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	<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>

<pb xlink:href="071/01/002.jpg" pagenum="311"></pb><p type="head">


<s>DISCOURSES <lb></lb>OF THE <lb></lb>MECHANICKS: <lb></lb>A MANVSCRIPT of <lb></lb>Monſieur Des-Cartes.</s></p><p type="head">

<s>The Explication.</s></p><p type="main">

<s><emph type="italics"></emph>Of Engines, by help of which we may raiſe a very great <lb></lb>weight with ſmall ſtrength.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>The Invention of all theſe Engines de<lb></lb>pends upon one ſole Principle, which is, <lb></lb>That the ſame Force that can lift up a <lb></lb>Weight, for example, of 100 pounds to <lb></lb>the height of one foot, can life up one of <lb></lb>200 pounds to the height of half a foot, <lb></lb>or one of 400 pounds to the height of a <lb></lb>fourth part of a foot, and ſo of the reſt, <lb></lb>be there never ſo much applyed to it: and <lb></lb>this Principle cannot be denied if we conſider, that the Effect <lb></lb>ought to be proportioned to the Action that is neceſſary for the <lb></lb>production of it; ſo that, if it be neceſſary to employ an Action by <lb></lb>which we may raiſe a Weight of 100 pounds to the height of two <lb></lb>foot, for to raiſe one ſuch to the height of one foot only this ſame <lb></lb>ought to weigh 200 pounds: for its the ſame thing to raiſe 100 <lb></lb>pounds to the height of one foot, and again yet another 100 <lb></lb>pounds to the height of one foot, as to raiſe one of 200 pounds to <lb></lb>the height of one foot, and the ſame, alſo, as to raiſe 100 pounds <lb></lb>to the height of two feet.</s></p><p type="main">

<s>Now, the Engines which ſerve to make this Application of a <lb></lb>Force which acteth at a great Space upon a <emph type="italics"></emph>W<emph.end type="italics"></emph.end>eight which it cau<pb xlink:href="071/01/003.jpg" pagenum="312"></pb>ſeth to be raiſed by a leſſer, are the Pulley, the Inclined Plane, the <lb></lb>Wedg, the Capſten, or Wheel, the Screw, the Leaver, and ſome <lb></lb>others, for if we will not apply or compare them one to another, <lb></lb>we cannot well number more, and if we will apply them we need <lb></lb>not inſtance in ſo many.</s></p><p type="head">

<s>The PVLLEY, <emph type="italics"></emph>Trochlea.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Let A B C be a Chord put about the Pulley D, to which let <lb></lb>the Weight E be faſtned; and firſt, ſuppoſing that two <lb></lb>men ſuſtain or pull up equally each of them one of the <lb></lb><figure id="id.071.01.003.1.jpg" xlink:href="071/01/003/1.jpg"></figure><lb></lb>ends of the ſaid Chord: <lb></lb>it is manifeſt, that if the <lb></lb>Weight weigheth 200 <lb></lb>pounds, each of thoſe <lb></lb>men ſhal employ but the <lb></lb>half thereof, that is to ſay, <lb></lb>the Force that is requiſite <lb></lb>for ſuſtaining or raiſing <lb></lb>of 100 pounds, for each <lb></lb>of them ſhal bear but the <lb></lb>half of it.</s></p><p type="main">

<s>Afterwards, let us ſup<lb></lb>poſe that A, one of the <lb></lb>ends of this Chord, being <lb></lb>made faſt to ſome Nail, <lb></lb>the other C be again ſu<lb></lb>ſtained by a Man; and it <lb></lb>is manifeſt, that this Man in C, needs not (no more than before) <lb></lb>for the ſuſtaining the Weight E, more Force than is requiſite for <lb></lb>the ſuſtaining of 100 pounds: becauſe the Nail at A doth the <lb></lb>ſame Office as the Man which we ſuppoſed there before. </s>

<s>In fine, <lb></lb>let us ſuppoſe that this Man in C do pull the Chord to make the <lb></lb>Weight E to riſe, and it is manifeſt, that if he there employeth <lb></lb>the Force which is requiſite for the raiſing of 100 pounds to the <lb></lb>height of two feet, he ſhall raiſe this Weight E of 200 pounds to <lb></lb>the height of one foot: for the Chord A B C being doubled, as it <lb></lb>is, it muſt be pull&#039;d two feet by the end C, to make the Weight E <lb></lb>riſe as much, as if two men did draw it, the one by the end A, <lb></lb>and the other by the end C, each of them the length of one foot <lb></lb>only.</s></p><p type="main">

<s>There is alwaies one thing that hinders the exactneſs of the Cal<lb></lb>culation, that is the ponderoſity of the Chord or Pulley, and the <lb></lb>difficulty that we meet with in making the Chord to ſlip, and in <lb></lb>bearing it: but this is very ſmall in compariſon of that which <pb xlink:href="071/01/004.jpg" pagenum="313"></pb>raiſeth it, and cannot be eſtimated ſave wthin a ſmall matter.</s></p><p type="main">

