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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Thu, 02 May 2013 11:14:40 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" ><archimedes> <info> <author>Roberval, Gilles Personne de</author> <title>Letter to Fermat</title> <date>1665</date> <place>London</place> <translator>Thomas Salusbury</translator> <lang>en</lang> <cvs_file>rober_ferma_072_en_1665.xml</cvs_file> <cvs_version></cvs_version> <locator>072.xml</locator> </info> <text> <front> </front> <body> <chap> <pb xlink:href="072/01/001.jpg"/><pb xlink:href="072/01/002.jpg" pagenum="325"/><p type="head"> <s>A <lb/>LETTER <lb/>OF <lb/>Mon&longs;ieur de Robberval <lb/>TO <lb/>Mon&longs;ieur de Fermates, <lb/>Coun&longs;ellour of <emph type="italics"/>THOULOUSE,<emph.end type="italics"/><lb/>Containing certain Propo&longs;itions in the <lb/>MECHANICKS.</s></p><p type="main"> <s>MONSIEUR,</s></p><p type="main"> <s>I have, according to my promi&longs;e, &longs;ent you the <lb/>Demon&longs;tration of the Fundamental Propo&longs;i­<lb/>tion of our Mechanicks, in which I follow the <lb/>common method of explaining, in the fir&longs;t <lb/>place, the Definitions and Principles of which <lb/>we make u&longs;e.</s></p><p type="main"> <s>We in general call that Quality a Force or <lb/>Power, by means of which any thing whatever <lb/>doth tend or a&longs;pire into another place than that in which it is, be it <lb/>downwards, upwards, or &longs;ide waies, whether this Quality naturally <lb/>belongeth to the Body, or be communicated to it from without. <lb/></s> <s>From which definition it followeth, that all Weights are a &longs;pecies <lb/>of Force, in regard that it is a Quality, by means whereof Bodies <lb/>do tend downwards. </s> <s>We often al&longs;o a&longs;&longs;ign the name of Force to <lb/>that very thing to which the Force belongeth, as a ponderous Bo­<lb/>dy is called a Weight, but with this pre-caution, that this is in re­<lb/>ference to the true Force, the which augmenting or dimini&longs;hing <lb/>&longs;hall be called a greater or le&longs;&longs;er Force, albeit that the thing to <lb/>which it belongeth do remain alwaies the &longs;ame.</s></p><p type="main"> <s>If a Force be &longs;u&longs;pended or fa&longs;tned to a Flexible Line that is <lb/>without Gravity, and that is made fa&longs;t by one end unto &longs;ome <emph type="italics"/>Ful­<lb/>ciment<emph.end type="italics"/> or &longs;tay, in &longs;uch &longs;ort as that it &longs;u&longs;tain the Force, drawing <pb xlink:href="072/01/003.jpg" pagenum="326"/>without impediment by this Line, the Force and the Line &longs;hall <lb/>take &longs;ome certain po&longs;ition in which they &longs;hall re&longs;t, and the Line <lb/>&longs;hall of nece&longs;&longs;ity be &longs;treight, let that Line be termed <emph type="italics"/>the Pendant,<emph.end type="italics"/><lb/>or <emph type="italics"/>Line of Direction of the Force.<emph.end type="italics"/> And let the Point by which it is <lb/>fa&longs;tned to the Fulciment be called <emph type="italics"/>the Point of Su&longs;pen&longs;ion<emph.end type="italics"/>: which <lb/>may &longs;ometimes be the Arm of a Leaver or Ballance; and then let <lb/>the Line drawn from the Center of the Fulciment of the Leaver <lb/>or Ballance to the Point of Su&longs;pen&longs;ion be named <emph type="italics"/>the Di&longs;tance<emph.end type="italics"/> or <lb/><emph type="italics"/>the Arm of the Force<emph.end type="italics"/>: which we &longs;uppo&longs;e to be a Line fixed, and <lb/>con&longs;idered without Gravity. </s> <s>Moreover, let the Angle comprehen­<lb/>ded betwixt the Arm of the Force and the Line of Direction be <lb/>termed <emph type="italics"/>the Angle of the Direction of the Force.