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DESpecs 2.0 Autumn 2009
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Thu, 02 May 2013 11:14:40 +0200 |
| parents | 22d6a63640c6 |
| children |
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<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" ><archimedes> <info> <author>Galilei, Galileo</author> <title>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"/><p type="head"> <s>GALILEUS, <lb/>HIS <lb/>MECHANICKS: <lb/>OF THE BENEFIT DERIVED <lb/>FROM THE SCIENCE OF MECHANICKS, <lb/>AND FROM ITS INSTRUMENTS.</s></p> </section> </front> <body> <chap> <p type="main"> <s>I judged it extreamly nece&longs;&longs;ary, before our <lb/>de&longs;cending to the Speculation of Mecha­<lb/>nick In&longs;truments, to con&longs;ider how I might, <lb/>as it were, &longs;et before your eyes in a gene­<lb/>ral Di&longs;cour&longs;e, the many benefits that are <lb/>derived from the &longs;aid In&longs;truments: and <lb/>this I have thought my &longs;elf the more ob­<lb/>liged to do, for that (if I am not mi&longs;taken) <lb/>I have &longs;een the generality of <emph type="italics"/>M<emph.end type="italics"/>echaniti­<lb/>ans deceive them&longs;elves in going about to apply Machines to many <lb/>operations of their own nature impo&longs;&longs;ible; by the &longs;ucce&longs;&longs;e where­<lb/>of they have been di&longs;appointed, and others likewi&longs;e fru&longs;trate of <lb/>the hope which they had conceived upon the promi&longs;e of tho&longs;e pre­<lb/>&longs;umptuous undertakers: of which mi&longs;takes I think I have found <lb/>the principall cau&longs;e to be the belief and con&longs;tant opinion the&longs;e <pb xlink:href="070/01/002.jpg" pagenum="272"/>Artificers had, and &longs;till have, that they are able with a &longs;mall force <lb/>to move and rai&longs;e great weights; (in a certain manner with their <lb/>Machines cozening nature, who&longs;e In&longs;tinct, yea mo&longs;t po&longs;itive con­<lb/>&longs;titution it is, that no Re&longs;i&longs;tance can be overcome, but by a Force <lb/>more potent then it:) which conjecture how fal&longs;e it is, I hope by <lb/>the en&longs;uing true and nece&longs;&longs;ary Demon&longs;trations to evince.</s></p><p type="main"> <s>In the mean time, &longs;ince I have hinted, that the benefit and help <lb/>derived from Machines is not, to be able with le&longs;&longs;e Force, by help <lb/>of the Machine to move tho&longs;e weights, which, without it, could <lb/>not be moved by the &longs;ame Force: it would not be be&longs;ides the <lb/>purpo&longs;e to declare what the Commodities be which are derived to <lb/>us from &longs;uch like faculties, for if no profit were to be hoped for, <lb/>all endeavours employed in the acqui&longs;t thereof will be but lo&longs;t <lb/>labour.</s></p><p type="main"> <s>Proceeding therefore according to the nature of the&longs;e Studies, <lb/>let us fir&longs;t propo&longs;e four things to be con&longs;idered. </s> <s>Fir&longs;t, the weight <lb/>to be transferred from place to place; and &longs;econdly, the Force <lb/>and Power which &longs;hould move it; thirdly, the Di&longs;tance between <lb/>the one and the other Term of the Motion; Fourthly, the Time <lb/>in which that mutation is to be made: which Time becometh the <lb/>&longs;ame thing with the Dexterity, and Velocity of the Motion; we <lb/>determining that Motion to be more &longs;wift then another, which in <lb/>le&longs;&longs;e Time pa&longs;&longs;eth an equal Di&longs;tance.</s></p><p type="main"> <s>Now, any determinate Re&longs;i&longs;tance and limited Force what&longs;oever <lb/>being a&longs;&longs;igned, and any Di&longs;tance given, there is no doubt to be <lb/>made, but that the given Force may carry the given Weight to the <lb/>determinate Di&longs;tance; for, although the Force were extream <lb/>&longs;mall, yet, by dividing the Weight into many &longs;mall parts, none <lb/>of which remain &longs;uperiour to the Force, and by transferring them <lb/>one by one, it &longs;hall at la&longs;t have carried the whole Weight to the <lb/>a&longs;&longs;igned Term: and yet one cannot at the end of the Work with <lb/>Rea&longs;on &longs;ay, that that great Weight hath been moved, and tran&longs;­<lb/>ported by a Force le&longs;&longs;e then it &longs;elf, howbeit indeed it was done <lb/>by a Force, that many times reiterated that Motion, and that <lb/>Space, which &longs;hall have been mea&longs;ured but only once by the whole <lb/>Weight. </s> <s>From whence it appears, that the Velocity of the Force <lb/>hath been as many times Superiour to the Re&longs;i&longs;tance of the weight, <lb/>as the &longs;aid Weight was &longs;uperiour to the Force; for that in the <lb/>&longs;ame Time that the moving Force hath many times mea&longs;ured the <lb/>intervall between the Terms of the Motion, the &longs;aid Moveable <lb/>happens to have pa&longs;t it onely once: nor therefore ought we to <lb/>affirm a great Re&longs;i&longs;tance to have been overcome by a &longs;mall Force, <lb/>contrary to the con&longs;titution of Nature. </s> <s>Then onely may we &longs;ay <lb/>the Natural Con&longs;titution is overcome, when the le&longs;&longs;er Force tran&longs;­<lb/>fers the greater Re&longs;i&longs;tance, with a Velocity of Motion like to that <pb xlink:href="070/01/003.jpg" pagenum="273"/>wherewith it &longs;elf doth move; which we affirm ab&longs;olutely to be <lb/>impo&longs;&longs;ible to be done with any Machine imaginable. </s> <s>But becau&longs;e <lb/>it may &longs;ometimes come to pa&longs;&longs;e, that having but little Force, it is <lb/>required to move a great Weight all at once, without dividing it <lb/>in pieces, on this occa&longs;ion it will be necei&longs;ary to have recour&longs;e to <lb/>the Machine, by means whereof the propo&longs;ed Weight may be <lb/>transferred to the a&longs;&longs;igned Space by the Force given. </s> <s>But yet <lb/>this doth not hinder, but that the &longs;ame Force is to move, mea&longs;uring <lb/>that &longs;ame Space, or another equall to it, as many &longs;everall times as <lb/>it is exceeded by the &longs;aid Weight. </s> <s>So that in the end of the a­<lb/>ction we &longs;hall &longs;ind that we have received from the Machine no <lb/>other benefit tnen only that of tran&longs;porting the &longs;aid Weight with <lb/>the given Force to the Term given, all at once. </s> <s>Which Weight, <lb/>being divided into parts, would without any Machine have been <lb/>carried by the &longs;ame Force, in the &longs;ame Time, through the &longs;ame <lb/>Intervall. </s> <s>And this ought to pa&longs;&longs;e for one of the benefits taken <lb/>from the Mechanicks: for indeed it frequently happens, that be­<lb/>ing &longs;canted in Force but not Time, we are put upon moving great <lb/>Weights unitedly or in gro&longs;&longs;e: but he that &longs;hould hope, and at­<lb/>tempt to do the &longs;ame by the help of Machines without increa&longs;e of <lb/>Tardity in the Moveable, would certainly be deceived, and would <lb/>declare his ignorance of the u&longs;e of Mechanick In&longs;truments, and <lb/>the rea&longs;on of their effects.</s></p><p type="main"> <s>Another benefit is drawn from the In&longs;truments, which depend­<lb/>eth on the place wherein the operation is to be made: for all In­<lb/>&longs;truments cannot be made u&longs;e of in all places with equall conve­<lb/>nience. </s> <s>And &longs;o we &longs;ee (to explain our &longs;elves by an example) that <lb/>for drawing of Water out of a Well, we make u&longs;e of onely a <lb/>Rope and a Bucket fitted to receive and hold Water, wherewith <lb/>we draw up a determinate quantity of Water, in a certain Time, <lb/>with our limited &longs;trength: and he that &longs;hould think he could with <lb/>a Machine of what&longs;oever Force, with the &longs;ame &longs;trength, and in <lb/>the &longs;ame Time, take up a great quantity of Water, is in a gro&longs;&longs;e <lb/>Errour. </s> <s>And he &longs;hall find him&longs;elf &longs;o much the more deceived, <lb/>the more he &longs;hall vary and multiply his Inventions: Yet never­<lb/>thele&longs;&longs;e we &longs;ee Water drawn up with other Engines, as with a Pump <lb/>that drinks up Water in the Hold of Ships; where you mu&longs;t note <lb/>that the Pump was not imployed in tho&longs;e Offices, for that it draws <lb/>up more Water in the &longs;ame Time, and with the &longs;ame &longs;trength <lb/>then that which a bare Bucket would do, but becau&longs;e in that place <lb/>the u&longs;e of the Bucket or any &longs;uch like Ve&longs;&longs;el could not effect what <lb/>is de&longs;ired, namely to keep the Hold of the Ship quite dry from e­<lb/>very little quantity of Water; which the Bucket cannot do, for <lb/>that it cannot dimerge and dive, where there is not a con&longs;iderable <lb/>depth of Water. </s> <s>And thus we &longs;ee the Holds of Ships by the <pb xlink:href="070/01/004.jpg" pagenum="274"/>&longs;aid In&longs;trument kept dry, when Water cannot but onely oblique­<lb/>ly be drawn up, which the ordinary u&longs;e of the Bucket would not <lb/>effect, which ri&longs;eth and de&longs;cends with its Rope perpendicu­<lb/>larly.</s></p><p type="main"> <s>The third is a greater benefit, haply, then all the re&longs;t that are <lb/>derived from Mechanick In&longs;truments, and re&longs;pects the a&longs;&longs;i&longs;tance <lb/>which is borrowed of &longs;ome Force exanimate, as of the &longs;tream of a <lb/>River, or el&longs;e animate, but of le&longs;&longs;e expence by far, then that which <lb/>would be nece&longs;&longs;ary for maintaining humane &longs;trength: as when to <lb/>turn Mills, we make u&longs;e of the Current of a River, or the &longs;trength <lb/>of a Hor&longs;e, to effect that, which would require the &longs;trength of five <lb/>or fix Men. </s> <s>And this we may al&longs;o advantage our &longs;elves in rai&longs;ing <lb/>Water, or making other violent Motions, which mu&longs;t have been <lb/>done by Men, if there were no other helps; becau&longs;e with one &longs;ole <lb/>Ve&longs;&longs;el we may take Water, and rai&longs;e, and empty it where occa&longs;ion <lb/>requires; but becau&longs;e the Hor&longs;e, or &longs;uch other Mover wanteth <lb/>Rea&longs;on, and tho&longs;e In&longs;truments which are requi&longs;ite for holding and <lb/>emptying the Ve&longs;&longs;el in due time, returning again to fill it, and one­<lb/>ly is endued with Force, therefore it's nece&longs;&longs;ary that the Mecha­<lb/>nitian &longs;upply the naturall defect of that Mover, furni&longs;hing it with <lb/>&longs;uch devices and inventions, that with the &longs;ole application of it's <lb/>Force the defired effect may follow. </s> <s>And therein is very great <lb/>advantage, not becau&longs;e that a Wheel or other Machine can enable <lb/>one to tran&longs;port the &longs;ame Weight with le&longs;&longs;e Force, and greater <lb/>Dexterity, or a greater Space than an equall Force, without tho&longs;e <lb/>In&longs;truments, but having Judgment and proper Organs, could have <lb/>done; but becau&longs;e that the &longs;tream of a River co&longs;teth little or <lb/>nothing, and the charge of keeping of an Hor&longs;e or other Bea&longs;t, <lb/>who&longs;e &longs;trength is greater then that of eight, or it may be more <lb/>Men, is far le&longs;&longs;e then what &longs;o many Men would be kept <lb/>for.</s></p><p type="main"> <s>The&longs;e then are the benefits that may be derived from Mecha­<lb/>nick In&longs;truments, and not tho&longs;e which ignorant Engineers dream <lb/>of, to their own di&longs;grace, and the abu&longs;e of &longs;o many Princes, <lb/>whil&longs;t they undertake impo&longs;&longs;ible enterprizes; of which, both <lb/>by the little which hath been hinted, and by the much which <lb/>&longs;hall be demon&longs;trated in the Progre&longs;&longs;e of this Treati&longs;e, we &longs;hall <lb/>come to a&longs;&longs;ure our &longs;elves, if we attentively heed that which &longs;hall <lb/>be &longs;poken.</s></p><pb xlink:href="070/01/005.jpg" pagenum="275"/><p type="head"> <s>DEFINITIONS.</s></p><p type="main"> <s>That which in all Demon&longs;trative Sciences is nece&longs;&longs;ary to be <lb/>ob&longs;erved, we ought al&longs;o to follow in this Di&longs;cour&longs;e, that is; <lb/>to propound the Definitions of the proper Terms of this <lb/>Art, and the primary Suppo&longs;itions, from which, as from &longs;eeds full <lb/>of fecundity, may of con&longs;equence &longs;pring and re&longs;ult the cau&longs;es, <lb/>and true Demon&longs;trations, of the Nature of all the Mechanick <lb/>Engines which are u&longs;ed, for the mo&longs;t part about the Motions of <lb/>Grave Matters, therefore we will determine, fir&longs;t, what is <emph type="italics"/>GRA­<lb/>VITIE.<emph.end type="italics"/></s></p><p type="main"> <s>We call <emph type="italics"/>GRAVITIE<emph.end type="italics"/> then, That propen&longs;ion of moving <lb/>naturally downwards, which is found in &longs;olid Bodies, cau&longs;ed by <lb/>the greater or le&longs;&longs;e quantity of matter, whereof they are con&longs;ti­<lb/>tuted.</s></p><p type="main"> <s><emph type="italics"/>MOMENT<emph.end type="italics"/> is the propen&longs;ion of de&longs;cending, cau&longs;ed not &longs;o <lb/>much by the Gravity of the moveable, as by the di&longs;po&longs;ure which <lb/>divers Grave Bodies have in relation to one another; by means of <lb/>whichMoment, we oft &longs;ee a Body le&longs;s Grave counterpoi&longs;e another <lb/>of greater Gravity: as in the Stiliard, a great Weight is rai&longs;ed by <lb/>a very &longs;mall counterpoi&longs;e, not through exce&longs;s of Gravity, but <lb/>through the remotene&longs;&longs;e from the point whereby the Beam is up­<lb/>held, which conjoyned to the Gravity of the le&longs;&longs;er weight adds <lb/>thereunto Moment, and <emph type="italics"/>Impetus<emph.end type="italics"/> of de&longs;cending, wherewith the <lb/>Moment of the other greater Gravity may be exceeded. <emph type="italics"/>MO­<lb/>MENT<emph.end type="italics"/> then is that IMPETUS of de&longs;cending, compounded <lb/>of Gravity, Po&longs;ition, and the like, whereby that propenfion may <lb/>be occa&longs;ioned</s></p><p type="main"> <s>The <emph type="italics"/>CENTER<emph.end type="italics"/> of <emph type="italics"/>GRAVITY<emph.end type="italics"/> we define to be that point <lb/>in every Grave Body, about which con&longs;i&longs;t parts of equall Moment: <lb/>&longs;o that, imagining &longs;ome Grave Body to be &longs;u&longs;pended and &longs;u&longs;tain­<lb/>ed by the &longs;aid point, the parts on the right hand will Equilibrate <lb/>tho&longs;e on the left, the Anteriour, the Po&longs;teriour, and tho&longs;e above <lb/>tho&longs;e below; &longs;o that be it in any what&longs;oever fite, and po&longs;ition, <lb/>provided it be &longs;u&longs;pended by the &longs;aid <emph type="italics"/>CENTER,<emph.end type="italics"/> it &longs;hall &longs;tand <lb/>&longs;till: and this is that point which would gladly unite with the <lb/>univer&longs;all Center of Grave Bodies, namely withthat of the Earth, <lb/>if it might thorow &longs;ome free <emph type="italics"/>Medium<emph.end type="italics"/> de&longs;cend thither. </s> <s>From <lb/>whence we take the&longs;e Suppo&longs;itions.</s></p><pb xlink:href="070/01/006.jpg" pagenum="276"/><p type="head"> <s>SUPPOSITIONS.</s></p><p type="main"> <s>Any Grave Body, (as to what belongeth to it's proper ver­<lb/>tue) moveth downwards, &longs;o that the Center of it's Gravity <lb/>never &longs;trayeth out of that Right Line which is produced <lb/>from the &longs;aid Center placed in the fir&longs;t Term of the Motion unto <lb/>the univer&longs;al Center of Grave Bodies. </s> <s>Which is a Suppo&longs;ition <lb/>very manife&longs;t, becau&longs;e that &longs;ingle Center being obliged to endea­<lb/>vour to unite with the common Center, it's nece&longs;&longs;ary, unle&longs;&longs;e &longs;ome <lb/>impediment intervene, that it go &longs;eeking it by the &longs;horte&longs;t Line, <lb/>which is the Right alone: And from hence may we &longs;econdarily <lb/>&longs;uppo&longs;e</s></p><p type="main"> <s>Every Grave Body putteth the greate&longs;t &longs;tre&longs;&longs;e, and weigheth <lb/>mo&longs;t on the Center of it's Gravity, and to it, as to its proper &longs;eat, <lb/>all <emph type="italics"/>Impetus,<emph.end type="italics"/> all Pondero&longs;ity, and, in &longs;ome, all Moment hath re­<lb/>cour&longs;e.