Cet article a pour objet de montrer la nouveauté du traitement varignonien du mouvement des projectiles dans les milieux résistants par rapport au traitement de ce problème présenté entre autres par Newton dans les Principia. Aussi, après avoir analysé cursivement différentes Propositions du Livre II des Principia, nous étudierons plus spécialement, dans les Mémoires présentés par Varignon à l'Académie Royale des Sciences entre 1707 et 1711, la mise en place de l'expression d'une ‘Proposition générale’. Nous montrerons ensuite sur quelques (...) cas particuliers en quel sens on peut dire que l'étude du mouvement des projectiles dans les milieux résistants se réduit alors, pour l'essentiel, à de simples questions d'intégration et de différentiation.SummaryThe aim of this article is to show the novelty of Varignon's treatment of the motion of projectiles in resisting media in comparison with the treatment of this problem presented, among others, by Newton in the Principia. Also, having briefly analysed the various propositions of Book II of the Principia, we shall study more particularly, in the memoirs presented to the Académie Royale des Sciences by Varignon between 1707 and 1711, the setting up of the expression of a ‘general proposition’. Next, on the basis of some particular cases, we show the sense in which we can say that the study of the motion of projectiles in resisting media is thereby reduced essentially to simple questions of integration and differentiation. (shrink)

: This paper first discusses the current historical and philosophical framework forged during the last century to account for both the history and the epistemic status of Newton's theory of general gravitation. It then examines the conflict surrounding this theory at the close of the seventeenth century and the first steps towards the revolutionary shift in rational mechanics in the eighteenth century. From a historical point of view, it shows the crucial contribution of the Cartesian mechanistic philosophy and Leibnizian analytic (...) methods to the emergence of so-called Newtonian mechanics which can also be fairly characterized as a synthetic theory of attraction. From a philosophical standpoint, the paper suggests that the reworking of Newton's theory in the 18th century is better understood in a theoretical framework that reconciles Kuhn's notion of "invisible revolution" rather than his notion of "normal science" with Whewell's ascription of the completion of dynamical studies to the post Newtonian period. (shrink)

Le 31 mai 1703, Étienne-François Geoffroy, qui était alors chimiste à l’Académie royale des sciences, soutint une thèse de médecine présidée par Guy-Crescent Fagon à la Faculté de médecine de Paris sous le titre An medicus, Philosophus Mechanico-Chymicus? S’appuyant notamment sur les travaux de Varignon pour la mécanique et de Homberg pour la chimie, Geoffroy montre que le corps humain n’est pas seulement une merveilleuse machine hydrostatique, mais aussi une machine d’un ordre supérieur, dont seule la chimie peut rendre (...) compte de certaines opérations. Comme Hippocrate lui-même l’aurait suggéré, selon une lecture quelque peu anachronique, le bon médecin, loin de devoir choisir entre mécanique et chimie, doit au contraire user de leur nécessaire complémentarité pour comprendre le fonctionnement du corps humain. (shrink)

Newton certainly regarded his second law of motion in the Principia as a fundamental axiom of mechanics. Yet the works that came after the Principia, the major treatises on the foundations of mechanics in the eighteenth century—by Varignon, Hermann, Euler, Maclaurin, d’Alembert, Euler (again), Lagrange, and Laplace—do not record, cite, discuss, or even mention the Principia’s statement of the second law. Nevertheless, the present study shows that all of these scientists do in fact assume the principle that the Principia’s (...) second law asserts as a fundamental axiom in their mechanics. (For what that second law asserts, we rely on Newton’s own testimony.) Some, like Varignon and Hermann, assume the axiom implicitly, apparently unaware that any assumption is being made, while others, like Maclaurin and Euler, assume the axiom explicitly, apparently unaware that the assertion assumed is the second law as Newton himself understood it. But in every case these scientists employ the principle asserted by the Principia’s second law fundamentally, unaware that they should be citing Neutonus, Prin., Phil. Nat. Math., Lex II. (shrink)

Summary This article intends to propose new hypotheses concerning the origin of the ?Principe général? of mechanics of Jean Le Rond D'Alembert expressed in its Traité de dynamique in 1743. The examination of the statics of Pierre Varignon and its inheritance suggests that D'Alembert retains, through a case of oblique collision on a hard surface, a method of decomposition and equilibrium of forces which is close to its principle. On the other hand, this principle requires a definition of the (...) equilibrium widely spread in the 17th and 18th centuries, D'Alembert seeming then more to innovate from an epistemological point of view that on a technical one. Finally, the determination of rules of collision based at the same time on decompositions of movements at the level of the centre of mass and on appeal to a principle of relativity constitutes a know-how at the beginning of the 18th century which can be moved closer to methods developed by D'Alembert. These three aspects try then to replace the science of D'Alembert in a context by insisting on the essential role of the collisions of body and on that of the notion of equilibrium concerning the birth of its principle. (shrink)

