more conſtrained to paſs along the Circumference C F L, is <lb></lb>no other, than if we ſhould imagine the ſame Circumference <lb></lb>C F L I to be a Superſicies ſo curved, and put under the ſame <lb></lb>Moveable: ſo that bearing it ſelf thereon it were conſtrained to <lb></lb>deſcend along thereby; for if in the one and other manner the <lb></lb>Moveable deſcribeth the ſame Courſe or Way, it will nothing im<lb></lb>port whether, if ſuſpended at the Center B, it is ſuſtained by the <lb></lb>Semidiameter of the Circle, or elſe, whether that Fulciment being <lb></lb>taken away, it proceed along the Circumference C F L I: So that <lb></lb>we may confidently affirm, that the Grave deſcending downwards <lb></lb>from the point C along the Circumference C F L I, its Moment <lb></lb>of Deſcent in the point C is total and entire, becauſe it is not in <lb></lb>any part ſuſtained by the Circumference: And there is not in that <lb></lb>firſt point C, any indiſpoſition to Motion different from that, which <lb></lb>being at liberty, it would make along the Perpendicular and Con<lb></lb>tingent Line D C E: But if the Moveable ſhall be placed in the <lb></lb>point F, then its Gravity is in part ſuſtained, and its Moment of <lb></lb>Deſcent is diminiſhed by the Circular Path or Way that is placed <lb></lb>under it, in that proportion wherewith the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine <emph type="italics"></emph>B<emph.end type="italics"></emph.end> K is overcome <lb></lb>by <emph type="italics"></emph>B<emph.end type="italics"></emph.end> C: But if when the Moveable is in F, at the firſt inſtant of <lb></lb>ſuch its Motion, it be as if it were in the Plane elevated according <lb></lb>to the Contingent <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine G F H, for that reaſon the inclination of the <lb></lb>Circumference in the point F differeth not from the inclination of <lb></lb>the Contingent <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine F G any more ſave the inſenſible Angle of <lb></lb>the Contact. </s> <s>And in the ſame manner we ſhall find the Moment <lb></lb>of the ſaid Moveable to diminiſh in the point <emph type="italics"></emph>L,<emph.end type="italics"></emph.end> as the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine BM <lb></lb>is diminiſhed by B C; ſo that in the Plane contingent to the Circle <lb></lb>in the point <emph type="italics"></emph>L,<emph.end type="italics"></emph.end> as for inſtance, according to the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine N <emph type="italics"></emph>L<emph.end type="italics"></emph.end> O, the <lb></lb>Moment of Deſcent diminiſheth in the Moveable with the ſame <lb></lb>proportion. </s> <s>If therefore ^{*} upon the Plane HG the Moment of the <lb></lb><arrow.to.target n="marg1117"></arrow.to.target><lb></lb>Moveable be diminiſhed by the total <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> which it hath in its <lb></lb>Perpendicular D C E, according to the proportion of the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine K B <lb></lb>to the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine B C, and B F, being by the Solicitude of the Triangles <lb></lb>K B F and K F H the ſame proportion betwixt the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ines K F and <lb></lb>F H, as betwixt the ſaid K B and <emph type="italics"></emph>B<emph.end type="italics"></emph.end> F, we will conclude that the <lb></lb>proportion of the entire and abſolute Moment, that the Moveable <lb></lb>hath in the Perpendicular to the Horizon to that which it hath up<lb></lb>on the Inclined Plane H F, hath the ſame proportion that the <lb></lb><emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine H F hath to the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine F K; that is, that the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ength of the <lb></lb>Inclined Plane hath to the Perpendicular which ſhall fall from it <lb></lb>unto the Horizon. </s> <s>So that paſſing to a more diſtinct Figure, ſuch <lb></lb>as this here preſent, the Moment of Deſcending which the Move<pb xlink:href="070/01/026.jpg" pagenum="296"></pb>able hath upon the inclined Plane C A hath to its total Moment <lb></lb>wherewith it gravitates in the Perpendicular to the Horizon C P the <lb></lb>ſame proportion that the ſaid Line P C hath to C A. </s> <s>And if thus it <lb></lb>be, it is manifeſt, that like as the Force that ſuſtai<lb></lb>neth the Weight in the Perpendiculation P C ought <lb></lb><figure id="id.070.01.026.1.jpg" xlink:href="070/01/026/1.jpg"></figure><lb></lb>to be equal to the ſame, ſo for ſuſtaining it in the <lb></lb>inclined Plane C A, it will ſuffice that it be ſo much <lb></lb>leſſer, by how much the ſaid Perpendicular C P wan<lb></lb>teth of the Line C A: and becauſe, as ſometimes we <lb></lb>ſce, it ſufficeth, that the Force for moving of the <lb></lb>Weight do inſenſibly ſuperate that which ſuſtaineth it, therefore <lb></lb>we will infer this univerſal Propoſition, [That upon an Elevated <lb></lb>Plane the Force hath to the Weight the ſame proportion, as the <lb></lb>Perpendicular let fall from the Plane unto the Horizon hath to the <lb></lb>Length of the ſaid Plane.]</s></p><p type="margin"> <s><margin.target id="marg1117"></margin.target>* Or along</s></p><p type="main"> <s>Returning now to our firſt Intention, which was to inveſtigate <lb></lb>the Nature of the Screw, we will conſider the Triangle A B C, of <lb></lb>which the Line A B is Horizontal, B C perpendicular to the ſaid <lb></lb>Horizon, and A C a Plane elevated; upon which the Moveable D <lb></lb>ſhall be drawn by a Force ſo much leſs than it, by how much the <lb></lb>Line B C is ſhorter than C A: But to elevate or raiſe the ſaid <lb></lb>Weight along the ſaid Plane A C, is as much as if the Triangle <lb></lb>C A B ſtanding ſtill, the Weight <lb></lb><figure id="id.070.01.026.2.jpg" xlink:href="070/01/026/2.jpg"></figure><lb></lb>D be moved towards C, which is <lb></lb>the ſame, as if the ſame Weight <lb></lb>never removing from the Perpen<lb></lb>dicular A E, the Triangle did <lb></lb>preſs forwards towards H. </s> <s>For if <lb></lb>it were in the Site F H G, the <lb></lb>Moveable would be found to <lb></lb>have mounted the height A I. <lb></lb>Now, in fine, the primary Form and Eſſence of the Screw is no<lb></lb>thing elſe but ſuch a Triangle A C B, which being forced for<lb></lb>wards, ſhall work it ſelf under the Grave Body to be raiſed, and <lb></lb>lifteth it