History of Science and History of Mathematization: The Example the Science of Motion at the Turn of the 17th and 18th Centuries

  • Michel Blay
Part of the Boston Studies in the Philosophy of Science book series (BSPS, volume 151)


For the sake of clarity I shall enounce, from the outset, the meaning I give to the expression “history of mathematization”:
  • It is a historical approach whose object is to examine the precise role of mathematics and its development, as a dynamic and creative factor, in the realm of mathematical physics. Or to put it differently:

  • It is an examination, based on historical examples, of the specifically mathematical impact of mathematical physics in conceptual coordination, coherence and genesis.


Supra Note Central Force Rectilinear Motion Rectilinear Trajectory Conceptual Coordination 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Liste des Oeuvres de Pierre Varignon“, Briefwechsel,II-1, 387–408.Google Scholar
  2. AM, since 1699 (1702).Google Scholar
  3. 3.
    Explicit references to Newton’s Principia appear quite early under Varignon’s pen: Briefwechsel,II-1, 96 and 229 (Letters to Johann I Bernoulli dated 24 May and 12 July 1699).Google Scholar
  4. 4.
    AM, 1700 (1703), 84.Google Scholar
  5. 5.
    AM, 1707 (1708), 382.Google Scholar
  6. 6.
    On this issue see, in particular, Michel Blay “Deux moments de la critique du calcul infinitésimal: Michel Rolle et George Berkeley”, Revue d’Histoire des Sciences (1986), 223–253.Google Scholar
  7. 7.
    I have edited these Memoirs under the title “Quatre Mémoires inédits de Pierre Varignon consacrés à la science du mouvement”, Archives Internationales d’Histoire des Sciences (1989), 218–248. For an indepth study of these matters, see my book La naissance de la mécanique analytique (Paris, PUF, 1992).Google Scholar
  8. 8.
    A.Ac.Sc. Registres, t. 17, fol. 298 v° — 305 r°.Google Scholar
  9. 9.
    AH, 1700 (1703), 85.Google Scholar
  10. 10.
    The term “variable” belongs to the language of the period. See Analyse,“Définition I”.Google Scholar
  11. 11.
    A.Ac.Sc. Registres, t. 17, fol. 298 v°.Google Scholar
  12. 12.
    I recall that in modern terms using the concept of function (x representing space, t time, and y velocity) motion is defined by its temporal equation x = f (t) Analysis of this function will give the behavior of the point on the trajectory. The graph of the function a = f’(t) is called the diagram of accelerations. Varignon also introduces, to use current terminology, the function y = g (x),whose graph we name the first diagram of velocities (the second being that of v = f(t)) Google Scholar
  13. 13.
    A.Ac.Sc. Registres, t. 17, fol. 298 v°.Google Scholar
  14. 14.
    Varignon does not define explicitely here the concept of instant. On the other hand, in his Traité du mouvement et de la mesure des eaux coulantes et jaillissantes. Avec un Traité préliminaire du Mouvement en général. Tiré des ouvrages manuscrits de feu Monsieur Varignon, par l’Abbé Pujol, published posthumously in Paris in 1725, one reads: “Définition V. Un instant est pris ici pour la plus petite portion de temps possible, et par conséquent doit être regardé comme un point indivisible d’une durée quelconque”, 3. In a Memoir dated July 6, 1607 under the title “Des mouvemens variés à volonté, comparés entr’eux et avec les uniformes”, the definition appears in a slightly different mathematical setting: “Définition I. Par le mot d’instant, nous entendons ici une particule de tems infiniment petites, c’est-à-dire, moindre que quelques grandeur assignable de tems infiniment petite, c’est-à-dire, moindre que quelque grandeur assignable de tems que ce soit: c’est ce qu’en langage des Anciens l’on appelleroit minor quavis quantitate data…”, AM, 1707 (1708), 222.Google Scholar
  15. 15.
    A.Ac.Sc. Registres, t. 17, fol. 298 v°.Google Scholar