<s>Moreover, it is neceſſary to obſerve, that it is nothing but the <lb></lb>redoubling of the Chord, and not the Pulley, that cauſeth this <lb></lb>Force: for if we faſten yet another Pulley towards A, about <lb></lb>which we paſs the Chord A B C H, there will be required no leſs <lb></lb>Force to draw H towards K, and ſo to lift up the Weight E, than <lb></lb>there was before to draw C towards G. </s>

<s>But if to theſe two Pul<lb></lb>leys we add yet another towards D, to which we faſten the Weight, <lb></lb>and in which we make the Chord to run or ſlip, juſt as we did in <lb></lb>the firſt, then we ſhall need no more Force to lift up this Weight <lb></lb>of 200 pounds than to lift up 50 pounds without the Pulley: be<lb></lb>cauſe that in drawing four feet of Chord we lift it up but one <lb></lb>foot. </s>

<s>And ſo in multiplying of the Pulleys one may raiſe the great<lb></lb>eſt Weights with the leaſt Forces. </s>

<s>It is requiſite alſo to obſerve, <lb></lb>that a little more Force is alwaies neceſſary for the raiſing of a <lb></lb>Weight than for the ſuſtaining of it: which is the reaſon why I <lb></lb>have ſpoken here diſtinctly of the one and of the other.</s></p><p type="head">

<s><emph type="italics"></emph>The Inclined<emph.end type="italics"></emph.end> PLANE.</s></p><p type="main">

<s>If not having more Force than ſufficeth to raiſe 100 pounds, one <lb></lb>would nevertheleſs raiſe this Body F, that weigheth 200 pounds, <lb></lb>to the height of the Line B A, there needs no more but to draw <lb></lb>or rowl it along the Inclined Plane C A, which I ſuppoſe to be <lb></lb>twice as long as the Line <lb></lb><figure id="id.071.01.004.1.jpg" xlink:href="071/01/004/1.jpg"></figure><lb></lb>A B, for by this means, <lb></lb>for to make it arrive at <lb></lb>the point A, we muſt <lb></lb>there employ the Force <lb></lb>that is neceſſary for the <lb></lb>raiſing 100 pounds twice <lb></lb>as high, and the more inclined this Plane ſhall be made, ſo much <lb></lb>the leſs Force ſhall there need to raiſe the Weight F. </s>

<s>But yet there <lb></lb>is to be rebated from this Calculation the difficulty that there is <lb></lb>in moving the Body F, along the Plane A C, if that Plane were <lb></lb>laid down upon the Line B C, all the parts of which I ſuppoſe to <lb></lb>be equidiſtant from the Center of the Earth.</s></p><p type="main">

<s>It is true, that this impediment being ſo much leſs as the Plane is <lb></lb>more united, more hard, more even, and more polite; it cannot <lb></lb>likewiſe be eſtimated but by gueſs, and it is not very conſide<lb></lb>rable.</s></p><p type="main">

<s>We need not neither much to regard that the Line B C being a <lb></lb>part of a Circle that hath the ſame Center with the Earth, the <lb></lb>Plane A C ought to be (though but very little) curved, and to <lb></lb>have the Figure of part of a Spiral, deſcribed between two Circles, <pb xlink:href="071/01/005.jpg" pagenum="314"></pb>which likewiſe have for their Center that of the Earth, for that it <lb></lb>is not any way ſenſible.</s></p><p type="head">

<s><emph type="italics"></emph>The<emph.end type="italics"></emph.end> WEDGE, <emph type="italics"></emph>Cuneus.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>The Force of the Wedge A B C D is eaſily underſtood after <lb></lb>that which hath been ſpoken above of the Inclined Plane, <lb></lb>for the Force wherewith we ſtrike downwards acts as if it <lb></lb>were to make it move according to the Line B D; and the Wood, <lb></lb>or other thing and Body that it cleaveth, openeth not, or the <lb></lb>Weight that it raiſeth doth not riſe, ſave only according to the <lb></lb><figure id="id.071.01.005.1.jpg" xlink:href="071/01/005/1.jpg"></figure><lb></lb>Line A C, inſomuch that the Force, <lb></lb>wherewith one driveth or ſtriketh this <lb></lb>Wedge, ought to have the ſame Pro<lb></lb>portion to the Reſiſtance of this <lb></lb>Wood or Weight, that A C hath to <lb></lb>A B. </s>

<s>Or elſe again, to be exact, it <lb></lb>would be convenient that B D were <lb></lb>a part of a Circle, and A D and <lb></lb>C D two portions of Spirals that had the ſame Center with the <lb></lb>Earth, and that the Wedge were of a Matter ſo perfectly hard <lb></lb>and polite, and of ſo ſmall weight, as that any little Force would <lb></lb>ſuffice to move it.</s></p><p type="head">