up, as we ſay, by the ^{*} head and ſhoulders. </s> <s>And this was <lb></lb><arrow.to.target n="marg1118"></arrow.to.target><lb></lb>its firſt Original: For its firſt Inventor (whoever he was) conſi<lb></lb>dering how that the Triangle A B C going forwards raiſeth the <lb></lb>Weight D, he might have framed an Inſtrument like to the ſaid <lb></lb>Triangle, of a very ſolid Matter, which being thruſt forwards did <lb></lb>raiſe up the propoſed Weight: But afterwards conſidering better, <lb></lb>how that that ſame Machine might be reduced into a much leſſer <lb></lb>and more commodious Form, taking the ſame Triangle he twined <lb></lb>and wound it about the Cylinder A B C D in ſuch a faſhion, that <lb></lb>the height of the ſaid Triangle, that is the Line C B, did make the <lb></lb>Height of the Cylinder, and the Aſcending Plane did beget upon <pb xlink:href="070/01/027.jpg" pagenum="297"></pb>the ſaid Cylinder the Helical Line deſcribed by the Line AEFGH, <lb></lb>which we vulgarly call the Wale of the Screw, which was produ<lb></lb>ced by the Line A C. </s> <s>And in this manner is the Inſtrument made, <lb></lb>which is by the Greeks called <foreign lang="grc">Κόχλος,</foreign> and by us a Screw; which <lb></lb><arrow.to.target n="marg1119"></arrow.to.target><lb></lb>winding about <lb></lb>cometh to work <lb></lb><figure id="id.070.01.027.1.jpg" xlink:href="070/01/027/1.jpg"></figure><lb></lb>and inſinu<lb></lb>ate with its <lb></lb>Wales under <lb></lb>the Weight, and <lb></lb>with facility rai<lb></lb>ſeth it. </s> <s>And we <lb></lb>having demon<lb></lb>ſtrated, That up<lb></lb>on [<emph type="italics"></emph>or along<emph.end type="italics"></emph.end>] <lb></lb>the elevated Plane the Force hath the ſame proportion to the <lb></lb>Weight, that the perpendicular Altitude of the ſaid Plane hath to <lb></lb>its Length; ſo, ſuppoſing that the Force in the Screw A B C D is <lb></lb>multiplied according to the proportion by which the Length of the <lb></lb>whole Wale exceedeth the Altitude C B, from hence we come <lb></lb>to know that making the Screw with its Helix's more thick or cloſe <lb></lb>together, it becometh ſo much the more forceable, as being begot <lb></lb>by a Plane leſs elevated, and whoſe Length regards its own Per<lb></lb>pendicular Altitude with greater proportion. </s> <s>But we will not <lb></lb>omit to advertiſe you, that deſiring to find the Force of a propo<lb></lb>ſed Screw, it will not be needful that we meaſure the Length of <lb></lb>all its Wales, and the Altitude of the whole Cylinder, but it <lb></lb>will be enough if we ſhall but examine how many times the Di<lb></lb>ſtance betwixt two ſingle and Contiguous terms do enter into one <lb></lb>ſole Turn of the ſame Wale, as for example, how many times <lb></lb>the Diſtance AF is contained in the Length of the Turn AEF: <lb></lb>For this is the ſame proportion that the Altitude CB hath to all <lb></lb>the Wale.</s></p><p type="margin"> <s><margin.target id="marg1118"></margin.target><emph type="italics"></emph>Levar in capo<emph.end type="italics"></emph.end><lb></lb> ſignfieth to lift <lb></lb>on high by force</s></p><p type="margin"> <s><margin.target id="marg1119"></margin.target>* <foreign lang="grc">Κόχλος,</foreign> in La<lb></lb>tine <emph type="italics"></emph>Cocblea,<emph.end type="italics"></emph.end> any <lb></lb>Screw winding <lb></lb>like the Shell of <lb></lb>a Snail.</s></p><p type="main"> <s>If all that be underſtood which we have hitherto ſpoken touch<lb></lb>ing the Nature of this Inſtrument, I do not doubt in the leaſt but <lb></lb>that all the other circumſtances may without difficulty be compre<lb></lb>hended: as for inſtance, that inſteed of making the Weight to <lb></lb>mount upon the Screw if one accommodates its Nut with <lb></lb>the Helix incavated or made hollow, into which the Male Screw <lb></lb>that is the Wale entring, & then being turned round it raiſeth and <lb></lb>lifteth up the Nut or Male Screw together with the Weight which <lb></lb>was hanged thereat. </s> <s>Laſtly, we are not to paſs over that Conſidera<lb></lb>tion with ſilence which at the beginning hath been ſaid to be neceſ<lb></lb>ſary for us to have in all Mechanick Inſtruments, to wit, That <lb></lb>what is gained in Force by their aſſiſtance, is loſt again in Time, <pb xlink:href="070/01/028.jpg" pagenum="298"></pb>and in the Velocity: which peradventure, might not have ſeemed <lb></lb>to ſome ſo true and manifeſt in the preſent Contemplation; nay, <lb></lb>rather it ſeems, that in this caſe the Force is multiplied without the <lb></lb>Movers moving a longer way than the Moveable: In regard, that <lb></lb>if we ſhall in the Triangle A B C ſuppoſe the Line A B to be the <lb></lb>Plane of the Horizon, A C the elevated Plane, whoſe Altitude is <lb></lb>meaſured by the Perpendicular C B, a Moveable placed upon the <lb></lb>Plane A C, and the Cord E D <emph type="italics"></emph>F<emph.end type="italics"></emph.end> tyed to it, and a <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce or Weight <lb></lb>applyed in <emph type="italics"></emph>F<emph.end type="italics"></emph.end> that hath to the <lb></lb>Gravity of the Weight E the <lb></lb><figure id="id.070.01.028.1.jpg" xlink:href="070/01/028/1.jpg"></figure><lb></lb>ſame proportion that the Line <lb></lb>B C hath to C A; by what <lb></lb>hath been demonſtrated, the <lb></lb>Weight <emph type="italics"></emph>F<emph.end type="italics"></emph.end> ſhall deſcend <lb></lb>downwards, drawing the <lb></lb>Moveable E along the eleva<lb></lb>ted Plane; nor ſhall the Move<lb></lb>able E meaſure a greater Space <lb></lb>when it ſhall have paſſed the <lb></lb>whole Line A <emph type="italics"></emph>C,<emph.end type="italics"></emph.end> than that which the ſaid Grave <emph type="italics"></emph>F<emph.end type="italics"></emph.end> meaſureth in its <lb></lb>deſcent downwards. </s> <s>But here yet it muſt be advertiſed, that al<lb></lb>though the Moveable E ſhall have paſſed the whole Line A C, in <lb></lb>the ſame Time that the other Grave <emph type="italics"></emph>F<emph.end type="italics"></emph.end> ſhall have been abaſed the <lb></lb>like Space, nevertheleſs the Grave E ſhall not have retired from the <lb></lb>common Center of things Grave more than the Space of the Per<lb></lb>pendicular <emph type="italics"></emph>C<emph.end type="italics"></emph.end> B. but yet the Grave <emph type="italics"></emph>F<emph.end type="italics"></emph.end> deſcending Perpendicularly ſhall <lb></lb>be abaſed a Space equal to the whole Line A <emph type="italics"></emph>C.