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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
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| date | Thu, 02 May 2013 11:38:23 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" ><archimedes> <info> <author>Descartes, Rene</author> <title>Mechanics</title> <date>1665</date> <place>London</place> <translator>Thomas Salusbury</translator> <lang>en</lang> <cvs_file>desca_mecha_071_en_1665.xml</cvs_file> <cvs_version></cvs_version> <locator>071.xml</locator> </info> <text> <front> </front> <body> <chap> <pb xlink:href="071/01/001.jpg"/> <pb xlink:href="071/01/002.jpg" pagenum="311"/><p type="head"> <s>DISCOURSES <lb/>OF THE <lb/>MECHANICKS: <lb/>A MANVSCRIPT of <lb/>Mon&longs;ieur Des-Cartes.</s></p><p type="head"> <s>The Explication.</s></p><p type="main"> <s><emph type="italics"/>Of Engines, by help of which we may rai&longs;e a very great <lb/>weight with &longs;mall &longs;trength.<emph.end type="italics"/></s></p><p type="main"> <s>The Invention of all the&longs;e Engines de­<lb/>pends upon one &longs;ole Principle, which is, <lb/>That the &longs;ame Force that can lift up a <lb/>Weight, for example, of 100 pounds to <lb/>the height of one foot, can life up one of <lb/>200 pounds to the height of half a foot, <lb/>or one of 400 pounds to the height of a <lb/>fourth part of a foot, and &longs;o of the re&longs;t, <lb/>be there never &longs;o much applyed to it: and <lb/>this Principle cannot be denied if we con&longs;ider, that the Effect <lb/>ought to be proportioned to the Action that is nece&longs;&longs;ary for the <lb/>production of it; &longs;o that, if it be nece&longs;&longs;ary to employ an Action by <lb/>which we may rai&longs;e a Weight of 100 pounds to the height of two <lb/>foot, for to rai&longs;e one &longs;uch to the height of one foot only this &longs;ame <lb/>ought to weigh 200 pounds: for its the &longs;ame thing to rai&longs;e 100 <lb/>pounds to the height of one foot, and again yet another 100 <lb/>pounds to the height of one foot, as to rai&longs;e one of 200 pounds to <lb/>the height of one foot, and the &longs;ame, al&longs;o, as to rai&longs;e 100 pounds <lb/>to the height of two feet.</s></p><p type="main"> <s>Now, the Engines which &longs;erve to make this Application of a <lb/>Force which acteth at a great Space upon a <emph type="italics"/>W<emph.end type="italics"/>eight which it cau­<pb xlink:href="071/01/003.jpg" pagenum="312"/>&longs;eth to be rai&longs;ed by a le&longs;&longs;er, are the Pulley, the Inclined Plane, the <lb/>Wedg, the Cap&longs;ten, or Wheel, the Screw, the Leaver, and &longs;ome <lb/>others, for if we will not apply or compare them one to another, <lb/>we cannot well number more, and if we will apply them we need <lb/>not in&longs;tance in &longs;o many.</s></p><p type="head"> <s>The PVLLEY, <emph type="italics"/>Trochlea.<emph.end type="italics"/></s></p><p type="main"> <s>Let A B C be a Chord put about the Pulley D, to which let <lb/>the Weight E be fa&longs;tned; and fir&longs;t, &longs;uppo&longs;ing that two <lb/>men &longs;u&longs;tain or pull up equally each of them one of the <lb/><figure id="id.071.01.003.1.jpg" xlink:href="071/01/003/1.jpg"/><lb/>ends of the &longs;aid Chord: <lb/>it is manife&longs;t, that if the <lb/>Weight weigheth 200 <lb/>pounds, each of tho&longs;e <lb/>men &longs;hal employ but the <lb/>half thereof, that is to &longs;ay, <lb/>the Force that is requi&longs;ite <lb/>for &longs;u&longs;taining or rai&longs;ing <lb/>of 100 pounds, for each <lb/>of them &longs;hal bear but the <lb/>half of it.</s></p><p type="main"> <s>Afterwards, let us &longs;up­<lb/>po&longs;e that A, one of the <lb/>ends of this Chord, being <lb/>made fa&longs;t to &longs;ome Nail, <lb/>the other C be again &longs;u­<lb/>&longs;tained by a Man; and it <lb/>is manife&longs;t, that this Man in C, needs not (no more than before) <lb/>for the &longs;u&longs;taining the Weight E, more Force than is requi&longs;ite for <lb/>the &longs;u&longs;taining of 100 pounds: becau&longs;e the Nail at A doth the <lb/>&longs;ame Office as the Man which we &longs;uppo&longs;ed there before. </s> <s>In fine, <lb/>let us &longs;uppo&longs;e that this Man in C do pull the Chord to make the <lb/>Weight E to ri&longs;e, and it is manife&longs;t, that if he there employeth <lb/>the Force which is requi&longs;ite for the rai&longs;ing of 100 pounds to the <lb/>height of two feet, he &longs;hall rai&longs;e this Weight E of 200 pounds to <lb/>the height of one foot: for the Chord A B C being doubled, as it <lb/>is, it mu&longs;t be pull'd two feet by the end C, to make the Weight E <lb/>ri&longs;e as much, as if two men did draw it, the one by the end A, <lb/>and the other by the end C, each of them the length of one foot <lb/>only.