</s></p><p type="main"> <s>We la&longs;tly &longs;uppo&longs;e the Center of the Gravity of two Bodies e­<lb/>qually Grave to be in the mid&longs;t of that Right Line which conjoyns <lb/>the &longs;aid two Centers; or that two equall weights, &longs;u&longs;pended in <lb/>equall di&longs;tence, &longs;hall have the point of <emph type="italics"/>Equilibrium<emph.end type="italics"/> in the common <lb/>Center, or meeting of tho&longs;e equal Di&longs;tances. </s> <s>As for Example, <lb/>the Di&longs;tance C E being equall to the Di&longs;tance E D, and there be­<lb/>ing by them two equall weights &longs;u&longs;pended, A and B, we &longs;uppo&longs;e <lb/>the point of <emph type="italics"/>Equilibrium<emph.end type="italics"/> to be in the point E, there being no <lb/>greater rea&longs;on for inclining to <lb/>one, then to the other part. </s> <s>But <lb/><figure id="id.070.01.006.1.jpg" xlink:href="070/01/006/1.jpg"/><lb/>here is to be noted, that the Di­<lb/>&longs;tances ought to be mea&longs;ured <lb/>with Perpendicular Lines, which <lb/>from the point of Su&longs;pen&longs;ion E, <lb/>fall on the Right Lines, that from <lb/>the Center of the Gravity of the <lb/>Weights A and B, are drawn to <lb/>the common Center of things <lb/>Grave; and therefore if the Di&longs;tance E D were tran&longs;ported into <lb/>E F, the weight B would not counterpoi&longs;e the weight A, becau&longs;e <lb/>drawing from the Centers of Gravity two Right Lines to the Cen­<lb/>ter of the Earth, we &longs;hall &longs;ee that which cometh from the Center <lb/>of the Weight I, to be nearer to the Center E, then the other <lb/>produced from the Center of the weight A. </s> <s>Therefore our &longs;aying <lb/>that equal Weights are &longs;u&longs;pended by [or at] equal Di&longs;tances, is <lb/>to be under&longs;tood to be meant when as the Right Lines that go from <lb/>their Centers & to &longs;eek out the common Center of Gravity, &longs;hall be <lb/>equidi&longs;ta nt from that Right Line, which is produced from the &longs;aid <pb xlink:href="070/01/007.jpg" pagenum="277"/>Term of tho&longs;e Di&longs;tances, that is from the point of Su&longs;pen&longs;ion, to <lb/>the &longs;ame Center of the Earrh.</s></p><p type="main"> <s>The&longs;e things determined and &longs;uppo&longs;ed, we come to the explica­<lb/>tion of a Principle, the mo&longs;t common and materiall of the greater <lb/>part of Mechanick In&longs;truments: demon&longs;trating, that unequall <lb/>Weights weigh equally when &longs;u&longs;pended by [or at] unequal Di&longs;tan­<lb/>ces, which have contrary proportion to that which tho&longs;e weights <lb/>are found to have, See the Demon&longs;tration in the beginning of the <lb/>&longs;econd Dialogue of Local-Motions.</s></p><p type="head"> <s><emph type="italics"/>Some Adverii&longs;ements about what hath been &longs;aid.<emph.end type="italics"/></s></p><p type="main"> <s>Now being that Weights unequall come to acquire equall <lb/>Moment, by being alternately &longs;u&longs;pended at Di&longs;tances that <lb/>have the &longs;ame proportion with them; I think it not fit to <lb/>over pa&longs;&longs;e with &longs;ilence another congruicy and probability, which <lb/>may confirm the &longs;ame truth; for let the Ballance A B, be con&longs;ide­<lb/>red, as it is divided into unequal parts in the point C, and let the <lb/>Weights be of the &longs;ame propor­<lb/><figure id="id.070.01.007.1.jpg" xlink:href="070/01/007/1.jpg"/><lb/>tion that is between the Di&longs;tan­<lb/>ces B C, and C A, alternately <lb/>&longs;u&longs;pended by the points A, and <lb/>B: It is already manife&longs;t, that <lb/>the one will counterpoi&longs;e the <lb/>other, and con&longs;equently, that <lb/>were there added to one of them <lb/>a very &longs;mall Moment of Gravity, it would preponderate, rai&longs;ing <lb/>the other, &longs;o that an in&longs;en&longs;ible Weight put to the Grave B, the <lb/>Ballance would move and de&longs;cend from the point B towards E, <lb/>and the other extream A would a&longs;cend into D, and in regard that <lb/>to weigh down B, every &longs;mall Gravity is &longs;ufficient, therefore not <lb/>keeping any accompt of this in&longs;en&longs;ible Moment, we will put no <lb/>difference between one Weights <emph type="italics"/>&longs;u&longs;taining,<emph.end type="italics"/> and one Weights <lb/><emph type="italics"/>moving<emph.end type="italics"/> another. </s> <s>Now, let us con&longs;ider the Motion which the <lb/>Weight B makes, de&longs;cending into E, and that which the other <lb/>A makes in a&longs;cending into D, we &longs;hall without doubt find the <lb/>Space B E to be &longs;o much greater than the Space A D, as the Di­<lb/>&longs;tance B C is greater than C A, forming in the Center C two an­<lb/>gles D C A, and E C B, equall as being at the Cock, and con&longs;e­<lb/>quently two Circumferences A D and B E alike; and to have the <lb/>&longs;ame proportion to one another, as have the Semidiameters B C, <lb/>and C A, by which they are de&longs;cribed: &longs;o that then the Velocity <lb/>of the Motion of the de&longs;cending Grave B cometh to be &longs;o much <lb/>Superiour to the Velocity of the other a&longs;cending Moveable A, as <lb/>the Gravity of this exceeds the Gravity of that; and it not being <pb xlink:href="070/01/008.jpg" pagenum="278"/>po&longs;&longs;ible that the Weight A &longs;hould be rai&longs;ed to D, although &longs;low­<lb/>ly, unle&longs;&longs;e the other Weight B do move to E &longs;wiftly, it will not <lb/>be &longs;trange, or incon&longs;i&longs;tent with the Order of Nature, that the <lb/>Velocity of the Motion of the Grave B, do compen&longs;ate the greater <lb/>Re&longs;i&longs;tance of the Weight A, &longs;o long as it moveth &longs;lowly to D, <lb/>and the other de&longs;cendeth &longs;wiftly to E, and &longs;o on the contrary, <lb/>the Weight A being placed in the point D, and the other B in <lb/>the point E, it will not be unrea&longs;onable that that falling lea&longs;urely <lb/>to A, &longs;hould be able to rai&longs;e the other ha&longs;tily to B, recovering by <lb/>its Gravity what it had lo&longs;t by it's Tardity of Motion. </s> <s>And by <lb/>this Di&longs;cour&longs;e we may come to know how the Velocity of the <lb/>Motion is able to encrea&longs;e Moment in the Moveable, according to <lb/>that &longs;ame proportion by which the &longs;aid Velocity of the Motion is <lb/>augmented.</s></p><p type="main"> <s>There is al&longs;o another thing, before we proceed any farther, to <lb/>be confidered; and this is touching the Di&longs;tances, whereat, or <lb/>wherein Weights do hang: for it much imports how we are to <lb/>under&longs;tand Di&longs;tances equall, and unequall; and, in &longs;um, in what <lb/>manner they ought to be mea­<lb/><figure id="id.070.01.008.1.jpg" xlink:href="070/01/008/1.jpg"/><lb/>&longs;ured: for that A B being the <lb/>Right Line, and two equall <lb/>Weights being &longs;u&longs;pended at <lb/>the very ends thereof, the point <lb/>C being taken in the mid&longs;t of <lb/>the &longs;aid Line, there &longs;hall be an <lb/><emph type="italics"/>Equilibrium<emph.end type="italics"/> upon the &longs;ame: <lb/>And the rea&longs;on is for that the <lb/>Di&longs;tance C B is equal to C A. <lb/></s> <s>But if elevating the Line C B, moving it about the point C, it <lb/>&longs;hall be transferred into CD, &longs;o that the Ballance &longs;tand according <lb/>to the two Lines A C, and C D, the two equall Weights hanging <lb/>at the Terms A and D, &longs;hall no longer weigh equally on that <lb/>point C, becau&longs;e the di&longs;tance of the Weight placed in D, is made <lb/>le&longs;&longs;e then it was when it hanged in B. </s> <s>For if we confider the Lines, <lb/>along [or by] which the &longs;aid Graves make their Impul&longs;e, and <lb/>would de&longs;cend, in ca&longs;e they were freely moved, there is no doubt <lb/>but that they would make or de&longs;cribe the Lines A G, D F, B H: <lb/>Therefore the Weight hanging on the point D, maketh it's Moment <lb/>and <emph type="italics"/>Impetus<emph.end type="italics"/> according to the Line D F: but when it hanged in <lb/>B, it made <emph type="italics"/>Impetus<emph.end type="italics"/> in the Line B H: and becau&longs;e the Line D F is <lb/>nearer to the Fulciment C, then is the Line B H Therefore we <lb/>are to under&longs;tand that the Weights hanging on the points A and D, <lb/>are not equi-di&longs;tant from the point C, as they be when they are <lb/>con&longs;tituted according to their Right Line A C B: And la&longs;tly, <lb/>we are to take notice, that the Di&longs;tance is to be mea&longs;ured by <pb xlink:href="070/01/009.jpg" pagenum="279"/>Lines, which fall at Right Angles on tho&longs;e whereon the Weights <lb/>hang, and would move, if &longs;o be they were permitted to de&longs;cend <lb/>freely.</s></p><p type="head"> <s>Of the BALLANCE and LEAVER.</s></p><p type="main"> <s>Having under&longs;tood by certain Demon&longs;tration, one of the <lb/>fir&longs;t Principles, from which, as from a plenti&longs;ul Fountain, <lb/>many of the Mechanical In&longs;truments are derived, we may <lb/>take occa&longs;ion without any difficulty to come to the knowledge of <lb/>the nature of them: and fir&longs;t &longs;peaking of the Stiliard, an In&longs;tru­<lb/>ment of mo&longs;t ordinary u&longs;e, with which divers Merchandizes are <lb/>weighed, &longs;u&longs;taining them, though very heavy, with a very &longs;mall <lb/>counterpoi&longs;e, which is com­<lb/>monly called the Roman or <lb/><figure id="id.070.01.009.1.jpg" xlink:href="070/01/009/1.jpg"/><lb/>Plummet, we &longs;hall prove that <lb/>there is no more to be done in <lb/>&longs;uch an operation, but to re­<lb/>duce into act and practice <lb/>what hath been above contemplated. </s> <s>For if we propo&longs;e the Bal­<lb/>lance A B, who&longs;e Fulciment or Lanquet is in the point C, by <lb/>which, at the &longs;mall Di&longs;tance C A, hangeth the heavy Weight D, <lb/>and if along the other greater C B, (which we call the Needle of <lb/>the Stiliard) we &longs;hould &longs;uppo&longs;e the Roman F, though of but little <lb/>weight in compari&longs;on of the Grave Body D to be &longs;lipped to and <lb/>fro, it &longs;hall be pof&longs;ible to place it &longs;o remotely from the Lanquet C, <lb/>that the &longs;ame proportion may be found between the two Weights <lb/>D and F, as is between the Di&longs;tances F C, and C A: and then &longs;hall <lb/>an <emph type="italics"/>Equilibrium<emph.end type="italics"/> &longs;ucceed; unequall Weights hanging at Di&longs;tances <lb/>alternately proportional to them.</s></p><p type="main"> <s>Nor is this In&longs;trument different from that other called <emph type="italics"/>Vectis,<emph.end type="italics"/><lb/><arrow.to.target n="marg1107"></arrow.to.target><lb/>and vulgarly the ^{*} Leaver, wherewith great Weights are moved <lb/>by &longs;mall Force; the application of which is according to the Fi­<lb/>gure prefixed; wherein the Leaver <lb/>is repre&longs;ented by the Bar of wood <lb/>or other &longs;olid matter, <emph type="italics"/>B<emph.end type="italics"/> C D, let <lb/><figure id="id.070.01.009.2.jpg" xlink:href="070/01/009/2.jpg"/><lb/>the heavy Weight to be rai&longs;ed be <lb/>A, and let the &longs;teadfa&longs;t &longs;upport <lb/>or Fulciment on which the Leaver <lb/>re&longs;ts and moves be &longs;uppo&longs;ed to be <lb/>E, and putting one end of the <lb/>Leaver under the Weight A, as <lb/>may be &longs;een in the point C, en­<lb/>crea&longs;ing the Weight or Force at the other end D, it will be able <lb/>to lift up the Weight A, though not much, whenever the Force in <pb xlink:href="070/01/010.jpg" pagenum="280"/>D hath the &longs;ame proportion to the Re&longs;i&longs;tance made by the Weight <lb/>A, in the point C: as the Di&longs;tance <emph type="italics"/>B<emph.end type="italics"/> C hath to the Di&longs;tance C D, <lb/>whereby it's clear, that the nearer the Fulciment E &longs;hall approach <lb/>to the Term B, encrea&longs;ing the proportion of the Di&longs;tance D C to <lb/>the Di&longs;tance C <emph type="italics"/>B,<emph.end type="italics"/> the more may one dimini&longs;h the Force in D which <lb/>is to rai&longs;e the Weight A. </s> <s>And here it is to be noted, which I &longs;hall <lb/>al&longs;o in its place remember you of, that the benefit drawn from all <lb/>Mechanical In&longs;truments, is not that which the vulgar Mechanitians <lb/>do per&longs;wade us, to wit, &longs;uch, that there by Nature is overcome, and <lb/>in a certain manner deluded, a &longs;mall Force over-powring a very <lb/>great Re&longs;i&longs;tance with help of the Leaver; for we &longs;hall demon&longs;trate, <lb/>that without the help of the length of the Leaver, the &longs;ame Force, <lb/>in the &longs;ame Time, &longs;hall work the &longs;ame effect. </s> <s>For taking the &longs;ame <lb/>Leaver B C D, who&longs;e re&longs;t or Fulci­<lb/>ment is in C, let the Di&longs;tance C D <lb/><figure id="id.070.01.010.1.jpg" xlink:href="070/01/010/1.jpg"/><lb/>be &longs;uppo&longs;ed, for example, to be <lb/>in quintuple proportion to the <lb/>Di&longs;tance C <emph type="italics"/>B,<emph.end type="italics"/> & the &longs;aid Leaver to <lb/>be moved till it come to I C G: In <lb/>the Time that the Force &longs;hall have <lb/>pa&longs;&longs;ed the Space D I, the Weight <lb/>&longs;hall have been moved from B <lb/>to G: and becau&longs;e the Di&longs;tance <lb/>D C, was &longs;uppo&longs;ed quintuple to the other C B, it is manife&longs;t from <lb/>the things demon&longs;trated, that the Weight placed in B may be five <lb/>times greater then the moving Force &longs;uppo&longs;ed to be in D: but now, <lb/><arrow.to.target n="marg1108"></arrow.to.target><lb/>if on the contrary, we take notice of the ^{*} Way pa&longs;&longs;ed by <lb/>the Force from D unto I, whil&longs;t the Weight is moved from B unto <lb/>G, we &longs;hall find likewi&longs;e the Way D I, to be quintuple to the Space <lb/>B G. </s> <s>Moreover if we take the Di&longs;tance C L, equal to the Di&longs;tance <lb/>C B, and place the &longs;ame Force that was in D, in the point L, and <lb/>in the point B the fifth part onely of the Weight that was put there <lb/>at fir&longs;t, there is no que&longs;tion, but that the Force in L being now <lb/>equal to this Weight in B, and the Di&longs;tances L C and C B being <lb/>equall, the &longs;aid Force &longs;hall be able, being moved along the Space LM <lb/>to transfer the Weight equall to it &longs;elf, thorow the other equall <lb/>Space B G: which five times reiterating this &longs;ame action, &longs;hall tran&longs;­<lb/>port all the parts of the &longs;aid Weight to the &longs;ame Term G: But <lb/>the repeating of the Space L M, is certainly nothing more nor le&longs;&longs;e <lb/>then the onely once mea&longs;uring the Space D I, quintuple to the <lb/>&longs;aid L M. </s> <s>Therefore the transferring of the Weight from B to G, <lb/>requireth no le&longs;&longs;e Force, nor le&longs;&longs;e Time, nor a &longs;horter Way if it <lb/>wee placed in D, than it would need if the &longs;ame were applied <lb/>in L: And, in &longs;hort, the benefit that is derived from the length of <lb/>the Leaver C D, is no other, &longs;ave the enabling us to move that <pb xlink:href="070/01/011.jpg" pagenum="281"/>Body all at once, which would not have been moved by the &longs;ame <lb/>Force, in the &longs;ame Time, with an equall Motion, &longs;ave onely in <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/>called a Crow, <lb/>if of wood, a Bar <lb/>or Hand-&longs;pike.</s></p><p type="margin"> <s><margin.target id="marg1108"></margin.target>Or Space.</s></p><p type="head"> <s><emph type="italics"/>Of the<emph.end type="italics"/> CAPSTEN <emph type="italics"/>and of the<emph.end type="italics"/> CRANE.