Roberval, Gilles Personne de Letter to Fermat 1665 London Thomas Salusbury en rober_ferma_01_en_1665 072.xml

A LETTER OF Mon&longs;ieur de Robberval TO Mon&longs;ieur de Fermates, Coun&longs;ellour of THOULOUSE,Containing certain Propo&longs;itions in the MECHANICKS.

MONSIEUR,

I have, according to my promi&longs;e, &longs;ent you the Demon&longs;tration of the Fundamental Propo&longs;i­tion of our Mechanicks, in which I follow the common method of explaining, in the fir&longs;t place, the Definitions and Principles of which we make u&longs;e.

We in general call that Quality a Force or Power, by means of which any thing whatever doth tend or a&longs;pire into another place than that in which it is, be it downwards, upwards, or &longs;ide waies, whether this Quality naturally belongeth to the Body, or be communicated to it from without. From which definition it followeth, that all Weights are a &longs;pecies of Force, in regard that it is a Quality, by means whereof Bodies do tend downwards. We often al&longs;o a&longs;&longs;ign the name of Force to that very thing to which the Force belongeth, as a ponderous Bo­dy is called a Weight, but with this pre-caution, that this is in re­ference to the true Force, the which augmenting or dimini&longs;hing &longs;hall be called a greater or le&longs;&longs;er Force, albeit that the thing to which it belongeth do remain alwaies the &longs;ame.

If a Force be &longs;u&longs;pended or fa&longs;tned to a Flexible Line that is without Gravity, and that is made fa&longs;t by one end unto &longs;ome Ful­ciment or &longs;tay, in &longs;uch &longs;ort as that it &longs;u&longs;tain the Force, drawing without impediment by this Line, the Force and the Line &longs;hall take &longs;ome certain po&longs;ition in which they &longs;hall re&longs;t, and the Line &longs;hall of nece&longs;&longs;ity be &longs;treight, let that Line be termed the Pendant,or Line of Direction of the Force. And let the Point by which it is fa&longs;tned to the Fulciment be called the Point of Su&longs;pen&longs;ion: which may &longs;ometimes be the Arm of a Leaver or Ballance; and then let the Line drawn from the Center of the Fulciment of the Leaver or Ballance to the Point of Su&longs;pen&longs;ion be named the Di&longs;tance or the Arm of the Force: which we &longs;uppo&longs;e to be a Line fixed, and con&longs;idered without Gravity. Moreover, let the Angle comprehen­ded betwixt the Arm of the Force and the Line of Direction be termed the Angle of the Direction of the Force.

AXIOM I.

After the&longs;e Definitions we lay down for a Principle, that in the Leaver, and in the Ballance, Equal Forces drawing by Arms that are equal, and at equall Angles of Direction, do draw equal­ly. And if in this Po&longs;ition they draw one again&longs;t the other they &longs;hall make an Equilibrium: but if they draw together, or towards the &longs;ame part, the Effect &longs;hall be double.

If the Forces being equal, and the Augles of Direction al&longs;o equal, the Arms be unequal, the Force that &longs;hall be &longs;u&longs;pended at the greater Arm &longs;hall work the greater Effect.

As in this Figure, the Center of the Ballance or Leaver being A,

if the Arms A B and A C are equal, as al&longs;o the Angles A B D, and A C E, the equal Forces D and E &longs;hall draw equally, and make an Equili­brium. So likewi&longs;e the Arm A F be­ing equal to A B, the Angle A F G to the Angle A B D, and the Force G to D, the&longs;e two Forces ^{*} G and D draw equally; and in regard that they draw both one way, the Effect &longs;hall be double.

* In the M. S. Copy it is C and D.

In the &longs;ame manner the Forces G and E &longs;hall make an Equilibri­um; as al&longs;o I and L &longs;hall counterpoi&longs;e, if (being equal) the Arms A K and A H, and the Angles A H T, and A K L be equal.

The &longs;ame &longs;hall befall in the Forces P and R, if all things be di&longs;po&longs;ed as before. And in this ca&longs;e we make no other di&longs;tinction betwixt Weights and other Forces &longs;ave only this, that Weights all tend towards the Center of Grave Bodies, and Forces may be un­der&longs;tood to tend all towards all parts of the Univer&longs;e, with &longs;o much greater or le&longs;&longs;er Impetus than Weights. So that Weights and their parts do draw by Lines of Direction, which all concur in one and the &longs;ame Point; and Forces and their parts may be under&longs;tood to draw in &longs;uch &longs;ort that all the Lines of Direction are parallel to each other.

AXIOM II.

In the &longs;econd place, we &longs;uppo&longs;e that a Force and its Line of Di­rection abiding alwaies in the &longs;ame po&longs;ition, as al&longs;o the Center of the Ballance or Leaver, be the Arm what it will that is drawn from the Center of the Ballance to the Line of Direction, the Force drawing alwaies in the &longs;ame fa&longs;hion, will alwaies produce the &longs;ame Effect.

