Chapter 21 entitled “Of the Force of Bodies” is not about force in the modern sense, but about living force and its conservation, i.e., a rudimentary concept of energy. The title has the following footnote:

Quoique l’Auteur des Institutions ait fait beaucoup de changemens à son Ouvrage pour cette seconde Edition, elle n’en fait aucun à ce Chapitre XXI. (On a seulement ajoure quelques mots au §560. pour l’éclaircir), asin que le Lecteur le trouve ici tel qu‘il étoit, lorsque la dispute publique qu’elle a eue avec Mr.de Mairan, au sujet des Forces vives, a commencé. (Amsterdam 1742)

Although the author of the Institutions has made many changes in the second edition of this book, no changes are made in this Chapter XXI. (Only a few words have been added to § 560. for clarification). So, the reader finds this chapter here as it stands, when her public dispute with Mr. de Mairan about the Living Forces began. (Copyright © 2009 BZ)

Chapter 21 of the Amsterdam edition 1742 is nearly identical to that of the Paris edition 1740. Both editions differ from the manuscript, which consists of two versions (two different hands). The second version contains many inserted pages from Du Châtelet’s hand, as well as numerous comments at the margin. A detailed comparison would be valuable, but this goes beyond the scope of this reader.

As already underlined in the previous chapter, two types of “force” can be distinguished: The first is the force that acts as a tendency when a body resists (without real effect). This force is the so-called dead force. The second type of force is the so-called living force, when the body is in the state of a real and finite motion (InstPhy, § 559). Both “forces” correspond to two different measures of force: (a) the measure of the product of mass and velocity for the dead force, and (b) the measure of the mass and the square of the velocity for living force.

Du Châtelet mentions several examples for the existence of dead force, among them the case when the body is kept from falling by an obstacle. Other examples are a lever on a balance (InstPhy, § 563, Fig. 71), or hydrostatic equilibrium (InstPhy, § 563, Fig. 72.). Further, Du Châtelet gives two examples for the living force, namely elasticity and gravity. Given three elastic springs equally strong and equally tense (InstPhy, § 567, Fig. 73), then a body receiving the force of three equal and similar springs will acquire three times the force. Another example: Given gravity as an infinite elastic spring, then living forces of the body descending by gravity are as the spaces AB to AR. These spaces are as the squares of the velocities (InstPhy, § 567, Fig. 74). Appealing to Galileo’s treatment of free fall and pendula, Du Châtelet argues that a fourfold increase in the height of fall results in a doubling of the speed, i.e., the force must be proportional to the product of the mass and the square of the veloctiy.

From today’s perspective it seems obvious that Du Châtelet describes a rudimentary expression for the conversion of potential energy to kinetic energy, at least if we concede that she used the word “living force,” or vis viva, to signify what we call “work” (see Fig. 71 and Fig. 72). From today’s perspective, the dispute about the question of whether the force of bodies in motion, striking each other, is proportional to the simple velocity of the motion or proportional to the square of the velocity, was wrongly posed. Both quantities are conserved, the product of mass and velocity as well as the quantity of the product of mass, multiplied by its velocity squared. The first refers to what is now called momentum, the second to the kinetic energy of a moving body.

In the 18th century there was no clear distinction between the change in velocity, which is directly proportional to time when acceleration is constant, and kinetic energy, which is proportional to the square of velocity. Du Châtelet was involved in a heated debate on the true measure of force (InstPhy, § 567):

Cette assertion parut d’abord une espèce d’Hérésie Physique. D’où viendroit ce quarré, disoit-on?