<emph.end type="italics"></emph.end> And becauſe Grave <lb></lb>Bodies make no Reſiſtance to Tranſverſal Motions, but only ſo <lb></lb>far as they happen to recede from the <emph type="italics"></emph>C<emph.end type="italics"></emph.end>enter of the Earth; There<lb></lb>fore the Moveable E in all the Motion A <emph type="italics"></emph>C<emph.end type="italics"></emph.end> being raiſed no more <lb></lb>than the length of the Line <emph type="italics"></emph>C<emph.end type="italics"></emph.end>B, but the other <emph type="italics"></emph>F<emph.end type="italics"></emph.end> being abaſed per<lb></lb>pendicularly the quantity of all the Line A <emph type="italics"></emph>C<emph.end type="italics"></emph.end>: Therefore we may <lb></lb>deſervedly affirm that Way of the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce E maintaineth the ſame <lb></lb>proportion to the <emph type="italics"></emph>F<emph.end type="italics"></emph.end>orce <emph type="italics"></emph>F<emph.end type="italics"></emph.end> that the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine A <emph type="italics"></emph>C<emph.end type="italics"></emph.end> hath to <emph type="italics"></emph>C<emph.end type="italics"></emph.end> B; that is, <lb></lb>the Weight E to the Weight <emph type="italics"></emph>F.<emph.end type="italics"></emph.end> It very much importeth, therefore, <lb></lb>to conſider by [<emph type="italics"></emph>or along<emph.end type="italics"></emph.end>] what <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ines the Motions are made, eſpe<lb></lb>cially in exanimate Grave Bodies, the Moments of which have their <lb></lb>total Vigour, and entire Reſiſtance in the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine Perpendicular to <lb></lb>the Horizon; and in the others tranſverſally Elevated and Inclined <lb></lb>they feel the more or leſs Vigour, <emph type="italics"></emph>Impetus,<emph.end type="italics"></emph.end> or Reſiſtance, the more <lb></lb>or leſs thoſe Inclinations approach unto the Perpendicular Inclina<lb></lb>tion.</s></p><pb xlink:href="070/01/029.jpg" pagenum="299"></pb><p type="head"> <s><emph type="italics"></emph>Of the SCREW of<emph.end type="italics"></emph.end> ARCHIMEDES <lb></lb><emph type="italics"></emph>to draw Waier.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>I Do not think it ſit in this place to paſs over with Silence the <lb></lb>Invention of <emph type="italics"></emph>Archimedes<emph.end type="italics"></emph.end> to raiſe Wa er with the Screw, which <lb></lb>is not only marvellous, but miraculous: for we ſhall find that <lb></lb>the Water aſcendeth in the Screw continually deſcending; and in <lb></lb>a given Time, with a given Force doth raiſe an unſpeakable quan<lb></lb>tity therof. </s> <s>But before we proceed any farther, let us declare the uſe <lb></lb>of the Screw in making Water to riſe: And in the enſuing Figure, <lb></lb>let us conſider the Line I L O P Q <lb></lb><figure id="id.070.01.029.1.jpg" xlink:href="070/01/029/1.jpg"></figure><lb></lb>R S H being wrapped or twined <lb></lb>about the Collumn M I K H, <lb></lb>which Line you are to ſuppoſe to <lb></lb>be a Chanel thorow which the <lb></lb>Water may run: If we ſhall put <lb></lb>the end I into the Water, making <lb></lb>the Screw to ſtand leaning, ſo as <lb></lb>the point L may be lower than <lb></lb>the firſt I, as the Diagram ſhew<lb></lb>eth, and ſhall turn it round about <lb></lb>on the two Axes, T and V, the Water ſhall run thorow the Cha<lb></lb>nel, till that in the end it ſhall diſcharge ſorth at the mouth H. <lb></lb></s> <s>Now I ſay, that the Water, in its conveyance from the point I to <lb></lb>the point H, doth go all the way deſcending, although the point H <lb></lb>be higher than the point I. </s> <s>Which that it is ſo, we will declare <lb></lb>in this manner. </s> <s>We will deſcribe the Triangle A C B, which is <lb></lb>that of which the Screw H I is generated, in ſuch ſort that the <lb></lb>Chanel of the Screw is repreſented by the Line A C, whoſe <lb></lb>Aſcent and Elevation is determined by the Angle C A B; that is <lb></lb>to ſay, if ſo be, that that Angle ſhall be the third or fourth part of a <lb></lb>Right Angle, then the Elevation of the Chanel A C ſhall be ac<lb></lb>cording to 1/3, or 1/4 of a Right Angle. </s> <s>And it is manifeſt; that the <lb></lb>Riſe of that ſame Chanel A C will be taken away debaſing the <lb></lb>point C as far as to B: for then the Chanel A C ſhall have no <lb></lb>Elevation. </s> <s>And debaſing the point C a little below B, the Water <lb></lb>will naturally run along the Chanel A C downwards from the <lb></lb>point A towards C. </s> <s>Let us therefore conclude, that the Angle A <lb></lb>being 1/3 of a Right Angle, the Chanel A C ſhall no longer have any <lb></lb>Riſe, debaſing it on the part <emph type="italics"></emph>C<emph.end type="italics"></emph.end> for 1/3 of a Right Angle.</s></p><p type="main"> <s>Theſe things underſtood, let us infold the Triangle about the <lb></lb>Column, and let us make the Screw B A E F G, &c. </s> <s>which if it <lb></lb>ſhall be placed at Right Angles with the end B in the Water, turn<lb></lb>ing it about, it ſhall not this way draw up the Water, the Chanel <lb></lb>about the Column being elevated, as may be ſeen by the part B A.</s> <pb xlink:href="070/01/030.jpg" pagenum="300"></pb><s>But although the Column ſtand erect at Right-Angles, yet for all <lb></lb>that, the Riſe along the Screw, folded about the Column, is not of <lb></lb>a greater Elevation than of 1/3 of a Right Angle, it being generated <lb></lb>by the Elevation of the Chanel A C: Therefore if we incline the <lb></lb>Column but 1/3 of the <lb></lb><figure id="id.070.01.030.1.jpg" xlink:href="070/01/030/1.jpg"></figure><lb></lb>ſaid Right Angle, and <lb></lb>a little more, as we ſee <lb></lb>I K H M, there is a <lb></lb>Tranſition and Moti<lb></lb>on along the Chanel <lb></lb>I L: Therefore the <lb></lb>Water from the point <lb></lb>I to the point L ſhall <lb></lb>move deſcending, and <lb></lb>the Screw being turned <lb></lb>about, the other parts <lb></lb>of it ſhall ſucceſſively <lb></lb>diſpoſe or preſent <lb></lb>themſelves to the Wa<lb></lb>ter in the ſame Poſition as the part I L: Whereupon the Water <lb></lb>ſhall go ſucceſſively deſcending, and in the end ſhall be found to <lb></lb>be aſcended from the point I to the point H. </s> <s>Which how admira<lb></lb>ble a thing it is, I leave ſuch to judge who ſhall perfectly have un<lb></lb>derſtood it. </s> <s>And by what hath been ſaid, we come to know, That <lb></lb>the Screw for raiſing of Water ought to be inclined a little more <lb></lb>than the quantity of the Angle of the Triangle by which the ſaid <lb></lb>Screw is deſcribed.