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Explaining the Sudden Rise of Methods of Indivisibles

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Seventeenth-Century Indivisibles Revisited

Part of the book series: Science Networks. Historical Studies ((SNHS,volume 49))

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Abstract

This book will analyse the principal methods of indivisibles developed over the course of the seventeenth century. These analyses will be preceded by an examination of the general philosophical arguments regarding the composition of continuous magnitudes with which mathematicians were faced, and followed by a comparison of the different narratives put forward by historians of mathematics with regard to the content of these methods and their role in the formulation of differential and integral calculus.

Translated from French by Sam Brightbart.

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Notes

  1. 1.

    Phys. VI, 231 a21–232 a22.

  2. 2.

    Leibniz, Math. Schr., II, p. 219. All quotations are the translator’s own translations, except where otherwise indicated.

  3. 3.

    Montucla (1799, II, p. 29).

  4. 4.

    Cantor (1901, II, p. 750).

  5. 5.

    To Galileo, 28 June 1639, in Galileo Galilei, Opere, Vol. 18, n. 3889.

  6. 6.

    Component parts (“des éléments composants”) differ from determinant and constituent parts (“des éléments déterminants” and “constitutifs”) here in that they are of the same nature as that of which they are component.

  7. 7.

    Koyré (1954), quotation is a translation from the reprint Etudes d’histoire de la pensée scientifique, Paris, PUF, 1966, pp. 303/4.

  8. 8.

    Marquis de l’Hôpital (1696), Analyse des infiniment petits….

  9. 9.

    Torricelli, Opere scelte, p. 383, quoted in De Gandt (1987, p. 152).

  10. 10.

    Giusti (1980–2, pp. 201–202).

  11. 11.

    See Chap. 14, n. 9.

  12. 12.

    See Chap. 14, p. 4.

  13. 13.

    Letter to Torricelli, 1647, in 1693, Roberval, pp. 283–284.

  14. 14.

    Pascal, Letter to Carcavy, 1658, in Œuvres Complètes, Lafuma, p. 135.

  15. 15.

    See Chap. 12 .

  16. 16.

    Leibniz (1675–1676), De quadratura arithmetica circuli…, prop VI, p. 51. See Chap. 15.

  17. 17.

    Carnot (1797).

  18. 18.

    De Gandt (1987, p. 151).

  19. 19.

    See Chap. 5.

  20. 20.

    Cavalieri (1635), Translation of the text (b2, recto) of the 1653 edition in Bologna by Malet (1996, p. 16).

  21. 21.

    Leibniz, Math. Schrift. IV, 92; 2 February 1702.

  22. 22.

    Whiteside (1960, p. 184).

  23. 23.

    Guldin (1635–1641), De centro gravitatis, Lib. IV, Cap. V, Prop II, num 4, p. 342.

  24. 24.

    See Chap. 4.

  25. 25.

    Arnauld et Nicole (1662), quoted in english ed. 1727.

  26. 26.

    Arnauld et Nicole (1662), quoted in english ed. 1727, p. 375.

  27. 27.

    Carnot (1797, p. 113).

  28. 28.

    We will not discuss here whether or not these four characteristics are strictly novel, or whether or not some were, in part, already present in the work of other writers; nor will we discuss the significant reworking that the work of Newton and Leibniz subsequently underwent. This will only be a general, qualitative evaluation.

  29. 29.

    Comte (1879, p. 42). See Chap. 17.

  30. 30.

    Boyer (1959, p. 236).

  31. 31.

    Larousse du XIXe siècle, tome 14, p. 523 c.

  32. 32.

    Boyer (1959, p. 238) quoted in Dahan and Peiffer (1986). In this book, Amy Dahan and Jeanne Peiffer provide very clear judgments: “Any attempt to find clear definitions in Newton’s work will be met with failure”, p. 191.

    “In order to explain his ‘final ratio’—which is roughly equivalent to a limit—Newton resorts to an analogy with mechanics, using the image of the final speed of a body arriving at a certain position. The method of fluxions, even when it is based on the method of first and final ratios, remains inadequate for placing differential and integral calculus on solid foundations”, p. 193.

    “The strength of Leibniz’s method lies in the simplicity of his algorithm, his elegant notation, and the formalism he employs in operations which allows him to make calculations almost automatically, masking the nature of the objects in question. Whilst Leibniz does add infinitely small magnitudes to the system of usual magnitudes, the status of the former remains extremely vague. Leibniz wavers between taking a formalist approach and relying on analogies with geometry”, pp. 196–197.

    “The concept of limits cannot be deduced from the theories of Newton and Leibniz” (p. 197).

    “Despite the generality of the methods used for them, and the addition of algebra to the calculations associated with them, differential and integral calculus were not built on solid foundations. Fundamental concepts such as limits, continuity, derivatives, and integrals were not defined”, p. 197.

  33. 33.

    Marquis de l’Hôpital (1696).

  34. 34.

    Marquis de l’Hôpital (1696), Preface.

  35. 35.

    Leibniz, Math. Schrift. VII, 361; an den Freiherrn von Bodenhausen 1690.

  36. 36.

    Math. Schr. V, p. 350.

  37. 37.

    Breger (1992, p. 80).

  38. 38.

    Dahan and Peiffer (1986, p. 198).

  39. 39.

    Whiteside (1960, p. 184).

  40. 40.

    d’Alembert (1784, t. II, 310a).

  41. 41.

    Id.

  42. 42.

    Lagrange, Œuvres, ed. Serret, t. IX, 1881, p. 18.

  43. 43.

    Quoted in Brunschvicg (1922, p. 247).

  44. 44.

    Dahan and Peiffer (1986, p. 204).

  45. 45.

    Leibniz (1710), Theodicy, p. 53.

  46. 46.

    Ibid., p. 54.

  47. 47.

    Breger (1992, p. 76).

  48. 48.

    Brunschvicg (1922, p. 215). See Leibniz, Phil. Schr., III, p. 52.

  49. 49.

    Whiteside (1960, p. 381).

  50. 50.

    Leibniz (A), VI, 6, first ed. 1765, Preface to New Essays on Human Understanding, English trad. Jonathan Bennett, perpus-fkip/Perpustakaan/Filsafat/Filsafat Barat Klasik/leibne.pdf, 2005 (cor. 2008), The metaphysical root of Leibniz’s analysis is perfectly clear in the following passage: “all the different forms of life, which together make up the universe, are, in the ideas of God, who knows their fundamental gradation, merely a number of coordinates on a curve, between which no other points could be placed, as this would signify disorder or imperfection…”, Apud Gurhauer, Phil. Schr., I, p. 32, quoted in Brunschvicg (1922. p. 229).

  51. 51.

    Chasles (1837, note XXIV, p. 359).

  52. 52.

    Dahan and Peiffer (1986, p. 198).

  53. 53.

    Malet (1996, p. 155).

  54. 54.

    Whiteside (1960, p. 384).

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  1. Département de Philosophie, Université de Nantes, Chemin de la Censive du Tertre, Nantes, France

    Vincent Jullien

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Jullien, V. (2015). Explaining the Sudden Rise of Methods of Indivisibles. In: Jullien, V. (eds) Seventeenth-Century Indivisibles Revisited. Science Networks. Historical Studies, vol 49. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-00131-9_1

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