New insight into the origins of the calculus war

ABSTRACT

The consensus today is that both Newton and Leibniz created calculus independently. Yet, this was not so clear at the beginning of the eighteenth century. A bitter controversy took place at that time, which came to be known as the ‘calculus war’, probably the greatest clash in the history of science. While it is accepted that the debate started when Fatio de Duillier publicly accused Leibniz of plagiarism in 1699, earlier evidence of its origins can be found in an exchange of letters between Leibniz and Huygens.

KEYWORDS:

• Leibniz
• Huygens
• Newton
• calculus
• Fatio de Duillier
• history of mathematics

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Acknowledgements

This work would not have been possible without the suggestions and commentaries of Prof. Jeffrey K. McDonough, Prof. Richard Arthur, and Dr Siegmund Probst. Prof. Bartolomé Segura also helped with a couple of Latin passages. Lastly, I thank the anonymous referees for their helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 S. Probst, ‘The Calculus’, in The Oxford Handbook of Leibniz, ed. by M. R. Antognazza (Oxford: Oxford University Press, 2018), pp. 212–13.

2 C. J. Scriba, ‘The Inverse Method of Tangents: A Dialogue between Leibniz and Newton (1675–1677)’, Archive for History of Exact Sciences, 2, no. 2 (1963), 113–37.

3 G. W. Leibniz, Sämtliche Schriften und Briefe, (Darmstadt and Berlin: Deutschen Akademie der Wissenschaften zu Berlin, 1923ff.), III 4, n.271, p. 548.

4 Ibid., III 4, n. 271, p. 548; III 4, n. 282, pp. 588–619; III 4, n. 283, pp. 619–24.

5 Ibid., III 4, n. 291, pp. 654–55.

6 R. Iliffe, ‘Servant of Two Masters: Fatio de Duillier between Christiaan Huygens and Isaac Newton’, in Newton and the Netherlands. How Isaac Newton Was Fashioned in the Dutch Republic, ed. by E. Jorink and A. Maas (Amsterdam: Leiden University Press, 2012), p. 69.

7 Ch. Huygens, Oeuvres complètes de Christiaan Huygens, ed. by the Société hollandaise des sciences, 22 vols. (The Hague: Martinus Nijhoff, 1888–1950), 22, pp. 126–51.

8 Ibid., 22, n. LXXIV, p. 149.

9 Leibniz, SämtlicheSchriften und Briefe, III 5, n. 41, pp. 181–89.

10 Ibid., III 4, n. 296, pp. 689–90.

11 Ibid., III 5, n. 6, p. 47.

12 Ibid., III 5, n. 8, pp. 56–57.

13 Ibid., III 5, n. 9, p. 59.

14 ‘Si M. Fatio trouve bon de me communiquer sa methode pour vos deux lignes[,] je luy communiqueray la mienne pour ces deux d'à present, oú il a trouvé de la difficulté’, Leibniz, Sämtliche Schriften und Briefe, III 5, n. 9, p. 60.

15 Ibid., III 5, n. 18, p. 104.

16 Ibid., III 5, n. 21, pp. 111–12.

17 ‘Quand j'auray respiré un peu des distractions du voyage dont les recherches dans les archives et Bibliotheques m'ont imposé la necessité, j'envoyeray ma methode en echange de celle de M. Facio’, Leibniz, Sämtliche Schriften und Briefe, III 5, n. 22, p. 114.

18 Ibid., III 5, n. 41, pp. 181–89.

19 J. G. O'Hara, ‘Huygens, Leibniz and the “petit demon”: Agreement and Dissension in their Mathematical Correspondence’, De Zeventiende Eeuw, 12 (1996), pp. 158–159.

20 Leibniz, Sämtliche Schriften und Briefe, III 5, n. 46, p. 201.

21 O’Hara, ‘Huygens, Leibniz and the “petit demon”’, p. 159.

22 Leibniz, Sämtliche Schriften und Briefe, III 5, n. 53, p. 237.

23 Ibid., III 5, n. 65, p. 280.

24 Ibid., III 5, n. 69, p. 290.

25 A. Pérez de Laborda, Leibniz y Newton I: La discusión sobre la invención del cálculo infinitesimal, (Salamanca: Universidad Pontificia de Salamanca, 1977), p. 170.