This assertion first appeared to be a kind of Physical Heresy. Where’d that square come from? (Copyright © 2009 BZ)

In 1686, Leibniz published the article “Brevis demonstratio” in the journal Acta Eruditorum. Leibniz aimed to demonstrate that the moving force (“vis motrix”) has to be distinguished from the quantity of motion (“quantitas motus”) in order to calculate the motion of a body (Leibniz, 1686, pp. 2027f.). Descartes had considered that the quantity conserved in collisions equals the combined sum of the products of size and speed of each impacting body. This conserved quantity, which Descartes refers to indiscriminately as “motion” or “quantity of motion,” is not always contained in the same parts of matter, but transferred from some parts to others depending on the ways in which they come into contact. Leibniz maintained that the conservation principle would be violated if the moving force was equivalent to the Cartesian quantity of motion; rather, the force of a body has to be estimated from the quantity of the effect which it can produce. Appealing to Galileo’s treatment of free fall and pendula, Leibniz argued that the force needed to lift a weight can be determined by the product of weight and height. So, according to Galileo’s law of falling bodies, the force must be proportional to mv². This measure of force was later expressed by Leibniz as living force (vis viva).

Du Châtelet offers at least three arguments for living forces and their measure. The first argument concerns the mathematical derivation of living forces from dead forces. According to Du Châtelet living force and dead force are heterogeneous quantities. Nevertheless both are comparable insofar as the living force must be infinitely greater than its element. Dead force “can become a finite living force if it is repeated an infinity of times and accumulated by an infinite number of successive pressures in the body which receives the motion”: «qui ne peut devenir une force vive finie, que lorsqu’elle est répétée une infinité de fois, et accumulée par une infinité de pressions successives dans le corps qui reçoit le mouvement» (InstPhy, § 561). Like Leibniz before her, Du Châtelet said that the dead force and living force are in the same ratio as points to straight lines or as a line to a surface (InstPhy, § 566):

Sans entrer encore dans la discussion de la mesure de cette force vive, on s’apperçoit aisément qu’elle est d’un autre genre que la force morte, qu’elle doit être infiniment plus grande que son  élement, & qu’elle doit lui être comme une ligne est à un point, ou comme une surface est à une ligne. Mr.de Leibnits qui a découvert le premier la véritable mesure de la force vive, a distingué avec beaucoup de soin ces deux forces, & il a si bien expliqué leurs différences qu’il eût été impossible de s’y méprendre. (Amsterdam 1742)

Without entering into the discussion of the measurement of this living force, it is easy to see that it is of a different kind than the dead force, that it must be infinitely greater than its element, and that it must be like a line to a point, or like a surface to a line. Mr. de Leibnits, who first discovered the true measure of the living force, distinguished these two forces with great care, and he explained their differences so well that it would have been impossible to misunderstand them. (Copyright © 2009 BZ)

This analogy expresses a difference in order, i.e., of dimensionality. A line can be constituted by points, although both are heterogeneous quantities. Du Châtelet then invalidates a weighty objection to the measurement of the living force. If living force is the integral of dead force, the question then arises whether living force should be integrated over time or over distance. Integration over time results in a quantity that is proportional to the chance of the quantity in motion, i.e. mv. Integration over distance is proportional to the chance in living force, i.e. mv². Du Châtelet argues that the factor time is irrelevant regarding the measurement of living force (InstPhy, § 569):

Les adversaires des forces vives ont crû pouvoir se dérober à cette conclusion par la considération du tems, lequel, disent-ils, doit toujours être la mesure commune de deux forces que l’on compare ; or les corps qui avec des vîtesses doubles sont des effets quadruples […]. Il me semble qu’il y a une reponse bien simple à cette à cette Objection ; car pouvoir produire plus d’effet, & agir pendant plus de tems, c’est là ce que j’appelle, & ce que je crois que tout le monde doit appeler, avoir plus de force, & la mesure totale de cette force doit être ce que le corps peut faire, depuis le tems qu’il commence à se mouvoir, jusqu’à celui ou il aura épuisé toute sa force, quelque soit le tems  qu’il employe, & le tems ne doit pas plus entrer dans cette considération que dans la mesure de la richesse d’un homme, qui doit avoir été toujours la même, soit qu’il ait dépensé son bien dans un jour, ou dans un an, ou dans cent ans. (Amsterdam 1742)