</s></p><p type="main"> <s>There is alwaies one thing that hinders the exactne&longs;s of the Cal­<lb/>culation, that is the pondero&longs;ity of the Chord or Pulley, and the <lb/>difficulty that we meet with in making the Chord to &longs;lip, and in <lb/>bearing it: but this is very &longs;mall in compari&longs;on of that which <pb xlink:href="071/01/004.jpg" pagenum="313"/>rai&longs;eth it, and cannot be e&longs;timated &longs;ave wthin a &longs;mall matter.</s></p><p type="main"> <s>Moreover, it is nece&longs;&longs;ary to ob&longs;erve, that it is nothing but the <lb/>redoubling of the Chord, and not the Pulley, that cau&longs;eth this <lb/>Force: for if we fa&longs;ten yet another Pulley towards A, about <lb/>which we pa&longs;s the Chord A B C H, there will be required no le&longs;s <lb/>Force to draw H towards K, and &longs;o to lift up the Weight E, than <lb/>there was before to draw C towards G. </s> <s>But if to the&longs;e two Pul­<lb/>leys we add yet another towards D, to which we fa&longs;ten the Weight, <lb/>and in which we make the Chord to run or &longs;lip, ju&longs;t as we did in <lb/>the fir&longs;t, then we &longs;hall need no more Force to lift up this Weight <lb/>of 200 pounds than to lift up 50 pounds without the Pulley: be­<lb/>cau&longs;e that in drawing four feet of Chord we lift it up but one <lb/>foot. </s> <s>And &longs;o in multiplying of the Pulleys one may rai&longs;e the great­<lb/>e&longs;t Weights with the lea&longs;t Forces. </s> <s>It is requi&longs;ite al&longs;o to ob&longs;erve, <lb/>that a little more Force is alwaies nece&longs;&longs;ary for the rai&longs;ing of a <lb/>Weight than for the &longs;u&longs;taining of it: which is the rea&longs;on why I <lb/>have &longs;poken here di&longs;tinctly of the one and of the other.</s></p><p type="head"> <s><emph type="italics"/>The Inclined<emph.end type="italics"/> PLANE.</s></p><p type="main"> <s>If not having more Force than &longs;ufficeth to rai&longs;e 100 pounds, one <lb/>would neverthele&longs;s rai&longs;e this Body F, that weigheth 200 pounds, <lb/>to the height of the Line B A, there needs no more but to draw <lb/>or rowl it along the Inclined Plane C A, which I &longs;uppo&longs;e to be <lb/>twice as long as the Line <lb/><figure id="id.071.01.004.1.jpg" xlink:href="071/01/004/1.jpg"/><lb/>A B, for by this means, <lb/>for to make it arrive at <lb/>the point A, we mu&longs;t <lb/>there employ the Force <lb/>that is nece&longs;&longs;ary for the <lb/>rai&longs;ing 100 pounds twice <lb/>as high, and the more inclined this Plane &longs;hall be made, &longs;o much <lb/>the le&longs;s Force &longs;hall there need to rai&longs;e the Weight F. </s> <s>But yet there <lb/>is to be rebated from this Calculation the difficulty that there is <lb/>in moving the Body F, along the Plane A C, if that Plane were <lb/>laid down upon the Line B C, all the parts of which I &longs;uppo&longs;e to <lb/>be equidi&longs;tant from the Center of the Earth.</s></p><p type="main"> <s>It is true, that this impediment being &longs;o much le&longs;s as the Plane is <lb/>more united, more hard, more even, and more polite; it cannot <lb/>likewi&longs;e be e&longs;timated but by gue&longs;s, and it is not very con&longs;ide­<lb/>rable.</s></p><p type="main"> <s>We need not neither much to regard that the Line B C being a <lb/>part of a Circle that hath the &longs;ame Center with the Earth, the <lb/>Plane A C ought to be (though but very little) curved, and to <lb/>have the Figure of part of a Spiral, de&longs;cribed between two Circles, <pb xlink:href="071/01/005.jpg" pagenum="314"/>which likewi&longs;e have for their Center that of the Earth, for that it <lb/>is not any way &longs;en&longs;ible.</s></p><p type="head"> <s><emph type="italics"/>The<emph.end type="italics"/> WEDGE, <emph type="italics"/>Cuneus.<emph.end type="italics"/></s></p><p type="main"> <s>The Force of the Wedge A B C D is ea&longs;ily under&longs;tood after <lb/>that which hath been &longs;poken above of the Inclined Plane, <lb/>for the Force wherewith we &longs;trike downwards acts as if it <lb/>were to make it move according to the Line B D; and the Wood, <lb/>or other thing and Body that it cleaveth, openeth not, or the <lb/>Weight that it rai&longs;eth doth not ri&longs;e, &longs;ave only according to the <lb/><figure id="id.071.01.005.1.jpg" xlink:href="071/01/005/1.jpg"/><lb/>Line A C, in&longs;omuch that the Force, <lb/>wherewith one driveth or &longs;triketh this <lb/>Wedge, ought to have the &longs;ame Pro­<lb/>portion to the Re&longs;i&longs;tance of this <lb/>Wood or Weight, that A C hath to <lb/>A B. </s> <s>Or el&longs;e again, to be exact, it <lb/>would be convenient that B D were <lb/>a part of a Circle, and A D and <lb/>C D two portions of Spirals that had the &longs;ame Center with the <lb/>Earth, and that the Wedge were of a Matter &longs;o perfectly hard <lb/>and polite, and of &longs;o &longs;mall weight, as that any little Force would <lb/>&longs;uffice to move it.