</s></p><p type="head"> <s><emph type="italics"></emph>Of the Force of the <lb></lb>HAMMER, MALLET, or BEETLE.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>The Inveſtigation of the cauſe of the Force of theſe Percuti<lb></lb>ents is neceſſary for many Reaſons: and firſt, becauſe that <lb></lb>there appeareth in it much more matter of admiration than <lb></lb>is obſerved in any other Mechanick Inſtrument whatſoever. </s> <s>For <lb></lb>ſtriking with the Hammer upon a Nail, which is to be driven into <lb></lb>a very tough Poſt, or with the Beetle upon a Stake that is to pene<lb></lb>trate into very ſtiffe ground, we ſee, that by the ſole vertue of the <lb></lb>blow of the Percutient both the one and the other is thruſt for<lb></lb>wards: ſo that without that, only laying the Beetle upon the <lb></lb>Nail or Stake it will not move then, nay, more, although you <lb></lb>ſhould lay upon them a Weight very much heavier than the ſaid <lb></lb>Beetle. </s> <s>An effect truly admirable, and ſo much the more worthy <lb></lb>of Contemplation, in that, as I conceive, none of thoſe who have <pb xlink:href="070/01/031.jpg" pagenum="301"></pb>hitherto diſcourſed upon it, have ſaid any thing that hits the mark; <lb></lb>which we may take for a certain Sign and Argument of the Obſcu<lb></lb>rity and difficulty of this <emph type="italics"></emph>S<emph.end type="italics"></emph.end>peculation. </s> <s>For <emph type="italics"></emph>Ariſtotle,<emph.end type="italics"></emph.end> or others, <lb></lb>who would reduce the cauſe of this admirable Effect unto the <lb></lb>length of the <emph type="italics"></emph>Manubrium,<emph.end type="italics"></emph.end> or Handle, may, in my judgement, be <lb></lb>made to ſee their miſtake in the effect of thoſe Inſtruments, which <lb></lb>having no Handle, yet percuſs, either in falling from on high <lb></lb>downwards, or by being thrown with Velocity ſidewaies. </s> <s>There<lb></lb>fore it is requiſite, that we have recourſe to ſome other Principle, if <lb></lb>we would find out the truth of this buſineſs; the cauſe of which, <lb></lb>although it be of its own nature ſomewhat obſcure, and of diffi<lb></lb>cult conſideration, yet nevertheleſs we will attempt with the grea<lb></lb>teſt perſpicuity poſſible to render it clear and obvious, ſhewing, for <lb></lb>a cloſe of all, that the Principle and Original of this Effect is deri<lb></lb>ved from no other Fountain than this, from which the reaſons of all <lb></lb>other Mechanick Effects do proceed: and this we will do, by ſetting <lb></lb>before your eyes that very thing which is ſeen to befall in every <lb></lb>other Mechanick Operation, <emph type="italics"></emph>ſcilicet,<emph.end type="italics"></emph.end> That the Force, the Reſiſtance, <lb></lb>and the Space by which the Motion is made, do go alternately <lb></lb>with ſuch proportion operating, and with ſuch a rate anſwering to <lb></lb>each other, that a Reſiſtance, equal to the Force, ſhall be moved by <lb></lb>the ſaid Force along an equal Space, with Velocity equal to that <lb></lb>with which it is moved. </s> <s>Likewiſe, That a Force that is leſs by half <lb></lb>than a Reſiſtance ſhall be able to move it, ſo that it be moved <lb></lb>with double Velocity, or, if you will, for a Diſtance twice as great <lb></lb>as that which the moved Reſiſtance ſhall paſs: and, in a word, it <lb></lb>hath been ſeen in all the other Inſtruments, that any, never ſo great, <lb></lb>Reſiſtance may be moved by every ſmall Force given, provided, <lb></lb>that the Space, along which the Reſiſtance ſhall move, have the <lb></lb>ſame proportion that is found to be betwixt the ſaid great Reſi<lb></lb>ſtance and the Force: and that this is according to the neceſſary <lb></lb>Order and Conſtitution of Nature: So that inverting the Diſcourſe, <lb></lb>and Arguing the contrary way, what wonder ſhall it be, if that <lb></lb>Power that ſhall move a ſmall Reſiſtance a great way, ſhall carry <lb></lb>one an hundred times bigger an hundredth part of that Diſtance? <lb></lb></s> <s>Certainly none at all: nay, it would be abſurd, yea, impoſſible <lb></lb>that it ſhould be otherwiſe. </s> <s>Let us therefore conſider, what the <lb></lb>Reſiſtance of the Beetle unto Motion may be in that point where <lb></lb>it is to ſtrike, and how far, if it do not ſtrike, it would be carryed <lb></lb>by the received Force beyond that point: and again, what Reſi<lb></lb>ſtance to Motion there is in him who ſtriketh, and how much by <lb></lb>that ſame Percuſſion he is moved: and, having found that this <lb></lb>great Reſiſtance goeth forwards by a percuſſion ſo much leſs than <lb></lb>the Beetle driven by the <emph type="italics"></emph>Impetus<emph.end type="italics"></emph.end> of him that moveth it would do, <lb></lb>by how much that ſame great Reſiſtance is greater than that of <pb xlink:href="070/01/032.jpg" pagenum="302"></pb>the Beetle; we ſhall ceaſe to wonder at the Effect, which doth not <lb></lb>in the leaſt exceed the terms of Natural Conſtitutions, and of <lb></lb>what hath been ſpoken. </s> <s>Let us, for better underſtanding, give an <lb></lb>example thereof in particular Terms. </s> <s>There is a Beetle, which ha<lb></lb>ving four degrees of Reſiſtance, is moved by ſuch a Force, that <lb></lb>being freed from it in that term where it maketh the Percuſſion, it <lb></lb>would, meeting with no ſtop, go ten Paces beyond it, and in that <lb></lb>term a great poſt being oppoſed to it, whoſe Reſiſtance to Moti<lb></lb>on is as four thouſand, that is, a thouſand times greater than that of <lb></lb>the Beetle, (but yet is not immoveable) ſo that it without mea<lb></lb>ſure or proportion exceeds the Reſiſtance of the Beetle, yet the <lb></lb>Percuſſion being made on it, it ſhall be driven forwards, though in<lb></lb>deed no more but the thouſandth part of the ten Paces which the <lb></lb>Beetle ſhall be moved: and thus in an inverted method, changing <lb></lb>that which hath been ſpoken touching the other Mechanical Effects, <lb></lb>we may inveſtigate the reaſon of the Force of the Percutient. </s> <s>I <lb></lb>know that here ariſe difficulties and objections unto ſome, which <lb></lb>they will not eaſily be removed from, but we will freely remit them <lb></lb><arrow.to.target n="marg1120"></arrow.to.target><lb></lb>to the ^{*} Problems Mechanical, which we ſhall adjoyn in the end of <lb></lb>this Diſcourſe.