The opponents of the living forces believed that they could avoid this conclusion by considering time, which, they say, must always be the common measure of two forces that are compared; but bodies that with double speeds are quadruple effects […]. It seems to me that there is a very simple answer to this objection; because being able to produce more effect, and acting for more time, that’s what I call, and what I believe everyone should call, have more force, and the total measure of that force should be what the body can do, from the time it starts moving until it has exhausted all its force, whatever time the body needs, & time must not be included in this consideration any more than in the measure of a man’s wealth, which must have always been the same, whether he has spent his property in a day, or in a year, or in a hundred years. (Copyright © 2009 BZ)

Du Châtelet quotes Johann I Bernoulli’s Discours sur les loix de la communication du mouvement (1724/26) in order to support the equation mv² for living force. In 1724, the Paris Academy of Science offered a prize for the best proposal discussing the laws of impact. Bernoulli’s essay based on the hypothesis of the elasticity of matter. He maintained that absolutely hard bodies did not exist and went on to model the force of collision by analogy with the compression and release of springs.

Bernoulli’s contribution was declined by the Jury because it had missed the point. The Academy’s question about hard-body collisions were not answered, but rejected. Colin Maclaurin won the prize. In his contribution Démonstration des loix du choc des corps (1724) he argued for a mechanics based on Newton’s laws of motion, with the force of bodies defined as mv. 10 In the following years the question of the measure of force became a controversial subject at the Paris Academy: The most animated debate took place in 1728 and 1729, when it turned out that Bernoulli had a significant number of allies within the institution who defended his model of elastic bodies and who promoted his distinctly Leibnizian mathematical methods for doing analytical mechanics.

In April and May of 1728, Jean-Jacques d’Ortous de Mairan gave a lecture at the Academy in which he rebuilt a Cartesian position by reducing acceleration to uniform inertial motion and argued that force was proportional to the simple velocity. Du Châtelet questioned de Mairan (InstPhys, § 567). She stated that, in determining the force of a body, the correct calculation was that of Bernoulli and Leibniz. To illustrate this, she gives a simple example: A courier who needs less time for the same distance as another courier is faster than his competitor. The faster a courier must walk to cover the same distance in less time, the more force he needs (InstPhy, § 575):

Cette doctrine peut être confirmée par raisonnement fort simple, & que: tout le monde fait naturellement quand l’occasion s’en présente: que deux voyageurs marchent, également vite, & que l’un marche pendant une heure, & fasse une lieue, & l’autre deux lieues pendant deux heures, tout le monde convient que le sécond a fait le double du chemin du premier, & que la force qu’il a employé à faire deux lieues, est double de celle que le premier a employée pour faire une lieue: or, supposànt maintenant qu’un troisiéme voyageur fasse ces deux lieues en une heure, c’est-à-dire, qu’il marche avec une vitesse double, il est envore évident que le trôisième voyageur, qui fait deux lieues dans une heure, employé deux fois autant de force que celui qui fait ces; deux lieues en deux heures: car оn sait ue plus un courier doit marcher vîte, & faire le même chemin en moins de terms, plus il lui faut de force, ce que tout courier sent si bien qu’il n’y en a point qui ne veuille être d’autant mieux paye, qu’il va plùs vite; or puisque le troisîéme voyageur employé deux fois plus de force ue le seconde, & que que le second en employed aux fois plus que le premier, il est évident que le voyageur qui marche avec nne double vîtesse pendant le même tems, employé quatre fois pus de force; & que par conséquent les forces que ces voyageurs auront dépensées, seront comme le quarré de leurs vîtesses. (Amsterdam 1742)

This doctrine can be confirmed by a very simple argument, which everyone makes naturally when the occasion arises: if two travelers walk equally fast, and one walks for one hour, and makes one lieue, and the other two lieues in two hours, everyone acknowledges that the second made double the distance of the first, and that the force he used to cover two lieues is double that which the fi rst used to walk one lieue. Now, supposing that a third traveler covers these two lieues in one hour, that is to say, that he walks at double the speed, it is evident again that the third traveler, who makes two lieues in one hour, uses two times the force used by the one who walked these two lieues in two hours. For we know that the faster a courier must walk to cover the same distance in less time, the more force he needs, which all couriers understand so well that they all want to be better paid the faster they go. Now, since the third traveler uses two times more force than the second, and the second uses two times more than the first, it is obvious that the traveler who walks at double the speed during the same time, uses four times more; and consequently the forces that these travelers expended will be as the square of their speeds. (Copyright © 2009 BZ)