</s></p><p type="head"> <s><emph type="italics"/>The<emph.end type="italics"/> CRANE, <emph type="italics"/>or the<emph.end type="italics"/> CAPSTEN, <lb/><emph type="italics"/>Axis in Peritrochio.<emph.end type="italics"/></s></p><p type="main"> <s>We &longs;ee al&longs;o very ea&longs;ily, that the Force wherewith the Wheel <lb/>A or Cogg B is turned, which make the Axis or Cylinder C <lb/>to move, about which a Chord is rolled, to which the <lb/>Weight D, which we would rai&longs;e, is fa&longs;tned, ought to have the <lb/><figure id="id.071.01.005.2.jpg" xlink:href="071/01/005/2.jpg"/><lb/>&longs;ame proportion to the &longs;aid <lb/>Weight, as the Circumference of <lb/>the Cylinder hath to the Cir­<lb/>cumference of a Circle which <lb/>that Force de&longs;cribeth, or that the <lb/>Diameter of the one hath unto <lb/>the Diameter of the other; for <lb/>that the Circumferences have the <lb/>&longs;ame proportion as the Diame­<lb/>ters: in&longs;omuch that the Cylinder C, having no more but one foot <lb/>in Diameter, if the Wheel AB be &longs;ix feet in its Diameter, and the <lb/>Weight D do weigh 600 pounds, it &longs;hall &longs;uffice that the Force in <lb/>B &longs;hall be capable to rai&longs;e 100 pounds, and &longs;o of others. </s> <s>One may <pb xlink:href="071/01/006.jpg" pagenum="315"/>al&longs;o in&longs;tead of the Chord that rolleth about the Cylinder C, place <lb/>there a &longs;mall Wheel with teeth or Coggs, that may turn another <lb/>greater, and by that means multiply the power of the Force as <lb/>much as one &longs;hall plea&longs;e, without having any thing to deduct of <lb/>the &longs;ame, &longs;ave only the difficulty of moving the Machine, as in the <lb/>others.</s></p><p type="head"> <s><emph type="italics"/>The<emph.end type="italics"/> SCREW, <emph type="italics"/>Cochlea.<emph.end type="italics"/></s></p><p type="main"> <s>When once the Force of the Cap&longs;ten and of the In­<lb/>clined Plane is under&longs;tood, that of the Screw is ea&longs;ie <lb/>to be computed, for it is compo&longs;ed only of a Plane <lb/>much inclined, which windeth about a Cylinder: and if this Plane <lb/>be in &longs;uch manner Inclined, as that the Cylinder ought to make <lb/><emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> ten turns to advance forwards the length of a foot in the <lb/>Screw, and that the bigne&longs;s of the Circumference of the Circle <lb/><figure id="id.071.01.006.1.jpg" xlink:href="071/01/006/1.jpg"/><lb/>which the Force that turneth it <lb/>about doth de&longs;cribe be of ten <lb/>feet; fora&longs;much as ten times ten <lb/>are one hundred, one Man alone <lb/>&longs;hall be able to pre&longs;s as &longs;trongly <lb/>with this In&longs;trument, or Screw, as <lb/>one hundred without it, provided <lb/>alwaies, that we rebate the Force <lb/>that is required to the turning <lb/>of it.</s></p><p type="main"> <s>Now I &longs;peak here of Pre&longs;&longs;ing rather than of Rai&longs;ing, or Remo­<lb/>ving, in regard that it is about this mo&longs;t commonly that the Screw <lb/>is employed, but when we would make u&longs;e of it for the rai&longs;ing of <lb/>Weights, in&longs;tead of making it to advance into a Female Screw, we <lb/>joyn or apply unto it a Wheel of many Coggs, in &longs;uch &longs;ort <lb/>made, that if <emph type="italics"/>v. </s> <s>gr.<emph.end type="italics"/> this <emph type="italics"/>W<emph.end type="italics"/>heel have thirty Coggs, whil&longs;t the Screw <lb/>maketh one entire turn, it &longs;hall not cau&longs;e the <emph type="italics"/>W<emph.end type="italics"/>heel to make more <lb/>than the thirtieth part of a turn, and if the <emph type="italics"/>W<emph.end type="italics"/>eight be fa&longs;tned to <lb/>a Chord that rowling about the Axis of this <emph type="italics"/>W<emph.end type="italics"/>heel &longs;hall rai&longs;e it but <lb/>one foot in the time that the <emph type="italics"/>W<emph.end type="italics"/>heel makes one entire revolution, <lb/>and that the greatne&longs;s of the Circumference of the Circle that is <lb/>de&longs;cribed by the Force that turneth the Screw about be al&longs;o of ten <lb/>&longs;eet, by rea&longs;on that 10 times 30 make 300, one &longs;ingle Man &longs;hall be <lb/>able to rai&longs;e a <emph type="italics"/>W<emph.end type="italics"/>eight of that bigne&longs;s with this In&longs;trument, which <lb/>is called the Perpetual Screw, as would require 300 men with­<lb/>out it.</s></p><p type="main"> <s>Provided, as before, that we thence deduct the difficulty that <lb/>we meet with in turning of it, which is not properly cau&longs;ed by the <lb/>Pondero&longs;ity of the <emph type="italics"/>W<emph.end type="italics"/>eight, but by the Force or Matter of the In­<pb xlink:href="071/01/007.jpg" pagenum="316"/>&longs;trument: which difficulty is more &longs;en&longs;ible in it than in tho&longs;e afore­<lb/>going, fora&longs;much as it hath greater Force.