<s><emph type="italics"></emph>The<emph.end type="italics"></emph.end> CRANE, <emph type="italics"></emph>or the<emph.end type="italics"></emph.end> CAPSTEN, <lb></lb><emph type="italics"></emph>Axis in Peritrochio.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>We ſee alſo very eaſily, that the Force wherewith the Wheel <lb></lb>A or Cogg B is turned, which make the Axis or Cylinder C <lb></lb>to move, about which a Chord is rolled, to which the <lb></lb>Weight D, which we would raiſe, is faſtned, ought to have the <lb></lb><figure id="id.071.01.005.2.jpg" xlink:href="071/01/005/2.jpg"></figure><lb></lb>ſame proportion to the ſaid <lb></lb>Weight, as the Circumference of <lb></lb>the Cylinder hath to the Cir<lb></lb>cumference of a Circle which <lb></lb>that Force deſcribeth, or that the <lb></lb>Diameter of the one hath unto <lb></lb>the Diameter of the other; for <lb></lb>that the Circumferences have the <lb></lb>ſame proportion as the Diame<lb></lb>ters: inſomuch that the Cylinder C, having no more but one foot <lb></lb>in Diameter, if the Wheel AB be ſix feet in its Diameter, and the <lb></lb>Weight D do weigh 600 pounds, it ſhall ſuffice that the Force in <lb></lb>B ſhall be capable to raiſe 100 pounds, and ſo of others. </s>

<s>One may <pb xlink:href="071/01/006.jpg" pagenum="315"></pb>alſo inſtead of the Chord that rolleth about the Cylinder C, place <lb></lb>there a ſmall Wheel with teeth or Coggs, that may turn another <lb></lb>greater, and by that means multiply the power of the Force as <lb></lb>much as one ſhall pleaſe, without having any thing to deduct of <lb></lb>the ſame, ſave only the difficulty of moving the Machine, as in the <lb></lb>others.</s></p><p type="head">

<s><emph type="italics"></emph>The<emph.end type="italics"></emph.end> SCREW, <emph type="italics"></emph>Cochlea.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>When once the Force of the Capſten and of the In<lb></lb>clined Plane is underſtood, that of the Screw is eaſie <lb></lb>to be computed, for it is compoſed only of a Plane <lb></lb>much inclined, which windeth about a Cylinder: and if this Plane <lb></lb>be in ſuch manner Inclined, as that the Cylinder ought to make <lb></lb><emph type="italics"></emph>v. </s>

<s>gr.<emph.end type="italics"></emph.end> ten turns to advance forwards the length of a foot in the <lb></lb>Screw, and that the bigneſs of the Circumference of the Circle <lb></lb><figure id="id.071.01.006.1.jpg" xlink:href="071/01/006/1.jpg"></figure><lb></lb>which the Force that turneth it <lb></lb>about doth deſcribe be of ten <lb></lb>feet; foraſmuch as ten times ten <lb></lb>are one hundred, one Man alone <lb></lb>ſhall be able to preſs as ſtrongly <lb></lb>with this Inſtrument, or Screw, as <lb></lb>one hundred without it, provided <lb></lb>alwaies, that we rebate the Force <lb></lb>that is required to the turning <lb></lb>of it.</s></p><p type="main">

<s>Now I ſpeak here of Preſſing rather than of Raiſing, or Remo<lb></lb>ving, in regard that it is about this moſt commonly that the Screw <lb></lb>is employed, but when we would make uſe of it for the raiſing of <lb></lb>Weights, inſtead of making it to advance into a Female Screw, we <lb></lb>joyn or apply unto it a Wheel of many Coggs, in ſuch ſort <lb></lb>made, that if <emph type="italics"></emph>v. </s>

<s>gr.<emph.end type="italics"></emph.end> this <emph type="italics"></emph>W<emph.end type="italics"></emph.end>heel have thirty Coggs, whilſt the Screw <lb></lb>maketh one entire turn, it ſhall not cauſe the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>heel to make more <lb></lb>than the thirtieth part of a turn, and if the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>eight be faſtned to <lb></lb>a Chord that rowling about the Axis of this <emph type="italics"></emph>W<emph.end type="italics"></emph.end>heel ſhall raiſe it but <lb></lb>one foot in the time that the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>heel makes one entire revolution, <lb></lb>and that the greatneſs of the Circumference of the Circle that is <lb></lb>deſcribed by the Force that turneth the Screw about be alſo of ten <lb></lb>ſeet, by reaſon that 10 times 30 make 300, one ſingle Man ſhall be <lb></lb>able to raiſe a <emph type="italics"></emph>W<emph.end type="italics"></emph.end>eight of that bigneſs with this Inſtrument, which <lb></lb>is called the Perpetual Screw, as would require 300 men with<lb></lb>out it.</s></p><p type="main">

<s>Provided, as before, that we thence deduct the difficulty that <lb></lb>we meet with in turning of it, which is not properly cauſed by the <lb></lb>Ponderoſity of the <emph type="italics"></emph>W<emph.end type="italics"></emph.end>eight, but by the Force or Matter of the In<pb xlink:href="071/01/007.jpg" pagenum="316"></pb>ſtrument: which difficulty is more ſenſible in it than in thoſe afore<lb></lb>going, foraſmuch as it hath greater Force.</s></p><p type="head">