  16. 16.
    Ibid., fol. 298 v° — 299 r°.Google Scholar
  17. 17.
    AM, 1707 (1708), 222–274.Google Scholar
  18. 18.
    Ibid., 223.Google Scholar
  19. 19.
    I will use either the expression “Leibnizian calculus” or “calculus of differences” rather than “differential calculus”, which could imply too modern an approach. In the Analyse des infiniment petits pour l’intelligence des lignes courbes (Paris, 1696) of L’Hospital, one reads: “Définition II. La portion infiniment petite dont une quantité variable augmente ou diminue continuellement, en est appelée la différence”, 1 Google Scholar
  20. 20.
    Analyse, 2.Google Scholar
  21. 21.
    Varignon will also call this concept “vitesse instantanée”, AM,1701 (1708), 224. I nevertheless prefer to speak of the velocity at each instant; for Varignon interprets the expression dx/dt not as a derivative but rather as a quotient. See especially A.AC.Sc. Registres,t. 17, fol. 386 r°.Google Scholar
  22. 22.
    Traité du mouvement…, 22.Google Scholar
  23. 23.
    A.Ac.Sc. Registres, t. 17, fol. 299 r°.Google Scholar
  24. 24.
    Ibid., t. 17, fol. 386 r° - 391 v°.Google Scholar
  25. 25.
    Ibid., fol. 387 r°.Google Scholar
  26. 26.
    See Analyse, 3.Google Scholar
  27. 27.
    KG represents the distance covered along its trajectory GG. This point K, the origin of the motion along the trajectory, will be very useful in the rest of the Memoir.Google Scholar
  28. 28.
    A.Ac.Sc. Registres, t. 17, fol. 387 r°.Google Scholar
  29. 29.
    In 1697 and 1698 Varignon already worked on the problem of isochronous fall in the case of the reversed cycloid, but this procedure remained geometrical and did not belong to an approach using the concept of velocity at each instant.Google Scholar
  30. 30.
    La courbe cc“ corresponds to the first diagram of velocities (see above note 12).Google Scholar
  31. 31.
    A.Ac.Sc. Registres, t. 17, fol. 387 v°.Google Scholar
  32. 32.
    Ibid., fol. 391 r° - 391 v°.Google Scholar
  33. 33.
    Supra Note 7.Google Scholar
  34. 34.
    AM, 1700 (1703), 22–27; A.Ac.Sc. Registres, t. 19, fol. 31 r° - 37 r°.Google Scholar
  35. 35.
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  36. 36.
    AM, 1700 (1703), 218–237; A.Ac.Sc. Registres, t. 19, fol. 360 v° - 364 v°.Google Scholar
  37. 37.
    AM, 1700 (1703), 22. The expression “force centrale” seems somewhat ambiguous; for Varignon, at no point, explicitly introduces any consideration of the mass.Google Scholar
  38. 38.
    A.Ac.Sc. Registres, t. 19, fol. 31 r°.Google Scholar
  39. 39.
    Ibid., fol. 31 r°.Google Scholar
  40. 40.
    Ibid., fol. 31 r°.Google Scholar
  41. 41.
    AM, 1700 (1703), 22.Google Scholar
  42. 42.
    Ibid., 23.Google Scholar
  43. 43.
    Varignon by “faisant dt constante” merely makes use of procedures current at the time of Leibnizian calculus. On this issue, see H. J. M. Bos, “Differentials, Higher-Order Differentials and the Derivative in the Leibnizian Calculus”, Archive for History of Exact Sciences (1974–1975), 3–90.Google Scholar
  44. 44.
    See also E. J. Aiton, “The Inverse Problem of Central Forces”, Annals of Science (1964), 86, note 17.Google Scholar
  45. 45.
    AM, 1700 (1703), 23.Google Scholar
  46. 46.
    Principia, translated into English by Andrew Motte in 1729 (University of California Press, 1934, 1962), 34–35.Google Scholar
  47. 47.
    AM, 1700 (1703), 23.Google Scholar
  48. 48.
    Auguste Comte, Philosophie première, cours de philosophie positive, leçons I à 45 (Paris, Hermann, 1975), dix-septième leçon, 268.Google Scholar
  49. 49.
    AM, 1700 (1703), 23.Google Scholar
  50. 50.
    Ibid., 26.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1994

Authors and Affiliations

  • Michel Blay
    • 1
  1. 1.Center National de la Recherche ScientifiqueParisFrance

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