Alexis Fontaine's integration of ordinary differential equations and the origins of the calculus of several variables

References

• November 1738 . L'Académie Royale des Sciences, registre de procés-verbaus November , 190r – 190r . 26 29 November 1738, 191r.
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• This has been published as Der Briefwechsel von Johann Bernoulli Spiess Otto Basel 1955 1 121 504
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• Ms. LIa 670 Bernoulli Archives Universitätsbibliothek Basel
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• Ms. LIa 665 Bernoulli Archives Universitätsbibliothek Basel
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• L'Hôpital's Analyse des infiniment petits (1696) was concerned exclusively with the differential calculus (in fact, it was the first text ever published on the subject). Meanwhile, ‘Varignon, who at one time earnestly but in vain sought instruction from [Johann I] Bernoulli, never gained great facility in the integral calculus’ Aiton E.J. The inverse problem of central forces Ann. sci. 1964 20 81 99 (p. 98)). Varignon's Eclaircissements sur l'analyse des infiniment petits, published posthumously in 1725, was limited to the differential calculus.
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• Letter from Montmort to Bernoulli, 27 February 1703, No. 1* Bernoulli Archives Universitätsbibliothek Basel fol. 1r.
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• Letter from Montmort to Bernoulli, 9 September 1712 Bernoulli Archives Universitätsbibliothek Basel fol. 1v; Bernard Le Bovier de Fontenelle, ‘Eloge de M. de Montmort’, Hist. Acad. Roy. Sci., (1719: publ. 1721), 83–93 (p. 85). For example, in one of the three variants of an important letter from J. Lelong to Leibniz, dated October 1707, which have been used in the past to date Malebranche's first encounter with Newton's Opticks (1704), to which De quadratura curvarum was appended, and of which excerpts of all three appear in Oeuvres de Malebranche, vol. 19, Correspondance et actes 1690–1715, (2nd ed., ed. André Robinet: 1978, Paris), 768-769, Lelong says: ‘Ce Père [Malebranche] est à la campagne avec un de ses amis. Ils ont emporté un ecrit de M. Newton imprimé à Londres en 1704, De Quadratura Curvarum …’ (p. 768). Who was Malebranche's mysterious ‘friend’? ‘Malebranche allait en vacances chez lui [Montmort] à Mareuil …’ (quoted in Oeuvres de Malebranche, vol. 20, Documents biographiques et bibliographiques (2nd ed., ed. André Robinet: 1978, Paris), 166). That Malebranche had first seen one of the reproductions made available by Montmort seems inescapable.
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• In addition to his correspondence with Bernoulli Johann I Bernoulli Archives Universitätsbibliothek Basel Montmort had a long and fruitful correspondence with Johann's extremely talented nephew, Nikolaus I Bernoulli, the early part of which was published in Montmort's Essay d'analyse sur les jeux de hazard (2nd ed., 1713, Paris), 283–414. The portion that appears in Montmort's book, however, is concerned almost exclusively with probability theory, as the title of Montmort's book would suggest. The bulk of Montmort's correspondence with Nikolaus I Bernoulli, including the part that deals with integral calculus, remains unpublished as mss. LIa 212 and LIa 222 (Bernoulli Archives, Universitätsbibliothek Basel).
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• Taylor , Brook . 1793 . Contemplatio Philosophica Edited by: Young , William . 81 – 150 . London
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• The most comprehensive study of the work of the ‘Malebranche group’ to date is Oeuvres de Malebranche, vol. 17, pt. 2, Mathematica , 2nd ed. Costabel Pierre Paris 1979 and vols. 19 and 20 of this same series (footnote 7). Perhaps the most ambitious, if unsuccessful, proposal by a Malbranchist to do Original work in integral calculus during this period came from Christophe-Bernard de Bragelongne, reputed to have been a child prodigy, who in 1711 ‘… a donna immediatement après sa réception [à l'Académie Royale des Sciences] un mémoire sur les quadratures des courbes. Il avait eu dessein, et il en fait mention lui-même au commencement de son mémoire, de donner une méthode pour intégrer les quantities différentielles à plusieurs variables [see “L'Académie Royale des Sciences, registre de procès-verbaux”, 15 July 1711, 302v–314r]; mais il n'a jamais lû que la premiere partie de cet ouvrage qui traite de la quadrature des courbes' (quoted in Jean-Paul Grandjean de Fouchy, ‘Eloge de M. l'Abbé de Bragelongne’, Hist. Acad. Roy. Sci., (1744: publ. 1748), 65–70 (p. 66)).