</s></p><p type="main"> <s>The In&longs;truments which we are now about to declare, have <lb/>immediate dependence upon the Leaver, nay, are no other <lb/>but a perpetual Vectis or Leaver. </s> <s>For if we &longs;hall &longs;uppo&longs;e the <lb/>Leaver B A C to be &longs;u&longs;tained in <lb/>the point A, and the Weight G to <lb/><figure id="id.070.01.011.1.jpg" xlink:href="070/01/011/1.jpg"/><lb/>hang at the point B, the Force be­<lb/>ing placed in C; It is manife&longs;t, <lb/>that transferring the Leaver unto <lb/>the points D A E, the Weight G <lb/>doth alter according to the Di­<lb/>&longs;tance B D, but cannot much far­<lb/>ther continue to rai&longs;e it, &longs;o that <lb/>if it were required to elevate it yet <lb/>higher, it would be nece&longs;&longs;ary to <lb/>&longs;tay it by &longs;ome other Fulciment <lb/>in this Po&longs;ition, and to remit or return the Leaver to its former Po­<lb/>&longs;ition B A C, and &longs;u&longs;pending the Weight anew thereat, to rai&longs;e it <lb/>once again to the like height B D; and in this manner repeating <lb/>the work, many times one &longs;hall come with an interrupted Motion <lb/>to effect the drawing up of the Weight, which for many re&longs;pects <lb/>will not prove very beneficial: whereupon this difficulty hath bin <lb/>thought on, and remedied, by finding out a way how to unite to­<lb/>gether almo&longs;t infinite Leavers, perpetuating the operation without <lb/>any interruption; and this hath been done by framing a Wheel <lb/>about the Center A, according to the Semidiameter A C, and an <lb/>Axis or Nave, about the &longs;ame Center, of which let the Line A B <lb/>be the Semidiameter; and all this of very tough wood, or of other <lb/>&longs;trong and &longs;olid matter, afterwards &longs;u&longs;taining the whole Machine <lb/>upon a Gudgeon or Pin of Iron planted in the point A, which <lb/>pa&longs;&longs;eth quite thorow, where it is held fa&longs;t by two fixed Fulciments, <lb/>and the Rope D B G, at which the weight G hangeth, being be-laid <lb/>or wound about the Axis or Barrell, and applying another Rope <lb/>about the greater Wheel, at which let the other Grave I be hang­<lb/>ed: It is manife&longs;t, that the length C A having to the other A B <lb/>the &longs;elf-&longs;ame proportion that the Weight G hath to the Weight I, <lb/>it may &longs;u&longs;tain the Grave G, and with any little Moment more &longs;hall <lb/>move it: and becau&longs;e the Axis turning round together with the <lb/>Wheel, the Ropes that &longs;u&longs;tain the Weights are alwaies pendent and <lb/>contingent with the extream Circumferences of that Wheel and <pb xlink:href="070/01/012.jpg" pagenum="282"/>Axis, &longs;o that they &longs;hall con&longs;tantly maintain alike Site and Po&longs;ition <lb/>in re&longs;pect of the Di&longs;tances B A and A C, the Motion &longs;hall be <lb/>perpetuated, the Weight I de&longs;cending, and forcing the other G <lb/>to a&longs;cend. </s> <s>Where we are to ob&longs;erve the nece&longs;&longs;ity of be-laying <lb/>or winding the Rope about the Wheel, that &longs;o the Weight I may <lb/>hang according to the Line that is tangent to the &longs;aid Wheel: for <lb/>if one &longs;hould &longs;u&longs;pend the &longs;aid Weight, &longs;o as that it did hang by the <lb/>point F, cutting the &longs;aid Wheel, as is &longs;een along the Line F N M, <lb/>the Motion would cea&longs;e, the Moment of the Weight M being di­<lb/>mini&longs;hed; which would weigh no more then if it did hang by the <lb/>point N: becau&longs;e the Di&longs;tance of its Su&longs;pen&longs;ion from the Center <lb/>A, cometh to be determined by the Line A N, which falleth per­<lb/>pendicularly upon the Rope F M, and is no longer terminated by <lb/>the Semidiameter of the Wheel A F, which falleth at unequall <lb/>Angles upon the &longs;aid Line F M. </s> <s>A violence therefore being offered <lb/>in the Circumference of the Wheel by a Grave and Exanimate <lb/>Body that hath no other <emph type="italics"/>Impetus<emph.end type="italics"/> then that of De&longs;cending, it is <lb/>nece&longs;&longs;ary that it be &longs;u&longs;tained by a Line that is contingent with <lb/>the Wheel, and not by one that cutteth it. </s> <s>But if in the &longs;ame <lb/>Circumference an Animate Force were employed, that had a Mo­<lb/>ment or Faculty of making an <emph type="italics"/>Impul&longs;e<emph.end type="italics"/> on all &longs;ides, the work might <lb/>be effected in any whatever place of the &longs;aid Circumference. </s> <s>And <lb/>thus being placed in F, it would draw up the Weight by turning <lb/>the Wheel about, pulling not according to the Line F M down­<lb/>wards, but &longs;ide-waies according to the Contingent Line F L, which <lb/>maketh a Right Angle, with that which is drawn from the Center <lb/>A unto the point of Contact F: &longs;o, that if in this manner one do <lb/>mea&longs;ure the Di&longs;tance from the Center A to the Force placed in <lb/>F, according to the Line A F perpendicular to F L, along which <lb/>the <emph type="italics"/>Impetus<emph.end type="italics"/> is made, a man &longs;hall not in any part have altered the <lb/>u&longs;e of the ordinary Leaver. </s> <s>And we mu&longs;t note, that the &longs;ame <lb/>would be po&longs;&longs;ible to be done likewi&longs;e with an Exanimate Force, <lb/>in ca&longs;e that a way were found out to cau&longs;e that its Moment might <lb/>make Impul&longs;e in the point F, drawing according to the Contingent <lb/>Line F L: which would be done by adjoyning beneath the Line F L <lb/>a turning Pulley, making the Rope wound about the Wheel to <lb/>pa&longs;&longs;e along upon it, as it is &longs;een to do by the Line F L X, &longs;u&longs;pending <lb/>at the end thereof the Weight X equall to the other I, which ex­<lb/>erci&longs;ing its Force according to the Line F L, &longs;hall alwaies keep a <lb/>Di&longs;tance from the Center A equall unto the Semidiameter of the <lb/>Wheel. </s> <s>And from what hath been declared we will gather for a <lb/>Conclu&longs;ion, That in this In&longs;trument the Force hath alwaies the <lb/>&longs;ame proportion to the Weight, as the Semidiameter of the Axis <lb/>or Barrell hath to the Semidiameter of the Wheel.</s></p><pb xlink:href="070/01/013.jpg" pagenum="283"/><p type="main"> <s>From the In&longs;trument la&longs;t de&longs;cribed, the other In&longs;trument which <lb/>we call the Crane is not much different, as to form, nay, differeth <lb/>nothing, &longs;ave in the way of applying or employing it: For that the <lb/>Cap&longs;ten moveth and is con&longs;tituted perpendicular to the Horizon, <lb/>and the Crane worketh with its Moment parallel to the &longs;ame Ho­<lb/><figure id="id.070.01.013.1.jpg" xlink:href="070/01/013/1.jpg"/><lb/>rizon. </s> <s>For if upon the Circle D A E we &longs;uppo&longs;e an Axis to be <lb/>placed Column-wi&longs;e, turning about the Center B, and about which <lb/>the Rope D H, fa&longs;tened to the Weight that is to be drawn, is be­<lb/>laid, and if the Bar F E B D be let into the &longs;aid Axis [<emph type="italics"/>by the Mor­<lb/>tace B<emph.end type="italics"/>] and the Force of a Man, of an Hor&longs;e, or of &longs;ome other <lb/>Animal apt to draw, be applyed at its end F, which moving round, <lb/>pa&longs;&longs;eth along the Circumference F G C, the Crane &longs;hall be framed <lb/>and fini&longs;hed, &longs;o that by carrying round the Bar F B D, the Barrell <lb/>or Axis E A D &longs;hall turn about, and the Rope which is twined a­<lb/>bout it, &longs;hall con&longs;train the Weight H to go forward: And becau&longs;e <lb/>the point of the Fulciment about which the Motion is made, is the <lb/>point B, and the Moment keeps at a Di&longs;tance from it according to <lb/>the Line B F, and the Re&longs;i&longs;tor at the Di&longs;tance B D, the Leaver <lb/>F B D is formed, by vertue of which the Force acquireth Moment <lb/>equall to the Re&longs;i&longs;tance, if &longs;o be, that it be in proportion to it, as <lb/>the Line B D is to B F, that is, as the Semidiameter of the Axis to <lb/>the Semidiameter of the Circle, along who&longs;e Circumference the <lb/>Force moveth. </s> <s>And both in this, and in the other In&longs;trument we <lb/>are to ob&longs;erve that which hath been frequently mentioned, that is, <lb/>That the benefit which is derived from the&longs;e Machines, is not that <lb/>which the generality of the Vulgar promi&longs;e them&longs;elves from the <lb/>Mechanicks; namely, that being too hard for Nature, its po&longs;&longs;ible <pb xlink:href="070/01/014.jpg" pagenum="284"/>with a Machine to overcome a Re&longs;i&longs;tance, though great, with a <lb/>&longs;mall Force, in regard, that we &longs;hall manife&longs;tly prove that the &longs;ame <lb/>Force placed in F, might in the &longs;ame Time conveigh the &longs;ame <lb/>Weight, with the &longs;ame Motion, unto the &longs;ame Di&longs;tance, without <lb/>any Machine at all: For &longs;uppo&longs;ing, for example, that the Re&longs;i&longs;tance <lb/>of the Grave H be ten times greater than the Force placed in F, it <lb/><figure id="id.070.01.014.1.jpg" xlink:href="070/01/014/1.jpg"/><lb/>will be requi&longs;ite for the mo­<lb/>ving of the &longs;aid Re&longs;i&longs;tance, <lb/>that the Line F B be decuple <lb/>to B D; and con&longs;equently, <lb/>that the Circumference of the <lb/>Circle F G C be al&longs;o decuple <lb/>to the Circumference E A D: <lb/>and becau&longs;e when the Force <lb/>&longs;hall be moved once along the <lb/>whole Circumference of the <lb/>Circle F G C, the Barrel EAD, <lb/>about which the Rope is be-laid which draweth the Weight, &longs;hall <lb/>likewi&longs;e have given one onely turn; it is manife&longs;t, that the Weight <lb/>H &longs;hall not have been moved more than the tenth part of that way <lb/>which the Mover &longs;hall have gone. </s> <s>If therefore the Force that is to <lb/>move a Re&longs;i&longs;tance that is greater than it &longs;elf, for &longs;uch an a&longs;&longs;igned <lb/>Space by help of this Machine, mu&longs;t of nece&longs;&longs;ity move ten times as <lb/>far, there is no doubt, but that dividing that Weight into ten parts, <lb/>each of them &longs;hall be equall to the Force, and con&longs;equently, might <lb/>have been tran&longs;ported one at a Time, as great a Space as that <lb/>which it &longs;elf did move, &longs;o that making ten journeys, each equal to <lb/>the Circumference E A D, it &longs;hall not have gone any farther than <lb/>if it did move but once alone about the Circumference F G C; <lb/>and &longs;hall have conveighed the &longs;ame Weight H to the &longs;ame Di­<lb/>&longs;tance. </s> <s>The benefit therefore that is to be derived from the&longs;e <lb/>Machines is, that they carry all the Weight together, but not with <lb/>le&longs;&longs;e Labour, or with greater Expedition, or a greater Way than <lb/>the &longs;ame Force might have done conveying it by parcels.</s></p><p type="head"> <s>Of PULLIES.</s></p><p type="main"> <s>The In&longs;truments, who&longs;e Natures are reducible unto the Bal­<lb/>lance, as to their Principle and Foundation, and others little <lb/>differing from them, have been already de&longs;cribed; now for <lb/>the under&longs;tanding of that which we have to &longs;ay touching Pullies, <lb/>it is requi&longs;ite, that we con&longs;ider in the fir&longs;t place another way to u&longs;e <lb/>the Leaver, which will conduce much towards the inve&longs;tigation of <lb/>the Force of Pullies, and towards the under&longs;tanding of other Me­<lb/>chanical Effects. </s> <s>The u&longs;e of the Leaver above declared &longs;uppo&longs;ed <pb xlink:href="070/01/015.jpg" pagenum="285"/>the Weight to be at one extream, and the Force at the other, and <lb/>the Fulciment placed in &longs;ome point between the extreams: but we <lb/>may make u&longs;e of the Leaver another way, yet, placing, as we &longs;ee, <lb/>the Fulciment in the extream A, the Force in the other extream C, <lb/>and &longs;uppo&longs;ing the Weight D to hang by &longs;ome point in the mid&longs;t, <lb/><figure id="id.070.01.015.1.jpg" xlink:href="070/01/015/1.jpg"/><lb/>as here we &longs;ee by the point B, in <lb/>this example it's manife&longs;t, that if <lb/>the Weight did hang at a point <lb/>Equi-di&longs;tant from the two ex­<lb/>treams A and C, as at the point F, <lb/>the labour of &longs;u&longs;taining it would <lb/>be equally divided betwixt the <lb/>two points A and C, &longs;o that half <lb/>the Weight would be felt by the <lb/>Force C, the other half being &longs;u­<lb/>&longs;tained by the Fulciment A: but if the Grave Body &longs;hall be hanged <lb/>at another place, as at B, we &longs;hall &longs;hew that the Force in C is &longs;uffi­<lb/>cient to &longs;u&longs;tain the Weight in B, as it hath the &longs;ame proportion <lb/>to it, that the Di&longs;tance, A B hath to the Di&longs;tance A C. </s> <s>For De­<lb/>mon&longs;tration of which, let us imagine the Line B A to be continued <lb/>right out unto G, and let the Di&longs;tance B A be equall to A G, and <lb/>let the Weight hanging at G, be &longs;uppo&longs;ed equall to D: It is ma­<lb/>nife&longs;t, that by rea&longs;on of the equality of the Weights D and E, and <lb/>of the Di&longs;tances G A and A B, the Moment of the Weight E <lb/>&longs;hall equalize the Moment of the Weight D, and is &longs;ufficient to <lb/>&longs;u&longs;tain it: Therefore whatever Force &longs;hall have Moment equall to <lb/>that of the Weight E, and that &longs;hall be able to &longs;u&longs;tain it, &longs;hall be <lb/>&longs;ufficient likewi&longs;e to &longs;u&longs;tain the Weight D: But for &longs;u&longs;taining the <lb/>Weight E, let there be placed in the point C &longs;uch a Force, who&longs;e <lb/>Moment hath that proportion to the Weight E, that the Di&longs;tance <lb/>G A hath to the Di&longs;tance A C, it &longs;hall be &longs;ufficient to &longs;u&longs;tain it: <lb/>Therefore the &longs;ame Force &longs;hall likewi&longs;e be able to &longs;u&longs;tain the <lb/>Weight D, who&longs;e Moment is equall to the of E: But look what <lb/>Proportion the Line G A hath to the Line A C; and A B al&longs;o hath <lb/>the &longs;ame to the &longs;aid A C, G A having been &longs;uppo&longs;ed equall to A B: <lb/>And becau&longs;e the Weights E and D are equall, each of them &longs;hall <lb/>have the &longs;ame proportion to the Force placed in C: Therefore the <lb/>Force in C is concluded to equall the Moment of the Weight D, <lb/>as often as it hath unto it the &longs;ame proportion that the Di&longs;tance B A <lb/>hath to the Di&longs;tance C A. </s> <s>And by moving the Weight, with the <lb/>Leaver u&longs;ed in this manner, it is gathered in this al&longs;o, as well as in <lb/>the other In&longs;truments, that what is gained in Force is lo&longs;t in Velo­<lb/>city: for the Force C rai&longs;ing the Leaver, and transferring it to A I, <lb/>the Weight is moved the Space B H, which is as much le&longs;&longs;er than <lb/>the Space C I pa&longs;&longs;ed by the Force, as the Di&longs;tance A B is le&longs;&longs;er <pb xlink:href="070/01/016.jpg" pagenum="286"/>than the Di&longs;tance A C; that is, as the Force is le&longs;&longs;e than the <lb/>Weight.</s></p><p type="main"> <s>The&longs;e Principles being declared, we will pa&longs;&longs;e to the Contem­<lb/>plation of Pullies, the compo&longs;ition and &longs;tructure of which, together <lb/>with their u&longs;e, &longs;hall be de&longs;cribed by us. </s> <s>And fir&longs;t let us &longs;uppo&longs;e the <lb/><arrow.to.target n="marg1109"></arrow.to.target><lb/>^{*} Little Pulley A B C, made of Mettall or hard Wood, voluble a­<lb/>bout it's Axis which pa&longs;&longs;eth thorow it's Center D, and about this <lb/><figure id="id.070.01.016.1.jpg" xlink:href="070/01/016/1.jpg"/><lb/>Pulley let the Rope E A B C be put, <lb/>at one end of whichlet the Weight E <lb/>hang, and at the other let us &longs;uppo&longs;e <lb/>the Force F. </s> <s>I &longs;ay, that the Weight <lb/>being &longs;u&longs;tained by a Force equall to <lb/>it &longs;elf in the upper Nut or Pulley <lb/>A B C, bringeth &longs;ome benefit, as the <lb/>moving or &longs;u&longs;taining of the &longs;aid <lb/>Weight with the Force placed in F: <lb/>For if we &longs;hall under&longs;tand, that from <lb/>the Center D, which is the place of the Fulciment, two Lines be <lb/>drawn out as far as the Circumference of the Pulley in the points <lb/>A and C, in which the pendent Cords touch the Circumference, we <lb/>&longs;hall have a Ballance of equal Arms which determine the Di&longs;tance <lb/>of the two Su&longs;pen&longs;ions from the Center and Fulciment D: Where­<lb/>upon it is manife&longs;t, that the Weight hanging at A cannot be &longs;u&longs;tain­<lb/>ed by a le&longs;&longs;er Weight hanging at G, but by one equal to it; &longs;uch <lb/>is the nature of equal Weights hanging at equal Di&longs;tances. </s> <s>And <lb/>although in moving downwards, the Force F cometh to turn about <lb/>the Pulley A B C, yet there followeth no alteration of the Alti­<lb/>tude or Re&longs;pect, that the Weight and Force have unto the two <lb/>Di&longs;tances A D and D C, nay, the Pulley encompa&longs;&longs;ed becometh a <lb/>Ballance equal to A C, but perpetuall. </s> <s>Whence we may learn, <lb/>how childi&longs;hly <emph type="italics"/>Ari&longs;totle<emph.end type="italics"/> deceiveth him&longs;elf, who holds, that by making <lb/>the &longs;mall Pulley A B C bigger, one might draw up the Weight with <lb/>a le&longs;&longs;er Force; he con&longs;idering that upon the enlargement of the <lb/>&longs;aid Pulley, the Di&longs;tance D C encrea&longs;ed, but not con&longs;idering that <lb/>there was as great an encrea&longs;e of the other Di&longs;tance of the Weight, <lb/>that is, the other Semidiameter D A. </s> <s>The benefit therefore that may <lb/>be drawn from the In&longs;trument above &longs;aid, is nothing at all as to the <lb/>diminution of the labour: and if any one &longs;hould ask how it hap­<lb/>pens, that on many occa&longs;ions of rai&longs;ing Weights, this means is made <lb/>u&longs;e of to help the Axis, as we &longs;ee, for example, in drawing up the <lb/>Water of