</s></p><p type="margin"> <s><margin.target id="marg1120"></margin.target>* Theſe Pro<lb></lb>blems he here <lb></lb>promiſeth were <lb></lb>never yet ex<lb></lb>tant.</s></p><pb xlink:href="070/01/033.jpg" pagenum="303"></pb><p type="head"> <s>THE <lb></lb>BALLANCE <lb></lb>OF <lb></lb><emph type="italics"></emph>Signeur GALILEO GALILEI<emph.end type="italics"></emph.end>;</s></p><p type="head"> <s>In which, in immitation of <emph type="italics"></emph>Archimedes<emph.end type="italics"></emph.end> in the <lb></lb>Problem of the Crown, he ſheweth how to <lb></lb>find the proportion of the Alloy of <lb></lb>Mixt-Metals; and how to make <lb></lb>the ſaid Inſtrument.</s></p><p type="main"> <s>As it is well known, by ſuch who take the pains to read <lb></lb>old Authors, that <emph type="italics"></emph>Archimedes<emph.end type="italics"></emph.end> detected the Cheat of <lb></lb>the Goldſmith in the Crown of ^{*} <emph type="italics"></emph>Hieron,<emph.end type="italics"></emph.end> ſo I think it <lb></lb><arrow.to.target n="marg1121"></arrow.to.target><lb></lb>hitherto unknown what method this Great Philoſo<lb></lb>pher obſerved in that Diſcovery: for the opinion, that he did per<lb></lb>form it by putting the Crown into the Water, having firſt put in<lb></lb><arrow.to.target n="marg1122"></arrow.to.target><lb></lb>to it ſuch another Maſs of pure Gold, and another of Silver ſeve<lb></lb>rally, and that from the differences in their making the Water <lb></lb>more or leſs riſe and run over, he came to know the Mixture or <lb></lb>Alloy of the Gold with the Silver, of which that Crown was <lb></lb>compounded; ſeems a thing (if I may ſpeak it) very groſs, and <lb></lb>far from exactneſs. </s> <s>And it will ſeem ſo much the more dull to <lb></lb>ſuch who have read and underſtood the exquiſite Inventions of ſo <lb></lb>Divine a Man amongſt the Memorials that are extant of him; by <lb></lb>which it is very manifeſt that all other Wits are inferiour to that <lb></lb>of <emph type="italics"></emph>Archimedes.<emph.end type="italics"></emph.end> Indeed I believe, that Fame divulging it abroad, <lb></lb>that <emph type="italics"></emph>Archimedes<emph.end type="italics"></emph.end> had diſcovered that ſame Fraud by means of the <lb></lb>Water, ſome Writer of thoſe Times committed the memory there<lb></lb>of to Poſterity, and that this perſon, that he might add ſomething <lb></lb>to that little which he had heard by common Fame, did relate that <lb></lb><emph type="italics"></emph>Archimedes<emph.end type="italics"></emph.end> had made uſe of the Water in that manner, as ſince <lb></lb>hath been by the generality of men believed.</s></p><p type="margin"> <s><margin.target id="marg1121"></margin.target>* King of <emph type="italics"></emph>Sicily,<emph.end type="italics"></emph.end><lb></lb>and Kinſman to <lb></lb>that Great Ma<lb></lb>thematician.</s></p><p type="margin"> <s><margin.target id="marg1122"></margin.target><emph type="italics"></emph>Plutarch in Vit. <lb></lb></s> <s>Marcel.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>But in regard I know, that that method is altogether fallacious, <lb></lb>and falls ſhort of that exactneſs which is required in Mathematical <lb></lb>Matters, I have often thought in what manner, by help of the <lb></lb>Water, one might exactly find the Mixture of two Metals, and <lb></lb>in the end, after I had diligently peruſed that which <emph type="italics"></emph>Archimedes<emph.end type="italics"></emph.end><lb></lb>demonſtrateth in his Books <emph type="italics"></emph>De inſidentibus aquæ,<emph.end type="italics"></emph.end> and thoſe others <pb xlink:href="070/01/034.jpg" pagenum="304"></pb><emph type="italics"></emph>De æquiponder antium,<emph.end type="italics"></emph.end> there came into my thoughts a Rule which <lb></lb>exquiſitely reſolveth our Queſtion; which Rule I believe to be <lb></lb>the ſame that <emph type="italics"></emph>Archimedes<emph.end type="italics"></emph.end> made uſe of, ſeeing that beſides the <lb></lb>uſe that is to be made of the Water, the exactneſs of the Work <lb></lb>dependeth alſo upon certain Demonſtrations found by the ſaid <lb></lb><emph type="italics"></emph>Archimedes.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>The way is by help of a Ballance, whoſe Conſtruction and Uſe <lb></lb>ſhall be ſhewn by and by, after we ſhall have declared what is <lb></lb>neceſſary for the knowledge thereof. </s> <s>You muſt know there<lb></lb>fore, that the Solid Bodies that ſink in the Water weigh ſo much <lb></lb>leſs in the Water than in the Air, as a Maſs of Water equal to <lb></lb>the ſaid Solid doth weigh in the Air: which hath been demon<lb></lb>ſtrated by <emph type="italics"></emph>Archimedes.<emph.end type="italics"></emph.end> But, in regard his Demonſtration is very <lb></lb>mediate, becauſe I would not be over long, laying it aſide, I ſhall <lb></lb>declare the ſame another way. </s> <s>Let us conſider, therefore, that <lb></lb>putting into the Water <emph type="italics"></emph>v. </s> <s>g.<emph.end type="italics"></emph.end> a Maſs of Gold, if that Maſs were <lb></lb>of Water it would have no weight at all: For the Water moveth <lb></lb>neither upwards, nor downwards in the Water: It remains, <lb></lb>therefore, that the Maſs of Gold weigheth in the Water only ſo <lb></lb>much as the Gravity of the Gold exceeds the Gravity of the Wa<lb></lb>ter. </s> <s>And the like is to be underſtood of other Metals. </s> <s>And be<lb></lb>cauſe the Metals are different from each other in Gravity, their <lb></lb>Gravity in the Water ſhall diminiſh according to ſeveral proporti<lb></lb>ons. </s> <s>As for example: Let us ſuppoſe that Gold weigheth twenty <lb></lb>times more than Water, it is manifeſt by that which hath been <lb></lb>ſpoken, that the Gold will weigh leſs in the Water than in the <lb></lb>Air by a twentieth part of its whole weight. </s> <s>Now, let us ſuppoſe <lb></lb>that Silver, as being leſs Grave than Gold, weigheth 12 times more <lb></lb>than Water: this then, being weighed in the Water, ſhall di<lb></lb>miniſh in Gravity the twelfth part of its whole weight. </s> <s>Therefore <lb></lb>the Gravity of Gold in the Water decreaſeth leſs than that of <lb></lb>Silver; for that diminiſheth a twentieth part, and this a twelfth. <lb></lb></s> <s>If therefore in an exquiſite Ballance we ſhall hang a Metal at the <lb></lb>one