PIERRE VARIGNON AND THE CALCULUS OF MOTION Michael S. Mahoney Princeton University The new currents of scientific thought that reshaped the Académie des Sciences in the 1690s flowed from sources outside France and met in Paris during the period 1689-92. From England came Newton's Principia, familiar in style perhaps to those who knew Huygens' work, but novel in substance surely to those schooled in Cartesian mechanics. From Germany1 by way of the Acta eruditorum came Leibniz' "Tentamen de motuum coelestium causis", familiar in its vortex cosmology but strange in its use of a new "differential calculus" to analyze the motion of bodies. From Basel came Johann Bernoulli, an unknown young mathematician skilled in that calculus and in the even newer and more mysterious "method of integrals", which his brother Jakob and he had recently be developing. Though the last of the new currents to arrive in Paris, the Bernoullis' infinitesimal analysis was the first to influence the direction of mathematics there. It opened the way to Leibniz' recent work on dynamics and thereby provided the tools with which a new generation of Academicians would recast and reorganize Newton's mechanics, integrating it into a program of research that would remain essentially Cartesian for another half-century. Hence, we shall consider the new currents in reverse order. Although Leibniz had worked out the essentials of his new calculus while still in Paris in the mid-1670s,2 he waited almost a decade before publishing an account of it in the May 1684 number of Acta eruditorum. The article, "Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractis, nec irrationales quantitates moratur, & singulare pro illis calculi, genus"3 set forth only the bare algorithms of differentiation, with no explanation or demonstration to speak of and with the sketchiest of illustrative examples. Articles that followed over the next few years shed a bit more light on the new method, especially on the ways in which it set the bases for a new, "transcendent" geometry, but they still offered little help or guidance to the reader who wanted to gain command over the calculus. Indeed, what constituted command remained itself unclear, as Leibniz failed to undertake a systematic exposition of the techniques of the calculus and of the range of its applications. 4 In particular, although he paid practically no attention at all to the inverse operation of integration and to its application to quadrature, rectification, and the inverse problem of tangents. It was instead Jakob Bernoulli and his younger brother Johann who assumed that responsibility in the late 1680s and who rapidly extended the purview of differential calculus to include all known curves and many new ones. 1More precisely, from Rome, where Leibniz was visiting when he first learned of the appearance of the Principia and hastened to put his own, similar thoughts to paper. See most recently Domenico Bertoloni Meli, Equivalence and Priority: The Newton-Leibniz Dispute on the World System (Oxford : Clarendon Press ; New York : Oxford University Press, 1993), together with the earlier studies of E.J. Aiton, "The Celestial Mechanics of Leibniz", Annals of Science 16(1960), 65-82; "The celestial mechanics of Leibniz in the light of Newtonian criticism", ibid. 18(1962), 31-41; and "The celestial mechanics of Leibniz: A new interpretation", ibid. 20(1964), 111-123. 2See J.E. Hofmann, Leibniz in Paris, 1672-1676: His Growth to Mathematical Maturity (London, New York: Cambridge University Press, 1974). 3pp.467-473; LMS.V.220-226 4See M.S. Mahoney, "Infinitesimals and Transcendent Relations", in D.C. Lindberg and R.S. Westfall (eds.), Reappraisals of the Scientific Revolution, Chap.12. ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 2 The notes on the calculus made by the Marquis de l'Hôpital for Nicolas Malebranche in 1689 show by contrast the necessity and the originality of the Bernoullis' efforts.5 L'Hôpital failed to discern in Leibniz' notation the operational nature of d when prefixed to the symbol of a quantity. Even though Leibniz had earlier spoken of dx as "a certain modification" (quaedam modificatio) of x l'Hôpital treated it as simply a variation on the e that Barrow had used to denote an infinitesimal increment.6 The examples l'Hôpital offered Malebranche scarcely went beyond what Fermat and others had been doing fifty years earlier, either in substance or in style. Two years later, when Johann Bernoulli first appeared in Paris, the situation had changed little. L'Hôpital, who counted as the leading mathematician in Paris, had but a tenuous hold on the new infinitesimal methods. Quick to recognize Bernoulli's superior skills upon first meeting him at one of Malebranche's Thursday soirées, l'Hôpital contracted with the young Swiss to learn the