This example clearly shows  that, from today’s point of view, power is meant, i.e. energy divided by time. Getting something done faster requires more power. This is also true for Du Châtelet’s reference to Jacob Hermann’s experiments on the elastic collision in De mensura virium corporum (in Comentarii Academiae Scientiarum Imperalis Petropolitanae 1726) (InstPhys, § 577, Fig. 75):

Mais M. Herman rapporte un cas qui ne laisse lieu à auçun subterfuge, & dans lequel on ne peut disputer que la force du corps n’ait été quadruple en vertu d’une double vîtesse; ce cas est çelui dans lequel une bôule A. qui a un de masse, par exemple, & deux de vîtesse, frappe successivement sur un plan Horisontal, supposé parfaitement poli, une boule B. en repos , qui à 3. dé masse, & un boule C. qui à 1. de masse; car ce corps A. donnera un degré de vîtesse à la boule B. dont la masse 3 & il donnera le dégré de vîtesse qui lui reste à la boule C. qu‘il rencontre ensuite, & dont la masse est un cest-à-dire, égale à la sienne, & ce corps A. ayant alors perdu toute la vîtesse restera en repos. Or, éxaminons quelle ést la force des corps B. & C. auxquels le corps A. a, communiqué toute sa force, & toute sa vîtesse,, certainement la masse du corps B. étant 3. sa & vîtesse un, sa force sera trois de l’aveu mème de ceux qui refusent d’admettre les forces vives, le corps C. dont la vîtesse est un, & la masseun, aura aussi un de force: donc le corps A. aura communiqué la forcé trois au corps B. & la force un au corps C. donc le corps A. avec 2. de vîtesse a donné 4. de force: donc il avoit cette force; car s’il ne l’avoit pas éue, il n’auroit pu la donner: donc la force du corps A. qui avoit 2. dè vîtesse & un de masse, étoit 4. c’est-à-dire, comme le quarré de cette vîtesse multiplié par sa masse. (Amsterdam 1742)

But M. Hermann reports a case that leaves no place for any subterfuge, and in which it cannot be disputed that the force of a body was squared by virtue of a doubled speed. This is the case in which, for example, a ball A which has 1 of mass, 2 of speed, successively hits on a horizontal plane, supposed to be perfectly smooth, a ball B at rest, which has 3 of mass, and a ball C that has 1 of mass; for this body A will give a degree of speed to ball B whose mass is 3, and it will give the remaining degree of speed to ball C, which it next encounters, and whose mass is 1, that is to say, equal to its own; and this body A, having then lost all its speed, will stay at rest. Now let us examine what the force will be of bodies B and C to which body A communicated all its force and all its speed; certainly the mass of body B being 3 and its speed 1, its force will be 3 even in the opinion of those who refuse to accept forces vives; body C, whose speed is 1 and mass 1 will also have 1 of force: thus body A will have communicated the force of 3 to body B and the force of 1 to body C. Thus body A with 2 of speed gave 4 of force. This means that it had this force; for, if it had not had it, it could not have given it; thus, the force of body A, which had 2 of speed and 1 of mass, was 4, that is to say, as the square of this speed multiplied by its mass. (Copyright © 2009 BZ)

De plus, la force est toujours la méme, soit qu‘elle ait été communiquée dans un petit tems ou dans un grand tems; le tems dans lequel les ressorts communiquent leur force, par exemple, dépend des circonstances dans lequelles ils se déployent; car il a des circonstances dans lesquelles la force d’un ressort peut sè. transmettre dans un même corps plus vite que dans d’autres circonstances,. cependant la force que ce ressort lui communique, est toujours la même: ainsi, quatre ressorts égaux communiqueront la même force au même corps, soit qu’ils la lui communiquent en une, en deux, ou en trois minutes, comme dans les Fig. 77. 78 & 79. & ce tems pourroit être varié à l’infini, selon qu’on laisseroit à ces ressorts plus ou moins de liberté d’agir. (Amsterdam 1742)

