The Absolute and Relative Force

PROPOSITION 22. THEOREM.

179. Let there be two parts of a point separated at b and d (Fig. 19); I say that these are to be joined together by a force of restitution in the point c, at the centre of gravity of the particular b and d.

DEMONSTRATION.

In the first place these parts are joined together at A, and these are then pulled apart by the forces AB and AD to the points b and d in the same element of time dt. Truly the force AC is equivalent to these two forces, which in the same element of time act on the point and pull the element to c from A. Therefore it is clear that the parts b and d must be drawn together by the force c, since the force AC has the same effect acting on the whole point A as both AB and AD acting on the two parts of this (149).

Hence it is therefore to be understood that c is the point of concurrence, into which the small parts b and d are driven together by the resisting force. By which moreover [p. 71] the small mass b is pulled forwards by the force AB for the element of time dt through the small distance Ab, ought to be Ab  n. ABb.dt 2 (159) or AB  Ab.b2 . Likewise on account of the ratio, AD  Ad .2d and AC  ndt ndt Ac.(b  d ) ndt 2 .

Hence truly AC is the diagonal of the parallelogram, that is constituted from the forces AB and AD, which is equivalent to these. Moreover from these equations it can be deduced that AB  AD  AC ; truly AB, AD, and AC are between themselves as the sines of the Ab Ad Ac DAC  sin BAC  sin BAD , from angles DAC, BAC, and BAD. On account of which sin Ab Ad Ac which property it follows that the points b, c , and d are in the directions given. From which, bc : cd  sin BAC. Ab : sin DAC. Ad  AD. Ab : AB. Ad . But AD : AB = Ad.d : Ab.b. Consequently it follows that bc:cd = d:b or b.bc = d.dc. From which it is understood that the point c is the centre of gravity of the particular b and d. Q. E. D.

Corollary 1.

180. Therefore the point A taken in any place, always falls upon the position of the same point of concurrence c, from which it is apparent that the constant restoring force does not depend either on the position of the point A nor on the particular particles of A b and d acted upon

Scholium.

181. This force of restitution agrees uncommonly well with the effect of an elastic force, that it is permitted to put in its place. For an elastic string bd can join together the small parts b and d, that by acting together make b and d meet in c. Moreover this force acts to draw the particles b and d together equally, since each is trying to contract equally.

For truly the distance, by which b and d are drawn together in the same time, vary inversely as the sizes of the particles themselves (159), since they are affected by the same force. Whereby if the point of concurrence is c, bc and dc will vary inversely as b to d or b.bc = d.cd. From which also it is understood that the point c is the centre of gravity of the particular masses b and d.

Corollary 2.

182. Therefore although the force of restitution is imaginary and only exists in the form of thoughts, yet the effect of this follows the real laws of motion. And from this we can be more sure with the aid of this principle of restitution to always arrive at the truth. [Note that this is not to be confused with the coefficient of restitution that is a more modern concept involving elastic and inelastic collisions.]

PROPOSITION 23. THEOREM.

183. Let a, b, c, and d be parts of a point mutually separated, which are to be joined together again by the force of restitution, and these are in agreement with a common centre of gravity g.

DEMONSTRATION.

Initially we put the whole mass at some point O (Fig. 20), from which these individual parts a, b, c, and d in the element of time dt by the forces OA, OB, OC, OD are to be drawn out into a, b, c, d.

OG is taken as the equivalent of these forces, which in the same element of time, pulls the whole point mass forwards from O to g; and g is the point at which all the parts a, b, c, and d will be drawn together by the force of restitution (149). [p. 73] Through the point O some line KN is drawn, and to that line perpendiculars are sent from the points A, a; B, b; C, c; D, d; G, g. Moreover, these are given by :

Og .( a  b  c  d ) OA  Oa.2a , OB  Ob.b2 , OC  Oc.c2 , OD  Od .2d and OA  (159). But from ndt 2 ndt ndt ndt ndt the similar triangles OAK, Oak; OBL, Obl, etc, it follows that : .ak  ak .a , BL  OB.bl  bl .b , CM  OC .cm  cm.c , AK  OA 2 2 2 Oa Ob Oc ndt ndt ndt . .(    ) OG gs gs a b c d .dn  dn.d , and GS  DN  OD  , and Od Og ndt 2 ndt 2 .Ok  Ok .a , OL  OB.Ol  Ol .b , OM  OC .om  Om.c , OK  OAOa Ob Oc ndt 2 ndt 2 ndt 2 .On  On.d , and OS  OG.Os  Os.( a  b  c  d ) . ON  OD Od Og ndt 2 ndt 2

But since OG is the force equivalent to the forces OA, OB, OC, OD, it is agreed from statics that AK + BL + CM + DN = GS, et OK + OL – OM – ON = OS. Hence we find ak .a  bl.b  cm.c  dn.d  gs.(a  b  c  d ) and Ok .a  Ol.b  Om.c  On.d  Os.(a  b  c  d ).

From which properties it is understood that the point g is the centre of gravity of the particles a, b, c, d. Hence the force restoring these particles agrees with the common centre of gravity g. Q. E. D.

Corollary 1.

184. Therefore the effect of the force of restitution is in agreement with this, that bodies separated into any number of parts can be brought together at the common centre of gravity. [p. 74]

Corollary 2.

185. Therefore in this manner, the motion of a point acted on by many forces can be determined without considering equivalent forces, provided that from the individual forces some number of parts are put in place to be affected for any short time interval, and to be brought together again by the restituting force. Scholium.

186. The demonstration of this theorem with the help of contracting elastic strings can be done in the same manner as we did before (181).

For let the separate particles (Fig. 21) be at a, b, c, and d, and in the first place we only put a string to that draws the particles a and b together, these fit together with the centre of gravity e.

We consider the particles a and b located at e to be joined with the particle at c, it will be the point of concurrence f, which is the centre of gravity of the three points a, b, and c. Now these three placed at f are joined to the fourth particle located at d, and it will be the point of concurrence g, the centre of gravity of the four points a, b, c and d. Whereby from the force of restitution all the particles come together at the common centre of gravity.

Corollary 3.

187. Therefore it is agreed once more correctly that the restoring force arising from the contraction of elastic fibers can be used to represent the join between each two particles.

General Scholium

188. Therefore from these principles put in place, from which the motion of a free point can be determined from the action of some number of forces, we will progress to the motion of free points that have to be investigated. Indeed it will be convenient to separate this tract into two parts, in the first of which only rectilinear motion will be examined, and in the other any kind of curvilinear motion.

The first rectilinear motion, as it is understood from what has been said, arises when the motion agrees with the direction of the force ; and truly the other, when these directions disagree.

Truly we will deal with each part in two ways for the two kinds of forces involved, and then for both the absolute and the relative forces taken together.

In place of the relative force we will substitute a medium with resistance, which as now we have reminded, are to be considered; for the relative forces are to be determined for the motion of bodies in fluids (116), on account of which these will be mainly for the kinds found in nature, and neither shall we say much about these which are only to be found in the imagination, and over which we will not tarry.

First therefore we will handle the rectilinear motion of a free point acted on by absolute forces. Then we will investigate the rectilinear motion of a free point in a resisting medium. Thirdly we set out the curvilinear motion of a free point acted on by any absolute forces. Fourthly we set out the curvilinear motion of a free point in a resisting medium.