calculus. Installed at the Marquis' summer home in Oucques, Bernoulli composed for l'Hôpital a short exposition of the differential calculus and a long treatise on the methods of the integral calculus.7 In a now notorious arrangement, Bernoulli ceded his rights of authorship to l'Hôpital, who published the work on the differential calculus, along with some material on the integral calculus, as his own Analyse des infiniment petits, pour l'intelligence des lignes courbes in 1696. Bernoulli had another eager student in Pierre Varignon, a friend of l'Hôpital and his sometime collaborator. At the time Professor of Mathematics at the Collège Mazarin, Varignon had established a reputation in the subject through his work in statics pursued in the traditional geometrical mode. Sharing in the lectures at Oucques, and more astute and talented as a mathematician than the Marquis, Varignon quickly made the calculus his own and was soon exchanging problems and solutions with Bernoulli, who recognized in him a colleague rather than a student and quasi-patron and who saw to it personally that Varignon received issues of the Acta eruditorum as they appeared, thus overcoming an embargo on trade with France.8 Hence, Varignon was able to make steady progress toward a command of the differential and integral calculus and to direct it to his own interests. In particular, Varignon strode out on his own precisely at the point at which l'Hôpital admitted having reached the limits of his understanding. Toward the end of the lectures on the integral calculus, Bernoulli offered a series of "physico-mechanical" problems, in which the conditions placed on quantities stemmed from the physics of static forces and of bodies moving under constraint. Bernoulli did not go all that far in his presentation, and he offered nothing like a system of analytic mechanics. His purpose was to illustrate applications of the integral calculus. Only two of the problems involved motion, namely the isochrone and the tautochrone, and both of them could be approached by means of Galileo's analysis of uniform motion and his laws of fall, in particular the law that ties the velocity of a falling body to the square root of the distance through which it has fallen from rest. To apply these 5Cf. Vol. 17,2 of Malebranche's Oeuvres|. 6"De geometria recondita et analysi indivisibilium atque infinitorum", Acta eruditorum (1686), pp. 292-300; LMS.V.226-33. 7/*references*/ 8/*documentation*/ ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 3 laws, Bernoulli made an assumption that became fundamental to analytic mechanics: over infinitesimal distances ds traversed in infinitesimal times dt a motion may be considered as uniform, whatever its actual nature in the finite realm.9 That is, by Galileo's law of uniform motion, ds = vdt. This assumption is revealing in two ways of the conceptual structure of the differential calculus as it was originally developed and applied to mechanics. First, it treats the velocity as a primitive, rather than a derived quantity. If, that is, the relation ds = vdt defines anything, it defines the distance traversed by a body moving at a given speed. The relation stands as it is, rather than as a consequence of what has become a definition of velocity, namely v = ds/dt. Second, recall that, when Huygens moved from indivisibles to infinitesimal segments in his demonstration of the tautochrony of the cycloid, he worried about variations in speed over those segments. So he took pains first to try to show that the mean speed and the speeds at either end of the segment were close enough to be treated as equal and then to follow the laborious path of a double reductio ad absurdum. When Bernoulli first came to Paris, Varignon had been engaged in an effort to capture the science of weights and Galileo's laws of motion in algebraic form. It did not take much for Bernoulli to convince him that in the calculus he had a tool better suited to his purpose. Moreover, through Bernoulli's lectures Varignon gained access to the mathematical details of Leibniz' concise and confusing treatment of planetary orbits in his "Tentamen de motuum coelestium causis", hastily written upon learning of the publication of the Principia, which in turn was opened to for Varignon a new reading.10 Following a generalized account of isochronic curves in 1699, Varignon turned in 1700 to a general theory of motion determined by central forces and in a series of memoirs rendered into the language of the calculus the mechanical substance of Book I, Sections 2 - 10, of the Principia. His first memoir, "Manière générale de déterminer les forces, les vitesses, les espaces, & les temps, une seule de 9Cf. Euler, Mechanica (1736), Book I, Prop.3: In motu quantumvis inaequabili minima spatii elementa motu aequabili percurri concipi possunt.