26 P. Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford: Oxford University Press, 1996), p. 63.

27 Ibid., p. 84.

28 Ibid., p. 164.

29 D. Crippa, The Impossibility of Squaring the Circle in the 17th Century. A Debate Among Gregory, Huygens and Leibniz (Basel: Birkhäuser Verlag, 2019), pp. 37–40.

30 M. G. Reyes, ‘The Rhetoric in Mathematics: Newton, Leibniz, the Calculus, and the Rhetorical Force of the Infinitesimal’, Quarterly Journal of Speech, 90, no. 2 (2004), pp. 168–69.

31 M. Nielsen, Reinventing Discovery: The New Era of Networked Science (Princeton, NJ: Princeton University Press, 2012), p. 174.

32 See A. Goldgar, Impolite Learning, Conduct and Community in the Republic of Letters, 1680–1750 (New Haven, CT: Yale University Press, 1995).

33 R. De Grijs, Time and Time Again: Determination of Longitude at Sea in the 17th Century (Bristol: IOP Publishing, 2017), p. 8.

34 Ibid., p. 9.

35 M. Gotti, ‘Scientific Interaction Within Henry Oldenburg’s Letter Network’, Journal of Early Modern Studies, no. 3 (2014), p. 152.

36 M. Heyd, Be Sober and Reasonable, The Critique of Enthusiasm in the Seventeenth and Early Eighteenth Centuries (Leiden: EJ Brill, 1995), p. 279.

37 ‘The fact that two of his protégés, Fatio de Duillier and William Whiston were punished for holding proscribed religious views speak for itself why during his life time Newton never approved publication of any of his non-scientific writings on theology’ P. K. Srivastava, ‘Remembering Newton’, Resonance, 12, no. 8 (2007), 86.

38 Huygens, Oeuvres complètes de Christiaan Huygens, 10, n. 2721, pp. 209–11.

39 E. J. Aiton, Leibniz: A Biography (Bristol: Adam Hilger, 1985), p. 172.

40 ‘Auch der in London weilende N. Fatio de Duillier hatte 1691 Einsicht in die berühmten epistolae Newtons erhalten, die 1676 an Leibniz übersandt worden waren. Dieser gebürtige Schweizer war es dann auch, der das über die Landesgrenzen hinaus bekanntmachte, was in England zwar viele Mathematiker dachten, aber nur hinter vorgehaltener Hand zu sagen wagten: dass Leibniz' Calculus von Newtons Fluxionsrechnung abhängig sei. Davon erfuhr Leibniz 1692 allerdings noch nichts', H.J. Hess, ‘Leibniz auf dem Höhepunkt seines mathematischen Ruhms’, Studia Leibnitiana, 37, 1 (2005), pp. 48–67.

41 Ch. Wahl, ‘Between Cosmopolitanism and Nationalism: The Role of Expatriates in the Dissemination of Leibniz’s Differential Calculus’, Almagest, 5, no. 2 (2014), 61.

42 Huygens, Oeuvres complètes de Christiaan Huygens, 10, n. 2723, p. 213.

43 Ibid., 10, n. 2745, p. 271.

44 Ch. Huygens, ‘Extraits de la correspondance de Nicolas Fatio’, Bibliotheque universelle des sciences, belles-lettres et arts, 22 (1823), 256–258.

45 Huygens, Oeuvres complètes de Christiaan Huygens, 10, n. 2733, p. 242.

46 Ibid., 10, n. 2739, p. 258.

47 J. Bernoulli, ‘Supplementun defectus geometriae cartesiane circa inventionem locorum’, Acta Eruditorum, June 1696 (1696), p. 269.

48 M. Palomo, ‘Describing Reality: Bernoulli’s Challenge of the Catenary Curve and its Mathematical Description by Leibniz and Huygens’, in Leibniz and the Dialogue Between Sciences, Philosophy and Engineering, 1646–2016. New Historical and Epistemological Insights, ed. by R. Pisano, M. Fichant, P. Bussotti, and A. Oliveira (London: The College’s Publications, 2017), p. 334.

49 De Grijs, Time and Time Again, p. 11.

50 I. Newton, ‘Epistola Missa ad Praenobilem Virum D.Carolum Mountague Armigerum, Scaccarii Regii apud Anglos Cancellarium, et Societatis Regiae Praesidem, in qua Solvuntur duo Problemata Mathematica a Johanne Barnoullo Mathematico Celeberrimo Proposita’, Philosophical Transactions, 19, 224 (1697), pp. 384–389.

