In Chapter 11, Du Châtelet defines motion as follows (InstPhy, § 211):
Motion is the passage of a body from the place it occupies to another place. (Copyright © 2009 BZ)
In § 212, Du Châtelet distinguishes three kinds of motion: (i) absolute motion (“le mouvement absolu”), (ii) relative common motion (“le mouvement rélatif commune”), and (iii) proper relative motion (“le mouvement rélatif propre”). Absolute (real) motion is the motion of a body relative to other bodies which are assumed to be immobile (InstPhy, § 213):
Absolute motion is the successive relation of a body to different bodies considered as immobile, and strictly speaking, this is real motion. (Copyright © 2009 BZ)
Relative common motion is defined as follows (InstPhy, § 214):
Le mouvement relatif commun est celui qu’un Corps éprouve, lorsqu’étant en repos, par rapport aux Corps qui l’entourent, il acquiert cependant avec eux des relations successives, par rapport à d’autres Corps, que l’on considère comme immobiles. (Amsterdam 1742)
A body experiences common relative motion when, being at rest in relation to the bodies that surround it, it nonetheless develops with them successive relationships, in relation to other bodies that are considered stationary [immobile]. (Copyright © 2009 BZ)
Du Châtelet adds that this is the case when the absolute place of the bodies changes, though their relative place stays the same; this is what happens to a pilot who sleeps at the tiller while his ship moves, or a dead fish carried along by the current.
Proper relative motion (motion relative to itself) means (InstPhy, § 215):
Le mouvement relatif propre est celui que l’on eprouve, lorsqu’etant transporte avec d’autres Corps d’un mouvement relative commun, on change cependant fa relation avec eux, comme lorfque je marche dans un Vaisseau qui fait voile. (Amsterdam 1742)
Motion relative to itself [relative proper motion] is the one experienced when, being transported with other bodies in a common relative motion, one nonetheless changes one’s relationship with them, as when I walk on a sailing ship; for I keep changing my relationship with the parts of this ship that transports me. (Copyright © 2009 BZ)
What follows is Du Châtelet’s version of Galileo’s ship (although she did not mention Galileo’s name). The his Dialogue Concerning the Two Chief World Systems (Dialogo sopra i due massimi sistemi del mondo, 1632; latin translation 1699) Galileo Galilei formulated was has become known as the principle of relativity according to which there is no internal observation (i.e., without looking out the window) by which one can distinguish between a system moving uniformly from one at rest. Hence, any two systems moving without acceleration are equivalent, and unaccelerated motion is relative, i.e., it dependens on the observer. To quote the thought experiment according to Du Châtelet (InstPhy, § 217/18):
Si un Vaisseau marchoit vers l’Orient, & qu’un homme se promenât dans ce Vaisseau de la proue à la poupe,c’est-a-dire, de l’Orient vers l’Occident avec la même vîtesse , dont le Vaisseau est emporté, cet homme auroit, pendant qu’il parcourt la longueur de ce Vaisseau , un mouvement rélatif propre, mais son mouvement absolu ne seroit qu’apparent, puisqu’en changeant à tout moment sa situation , par rapport aux parties de ce Vaisseau, vil répondroit cependant toujours aux mêmes points hors du Vaisseau. Si au contraire cet homme marchoit dans ce Vaisseau de la poupe à la proue, c’est-à-dire, dans la même direction que le Vaisseau qui le porte, il auroit en même tems un mouvement rélatif commun avec le Vaisseau, & un mouvement rélatif propre; car il changeroit à tout moment fa situation avec les parties de ce Vaisseau, & avec les Corps hors du Vaisseau: c’est qui marchent sur la Terre éprouvent, car la Terre marche sans cesse. Si au-lieu de cet homme, on imagine une pierre jettée horisontalement dans ce Vaisseau, dans un sens contraire à celui dans lequel le Vaisseau marche, mais avec une vitesse égale à celle dont il est emporté, cette pierre paraîtra à ceux qui sont dans le Vaisseau avoir mouvement rélatif propre, dans le sens dans lequel on l’a jettée, mais ceux qui sont sur le rivage la verront dans un repos absolu par rapport à son mouvement horisontal, & ce repos est son état réel. Cette pierre est dans un repos absolu par rapport à son mouvement horisontal, parce qu’étant transportée avec ce Vaisseau, elle avoit aquis dans la direction dans laquelle ce Vaisseau marche, une force égale à celle dont le Vaisseau étoit emporté; or, comme on suppose qu’elle a été jettée en sens contraire par une force égale à celle qui emporte le Vaisseau, ces deux forces égales & opposées se détruisent mutuellement, & la pierre reste dans un repos absolu par rapport au mouvement horisontal; car la main qui l’a jettée, à trouvé en elle une force réelle, & celle qu’-elle lui a imprimée, a été employée toute entière à la détruire. Il en arriveroit tout autrement, si cette pierre étoit jettée dans le Vaisseau par une main qui fût hors du Vaisseau; ar alors la pierre auroit réellement un mouvement rélatif propre de l’Orient vers l’Occident, & elle tomberoit dans la mer hors du Vaisseau. (Amsterdam 1742)