Arm, and at the other a Counterpoiſe that weigheth equally <lb></lb>with the ſaid Metal in the Water, leaving the Counterpoiſe in the <lb></lb>Air, to the end that it may equivalate and compenſate the Me<lb></lb>tal, it will be neceſſary to hang it nearer the Perpendicular or <lb></lb>Cook. </s> <s>As for example, Let the Ballance be A B, its Perpendicu<lb></lb><figure id="id.070.01.034.1.jpg" xlink:href="070/01/034/1.jpg"></figure><lb></lb>lar C, and let a <lb></lb>Maſs of ſome <lb></lb>Metal be ſu<lb></lb>ſpended at B, <lb></lb>counterpoiſedby <lb></lb>the Weight D: putting the Weight B into the Water, the <lb></lb>Weight D in A would weigh more: therefore that they may <pb xlink:href="070/01/035.jpg" pagenum="305"></pb>weigh equally it would be neceſſary to hang it nearer to the <lb></lb>Perpendicular C, as <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> in E: and look how many times the Di<lb></lb>ſtance C A ſhall contain A E, ſo many times ſhall the Metal <lb></lb>weigh more than the Water. </s> <s>Let us therefore ſuppoſe that the <lb></lb>Weight in B be Gold, and that weighed in the Water it with<lb></lb>draws the Counterpoiſe D into E; and then doing the ſame with <lb></lb>pure Silver, let us ſuppoſe that its Counterpoiſe, when afterwards <lb></lb>it is weighed in the Water, returneth to F: which point ſhall be <lb></lb>nearer to the point C, as Experience ſheweth, becauſe the Silver <lb></lb>is leſs grave than the Gold: And the Diſtance that is between <lb></lb>A and F ſhall have the ſame Difference with the Diſtance A E, <lb></lb>that the Gravity of the Gold hath with that of the Silver. </s> <s>But if <lb></lb>we have a Mixture of Gold and Silver, it is clear, that by reaſon it <lb></lb>participates of Silver, it ſhall weigh leſs than the pure Gold, and <lb></lb>by reaſon it participates of Gold, it ſhall weigh more than the <lb></lb>pure Silver: and therefore being weighed in the Air, and deſiring <lb></lb>that the ſame Counterpoiſe ſhould counterpoiſe it, when that <lb></lb>Mixture ſhall be put into the Water it will be neceſſary to draw <lb></lb>the ſaid Counterpoiſe more towards the Perpendicular C, than the <lb></lb>point E is, which is the term of the Gold; and more from C <lb></lb>than F is, which is the term of the pure Silver; Therefore it ſhall <lb></lb>fall between the points E and F: And the proportion into which <lb></lb>the Diſtance EF ſhall be divided, ſhall exactly give the proportion <lb></lb>of the two Metals which compound that Mixture. </s> <s>As for exam<lb></lb>ple: Let us ſuppoſe the Mixture of Gold and Silver to be in B, <lb></lb><figure id="id.070.01.035.1.jpg" xlink:href="070/01/035/1.jpg"></figure><lb></lb>counterpoiſed in <lb></lb>the Air by D, <lb></lb>which Counter<lb></lb>poiſe when the <lb></lb>Compound Me<lb></lb>tal is put into the Water returneth into G: I ſay now, that the <lb></lb>Gold and the Silver which compound this Mixture are to one ano<lb></lb>ther in the ſame proportion, as the Diſtance F G is to the Diſtance <lb></lb>G E. </s> <s>But you muſt know that the Diſtance G F terminated in <lb></lb>the mark of the Silver, ſhall denote unto us the quantity of the <lb></lb>Gold, and the Diſtance G E, terminated in the mark of the Gold, <lb></lb>ſhall ſhew us the quantity of the Silver: inſomuch that if F G <lb></lb>ſhall prove double to G E, then that Mixture ſhall be two parts <lb></lb>Gold, and one part Silver: and in the ſame method proceeding in<lb></lb>the examination of other Mixtures, one ſhall exactly find the <lb></lb>quantity of the ſimple Metals.</s></p><p type="main"> <s>To compoſe the Ballance, therefore, take a Rod at leaſt a yard <lb></lb>long, (and the longer it is, the exacter the Inſtrument ſhall be) <lb></lb>and divide it in the midſt, where place the Perpendicular: then <lb></lb>adjuſt the Arms that they may ſtand in <emph type="italics"></emph>Equilibrium,<emph.end type="italics"></emph.end> by filing or <pb xlink:href="070/01/036.jpg" pagenum="306"></pb>ſhaving that leſs which weigheth moſt; and upon one of the Arms <lb></lb>note the terms to which the Counterpoiſes of ſimple Metals return <lb></lb>when they ſhall be weighed in the Water: taking care to weigh the <lb></lb>pureſt Metals that can be found. </s> <s>This being done, it remaineth <lb></lb>that we find out a way, how we may with facility diſcover the <lb></lb>proportion, according to which, the Diſtances between the terms <lb></lb>of the ſimple and pure Metals are divided by the Marks of the <lb></lb>Mixt Metals: Which ſhall be effected in this manner.</s></p><p type="main"> <s>We are to have two very ſmall Wires drawn thorow the ſame <lb></lb>drawing-Iron, one of Steel, the other of Braſs, and above the <lb></lb>terms of the ſimple Metals we muſt wind the Steel Wyer; as for <lb></lb>example: above the point E, the term of the pure Gold, we are <lb></lb>to wind the Steel Wyer, and under it the other Braſs Wyre, and <lb></lb>having made ten folds of the Steel Wyer, we muſt make ten <lb></lb>more with that of Braſs, and thus we are to continue to do with <lb></lb>ten of Steel, and ten of Braſs, until that the whole Space be<lb></lb>tween the points E and F, the terms of the pure Metals, be full; <lb></lb>cauſing thoſe two terms to be alwaies viſible and perſpicuous: <lb></lb>and thus the Diſtance E F ſhall be divided into many equal parts, <lb></lb>and numbred by ten and ten. </s> <s>And if at any time we would know <lb></lb>the proportion that is between F G and G E, we muſt count the <lb></lb>Wyers F G, and the Wyers G E: and finding the Wyers F G <lb></lb>to be, for example, 40, and the Wyers G E, 21: we will ſay that <lb></lb>there is in the mixt Metal 40 parts of Gold, and 21 of Silver. </s> <s>But <lb></lb>here you muſt note, that there is ſome difficulty in the counting, <lb></lb>for thoſe Wyers being very ſmall, as it is requiſite for exactneſs <lb></lb>ſake, it is not poſſible with the eye to tell them, becauſe the <lb></lb>ſmalneſs of the Spaces dazleth & confoundeth the Sight. </s> <s>Therefore <lb></lb>to number them with facility, take a Bodkin as ſharp as a Needle <lb></lb>and ſet it into an handle, or a very fine pointed Pen-knife, with <lb></lb>which we may eaſily run over all the ſaid Wyers, and this way <lb></lb>partly by help of hearing, partly by the impediments the hand <lb></lb>ſhall feel at every Wyer, thoſe Wyers ſhall be counted; <lb></lb>the number of which, as I ſaid before, ſhall give us the exact <lb></lb>quantity of the ſunple Metals, of which the Mixt-Metal is com<lb></lb>pounded: taking notice that the Simple anſwer alternately to the <lb></lb>Diſtances. </s> <s>As for example, in a Mixture of Gold and Silver, <lb></lb>the Wyers that ſhall be towards the term of Gold ſhall ſhew us <lb></lb>the quantity of the Silver: And the ſame is to be underſtood of <lb></lb>other Metals.