VIS VIVA Revisited

Abbreviations.

AS: Académie Royale des Sciences (Paris).

BEB: Bernoulli-Edition, Basel.

HAS: Histoire de l'Académie Royale des Sciences (Paris).

MAS: Mémoires de l'Académie Royale des Sciences (Paris).

“The mathematical expressions by themselves were not sufficient to remove existing confusions.” Thomas Hankins, “Eighteenth-century attempts to resolve the vis viva controversy”, Isis, lvi (1965), 281–97, p. 286. All of the following relations can be found in the mathematics literature of the late seventeenth and early eighteenth centuries: Fdx = mv dv; F dt = mdv; Fdx = ½v² (for unit mass). See also Hankins T. , Jean d'Alembert: Science and the Enlightenment (Oxford, 1970), 204–13.

On Varignon, see Blay Michel , La naissance de la mécanique analytique: La science du mouvement au tournant des XVIIe et XVIIIe siècles (Paris, 1992); Guicciardini Niccolo , Reading the Principia: The debate on Newton's mathematical methods for natural philosophy from 1687 to 1736 (Cambridge and New York, 1999), 201–5; and Shank J. B. , “Before Voltaire: Newtonianism and the origins of the Enlightenment in France, 1687–1734”, Ph.D. dissertation, Stanford University, 2000. See also Guerlac Henry , Newton on the Continent (Cornell, 1981).

On the reception of Newton, see Meli Domenico Bertoloni , Equivalence and priority: Newton versus Leibniz (Oxford, 1993), chap. 9, and Gingras Yves , “What did mathematics do to physics?”, History of science, xxxix (2001), 383–416. For a contemporary French account of Newton's accomplishments, see de Fontenelle B. , “Éloge de M. le Chevalier Newton”, HAS, 1727, 151–72.

Leibniz G. W. , “Brevis demonstratio erroris memorabilis Cartesii et aliorum circa legem naturalem”, Acta eruditorum, 1686, 161–3. Leibniz developed his views on force in his other dynamical works, especially Specimen dynamicum. See Iltis Carolyn , “Leibniz and the vis viva controversy”, Isis, lxii (1971), 21–35, and “The decline of Cartesianism in mechanics”, Isis, lxiv (1973), 356–73.

'Gravesande was famous for having changed his mind about the measure of force after conducting experiments by dropping hard balls into soft clay and measuring the depth of the impressions in a series of papers in Journal littéraire de la Haye, starting in 1722. Samuel Clarke sent a letter to the Philosophical transactions in 1728 (after Newton's death) arguing that a vote for vis viva was a vote against the probity of the great man, and 'sGravesande responded. See Hankins , “Eighteenth-century attempts” (ref. 1).

David Papineau refers to the Continental mathematicians as “Dutch, German and Italian”, presumably referring to their native language rather than geographical origin, as the Swiss were key protagonists in the ongoing dispute. Papineau David , “The vis viva controversy: Do meanings matter?”, Studies in the history and philosophy of science, viii (1977), 111–37, p. 111.

Some salient examples from the rich literature on this subject, in addition to works cited above (ref. 4), are: Iltis C. , “D'Alembert and the vis viva controversy”, Studies in the history and philosophy of science, i (1970), 135–44; idem, “Madame du Châtelet's metaphysics and mechanics”, Studies in the history and philosophy of science, vii (1977), 29–48; Hankins , Jean d'Alembert (ref. 1), 204 ff.; Papineau , “Vis viva controversy” (ref. 6); and Costabel Pierre , La signification d'un débat sur trente ans (1728–1758): La question des forces vives (Paris, 1984).

de Fontenelle B. , “Sur la force des corps en mouvement”, HAS, 1728, 73–96. Costabel , “La question des forces vives” (ref. 7), analyses the published mémoires to determine by modern standards where the various participants were right and/or wrong. The extent of discussion is only evident from private correspondence and the minutes of Academy meetings. See Terrall Mary , The man who flattened the Earth: Maupertuis and the sciences in the Enlightenment (Chicago, 2002), chap. 3. On Fontenelle's policy of smoothing over academic disputes for public consumption, see Greenberg John , The problem of the Earth's shape from Newton to Clairaut (Cambridge, 1995), 114–17.

de Fontenelle B. , “Sur la force des corps en mouvement”, HAS, 1721, 81.

“Quelles sont les loix suivant lesquelles un corps parfaitement dur, mis en mouvement, en meut un autre de même nature, soit en repos, soit en mouvement, qu'il rencontre, soit dans le vuide, soit dans le plein?” Maindron Ernest , Les fondations des prix à l'Académie des sciences (Paris, 1881), 18.