51 G. W. Leibniz, ‘Communicatio suae pariter duarumque alienarum […] solutionum problematis curvae celerrimi descensus’, Acta Eruditorum, May 1697 (1697), pp. 203–204.

52 N. Fatio de Duillier, Lineae brevissimi descensus investigatio geometrica duplex, (London: R. Everingham, 1699), p. 18.

53 Wahl, ‘Between Cosmopolitanism and Nationalism’, p. 61

54 Fatio de Duillier, Lineae brevissimi descensus investigatio geometrica duplex, p. 18.

55 Leibniz, Sämtliche Schriften und Briefe, III 6, n.14, pp. 44–48.

56 Ibid., III 6, n. 34, pp. 85–86.

57 D. Goodman, The Republic of Letters: A Cultural History of the French Enlightenment (Ithaca, NY: Cornell University Press, 1994), p. 143.

58 Gotti, ‘Scientific Interaction Within Henry Oldenburg’s Letter Network’, p. 153.

59 Leibniz, SämtlicheSchriften und Briefe, III 8, n. 182, pp. 463–64.

60 Hess, ‘Leibniz auf dem Höhepunkt seines mathematischen Ruhms’, p. 65, note 84.

61 Leibniz, Sämtliche Schriften und Briefe, III 8, n. 196, p. 504.

62 Fatio de Duillier, Lineae brevissimi descensus investigatio geometrica duplex, p. 18.

63 E. Knobloch, ‘Leibniz and the Brachistochrone’, Documenta Mathematica, Extra Volume ISMP (2012), p. 18.

64 J. Echevarría, Valores contrapuestos en la controversia Newton-Leibniz, Matemáticas y matemáticos, edited by A.J. Durán and J. Ferreirós, (Sevilla: Secretariado de Publicaciones de la Universidad de Sevilla, 2004), p. 87.

65 A. Rupert Hall, Philosophers at War. The Quarrel Between Newton and 
Leibniz (Cambridge: Cambridge University Press, 1980), p. 118.

66 J. de Lorenzo, ‘Estudio preliminar’, in Análisis infinitesimal, ed. by J. de Lorenzo (Madrid: Tecnos, 1987), p. LXXIII.

67 J. Babini, El cálculo infinitesimal: Origen-polémica (Buenos Aires: Editorial 
universitaria de Buenos Aires, 1972), p. 59.

68 G. W. Leibniz, Obras filosóficas y científicas, 20 vols., (Granada: Comares, 2007ff.), 7A, p. 341.

69 Ibid., 7A, p. 355.

70 A. De Morgan, Essays on the Life and Work of Newton (Chicago: Open Court, 1914).

71 C. B. Boyer, The History of the Calculus and its Conceptual Development (New York: Dover Publications, 1959).

72 M. E. Baron, The Origins of the Infinitesimal Calculus (Cambridge: Cambridge University Press, 1969).

73 Leibniz, Sämtliche Schriften und Briefe, III 8.

74 Wahl, ‘Between Cosmopolitanism and Nationalism’; O’Hara, ‘Huygens, Leibniz and the “petit demon”’.

75 R. S. Westfall, Never at Rest: A Biography of Isaac Newton (Cambridge University Press, 2005), p. 539.

76 Ibid., p. 512.

77 Aiton, Leibniz: A Biography, p. 172; Huygens, Oeuvres complètes de Christiaan Huygens, 10, n. 2723, pp. 213–15.

78 Probst, The Calculus, p. 219.

79 Westfall, Never at Rest: A Biography of Isaac Newton, p. 514.

80 A. J. Durán, La Polémica sobre la invención del cálculo infinitesimal: Escritos y documentos, (Barcelona: Crítica, 2006), p. 94.

81 M. R. Antognazza, Leibniz: An Intellectual Biography (Cambridge: Cambridge University Press, 2009), p. 356.

82 Pérez de Laborda, Leibniz y Newton I: La discusión sobre la invención del cálculo infinitesimal, p. 144.

83 Ibid., pp. 147–50.

84 Iliffe, ‘Servant of Two Masters’, p. 79.

85 Ibid., p. 68.