Wells; it is an&longs;wered, that that is done, becau&longs;e that <lb/>by this means the manner of employing the Force is found more <lb/>commodious: for being to pull downwards, the proper Gravity of <lb/>our Arms and other parts help us, whereas if we were to draw <lb/>the fame Weight upwards with a meer Rope, by the &longs;ole &longs;trength <pb xlink:href="070/01/017.jpg" pagenum="287"/>of the Members and Mu&longs;cles, and as we u&longs;e to &longs;ay, by Force of <lb/>Armes, be&longs;ides the extern Weight, we are to lift up the Weight of <lb/>our own Armes, in which greater pains is required. </s> <s>Conclude we, <lb/>therefore, that this upper Pulley doth not bring any Facility to the <lb/>Force &longs;imply con&longs;idered, but onely to the manner of applying it: <lb/>but if we &longs;hall make u&longs;e of the like Machine <lb/><figure id="id.070.01.017.1.jpg" xlink:href="070/01/017/1.jpg"/><lb/>in another manner, as we are now about to <lb/>declare; we may rai&longs;e the Weight with di­<lb/>minution of Forces: For let the Pulley <lb/>B D C be voluble about the Center E placed <lb/>in it's Frame B L C, at which hang the <lb/>Grave G; and let the Rope A B D C F <lb/>pa&longs;&longs;e about the Pulley; of which let the end <lb/>A be fa&longs;tned to &longs;ome fixed &longs;tay, and in the <lb/>other F let the Force be placed; which <lb/>moving to wards H &longs;hall rai&longs;e the Machine <lb/>B L C, and con&longs;equently the Weight G: <lb/>and in this operation I &longs;ay, that the Force in <lb/>F is the half of the Weight &longs;u&longs;tained by it. <lb/></s> <s>For the &longs;aid Weight being kept to Rights by the two ^{*} Ropes A B <lb/><arrow.to.target n="marg1110"></arrow.to.target><lb/>and F C, it is manife&longs;t, that the Labour is equally &longs;hared betwixt <lb/>the Force F and the Fulciment A: and more &longs;ubtilly examining the <lb/>nature of this In&longs;trument, if we but continue forth the Diameter <lb/>B E C, we &longs;hall &longs;ee a Leaver to be made, at the mid&longs;t of which, that <lb/>is at the point E, the Grave doth hang, and the Fulciment cometh <lb/>to be at the end B, and the Force in the Term C: whereupon, by <lb/>what hath been above demon&longs;trated, the Force &longs;hall have the &longs;ame <lb/>proportion to the Weight, that the Di&longs;tance E B hath to the Di­<lb/>&longs;tance; Therefore it &longs;hall be the half of the &longs;aid Weight: And <lb/>becau&longs;e the Force ri&longs;ing towards A, the Pulley turneth round, <lb/>therefore that Re&longs;pect or Con&longs;titution which the Fulciment B and <lb/>Center E, on which the Weight and Term C, in which the Force <lb/>is employed do depend, &longs;hall not change all the while; but yet in <lb/>the Circuinduction the Terms B and C happen to vary in number, <lb/>but not in vertue, others and others continually &longs;ucceeding in their <lb/>place, whereby the Leaver B C cometh to be perpetuated. </s> <s>And <lb/>here (as hath been done in the other In&longs;truments, and &longs;hall be in <lb/>tho&longs;e that follow) we will not pa&longs;&longs;e without con&longs;idering how that <lb/>the journey that the Force maketh, is double to the Moment of the <lb/>Weight. </s> <s>For in ca&longs;e the Weight &longs;hall be moved &longs;o far, till that <lb/>the Line B C come to arrive with it's points B and C, at the points <lb/>A and F, it is nece&longs;&longs;ary that the two equal Ropes be di&longs;tended in <lb/>one &longs;ole Line F H, and con&longs;equently, when the Weight &longs;hall have <lb/>a&longs;cended along the Intervall B A, the Force &longs;hall have been moved <lb/>twice as far, that is, from <emph type="italics"/>F<emph.end type="italics"/> unto H. </s> <s>Then con&longs;idering that the <pb xlink:href="070/01/018.jpg" pagenum="288"/>Force in <emph type="italics"/>F,<emph.end type="italics"/> that it may rai&longs;e the Weight, mu&longs;t move upwards, which <lb/>to exanimate Movers, as being for the mo&longs;t part Grave Bodies, is al­<lb/><figure id="id.070.01.018.1.jpg" xlink:href="070/01/018/1.jpg"/><lb/>together impo&longs;&longs;ible, or at lea&longs;t more laborious, <lb/>than the making of the &longs;ame <emph type="italics"/>F<emph.end type="italics"/>orce down­<lb/>wards: Therefore to help this inconvenience, <lb/>a Remedy hath been found by adjoyning an­<lb/>other Nut or Pulley above, as in the adjacent <lb/><emph type="italics"/>F<emph.end type="italics"/>igure is &longs;een, where the Rope C E <emph type="italics"/>F<emph.end type="italics"/> hath <lb/>been made to pa&longs;s about the upper Pulley <emph type="italics"/>F<emph.end type="italics"/> G <lb/>upheld by the Hook L, &longs;o that the Rope pa&longs;&longs;ing <lb/>to H, and thither transferring the <emph type="italics"/>F<emph.end type="italics"/>orce E, it <lb/>&longs;hall be able to move the Weight X by pulling <lb/>downwards, but not that it may be le&longs;&longs;er than <lb/>it was in E: <emph type="italics"/>F<emph.end type="italics"/>or the Motions of the <emph type="italics"/>F<emph.end type="italics"/>orce <lb/><emph type="italics"/>F<emph.end type="italics"/> H, hanging at the equal Di&longs;tances <emph type="italics"/>F<emph.end type="italics"/> D and <lb/>D G of the upper Pulley, do alwaies continue <lb/>equal; nor doth that upper Pulley (as hath <lb/>been &longs;hewn above) come to produce any di­<lb/>minution in the Labour. </s> <s>Moreover it having been nece&longs;&longs;ary by <lb/>the addition of the upper Pulley to introduce the Appendix B, by <lb/>which it is &longs;u&longs;tained, it will prove of &longs;ome benefit to us to rai&longs;e <lb/>the other A, to which one end of the Rope was fa&longs;tned, transferring <lb/>it to a Ring annexed to the lower part of the <emph type="italics"/>F<emph.end type="italics"/>rame of the upper <lb/>Pulley, as we &longs;ee it done in M. </s> <s>Now finally, this Machine com­<lb/>pounded of upper and lower Pullies, is that which the Greeks call <lb/><arrow.to.target n="marg1111"></arrow.to.target><lb/><foreign lang="greek">*tpoxi/lion.</foreign></s></p><p type="margin"> <s><margin.target id="marg1109"></margin.target>*Called by &longs;ome <lb/>a Nut.</s></p><p type="margin"> <s><margin.target id="marg1110"></margin.target>* Or two ends of <lb/>the &longs;ame Rope.</s></p><p type="margin"> <s><margin.target id="marg1111"></margin.target>In Latine <emph type="italics"/>Tro­<lb/>chlea.<emph.end type="italics"/></s></p><p type="main"> <s>We have hitherto explained, how by help of Pullies one may <lb/>double the <emph type="italics"/>F<emph.end type="italics"/>orce, it remaineth that with the greate&longs;t brevity po&longs;­<lb/>&longs;ible, we &longs;hew the way how to encrea&longs;e it according to any Multi­<lb/>plicity. </s> <s>And fir&longs;t we will &longs;peak of the Multiplicity according to <lb/>the even numbers, and then the odde: To &longs;hew how we may mul­<lb/>tiply the <emph type="italics"/>F<emph.end type="italics"/>orce in a quadruple Proportion, we will propound the <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"/>F<emph.end type="italics"/>ulciments in the ex­<lb/><figure id="id.070.01.018.2.jpg" xlink:href="070/01/018/2.jpg"/><lb/>treams A and C; and at the middles <lb/>of each of them let the Grave G hang, <lb/>&longs;u&longs;tained by two <emph type="italics"/>F<emph.end type="italics"/>orces of equal Mo­<lb/>ment placed in B and D. </s> <s>I &longs;ay, that <lb/>the Moment of each of them will <lb/>equal the Moment of the fourth part <lb/>of the Weight G. <emph type="italics"/>F<emph.end type="italics"/>or the two <emph type="italics"/>F<emph.end type="italics"/>or­<lb/>ces B and D bearing equally, it is <lb/>manife&longs;t, that the <emph type="italics"/>F<emph.end type="italics"/>orce D hath not <lb/>contra&longs;ted with more then one half of the Weight G: But if the <lb/><emph type="italics"/>F<emph.end type="italics"/>orce D do by benefit of the Leaver D C &longs;u&longs;tain the half of the <pb xlink:href="070/01/019.jpg" pagenum="289"/>Weight G hanging at <emph type="italics"/>F,<emph.end type="italics"/> it hath been already demon&longs;trated, that <lb/>the &longs;aid <emph type="italics"/>F<emph.end type="italics"/>orce D hath to the Weight &longs;o by it &longs;u&longs;tained, that &longs;ame <lb/>proportion which the Di&longs;tance <emph type="italics"/>F<emph.end type="italics"/> C hath to the Di&longs;tance C D: <lb/>Which is &longs;ubduple proportion: Therefore the Moment D is &longs;ub­<lb/>duple to the Moment of half of the Weight G &longs;u&longs;tained by it: <lb/>Wherefore it followeth, that it is the fourth part of the Moment <lb/>of the whole Weight. </s> <s>And in the &longs;ame manner the &longs;ame thing is <lb/>demon&longs;trated, of the Moment <emph type="italics"/>B<emph.end type="italics"/>; and it is but rea&longs;onable, that the <lb/>Weight G being &longs;u&longs;tained by the four points, A, <emph type="italics"/>B,<emph.end type="italics"/> C, D, each of <lb/>them &longs;hould feel an equall part of the Labour.</s></p><p type="main"> <s>Let us come now to apply this Con&longs;ideration to Pullies, and let <lb/>the Weight X be &longs;uppo&longs;ed to hang at the two Pullies A B and D E <lb/>entwining about them, and about the uppermo&longs;t Pulley G H, the <lb/>Rope, as we &longs;ee, I D E H G A B, &longs;u&longs;taining the whole Machine in <lb/>the point K. </s> <s>Now I &longs;ay, that placing the Force in L, it &longs;hall be able <lb/>to &longs;u&longs;tain the Weight X, if &longs;o be, it be equal to the fourth part of <lb/>it. </s> <s>For if we do imagine the two Diameters D E and A B, and the <lb/>Weights hanging at the middle points F and C, we &longs;hall have two <lb/>Leavers like to tho&longs;e before de&longs;cribed, the Fulciments of which an­<lb/>&longs;wer to the points D and A. </s> <s>Whereupon the Force placed in B, <lb/><figure id="id.070.01.019.1.jpg" xlink:href="070/01/019/1.jpg"/><lb/>or if you will, in L, &longs;hall be able to &longs;u­<lb/>&longs;tain the Weight X, being the fourth <lb/>part of it: And if we adde another Pul­<lb/>ley above the other two, making the <lb/>Rope or Cord to pa&longs;s along L M N, trans­<lb/>ferring the Force L into N, it &longs;hall be <lb/>able to bear the &longs;ame Weight gravitating <lb/>downwards, the upper Pulley neither aug­<lb/>menting or dimini&longs;hing the Force, as hath <lb/>been declared. </s> <s>And we will likewi&longs;e <lb/>note, that to make the: Weight a&longs;cend the <lb/><arrow.to.target n="marg1112"></arrow.to.target><lb/>four Ropes B L, E H, D I, and A G <lb/>ought to pa&longs;s, whereupon the Mover will <lb/>be to begin, as much as tho&longs;e Ropes are <lb/>long; and yet neverthele&longs;s the Weight <lb/>&longs;hall move but only as much as the length <lb/>of one of them: So that we may &longs;ay by <lb/>way of adverti&longs;ement, and for confirma­<lb/>tion of what hatn been many times &longs;po­<lb/>ken, namely, that look with what proportion the Labour of the <lb/><arrow.to.target n="marg1113"></arrow.to.target><lb/>Mover is dimini&longs;hed, the length of the Way, on the contrary, is <lb/>encrea&longs;ed with the &longs;ame proportion</s></p><p type="margin"> <s><margin.target id="marg1112"></margin.target>* Or four parts <lb/>of the &longs;ame Rope</s></p><p type="margin"> <s><margin.target id="marg1113"></margin.target>* The word <emph type="italics"/>Gy­<lb/>rilla<emph.end type="italics"/> &longs;ignifieth a <lb/>Shiver, Rundle, <lb/>or &longs;mall Wheel <lb/>of a Pulley, tran­<lb/>&longs;lated by we <lb/>&longs;ometimes Pul­<lb/>ley, &longs;ometimes <lb/>Nut or Girill.</s></p><p type="main"> <s>But if we would encrea&longs;e the Force in &longs;excuple proportion, it <lb/>will be requi&longs;ite that we adjoyn another ^{*} &longs;mall Pulley or Gyrill <lb/>to the inferiour Pulley which that you may the better under&longs;tand <pb xlink:href="070/01/020.jpg" pagenum="290"/>we will &longs;et before you the pre&longs;ent Contemplation. </s> <s>Suppo&longs;e, there­<lb/>fore, that A B, C D, and E F are three Leavers; and that on the <lb/>middle points of them G, H, and I the Weight K doth hang in <lb/>common, &longs;o that every one of them &longs;hall &longs;u&longs;tain the third part of <lb/><figure id="id.070.01.020.1.jpg" xlink:href="070/01/020/1.jpg"/><lb/>it: And becau&longs;e the Power in <lb/>B, &longs;u&longs;taining with the Leaver <lb/>B A thependent Weight in G, <lb/>hapneth to be the half of the <lb/>&longs;aid Weight, and it hath been <lb/>already &longs;aid, that it &longs;u&longs;taineth <lb/>the third part of the Weight <lb/>K: Therefore the Moment of <lb/>the Force B is equal to half of <lb/>the third part of the Weight K; that is, to the &longs;ixth part of it: <lb/>And the &longs;ame &longs;hall be demon&longs;trated of the other Forces D and F: <lb/>From whence we may ea&longs;ily gather, that putting three Gyrils or <lb/>Rundles into the inferiour Pulley, and two or three into the upper­<lb/><figure id="id.070.01.020.2.jpg" xlink:href="070/01/020/2.jpg"/><lb/>mo&longs;t, we may multiply the Force accor­<lb/><arrow.to.target n="marg1114"></arrow.to.target><lb/>ding to our ^{*} <emph type="italics"/>Senarius.<emph.end type="italics"/> And if we would <lb/>encrea&longs;e it according to any other even <lb/>Number, the Gyrils of the Pulley below <lb/>mu&longs;t be multiplyed according to the half <lb/>of that Number, according to which the <lb/>Force is to be multiplyed, circumpo&longs;ing <lb/>the Rope about the Pulleys, &longs;o as that one <lb/>of the ends be fa&longs;tned to the upper Pul­<lb/>ley, and let the Force be in the other; as <lb/>in this Figure adjoyning may manife&longs;tly <lb/>be gathered.</s></p><p type="margin"> <s><margin.target id="marg1114"></margin.target>* Or in Sexcuple <lb/>proportion.</s></p><p type="main"> <s>Now pa&longs;&longs;ing to the Declaration of the <lb/>manner how to multiply the Force ac­<lb/>cording to the odd Numbers, and begin­<lb/><figure id="id.070.01.020.3.jpg" xlink:href="070/01/020/3.jpg"/><lb/>ning at the triple proportion: fir&longs;t, let us <lb/>propo&longs;e the pre&longs;ent Contemplation, as <lb/>that, on the under&longs;tanding of which the <lb/>knowledge of all the Work in hand <lb/>doth depend. </s> <s>Let therefore the Leaver <lb/>be A B, its Fulciment A, and from the <lb/>middle of it, that is, at the point C let <lb/>the Grave D be hanged; and let it be &longs;u­<lb/>&longs;tained by two equal Forces; and let one of them be applied to the <lb/>point C, and the other to the term B. </s> <s>I &longs;ay, that each of tho&longs;e Powers <lb/>have Moment equal to the third part of the Weight D. </s> <s>For the <lb/>Force in C &longs;u&longs;taineth a Weight equal to it &longs;elf, being placed in the <lb/>&longs;ame Line in which the Weight D doth hang & Gravitate: But the <pb xlink:href="070/01/021.jpg" pagenum="291"/>Force in B &longs;u&longs;taineth a part of the Weight D double to it &longs;elf, its <lb/>Di&longs;tance from the Fulciment A, that is, the Line B A being dou­<lb/>ble to the Di&longs;tance A C at which the Grave hangeth: But becau&longs;e <lb/>the two Forces in B and C are &longs;uppo&longs;ed to be equal to each other: <lb/>Therefore the part of the Weight D, which is &longs;u&longs;tained by the <lb/>Force in B, is double to the part &longs;u&longs;tained by the Force in C. </s> <s>If <lb/>therefore, of the Grave D two parts be made, the one double to <lb/>the remainder, the greater is &longs;u&longs;tained by the Force in B, and the <lb/>le&longs;&longs;er by the Force in C: But this le&longs;&longs;er is the third part of the <lb/>Weight D: Therefore the Moment of the Force in C is equal to <lb/>the Moment of the third part of the Weight D; to which, of <lb/>con&longs;equence, the Force B &longs;hall be equal, we having &longs;uppo&longs;ed it <lb/>equal to the other Force C: Wherefore our intention is manifell, <lb/>which we were to demon&longs;trate, how that each of the two Powers <lb/>C and B is equal to the third part of the Weight D. </s> <s>Which be­<lb/>ing demon&longs;trated, we will pa&longs;s forwards to the Pulleys, and will <lb/>de&longs;cribe the inferiour Gyrils of A C B, voluble about the Center <lb/>G, and the Weight H hanging thereat, we will draw the other up­<lb/>per one E F, winding about them both the Rope D F E A C B I, <lb/>of which let the end D be fa&longs;tned to the inferiour Pulley, and to <lb/><figure id="id.070.01.021.1.jpg" xlink:href="070/01/021/1.jpg"/><lb/>the other I let the Force be applyed: <lb/>Which, I &longs;ay, &longs;u&longs;taining or moving the <lb/>Weight H, &longs;hall feele no more than the <lb/>third part of the Gravity of the &longs;ame. </s> <s>For <lb/>con&longs;idering the contrivance of this Ma­<lb/>chine, we &longs;hall find that the Diameter A B <lb/>&longs;upplieth the place of a Leaver, in who&longs;e <lb/>term B the Force I is applied, and in the <lb/>other A the <emph type="italics"/>F<emph.end type="italics"/>uiciment is placed, at the mid­<lb/>dle G the Grave H is hanged, and another <lb/><emph type="italics"/>F<emph.end type="italics"/>orce D applied at the &longs;ame place: &longs;o that <lb/><arrow.to.target n="marg1115"></arrow.to.target><lb/>the Weight is fa&longs;tned to the ^{*} three Ropes <lb/>I B, <emph type="italics"/>F<emph.end type="italics"/> D, and E A, which with equal Labour <lb/>&longs;u&longs;tain the Weight. </s> <s>Now, by what hath <lb/>already been contemplated, the two <emph type="italics"/>F<emph.end type="italics"/>orces <lb/>D and B being applied, one, to the mid&longs;t of the Leaver A B, and <lb/>the other to the extream term B, it is manife&longs;t, that each of them <lb/>holdeth no more but the third part of the Weight H: Therefore <lb/>the Power I, having a Moment equal to the third part of the <lb/>Weight H, &longs;hall be able to &longs;u&longs;tain and move it: but yet the Way <lb/>of the <emph type="italics"/>F<emph.end type="italics"/>orce in I &longs;hall be triple to the Way that the Weight &longs;hall <lb/>pa&longs;s; the &longs;aid Force being to di&longs;tend it &longs;elf according to the <lb/>Length of the three Ropes I B, <emph type="italics"/>F<emph.end type="italics"/> D, and E A, of which one alone <lb/>mea&longs;ureth the Way of the Weight H.