</s></p><pb xlink:href="070/01/037.jpg" pagenum="307"></pb><p type="head"> <s>Annotations of <emph type="italics"></emph>Dominico Mantovani<emph.end type="italics"></emph.end> upon the Bal<lb></lb>lance of <emph type="italics"></emph>Signore Galileo Galilei.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Firſt, I conceive that the difficulty of Numbring the Wyres <lb></lb>is removed by wrapping about the Ballance ten of Steel, <lb></lb>and then ten of Braſs, which being divided by tens, there <lb></lb>only remains that tenth part to be numbred, in which the term <lb></lb>of the Mixt Metal falleth. </s> <s>For although <emph type="italics"></emph>Signore Galileo,<emph.end type="italics"></emph.end> who is <lb></lb>Author of this Invention, makes mention of two Wyres, one of <lb></lb>Steel, the other of Braſs, yet he doth not ſay, that we are to <lb></lb>take ^{*} ten of the one, and ten of the other: which it may be <lb></lb><arrow.to.target n="marg1123"></arrow.to.target><lb></lb>hapneth by the negligence of him that hath tranſcribed it; al<lb></lb>though I muſt confeſs that the Copy which came to my hands was <lb></lb>of his own writing.</s></p><p type="margin"> <s><margin.target id="marg1123"></margin.target>* <emph type="italics"></emph>Galileus<emph.end type="italics"></emph.end> ſaith it <lb></lb>expreſly in this <lb></lb>Copy which I fol<lb></lb>low, but might <lb></lb>omit it in the Co<lb></lb>py which came to <lb></lb>the hands of <emph type="italics"></emph>Man<lb></lb>tovani.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Secondly, it is ſuppoſed in this Problem that the Compoſition <lb></lb>of two Metals do retain the ſame proportion of Maſs in the <lb></lb>Mixture as the two Simple Metals, of which it is compounded, <lb></lb>had at firſt. </s> <s>I mean, that the Simple Metals retain and keep in <lb></lb>the Compoſition (after that they are incorporated and commix<lb></lb>ed) the ſame proportion in Maſs that the Simple Metals had <lb></lb>when they were ſeparated: Which in the Caſe of <emph type="italics"></emph>Signore Gali<lb></lb>leo,<emph.end type="italics"></emph.end> touching the Commixtion of Gold and Silver, I do neither <lb></lb>deny, nor particularly confeſs. </s> <s>But if one would, for example, <lb></lb>unite 101 pounds of Copper with 21 pounds of Tin, to make <lb></lb>thereof 120 pounds of Bell-Metal, (I abate two pounds, <lb></lb>ſuppoſed to be waſted in the Melting) I do think that 120 <lb></lb>pounds of Compound Metal will have a leſs Bulk than the 100 <lb></lb>pounds of pure Copper, and the 20 pounds of Tin unmixt, that <lb></lb>is, before they were incorporated and melted into one Maſs, and <lb></lb>that the Compoſition is more grave <emph type="italics"></emph>in Specie<emph.end type="italics"></emph.end> than the ſingle Cop<lb></lb>per, and the ſingle Braſs: and in the Caſe of <emph type="italics"></emph>Signore Galileo<emph.end type="italics"></emph.end> the <lb></lb>Compoſition of Gold and Silver is ſuppoſed to be lighter <emph type="italics"></emph>in Specie<emph.end type="italics"></emph.end><lb></lb>than the pure Gold, and heavier <emph type="italics"></emph>in Specie<emph.end type="italics"></emph.end> than the pure Silver. </s> <s>Of <lb></lb>which it would be eaſie to make ſome ſuch like experiment, melt<lb></lb>ing together, <emph type="italics"></emph>v. </s> <s>gr.<emph.end type="italics"></emph.end> 10 pounds of Lead with 5 pounds of Tin, <lb></lb>and obſerving whether thoſe 15 pounds, or whatever the Mixture <lb></lb>maketh, do give the difference betwixt the weight in the Water <lb></lb>to the weight in the Air, in the proportion that the 15 pounds of <lb></lb>the two Metals diſ-united gave before: I do not ſay, the ſame diffe<lb></lb>rence, becauſe I pre ſuppoſe that they will waſte in melting down, <lb></lb>and that the Compound will be leſs than 15 pounds, therefore I <lb></lb>ſay in proportion.</s></p><p type="main"> <s>Thirdly, He doth alſo ſuppoſe, that one ought to take the <pb xlink:href="070/01/038.jpg" pagenum="308"></pb>Simple Metals, that is, the Gold and the Silver, each of the ſame <lb></lb>weight as the Mixture, although he doth not ſay ſo; which may <lb></lb>be collected in that he marketh the ballance only betwixt the <lb></lb>Terms of the Gold and the Silver, which is the cauſe of the great <lb></lb>facility in reſolving the Problem by only counting the <lb></lb>Wyers.</s></p><p type="main"> <s>One might take the pure Gold, and pure Silver of the ſame <lb></lb>weight, in reſpect of one another, but yet different from the <lb></lb>weight of the Mixture, that is, either more or leſs grave than the <lb></lb>Mixt Metal: and being equal in weight to one another they <lb></lb>might ſhew the proportion of the Maſs of the Gold to that of the <lb></lb>Silver; but yet with this difference, that the more grave will ſhew <lb></lb>the ſaid proportion more exactly than the ſmall and leſs grave. <lb></lb></s> <s>But the Simple and pure Metals not being of the ſame weight as <lb></lb>the Compound, it will be neceſſary, having found the proportion <lb></lb>of the Maſs of the Gold to that of the Silver; to find by numbers <lb></lb>proportionally the exact quantity of each of the two Metals com<lb></lb>pounding the Mixture.</s></p><p type="main"> <s>A man may likewiſe uſe the quantity of the ſimple Metals ac<lb></lb>cording to neceſſity and convenience, although of different <lb></lb>Weights, both as to each other, and to the Mixture, provided that <lb></lb>each of them be pure in its kind: but then we muſt after<lb></lb>wards by numbers find the proportion of the Maſſes of the two <lb></lb>Simple ones of equal weight (which is ſoon done, taking them of <lb></lb>equal weight as was ſaid before) and then according to this pro<lb></lb>portion to find, by means of the Weight, and of the Maſs of the <lb></lb>Compound Metal, the diſtinct quantity of each of the two Sim<lb></lb>ple ones that make the Compoſition: of each of which Caſes <lb></lb>examples might be given. </s> <s>But to conclude, if the pure Gold, <lb></lb>and pure Silver, and the Mixt Metal ſhould be of equal Maſs, <lb></lb>they would be unequal in Weight, and it would not need to <lb></lb>weigh them in the Water, for being of equal Bulk, the differen<lb></lb>ces of their Weights in the Air and in the Water would be alſo <lb></lb>equal: for the difference of the weight of any Body in the Air <lb></lb>to its weight in the Water, is alwaies equal to the Weight of ſo <lb></lb>much Water as equalleth the ſame Body in Maſs, by <emph type="italics"></emph>Archimedes<emph.end type="italics"></emph.end><lb></lb>his fifth Propoſition, <emph type="italics"></emph>De ijs quæ vehuntur in aqua.