</s></p><p type="head"> <s>The LEAVER, <emph type="italics"/>Vectis.<emph.end type="italics"/></s></p><p type="main"> <s>I Have deferred to &longs;peak of the Leaver until the la&longs;t, in regard <lb/>that it is of all Engines for rai&longs;ing of Weights, the mo&longs;t diffi­<lb/>cult to be explained.</s></p><p type="main"> <s>Let us &longs;uppo&longs;e that C H is a Leaver, in &longs;uch manner &longs;upported <lb/>at the point O, (by means of an Iron Pin that pa&longs;&longs;eth thorow it <lb/>acro&longs;s, or otherwi&longs;e) that it may turn about on this point O, its <lb/>part C de&longs;cribing the Semicircle A B C D E, and its part H the <lb/><figure id="id.071.01.007.1.jpg" xlink:href="071/01/007/1.jpg"/><lb/>Semicircle F G H I K; and that <lb/>the Weight which we would <lb/>rai&longs;e by help of it were in H, <lb/>and the Force in C, the Line <lb/>C O being &longs;uppo&longs;ed triple of <lb/>O H. </s> <s>Then let us con&longs;ider that <lb/>in the Time whil&longs;t the Force <lb/>that moveth this Leaver de&longs;cri­<lb/>beth the whole Semicircle <lb/>A B C D E, and acteth accord­<lb/>ing to the Line A B C D E, al­<lb/>though that the Weight de&longs;cri­<lb/>beth likewi&longs;e the Semicircle <lb/>F G H I K, yet it is not rai&longs;ed to <lb/>the length of this curved Line <lb/>F G H I K, but only to that of the Line F O K; in&longs;omuch that the <lb/>Proportion that the Force which moveth this Weight ought to <lb/>have to its Pondero&longs;ity ought not to be mea&longs;ured by that which is <lb/>between the two Diameters of the&longs;e Circles, or between their two <lb/>Circumferences, as it hath been &longs;aid above of the Wheel, but ra­<lb/>ther by that which is betwixt the Circumference of the greater, <lb/>and the Diameter of the le&longs;&longs;er. </s> <s>Furthermore let us con&longs;ider, that <lb/>there is a nece&longs;&longs;ity that this Force needeth not to be &longs;o great, at <lb/>&longs;uch time as it is near to A, or near to E, for the turning of the <lb/>Leaver, as then when it is near to B, or to D; nor &longs;o great when <lb/>it is near to B or D, as then when it is near to C: of which the rea­<lb/>&longs;on is, that the Weights do there mount le&longs;s: as it is ea&longs;ie to un­<lb/>der&longs;tand, if having &longs;uppo&longs;ed that the Line C O H is parallel to the <lb/>Horizon, and that A O F cutteth it at Right Angles, we take the <lb/>point G equidi&longs;tant from the points F and H, and the point B equi­<lb/>di&longs;tant from A and C; and that having drawn G S perpendicular <lb/>to F O, we ob&longs;erve that the Line F S (which &longs;heweth how much <lb/>the Weight mounteth in the Time that the Force operates along <pb xlink:href="071/01/008.jpg" pagenum="317"/>the Line A B) is much le&longs;&longs;er than the Line S O, which &longs;heweth <lb/>how much it mounteth in the Time that the Force opperates along <lb/>the Line B C.</s></p><p type="main"> <s>And to mea&longs;ure exactly what his Force ought to be in each Point <lb/>of the curved Line A B C D E, it is requi&longs;ite to know that it ope­<lb/>rates there ju&longs;t in the &longs;ame manner as if it drew the Weight along <lb/>a Plane Circularly Inclined, and that the Inclination of each of the <lb/>Points of this circular Plane were to be mea&longs;ured by that of the <lb/>right Line that toucheth the Circle in this Point. </s> <s>As for example, <lb/>when the Force is at the Point B, for to find the proportion that it <lb/>ought to have with the pondero&longs;ity of the Weight which is at that <lb/>time at the Point G, it is nece&longs;&longs;ary to draw the Contingent Line <lb/>G M, and to account that the pondero&longs;ity of the Weight is to the <lb/>Force which is required to draw it along this Plane, and con&longs;e­<lb/>quently to rai&longs;e it, according to the Circle F G H, as the Line G M <lb/>is to SM Again, for as much as B O is triple of O G, the Force <lb/>in B needs to be to the Weight in G but as the third part of the <lb/>Line SM is unto the whole Line G M. </s> <s>In the &longs;elf &longs;ame manner, <lb/>when the Force is at the Point D, to know how much the Weight <lb/>weigheth at I, it is nece&longs;&longs;ary to draw the Contingent Line betwixt <lb/>I and P, and the right Line I N perpendicular upon the Horizon, <lb/>and from the Point P taken at di&longs;cretion in the Line I P, provided <lb/>that it be below the Point I, you mu&longs;t draw P N parallel to the <lb/>&longs;ame Horizon, to the end you may have the proportion that is be­<lb/>twixt the Line I P and the third part of the Line I N, for that which <lb/>betwixt the pondero&longs;ity of the Weight, and the Force that ought