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• Letter from Bernoulli to Montmort, 15 June 1719, No. 14 Bernoulli Archives Universitätsbibliothek Basel 5r 5r
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• Castel , Louis-Bertrand . 1735 . “ Discours préliminaire, servant de supplément à la préface qui est à la tête de L' Analyse des infiniment petits de M. le Marquis de L'Hôpital ” . In Analise des infiniment petits, comprenant le calcul intégral, dans toute son etendue … servant de suite aux infiniment petits de M. le Marquis de L'Hôpital Edited by: Stone , Edmund . xcviii – xcviii . Paris in trans. Rondet found the second volume of Reyneau's work to be full of obstacles, and he referred the reader instead to the works of Wallis, Bernoulli, L'Hôpital, Newton, Stone, and Gregory of St. Vincent—at least as preliminary reading to Reyneau's volume!
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• Until Clairaut corrected some errors in the treatise Journal de Trévour 1728 2164 2166 and D'Alembert corrected some others that lingered in the second edition (1736) (‘L'Académie Royale des Sciences, registre de procès-verbaux’, 29 July 1739, 145r–146r), which is evidence that Clairaut and D'Alembert had a thorough understanding of its contents, evidence of specific impact of the work is lacking. The impact made by the first edition of Reyneau's treatise has never been investigated.
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• Other by-products were the Royal Academy expeditions of the mid-1730s to measure degrees of meridian in Peru and in Lapland. See Brown Harcourt From London to Lapland: Maupertuis, Johann Bernoulli I and La Terre applatie, 1728–1738 Literature and history in the age of ideas Williams Charles G.S. Columbus 1975 69 94 in and Harcourt Brown, ‘From London to Lapland and Berlin’, in Brown's Science and the human comedy (1976, Toronto and Buffalo), 167–206.
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• While Fontaine was still vying for membership in the Royal Academy, Maupertuis described him to Bernoulli as ‘a fine mathematician and a good friend’ (fort habile en géométrie et fort de mes amis). See the letter from Maupertuis to Johann I Bernoulli, 31 March 1733. No. 30* Bernoulli Archives Universitätsbibliothek Basel 2r 2r
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• Letter from Maupertuis to Johann I Bernoulli, October 1730, No. 5* Bernoulli Archives Universitätsbibliothek Basel 2r 2r where Manupertuis gives this as an excuse for not reading James Stirling's Methodus differentialis, sive tractatus de summatione et interpolatione serierum infinitarum (1731).
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• A bibliography can be found in Leonhardi Euleri opera omnia Taton René Youschkevitch A.P. Basel 1980 5 77 77 ser. 4A (annotation [4] to Euler's letter to Clairaut of (30) 19 October 1740). All of the relevant memoirs were republished in Johann I Bernoulli's Opera omnia (4 vols., 1742, Lausanne and Geneva), vol. 2, 275–281, 286–314, 423–472.
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• On the difficulties of obtaining the Acta eruditorum in France, see Greenberg Alexis Fontaine's “Fluxio-differential method” and the origins of the calculus of several variables Annals of science 1981 38 284 284 See also the letter from d'Ortous de Mairan to Johann I Bernoulli, 9 January 1724, No. 6* (footnote 16), 2r; and the letter from Johann I Bernoulli to d'Ortous de Mairan, 1 March 1724, No. 7 (footnote 16), 5v. I have not been able to locate any book reviews of the relevant volumes of the Acta in French journals.