For Bernoulli's role in training l'Hôpital to use the calculus, see Feigenbaum Lenore , “The fragmentation of the European mathematical community”, in The investigation of difficult things, ed. by Harman P. M. Shapiro Alan (Cambridge, 1992). See also Die Briefwechsel von Johann I Bernoulli, ii: Der Briefwechsel mit Pierre Varignon, 1692–1702 (Basel, 1988), and Meli Bertoloni , Equivalence and priority (ref. 3).

Johann Bernoulli to Dortous de Mairan, 21 January 1724, BEB.

Mairan to Bernoulli , 5 February 1724, BEB.

Bernoulli Johann , Discours sur les loix de la communication du mouvement, in his Opera omnia (Lausanne and Geneva, 1742), iii, 47. On Bernoulli's treatment of vis viva, see Harman P. M. , “Dynamics and intelligibility: Bernoulli and Maclaurin”, in Woolhouse R. S. (ed.), Metaphysics and philosophy of science in the seventeenth and eighteenth centuries (Amsterdam, 1988), 213–25, and Shea William , “The unfinished revolution: Johann Bernoulli (1667–1748) and the debate between the Cartesians and the Newtonians”, in Shea W. (ed.). Revolutions in science (Canton, MA, 1988), 70–92.

de Fontenelle B. , “Sur la force des corps en mouvement”, HAS, 1721, 82. This remark appeared in a brief article summarizing Louville's first comments on living force in the Academy.

Maclaurin Colin , “Démonstration des loix du choc des corps”, Pièce qui a remporté le prix de l'Académie Royale des Sciences (Paris, 1724). Bernoulli's essay was published subsequently as Discours sur les loix de la communication du mouvement (Paris, 1727); it also appeared as runner-up in the Academy's Receuil des pièces qui ont remporté les prix de l'Académie (Paris, 1732), and in Bernoulli , Opera omnia (ref. 14), iii.

Mairan to Bernoulli , 26 April 1724; Bernoulli to Mairan , 30 September 1724; Mairan to Bernoulli , 27 November 1724; Bernoulli to Mairan, 18 December 1724, BEB. Bernoulli sent his scathing comments on the inadequacies of Maclaurin's essay to Mairan, who refused to circulate them to other academicians.

Mairan to Bernoulli , 26 April 1724, BEB.

For the prizewinning essays, and some runners-up (including the essay by Bernoulli), see Receuil des pièces (ref. 16).

Bernoulli to Mairan , 26 December 1724, BEB.

Fontenelle , “Sur la force des corps” (ref. 8), 73: “On se réveilla dans l'Académie sur ces forces vives, auxquelles on ne pensoit plus, on examina cette matiere avec plus de soin, et on se partagea.”.

The most antagonistic were papers by Bouguer and Saurin; the latter openly accused Bouguer of “paralogisms”. Saurin , “Défense de la démonstration a priori de M. Bernoulli sur les forces vives, contre le mémoire de M. Bouguer sur les mêmes forces”, AS, 6 March 1728, fol. 93v. (This piece was never published.) For the official version, see Fontenelle , “Sur la force des corps en mouvement” (ref. 8).

Bouguer was not yet a member of the Academy; he was elected only in 1731.

de Fontenelle B. , “Éloge de M. Saurin”, HAS, 1737, 110–20, and de Fouchy Grandjean , “Éloge de M. Camus”, HAS, 1768, 144–54.

Saurin Joseph , “Défense de la démonstration a priori de Mr. Bernoulli sur les forces vives, contre le Mémoire de M. Bouguer sur les mêmes forces”, AS procès-verbaux, 6 March 1728, fols. 89–93v. Mairan objected to the trivializing implication of “dispute de mots”: Mairan to Bernoulli , 27 April 1728, BEB.

Camus gave two papers on vis viva in the winter of 1728, one on 4, 11 and 14 February and another on 10, 13, and 17 March (AS procès-verbaux). See Camus Charles-Étienne , “Du mouvement acceleré par des ressorts, et des forces qui resident dans les corps en mouvement”, MAS, 1728, 159–96. Fontenelle decided to publish just two papers of the many on vis viva read to the Academy in 1728: Camus in favour and Mairan against (see below, ref. 34). Louville's paper was not printed because it was too direct an attack on Bernoulli.

Johann Bernoulli to Thiancourt, 29 May 1729, BEB. Bernoulli had probably heard about the controversy in the Academy from Gabriel Cramer, a Swiss mathematician travelling in England at this time, but I have not found the letter where it was first mentioned. See Bernoulli to Cramer, 4 September 1728, BEB: “Je plains surtout le pauvre Abbé le Camus contre lequel il s'eleva un murmure de ce qu'il attaquoit une Opinion approuvée par l'Académie.”.

Mairan to Bernoulli , 23 February 1728, BEB.