to <lb/>be at the Point D for the moving of it: and &longs;o of others. </s> <s>Where, <lb/>neverthele&longs;s, you mu&longs;t except the Point H, at which the Contin­<lb/>gent Line being perpendicular upon the Horizon, the Weight can <lb/>be no other than triple the Force which ought to be in C for the <lb/>moving of it: in the Points F and K, at which the Contingent <lb/>Line being parallel unto the Horizon it &longs;elf, the lea&longs;t Force that <lb/>one can a&longs;&longs;ign is &longs;ufficient to move the Weight. </s> <s>Moreover, that you <lb/>may be perfectly exact, you mu&longs;t ob&longs;erve that the Lines S G and <lb/>P N ought to be parts of a Circle that have for their Center that <lb/>of the Earth; and GM and I P parts of Spirals drawn between two <lb/>&longs;uch Circles; and, la&longs;tly, that the right Lines S M and I N both <lb/>tending towards the Center of the Earth are not exactly Paral­<lb/>lels: and furthermore, that the Point H where I &longs;uppo&longs;e the <lb/>Contingent Line to be perpendicular unto the Horizon ought <lb/>to be &longs;ome &longs;mall matter nearer to the Point F than to K, at the <lb/>which F and K the Contingent Lines are Parallels unto the &longs;aid <lb/>Horizon.</s></p><p type="main"> <s>This done, we may ea&longs;ily re&longs;olve all the difficulties of the Ba­<lb/>lance, and &longs;hew, That then when it is mo&longs;t exact, and for in&longs;tance, <pb xlink:href="071/01/009.jpg" pagenum="318"/>&longs;uppo&longs;ing it's Centre at O by which it is &longs;u&longs;tained to be no more <lb/>but an indivi&longs;ible Point, like as I have &longs;uppo&longs;ed here for the Leaver, <lb/>if the Armes be declined one way or the other, that which &longs;hall be <lb/>the lowermo&longs;t ought evermore to be adjudged the heavier; &longs;o that <lb/>the Centre of Gravity is not &longs;ixed and immoveable in each &longs;everal <lb/>Body, as the Ancients have &longs;uppo&longs;ed, which no per&longs;on, that I <lb/>know of, hath hitherto ob&longs;erved.</s></p><p type="main"> <s>But the&longs;e la&longs;t Con&longs;iderations are of no moment in Practice, and <lb/>it would be good for tho&longs;e who &longs;et them&longs;elves to invent new <lb/>Machines, that they knew nothing more of this bu&longs;i­<lb/>ne&longs;&longs;e than this little which I have now writ thereof, <lb/>for then they would not be in danger of decei­<lb/>ving them&longs;elves in their Computation, <lb/>as they frequently do in &longs;uppo&longs;ing <lb/>other Principles.</s></p><p type="head"> <s><emph type="italics"/>FINIS.<emph.end type="italics"/></s></p><figure id="id.071.01.009.1.jpg" xlink:href="071/01/009/1.jpg"/><pb xlink:href="071/01/010.jpg" pagenum="319"/><p type="head"> <s>A <lb/>LETTER <lb/>OF <lb/>Mon&longs;ieur Des-Cartes <lb/>TO THE <lb/>REVEREND FATHER <lb/><emph type="italics"/>MARIN MERSENNE.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Reverend Father,<emph.end type="italics"/></s></p><p type="main"> <s>I Did think to have deferred writing unto you <lb/>yet eight or fifteen dayes, to the end I might <lb/>not trouble you too often with my Letters, <lb/>but I have received yours of the fir&longs;t of <emph type="italics"/>Sept.<emph.end type="italics"/><lb/>which giveth me to under&longs;tand that it is an <lb/>hard matter to admit the Principle which I <lb/>have &longs;uppo&longs;ed in my Examination of the <lb/>Geo&longs;tatick Que&longs;tion, and in regard that if it <lb/>be not true, all the re&longs;t that I have inferred from it would be yet <lb/>le&longs;&longs;e true: I would not one onely day defer &longs;ending you a more <lb/>particular Explication. </s> <s>It is requi&longs;ite above all things to con&longs;ider <lb/>that I did &longs;peak of the Force that &longs;erveth to rai&longs;e a Weight to &longs;ome <lb/>heighth, the which Force hath evermore two Dimen&longs;ions, and not <lb/>of that which &longs;erveth in each point to &longs;u&longs;tain it, which hath never <lb/>more than one Dimen&longs;ion, in&longs;omuch that the&longs;e two Forces differ <lb/>as much the one from the other, as a Superficies differs from a Line: <lb/>for the &longs;ame Force which a Nail ought to have for the &longs;u&longs;taining of <lb/>a Weight of 100 pound one moment of time, doth al&longs;o &longs;uffice for <lb/>to &longs;u&longs;tain it the &longs;pace of a year, provided that it do not dimini&longs;h, <lb/>but the &longs;ame Quantity of this Force which &longs;erveth to rai&longs;e the <lb/>Weight to the heighth of one foot, &longs;ufficeth not <emph type="italics"/>(eadem numero)<emph.end type="italics"/><lb/>to rai&longs;e it two feet; and it is not more manife&longs;t that two and two <lb/>make four, than it's manife&longs;t that we are to employ double as much <lb/>therein.