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• Letter from d'Ortous de Mairan to Johann I Bernoulli, c. 8 August 1724, No. 9* Bernoulli Archives Universitätsbibliothek Basel 4r 4v
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• Letter from Johann I Bernoulli to Fontenelle, 4 March 1728, No. 5 Bernoulli Archives Universitätsbibliothek Basel 1v 2v
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• Delambre , J.-B.-J. and Maurice , F. 1820 . “ Maupertuis (Pierre-Louis Moreau de) ” . In Biographie universelle, ancienne et moderne Edited by: Michaud , L.G. Vol. 27 , 529 – 537 . Paris (p. 529); L. Angliviel de La Beaumelle, Vie de Maupertuis (1856, Paris), 14; Pierre Brunet, Maupertuis. Etude biographique (1929, Paris), 12.
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• Nicole's friend and one-time patron Pierre-Rémond de Montmort was himself a close friend of Brook Taylor. This was probably a contributing factor to Nicole's falling under Taylor's influence. Taylor was Montmort's house guest on several occasions, including Taylor's period of convalescence in France following a serious illness. Meanwhile, as of 1716 Montmort had already made three trips to England, two of them before 1700, at which time he met Newton, despite the fact that Anglo-French relations were strained to the hilt at that moment. Nicole, too, visited England and knew Newton personally. Both Montmort and Nicole made use of infinite series after Taylor's fashion. Together they make up part of the ‘British connection’ in France which preceded the widespread assimilation of Newtonian natural philosophy by the French during the 1730s and 1740s. For further discussion of the early Anglophiles in France, see Cohen I. Bernard Isaac Newton, Hans Sloane and the Académie Royale des Sciences L'aventure de la science: Melanges Alexandre Koyré Paris 1964 1 61 116 in Dennis J. Fletcher, ‘Bolingbroke and the diffusion of Newtonianism in France’, Studies on Voltaire and the eighteenth century, 53 (1967), 29–46; A. Rupert Hall, ‘Newton in France: a new view’, History of Science, 13 (1975), 233–250; A. Rupert Hall, ‘Les liens publics et privés dans les relations franco-anglaises (1660–1720) II: d'après la correspondance de Newton’, Revue de synthèse, (3), Nos. 81–82 (1976), 60–70; Henry Guerlac, ‘Newton in France—two minor episodes’, Isis, 53 (1962), 219–221; his ‘The Newtonianism of Dortous de Mairan’, in his Essays and papers in the history of modern science (1977, Baltimore and London), 479–490: and his ‘Some areas or further Newtonian studies’, History of science, 17 (1979), 75–101.
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• See my Alexis Fontaine's “Fluxio-differential method” and the origins of the calculus of several variables Annals of science 1981 38 251 290
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• Caritat , M.-J.-A.-N. and de Condorcet , Marquis . 1771 . Eloge de M. Fontaine . Hist. Acad. Roy. Sci. , : 105 – 130 . publ. 1774 (p. 105); J.-L. Boucharlat, ‘Fontaine des Bertins (Alexis)’, Biographie universelle, ancienne et moderne, vol. 15 (ed. L. G. Michaud: 1816, Paris), 179–183 (pp. 179–180).
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• Castel , Louis-Bertrand . 1728 . Mathématique universelle, abrégée à l'usage et à la portée de tout le monde 554 – 556 . Paris 563–564. For example, ‘On pourrait aussi s'y servir des séries indéterminées, comme dans toutes les autres approximations. Cette manière est même assez nécessaire, lorsqu'il s'agit de résoudre une équation indéterminée qui contient diverses differences; comme dx+ydx-dy=0…’ (p. 556).
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• Castel . 1735 . “ Discours préliminaire, servant de supplément à la préface qui est à la tête de L' Analyse des infiniment petits de M. le Marquis de L'Hôpital ” . In Analise des infiniment petits, comprenant le calcul intégral, dans toute son etendue … servant de suite aux infiniment petits de M. le Marquis de L'Hôpital Edited by: Stone , Edmund . vi – vi . Paris in trans. Rondet See also Castel's review of Stone's The method of fluxions, both direct and inverse, in the Journal de Trévoux, (1732), 103–113. Castel's ‘Discours préliminaire’ was, I might add, basically an attempt to trace historically the roots of the differential and integral calculus to the applications of infinite series by Gregory of St. Vincent.