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>And laſt of all, the Simple and pure Metals may have the ſame <lb></lb>proportion in Gravity, mutually or reciprocally, as their Bodies <lb></lb>have in Bulk: In which caſe, as well the Maſs, found by help of <lb></lb>the weight in Water, or by any other meanes, as their Weight in <lb></lb>the Air ſhall ſhew the proportion of their Specifical Gravities; as <lb></lb>their Weights in the Water do when their Weights in the Air <lb></lb>are equal; but yet alternately weighed: that is to ſay, the Spe<lb></lb>cifical Gravity of the Gold ſhall have ſuch proportion to the <pb xlink:href="070/01/039.jpg" pagenum="309"></pb>Specifical Gravity of the Silver, as the Maſs of the Silver hath to <lb></lb>the Maſs of the Gold; that is, as the difference betwixt the <lb></lb>Weight in Water and Weight in Air of the Silver, hath to the <lb></lb>difference betwixt the Weight in Water and Weight in Air of <lb></lb>the Gold.</s></p><p type="main"> <s>With this ſame Ballance one may with facility meaſure the <lb></lb>Maſs or Magnitude of any Body, in any manner whatſoever Irre<lb></lb>gular in manner following, namely:</s></p><p type="main"> <s>We will have at hand a Solid Body of a ſubſtance more grave <lb></lb><emph type="italics"></emph>in Specie<emph.end type="italics"></emph.end> than the Water; as for inſtance of Lead; or if it were <lb></lb>of Wood, or other matter more light <emph type="italics"></emph>in Specie<emph.end type="italics"></emph.end> than the Water, <lb></lb>it may be made heavier by faſtning unto it Lead, or ſome other <lb></lb>thing that makes it ſink in the Water, and let us take ſome <lb></lb>known Meaſure, and with it meaſure the Irregular Solid; as for <lb></lb>inſtance, the Roman Palm, the Geometrical Foot, or any other <lb></lb>known meaſure, or part of the ſame, as the half Foot, the quar<lb></lb>ter of a Foot, or any ſuch like part known; then let it be weighed <lb></lb>in the Air, and ſuppoſe that it weigh 10 pounds; let the ſame <lb></lb>Meaſure be weighed in the Air, and ſuppoſe that it weigh 8 <lb></lb>pounds: and ſubſtract 8 pounds, the Weight in the Water, from <lb></lb>10 pounds, the Weight in the Air, and there remaineth 2 pounds <lb></lb>for the Weight of a Body of Water equal in Magnitude to the <lb></lb>Meaſure known. </s> <s>Now, if we would meaſure a Statue of Mar<lb></lb>ble, let it be weighed firſt in the Air, and then in the Water, and <lb></lb>ſubſtract the Weight in the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>ater from the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>eight in the Air, and <lb></lb>the remainder ſhall be the weight of ſo much <emph type="italics"></emph>W<emph.end type="italics"></emph.end>ater as equalleth <lb></lb>the Statue in Maſs; which being divided by the difference betwixt <lb></lb>the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>eight in <emph type="italics"></emph>W<emph.end type="italics"></emph.end>ater and the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>eight in Air of the Meaſure known, <lb></lb>the Quotient will give how many times the Statue containeth the <lb></lb>ſame given Meaſure. </s> <s>As for example; if the Statue in Air weigh <lb></lb>100 pounds, and in the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>ater 80 pounds, 80 pounds being ſub<lb></lb>ſtracted from 100 there reſteth 20 pounds for the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>eight of ſo <lb></lb>much <emph type="italics"></emph>W<emph.end type="italics"></emph.end>ater in Maſs as equalleth the Statue. </s> <s>But becauſe the <lb></lb>difference betwixt the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>eight in <emph type="italics"></emph>W<emph.end type="italics"></emph.end>ater, and the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>eight in Air <lb></lb>equal in Magnitude to the Meaſure known, was ſuppoſed to be <lb></lb>2 pounds; divide 18 pounds by two pounds, and the Quotient <lb></lb>is 9, for the number of times that the propoſed Statue containeth <lb></lb>the given Meaſure. </s> <s>The ſame Method may be obſerved, if it <lb></lb>were required, to meaſure a Statue, or other Maſs of any kind of <lb></lb>Metal: only it muſt be advertiſed, that all the holes muſt be <lb></lb>ſtopt, that the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>ater may not enter into the Body of the Statue: <lb></lb>but he that deſireth only the Solid content of the Metal of the <lb></lb>ſaid Statue muſt open the holes, and with Tunnels fill the whole <lb></lb>cavity of the Statue with <emph type="italics"></emph>W<emph.end type="italics"></emph.end>ater. </s> <s>And if the Statue were of a <lb></lb>Subſtance lighter <emph type="italics"></emph>in Specie<emph.end type="italics"></emph.end> than the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>ater; as, for example, of <pb xlink:href="070/01/040.jpg" pagenum="310"></pb>Wax, it will be requiſite to add unto the Statue ſome Counter<lb></lb>poiſe, that maketh it ſink in the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>ater, and then to meaſure the <lb></lb>Counterpoiſe, as above, and to ſubſtract its meaſure from the <lb></lb>Compound Body, and there will remain the Meaſure of the <lb></lb>Statue of <emph type="italics"></emph>W<emph.end type="italics"></emph.end>ax. </s> <s>And laſtly, to make uſe of the ſaid Ballance, <lb></lb>inſtead of ſeeking the numbers of the pounds of the Differences <lb></lb>of the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>eights of the Meaſure known, and of the Solid <lb></lb>to be meaſured in <emph type="italics"></emph>W<emph.end type="italics"></emph.end>ater, and in Air, we may count the <lb></lb><emph type="italics"></emph>W<emph.end type="italics"></emph.end>yers of the Arm of the Ballance, which <lb></lb>being very ſmall will give the <lb></lb>Meaſure exactly.</s></p><p type="head"> <s><emph type="italics"></emph>FINIS.<emph.end type="italics"></emph.end></s></p></chap> </body> <back></back> </text></archimedes>