</s></p><p type="main"> <s>Now, fora&longs;much as that this is nothing but the &longs;ame thing that <lb/>I have &longs;uppo&longs;ed for a Principle, I cannot gue&longs;&longs;e on what the Scruple <lb/>&longs;hould be grounded that men make of receiving it; but I &longs;hall in <pb xlink:href="071/01/011.jpg" pagenum="320"/>this place &longs;peak of all &longs;uch as I &longs;u&longs;pect, which for the mo&longs;t part <lb/>ari&longs;e onely from this, that men are before-hand over-knowing in <lb/>the Mechanicks; that is to &longs;ay, that they are pre-occupied with <lb/>Principles that others prove touching the&longs;e matters, which not being <lb/>ab&longs;olutely true, they deceive the more, the more true they &longs;eem to <lb/>be.</s></p><p type="main"> <s>The fir&longs;t thing wherewith a man may be pre-occupied in this <lb/>bu&longs;ine&longs;&longs;e, is, that they many times confound the Con&longs;ideration of <lb/><figure id="id.071.01.011.1.jpg" xlink:href="071/01/011/1.jpg"/><lb/>Space, with that of Time, or of the Ve­<lb/>locity, &longs;o that, for Example, in the <lb/><emph type="italics"/>L<emph.end type="italics"/>eaver, or (which is the &longs;ame) the Ba­<lb/>llance A B C D having &longs;uppo&longs;ed that <lb/>the Arm A B is double to B C, and the <lb/>Weight in C double to the Weight <lb/>in A, and al&longs;o that they are in <emph type="italics"/>Equilibrium,<emph.end type="italics"/> in&longs;tead of &longs;aying, that <lb/>that which cau&longs;eth this <emph type="italics"/>Equilibrium<emph.end type="italics"/> is, that if the Weight C did <lb/>&longs;u&longs;tain, or was rai&longs;ed up by the Weight A, it did not pa&longs;&longs;e more <lb/>than half &longs;o much Space as it, they &longs;ay that it did move &longs;lower by <lb/>the half: which is a fault &longs;o much the more prejudicial, in that it is <lb/>very difficult to be known: for it is not the difference of <lb/><figure id="id.071.01.011.2.jpg" xlink:href="071/01/011/2.jpg"/><lb/>the Velocity that is the cau&longs;e why the&longs;e Weights are to be <lb/>one double to the other, but the difference of the Space, as <lb/>appeareth by this, that to rai&longs;e, for Example, the Weight F <lb/>with the hand unto G, it is not nece&longs;&longs;ary to employ a Force <lb/>that is preci&longs;ely double to that which one &longs;hould have <lb/>therein employed the fir&longs;t bout, to rai&longs;e it twice as quick­<lb/>ly, but it is requi&longs;ite to employ therein either more or le&longs;s <lb/>than the double, according to the different proportion that <lb/>this Velocity may have unto the Cau&longs;es that re&longs;i&longs;t it.</s></p><p type="main"> <s>In&longs;tead of requiring a Force ju&longs;t double for the rai&longs;ing of it with <lb/>the &longs;ame Velocity twice as high, unto H, I &longs;ay that it is ju&longs;t dou­<lb/>ble in counting (as two and two make four) that one and one make <lb/>two, for it is requi&longs;ite to employ a certain quantity of this Force <lb/>to rai&longs;e the Weight from F to G, and again al&longs;o, as much more of <lb/>the &longs;ame Force to rai&longs;e it from G to H.</s></p><p type="main"> <s>For if I had had a mind to have joyned the Con&longs;ideration of the <lb/>Velocity with that of the Space, it had been nece&longs;&longs;ary to have <lb/>a&longs;&longs;igned three Dimen&longs;ions to the Force, whereas I have a&longs;&longs;igned it <lb/>no more but two, on purpo&longs;e to exclude it. </s> <s>And if I have te&longs;tified <lb/>that there is &longs;o little of worth in any part of this &longs;mall Tract of the <lb/>Staticks, yet I de &longs;ire that men &longs;hould know, that there is more in <lb/>this alone than in all the re&longs;t: for it's impo&longs;&longs;ible to &longs;ay any thing <lb/>that is good and &longs;olid touching Velocity, without having rightly <lb/>explained what we are to under&longs;tand by Gravity, as al&longs;o the whole <lb/>Sy&longs;teme of the World. </s> <s>Now becau&longs;e I would not under take it, <pb xlink:href="071/01/012.jpg" pagenum="321"/>I have thought good to omit this Con&longs;ideration, and in this manner <lb/>to &longs;ingle out the&longs;e others that I could explain without it: for <lb/>though there be no Motion but hath &longs;ome Velocity, neverthele&longs;s <lb/>it is onely the Augmentations and Diminutions of this Velocity <lb/>that are con&longs;iderable. </s> <s>And now that &longs;peaking of the Motion of a <lb/>Body, we &longs;uppo&longs;e that it is made according to the Velocity which <lb/>is mo&longs;t naturall to it, which is the &longs;ame as if we did not con&longs;ider it <lb/>at all.