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• In the editor's ‘Preface’ to the posthumously published Commentaires sur la Géométrie de M. Descartes by Claude Rabuel Lyon 1730 there is mention of the imminent publication of a sequel to Carré's treatise by Rabuel, who was professor of mathematics at the Grand Collège du Lyon (p. 15). In fact, there was apparently such a work in manuscript form, which was housed with the publisher Marcellin Duplain (see Journal de Trévoux, (1730), 1106), but the work was never published.
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• Johann I , Bernoulli . Remarques sur le livre intitulé Analyse des infiniment petits, comprenant le calcul intégral, dans toute son etendue, etc., par Mr. Stone, de la Société Royale de Londres . Opera omnia , 4 169 – 192 . in Bernoulli's (especially pp. 170–175).
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• Greenberg . 1981 . Alexis Fontaine's “Fluxio-differential method” and the origins of the calculus of several variables . Annals of science , 38 : 264 – 264 .
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• Greenberg . 1981 . Alexis Fontaine's “Fluxio-differential method” and the origins of the calculus of several variables . Annals of science , 38 : 274 – 283 .
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• February 1739 . L'Académie Royale des Sciences, registre de procès-verbaux February , 20r – 20v . 4 The original version of the report was sold to a private purchaser, because the sale is listed in the ‘Fichier Charavay’, 45, cic-cle. No. 26805 (89) (Salle des Manuscrits, Bibliothèque Nationale (Paris)).
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• Johann I , Bernoulli . 1726 . De integrationibus aequationum differentialum, ubi traditur methodi alicujus specimen integrandi sine pravia separatione indeterminatarum . Comment. Acad. Sci. Petrop. , 1 : 167 – 184 . publ. 1728) (especially pp. 175–176) (Opera omnia, vol. 3, 108–124 (especially pp. 115–116)).
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• Maupertuis . April 1731 . “ Sur la séparation des indéterminées ” . In L'Académie Royale des Sciences, registre de procès-verbaux April , 82v – 86v . in 28
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• Alexis-Claude , Clairaut . 1739 . Recherches générales sur le calcul intégral . Mém. Acad. Roy. Sci. , : 425 – 436 . publ. 1741) (p. 435).
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• Alexis-Claude , Clairaut . 1739 . Recherches générales sur le calcul intégral . Mém. Acad. Roy. Sci. , : 434 – 434 . publ. 1741)
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• See Greenberg Alexis Fontaine's “Fluxio-differential method” and the origins of the calculus of several variables Annals of science 1981 38 274 283
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• In the summer of 1741 Fontaine sent a substantial manuscript on integral calculus to the Royal Academy in the form of a pli cachéte. I have used a verbal account of this manuscript, given by D'Alembert and by Jean-Paul de Gua de Malves in their referees' report, to reconstruct Fontaine's special technique for dealing with this case. See L'Académie Royale des Sciences, registre de procès-verbaux 1742 January 14 21 17 (p. 16). I have not been able to find the contents of the pli cacheté itself. Fontaine's letter to d'Ortous de Mairan, dated 15 June 1741, announcing the imminent despatch of the pli cacheté, has been sold, as it is listed in the ‘Fichier Charavay’, 74, flo-for, No. 236 (191) (Salle des Manuscrits, Bibliothèque Nationale (Paris)).
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• Clairaut . 1739 . Recherches générales sur le calcul intégral . Mém. Acad. Roy. Sci. , : 434 – 435 . publ. 1741)
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• Greenberg . 1981 . Alexis Fontaine's “fluxio-differential method” and the origins of the calculus of several variables . Annals of science , 38 : 251 – 290 .
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• Greenberg . 1981 . Alexis Fontaine's “fluxio-differential method” and the origins of the calculus of several variables . Annals of science , 38 : 259 – 263 .