<s>The LEAVER, <emph type="italics"></emph>Vectis.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>I Have deferred to ſpeak of the Leaver until the laſt, in regard <lb></lb>that it is of all Engines for raiſing of Weights, the moſt diffi<lb></lb>cult to be explained.</s></p><p type="main">

<s>Let us ſuppoſe that C H is a Leaver, in ſuch manner ſupported <lb></lb>at the point O, (by means of an Iron Pin that paſſeth thorow it <lb></lb>acroſs, or otherwiſe) that it may turn about on this point O, its <lb></lb>part C deſcribing the Semicircle A B C D E, and its part H the <lb></lb><figure id="id.071.01.007.1.jpg" xlink:href="071/01/007/1.jpg"></figure><lb></lb>Semicircle F G H I K; and that <lb></lb>the Weight which we would <lb></lb>raiſe by help of it were in H, <lb></lb>and the Force in C, the Line <lb></lb>C O being ſuppoſed triple of <lb></lb>O H. </s>

<s>Then let us conſider that <lb></lb>in the Time whilſt the Force <lb></lb>that moveth this Leaver deſcri<lb></lb>beth the whole Semicircle <lb></lb>A B C D E, and acteth accord<lb></lb>ing to the Line A B C D E, al<lb></lb>though that the Weight deſcri<lb></lb>beth likewiſe the Semicircle <lb></lb>F G H I K, yet it is not raiſed to <lb></lb>the length of this curved Line <lb></lb>F G H I K, but only to that of the Line F O K; inſomuch that the <lb></lb>Proportion that the Force which moveth this Weight ought to <lb></lb>have to its Ponderoſity ought not to be meaſured by that which is <lb></lb>between the two Diameters of theſe Circles, or between their two <lb></lb>Circumferences, as it hath been ſaid above of the Wheel, but ra<lb></lb>ther by that which is betwixt the Circumference of the greater, <lb></lb>and the Diameter of the leſſer. </s>

<s>Furthermore let us conſider, that <lb></lb>there is a neceſſity that this Force needeth not to be ſo great, at <lb></lb>ſuch time as it is near to A, or near to E, for the turning of the <lb></lb>Leaver, as then when it is near to B, or to D; nor ſo great when <lb></lb>it is near to B or D, as then when it is near to C: of which the rea<lb></lb>ſon is, that the Weights do there mount leſs: as it is eaſie to un<lb></lb>derſtand, if having ſuppoſed that the Line C O H is parallel to the <lb></lb>Horizon, and that A O F cutteth it at Right Angles, we take the <lb></lb>point G equidiſtant from the points F and H, and the point B equi<lb></lb>diſtant from A and C; and that having drawn G S perpendicular <lb></lb>to F O, we obſerve that the Line F S (which ſheweth how much <lb></lb>the Weight mounteth in the Time that the Force operates along <pb xlink:href="071/01/008.jpg" pagenum="317"></pb>the Line A B) is much leſſer than the Line S O, which ſheweth <lb></lb>how much it mounteth in the Time that the Force opperates along <lb></lb>the Line B C.</s></p><p type="main">

<s>And to meaſure exactly what his Force ought to be in each Point <lb></lb>of the curved Line A B C D E, it is requiſite to know that it ope<lb></lb>rates there juſt in the ſame manner as if it drew the Weight along <lb></lb>a Plane Circularly Inclined, and that the Inclination of each of the <lb></lb>Points of this circular Plane were to be meaſured by that of the <lb></lb>right Line that toucheth the Circle in this Point. </s>

<s>As for example, <lb></lb>when the Force is at the Point B, for to find the proportion that it <lb></lb>ought to have with the ponderoſity of the Weight which is at that <lb></lb>time at the Point G, it is neceſſary to draw the Contingent Line <lb></lb>G M, and to account that the ponderoſity of the Weight is to the <lb></lb>Force which is required to draw it along this Plane, and conſe<lb></lb>quently to raiſe it, according to the Circle F G H, as the Line G M <lb></lb>is to SM Again, for as much as B O is triple of O G, the Force <lb></lb>in B needs to be to the Weight in G but as the third part of the <lb></lb>Line SM is unto the whole Line G M. </s>

<s>In the ſelf ſame manner, <lb></lb>when the Force is at the Point D, to know how much the Weight <lb></lb>weigheth at I, it is neceſſary to draw the Contingent Line betwixt <lb></lb>I and P, and the right Line I N perpendicular upon the Horizon, <lb></lb>and from the Point P taken at diſcretion in the Line I P, provided <lb></lb>that it be below the Point I, you muſt draw P N parallel to the <lb></lb>ſame Horizon, to the end you may have the proportion that is be<lb></lb>twixt the Line I P and the third part of the Line I N, for that which <lb></lb>betwixt the ponderoſity of the Weight, and the Force that ought to <lb></lb>be at the Point D for the moving of it: and ſo of others. </s>