</s></p><p type="main"> <s>The other rea&longs;on that may have hindred men from rightly un­<lb/>der&longs;tanding my Principle is, that they have thought that they could <lb/>demon&longs;trate without it &longs;ome of tho&longs;e things which I demon&longs;trate <lb/>not without it: As, for example, touching the Pulley A B C, they <lb/>have thought that it was enough to know that the Nail in A did <lb/><figure id="id.071.01.012.1.jpg" xlink:href="071/01/012/1.jpg"/><lb/>&longs;u&longs;tain the half of the Weight B; to conclude <lb/>that the Hand in C had need but of half &longs;o much <lb/>Force to &longs;u&longs;tain or rai&longs;e the Weight, thus wound <lb/>about the Pulley, as it would need for to &longs;u&longs;tain <lb/>or rai&longs;e it without it. </s> <s>But howbeit that this ex­<lb/>plaineth very well, how the application of the <lb/>Force at C is made unto a Weight double to that <lb/>which it could rai&longs;e without a Pulley, and that I <lb/>my &longs;elf did make u&longs;e thereof, yet I deny that <lb/>this is &longs;imply, becau&longs;e that that the Nail A &longs;u­<lb/>&longs;taineth one part of the Weight B, that the Force <lb/>in C, which &longs;u&longs;taineth it, might be le&longs;s than if it <lb/>had been &longs;o &longs;u&longs;tained. </s> <s>For if that had been true, the Rope C E be­<lb/>ing wound about the Pulley D, the Force in E might by the &longs;ame <lb/>rea&longs;on be le&longs;s than the Force in C: for that the Nail A doth not <lb/>&longs;u&longs;tain the Weight le&longs;s than it did before, and that there is al&longs;o <lb/>another Nail that &longs;u&longs;tains it, to wit, that to wich the Pulley D is <lb/>fa&longs;tned. </s> <s>Thus therefore, that we may not be mi&longs;taken in this, that <lb/>the Nail A &longs;u&longs;taineth the half of the Weight B, we ought to con­<lb/>clude no more but this, that by this application the one of the Di­<lb/>men&longs;ions of the Force that ought to be in C <lb/><figure id="id.071.01.012.2.jpg" xlink:href="071/01/012/2.jpg"/><lb/>to rai&longs;e up this Weight is dimini&longs;hed the one <lb/>half; and that the other, of con&longs;equence, be­<lb/>cometh double, in &longs;uch &longs;ort that if the Line <lb/>F G repre&longs;ent the Force that is required for <lb/>the &longs;u&longs;taining the Weight B in a point, with­<lb/>out the help of any Machine, and the <lb/>Quadrangle G H that which is required for <lb/>the rai&longs;ing of it to the height of a foot, the <lb/>&longs;upport of the Nail A dimini&longs;heth the Di­<lb/>men&longs;ion which is repre&longs;ented by the Line F G the one half, and the <lb/>redoubling of the Rope A B C maketh the other Dimen&longs;ion to <pb xlink:href="071/01/013.jpg" pagenum="322"/>double, which is repre&longs;ented by the Line FH; and &longs;o the Force <lb/>that ought to be in C for the rai&longs;ing of the Weight B to the height <lb/>of one foot is repre&longs;ented by the Quadrangle IK; and, as we know <lb/>in Geometry, that a Line being added to, or taken from a Superfi­<lb/>cies, neither augmenteth, nor dimini&longs;heth it in the lea&longs;t, &longs;o the <lb/>Force where with the Nail A &longs;u&longs;tains the Weight B, having but one <lb/>&longs;ole Dimen&longs;ion, cannot cau&longs;e that the Force in C, con&longs;idered ac­<lb/>cording to its two Dimen&longs;ions, ought to be le&longs;s for the rai&longs;ing in <lb/>like manner the Weight E, than for the rai&longs;ing it without any <lb/>Pulley.</s></p><p type="main"> <s>The third thing which may make men imagine &longs;ome Ob&longs;curity <lb/>in my Principle is, that they, it may be, have not had regard to all <lb/>the words by which I explain it; for I do not &longs;ay &longs;imply that the <lb/>Force that can rai&longs;e a Weight of 50 pounds to the height of four <lb/>feet can rai&longs;e one of 200 pounds to the height of one foot; but I <lb/>&longs;ay that it may do it, if &longs;o be that it be applyed to it: now it is <lb/>impo&longs;&longs;ible to apply the &longs;ame thereto, but by the means of &longs;ome Ma­<lb/>chine, or other Invention that &longs;hall cau&longs;e this Weight to a&longs;cend <lb/>but one, in the time whil&longs;t the Force pa&longs;&longs;eth the whole length <lb/>of four feet, and &longs;o that it do transform the Quandrangle, by <lb/>which the Force is repre&longs;ented that is required to rai&longs;e this <lb/>Weight of 400 pounds to the height of one foot into another <lb/>that is equall and like to that which repre&longs;ents the Force that is <lb/>required for to rai&longs;e a Weight of 50 pounds to the height of four <lb/>feet.