</s></p><pb xlink:href="070/01/022.jpg" pagenum="292"/><p type="margin"> <s><margin.target id="marg1115"></margin.target>* Or three parts <lb/>of one Rope.</s></p><p type="head"> <s><emph type="italics"/>Of the<emph.end type="italics"/> SCREW.</s></p><p type="main"> <s>Among&longs;t the re&longs;t of Mechanick In&longs;truments for &longs;undry u&longs;es <lb/>found out by the Wit of Man, the Screw doth, in my opi­<lb/>nion, both for Invention and for Utility, hold the fir&longs;t <lb/>place, as that which is appo&longs;itely accommodated, and &longs;o contrived <lb/>not only to move, but al&longs;o to &longs;tay and pre&longs;s with very great Force, <lb/>that taking up but little room, it worketh tho&longs;e effects which other <lb/>In&longs;truments cannot, unle&longs;s they were reduced to a great Machine. <lb/></s> <s>The Screw therefore being of mo&longs;t ingenious and commodious <lb/>contrivance, we ought de&longs;ervedly to be at &longs;ome pains in explaining, <lb/>with all the plainne&longs;s that is po&longs;&longs;ible, the Original and Nature of <lb/>it. </s> <s>The which that we may do, we will begin at a Speculation, <lb/>which, though at fir&longs;t blu&longs;h it may appear &longs;omewhat remote from <lb/>the con&longs;ideration of this In&longs;trument, yet is the <emph type="italics"/>Ba&longs;is<emph.end type="italics"/> and Founda­<lb/>tion thereof.</s></p><p type="main"> <s>No doubt, but that Natures operation in the Motions of Grave <lb/>Bodies is &longs;uch, that any whatever Body that hath a Gravity in it <lb/>hath a propen&longs;ion of moving, being at liberty, towards the Cen­<lb/><arrow.to.target n="marg1116"></arrow.to.target><lb/>ter, and that not only ^{*} by the Right Line perpendicularly, but al­<lb/>&longs;o (when it cannot do otherwi&longs;e) by any other Line, which ha­<lb/>ving &longs;ome inclination towards the Center goeth more and more <lb/>aba&longs;ing. </s> <s>And thus we &longs;ee the Water not only to fall downwards <lb/>along the Perpendicular from &longs;ome eminent place, but al&longs;o to run <lb/>about the Surface of the Earth along Lines though very little en­<lb/>clined; as we &longs;ee in the Cour&longs;e of Rivers, the Waters of which, if &longs;o <lb/>be that the Bed have any the lea&longs;t declivity, go freely declining <lb/>downwards. </s> <s>Which very effect, like as it is di&longs;cerned in all Fluid <lb/>Bodies, would appear al&longs;o in hard Bodies, if &longs;o be, that their Fi­<lb/>gure and other Accidental and Extern Impediments did not hinder <lb/>it. </s> <s>So that we, having a Superficies very well &longs;moothed and poli­<lb/>&longs;hed, as for in&longs;tance, that of a Looking-gla&longs;s, and a Ball exactly <lb/>rotund and &longs;leek, either of Marble, or of Gla&longs;s, or of any other <lb/>Matter apt to be poli&longs;hed, this being placed upon that Superficies <lb/>&longs;hall trundle along, in ca&longs;e that this have any, though very &longs;mall, <lb/>inclination; and &longs;hall lie &longs;till only upon that Superficies which is <lb/>exactly levelled and parallel to the Plane of the Horizon: as is <lb/>that, for example, of a Lake or &longs;tanding Water being frozen, up­<lb/>on which the &longs;aid Spherical Body would &longs;tand &longs;till, but in a con­<lb/>dition of being moved by every &longs;mall Force. </s> <s>For we having &longs;up­<lb/>po&longs;ed that if that Plane did incline but an hairs breadth only, the <lb/>&longs;aid Ball would move along it &longs;pontaneou&longs;ly towards the part de­<lb/>clining, and on the oppo&longs;ite would have a Re&longs;i&longs;tance, nay, would <lb/>not be able without &longs;ome Violence to move towards the part <pb xlink:href="070/01/023.jpg" pagenum="293"/>ri&longs;ing or a&longs;cending: it of nece&longs;&longs;ity remaineth manife&longs;t, that in the <lb/>Superficies which is exactly equilibrated, the &longs;aid Ball remaineth in­<lb/>different and dubious between Motion and Re&longs;t, &longs;o that every &longs;mall <lb/>Force is &longs;ufficient to move it, as on the contrary, every &longs;mall Re&longs;i­<lb/>&longs;tance, and no greater than that of the meer Air that environs it, is <lb/>able to hold it &longs;till.</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 Conclu&longs;ion for indubitable, That <lb/>Crave Bodies, all Extern and Adventitious Impediments being re­<lb/>moved, may be moved along the Plane of the Horizon by any ne­<lb/>ver &longs;o &longs;mall Force: but when the &longs;ame Grave is to be thrown along <lb/>an A&longs;cending Plane, then, it beginning to &longs;trive again&longs;t that a&longs;cent, <lb/>having an inclination to the contrary Motion, there &longs;hall be requi­<lb/>red greater Violence, and &longs;till greater the more Elevation that &longs;ame <lb/>Plane &longs;hall have. </s> <s>As for example, the Moveable G, being po&longs;ited <lb/>upon the Line A B parallel to the Horizon, it &longs;hall, as hath been <lb/>&longs;aid, be indifferent on it either to Motion or Re&longs;t, &longs;o that it may <lb/>be moved by a very &longs;mall Force: But if we &longs;hall have the Planes <lb/>Elevated, they &longs;hall not be driven along without Violence; which <lb/><figure id="id.070.01.023.1.jpg" xlink:href="070/01/023/1.jpg"/><lb/>Violence will be required to be <lb/>greater to move it along the Line <lb/>A D, than along A C; and &longs;till <lb/>greater along A E than along A D: <lb/>The which hapneth, becau&longs;e it hath <lb/>greater <emph type="italics"/>Impetus<emph.end type="italics"/> of going down­<lb/>wards along A E than along A D, <lb/>and along A D than along A C. </s> <s>So <lb/>that we may likewi&longs;e conclude <lb/>Grave Bodies to have greater Re&longs;i&longs;tance upon Planes differently <lb/>Elevared, to their being moved along the &longs;ame, according as one <lb/>&longs;hall be more or le&longs;s elevated than the other; and, in fine, that the <lb/>greate&longs;t Re&longs;i&longs;tance of the &longs;ame Grave to its being rai&longs;ed is in the <lb/>Perpendicular A F. </s> <s>But it will be nece&longs;&longs;ary to declare exactly what <lb/>proportion the Force mu&longs;t have to the Weight, that it may be able <lb/>to carry it along &longs;everal elevated Planes, before we proceed any <lb/>farther, to the end that we may perfectly under&longs;tand all that which <lb/>remains to be &longs;poken.</s></p><p type="main"> <s>Letting, therefore, Perpendiculars fall from the points C, D, <lb/>and E unto the Horizontal Line A B, which let be C H, D I, and <lb/>E K: it &longs;hall be demon&longs;trated that the &longs;ame Weight &longs;hall be mo­<lb/>ved along the Plane A C with le&longs;&longs;er Force than along the Perpendi­<lb/>cular A F, (where it is rai&longs;ed by a Force equal to it &longs;elf) accor­<lb/>ding to the proportion by which the Perpendicular C H is le&longs;s than <lb/>A C: and that along the Plane A D, the Force hath the &longs;ame pro­<lb/>portion to the Weight, that the Perpendicular I D hath to D A: <lb/>and, la&longs;tly, that in the Plane A E the <emph type="italics"/>F<emph.end type="italics"/>orce to the Weight ob&longs;er­<lb/>veth the proportion of E K and E A.</s></p><pb xlink:href="070/01/024.jpg" pagenum="294"/><p type="main"> <s>The pre&longs;ent Speculation hath been attempted by <emph type="italics"/>Pappus Alex­<lb/>andrinus<emph.end type="italics"/> in <emph type="italics"/>Lib.<emph.end type="italics"/> 8. <emph type="italics"/>de Collection. </s> <s>Mathemat.<emph.end type="italics"/> but, if I be in the <lb/>right, he hath not hit the mark, and was over&longs;een in the A&longs;&longs;umpti­<lb/>on that he maketh, where he &longs;uppo&longs;eth that the Weight ought to <lb/>be moved along the Horizontal Line by a <emph type="italics"/>F<emph.end type="italics"/>orce given; which is <lb/>fal&longs;e: there needing no &longs;en&longs;ible <emph type="italics"/>F<emph.end type="italics"/>orce (removing the Accidental <lb/>Impediments, which in the Theory are not regarded) to move the <lb/>given Weight along the Horizon, &longs;o that he goeth about in vain <lb/>afterwards to &longs;eek with what <emph type="italics"/>F<emph.end type="italics"/>orce it is to be moved along the <lb/>elevated Plane. </s> <s>It will be therefore better, the <emph type="italics"/>F<emph.end type="italics"/>orce that moveth <lb/>the Weight upwards perpendicularly, (which equalizeth the Gra­<lb/>vity of that Weight which is to be moved) being given, to <lb/>&longs;eek the <emph type="italics"/>F<emph.end type="italics"/>orce that moveth it along the Elevated Plane: Which <lb/>we will endeavour to do in a Method different from that of <lb/><emph type="italics"/>Pappus.<emph.end type="italics"/></s></p><p type="main"> <s>Let us therefore &longs;uppo&longs;e the Circle A I C, and in it the Diame­<lb/>ter A B C, and the Center B, and two Weights of equal Moment <lb/>in the extreams B and C; &longs;o that the Line A C being a Leaver, <lb/>or Ballance moveable about the Center B, the Weight C &longs;hall <lb/>come to be &longs;u&longs;tained by the Weight A. </s> <s>But if we &longs;hall imagine <lb/>the Arm of the Ballance B C to be inclined downwards according <lb/>to the Line B F, but yet in &longs;uch a manner that the two Lines <emph type="italics"/>A B<emph.end type="italics"/><lb/>and <emph type="italics"/>B F<emph.end type="italics"/> do continue &longs;olidly conjoyned in the point <emph type="italics"/>B,<emph.end type="italics"/> in this ca&longs;e <lb/>the Moment of the Weight C &longs;hall not be equal to the Moment <lb/><figure id="id.070.01.024.1.jpg" xlink:href="070/01/024/1.jpg"/><lb/>of the Weight <emph type="italics"/>A,<emph.end type="italics"/> for that the Di­<lb/>&longs;tance of the point <emph type="italics"/>F<emph.end type="italics"/> from the Line <lb/>of Direction, which goeth accord­<lb/>ing to B I, from the <emph type="italics"/>F<emph.end type="italics"/>ulciment B un­<lb/>to the Center of the Earth, is dimi­<lb/>ni&longs;hed: But if from the point <emph type="italics"/>F<emph.end type="italics"/> we <lb/>erect a Perpendicular unto B C, as is <lb/><emph type="italics"/>F<emph.end type="italics"/> K, the Moment of the Weight in <lb/><emph type="italics"/>F<emph.end type="italics"/> &longs;hall be as if it did hang by the <lb/>Line K <emph type="italics"/>F,<emph.end type="italics"/> and look how much the <lb/>Di&longs;tance K B is dimini&longs;hed by the <lb/>Di&longs;tance B <emph type="italics"/>A,<emph.end type="italics"/> &longs;o much is the Moment of the Weight <emph type="italics"/>F<emph.end type="italics"/> dimini&longs;hed <lb/>by the Moment of the <emph type="italics"/>W<emph.end type="italics"/>eight <emph type="italics"/>A. A<emph.end type="italics"/>nd in this fa&longs;hion inclining <lb/>the <emph type="italics"/>W<emph.end type="italics"/>eight more, as for in&longs;tance, according to B L, its Moment &longs;hall <lb/>&longs;till dimini&longs;h and &longs;hall be as if it did hang at the Di&longs;tance <emph type="italics"/>B<emph.end type="italics"/> M, ac­<lb/>cording to the <emph type="italics"/>L<emph.end type="italics"/>ine M <emph type="italics"/>L,<emph.end type="italics"/> in which point <emph type="italics"/>L<emph.end type="italics"/> it &longs;hall be &longs;u&longs;tained by <lb/>a <emph type="italics"/>W<emph.end type="italics"/>eight placed in <emph type="italics"/>A,<emph.end type="italics"/> &longs;o much le&longs;s than it &longs;elf, by how much the <lb/>Di&longs;tance B <emph type="italics"/>A<emph.end type="italics"/> is greater than the Di&longs;tance <emph type="italics"/>B<emph.end type="italics"/> M. </s> <s>See therefore that <lb/>the <emph type="italics"/>W<emph.end type="italics"/>eight placed in the extream of the <emph type="italics"/>L<emph.end type="italics"/>eaver B C, in inclining <lb/>downwards along the Circumference C <emph type="italics"/>F L<emph.end type="italics"/> I, cometh to dimini&longs;h <lb/>its Moment and <emph type="italics"/>Impetus<emph.end type="italics"/> of going downwards from time to time, <pb xlink:href="070/01/025.jpg" pagenum="295"/>more and le&longs;s, as it is more or le&longs;s &longs;u&longs;tained by the Lines B F and <lb/>B L: But the con&longs;idering that this Grave de&longs;cending, and &longs;u&longs;tained <lb/>by the Semidiameters B F and B L is one while le&longs;s, and another <lb/>while more con&longs;trained to pa&longs;s along the Circumference C F L, is <lb/>no other, than if we &longs;hould imagine the &longs;ame Circumference <lb/>C F L I to be a Super&longs;icies &longs;o curved, and put under the &longs;ame <lb/>Moveable: &longs;o that bearing it &longs;elf thereon it were con&longs;trained to <lb/>de&longs;cend along thereby; for if in the one and other manner the <lb/>Moveable de&longs;cribeth the &longs;ame Cour&longs;e or Way, it will nothing im­<lb/>port whether, if &longs;u&longs;pended at the Center B, it is &longs;u&longs;tained by the <lb/>Semidiameter of the Circle, or el&longs;e, whether that Fulciment being <lb/>taken away, it proceed along the Circumference C F L I: So that <lb/>we may confidently affirm, that the Grave de&longs;cending downwards <lb/>from the point C along the Circumference C F L I, its Moment <lb/>of De&longs;cent in the point C is total and entire, becau&longs;e it is not in <lb/>any part &longs;u&longs;tained by the Circumference: And there is not in that <lb/>fir&longs;t point C, any indi&longs;po&longs;ition to Motion different from that, which <lb/>being at liberty, it would make along the Perpendicular and Con­<lb/>tingent Line D C E: But if the Moveable &longs;hall be placed in the <lb/>point F, then its Gravity is in part &longs;u&longs;tained, and its Moment of <lb/>De&longs;cent is dimini&longs;hed by the Circular Path or Way that is placed <lb/>under it, in that proportion wherewith the <emph type="italics"/>L<emph.end type="italics"/>ine <emph type="italics"/>B<emph.end type="italics"/> K is overcome <lb/>by <emph type="italics"/>B<emph.end type="italics"/> C: But if when the Moveable is in F, at the fir&longs;t in&longs;tant of <lb/>&longs;uch its Motion, it be as if it were in the Plane elevated according <lb/>to the Contingent <emph type="italics"/>L<emph.end type="italics"/>ine G F H, for that rea&longs;on the inclination of the <lb/>Circumference in the point F differeth not from the inclination of <lb/>the Contingent <emph type="italics"/>L<emph.end type="italics"/>ine F G any more &longs;ave the in&longs;en&longs;ible Angle of <lb/>the Contact. </s> <s>And in the &longs;ame manner we &longs;hall find the Moment <lb/>of the &longs;aid Moveable to dimini&longs;h in the point <emph type="italics"/>L,<emph.end type="italics"/> as the <emph type="italics"/>L<emph.end type="italics"/>ine BM <lb/>is dimini&longs;hed by B C; &longs;o that in the Plane contingent to the Circle <lb/>in the point <emph type="italics"/>L,<emph.end type="italics"/> as for in&longs;tance, according to the <emph type="italics"/>L<emph.end type="italics"/>ine N <emph type="italics"/>L<emph.end type="italics"/> O, the <lb/>Moment of De&longs;cent dimini&longs;heth in the Moveable with the &longs;ame <lb/>proportion. </s> <s>If therefore ^{*} upon the Plane HG the Moment of the <lb/><arrow.to.target n="marg1117"></arrow.to.target><lb/>Moveable be dimini&longs;hed by the total <emph