If a ship went toward the orient [east], and a man walked in this ship from the prow to the poop deck, that is to say, from the orient toward the occident [west], with the same speed as the boat, this man would have, while he traverses the length of this boat, a motion relative to himself, but his absolute motion would only be apparent, since in constantly changing his position in relation to the parts of this ship, he would remain in correspondence to the same points outside the ship. If, on the contrary, this man walked on this ship from the poop deck to the prow, that is to say, in the same direction as the ship carrying him, he would have at the same time a common relative motion with the ship, and a motion relative to himself; for he would constantly change his position with the parts of this ship and with the bodies outside of the ship. It is this kind of motion that all bodies walking on Earth experience, for the Earth moves ceaselessly.. If instead of this man, one imagines a stone thrown horizontally in this ship, in a direction contrary to that in which the ship goes, but with a speed equal to that at which it is carried, this stone will appear to those who are on the ship to have a motion relative to itself, in the direction in which it was thrown; but those who are on the shore will see it in absolute rest, in relationship to its horizontal direction, and this rest is its real state. This stone is in absolute rest in relation to its horizontal motion, because, moving with this ship, it acquired in the direction in which this ship goes a force equal to that by which the ship was carried. Now, supposing that it is thrown in a contrary direction by a force equal to that which carries the ship; these two equal and opposite forces cancel each other, and the stone stays in absolute rest in relation to the horizontal motion; for the hand that threw it found in it a real force, and the one the hand imparted to it was consumed, canceling that force. It would be otherwise if this stone were thrown into the ship by a hand outside the ship; for then the stone would really have motion relative to itself from the orient toward the occident, and it would fall into the sea surrounding the ship. (Copyright © 2009 BZ)
[Compare the entry “Mouvement” in the Dictionnaire universel de mathématique et de physique 1753, ed. by Alexandre Savérien, Jacques Rollin, and Ch.-A. Jombert.]
In a next step, Du Châtelet distinguishes motion from rest: Rest is not the change in position of a body, but its continuation, i.e., its continuous existence in the same place (InstPhy, § 220):
Le Repos est l’éxistence continuë d’un Corps dans le même lieu. (Amsterdam 1742)
Rest is the continuous existence of a body in the same place. (Copyright © 2009 BZ)
Relative rest is defined as follows (InstPhy, § 222):
Relative rest is the continuation of the same relations of the Body, as we consider, to the bodies which surround it, although these bodies move with it. (Copyright © 2009 BZ)
And absolute rest is defined as follows (InstPhys, § 223):
Le Repos absolu est la permanence du Corps dans le même lieu absolu, c’est-à-dire, la continuation des mêmes rapports du Corps, que l’on considére, aux Corps qui l’entourent, considérés comme immobiles. (Amsterdam 1742)
Absolute rest is the permanence of a body in the same absolute place, this is to say, the continuation of the same relationships of the body being considered to the bodies that surround it, considered as immobile. (Copyright © 2009 BZ)
Du Châtelet’s classification of different kinds of motion is based on the foundational problem how one should be able to distinguish between real and apparent states of motion (and rest). Du Châtelet obviously struggled with this problem. This becomes evident from the comparison of the manuscript with the Paris edition of 1740 and the Amsterdam editionof 1742. There are two manuscript versions of Chapter 10 (Chapter 11 in the published editions). The first version begins by hand (corresponding to Chapter 11, §§ 211-228 in the published editions), followed by some pages from an early printed version with handwritten remarks on the margin and corrections in the text. This printed part is the result of the comprehensive correction of the subsequent handwritten version. The handwritten version is from another hand with Du Chatelet’s notes on the margin.
In the Amsterdam edition the laws of motion are formulated as follows (InstPhy, § 229):
TROISIEME LOI: La réaction est toujours égale à l’action; car un Corps ne pourroit agir sur un autre Corps, si cet autre Corps ne lui resistoit: ainsi, l’action & la réaction sont toujours égales & opposées. (Amsterdam 1742)
The active force and the passive force of the bodies are modified by their impact according to certain laws, which can be reduced to three principles.
FIRST LAW. A body perseveres in the state it is in, be it rest or motion, unless some cause brings an end to its motion or to its rest. [A Body perseveres in its state, be it rest or motion, unless it is compelled to change its state of motion or rest by some cause.]
SECOND LAW: The change that happens in the motion of a body is always proportional to the motor force that acts on it; and no change can happen to the speed and the direction of the moving body except by an exterior force; for without that, this change would happen without suffi cient reason. [The changes of the motion of a body are always proportional to the moving force which acts upon it; for otherwise this change would happen without sufficient reason.]