Bernoulli to Cramer , 13 May 1728, BEB.

Bernoulli to Mairan , 14 March 1728, BEB.

E.g., Bernoulli to Mairan , 14 March 1728, BEB.

Mairan to Bernoulli , 27 April 1728, BEB.

On Mairan's physics, see Kleinbaum A. R. , “Dortous de Mairan”, Ph.D. dissertation, Columbia University, 1970; Hine Ellen McNiven , “Dortous de Mairan, the ‘Cartonian’”, Studies on Voltaire and the eighteenth century, cclxvi (1989), 63–179; and Costabel , “La question des forces vives” (ref. 7), 37–43. In the secondary literature, Mairan's 1728 mémoire on mechanics has not been read in light of the lively dispute that took place in the closed meetings of the Academy in 1728–29.

de Mairan J.-J. Dortous , “Dissertation sur l'estimation et la mesure des forces motrices des corps”, MAS, 1728, 1–50, p. 40.

Ibid., 41–42, emphasis in original.

Ibid., 47.

Mairan mentions his studies of l'Hôpital in Mairan to Malebranche , 17 September 1713, Correspondance de Malebranche avec J.-J. Dortous de Mairan (Paris, 1947), 8. Mairan studied physics and mathematics as a young man in Paris at the turn of the eighteenth century; he then returned to his birthplace in Béziers, where he pursued various physics projects (on ice, phosphors, and light) while he worked as an assistant to the bishop there. He came back to Paris only in 1719, at the age of 43.

On Mairan's phenomenological approach to geodesy, see Greenberg , Problem of the Earth's shape (ref. 8), chap. 2.

Quotation from Bernoulli to Thiancourt , 29 May 1729, BEB.

See Meli Bertoloni , Equivalence and priority (ref. 3), 192–4.

Bernoulli to Cramer (in London), 4 September 1728, BEB.

Bernoulli to Maupertuis , 10 August 1730, BEB. Bernoulli had several young Swiss protégés, notably Gabriel Cramer and Samuel Klingenstierna, who travelled to Paris in this period with instructions to try to convert Mairan. See Cramer to Bernoulli, 6 May 1729, and Bernoulli to Cramer, 1 September 1731, BEB, where he encourages Cramer to reply to Mairan: “… vous serés plus capable que tout autre personne de Vous en acquiter dignement. … Il est vrai que Mr. de Maupertuis pourroit aussi le faire, mais étant son Compatriote et son Collegue et présentement tous deux dans la classe des Pensionnaires, il aura des raisons politiques de ne pas se mettre en lice contre Mr. de Mairan.”.

Bernoulli to Cramer , 3 August 1729, BEB.

Maupertuis to Bernoulli , 11 March 1731, BEB.

Bernoulli to Maupertuis , 11 October 1731, BEB.

Maupertuis to Bernoulli , 10 March 1732, BEB.

Maupertuis to Bernoulli , March 1731, BEB.

Bernoulli developed this line of attack in Bernoulli to Maupertuis , 13 April 1732, as well as in other letters from 1731–32.

Maupertuis to Bernoulli , 20 October 1732, BEB. Maupertuis's paper was never read to the Academy, and the manuscript does not survive.

Maupertuis to Bernoulli , 10 March 1732, BEB.

“Je crois comme vous, que ce n'est qu'une dispute de mots, que Mr. de Mairan auroit pu terminer tout d'un coup la dispute & qu'il semble qu'il ait voulu allonger”, Châtelet to Maupertuis, February 1738, in Besterman T. (ed.), Les lettres de la Marquise du Châtelet (Geneva, 1958), i, 217.

Châtelet to Maupertuis , 1 September 1738, ibid., i, 254–5.

Gabrielle-Émilie Breteuil du Châtelet, Institutions de physique (Paris, 1740). On the composition of this work, see Janik Linda Gardiner , “Searching for the metaphysics of science: The structure and composition of Mme. du Châtelet's Institutions de physique, 1737–1740”, Studies on Voltaire and the eighteenth century, cci (1982), 85–113, and Barber W. H. , “Mme du Châtelet and Leibnizianism: The genesis of the Institutions de physique”, in The Age of Enlightenment: Studies presented to Theodore Besterman, ed. by Barber W. H. (Edinburgh and London, 1967), 200–22.

Châtelet to Maupertuis, 20 January 1739, in Besterman , op. cit. (ref. 51), i, 310–11. “Si vous pouvez déterminer un de mssrs. Bernoulli à apporter la lumière dans mes ténèbres j'espere qu'il sera content de la docilité, de l'application, et de la reconnaissance de son écolière. Je … sens avec douleur que je me donne autant de peine que si j'apprenais le calcul, et que je n'avance point, parce que je manque de guide.” She continued to consult Maupertuis and Clairaut by letter; Clairaut worked with her subsequently on her translation and commentary on Newton's Principia. See Zinsser Judith , “Translating Newton's ‘Principia’: The Marquise du Châtelet's revisions and additions for a French audience”, Notes and records of the Royal Society of London, lv (2001), 227–45.