In addition, the force is always the same, whether it has been communicated in a short time or a long time. The time in which springs communicate their force, for example, depends on the circumstances in which they are deployed; for there are circumstances in which the force of a spring can be transmitted in the same body faster than in other circumstances. Yet the force that this spring communicates is always the same. Thus, four equal springs will communicate the same force to the same body, whether they communicate it in one, two, or three minutes, as in Figs. 77, 78, and 79, and this time could be infinitely varied, depending on whether these springs were more or less at liberty to act. (Copyright © 2009 BZ)

Du Châtelet was well acquainted with the past and present discussions about how to measure force, from Leibniz’s controversy with Abbe Catalan and Denis Papin (InstPhys, § 579) to Henry Pemperton’, John Theophilus Desaguliers’ and John Eames’ attack on the Leibnizians. Du Châtelet  herself was involved in a dispute with James Jurin, who had argued in several papers against vis viva.

Du Châtelet quotes from Jurin’s Dissertationes physico-mathematicae (1732): “Id si facere dignati fuerint, me ipsis discipulum, parum quidem illud est, ad multos, egregios viros, ausim promittere” (InstPhys, § 579) and then turns to Jurin’s principle that equal impressions sustained over equal times produce equal effects and consequently equal forces. She repeates that the time factor does not play any role, illustrating her argument with a drawing of springs.

Willem Jacob ’s Gravesande conducted experiments in which he found out that if dents in clay were used to measure the force of a body in motion, the height from which a ball falls is as the square of the velocity acquired in falling and with which the body strikes the clay (’s Gravesande, 1720; 1722). Thus, ’s Gravesande experiments would confirm that the force of motion is proportional to the square of the velocity (InstPhys, § 584):

M. de s’Gravesande a imaginé une éxpérience qui confirme merveilleusèment cette théorie, il affermit dans la Machine de Mariotte une boule de terre glaise, & la fit choquer successivement par une boule de cuivre, dont la masse étoit trois & la vîtesse un, & par une autre boule de même métal dont la vîtesse étoit 3. & la masse un, & il arriva que l’enfoncement fait par la boule un, dont la vîtesse étoit trois, fut toujours beaucoup plus grand que celui que faisoit la boule 3. avec la vîtesse un, ce qui marque l’inégalité des forces; mais quand ces deux boules avec les mêmes vîtesses. Que ci-devant choquoient en même tems, la boule de terre glaise suspendue librement à un fil, alors la boule de terre glaise n’éroit point ébranlée, & les deux boules de cuivre restoient en repos & également enfoncées dans la terre glaise, & ces enfoncemens égaux ayant été mesurés, ils se trouvèrent plus grands, que l’enfoncement que la boule trois avec la vîtesse un avoit fait, lorsqu’elle avoit frappé seule la boule de terre glaise affermie, & moindre que celui qui y avoit été fait par la boule 1. avec la vîtesse 3. car la boule 3. avoit employé la force à enfoncer la terre glaise, & son enfoncement avoit été augmenté par l’effort de la boule 1 qui a pressé la boule de terre glaise contre la boule 3. ce qui a diminué l’enfoncement de cette boule un; ainsi, les corps mous qui se rencontrent avec des vîtesses en raison inverse de leurs masses, restent en repos après le choc, parce qu’ils employent leurs forces à enfoncer mutuellement leurs parties; car ce n’est pas un simple repos qui joint ces parties, mais une véritable force, & pour aplatir un corps & enfoncer les parties, il faut surmonter cette force qu’on appelle cohérence, ou cohésion, & il ne se consume dans le choc que la force qui est employée à enfoncer ces parties. (Amsterdam 1742)