<emph.end type="italics"/></s></p><p type="head"> <s>AXIOM I.</s></p><p type="main"> <s>After the&longs;e Definitions we lay down for a Principle, that in the <lb/>Leaver, and in the Ballance, Equal Forces drawing by Arms <lb/>that are equal, and at equall Angles of Direction, do draw equal­<lb/>ly. </s> <s>And if in this Po&longs;ition they draw one again&longs;t the other they <lb/>&longs;hall make an <emph type="italics"/>Equilibrium<emph.end type="italics"/>: but if they draw together, or towards <lb/>the &longs;ame part, the Effect &longs;hall be double.</s></p><p type="main"> <s>If the Forces being equal, and the Augles of Direction al&longs;o <lb/>equal, the Arms be unequal, the Force that &longs;hall be &longs;u&longs;pended at <lb/>the greater Arm &longs;hall work the greater Effect.</s></p><p type="main"> <s>As in this Figure, the Center of the Ballance or Leaver being A, <lb/><figure id="id.072.01.003.1.jpg" xlink:href="072/01/003/1.jpg"/><lb/>if the Arms A B and A C are equal, <lb/>as al&longs;o the Angles A B D, and A C E, <lb/>the equal Forces D and E &longs;hall <lb/>draw equally, and make an <emph type="italics"/>Equili­<lb/>brium.<emph.end type="italics"/> So likewi&longs;e the Arm A F be­<lb/>ing equal to A B, the Angle A F G <lb/>to the Angle A B D, and the Force <lb/>G to D, the&longs;e two Forces ^{*} G and D <lb/><arrow.to.target n="marg1124"></arrow.to.target><lb/>draw equally; and in regard that <lb/>they draw both one way, the Effect <lb/>&longs;hall be double.</s></p><p type="margin"> <s><margin.target id="marg1124"></margin.target>* In the M. S. <lb/></s> <s>Copy it is <emph type="italics"/>C and <lb/>D.<emph.end type="italics"/></s></p><p type="main"> <s>In the &longs;ame manner the Forces G and E &longs;hall make an <emph type="italics"/>Equilibri­<lb/>um<emph.end type="italics"/>; as al&longs;o I and L &longs;hall counterpoi&longs;e, if (being equal) the Arms <lb/>A K and A H, and the Angles A H T, and A K L be equal.</s></p><p type="main"> <s>The &longs;ame &longs;hall befall in the Forces P and R, if all things be <lb/>di&longs;po&longs;ed as before. </s> <s>And in this ca&longs;e we make no other di&longs;tinction <lb/>betwixt Weights and other Forces &longs;ave only this, that Weights all <lb/>tend towards the Center of Grave Bodies, and Forces may be un­<lb/>der&longs;tood to tend all towards all parts of the Univer&longs;e, with &longs;o <lb/>much greater or le&longs;&longs;er <emph type="italics"/>Impetus<emph.end type="italics"/> than Weights. </s> <s>So that Weights and <pb xlink:href="072/01/004.jpg" pagenum="327"/>their parts do draw by Lines of Direction, which all concur in one <lb/>and the &longs;ame Point; and Forces and their parts may be under&longs;tood <lb/>to draw in &longs;uch &longs;ort that all the Lines of Direction are parallel to <lb/>each other.</s></p><p type="head"> <s>AXIOM II.</s></p><p type="main"> <s>In the &longs;econd place, we &longs;uppo&longs;e that a Force and its Line of Di­<lb/>rection abiding alwaies in the &longs;ame po&longs;ition, as al&longs;o the Center <lb/>of the Ballance or Leaver, be the Arm what it will that is drawn <lb/>from the Center of the Ballance to the Line of Direction, the <lb/>Force drawing alwaies in the &longs;ame fa&longs;hion, will alwaies produce <lb/>the &longs;ame Effect.