type="italics"/>Impetus<emph.end type="italics"/> which it hath in its <lb/>Perpendicular D C E, according to the proportion of the <emph type="italics"/>L<emph.end type="italics"/>ine K B <lb/>to the <emph type="italics"/>L<emph.end type="italics"/>ine B C, and B F, being by the Solicitude of the Triangles <lb/>K B F and K F H the &longs;ame proportion betwixt the <emph type="italics"/>L<emph.end type="italics"/>ines K F and <lb/>F H, as betwixt the &longs;aid K B and <emph type="italics"/>B<emph.end type="italics"/> F, we will conclude that the <lb/>proportion of the entire and ab&longs;olute Moment, that the Moveable <lb/>hath in the Perpendicular to the Horizon to that which it hath up­<lb/>on the Inclined Plane H F, hath the &longs;ame proportion that the <lb/><emph type="italics"/>L<emph.end type="italics"/>ine H F hath to the <emph type="italics"/>L<emph.end type="italics"/>ine F K; that is, that the <emph type="italics"/>L<emph.end type="italics"/>ength of the <lb/>Inclined Plane hath to the Perpendicular which &longs;hall fall from it <lb/>unto the Horizon. </s> <s>So that pa&longs;&longs;ing to a more di&longs;tinct Figure, &longs;uch <lb/>as this here pre&longs;ent, the Moment of De&longs;cending which the Move­<pb xlink:href="070/01/026.jpg" pagenum="296"/>able hath upon the inclined Plane C A hath to its total Moment <lb/>wherewith it gravitates in the Perpendicular to the Horizon C P the <lb/>&longs;ame proportion that the &longs;aid Line P C hath to C A. </s> <s>And if thus it <lb/>be, it is manife&longs;t, that like as the Force that &longs;u&longs;tai­<lb/>neth the Weight in the Perpendiculation P C ought <lb/><figure id="id.070.01.026.1.jpg" xlink:href="070/01/026/1.jpg"/><lb/>to be equal to the &longs;ame, &longs;o for &longs;u&longs;taining it in the <lb/>inclined Plane C A, it will &longs;uffice that it be &longs;o much <lb/>le&longs;&longs;er, by how much the &longs;aid Perpendicular C P wan­<lb/>teth of the Line C A: and becau&longs;e, as &longs;ometimes we <lb/>&longs;ce, it &longs;ufficeth, that the Force for moving of the <lb/>Weight do in&longs;en&longs;ibly &longs;uperate that which &longs;u&longs;taineth it, therefore <lb/>we will infer this univer&longs;al Propo&longs;ition, [That upon an Elevated <lb/>Plane the Force hath to the Weight the &longs;ame proportion, as the <lb/>Perpendicular let fall from the Plane unto the Horizon hath to the <lb/>Length of the &longs;aid 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 fir&longs;t Intention, which was to inve&longs;tigate <lb/>the Nature of the Screw, we will con&longs;ider the Triangle A B C, of <lb/>which the Line A B is Horizontal, B C perpendicular to the &longs;aid <lb/>Horizon, and A C a Plane elevated; upon which the Moveable D <lb/>&longs;hall be drawn by a Force &longs;o much le&longs;s than it, by how much the <lb/>Line B C is &longs;horter than C A: But to elevate or rai&longs;e the &longs;aid <lb/>Weight along the &longs;aid Plane A C, is as much as if the Triangle <lb/>C A B &longs;tanding &longs;till, the Weight <lb/><figure id="id.070.01.026.2.jpg" xlink:href="070/01/026/2.jpg"/><lb/>D be moved towards C, which is <lb/>the &longs;ame, as if the &longs;ame Weight <lb/>never removing from the Perpen­<lb/>dicular A E, the Triangle did <lb/>pre&longs;s forwards towards H. </s> <s>For if <lb/>it were in the Site F H G, the <lb/>Moveable would be found to <lb/>have mounted the height A I. <lb/>Now, in fine, the primary Form and E&longs;&longs;ence of the Screw is no­<lb/>thing el&longs;e but &longs;uch a Triangle A C B, which being forced for­<lb/>wards, &longs;hall work it &longs;elf under the Grave Body to be rai&longs;ed, and <lb/>lifteth it up, as we &longs;ay, by the ^{*} head and &longs;houlders. </s> <s>And this was <lb/><arrow.to.target n="marg1118"></arrow.to.target><lb/>its fir&longs;t Original: For its fir&longs;t Inventor (whoever he was) con&longs;i­<lb/>dering how that the Triangle A B C going forwards rai&longs;eth the <lb/>Weight D, he might have framed an In&longs;trument like to the &longs;aid <lb/>Triangle, of a very &longs;olid Matter, which being thru&longs;t forwards did <lb/>rai&longs;e up the propo&longs;ed Weight: But afterwards con&longs;idering better, <lb/>how that that &longs;ame Machine might be reduced into a much le&longs;&longs;er <lb/>and more commodious Form, taking the &longs;ame Triangle he twined <lb/>and wound it about the Cylinder A B C D in &longs;uch a fa&longs;hion, that <lb/>the height of the &longs;aid Triangle, that is the Line C B, did make the <lb/>Height of the Cylinder, and the A&longs;cending Plane did beget upon <pb xlink:href="070/01/027.jpg" pagenum="297"/>the &longs;aid Cylinder the Helical Line de&longs;cribed by the Line AEFGH, <lb/>which we vulgarly call the Wale of the Screw, which was produ­<lb/>ced by the Line A C. </s> <s>And in this manner is the In&longs;trument made, <lb/>which is by the Greeks called <foreign lang="greek">*ko/xlos,</foreign> and by us a Screw; which <lb/><arrow.to.target n="marg1119"></arrow.to.target><lb/>winding about <lb/>cometh to work <lb/><figure id="id.070.01.027.1.jpg" xlink:href="070/01/027/1.jpg"/><lb/>and in&longs;inu­<lb/>ate with its <lb/>Wales under <lb/>the Weight, and <lb/>with facility rai­<lb/>&longs;eth it. </s> <s>And we <lb/>having demon­<lb/>&longs;trated, That up­<lb/>on [<emph type="italics"/>or along<emph.end type="italics"/>] <lb/>the elevated Plane the Force hath the &longs;ame proportion to the <lb/>Weight, that the perpendicular Altitude of the &longs;aid Plane hath to <lb/>its Length; &longs;o, &longs;uppo&longs;ing that the Force in the Screw A B C D is <lb/>multiplied according to the proportion by which the Length of the <lb/>whole Wale exceedeth the Altitude C B, from hence we come <lb/>to know that making the Screw with its Helix's more thick or clo&longs;e <lb/>together, it becometh &longs;o much the more forceable, as being begot <lb/>by a Plane le&longs;s elevated, and who&longs;e Length regards its own Per­<lb/>pendicular Altitude with greater proportion. </s> <s>But we will not <lb/>omit to adverti&longs;e you, that de&longs;iring to find the Force of a propo­<lb/>&longs;ed Screw, it will not be needful that we mea&longs;ure the Length of <lb/>all its Wales, and the Altitude of the whole Cylinder, but it <lb/>will be enough if we &longs;hall but examine how many times the Di­<lb/>&longs;tance betwixt two &longs;ingle and Contiguous terms do enter into one <lb/>&longs;ole Turn of the &longs;ame Wale, as for example, how many times <lb/>the Di&longs;tance AF is contained in the Length of the Turn AEF: <lb/>For this is the &longs;ame proportion that the Altitude CB hath to all <lb/>the Wale.</s></p><p type="margin"> <s><margin.target id="marg1118"></margin.target><emph type="italics"/>Levar in capo<emph.end type="italics"/><lb/> &longs;ignfieth to lift <lb/>on high by force</s></p><p type="margin"> <s><margin.target id="marg1119"></margin.target>* <foreign lang="greek">*ko/xlos,</foreign> in La­<lb/>tine <emph type="italics"/>Cocblea,<emph.end type="italics"/> any <lb/>Screw winding <lb/>like the Shell of <lb/>a Snail.</s></p><p type="main"> <s>If all that be under&longs;tood which we have hitherto &longs;poken touch­<lb/>ing the Nature of this In&longs;trument, I do not doubt in the lea&longs;t but <lb/>that all the other circum&longs;tances may without difficulty be compre­<lb/>hended: as for in&longs;tance, that in&longs;teed of making the Weight to <lb/>mount upon the Screw if one accommodates its Nut with <lb/>the Helix incavated or made hollow, into which the Male Screw <lb/>that is the Wale entring, & then being turned round it rai&longs;eth and <lb/>lifteth up the Nut or Male Screw together with the Weight which <lb/>was hanged thereat. </s> <s>La&longs;tly, we are not to pa&longs;s over that Con&longs;idera­<lb/>tion with &longs;ilence which at the beginning hath been &longs;aid to be nece&longs;­<lb/>&longs;ary for us to have in all Mechanick In&longs;truments, to wit, That <lb/>what is gained in Force by their a&longs;&longs;i&longs;tance, is lo&longs;t again in Time, <pb xlink:href="070/01/028.jpg" pagenum="298"/>and in the Velocity: which peradventure, might not have &longs;eemed <lb/>to &longs;ome &longs;o true and manife&longs;t in the pre&longs;ent Contemplation; nay, <lb/>rather it &longs;eems, that in this ca&longs;e the Force is multiplied without the <lb/>Movers moving a longer way than the Moveable: In regard, that <lb/>if we &longs;hall in the Triangle A B C &longs;uppo&longs;e the Line A B to be the <lb/>Plane of the Horizon, A C the elevated Plane, who&longs;e Altitude is <lb/>mea&longs;ured by the Perpendicular C B, a Moveable placed upon the <lb/>Plane A C, and the Cord E D <emph type="italics"/>F<emph.end type="italics"/> tyed to it, and a <emph type="italics"/>F<emph.end type="italics"/>orce or Weight <lb/>applyed in <emph type="italics"/>F<emph.end type="italics"/> that hath to the <lb/>Gravity of the Weight E the <lb/><figure id="id.070.01.028.1.jpg" xlink:href="070/01/028/1.jpg"/><lb/>&longs;ame proportion that the Line <lb/>B C hath to C A; by what <lb/>hath been demon&longs;trated, the <lb/>Weight <emph type="italics"/>F<emph.end type="italics"/> &longs;hall de&longs;cend <lb/>downwards, drawing the <lb/>Moveable E along the eleva­<lb/>ted Plane; nor &longs;hall the Move­<lb/>able E mea&longs;ure a greater Space <lb/>when it &longs;hall have pa&longs;&longs;ed the <lb/>whole Line A <emph type="italics"/>C,<emph.end type="italics"/> than that which the &longs;aid Grave <emph type="italics"/>F<emph.end type="italics"/> mea&longs;ureth in its <lb/>de&longs;cent downwards. </s> <s>But here yet it mu&longs;t be adverti&longs;ed, that al­<lb/>though the Moveable E &longs;hall have pa&longs;&longs;ed the whole Line A C, in <lb/>the &longs;ame Time that the other Grave <emph type="italics"/>F<emph.end type="italics"/> &longs;hall have been aba&longs;ed the <lb/>like Space, neverthele&longs;s the Grave E &longs;hall not have retired from the <lb/>common Center of things Grave more than the Space of the Per­<lb/>pendicular <emph type="italics"/>C<emph.end type="italics"/> B. but yet the Grave <emph type="italics"/>F<emph.end type="italics"/> de&longs;cending Perpendicularly &longs;hall <lb/>be aba&longs;ed a Space equal to the whole Line A <emph type="italics"/>C.<emph.end type="italics"/> And becau&longs;e Grave <lb/>Bodies make no Re&longs;i&longs;tance to Tran&longs;ver&longs;al Motions, but only &longs;o <lb/>far as they happen to recede from the <emph type="italics"/>C<emph.end type="italics"/>enter of the Earth; There­<lb/>fore the Moveable E in all the Motion A <emph type="italics"/>C<emph.end type="italics"/> being rai&longs;ed no more <lb/>than the length of the Line <emph type="italics"/>C<emph.end type="italics"/>B, but the other <emph type="italics"/>F<emph.end type="italics"/> being aba&longs;ed per­<lb/>pendicularly the quantity of all the Line A <emph type="italics"/>C<emph.end type="italics"/>: Therefore we may <lb/>de&longs;ervedly affirm that Way of the <emph type="italics"/>F<emph.end type="italics"/>orce E maintaineth the &longs;ame <lb/>proportion to the <emph type="italics"/>F<emph.end type="italics"/>orce <emph type="italics"/>F<emph.end type="italics"/> that the <emph type="italics"/>L<emph.end type="italics"/>ine A <emph type="italics"/>C<emph.end type="italics"/> hath to <emph type="italics"/>C<emph.end type="italics"/> B; that is, <lb/>the Weight E to the Weight <emph type="italics"/>F.<emph.end type="italics"/> It very much importeth, therefore, <lb/>to con&longs;ider by [<emph type="italics"/>or along<emph.end type="italics"/>] what <emph type="italics"/>L<emph.end type="italics"/>ines the Motions are made, e&longs;pe­<lb/>cially in exanimate Grave Bodies, the Moments of which have their <lb/>total Vigour, and entire Re&longs;i&longs;tance in the <emph type="italics"/>L<emph.end type="italics"/>ine Perpendicular to <lb/>the Horizon; and in the others tran&longs;ver&longs;ally Elevated and Inclined <lb/>they feel the more or le&longs;s Vigour, <emph type="italics"/>Impetus,<emph.end type="italics"/> or Re&longs;i&longs;tance, the more <lb/>or le&longs;s tho&longs;e Inclinations approach unto the Perpendicular Inclina­<lb/>tion.</s></p><pb xlink:href="070/01/029.jpg" pagenum="299"/><p type="head"> <s><emph type="italics"/>Of the SCREW of<emph.end type="italics"/> ARCHIMEDES <lb/><emph type="italics"/>to draw Waier.<emph.end type="italics"/></s></p><p type="main"> <s>I Do not think it &longs;it in this place to pa&longs;s over with Silence the <lb/>Invention of <emph type="italics"/>Archimedes<emph.end type="italics"/> to rai&longs;e Wa er with the Screw, which <lb/>is not only marvellous, but miraculous: for we &longs;hall find that <lb/>the Water a&longs;cendeth in the Screw continually de&longs;cending; and in <lb/>a given Time, with a given Force doth rai&longs;e an un&longs;peakable quan­<lb/>tity therof. </s> <s>But before we proceed any farther, let us declare the u&longs;e <lb/>of the Screw in making Water to ri&longs;e: And in the en&longs;uing Figure, <lb/>let us con&longs;ider the Line I L O P Q <lb/><figure id="id.070.01.029.1.jpg" xlink:href="070/01/029/1.jpg"/><lb/>R S H being wrapped or twined <lb/>about the Collumn M I K H, <lb/>which Line you are to &longs;uppo&longs;e to <lb/>be a Chanel thorow which the <lb/>Water may run: If we &longs;hall put <lb/>the end I into the Water, making <lb/>the Screw to &longs;tand leaning, &longs;o as <lb/>the point L may be lower than <lb/>the fir&longs;t I, as the Diagram &longs;hew­<lb/>eth, and &longs;hall turn it round about <lb/>on the two Axes, T and V, the Water &longs;hall run thorow the Cha­<lb/>nel, till that in the end it &longs;hall di&longs;charge &longs;orth at the mouth H. <lb/></s> <s>Now I &longs;ay, that the Water, in its conveyance from the point I to <lb/>the point H, doth go all the way de&longs;cending, although the point H <lb/>be higher than the point I. </s> <s>Which that it is &longs;o, we will declare <lb/>in this manner. </s> <s>We will de&longs;cribe the Triangle A C B, which is <lb/>that of which the Screw H I is generated, in &longs;uch &longs;ort that the <lb/>Chanel of the Screw is repre&longs;ented by the Line A C, who&longs;e <lb/>A&longs;cent and Elevation is determined by the Angle C A B; that is <lb/>to &longs;ay, if &longs;o be, that that Angle &longs;hall be the third or fourth part of a <lb/>Right Angle, then the Elevation of the Chanel A C &longs;hall be ac­<lb/>cording to 1/3, or 1/4 of a Right Angle. </s> <s>And it is manife&longs;t; that the <lb/>Ri&longs;e of that &longs;ame Chanel A C will be taken away deba&longs;ing the <lb/>point C as far as to B: for then the Chanel A C &longs;hall have no <lb/>Elevation. </s> <s>And deba&longs;ing the point C a little below B, the Water <lb/>will naturally run along the Chanel A C downwards from the <lb/>point A towards C. </s> <s>Let us therefore conclude, that the Angle A <lb/>being 1/3 of a Right Angle, the Chanel A C &longs;hall no longer have any <lb/>Ri&longs;e, deba&longs;ing it on the part <emph type="italics"/>C<emph.end type="italics"/> for 1/3 of a Right Angle.</s></p><p type="main"> <s>The&longs;e things under&longs;tood, let us infold the Triangle about the <lb/>Column, and let us make the Screw B A E F G, &c. </s> <s>which if it <lb/>&longs;hall be placed at Right Angles with the end B in the Water, turn­<lb/>ing it about, it &longs;hall not this way draw up the Water, the Chanel <lb/>about the Column being elevated, as may be &longs;een by the part B A.</s> <pb xlink:href="070/01/030.jpg" pagenum="300"/><s>But although the Column &longs;tand erect at Right-Angles, yet for all <lb/>that, the Ri&longs;e along the Screw, folded about the Column, is not of <lb/>a greater Elevation than of 1/3 of a Right Angle, it being generated <lb/>by the Elevation of the Chanel A C: Therefore if we incline the <lb/>Column but 1/3 of the <lb/><figure id="id.070.01.030.1.jpg" xlink:href="070/01/030/1.jpg"/><lb/>&longs;aid Right Angle, and <lb/>a little more, as we &longs;ee <lb/>I K H M, there is a <lb/>Tran&longs;ition and Moti­<lb/>on along the Chanel <lb/>I L: Therefore the <lb/>Water from the point <lb/>I to the point L &longs;hall <lb/>move de&longs;cending, and <lb/>the Screw being turned <lb/>about, the other parts <lb/>of it &longs;hall &longs;ucce&longs;&longs;ively <lb/>di&longs;po&longs;e or pre&longs;ent <lb/>them&longs;elves to the Wa­<lb/>ter in the &longs;ame Po&longs;ition as the part I L: Whereupon the Water <lb/>&longs;hall go &longs;ucce&longs;&longs;ively de&longs;cending, and in the end &longs;hall be found to <lb/>be a&longs;cended from the point I to the point H. </s> <s>Which how admira­<lb/>ble a thing it is, I leave &longs;uch to judge who &longs;hall perfectly have un­<lb/>der&longs;tood it. </s> <s>And by what hath been &longs;aid, we come to know, That <lb/>the Screw for rai&longs;ing of Water ought to be inclined a little more <lb/>than the quantity of the Angle of the Triangle by which the &longs;aid <lb/>Screw is de&longs;cribed.