For her frustrations with learning the algorithms of calculus (with König), see Châtelet to Maupertuis , June 1739, in Besterman , Lettres de la Marquise du Châtelet (ref. 51), i, 369.

Vis viva is covered in Institutions de physique, chap. 21. On gravity as an infinite spring, ibid. , 421–22, and fig. 74; attack on Mairan , ibid., 429–33.

On Châtelet's dispute with Mairan, and her efforts to be taken seriously as a contributor to the discourse of science, see Terrall M. , “Émilie du Châtelet and the gendering of science”, History of science, xxxiii (1995), 283–310. See also litis, “Madame du Châtelet's metaphysics and mechanics” (ref. 7), and Kawashima Keiko , “La participation de Madame du Châtelet à la querelle sur les forces vives”, Historia scientiarum, xl (1990), 9–28.

de Mairan J.-J. Dortous , Lettre à Mme sur la question des forces vives (Paris, 1741); Châtelet , Réponse de Mme la Marquise du Châtelet à la lettre que M. de Mairan, secretaire perpetuel de l'Académie Royale des Sciences, lui à écrite le 18 février, 1741, sur la question des forces vives (Brussels, 1741); and Deidier Abbé , Nouvelle réfutation de l'hypothèse des forces vives (Paris, 1741). Deidier's book was published (and often bound) with Mairan , Dissertation sur l'estimation et la mesure des forces motrices des corps, and with Mairan , Lettre à Mme . Châtelet, Institutions de physique (Amsterdam, 1742); she also reprinted the exchange with Mairan as an appendix to her Dissertation sur le feu (Amsterdam, 1744).

Clairaut to Châtelet, n.d. [May 1741?], in Besterman T. (ed.), Correspondance de Voltaire (Geneva, 1968–2001), xcii, 29.

Châtelet to Maupertuis, [10 February 1738], in Besterman , Lettres de la Marquise du Châtelet (ref. 51), i, 216–17.

Voltaire to Mairan , 9 November 1736, in Correspondance de Voltaire (ref. 59), lxxxviii, 113.

Voltaire had adopted Newton's natural philosophy in his controversial Elements of Newton's philosophy (first edn, 1738). To counter Châtelet's Institutions de physique, Voltaire wrote Métaphysique de Newton (Amsterdam, 1740), which became the first section of Elémens de la philosophie de Newton starting with the 1741 edition. See Walters R. L. Barber W. H. , “Introduction” to Voltaire, Elémens de la philosophie de Newton, in Oeuvres complètes de Voltaire (Oxford, 1992), xv, 98–118.

Voltaire to Mairan , 9 November 1736, in Correspondance de Voltaire (ref. 59), lxxxviii, 113.

Ibid.

Voltaire , “Doutes sur la mesure des forces motrices”, in Beuchot (ed.), Oeuvres de Voltaire (Paris, 1829), xxxviii, 490–501, p. 491.

Ibid., 500.

Voltaire to Mairan , 1 April 1741, in Correspondance de Voltaire (ref. 59), xci, 459.

de Mairan J-J. D. , “Sur les forces motrices des corps”, HAS, 1741, 149–53. Voltaire told Mairan that Pitot had “converted” to Mairan's position on forces after reading the 1728 paper. Voltaire to Mairan , 9 November 1736, Correspondance de Voltaire (ref. 59), lxxxviii, 113.

Clairaut A. Pitot H. , “Rapport sur le mémoire de Mr. de Voltaire sur les forces vives”, AS procès-verbaux, 1741, fols 124–6.

Voltaire , “Doutes sur la mesure des forces motrices et sur leur nature, présentés à l'Académie des Sciences de Paris en 1741”, Nouvelle bibliothèque, 1741, 219–33.

Elisabeth Badinter suggests that Voltaire's contribution, and his flattery of Mairan, was motivated by his wish to be elected to the Academy of Sciences, though the evidence is circumstantial at best. Badinter E. , Les passions intellectuels: Désirs de gloire (Paris, 1999), 226–7.

On this process, see Gingras , “What did mathematics do to physics?” (ref. 3), and Terrall Mary , “Metaphysics, mathematics and the gendering of science in eighteenth-century France”, in The sciences in enlightened Europe, ed. by Clark William Golinski Jan Schaffer Simon (Chicago, 1999), 246–71.

Papineau , “The vis viva controversy” (ref. 6).