</s></p><p type="main"> <s>As, in this &longs;econd Figure, the Center of the Ballance being A, <lb/>the Force B, and the Line of Direction <lb/><figure id="id.072.01.004.1.jpg" xlink:href="072/01/004/1.jpg"/><lb/>B <emph type="italics"/>F<emph.end type="italics"/> prolonged, as occa&longs;ion &longs;hall re­<lb/>quire, in which the Arms A G, A C, and <lb/>A <emph type="italics"/>F<emph.end type="italics"/> do determine, in this po&longs;ition let <lb/>the Line B <emph type="italics"/>F<emph.end type="italics"/> be fa&longs;tned to the Arm <lb/>A <emph type="italics"/>F,<emph.end type="italics"/> or A C, or to another Arm drawn <lb/>from the Center A to the Line of Di­<lb/>rection ^{*} B <emph type="italics"/>F<emph.end type="italics"/>: we &longs;uppo&longs;e that this <lb/><arrow.to.target n="marg1125"></arrow.to.target><lb/><emph type="italics"/>F<emph.end type="italics"/>orce B &longs;hall alwaies work the &longs;ame <lb/>Effect upon the Ballance. </s> <s>And if <lb/>drawing by the Arm A C it make an <lb/><emph type="italics"/>Equilibrium<emph.end type="italics"/> with the <emph type="italics"/>F<emph.end type="italics"/>orce <emph type="italics"/>D<emph.end type="italics"/> drawing by the <emph type="italics"/>A<emph.end type="italics"/>rm <emph type="italics"/>A<emph.end type="italics"/> E, when <lb/>ever it &longs;hall draw by the <emph type="italics"/>A<emph.end type="italics"/>rms <emph type="italics"/>A F<emph.end type="italics"/> or <emph type="italics"/>A<emph.end type="italics"/> G, it &longs;hall likewi&longs;e make <lb/>an <emph type="italics"/>Equilibrium<emph.end type="italics"/> with the <emph type="italics"/>F<emph.end type="italics"/>orce D drawing by the <emph type="italics"/>A<emph.end type="italics"/>rm <emph type="italics"/>A E.<emph.end type="italics"/> This <lb/>Principle although it be not expre&longs;ly found in <emph type="italics"/>A<emph.end type="italics"/>uthors, yet it is <lb/>tacitly &longs;uppo&longs;ed by all tho&longs;e that have writ on this <emph type="italics"/>A<emph.end type="italics"/>rgument, and <lb/>Experience con&longs;tantly confirmeth it.</s></p><p type="margin"> <s><margin.target id="marg1125"></margin.target>* In the Original <lb/>it is writ, but by <lb/>the mi&longs;take of <lb/>the Tran&longs;criber, <lb/><emph type="italics"/>a la ligue de<emph.end type="italics"/> di­<lb/>rection A F.</s></p><p type="head"> <s>AXIOM III.</s></p><p type="main"> <s>I<emph type="italics"/>f<emph.end type="italics"/> the Arms of a Ballance or Leaver are directly placed the one to <lb/>the other, and that being equal they &longs;u&longs;tain equal <emph type="italics"/>F<emph.end type="italics"/>orces, of which <lb/>the Angles of Direction are Right An­<lb/><figure id="id.072.01.004.2.jpg" xlink:href="072/01/004/2.jpg"/><lb/>gles, the&longs;e <emph type="italics"/>F<emph.end type="italics"/>orces do alwaies weigh <lb/>equally upon the Center of the Bal­<lb/>lance, whether that they be near to the <lb/>&longs;ame Center, or far di&longs;tant, or both <lb/>conjoyned in the Center it &longs;elf; as in <lb/>this <emph type="italics"/>F<emph.end type="italics"/>igure the Ballance being E D, <lb/>the Center A, the equal Arms A D <lb/>and <emph type="italics"/>A<emph.end type="italics"/> E, let us &longs;u&longs;tain equal <emph type="italics"/>F<emph.end type="italics"/>orces H and I, of which the <emph type="italics"/>A<emph.end type="italics"/>ngles <pb xlink:href="072/01/005.jpg" pagenum="328"/>of Direction <emph type="italics"/>A<emph.end type="italics"/> D H and <emph type="italics"/>A<emph.end type="italics"/> E I are Right <emph type="italics"/>A<emph.end type="italics"/>ngles, we &longs;uppo&longs;e that <lb/>the&longs;e two <emph type="italics"/>F<emph.end type="italics"/>orces I and H weigh alike upon the Center <emph type="italics"/>A<emph.end type="italics"/> as if they <lb/>were nearer to the Center, at the equal Di&longs;tances <emph type="italics"/>A<emph.end type="italics"/> B and A C, <lb/>and we al&longs;o &longs;uppo&longs;e the &longs;ame if the&longs;e very <emph type="italics"/>F<emph.end type="italics"/>orces were &longs;u&longs;pended <lb/>both together in <emph type="italics"/>A,<emph.end type="italics"/> the <emph type="italics"/>A<emph.end type="italics"/>ngles of Directions being &longs;till Right <lb/><emph type="italics"/>A<emph.end type="italics"/>ngles.