</s></p><p type="main"> <s>In fine, it may be that men may have thought the wor&longs;e of my <lb/>Principle, becau&longs;e they have imagined that I have alledged the Ex­<lb/>amples of the Pulley, of the Inclined Plane, and of the Leaver, to <lb/>the end that I might better per&longs;warde the truth thereof, as if it had <lb/>been dubious, or el&longs;e that I had &longs;o ill di&longs;cour&longs;ed as to offer to a&longs;&longs;ume <lb/>from thence a Principle, which ought of it felf to be &longs;o clear, as not <lb/>to need any proof by things that are &longs;o difficult to comprehend as <lb/>that; it may be, they have never been well demon&longs;trated by any <lb/>man: but neither have I made u&longs;e of them, &longs;ave only with a de&longs;ign <lb/>to &longs;hew that this Principle extends it &longs;elf to all matters of which <lb/>one treateth in the Staticks: or, rather, I have made u&longs;e of this oc­<lb/>ca&longs;ion for to in&longs;ert them into my Treati&longs;e, for that I conceived <lb/>that it would have been too dry and barren if I had therein &longs;po­<lb/>ken of nothing el&longs;e but of this Que&longs;tion, that is of no u&longs;e, as of <lb/>that of the Geo&longs;taticks, which I purpo&longs;ed to examine.</s></p><p type="main"> <s>Now one may perceive, by what hath already been &longs;aid, how <lb/>the Forces of the Leaver and Pulley are demon&longs;trated by my <lb/>Principle &longs;o well, that there only remains the Inclined Plane, of <lb/>which you &longs;hall clearly &longs;ee the Demon&longs;tration by this Figure; in <lb/>which G F repre&longs;ents the fir&longs;t Dimen&longs;ion of the Force that the <pb xlink:href="071/01/014.jpg" pagenum="323"/>Rectangle F H de&longs;cribeth whil&longs;t it draweth the Weight D along <lb/>the Plane B A, by the means of a Chord parallel to this Plane, and <lb/>pa&longs;&longs;ing about the Pulley E, in &longs;uch &longs;ort, that H G, that is the height <lb/>of this Rectangle, is equal to B A, along which the Weight D is to <lb/>move, whil&longs;t it mounteth to the height of the Line C A. </s> <s>And N O <lb/>repre&longs;ents the fir&longs;t Dimen&longs;ion of &longs;uch another Force, that is de­<lb/>&longs;cribed by the Rectan­<lb/>gle N P, in the time that <lb/><figure id="id.071.01.014.1.jpg" xlink:href="071/01/014/1.jpg"/><lb/>it is rai&longs;ing the Weight <lb/>L to M. </s> <s>And I &longs;uppo&longs;e <lb/>that L M is equal to B A, <lb/>or double to C A; and <lb/>that N O is to F G, as <lb/>O P is to G H. </s> <s>This <lb/>done, I con&longs;ider that at <lb/>&longs;uch time as the Weight <lb/>D is moved from B to­<lb/>wards A, one may ima­<lb/>gine its Motion to be <lb/>compo&longs;ed of two others, of which the one carrieth it from B R to­<lb/>wards C A, (to which operation there is no Force required, as all <lb/>tho&longs;e &longs;uppo&longs;e who treat of the Mechanicks) and the other rai&longs;eth <lb/>it from B C towards R A, for which alone the Force is required: <lb/>in&longs;omuch that it needs neither more nor le&longs;s Force to move it <lb/>along the Inclined Plane B A, than along the Perpendicular C A. <lb/></s> <s>For I &longs;uppo&longs;e that the unevenne&longs;&longs;es, <emph type="italics"/>&c.<emph.end type="italics"/> of the Plane do not <lb/>at all hinder it, like as it is alwaies &longs;uppo&longs;ed in treating of this <lb/>matter.</s></p><p type="main"> <s>So then the whole Force F H is employed only about the rai&longs;ing <lb/>of D to the height of C A: and fora&longs;much as it is exactly equal to <lb/>the Force N P, that is required for the rai&longs;ing of L to the Height <lb/>of L M, double to C A, I conclude by my Principle that the <lb/>Weight D is double to the Weight L. </s> <s>For in regard that it is <lb/>nece&longs;&longs;ary to employ as much Force for the one as for the other, <lb/>there is as much to be rai&longs;ed in the one as in the other; and no <lb/>more knowledge is required than to count unto two for the <lb/>knowing that it is alike facile to rai&longs;e 200 pounds from C to A, <lb/>as to rai&longs;e 100 pounds from L to M: &longs;ince that L M is double <lb/>to C A.</s></p><p type="main"> <s>You tell me, moreover, that I ought more particularly to ex­<lb/>plain the nature of the Spiral Line that repre&longs;enteth the Plane <lb/>equally enclined, which hath many qualities that render it &longs;uffi­<lb/>ciently knowable.</s></p><pb xlink:href="071/01/015.jpg" pagenum="324"/><p type="main"> <s>For if A be the Center of the Earth, <lb/><figure id="id.071.01.015.1.jpg" xlink:href="071/01/015/1.jpg"/><lb/>and A N B C D the Spiral Line, having <lb/>drawn the Right Lines A B, A D, and the <lb/>like, there is the &longs;ame proportion betwixt <lb/>the Curved Line A N B and the Right Line <lb/>AB, as is betwixt the Curved Line A N B C, <lb/>and the Right Line A C; or betwixt <lb/>A N B C D and A D: and &longs;o of the <lb/>re&longs;t.</s></p><p type="main"> <s>And if one draw the Tangents D E, C F, <lb/>and B G, the Angles A D E, A C F, A B G, &c. <lb/></s> <s>&longs;hall be equal. </s> <s>As for the re&longs;t I will, &c.----</s></p><p type="main"> <s>Reverend Father,</s></p><p type="main"> <s>Your very humble Servant</s></p><p type="main"> <s><emph type="italics"/>DES-CARTES.<emph.end type="italics"/></s></p></chap> </body> <back></back> </text></archimedes>