<s>Where, <lb></lb>nevertheleſs, you muſt except the Point H, at which the Contin<lb></lb>gent Line being perpendicular upon the Horizon, the Weight can <lb></lb>be no other than triple the Force which ought to be in C for the <lb></lb>moving of it: in the Points F and K, at which the Contingent <lb></lb>Line being parallel unto the Horizon it ſelf, the leaſt Force that <lb></lb>one can aſſign is ſufficient to move the Weight. </s>

<s>Moreover, that you <lb></lb>may be perfectly exact, you muſt obſerve that the Lines S G and <lb></lb>P N ought to be parts of a Circle that have for their Center that <lb></lb>of the Earth; and GM and I P parts of Spirals drawn between two <lb></lb>ſuch Circles; and, laſtly, that the right Lines S M and I N both <lb></lb>tending towards the Center of the Earth are not exactly Paral<lb></lb>lels: and furthermore, that the Point H where I ſuppoſe the <lb></lb>Contingent Line to be perpendicular unto the Horizon ought <lb></lb>to be ſome ſmall matter nearer to the Point F than to K, at the <lb></lb>which F and K the Contingent Lines are Parallels unto the ſaid <lb></lb>Horizon.</s></p><p type="main">

<s>This done, we may eaſily reſolve all the difficulties of the Ba<lb></lb>lance, and ſhew, That then when it is moſt exact, and for inſtance, <pb xlink:href="071/01/009.jpg" pagenum="318"></pb>ſuppoſing it&#039;s Centre at O by which it is ſuſtained to be no more <lb></lb>but an indiviſible Point, like as I have ſuppoſed here for the Leaver, <lb></lb>if the Armes be declined one way or the other, that which ſhall be <lb></lb>the lowermoſt ought evermore to be adjudged the heavier; ſo that <lb></lb>the Centre of Gravity is not ſixed and immoveable in each ſeveral <lb></lb>Body, as the Ancients have ſuppoſed, which no perſon, that I <lb></lb>know of, hath hitherto obſerved.</s></p><p type="main">

<s>But theſe laſt Conſiderations are of no moment in Practice, and <lb></lb>it would be good for thoſe who ſet themſelves to invent new <lb></lb>Machines, that they knew nothing more of this buſi<lb></lb>neſſe than this little which I have now writ thereof, <lb></lb>for then they would not be in danger of decei<lb></lb>ving themſelves in their Computation, <lb></lb>as they frequently do in ſuppoſing <lb></lb>other Principles.</s></p><p type="head">

<s><emph type="italics"></emph>FINIS.<emph.end type="italics"></emph.end></s></p><figure id="id.071.01.009.1.jpg" xlink:href="071/01/009/1.jpg"></figure><pb xlink:href="071/01/010.jpg" pagenum="319"></pb><p type="head">

<s>A <lb></lb>LETTER <lb></lb>OF <lb></lb>Monſieur Des-Cartes <lb></lb>TO THE <lb></lb>REVEREND FATHER <lb></lb><emph type="italics"></emph>MARIN MERSENNE.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Reverend Father,<emph.end type="italics"></emph.end></s></p><p type="main">

<s>I Did think to have deferred writing unto you <lb></lb>yet eight or fifteen dayes, to the end I might <lb></lb>not trouble you too often with my Letters, <lb></lb>but I have received yours of the firſt of <emph type="italics"></emph>Sept.<emph.end type="italics"></emph.end><lb></lb>which giveth me to underſtand that it is an <lb></lb>hard matter to admit the Principle which I <lb></lb>have ſuppoſed in my Examination of the <lb></lb>Geoſtatick Queſtion, and in regard that if it <lb></lb>be not true, all the reſt that I have inferred from it would be yet <lb></lb>leſſe true: I would not one onely day defer ſending you a more <lb></lb>particular Explication. </s>

<s>It is requiſite above all things to conſider <lb></lb>that I did ſpeak of the Force that ſerveth to raiſe a Weight to ſome <lb></lb>heighth, the which Force hath evermore two Dimenſions, and not <lb></lb>of that which ſerveth in each point to ſuſtain it, which hath never <lb></lb>more than one Dimenſion, inſomuch that theſe two Forces differ <lb></lb>as much the one from the other, as a Superficies differs from a Line: <lb></lb>for the ſame Force which a Nail ought to have for the ſuſtaining of <lb></lb>a Weight of 100 pound one moment of time, doth alſo ſuffice for <lb></lb>to ſuſtain it the ſpace of a year, provided that it do not diminiſh, <lb></lb>but the ſame Quantity of this Force which ſerveth to raiſe the <lb></lb>Weight to the heighth of one foot, ſufficeth not <emph type="italics"></emph>(eadem numero)<emph.end type="italics"></emph.end><lb></lb>to raiſe it two feet; and it is not more manifeſt that two and two <lb></lb>make four, than it&#039;s manifeſt that we are to employ double as much <lb></lb>therein.</s></p><p type="main">