M. ’s Gravesande created an experiment that wonderfully confirms this theory. He took a firm ball of clay and, using Mariotte’s Machine, he made it collide successively with a copper ball, whose mass was 3 and speed 1, and with another ball of the same metal whose speed was 3 and mass 1, and it happened that the impression made by ball one, whose speed was 3, was always much greater than that made by ball three with the speed of 1, which indicates the inequality of the forces. But when these two balls with the same speeds as before collided at the same time with the clay ball freely suspended from a thread, then the clay ball was not set in motion and the two copper balls stayed at rest and equally depressed the clay; and these equal impressions having been measured, they were found to be much greater than the impression that ball three with the speed of 1 had made when it only hit the firmed clay ball, and less than that which had been made by ball one with the speed 3. For ball three had used its force to make an impression on the clay ball, and its impression having been augmented by the effort of ball one that pressed the clay ball against ball three, diminished the impression of this ball one. Thus soft bodies that collide with speeds in inverse proportion to their masses, stay at rest after the collision, because they use all their force to mutually impress their parts. For it is not simple rest that holds these parts together, but a real force, and in order to flatten a body and drive into its parts, this force, named coherence, must be overcome, and in the collision the force used to drive into and impress these parts is consumed. (Copyright © 2009 BZ)

[Manuscript Bibliotheque nationale de France (Paris), Fonds français 12265 ]

In the following paragraph Du Châtelet comes back to Jurin’s argument against living forces (InstPhy, § 585, Fig. 81):

Ainsi, ce cas que Mr. Jurin défie tous les Philosophes de concilier avec la doctrine des forces vives, n’est fondé que sur cette fausse supposition que le ressort R. communiquera au corps P. transporté sur un plan mobile ou dans un bateau, la même force qu’il lui communiqueroit si le ressort étoit appuié contre un obstacle inébranlable, & en repos, mais c’est ce qui n’est point, & ce qui ne peut point être, que dans le seul cas ou la masse du vaissèau seroit infinie par rapport à celle du corps. (Amsterdam 1742)

Thus, this case that M. Jurin defies all philosophers to reconcile with the doctrine of forces vives is only founded on this false supposition that the spring R will communicate to body P, carried on a moving plane or in a boat, the same force that it would communicate to it if the spring were pressing against an immoveable obstacle and at rest, but this is not the case, and cannot be, except in the case when the mass of the boat is infinite in relation to that of the body. (Copyright © 2009 BZ)

[ Manuscript Bibliotheque nationale de France (Paris), Fonds français 12265 ]

Du Châtelet claims that Jurin’s experiment confirms what he is trying to disprove. Jurin’s case based on the false assumption that the elastic spring R will commuicate to the body P (transported on a movable plane) the same velocity that it communicated to it when the spring was supported by a immovable obstacle at rest (Fig. 81). (Today, one would say that one has to consider the enery exchange of the moving body and the moving plane, or boat.) Not only Jurin, but Newton himself disagreed with Leibniz regarding the measure and concept of force. For Newton, Galileo’s law provided the basis for his second law of motion, which states that the force of bodies is proportional to the quantity of their motion (InstPhys, § 585, Fig. 82-83). Newton accused Leibniz for neglecting that the the distance traveled in free fall is proportional to the square of the elapsed time. In his Opticks (see Pierre Coste’s French translation (1729)), Newton argued that the amount of motion in the world is not constant. He considered two identical globes in empty space attached by a slender rod and revolving with angular speed about their combined center of mass. To quote from Du Châtelet (InstPhy, § 586, Fig. 83):