</s></p><p type="head"> <s>PROPOSITION I.</s></p><p type="main"> <s>The&longs;e Principles agreed upon, we will ea&longs;ily demon&longs;trate, <lb/>in Imitation of <emph type="italics"/>Archimedes,<emph.end type="italics"/> that upon a &longs;traight Balance <lb/>the <emph type="italics"/>F<emph.end type="italics"/>orces, of which and of all their parts the Lines of Dire­<lb/>ction are parallel to one another, and perpendicular to the Balance, <lb/>&longs;hall couuterpoi&longs;e and make an <emph type="italics"/>Equilibrium,<emph.end type="italics"/> when the &longs;aid <emph type="italics"/>F<emph.end type="italics"/>orces <lb/>&longs;hall be to one another in Reciprocal proportion of their Arms, <lb/>which we think to be &longs;o manife&longs;t to you, that we thence &longs;hall de­<lb/>rive the Demon&longs;tration of this Univer&longs;al Propo&longs;ition to which we <lb/>ha&longs;ten.</s></p><p type="head"> <s>PROPOS. II.</s></p><p type="main"> <s>In every Balance or Leaver, if the proportion of the <emph type="italics"/>F<emph.end type="italics"/>orces is <lb/>reciprocal to that of the Perpendicular Lines drawn from the <lb/>Center or Point of the <emph type="italics"/>F<emph.end type="italics"/>ulciment unto the Lines of Direction <lb/>of the <emph type="italics"/>F<emph.end type="italics"/>orces, drawing the one again&longs;t the other, they &longs;hall make <lb/>an <emph type="italics"/>Equilibrium,<emph.end type="italics"/> and drawing on one and the &longs;ame &longs;ide, they &longs;hall <lb/>have a like Effect, that is to &longs;ay, that they &longs;hall have as much <emph type="italics"/>F<emph.end type="italics"/>orce <lb/>the one as the other, to move the Balance.</s></p><p type="main"> <s>In this <emph type="italics"/>F<emph.end type="italics"/>igure let the Center of the Balance be <emph type="italics"/>A,<emph.end type="italics"/> the <emph type="italics"/>A<emph.end type="italics"/>rm <emph type="italics"/>A<emph.end type="italics"/> B, <lb/>bigger than <emph type="italics"/>A<emph.end type="italics"/> C, and fir&longs;t let the <emph type="italics"/>L<emph.end type="italics"/>ines of Direction B D, and E C <lb/>be perpendicular to the <emph type="italics"/>A<emph.end type="italics"/>rms <emph type="italics"/>A<emph.end type="italics"/> B and <emph type="italics"/>A<emph.end type="italics"/> C, by which Lines the <lb/><emph type="italics"/>F<emph.end type="italics"/>orces D and E (which may be made of Weights if one will) do <lb/>draw; and that there is the &longs;ame rate <lb/><figure id="id.072.01.005.1.jpg" xlink:href="072/01/005/1.jpg"/><lb/>of the <emph type="italics"/>F<emph.end type="italics"/>orce D to the Force E as there <lb/>is betwixt the <emph type="italics"/>A<emph.end type="italics"/>rm <emph type="italics"/>A<emph.end type="italics"/> C to the Arm <lb/><emph type="italics"/>A<emph.end type="italics"/> B: the Forces drawing one again&longs;t <lb/>the other, I &longs;ay, that they will make an <lb/><emph type="italics"/>Equilibrium<emph.end type="italics"/> upon the Balance <emph type="italics"/>C<emph.end type="italics"/> A B. <lb/></s> <s>For let the <emph type="italics"/>A<emph.end type="italics"/>rm C <emph type="italics"/>A<emph.end type="italics"/> be prolonged <lb/>unto F, &longs;o as that <emph type="italics"/>A<emph.end type="italics"/>F may be equal to <lb/><emph type="italics"/>A<emph.end type="italics"/> B: and let C <emph type="italics"/>A<emph.end type="italics"/> F be con&longs;idered as a <lb/>&longs;treight