The translation of Newton’s geometrical Propositions in the Principia into the language of the differential calculus in the form developed by Leibniz and his followers has been the subject of many scholarly articles and books. One of the most vexing problems in this translation concerns the transition from the discrete polygonal orbits and force impulses in Prop. 1 to the continuous orbits and forces in Prop. 6. Newton justified this transition by lemma 1 on prime and ultimate ratios which was (...) a concrete formulation of a limit, but it took another century before this concept was established on a rigorous mathematical basis. This difficulty was mirrored in the newly developed calculus which dealt with differentials that vanish in this limit, and therefore were considered to be fictional quantities by some mathematicians. Despite these problems, early practitioners of the differential calculus like Jacob Hermann, Pierre Varignon, and Johann Bernoulli succeeded without apparent difficulties in applying the differential calculus to the solution of the fundamental problem of orbital motion under the action of inverse square central forces. By following their calculations and describing some essential details that have been ignored in the past, I clarify the reason why the lack of rigor in establishing the continuum limit was not a practical problem. (shrink)

In this paper I attempt to trace the development of Gottfried Leibniz’s early thought on the status of the actually infinitely small in relation to the continuum. I argue that before he arrived at his mature interpretation of infinitesimals as fictions, he had advocated their existence as actually existing entities in the continuum. From among his early attempts on the continuum problem I distinguish four distinct phases in his interpretation of infinitesimals: (i) (1669) the continuum consists of assignable points separated (...) by unassignable gaps; (ii) (1670-71) the continuum is composed of an infinity of indivisible points, or parts smaller than any assignable, with no gaps between them; (iii) (1672-75) a continuous line is composed not of points but of infinitely many infinitesimal lines, each of which is divisible and proportional to a generating motion at an instant (conatus); (iv) (1676 onward) infinitesimals are fictitious entities, which may be used as compendia loquendi to abbreviate mathematical reasonings; they are justifiable in terms of finite quantities taken as arbitrarily small, in such a way that the resulting error is smaller than any pre-assigned margin. Thus according to this analysis Leibniz arrived at his interpretation of infinitesimals as fictions already in 1676, and not in the 1700's in response to the controversy between Nieuwentijt and Varignon, as is often believed. (shrink)

In Book 1 of the Principia, Newton presented two different descriptions of orbital motion under the action of a central force. In Prop. 1, he described this motion as a limit of the action of a sequence of periodic force impulses, while in Prop. 6, he described it by the deviation from inertial motion due to a continuous force. From the start, however, the equivalence of these two descriptions has been the subject of controversies. Perhaps the earliest one was the (...) famous discussion from December 1704 to 1706 between Leibniz and the French mathematician Pierre Varignon. But confusion about this subject has remained up to the present time. Recently, Pourciau has rekindled these controversies in an article in this journal, by arguing that “Newton never tested the validity of the equivalency of his two descriptions because he does not see that his assumption could be questioned. And yet the validity of this unseen and untested equivalence assumption is crucial to Newton’s most basic conclusions concerning one-body motion” (Pourciau in Arch Hist Exact Sci 58:283–321, 2004, 295). But several revisions of Props. 1 and 6 that Newton made after the publication in 1687 of the first edition of the Principia reveal that he did become concerned to provide mathematical proof for the equivalence of his seemingly different descriptions of orbital motion in these two propositions. In this article, we present the evidence that in the second and third edition of the Principia, Newton gave valid demonstrations of this equivalence that are encapsulated in a novel diagram discussed in Sect. 4. (shrink)

During the querelle des infiniment petits Leibniz wrote several texts addressed to Parisian savants to justify the use of the differential calculus, but only three of them were made public. One of the three, the “Sentiment de Monsieur Leibnitz”, was published without authorization in 1706 at the peak of the quarrel, together with the writings of other mathematicians united in the defence of the new calculus ( Joseph Saurin, Jacob Hermann and the Bernoulli brothers). However, Jean-Paul Bignon, director of the (...) Académie Royale des Sciences, confiscated the prints. All copies of this publication were assumed to be lost, but we have recently identified one copy of it in the British Library catalogue. In this article, we analyse the “Sentiment de Monsieur Leibnitz” by contextualizing it not only from an epistemological approach but also from a political perspective. We will examine the institutional issues at stake by relying on recently published correspondence. (shrink)

The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature, rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, (...) contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part–whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals—as well as imaginary roots and other well-founded fictions—may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality. (shrink)