THIRD LAW: The reaction is always equal to the action; for a body could not act on another body if this other body did not resist it. Thus the action and the reaction are always equal and opposite. (Copyright © 2009 BZ)
Given the principle of sufficient reason every motion must have a cause. A body at rest will never begin to move by itself. By the same principle, a moving body would never cease to move if some cause did not stop its motion (InstPhy, § 227f.). In accordance with these considerations, Du Châtelet incorporates the principle of sufficient reason into her formulation of the laws of motion and replaces the concept of “impressed force” by “cause:” every body perseveres in its state, except insofar as it is compelled to change its state by a cause. The word “impressed” is omitted. The same holds for “moving uniformly in a straight line.” Du Châtelet does not mention that “state of motion” also refers to the direction of the motion of a body, i.e., moving uniformly straight forward. Du Châtelet speaks about moving force («force motrice»). Otherwise, the change of motion of a body would happen without sufficient reason. Newton’s first and second axioms reveal that impressed forces come in pairs: To any action there is always an opposite and equal reaction; the actions of two bodies upon each other are always equal and always opposite in direction. Du Châtelet’s third law exactly repeats and explains Newton’s action-reaction principle. Also in line with Newton, Du Châtelet distinguishes between active forces and passive forces. Newton characterized inertia as an innate force and passive principle by which bodies persist in their motion or rest. Inertia tends to preserve existing states of motion. For every change in motion an active principle is necessary, i.e., impressed moving force. Moving force means, as Du Châtelet rightly states, a force externally impressed upon the body, and such an impressed force is an action exerted on a body, and hence from outside the body.
The introductory note of the manuscript explicitly mentions Newton’s name: “Mr. Newton has reduced the laws of motion to three fundamental principles, by which we may explain all the effects of motion which are known to us.” In both published editions of the Institutions this sentence is replaced by the distinction between active and passive force without mentioning Newton’s name. The first and third law is the same in all three versions. All the more remarkable are the modifications of the second law. The sentence “this change always takes place along the straight line in which this force is directed” in the manuscript was cancelled out by hand and replaced by “There can be no change in the velocity and direction of the body in motion except by an external force; for otherwise this change would happen without sufficient reason.” In the final version of Amsterdam 1742, the reference to “no change in the velocity and direction of the body in motion except by an external force” is left out.
Subsequently, Du Châtelet explains how to measure and calculate the motion of a body (InstPhy, §§ 239-253). She distinguishes between velocity and acceleration. The velocity of an object is the rate of change (time derivative) of its position. Velocity can be classified into uniform and non-uniform velocity. We also know the relation of acceleration and velocity, a body is said to be accelerating if its speed or its direction is changing with respect to time or both speed as well as direction are changing with respect to time. In the case of uniform motion neither the direction nor the magnitude of the velocity changes. That is, the direction and amount remain the same. This is called constant speed. A uniformly accelerated movement is a movement in which the acceleration is constant with regard to strength and direction (InstPhy, § 240):
Motion, that is to say, its speed, can be uniform or not uniform, accelerated or decelerated, equally unequally accelerated or decelerated. (Copyright © 2009 BZ)
Du Châtelet uses this classification in order to argue against a perpetuum mobile. For a perpetual mechanical motion, it would be necessary to find a body that was exempt from friction or had received an infinite force from the Creator, since it would be necessary to make this force surmount constantly repeated resistance, and without it ever running out, which is impossible, since real physical effects are always finite (InstPhy, § 244).
An outlook on hard, soft and elastic bodies follows. Completely hard means incapable of compression; completely soft means incapable of reconstitution after the compression of their particles; elastic, that is to say, capable of regaining their original shape after the compression (InstPhy, § 267). Du Châtelet emphasizes that we do not know any body that is completely hard, or completely soft, or perfectly elastic. These are idealizations as Fontenelle noticed (probably Du Châtelet refers here to Bernard de Fontenelle’s Elements de la geometrie de l’infini, first published in 1727).
At the end of the chapter, Du Châtelet classifies the concept of moving force by distinguishing between dead force and living force (InstPhy, § 268): dead force is the force of the potentiality of motion; it does not produce any effect, but it only tends to produce one. Living force is the force expressed in actual motion; it produces a real effect. Du Châtelet will discuss the dispute among philosophers about whether living and dead force must be estimated differently, later, in Chapter 21. Finally, Du Châtelet explains the consequence of inertial force which acts on the object in an opposite direction (giving, as an example, seasickness).
See also the entry MOUVEMENT, s. m. (Méchan.) and REPOS, s. m. (Physique.) in d’Alembert’s and Diderot’s Encyclopédie. See also the entry “Mouvement”, and “Repos” in: Johann Heinrich Samuel Formey (ed.): Dictionnaire instructif, où l’on trouve les principaux termes des sciences et des arts dont l’explication peut être utile ou agréable aux personnes qui n’ont pas fait des études approfondies. Halle: Gebauer 1767.