As, in this &longs;econd Figure, the Center of the Ballance being A, the Force B, and the Line of Direction

B F prolonged, as occa&longs;ion &longs;hall re­quire, in which the Arms A G, A C, and A F do determine, in this po&longs;ition let the Line B F be fa&longs;tned to the Arm A F, or A C, or to another Arm drawn from the Center A to the Line of Di­rection ^{*} B F: we &longs;uppo&longs;e that this Force B &longs;hall alwaies work the &longs;ame Effect upon the Ballance. And if drawing by the Arm A C it make an Equilibrium with the Force D drawing by the Arm A E, when ever it &longs;hall draw by the Arms A F or A G, it &longs;hall likewi&longs;e make an Equilibrium with the Force D drawing by the Arm A E. This Principle although it be not expre&longs;ly found in Authors, yet it is tacitly &longs;uppo&longs;ed by all tho&longs;e that have writ on this Argument, and Experience con&longs;tantly confirmeth it.

* In the Original it is writ, but by the mi&longs;take of the Tran&longs;criber, a la ligue de di­rection A F.

AXIOM III.

If the Arms of a Ballance or Leaver are directly placed the one to the other, and that being equal they &longs;u&longs;tain equal Forces, of which the Angles of Direction are Right An­

gles, the&longs;e Forces do alwaies weigh equally upon the Center of the Bal­lance, whether that they be near to the &longs;ame Center, or far di&longs;tant, or both conjoyned in the Center it &longs;elf; as in this Figure the Ballance being E D, the Center A, the equal Arms A D and A E, let us &longs;u&longs;tain equal Forces H and I, of which the Angles of Direction A D H and A E I are Right Angles, we &longs;uppo&longs;e that the&longs;e two Forces I and H weigh alike upon the Center A as if they were nearer to the Center, at the equal Di&longs;tances A B and A C, and we al&longs;o &longs;uppo&longs;e the &longs;ame if the&longs;e very Forces were &longs;u&longs;pended both together in A, the Angles of Directions being &longs;till Right Angles.

PROPOSITION I.

The&longs;e Principles agreed upon, we will ea&longs;ily demon&longs;trate, in Imitation of Archimedes, that upon a &longs;traight Balance the Forces, of which and of all their parts the Lines of Dire­ction are parallel to one another, and perpendicular to the Balance, &longs;hall couuterpoi&longs;e and make an Equilibrium, when the &longs;aid Forces &longs;hall be to one another in Reciprocal proportion of their Arms, which we think to be &longs;o manife&longs;t to you, that we thence &longs;hall de­rive the Demon&longs;tration of this Univer&longs;al Propo&longs;ition to which we ha&longs;ten.

PROPOS. II.

In every Balance or Leaver, if the proportion of the Forces is reciprocal to that of the Perpendicular Lines drawn from the Center or Point of the Fulciment unto the Lines of Direction of the Forces, drawing the one again&longs;t the other, they &longs;hall make an Equilibrium, and drawing on one and the &longs;ame &longs;ide, they &longs;hall have a like Effect, that is to &longs;ay, that they &longs;hall have as much Force the one as the other, to move the Balance.

In this Figure let the Center of the Balance be A, the Arm A B, bigger than A C, and fir&longs;t let the Lines of Direction B D, and E C be perpendicular to the Arms A B and A C, by which Lines the Forces D and E (which may be made of Weights if one will) do draw; and that there is the &longs;ame rate

of the Force D to the Force E as there is betwixt the Arm A C to the Arm A B: the Forces drawing one again&longs;t the other, I &longs;ay, that they will make an Equilibrium upon the Balance C A B. For let the Arm C A be prolonged unto F, &longs;o as that AF may be equal to A B: and let C A F be con&longs;idered as a &longs;treight Balance, of which let the Center be A: and let there be &longs;uppo&longs;ed two Forces G and H, of which and of all their parts the Lines of Direction are parallel to the Line C E, and that the Force G be equal to the Force D, and H to E, the one, to wit G, drawing upon the Arm A F, and the other, to wit H, upon the Arm A C: now, by the fir&longs;t Propo&longs;ition, G and H &longs;hall make an Equili­brium upon the Balance C A F: But, by the fir&longs;t Principle, the Force D upon the Arm A B worketh the &longs;ame effect as the Force G on the Arm A F: Therefore the Force D upon the Arm A B maketh an Equilibrium with the Force H upon A C: And the Force H drawing in the &longs;ame manner upon the Arm A C as the Force E, by the &longs;ame fir&longs;t Axiom, the Force D upon the Arm A B &longs;hall make an Equilibrium with the Force E upon the Arm A C.