Balance, of which let the Center be <emph type="italics"/>A<emph.end type="italics"/>: and let there be <lb/>&longs;uppo&longs;ed two Forces G and H, of which and of all their parts the <lb/>Lines of Direction are parallel to the Line C E, and that the <lb/>Force G be equal to the Force D, and H to E, the one, to wit G, <pb xlink:href="072/01/006.jpg" pagenum="329"/>drawing upon the Arm A <emph type="italics"/>F,<emph.end type="italics"/> and the other, to wit H, upon the Arm <lb/>A C: now, by the fir&longs;t Propo&longs;ition, G and H &longs;hall make an <emph type="italics"/>Equili­<lb/>brium<emph.end type="italics"/> upon the Balance C A F: But, by the fir&longs;t Principle, the Force <lb/>D upon the Arm A B worketh the &longs;ame effect as the Force G on <lb/>the Arm A F: Therefore the Force D upon the Arm A B maketh <lb/>an <emph type="italics"/>Equilibrium<emph.end type="italics"/> with the Force H upon A C: And the Force H <lb/>drawing in the &longs;ame manner upon the Arm <emph type="italics"/>A<emph.end type="italics"/> C as the Force E, by <lb/>the &longs;ame fir&longs;t <emph type="italics"/>A<emph.end type="italics"/>xiom, the Force D upon the Arm <emph type="italics"/>A<emph.end type="italics"/> B &longs;hall make an <lb/><emph type="italics"/>Equilibrium<emph.end type="italics"/> with the Force E upon the Arm <emph type="italics"/>A<emph.end type="italics"/> C.</s></p><p type="main"> <s>Now, in the following Figure, let the Center of the Balance be <lb/><emph type="italics"/>A,<emph.end type="italics"/> the Arms A B and A C, the Lines of Direction B D and C E <lb/>which are not Perpendicular to the Arms, and the Forces D and E <lb/>drawing likewi&longs;e by the Lines of Direction, upon which Perpen­<lb/>diculars are erected unto the Center A, that is A F upon B D, and <lb/>A G upon E C, and that as A F is to A G, &longs;o is the Force E to the <lb/>Force D: which Forces draw one <lb/><figure id="id.072.01.006.1.jpg" xlink:href="072/01/006/1.jpg"/><lb/>again&longs;t the other: I &longs;ay, that they will <lb/>make an <emph type="italics"/>Equilibrium<emph.end type="italics"/> upon the Balance <lb/>C A B: For let the Lines A F and A G <lb/>be under&longs;tood to be the two Arms of <lb/>a Balance G A F, upon which the For­<lb/>ces D and E do draw by the Lines of <lb/>Direction F D and G E: The&longs;e Forces <lb/>&longs;hall make an <emph type="italics"/>Equilibrium,<emph.end type="italics"/> by the fir&longs;t <lb/>part of this &longs;econd Propo&longs;ition; but, by the &longs;econd Axiom, the Force <lb/>D upon the Arm A F hath the &longs;ame Effect as upon the Arm A B: <lb/>Therefore the Force D upon the Arm A B maketh an <emph type="italics"/>Equilibrium<emph.end type="italics"/><lb/>with the Force E upon the Arm A C.</s></p><p type="main"> <s>There are many Ca&longs;es, according to the Series of Perpendicu­<lb/>lars, but it will be ea&longs;ie for you to &longs;ee that they have all but one <lb/>and the &longs;ame Demon&longs;tration.</s></p><p type="main"> <s>It is al&longs;o ea&longs;ie to demon&longs;trate, that if the Forces draw both on <lb/>one &longs;ide they &longs;hall make the &longs;ame Effect one as another, and that <lb/>the Effect of two together &longs;hall be double to that of one alone.</s></p><p type="head"> <s>OF THE <lb/>GEOSTATICKS.</s></p><p type="main"> <s>The Principle which you demand for the <emph type="italics"/>Geo&longs;taticks<emph.end type="italics"/> is, <lb/>That if two equal Weights are conjoyned by a right <lb/>Line fixed and void of Gravity, and that being &longs;o di­<lb/>&longs;po&longs;ed they may de&longs;cend freely, they will never re&longs;t till <lb/>that the middle of the Line, that is the Center of Gravitation of <lb/>the Ancients, unites it &longs;elf to the common Center of Grave Bodies.