Quoique l’autorité ne doive point être comptée lorsqu’il s’agit de la vérité, ce pendant je me crois obligé de vous dire que Mr. Newton n’admettoit point les forces vives, car le nom de Mr. Newton vaut presque une objection: ce Philosophe examine dans la derniére Question de son Optique le mouvement d’un bâton inflexible АB. aux bouts duquel on a attaché les corps A. & B. & il suppose que le centre de gravité de ce bâton AB. qu’il ne çonsidére que comme une ligne, se meuve le long de la droite С D. tandis que les corps A. & B. tournent sans cesse autour de ce centre, il arrive que lorsque la ligne AB. est perpendiculaire à CD. (comme dans la Figure 82.) la vîtesse je du corps A. est nulle, & celle du corps B. est deux; ainsi, le mouvement de ces corps est alors deux; mais quand cette ligne AB. est coïncidente ou presque coïncidente avec la ligne С D.( comme dans la Figure 83.) alors la somme des mouvemens des corps A. & B. devient 4. Mr. Newton conclut de cette considération, & de celle de l’inertie de la matière que le mouvement va sans cesse en diminuant dans l’Univers; & qu’ensin notre Systême aura besoin quelque jour d’être reformé par son Auteur, & cette conclusion étoit une-suite né,с cessaire de l’inertie de la matiére, & de l’opinion dans laquelle:étoit Mr. Newton, que la quantité de la force étoit égale à la quantité du movement; mais quand on prend pour force le produit de la masse par le quarré de ia vîtesse, il est aisé de prouver que la forcé vive demeure toujours la même , quoique la quantité du mouvement: varié peut-être à caque instant dans l’Univers, & que dan tous les cas, & spécialement dans celui que je viens de citer d’après Mr. Newton, la force vive demeure invariable,  quelque soit la position de la ligne AB. par rapport à la ligne CD. que parcourt son centre dé gravité. Ainsi les miracles continuels qui résultent de la position de cette ligne AB. n’ont plus lieu dans la doctrine des forces vives. (Amsterdam 1742)

Although authority must be counted when truth is at issue, I feel obliged to tell you that M. Newton did not acknowledge forces vives, for the name of M. Newton is in itself nearly an objection. In the last question of his Opticks this philosopher examines the movement of an inflexible stick AB, at both ends of which have been attached bodies A and B, and he supposes that the center of gravity of this stick AB that he only considers as a line, moves the length of the straight line CD, while the bodies A and B turn continuously around this center, when the line AB is perpendicular to CD (as in figure 82) the speed of body A is zero, and that of body B is 2. Thus the motion of these bodies is then 2; but when this line AB is coincident or almost coincident with line CD (as in figure 83) then the sum of the motions of bodies A and B becomes 4. M. Newton concludes from this consideration and that of the inertia of matter that motion is constantly diminishing in the universe; and lastly that our system will some day need to be formed anew by its Author, and this conclusion was a necessary consequence of the inertia of matter, and the opinion held by M. Newton that the quantity of force was equal to the quantity of motion.118 But when the product of the mass by the square of the speed is taken as force, it is easy to prove that the forces vives always remain the same, although the quantity of motion varies perhaps at each instant in the universe, and in all the cases, and especially in that which I have just cited from M. Newton, the forces vives stay invariable; whatever the position of the line AB in relation to line CD described by its center of gravity. Thus, the continual miracles, which result from the position of this line AB have no place in the doctrine of forces vives. (Copyright © 2009 BZ)

[ Manuscript Bibliotheque nationale de France (Paris), Fonds français 12265 ]

From the description of the motion of globes on the balance Newton concluded that the whole quantity of motion is lost in the universe. Du Châtelet objected that Newton’s statement violates the principle of conservation. The universe is not in need of an intervention by God preventing the world from running down (InstPhy, § 586). In fact, the vector sum of the momenta is constant, as is the “kinetic energy”, i.e. Leibniz’s vis viva. Further, the apparent loss of force in cases of inelastic collision, which appear to violate the conservation of vis viva, could be explained if one hypothesizes that living forces are being transformed to the smaller parts of the larger bodies (InstPhy, § 590):

Mais ce qui est certain, c’est que la force ne périt point, elle peut à la vérité paroître perdue, mais on la retrouveroit toujours dans les effets qu’elle a produits, si l’on pouvoit toujours appercevoir ces effets. (Amsterdam 1742)

But what is certain is that force does not perish; it may seem that it is lost; but force would always be found in the effects produced by it, if these effects were always observable. (Copyright © 2009 BZ)

Du Châtelet’s Institutions physiques end with these sentences which demonstrate once more her familiarity with the Leibniz-Clarke-correspondence. Du Châtelet was strongly convinced that the hypothesis of the conservation of living force will be confirmed even in the case of inelastic collisions. We know today that inelastic collisions conserve energy, but kinetic energy is lost in such collisions as it transforms into other forms of energy.

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Compare the entry Force vive, ou Force des Corps en mouvement  in d’Alembert’s and Diderot’s Encyclopédie.

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