<s>Now, foraſmuch as that this is nothing but the ſame thing that <lb></lb>I have ſuppoſed for a Principle, I cannot gueſſe on what the Scruple <lb></lb>ſhould be grounded that men make of receiving it; but I ſhall in <pb xlink:href="071/01/011.jpg" pagenum="320"></pb>this place ſpeak of all ſuch as I ſuſpect, which for the moſt part <lb></lb>ariſe onely from this, that men are before-hand over-knowing in <lb></lb>the Mechanicks; that is to ſay, that they are pre-occupied with <lb></lb>Principles that others prove touching theſe matters, which not being <lb></lb>abſolutely true, they deceive the more, the more true they ſeem to <lb></lb>be.</s></p><p type="main">

<s>The firſt thing wherewith a man may be pre-occupied in this <lb></lb>buſineſſe, is, that they many times confound the Conſideration of <lb></lb><figure id="id.071.01.011.1.jpg" xlink:href="071/01/011/1.jpg"></figure><lb></lb>Space, with that of Time, or of the Ve<lb></lb>locity, ſo that, for Example, in the <lb></lb><emph type="italics"></emph>L<emph.end type="italics"></emph.end>eaver, or (which is the ſame) the Ba<lb></lb>llance A B C D having ſuppoſed that <lb></lb>the Arm A B is double to B C, and the <lb></lb>Weight in C double to the Weight <lb></lb>in A, and alſo that they are in <emph type="italics"></emph>Equilibrium,<emph.end type="italics"></emph.end> inſtead of ſaying, that <lb></lb>that which cauſeth this <emph type="italics"></emph>Equilibrium<emph.end type="italics"></emph.end> is, that if the Weight C did <lb></lb>ſuſtain, or was raiſed up by the Weight A, it did not paſſe more <lb></lb>than half ſo much Space as it, they ſay that it did move ſlower by <lb></lb>the half: which is a fault ſo much the more prejudicial, in that it is <lb></lb>very difficult to be known: for it is not the difference of <lb></lb><figure id="id.071.01.011.2.jpg" xlink:href="071/01/011/2.jpg"></figure><lb></lb>the Velocity that is the cauſe why theſe Weights are to be <lb></lb>one double to the other, but the difference of the Space, as <lb></lb>appeareth by this, that to raiſe, for Example, the Weight F <lb></lb>with the hand unto G, it is not neceſſary to employ a Force <lb></lb>that is preciſely double to that which one ſhould have <lb></lb>therein employed the firſt bout, to raiſe it twice as quick<lb></lb>ly, but it is requiſite to employ therein either more or leſs <lb></lb>than the double, according to the different proportion that <lb></lb>this Velocity may have unto the Cauſes that reſiſt it.</s></p><p type="main">

<s>Inſtead of requiring a Force juſt double for the raiſing of it with <lb></lb>the ſame Velocity twice as high, unto H, I ſay that it is juſt dou<lb></lb>ble in counting (as two and two make four) that one and one make <lb></lb>two, for it is requiſite to employ a certain quantity of this Force <lb></lb>to raiſe the Weight from F to G, and again alſo, as much more of <lb></lb>the ſame Force to raiſe it from G to H.</s></p><p type="main">

<s>For if I had had a mind to have joyned the Conſideration of the <lb></lb>Velocity with that of the Space, it had been neceſſary to have <lb></lb>aſſigned three Dimenſions to the Force, whereas I have aſſigned it <lb></lb>no more but two, on purpoſe to exclude it. </s>

<s>And if I have teſtified <lb></lb>that there is ſo little of worth in any part of this ſmall Tract of the <lb></lb>Staticks, yet I de ſire that men ſhould know, that there is more in <lb></lb>this alone than in all the reſt: for it&#039;s impoſſible to ſay any thing <lb></lb>that is good and ſolid touching Velocity, without having rightly <lb></lb>explained what we are to underſtand by Gravity, as alſo the whole <lb></lb>Syſteme of the World. </s>

<s>Now becauſe I would not under take it, <pb xlink:href="071/01/012.jpg" pagenum="321"></pb>I have thought good to omit this Conſideration, and in this manner <lb></lb>to ſingle out theſe others that I could explain without it: for <lb></lb>though there be no Motion but hath ſome Velocity, nevertheleſs <lb></lb>it is onely the Augmentations and Diminutions of this Velocity <lb></lb>that are conſiderable. </s>

<s>And now that ſpeaking of the Motion of a <lb></lb>Body, we ſuppoſe that it is made according to the Velocity which <lb></lb>is moſt naturall to it, which is the ſame as if we did not conſider it <lb></lb>at all.</s></p><p type="main">

<s>The other reaſon that may have hindred men from rightly un<lb></lb>derſtanding my Principle is, that they have thought that they could <lb></lb>demonſtrate without it ſome of thoſe things which I demonſtrate <lb></lb>not without it: As, for example, touching the Pulley A B C, they <lb></lb>have thought that it was enough to know that the Nail in A did <lb></lb><figure id="id.071.01.012.1.jpg" xlink:href="071/01/012/1.jpg"></figure><lb></lb>ſuſtain the half of the Weight B; to conclude <lb></lb>that the Hand in C had need but of half ſo much <lb></lb>Force to ſuſtain or raiſe the Weight, thus wound <lb></lb>about the Pulley, as it would need for to ſuſtain <lb></lb>or raiſe it without it. </s>