Now, in the following Figure, let the Center of the Balance be A, the Arms A B and A C, the Lines of Direction B D and C E which are not Perpendicular to the Arms, and the Forces D and E drawing likewi&longs;e by the Lines of Direction, upon which Perpen­diculars are erected unto the Center A, that is A F upon B D, and A G upon E C, and that as A F is to A G, &longs;o is the Force E to the Force D: which Forces draw one

again&longs;t the other: I &longs;ay, that they will make an Equilibrium upon the Balance C A B: For let the Lines A F and A G be under&longs;tood to be the two Arms of a Balance G A F, upon which the For­ces D and E do draw by the Lines of Direction F D and G E: The&longs;e Forces &longs;hall make an Equilibrium, by the fir&longs;t part of this &longs;econd Propo&longs;ition; but, by the &longs;econd Axiom, the Force D upon the Arm A F hath the &longs;ame Effect as upon the Arm A B: Therefore the Force D upon the Arm A B maketh an Equilibriumwith the Force E upon the Arm A C.

There are many Ca&longs;es, according to the Series of Perpendicu­lars, but it will be ea&longs;ie for you to &longs;ee that they have all but one and the &longs;ame Demon&longs;tration.

It is al&longs;o ea&longs;ie to demon&longs;trate, that if the Forces draw both on one &longs;ide they &longs;hall make the &longs;ame Effect one as another, and that the Effect of two together &longs;hall be double to that of one alone.

OF THE GEOSTATICKS.

The Principle which you demand for the Geo&longs;taticks is, That if two equal Weights are conjoyned by a right Line fixed and void of Gravity, and that being &longs;o di­&longs;po&longs;ed they may de&longs;cend freely, they will never re&longs;t till that the middle of the Line, that is the Center of Gravitation of the Ancients, unites it &longs;elf to the common Center of Grave Bodies.

This Principle &longs;eems at the fir&longs;t very plau&longs;ible, but when the Que&longs;tion concerneth a Principle, you know what Conditions are required to it, that it may be received, the principal of which are wanting in the Principle now in controver&longs;ie: &longs;cil. that we do not know what is the radical Cau&longs;e why Grave Bodies de&longs;cend; and whence the Original of this Gravity ari&longs;eth: as al&longs;o that we are to­tally ignorant of that which would arrive at the Center whither Grave Bodies do tend, nor to other places without the Surface of the Earth, of which, in regard we inhabit upon it, we have &longs;ome Expe­riments upon which we ground our Principles.

For it may be, that Gravity is a Quality that re&longs;ides in the Body it &longs;elf that falleth; it may be that it is in another that attracteth that which de&longs;cends, as in the Earth: It may be, and it is very likely that it is a Natural Attraction, or a Natural De&longs;ire of two Bodies to unite together, as in the Iron and Load&longs;tone, which are &longs;uch, that if the Load&longs;tone be &longs;taid, the Iron, if nothing hinder it, will go find it out; and if the Iron be &longs;taid the Load&longs;tone will go towards it; and if they be both at liberty, they will reciprocally approach one another, yet after &longs;uch a fa&longs;hion, that the &longs;tronge&longs;t of the two will move the lea&longs;t way.

If the fir&longs;t be true, according to the common opinion, we &longs;ee not how your Principle can &longs;ub&longs;i&longs;t, for Common Sen&longs;e tells us, that in whatever place a Weight is, it alwaies weigheth alike, having ever­more the &longs;ame Quality that maketh it to weigh, and that then a Bo­dy will repo&longs;e at the Common Center of things Grave when the parts of the Body which &longs;hall be on each part of the &longs;aid Center &longs;hall be of equal Pondero&longs;ity to counterpoi&longs;e one another, without having any regard whether they be little or much removed from the Center. Since therefore that of the&longs;e three po&longs;&longs;ible Cau&longs;es of Gra­vitation, we know not which is the right, nay, that we are not cer­tain that it is any of them, it being po&longs;&longs;ibly that there is a fourth from which one may draw Conclu&longs;ions very different, it &longs;eemeth to me impo&longs;&longs;ible for us to lay down other Principles in this bufine&longs;s than tho&longs;e of which we are a&longs;&longs;ured by a continual Experience, and a &longs;ound Judgment. As for our parts, we call tho&longs;e Bodies equally or unequally Grave which have an equal or unequal Force of mo­ving towards the Common Center: and a Body is &longs;aid to have the &longs;ame Weight when it alwaies hath this &longs;ame Force: but if this Force augmenteth or dimini&longs;heth, then, although it be the &longs;ame Bo­dy, we con&longs;ider it no longer as the &longs;ame Weight: Now &longs;ince that this hapneth to Bodies that recede or approach to the Common Center, this is it which we de&longs;ire to know, but finding nothing that giveth me content upon this Subject, I will leave the Que&longs;tion un­determined and unde&longs;cribed.

>FINIS.