</s></p><p type="head"> <s><emph type="italics"/>Of the Force of the <lb/>HAMMER, MALLET, or BEETLE.<emph.end type="italics"/></s></p><p type="main"> <s>The Inve&longs;tigation of the cau&longs;e of the Force of the&longs;e Percuti­<lb/>ents is nece&longs;&longs;ary for many Rea&longs;ons: and fir&longs;t, becau&longs;e that <lb/>there appeareth in it much more matter of admiration than <lb/>is ob&longs;erved in any other Mechanick In&longs;trument what&longs;oever. </s> <s>For <lb/>&longs;triking with the Hammer upon a Nail, which is to be driven into <lb/>a very tough Po&longs;t, or with the Beetle upon a Stake that is to pene­<lb/>trate into very &longs;tiffe ground, we &longs;ee, that by the &longs;ole vertue of the <lb/>blow of the Percutient both the one and the other is thru&longs;t for­<lb/>wards: &longs;o that without that, only laying the Beetle upon the <lb/>Nail or Stake it will not move then, nay, more, although you <lb/>&longs;hould lay upon them a Weight very much heavier than the &longs;aid <lb/>Beetle. </s> <s>An effect truly admirable, and &longs;o much the more worthy <lb/>of Contemplation, in that, as I conceive, none of tho&longs;e who have <pb xlink:href="070/01/031.jpg" pagenum="301"/>hitherto di&longs;cour&longs;ed upon it, have &longs;aid any thing that hits the mark; <lb/>which we may take for a certain Sign and Argument of the Ob&longs;cu­<lb/>rity and difficulty of this <emph type="italics"/>S<emph.end type="italics"/>peculation. </s> <s>For <emph type="italics"/>Ari&longs;totle,<emph.end type="italics"/> or others, <lb/>who would reduce the cau&longs;e of this admirable Effect unto the <lb/>length of the <emph type="italics"/>Manubrium,<emph.end type="italics"/> or Handle, may, in my judgement, be <lb/>made to &longs;ee their mi&longs;take in the effect of tho&longs;e In&longs;truments, which <lb/>having no Handle, yet percu&longs;s, either in falling from on high <lb/>downwards, or by being thrown with Velocity &longs;idewaies. </s> <s>There­<lb/>fore it is requi&longs;ite, that we have recour&longs;e to &longs;ome other Principle, if <lb/>we would find out the truth of this bu&longs;ine&longs;s; the cau&longs;e of which, <lb/>although it be of its own nature &longs;omewhat ob&longs;cure, and of diffi­<lb/>cult con&longs;ideration, yet neverthele&longs;s we will attempt with the grea­<lb/>te&longs;t per&longs;picuity po&longs;&longs;ible to render it clear and obvious, &longs;hewing, for <lb/>a clo&longs;e of all, that the Principle and Original of this Effect is deri­<lb/>ved from no other Fountain than this, from which the rea&longs;ons of all <lb/>other Mechanick Effects do proceed: and this we will do, by &longs;etting <lb/>before your eyes that very thing which is &longs;een to befall in every <lb/>other Mechanick Operation, <emph type="italics"/>&longs;cilicet,<emph.end type="italics"/> That the Force, the Re&longs;i&longs;tance, <lb/>and the Space by which the Motion is made, do go alternately <lb/>with &longs;uch proportion operating, and with &longs;uch a rate an&longs;wering to <lb/>each other, that a Re&longs;i&longs;tance, equal to the Force, &longs;hall be moved by <lb/>the &longs;aid Force along an equal Space, with Velocity equal to that <lb/>with which it is moved. </s> <s>Likewi&longs;e, That a Force that is le&longs;s by half <lb/>than a Re&longs;i&longs;tance &longs;hall be able to move it, &longs;o that it be moved <lb/>with double Velocity, or, if you will, for a Di&longs;tance twice as great <lb/>as that which the moved Re&longs;i&longs;tance &longs;hall pa&longs;s: and, in a word, it <lb/>hath been &longs;een in all the other In&longs;truments, that any, never &longs;o great, <lb/>Re&longs;i&longs;tance may be moved by every &longs;mall Force given, provided, <lb/>that the Space, along which the Re&longs;i&longs;tance &longs;hall move, have the <lb/>&longs;ame proportion that is found to be betwixt the &longs;aid great Re&longs;i­<lb/>&longs;tance and the Force: and that this is according to the nece&longs;&longs;ary <lb/>Order and Con&longs;titution of Nature: So that inverting the Di&longs;cour&longs;e, <lb/>and Arguing the contrary way, what wonder &longs;hall it be, if that <lb/>Power that &longs;hall move a &longs;mall Re&longs;i&longs;tance a great way, &longs;hall carry <lb/>one an hundred times bigger an hundredth part of that Di&longs;tance? <lb/></s> <s>Certainly none at all: nay, it would be ab&longs;urd, yea, impo&longs;&longs;ible <lb/>that it &longs;hould be otherwi&longs;e. </s> <s>Let us therefore con&longs;ider, what the <lb/>Re&longs;i&longs;tance of the Beetle unto Motion may be in that point where <lb/>it is to &longs;trike, and how far, if it do not &longs;trike, it would be carryed <lb/>by the received Force beyond that point: and again, what Re&longs;i­<lb/>&longs;tance to Motion there is in him who &longs;triketh, and how much by <lb/>that &longs;ame Percu&longs;&longs;ion he is moved: and, having found that this <lb/>great Re&longs;i&longs;tance goeth forwards by a percu&longs;&longs;ion &longs;o much le&longs;s than <lb/>the Beetle driven by the <emph type="italics"/>Impetus<emph.end type="italics"/> of him that moveth it would do, <lb/>by how much that &longs;ame great Re&longs;i&longs;tance is greater than that of <pb xlink:href="070/01/032.jpg" pagenum="302"/>the Beetle; we &longs;hall cea&longs;e to wonder at the Effect, which doth not <lb/>in the lea&longs;t exceed the terms of Natural Con&longs;titutions, and of <lb/>what hath been &longs;poken. </s> <s>Let us, for better under&longs;tanding, give an <lb/>example thereof in particular Terms. </s> <s>There is a Beetle, which ha­<lb/>ving four degrees of Re&longs;i&longs;tance, is moved by &longs;uch a Force, that <lb/>being freed from it in that term where it maketh the Percu&longs;&longs;ion, it <lb/>would, meeting with no &longs;top, go ten Paces beyond it, and in that <lb/>term a great po&longs;t being oppo&longs;ed to it, who&longs;e Re&longs;i&longs;tance to Moti­<lb/>on is as four thou&longs;and, that is, a thou&longs;and times greater than that of <lb/>the Beetle, (but yet is not immoveable) &longs;o that it without mea­<lb/>&longs;ure or proportion exceeds the Re&longs;i&longs;tance of the Beetle, yet the <lb/>Percu&longs;&longs;ion being made on it, it &longs;hall be driven forwards, though in­<lb/>deed no more but the thou&longs;andth part of the ten Paces which the <lb/>Beetle &longs;hall be moved: and thus in an inverted method, changing <lb/>that which hath been &longs;poken touching the other Mechanical Effects, <lb/>we may inve&longs;tigate the rea&longs;on of the Force of the Percutient. </s> <s>I <lb/>know that here ari&longs;e difficulties and objections unto &longs;ome, which <lb/>they will not ea&longs;ily be removed from, but we will freely remit them <lb/><arrow.to.target n="marg1120"></arrow.to.target><lb/>to the ^{*} Problems Mechanical, which we &longs;hall adjoyn in the end of <lb/>this Di&longs;cour&longs;e.</s></p><p type="margin"> <s><margin.target id="marg1120"></margin.target>* The&longs;e Pro­<lb/>blems he here <lb/>promi&longs;eth were <lb/>never yet ex­<lb/>tant.</s></p><pb xlink:href="070/01/033.jpg" pagenum="303"/><p type="head"> <s>THE <lb/>BALLANCE <lb/>OF <lb/><emph type="italics"/>Signeur GALILEO GALILEI<emph.end type="italics"/>;</s></p><p type="head"> <s>In which, in immitation of <emph type="italics"/>Archimedes<emph.end type="italics"/> in the <lb/>Problem of the Crown, he &longs;heweth how to <lb/>find the proportion of the Alloy of <lb/>Mixt-Metals; and how to make <lb/>the &longs;aid In&longs;trument.</s></p><p type="main"> <s>As it is well known, by &longs;uch who take the pains to read <lb/>old Authors, that <emph type="italics"/>Archimedes<emph.end type="italics"/> detected the Cheat of <lb/>the Gold&longs;mith in the Crown of ^{*} <emph type="italics"/>Hieron,<emph.end type="italics"/> &longs;o I think it <lb/><arrow.to.target n="marg1121"></arrow.to.target><lb/>hitherto unknown what method this Great Philo&longs;o­<lb/>pher ob&longs;erved in that Di&longs;covery: for the opinion, that he did per­<lb/>form it by putting the Crown into the Water, having fir&longs;t put in­<lb/><arrow.to.target n="marg1122"></arrow.to.target><lb/>to it &longs;uch another Ma&longs;s of pure Gold, and another of Silver &longs;eve­<lb/>rally, and that from the differences in their making the Water <lb/>more or le&longs;s ri&longs;e and run over, he came to know the Mixture or <lb/>Alloy of the Gold with the Silver, of which that Crown was <lb/>compounded; &longs;eems a thing (if I may &longs;peak it) very gro&longs;s, and <lb/>far from exactne&longs;s. </s> <s>And it will &longs;eem &longs;o much the more dull to <lb/>&longs;uch who have read and under&longs;tood the exqui&longs;ite Inventions of &longs;o <lb/>Divine a Man among&longs;t the Memorials that are extant of him; by <lb/>which it is very manife&longs;t that all other Wits are inferiour to that <lb/>of <emph type="italics"/>Archimedes.<emph.end type="italics"/> Indeed I believe, that Fame divulging it abroad, <lb/>that <emph type="italics"/>Archimedes<emph.end type="italics"/> had di&longs;covered that &longs;ame Fraud by means of the <lb/>Water, &longs;ome Writer of tho&longs;e Times committed the memory there­<lb/>of to Po&longs;terity, and that this per&longs;on, that he might add &longs;omething <lb/>to that little which he had heard by common Fame, did relate that <lb/><emph type="italics"/>Archimedes<emph.end type="italics"/> had made u&longs;e of the Water in that manner, as &longs;ince <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"/>Sicily,<emph.end type="italics"/><lb/>and Kin&longs;man to <lb/>that Great Ma­<lb/>thematician.</s></p><p type="margin"> <s><margin.target id="marg1122"></margin.target><emph type="italics"/>Plutarch in Vit. <lb/></s> <s>Marcel.<emph.end type="italics"/></s></p><p type="main"> <s>But in regard I know, that that method is altogether fallacious, <lb/>and falls &longs;hort of that exactne&longs;s which is required in Mathematical <lb/>Matters, I have often thought in what manner, by help of the <lb/>Water, one might exactly find the Mixture of two Metals, and <lb/>in the end, after I had diligently peru&longs;ed that which <emph type="italics"/>Archimedes<emph.end type="italics"/><lb/>demon&longs;trateth in his Books <emph type="italics"/>De in&longs;identibus aquæ,<emph.end type="italics"/> and tho&longs;e others <pb xlink:href="070/01/034.jpg" pagenum="304"/><emph type="italics"/>De æquiponder antium,<emph.end type="italics"/> there came into my thoughts a Rule which <lb/>exqui&longs;itely re&longs;olveth our Que&longs;tion; which Rule I believe to be <lb/>the &longs;ame that <emph type="italics"/>Archimedes<emph.end type="italics"/> made u&longs;e of, &longs;eeing that be&longs;ides the <lb/>u&longs;e that is to be made of the Water, the exactne&longs;s of the Work <lb/>dependeth al&longs;o upon certain Demon&longs;trations found by the &longs;aid <lb/><emph type="italics"/>Archimedes.<emph.end type="italics"/></s></p><p type="main"> <s>The way is by help of a Ballance, who&longs;e Con&longs;truction and U&longs;e <lb/>&longs;hall be &longs;hewn by and by, after we &longs;hall have declared what is <lb/>nece&longs;&longs;ary for the knowledge thereof. </s> <s>You mu&longs;t know there­<lb/>fore, that the Solid Bodies that &longs;ink in the Water weigh &longs;o much <lb/>le&longs;s in the Water than in the Air, as a Ma&longs;s of Water equal to <lb/>the &longs;aid Solid doth weigh in the Air: which hath been demon­<lb/>&longs;trated by <emph type="italics"/>Archimedes.<emph.end type="italics"/> But, in regard his Demon&longs;tration is very <lb/>mediate, becau&longs;e I would not be over long, laying it a&longs;ide, I &longs;hall <lb/>declare the &longs;ame another way. </s> <s>Let us con&longs;ider, therefore, that <lb/>putting into the Water <emph type="italics"/>v. </s> <s>g.<emph.end type="italics"/> a Ma&longs;s of Gold, if that Ma&longs;s were <lb/>of Water it would have no weight at all: For the Water moveth <lb/>neither upwards, nor downwards in the Water: It remains, <lb/>therefore, that the Ma&longs;s of Gold weigheth in the Water only &longs;o <lb/>much as the Gravity of the Gold exceeds the Gravity of the Wa­<lb/>ter. </s> <s>And the like is to be under&longs;tood of other Metals. </s> <s>And be­<lb/>cau&longs;e the Metals are different from each other in Gravity, their <lb/>Gravity in the Water &longs;hall dimini&longs;h according to &longs;everal proporti­<lb/>ons. </s> <s>As for example: Let us &longs;uppo&longs;e that Gold weigheth twenty <lb/>times more than Water, it is manife&longs;t by that which hath been <lb/>&longs;poken, that the Gold will weigh le&longs;s in the Water than in the <lb/>Air by a twentieth part of its whole weight. </s> <s>Now, let us &longs;uppo&longs;e <lb/>that Silver, as being le&longs;s Grave than Gold, weigheth 12 times more <lb/>than Water: this then, being weighed in the Water, &longs;hall di­<lb/>mini&longs;h in Gravity the twelfth part of its whole weight. </s> <s>Therefore <lb/>the Gravity of Gold in the Water decrea&longs;eth le&longs;s than that of <lb/>Silver; for that dimini&longs;heth a twentieth part, and this a twelfth. <lb/></s> <s>If therefore in an exqui&longs;ite Ballance we &longs;hall hang a Metal at the <lb/>one Arm, and at the other a Counterpoi&longs;e that weigheth equally <lb/>with the &longs;aid Metal in the Water, leaving the Counterpoi&longs;e in the <lb/>Air, to the end that it may equivalate and compen&longs;ate the Me­<lb/>tal, it will be nece&longs;&longs;ary to hang it nearer the Perpendicular or <lb/>Cook. </s> <s>As for example, Let the Ballance be A B, its Perpendicu­<lb/><figure id="id.070.01.034.1.jpg" xlink:href="070/01/034/1.jpg"/><lb/>lar C, and let a <lb/>Ma&longs;s of &longs;ome <lb/>Metal be &longs;u­<lb/>&longs;pended at B, <lb/>counterpoi&longs;edby <lb/>the Weight D: putting the Weight B into the Water, the <lb/>Weight D in A would weigh more: therefore that they may <pb xlink:href="070/01/035.jpg" pagenum="305"/>weigh equally it would be nece&longs;&longs;ary to hang it nearer to the <lb/>Perpendicular C, as <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> in E: and look how many times the Di­<lb/>&longs;tance C A &longs;hall contain A E, &longs;o many times &longs;hall the Metal <lb/>weigh more than the Water. </s> <s>Let us therefore &longs;uppo&longs;e that the <lb/>Weight in B be Gold, and that weighed in the Water it with­<lb/>draws the Counterpoi&longs;e D into E; and then doing the &longs;ame with <lb/>pure Silver, let us &longs;uppo&longs;e that its Counterpoi&longs;e, when afterwards <lb/>it is weighed in the Water, returneth to F: which point &longs;hall be <lb/>nearer to the point C, as Experience &longs;heweth, becau&longs;e the Silver <lb/>is le&longs;s grave than the Gold: And the Di&longs;tance that is between <lb/>A and F &longs;hall have the &longs;ame Difference with the Di&longs;tance A E, <lb/>that the Gravity of the Gold hath with that of the Silver. </s> <s>But if <lb/>we have a Mixture of Gold and Silver, it is clear, that by rea&longs;on it <lb/>participates of Silver, it &longs;hall weigh le&longs;s than the pure Gold, and <lb/>by rea&longs;on it participates of Gold, it &longs;hall weigh more than the <lb/>pure Silver: and therefore being weighed in the Air, and de&longs;iring <lb/>that the &longs;ame Counterpoi&longs;e &longs;hould counterpoi&longs;e it, when that <lb/>Mixture &longs;hall be put into the Water it will be nece&longs;&longs;ary to draw <lb/>the &longs;aid Counterpoi&longs;e more towards the Perpendicular C, than the <lb/>point E is, which is the term of the Gold; and more from C <lb/>than F is, which is the term of the pure Silver; Therefore it &longs;hall <lb/>fall between the points E and F: And the proportion into which <lb/>the Di&longs;tance EF &longs;hall be divided, &longs;hall exactly give the proportion <lb/>of the two Metals which compound that Mixture. </s> <s>As for exam­<lb/>ple: Let us &longs;uppo&longs;e the Mixture of Gold and Silver to be in B, <lb/><figure id="id.070.01.035.1.jpg" xlink:href="070/01/035/1.jpg"/><lb/>counterpoi&longs;ed in <lb/>the Air by D, <lb/>which Counter­<lb/>poi&longs;e when the <lb/>Compound Me­<lb/>tal is put into the Water returneth into G: I &longs;ay now, that the <lb/>Gold and the Silver which compound this Mixture are to one ano­<lb/>ther in the &longs;ame proportion, as the Di&longs;tance F G is to the Di&longs;tance <lb/>G E. </s> <s>But you mu&longs;t know that the Di&longs;tance G F terminated in <lb/>the mark of the Silver, &longs;hall denote unto us the quantity of the <lb/>Gold, and the Di&longs;tance G E, terminated in the mark of the Gold, <lb/>&longs;hall &longs;hew us the quantity of the Silver: in&longs;omuch that if F G <lb/>&longs;hall prove double to G E, then that Mixture &longs;hall be two parts <lb/>Gold, and one part Silver: and in the &longs;ame method proceeding in<lb/>the examination of other Mixtures, one &longs;hall exactly find the <lb/>quantity of the &longs;imple Metals.