<s>But howbeit that this ex<lb></lb>plaineth very well, how the application of the <lb></lb>Force at C is made unto a Weight double to that <lb></lb>which it could raiſe without a Pulley, and that I <lb></lb>my ſelf did make uſe thereof, yet I deny that <lb></lb>this is ſimply, becauſe that that the Nail A ſu<lb></lb>ſtaineth one part of the Weight B, that the Force <lb></lb>in C, which ſuſtaineth it, might be leſs than if it <lb></lb>had been ſo ſuſtained. </s>

<s>For if that had been true, the Rope C E be<lb></lb>ing wound about the Pulley D, the Force in E might by the ſame <lb></lb>reaſon be leſs than the Force in C: for that the Nail A doth not <lb></lb>ſuſtain the Weight leſs than it did before, and that there is alſo <lb></lb>another Nail that ſuſtains it, to wit, that to wich the Pulley D is <lb></lb>faſtned. </s>

<s>Thus therefore, that we may not be miſtaken in this, that <lb></lb>the Nail A ſuſtaineth the half of the Weight B, we ought to con<lb></lb>clude no more but this, that by this application the one of the Di<lb></lb>menſions of the Force that ought to be in C <lb></lb><figure id="id.071.01.012.2.jpg" xlink:href="071/01/012/2.jpg"></figure><lb></lb>to raiſe up this Weight is diminiſhed the one <lb></lb>half; and that the other, of conſequence, be<lb></lb>cometh double, in ſuch ſort that if the Line <lb></lb>F G repreſent the Force that is required for <lb></lb>the ſuſtaining the Weight B in a point, with<lb></lb>out the help of any Machine, and the <lb></lb>Quadrangle G H that which is required for <lb></lb>the raiſing of it to the height of a foot, the <lb></lb>ſupport of the Nail A diminiſheth the Di<lb></lb>menſion which is repreſented by the Line F G the one half, and the <lb></lb>redoubling of the Rope A B C maketh the other Dimenſion to <pb xlink:href="071/01/013.jpg" pagenum="322"></pb>double, which is repreſented by the Line FH; and ſo the Force <lb></lb>that ought to be in C for the raiſing of the Weight B to the height <lb></lb>of one foot is repreſented by the Quadrangle IK; and, as we know <lb></lb>in Geometry, that a Line being added to, or taken from a Superfi<lb></lb>cies, neither augmenteth, nor diminiſheth it in the leaſt, ſo the <lb></lb>Force where with the Nail A ſuſtains the Weight B, having but one <lb></lb>ſole Dimenſion, cannot cauſe that the Force in C, conſidered ac<lb></lb>cording to its two Dimenſions, ought to be leſs for the raiſing in <lb></lb>like manner the Weight E, than for the raiſing it without any <lb></lb>Pulley.</s></p><p type="main">

<s>The third thing which may make men imagine ſome Obſcurity <lb></lb>in my Principle is, that they, it may be, have not had regard to all <lb></lb>the words by which I explain it; for I do not ſay ſimply that the <lb></lb>Force that can raiſe a Weight of 50 pounds to the height of four <lb></lb>feet can raiſe one of 200 pounds to the height of one foot; but I <lb></lb>ſay that it may do it, if ſo be that it be applyed to it: now it is <lb></lb>impoſſible to apply the ſame thereto, but by the means of ſome Ma<lb></lb>chine, or other Invention that ſhall cauſe this Weight to aſcend <lb></lb>but one, in the time whilſt the Force paſſeth the whole length <lb></lb>of four feet, and ſo that it do transform the Quandrangle, by <lb></lb>which the Force is repreſented that is required to raiſe this <lb></lb>Weight of 400 pounds to the height of one foot into another <lb></lb>that is equall and like to that which repreſents the Force that is <lb></lb>required for to raiſe a Weight of 50 pounds to the height of four <lb></lb>feet.</s></p><p type="main">

<s>In fine, it may be that men may have thought the worſe of my <lb></lb>Principle, becauſe they have imagined that I have alledged the Ex<lb></lb>amples of the Pulley, of the Inclined Plane, and of the Leaver, to <lb></lb>the end that I might better perſwarde the truth thereof, as if it had <lb></lb>been dubious, or elſe that I had ſo ill diſcourſed as to offer to aſſume <lb></lb>from thence a Principle, which ought of it felf to be ſo clear, as not <lb></lb>to need any proof by things that are ſo difficult to comprehend as <lb></lb>that; it may be, they have never been well demonſtrated by any <lb></lb>man: but neither have I made uſe of them, ſave only with a deſign <lb></lb>to ſhew that this Principle extends it ſelf to all matters of which <lb></lb>one treateth in the Staticks: or, rather, I have made uſe of this oc<lb></lb>caſion for to inſert them into my Treatiſe, for that I conceived <lb></lb>that it would have been too dry and barren if I had therein ſpo<lb></lb>ken of nothing elſe but of this Queſtion, that is of no uſe, as of <lb></lb>that of the Geoſtaticks, which I purpoſed to examine.</s></p><p type="main">