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• By far the best discussion of these aspects of the Leibnizian differential calculus of a single differential operation is to be found in Bos H.J.M. Differentials, higher-order differentials and the derivative in the Leibnizian calculus Arch. hist. exact sci. 1974–75 14 1 90
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• Bos , H.J.M. 1974–75 . Differentials, higher-order differentials and the derivative in the Leibnizian calculus . Arch. hist. exact sci. , 14 : 66 – 77 .
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• In fact, according to Bos Differentials, higher-order differentials and the derivative in the Leibnizian calculus Arch. hist. exact sci. 1974–75 14 59 62 Leibniz made an attempt to introduce non-zero quantities ddx in ways that were fundamentally incompatible with his calculus of a single differential operation d, hence was unsuccessful as a result.
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• Greenberg . 1981 . Alexis Fontaine's “fluxio-differential method” and the origins of the calculus of several variables . Annals of science , 38 : 265 – 266 .
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• Greenberg . 1981 . Alexis Fontaine's “fluxio-differential method” and the origins of the calculus of several variables . Annals of science , 38 : 280 – 282 .
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• Greenberg . 1981 . Alexis Fontaine's “fluxio-differential method” and the origins of the calculus of several variables . Annals of science , 38 : 272 – 274 .
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• Clairaut . 1739 . Recherches générales sur le calcul intégral . Mém. Acad. Roy. Sci. , : 436 – 436 . publ. 1741)
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• In the example here, if we let φ(x, y, p)≡x 3+y 2-p 2, then φ (x, y, p)=k is an integral for Sur l'intégration ou la construction des équations différentielles du premier ordre Mém. Acad. Roy. Sci. 1740 293 323 Dividing through by 2x, we see that φ(x, y, p)=k is an integral for as well. Meanwhile, φ(x, y, p)=k is not a one-parameter family of curves; it is a surface in rectangular coordinates x, y and p. Clairaut was apparently the first to interpret the integral (3.3) of an integrable equation (3.1) as an integral surface in rectangular coordinates x, y and p in his publ. 1742) (pp. 308–313).
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• Compare the argument here with the derivation of the commutation result in my previous paper Greenberg Alexis Fontaine's “fluxio-differential method” and the origins of the calculus of several variables Annals of science 1981 38 265 266
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• Clairaut . 1739 . Recherches générales sur le calcul intégral . Mém. Acad. Roy. Sci. , : 434 – 434 . publ. 1741)
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• See Clairaut Recherches générales sur le calcul intégral Mém. Acad. Roy. Sci. 1739 436 436 publ. 1741)
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• Clairaut . 1739 . Recherches générales sur le calcul intégral . Mém. Acad. Roy. Sci. , : 427 – 428 . publ. 1741)
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• Clairaut . 1740 . Sur l'intégration ou la construction des équations différentielles du premier ordre . Mém. Acad. Roy. Sci. , : 295 – 297 . publ. 1742)
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• See the letter from Clairaut to Euler, 17 September 1740, in Euler Leonhardi Euleri opera omnia Taton René Youschkevitch A.P. Basel 1980 5 68 69 ser. 4A also Greenberg (footnote 33), 267.
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• Letter from Euler to Clairaut, (30) 19 October 1740, in Euler Leonhardi Euleri opera omnia Taton René Youschkevitch A.P. Basel 1980 5 71 76 ser. 4A (pp. 74–76).
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• Clairaut . 1739 . Recherches générales sur le calcul intégral . Mém. Acad. Roy. Sci. , : 427 – 431 . publ. 1741)
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• Clairaut . 1740 . Sur l'intégration ou la construction des équations différentielles du premier ordre . Mém. Acad. Roy. Sci. , : 303 – 308 . publ. 1742) 313–320.
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• See Johann I Bernoulli De integrationibus aequationum differentialum, ubi traditur methodi alicujus specimen integrandi sine pravia separatione indeterminatarum Comment. Acad. Sci. Petrop. 1726 1 167 184 publ. 1728)
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• Clairaut . 1740 . Sur l'intégration ou la construction des équations différentielles du premier ordre . Mém. Acad. Roy. Sci. , : 320 – 323 . publ. 1742)
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• Clairaut . 1740 . Sur l'intégration ou la construction des équations différentielles du premier ordre . Mém. Acad. Roy. Sci. , : 425 – 425 . publ. 1742)
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• Clairaut , Alexis-Claude . 1731 . Recherches sur les courbes à double courbure 14 – 14 . Paris (§ 30).