</s></p><p type="main"> <s>To compo&longs;e the Ballance, therefore, take a Rod at lea&longs;t a yard <lb/>long, (and the longer it is, the exacter the In&longs;trument &longs;hall be) <lb/>and divide it in the mid&longs;t, where place the Perpendicular: then <lb/>adju&longs;t the Arms that they may &longs;tand in <emph type="italics"/>Equilibrium,<emph.end type="italics"/> by filing or <pb xlink:href="070/01/036.jpg" pagenum="306"/>&longs;having that le&longs;s which weigheth mo&longs;t; and upon one of the Arms <lb/>note the terms to which the Counterpoi&longs;es of &longs;imple Metals return <lb/>when they &longs;hall be weighed in the Water: taking care to weigh the <lb/>pure&longs;t Metals that can be found. </s> <s>This being done, it remaineth <lb/>that we find out a way, how we may with facility di&longs;cover the <lb/>proportion, according to which, the Di&longs;tances between the terms <lb/>of the &longs;imple and pure Metals are divided by the Marks of the <lb/>Mixt Metals: Which &longs;hall be effected in this manner.</s></p><p type="main"> <s>We are to have two very &longs;mall Wires drawn thorow the &longs;ame <lb/>drawing-Iron, one of Steel, the other of Bra&longs;s, and above the <lb/>terms of the &longs;imple Metals we mu&longs;t wind the Steel Wyer; as for <lb/>example: above the point E, the term of the pure Gold, we are <lb/>to wind the Steel Wyer, and under it the other Bra&longs;s Wyre, and <lb/>having made ten folds of the Steel Wyer, we mu&longs;t make ten <lb/>more with that of Bra&longs;s, and thus we are to continue to do with <lb/>ten of Steel, and ten of Bra&longs;s, until that the whole Space be­<lb/>tween the points E and F, the terms of the pure Metals, be full; <lb/>cau&longs;ing tho&longs;e two terms to be alwaies vi&longs;ible and per&longs;picuous: <lb/>and thus the Di&longs;tance E F &longs;hall be divided into many equal parts, <lb/>and numbred by ten and ten. </s> <s>And if at any time we would know <lb/>the proportion that is between F G and G E, we mu&longs;t count the <lb/>Wyers F G, and the Wyers G E: and finding the Wyers F G <lb/>to be, for example, 40, and the Wyers G E, 21: we will &longs;ay that <lb/>there is in the mixt Metal 40 parts of Gold, and 21 of Silver. </s> <s>But <lb/>here you mu&longs;t note, that there is &longs;ome difficulty in the counting, <lb/>for tho&longs;e Wyers being very &longs;mall, as it is requi&longs;ite for exactne&longs;s <lb/>&longs;ake, it is not po&longs;&longs;ible with the eye to tell them, becau&longs;e the <lb/>&longs;malne&longs;s of the Spaces dazleth & confoundeth the Sight. </s> <s>Therefore <lb/>to number them with facility, take a Bodkin as &longs;harp as a Needle <lb/>and &longs;et it into an handle, or a very fine pointed Pen-knife, with <lb/>which we may ea&longs;ily run over all the &longs;aid Wyers, and this way <lb/>partly by help of hearing, partly by the impediments the hand <lb/>&longs;hall feel at every Wyer, tho&longs;e Wyers &longs;hall be counted; <lb/>the number of which, as I &longs;aid before, &longs;hall give us the exact <lb/>quantity of the &longs;unple Metals, of which the Mixt-Metal is com­<lb/>pounded: taking notice that the Simple an&longs;wer alternately to the <lb/>Di&longs;tances. </s> <s>As for example, in a Mixture of Gold and Silver, <lb/>the Wyers that &longs;hall be towards the term of Gold &longs;hall &longs;hew us <lb/>the quantity of the Silver: And the &longs;ame is to be under&longs;tood of <lb/>other Metals.</s></p><pb xlink:href="070/01/037.jpg" pagenum="307"/><p type="head"> <s>Annotations of <emph type="italics"/>Dominico Mantovani<emph.end type="italics"/> upon the Bal­<lb/>lance of <emph type="italics"/>Signore Galileo Galilei.<emph.end type="italics"/></s></p><p type="main"> <s>Fir&longs;t, I conceive that the difficulty of Numbring the Wyres <lb/>is removed by wrapping about the Ballance ten of Steel, <lb/>and then ten of Bra&longs;s, which being divided by tens, there <lb/>only remains that tenth part to be numbred, in which the term <lb/>of the Mixt Metal falleth. </s> <s>For although <emph type="italics"/>Signore Galileo,<emph.end type="italics"/> who is <lb/>Author of this Invention, makes mention of two Wyres, one of <lb/>Steel, the other of Bra&longs;s, yet he doth not &longs;ay, that we are to <lb/>take ^{*} ten of the one, and ten of the other: which it may be <lb/><arrow.to.target n="marg1123"></arrow.to.target><lb/>hapneth by the negligence of him that hath tran&longs;cribed it; al­<lb/>though I mu&longs;t confe&longs;s that the Copy which came to my hands was <lb/>of his own writing.</s></p><p type="margin"> <s><margin.target id="marg1123"></margin.target>* <emph type="italics"/>Galileus<emph.end type="italics"/> &longs;aith it <lb/>expre&longs;ly in this <lb/>Copy which I fol­<lb/>low, but might <lb/>omit it in the Co­<lb/>py which came to <lb/>the hands of <emph type="italics"/>Man­<lb/>tovani.<emph.end type="italics"/></s></p><p type="main"> <s>Secondly, it is &longs;uppo&longs;ed in this Problem that the Compo&longs;ition <lb/>of two Metals do retain the &longs;ame proportion of Ma&longs;s in the <lb/>Mixture as the two Simple Metals, of which it is compounded, <lb/>had at fir&longs;t. </s> <s>I mean, that the Simple Metals retain and keep in <lb/>the Compo&longs;ition (after that they are incorporated and commix­<lb/>ed) the &longs;ame proportion in Ma&longs;s that the Simple Metals had <lb/>when they were &longs;eparated: Which in the Ca&longs;e of <emph type="italics"/>Signore Gali­<lb/>leo,<emph.end type="italics"/> touching the Commixtion of Gold and Silver, I do neither <lb/>deny, nor particularly confe&longs;s. </s> <s>But if one would, for example, <lb/>unite 101 pounds of Copper with 21 pounds of Tin, to make <lb/>thereof 120 pounds of Bell-Metal, (I abate two pounds, <lb/>&longs;uppo&longs;ed to be wa&longs;ted in the Melting) I do think that 120 <lb/>pounds of Compound Metal will have a le&longs;s Bulk than the 100 <lb/>pounds of pure Copper, and the 20 pounds of Tin unmixt, that <lb/>is, before they were incorporated and melted into one Ma&longs;s, and <lb/>that the Compo&longs;ition is more grave <emph type="italics"/>in Specie<emph.end type="italics"/> than the &longs;ingle Cop­<lb/>per, and the &longs;ingle Bra&longs;s: and in the Ca&longs;e of <emph type="italics"/>Signore Galileo<emph.end type="italics"/> the <lb/>Compo&longs;ition of Gold and Silver is &longs;uppo&longs;ed to be lighter <emph type="italics"/>in Specie<emph.end type="italics"/><lb/>than the pure Gold, and heavier <emph type="italics"/>in Specie<emph.end type="italics"/> than the pure Silver. </s> <s>Of <lb/>which it would be ea&longs;ie to make &longs;ome &longs;uch like experiment, melt­<lb/>ing together, <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> 10 pounds of Lead with 5 pounds of Tin, <lb/>and ob&longs;erving whether tho&longs;e 15 pounds, or whatever the Mixture <lb/>maketh, do give the difference betwixt the weight in the Water <lb/>to the weight in the Air, in the proportion that the 15 pounds of <lb/>the two Metals di&longs;-united gave before: I do not &longs;ay, the &longs;ame diffe­<lb/>rence, becau&longs;e I pre &longs;uppo&longs;e that they will wa&longs;te in melting down, <lb/>and that the Compound will be le&longs;s than 15 pounds, therefore I <lb/>&longs;ay in proportion.</s></p><p type="main"> <s>Thirdly, He doth al&longs;o &longs;uppo&longs;e, that one ought to take the <pb xlink:href="070/01/038.jpg" pagenum="308"/>Simple Metals, that is, the Gold and the Silver, each of the &longs;ame <lb/>weight as the Mixture, although he doth not &longs;ay &longs;o; which may <lb/>be collected in that he marketh the ballance only betwixt the <lb/>Terms of the Gold and the Silver, which is the cau&longs;e of the great <lb/>facility in re&longs;olving the Problem by only counting the <lb/>Wyers.</s></p><p type="main"> <s>One might take the pure Gold, and pure Silver of the &longs;ame <lb/>weight, in re&longs;pect of one another, but yet different from the <lb/>weight of the Mixture, that is, either more or le&longs;s grave than the <lb/>Mixt Metal: and being equal in weight to one another they <lb/>might &longs;hew the proportion of the Ma&longs;s of the Gold to that of the <lb/>Silver; but yet with this difference, that the more grave will &longs;hew <lb/>the &longs;aid proportion more exactly than the &longs;mall and le&longs;s grave. <lb/></s> <s>But the Simple and pure Metals not being of the &longs;ame weight as <lb/>the Compound, it will be nece&longs;&longs;ary, having found the proportion <lb/>of the Ma&longs;s of the Gold to that of the Silver; to find by numbers <lb/>proportionally the exact quantity of each of the two Metals com­<lb/>pounding the Mixture.</s></p><p type="main"> <s>A man may likewi&longs;e u&longs;e the quantity of the &longs;imple Metals ac­<lb/>cording to nece&longs;&longs;ity and convenience, although of different <lb/>Weights, both as to each other, and to the Mixture, provided that <lb/>each of them be pure in its kind: but then we mu&longs;t after­<lb/>wards by numbers find the proportion of the Ma&longs;&longs;es of the two <lb/>Simple ones of equal weight (which is &longs;oon done, taking them of <lb/>equal weight as was &longs;aid before) and then according to this pro­<lb/>portion to find, by means of the Weight, and of the Ma&longs;s of the <lb/>Compound Metal, the di&longs;tinct quantity of each of the two Sim­<lb/>ple ones that make the Compo&longs;ition: of each of which Ca&longs;es <lb/>examples might be given. </s> <s>But to conclude, if the pure Gold, <lb/>and pure Silver, and the Mixt Metal &longs;hould be of equal Ma&longs;s, <lb/>they would be unequal in Weight, and it would not need to <lb/>weigh them in the Water, for being of equal Bulk, the differen­<lb/>ces of their Weights in the Air and in the Water would be al&longs;o <lb/>equal: for the difference of the weight of any Body in the Air <lb/>to its weight in the Water, is alwaies equal to the Weight of &longs;o <lb/>much Water as equalleth the &longs;ame Body in Ma&longs;s, by <emph type="italics"/>Archimedes<emph.end type="italics"/><lb/>his fifth Propo&longs;ition, <emph type="italics"/>De ijs quæ vehuntur in aqua.<emph.end type="italics"/></s></p><p type="main"> <s>And la&longs;t of all, the Simple and pure Metals may have the &longs;ame <lb/>proportion in Gravity, mutually or reciprocally, as their Bodies <lb/>have in Bulk: In which ca&longs;e, as well the Ma&longs;s, found by help of <lb/>the weight in Water, or by any other meanes, as their Weight in <lb/>the Air &longs;hall &longs;hew the proportion of their Specifical Gravities; as <lb/>their Weights in the Water do when their Weights in the Air <lb/>are equal; but yet alternately weighed: that is to &longs;ay, the Spe­<lb/>cifical Gravity of the Gold &longs;hall have &longs;uch proportion to the <pb xlink:href="070/01/039.jpg" pagenum="309"/>Specifical Gravity of the Silver, as the Ma&longs;s of the Silver hath to <lb/>the Ma&longs;s of the Gold; that is, as the difference betwixt the <lb/>Weight in Water and Weight in Air of the Silver, hath to the <lb/>difference betwixt the Weight in Water and Weight in Air of <lb/>the Gold.</s></p><p type="main"> <s>With this &longs;ame Ballance one may with facility mea&longs;ure the <lb/>Ma&longs;s or Magnitude of any Body, in any manner what&longs;oever Irre­<lb/>gular in manner following, namely:</s></p><p type="main"> <s>We will have at hand a Solid Body of a &longs;ub&longs;tance more grave <lb/><emph type="italics"/>in Specie<emph.end type="italics"/> than the Water; as for in&longs;tance of Lead; or if it were <lb/>of Wood, or other matter more light <emph type="italics"/>in Specie<emph.end type="italics"/> than the Water, <lb/>it may be made heavier by fa&longs;tning unto it Lead, or &longs;ome other <lb/>thing that makes it &longs;ink in the Water, and let us take &longs;ome <lb/>known Mea&longs;ure, and with it mea&longs;ure the Irregular Solid; as for <lb/>in&longs;tance, the Roman Palm, the Geometrical Foot, or any other <lb/>known mea&longs;ure, or part of the &longs;ame, as the half Foot, the quar­<lb/>ter of a Foot, or any &longs;uch like part known; then let it be weighed <lb/>in the Air, and &longs;uppo&longs;e that it weigh 10 pounds; let the &longs;ame <lb/>Mea&longs;ure be weighed in the Air, and &longs;uppo&longs;e that it weigh 8 <lb/>pounds: and &longs;ub&longs;tract 8 pounds, the Weight in the Water, from <lb/>10 pounds, the Weight in the Air, and there remaineth 2 pounds <lb/>for the Weight of a Body of Water equal in Magnitude to the <lb/>Mea&longs;ure known. </s> <s>Now, if we would mea&longs;ure a Statue of Mar­<lb/>ble, let it be weighed fir&longs;t in the Air, and then in the Water, and <lb/>&longs;ub&longs;tract the Weight in the <emph type="italics"/>W<emph.end type="italics"/>ater from the <emph type="italics"/>W<emph.end type="italics"/>eight in the Air, and <lb/>the remainder &longs;hall be the weight of &longs;o much <emph type="italics"/>W<emph.end type="italics"/>ater as equalleth <lb/>the Statue in Ma&longs;s; which being divided by the difference betwixt <lb/>the <emph type="italics"/>W<emph.end type="italics"/>eight in <emph type="italics"/>W<emph.end type="italics"/>ater and the <emph type="italics"/>W<emph.end type="italics"/>eight in Air of the Mea&longs;ure known, <lb/>the Quotient will give how many times the Statue containeth the <lb/>&longs;ame given Mea&longs;ure. </s> <s>As for example; if the Statue in Air weigh <lb/>100 pounds, and in the <emph type="italics"/>W<emph.end type="italics"/>ater 80 pounds, 80 pounds being &longs;ub­<lb/>&longs;tracted from 100 there re&longs;teth 20 pounds for the <emph type="italics"/>W<emph.end type="italics"/>eight of &longs;o <lb/>much <emph type="italics"/>W<emph.end type="italics"/>ater in Ma&longs;s as equalleth the Statue. </s> <s>But becau&longs;e the <lb/>difference betwixt the <emph type="italics"/>W<emph.end type="italics"/>eight in <emph type="italics"/>W<emph.end type="italics"/>ater, and the <emph type="italics"/>W<emph.end type="italics"/>eight in Air <lb/>equal in Magnitude to the Mea&longs;ure known, was &longs;uppo&longs;ed to be <lb/>2 pounds; divide 18 pounds by two pounds, and the Quotient <lb/>is 9, for the number of times that the propo&longs;ed Statue containeth <lb/>the given Mea&longs;ure. </s> <s>The &longs;ame Method may be ob&longs;erved, if it <lb/>were required, to mea&longs;ure a Statue, or other Ma&longs;s of any kind of <lb/>Metal: only it mu&longs;t be adverti&longs;ed, that all the holes mu&longs;t be <lb/>&longs;topt, that the <emph type="italics"/>W<emph.end type="italics"/>ater may not enter into the Body of the Statue: <lb/>but he that de&longs;ireth only the Solid content of the Metal of the <lb/>&longs;aid Statue mu&longs;t open the holes, and with Tunnels fill the whole <lb/>cavity of the Statue with <emph type="italics"/>W<emph.end type="italics"/>ater. </s> <s>And if the Statue were of a <lb/>Sub&longs;tance lighter <emph type="italics"/>in Specie<emph.end type="italics"/> than the <emph type="italics"/>W<emph.end type="italics"/>ater; as, for example, of <pb xlink:href="070/01/040.jpg" pagenum="310"/>Wax, it will be requi&longs;ite to add unto the Statue &longs;ome Counter­<lb/>poi&longs;e, that maketh it &longs;ink in the <emph type="italics"/>W<emph.end type="italics"/>ater, and then to mea&longs;ure the <lb/>Counterpoi&longs;e, as above, and to &longs;ub&longs;tract its mea&longs;ure from the <lb/>Compound Body, and there will remain the Mea&longs;ure of the <lb/>Statue of <emph type="italics"/>W<emph.end type="italics"/>ax. </s> <s>And la&longs;tly, to make u&longs;e of the &longs;aid Ballance, <lb/>in&longs;tead of &longs;eeking the numbers of the pounds of the Differences <lb/>of the <emph type="italics"/>W<emph.end type="italics"/>eights of the Mea&longs;ure known, and of the Solid <lb/>to be mea&longs;ured in <emph type="italics"/>W<emph.end type="italics"/>ater, and in Air, we may count the <lb/><emph type="italics"/>W<emph.end type="italics"/>yers of the Arm of the Ballance, which <lb/>being very &longs;mall will give the <lb/>Mea&longs;ure exactly.</s></p><p type="head"> <s><emph type="italics"/>FINIS.<emph.end type="italics"/></s></p></chap> </body> <back></back> </text></archimedes>