<s>Now one may perceive, by what hath already been ſaid, how <lb></lb>the Forces of the Leaver and Pulley are demonſtrated by my <lb></lb>Principle ſo well, that there only remains the Inclined Plane, of <lb></lb>which you ſhall clearly ſee the Demonſtration by this Figure; in <lb></lb>which G F repreſents the firſt Dimenſion of the Force that the <pb xlink:href="071/01/014.jpg" pagenum="323"></pb>Rectangle F H deſcribeth whilſt it draweth the Weight D along <lb></lb>the Plane B A, by the means of a Chord parallel to this Plane, and <lb></lb>paſſing about the Pulley E, in ſuch ſort, that H G, that is the height <lb></lb>of this Rectangle, is equal to B A, along which the Weight D is to <lb></lb>move, whilſt it mounteth to the height of the Line C A. </s>

<s>And N O <lb></lb>repreſents the firſt Dimenſion of ſuch another Force, that is de<lb></lb>ſcribed by the Rectan<lb></lb>gle N P, in the time that <lb></lb><figure id="id.071.01.014.1.jpg" xlink:href="071/01/014/1.jpg"></figure><lb></lb>it is raiſing the Weight <lb></lb>L to M. </s>

<s>And I ſuppoſe <lb></lb>that L M is equal to B A, <lb></lb>or double to C A; and <lb></lb>that N O is to F G, as <lb></lb>O P is to G H. </s>

<s>This <lb></lb>done, I conſider that at <lb></lb>ſuch time as the Weight <lb></lb>D is moved from B to<lb></lb>wards A, one may ima<lb></lb>gine its Motion to be <lb></lb>compoſed of two others, of which the one carrieth it from B R to<lb></lb>wards C A, (to which operation there is no Force required, as all <lb></lb>thoſe ſuppoſe who treat of the Mechanicks) and the other raiſeth <lb></lb>it from B C towards R A, for which alone the Force is required: <lb></lb>inſomuch that it needs neither more nor leſs Force to move it <lb></lb>along the Inclined Plane B A, than along the Perpendicular C A. <lb></lb></s>

<s>For I ſuppoſe that the unevenneſſes, <emph type="italics"></emph>&amp;c.<emph.end type="italics"></emph.end> of the Plane do not <lb></lb>at all hinder it, like as it is alwaies ſuppoſed in treating of this <lb></lb>matter.</s></p><p type="main">

<s>So then the whole Force F H is employed only about the raiſing <lb></lb>of D to the height of C A: and foraſmuch as it is exactly equal to <lb></lb>the Force N P, that is required for the raiſing of L to the Height <lb></lb>of L M, double to C A, I conclude by my Principle that the <lb></lb>Weight D is double to the Weight L. </s>

<s>For in regard that it is <lb></lb>neceſſary to employ as much Force for the one as for the other, <lb></lb>there is as much to be raiſed in the one as in the other; and no <lb></lb>more knowledge is required than to count unto two for the <lb></lb>knowing that it is alike facile to raiſe 200 pounds from C to A, <lb></lb>as to raiſe 100 pounds from L to M: ſince that L M is double <lb></lb>to C A.</s></p><p type="main">

<s>You tell me, moreover, that I ought more particularly to ex<lb></lb>plain the nature of the Spiral Line that repreſenteth the Plane <lb></lb>equally enclined, which hath many qualities that render it ſuffi<lb></lb>ciently knowable.</s></p><pb xlink:href="071/01/015.jpg" pagenum="324"></pb><p type="main">

<s>For if A be the Center of the Earth, <lb></lb><figure id="id.071.01.015.1.jpg" xlink:href="071/01/015/1.jpg"></figure><lb></lb>and A N B C D the Spiral Line, having <lb></lb>drawn the Right Lines A B, A D, and the <lb></lb>like, there is the ſame proportion betwixt <lb></lb>the Curved Line A N B and the Right Line <lb></lb>AB, as is betwixt the Curved Line A N B C, <lb></lb>and the Right Line A C; or betwixt <lb></lb>A N B C D and A D: and ſo of the <lb></lb>reſt.</s></p><p type="main">

<s>And if one draw the Tangents D E, C F, <lb></lb>and B G, the Angles A D E, A C F, A B G, &amp;c. <lb></lb></s>

<s>ſhall be equal. </s>

<s>As for the reſt I will, &amp;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"></emph>DES-CARTES.<emph.end type="italics"></emph.end></s></p></chap>		</body>		<back></back>	</text></archimedes>
author	Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date	Thu, 02 May 2013 11:08:12 +0200
parents	22d6a63640c6
children