A second guest post by Hanna Szabelska.
Hanna Szabelska writes …
As I indicated in my previous post, the fatal destiny (fatalité), about which Voltaire complained in a letter to Jean-Jacques d’Ortous de Mairan , made Madame du Châtelet’s mind more and more prone to the allure of Leibniz’s metaphysics, in particular his concept of vis viva.
For example, the comparison of fire to living force notwithstanding, the first edition of her essay on heat shows the traces of the influence of de Mairan’s Dissertation sur l’estimation et la mesure des forces motrices des corps. One possible reason for this inconsistency being that de Mairan distanced himself from metaphysics and concentrated on pure laws of motion . In the version submitted for the Academy’s prize competition, du Châtelet added a note criticising Leibniz and praising de Mairan as an advocate of the Cartesian measure of force (mv). Afterwards the Marquise desperately fought for permission to remove it before publication. She argued that this insipid compliment (fadeur) had resulted from her ignorance and was not related to the main theme. But she was unsuccessful .
The Leibnizian measure – mv² – was incorporated only in the second version together with a remarkable passage that unravels a complex interplay between the experimental and the speculative approach in du Châtelet. Having discussed the hypothesis that the Sun is a solid body containing fire and emanating it to the Earth, she concludes:
But this emanation of light is subject to far greater difficulties, and seems impossible to be assumed despite the modern observations that apparently speak in its favour: certain observations are enough to destroy a superstition when they seem contrary to it, but they are not enough to establish it and physical and metaphysical difficulties undermining the [hypothesis of] emanation of light seem so insuperable that without them being removed there are no observations that can induce one to assume it. But this is not the place to discuss them. 
The moral of this digression is that observational data are not enough to establish a hypothesis if there are strong metaphysical objections against it. This is the assumption, although not always articulated, that remains at the core of du Châtelet’s rhetorical vein in the heat of debates, e.g. her discussion with de Mairan about one of Jacob Hermann’s experiments and the measure of force. Remarkably, the exchange with de Mairan was published not only together with the Institutions physiques (1742), the second edition of du Châtelet’s manual of physics, but also with the revised version of her essay on fire.
The experiment in question is as follows :
Let the ball A move with the velocity 2 on a horizontal plane and collide with another ball B=3A, being at rest. The ball A will give the velocity 1 to the ball B and move backwards with the velocity 1. Afterwards, let the ball A with the velocity 1 collide with another body at rest C=A. The ball A will also give to the ball C the velocity 1 and as a result of the second collision, it will come to halt. All this can be easily derived from the very well known rules of the motion of elastic bodies. 
To disprove du Châtelet, de Mairan adds scalar magnitudes (m|v|), and then he goes on to directed ones, i.e. applies the measure he accepted. 
His calculation could be interpreted as a correct addition of momenta , but du Châtelet does not consider it either as an alternative of force measure or a different concept. Here comes into play her rhetorical impetus:
To tell the truth, it is remarkable with what ease this small bar you put in front of the formula for the force of the body A rid you of this 8 of force that even your own calculation gave you after collision instead of 4 that you had expected from it; but, tell me, I beg you, you certainly do not think that this sign minus and this subtraction would take away some part of force from the bodies A and B, and that the effects exercised by these bodies on any obstacles would be diminished by it. I also doubt that you would like to either experience it or find yourself in the path of a body that would bounce back affected by this minus sign with 500 or 1000 of force. 
One may think that du Châtelet did not understand the concept of directed magnitudes but was this really the case? After all, she was a very attentive reader of Willem Jacob ‘s Gravesande, who analyses the paradoxical cases of bodies moving in the opposite directions and compares the effectiveness of Leibnizian force measure with the Cartesian one.
This is the description of ‘s Gravesande’s experiment, somewhat simplified by du Châtelet: 
‘s Gravesande devised an experiment that wonderfully confirms this theory. He fastened a ball of clay in Mariotte’s Machine and he made it collide successively with a copper ball, whose mass was three and velocity one, and with another ball of the same metal, whose velocity was three and mass one, and it happened that the impression made by ball one, whose velocity was three, was always much greater than that made by ball three with the velocity of 1, which testifies to the inequality of the forces. But when these two balls with the same velocities as before collided at the same time with the clay ball freely suspended from a thread, the clay ball was not shaken and the two copper balls stayed at rest and equally sunk in the clay and after measurement these equal impressions were found to be much greater than the impression that ball three with the velocity of one had made, having hit only the fastened clay ball and less than that which had been made by ball 1 with the velocity three. For ball 3 consumed its force to make an impression on clay, and its impression having been augmented by the effort of ball one that pressed the clay ball against ball three, reduced the impression of this ball one. Therefore, soft bodies that collide with velocities in inverse proportion to their masses, stay at rest after the collision, because they consume all their forces 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 or cohesion, must be overcome, and nothing but the force used to drive into these parts is consumed in the collision. 
For both ‘s Gravesande and du Châtelet force is a positive magnitude . Besides, she obviously agrees with ‘s Gravesande that opposite forces do not destroy each other in a direct manner but their interaction is much more complicated: in the collision of two bodies whose forces are opposite there are two actions and two reactions. 