</s></p><pb xlink:href="072/01/007.jpg" pagenum="330"/><p type="main"> <s>This Principle &longs;eems at the fir&longs;t very plau&longs;ible, but when <lb/>the Que&longs;tion concerneth a Principle, you know what Conditions <lb/>are required to it, that it may be received, the principal of which are <lb/>wanting in the Principle now in controver&longs;ie<emph type="italics"/>: &longs;cil.<emph.end type="italics"/> that we do not <lb/>know what is the radical Cau&longs;e why Grave Bodies de&longs;cend; and <lb/>whence the Original of this Gravity ari&longs;eth: as al&longs;o that we are to­<lb/>tally ignorant of that which would arrive at the Center whither <lb/>Grave Bodies do tend, nor to other places without the Surface of the <lb/>Earth, of which, in regard we inhabit upon it, we have &longs;ome Expe­<lb/>riments upon which we ground our Principles.</s></p><p type="main"> <s>For it may be, that Gravity is a Quality that re&longs;ides in the Body <lb/>it &longs;elf that falleth; it may be that it is in another that attracteth <lb/>that which de&longs;cends, as in the Earth: It may be, and it is very likely <lb/>that it is a Natural Attraction, or a Natural De&longs;ire of two Bodies to <lb/>unite together, as in the Iron and Load&longs;tone, which are &longs;uch, that <lb/>if the Load&longs;tone be &longs;taid, the Iron, if nothing hinder it, will go find <lb/>it out; and if the Iron be &longs;taid the Load&longs;tone will go towards it; <lb/>and if they be both at liberty, they will reciprocally approach one <lb/>another, yet after &longs;uch a fa&longs;hion, that the &longs;tronge&longs;t of the two <lb/>will move the lea&longs;t way.</s></p><p type="main"> <s>If the fir&longs;t be true, according to the common opinion, we &longs;ee not <lb/>how your Principle can &longs;ub&longs;i&longs;t, for Common Sen&longs;e tells us, that in <lb/>whatever place a Weight is, it alwaies weigheth alike, having ever­<lb/>more the &longs;ame Quality that maketh it to weigh, and that then a Bo­<lb/>dy will repo&longs;e at the Common Center of things Grave when the <lb/>parts of the Body which &longs;hall be on each part of the &longs;aid Center <lb/>&longs;hall be of equal Pondero&longs;ity to counterpoi&longs;e one another, without <lb/>having any regard whether they be little or much removed from the <lb/>Center. </s> <s>Since therefore that of the&longs;e three po&longs;&longs;ible Cau&longs;es of Gra­<lb/>vitation, we know not which is the right, nay, that we are not cer­<lb/>tain that it is any of them, it being po&longs;&longs;ibly that there is a fourth <lb/>from which one may draw Conclu&longs;ions very different, it &longs;eemeth to <lb/>me impo&longs;&longs;ible for us to lay down other Principles in this bufine&longs;s <lb/>than tho&longs;e of which we are a&longs;&longs;ured by a continual Experience, and <lb/>a &longs;ound Judgment. </s> <s>As for our parts, we call tho&longs;e Bodies equally <lb/>or unequally Grave which have an equal or unequal Force of mo­<lb/>ving towards the Common Center: and a Body is &longs;aid to have the <lb/>&longs;ame Weight when it alwaies hath this &longs;ame Force: but if this <lb/>Force augmenteth or dimini&longs;heth, then, although it be the &longs;ame Bo­<lb/>dy, we con&longs;ider it no longer as the &longs;ame Weight: Now &longs;ince that <lb/>this hapneth to Bodies that recede or approach to the Common <lb/>Center, this is it which we de&longs;ire to know, but finding nothing that <lb/>giveth me content upon this Subject, I will leave the Que&longs;tion un­<lb/>determined and unde&longs;cribed.</s></p> <p type="head"> <s>>FINIS.</s></p> </chap> </body> <back></back> </text></archimedes>