But there is one crucial difference between them: ‘s Gravesande, a Newtonian converted to the Leibnizian force measure by his experiments, was particularly sensitive to difficulties involved in theorizing observational data. For him, the concept of force is vague and leaves room for alternative measurements:
If the word ‘force’ is given a different meaning, if this different meaning is said to be more natural, I do not object: all I wanted to claim is that this what I have called ‘force’ must be measured by the product of mass and velocity squared. In order to claim that it is possible to assume a different measure of force as considered under a different aspect it is necessary to explain all the experiments conducted with respect to force and collision. This is what we do on our part; and I assure you that this has not been done yet by those who have adopted the contrary opinion. 
Not so Madame du Châtelet. The Marquise’s irony towards de Mairan, sardonic despite her capacity to grasp counterarguments, tempts one to suppose that it is one of the aforementioned difficultés métaphysiques that underlies it. Should the Cartesian force be posited as a metaphysical principle of the Universe, the Universe could potentially be left with a metaphysically embarrassing zero value (like in the case of two moving bodies whose momenta are equal but opposite: p and –p). In this respect, velocity squared in the vis viva formula guarantees its superiority.
What follows from this is that the relationship between the speculative and the experimental in du Châtelet’s arguments is far from being straightforward. On the one hand, rigorous conceptualization of experiments like that of Boerhaave can serve to build up metaphysical principles, e.g. weightless fire as one of the springs of the Creator. On the other, there is sometimes hidden metaphysical bias in interpreting experiments as the example of Hermann’s balls proves. This complex mix is certainly incommensurable with mathematized classical mechanics as taught today. The question that imposes itself here is: are we really able to pin down the slippery Proteus of experimentalism with a Leibnizian tinge?
- MLXXXIV – A M. de Mairan, à Bruxelles, le 1er avril 1741, in Oeuvres complètes de Voltaire, ed. Ch. Lahure, vol. 25 [Paris: Librairie de L. Hachette, 1861], p. 522.
- de Mairan, Dissertation sur l’estimation et la mesure des forces motrices des corps, Nouvelle édition, ed. Deidier [Paris, 1741], pp. 7-8.
- Mary Terrall, “Vis viva Revisited,” History of Science 42 (2004): 189-209.
- cf. Letter 148. To Pierre Louis Moreau de Maupertuis, Les lettres de la Marquise du Châtelet, ed. Theodore Besterman [Genève: Institut et Musée Voltaire, 1958], vol. 1, pp. 266-267; the errata allowed by the Academy contains nothing but a stylistic improvement; note a factual mistake in Du Châtelet, Selected Philosophical and Scientific Writings, ed. Judith P. Zinsser [Chicago: University of Chicago Press, 2009], p. 77, note 54 and p. 110, note 10: “In the errata that she was allowed to submit, she changed a reference to Dortous de Mairan’s formula for force to that of Bernoulli. She had been reading Bernoulli and Leibniz on the nature of collisions and had changed her mind.”
- Dissertation, p. 128.
- du Châtelet describes it on page 459 ff. of the Institutions physiques.
- Jakob Hermann, “De mensura virium corporum,” Commentarii Academiae Scientiarum Imperialis Petropolitanae 1 (1726, published 1728): 14.
- de Mairan, “Lettre sur la question des forces vives,” in du Châtelet, Institutions Physiques, p. 487 ff.
- cf. Leibniz’s “Essay de Dynamique sur les loix du mouvement,” unpublished at the time, in Leibnizens Mathematische Schriften, ed. Carl Immanuel Gerhardt, Bd. 6 [Halle: H. W. Schmidt, 1860], p. 215.
- Institutions physiques, p. 529.
- cf. Boudri’s interesting interpretation. However, ‘s Gravesande mentions this experiment in Essai d’une nouvelle théorie du choc des corps and not in Nouvelles expériences, as Boudri claims. Christiaan Boudri, What Was Mechanical about Mechanics: The Concept of Force between Metaphysics and Mechanics from Newton to Lagrange, trans. Sen McGlinn [Dordrecht: Kluwer Academic Publishers, 2002], p. 108.
- Institutions physiques, pp. 466-467. For this passage, I consulted the translation by I. Bour and J. P. Zinsser; Du Châtelet, Selected Philosophical…, pp. 196-197. There are, however, small inaccuracies. E.g. “He took a firm ball of clay and, using Mariotte’s Machine…” See ‘s Gravesande’s description on p. 236: “…une pièce de bois bien affermie par des vis, dans laquelle il y avoit de chaque côté une cavité en demi-sphère, qui servoit à affermir une boule de terre glaise…” ‘s Gravesande, “Essai d’une nouvelle théorie du choc des corps,” in Oeuvres philosophiques et mathématiques, ed. J. N. S. Allamand [Amsterdam: Rey, 1774], Première Partie, pp. 235-236.
- cf. ‘s Gravesande, Essai d’une nouvelle théorie du choc, p. 219, definition II and du Châtelet’s malicious remark that de Mairan would not like to be hit by a body moving with a considerable force either from the left or from the right side.
- cf. the combination of the loss of velocity and indentation in ‘s Gravesande’s experiment discussed above.
- “Nouvelles expériences,” in Oeuvres philosophiques et mathématiques, Première Partie, p. 284.