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• See Sur l'intégration ou la construction des équations différentielles du premier ordre Mém. Acad. Roy. Sci. 1740 293 323 publ. 1742)
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• See Euler Leonhardi Euleri opera omnia Taton René Youschkevitch A.P. Basel 1980 5 68 69 ser. 4A In his ‘Solution de plusieurs problèmes ou il s'agit de trouver des courbes dont la propriété consiste dans une certaine relation entre leurs branches, exprimée par une équation donnée’, Mém. Acad. Roy. Sci., (1734: publ. 1736), 196–215, Clairaut formulated certain trajectory-like problems of a type that he claimed ‘were rare in the mathematical literature’, and he cited (pp. 196–197) the problems of ‘reciprocal (or isogonal) trajectories’, as discussed by Johann I Bernoulli, Euler, and Henry Pemberton in Acta eruditorum for 1718–1720 and in vol. 2 of the St. Petersburg Academy Commentarii, as problems of a similar nature. Yet he appears to have overlooked the crucial literature on ‘orthogonal trajectories’ in Acta eruditorum for those years. In fact, while he was still preparing this memoir, Clairaut thought he had solved some new problems that, in reality, had been solved at the turn of the century, there by betraying certain lacuna in his knowledge of the Acta eruditorum literature of the 1690s concern ing antecedents of the ‘trajectory’ problems. It was Fontaine who corrected him (see ‘L'Académie Royale des Sciences, registre de procès-verbaux’, 21 January 1735, 17v–18r), and this precipitated another long, drawn out dispute between the two. Fontaine appears to have been more familiar with the literature than was Clairaut—at least, where precursors of the ‘trajectory’ problems are concerned.
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• Bos . 1974–75 . Differentials, higher-order differentials and the derivative in the Leibnizian calculus . Arch. hist. exact sci. , 14 : 7 – 8 .
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• Bos . 1974–75 . Differentials, higher-order differentials and the derivative in the Leibnizian calculus . Arch. hist. exact sci. , 14 : 6 – 9 .
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• In this regard, see, for example, Johann I Bernoulli's letter to Montmort, 29 September 1718, No. 12 Bernoulli Archives Universitätsbibliothek Basel 1r 18v (p. 18v), in which Bernoulli generalizes from ‘similar’ curves to ‘non-similar’ curves in the problem of ‘orthogonal trajectories’ by means of the parametric ‘module’.
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• Bos . 1974–75 . Differentials, higher-order differentials and the derivative in the Leibnizian calculus . Arch. hist. exact sci. , 14 : 40 – 41 .
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• ‘Variations of parameters’ in this context are discussed in Engelsman S.B. Lagrange's early contributions to the theory of first-order partial differential equations Historia mathematica 1980 7 7 23 (pp. 15–19). ‘Envelopes’ in this context are discussed in René Taton, L'oeuvre scientifique de Gaspard Monge (1951, Paris), chapters 4 and 6.
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• This equation with ‘singular solution’ appears in Clairaut Solution de plusieurs problèmes ou il s'agit de trouver des courbes dont la propriété consiste dans une certaine relation entre leurs branches, exprimée par une équation donnée Mém. Acad. Roy. Sci. 1734 196 215 publ. 1736
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• Montucla , J.-F. 1802 . Histoire des mathématiques Edited by: de Lalande , Jérôme . Vol. 4 , Paris facsimile reprint 1968, Paris), vol. 3, 344 (annotation by Sylvestre-François Lacroix).
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• See L'Académie Royale des Sciences, registre de procès-verbaux 1742 January 14 21 17 A resumé of the D'Alembert-De Gua referees' report appears as ‘Algèbre: calcul intégral’, Hist. Acad. Roy. Sci., (1742: publ. 1745), 55–56.
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• Fontaine , Alexis . 1764 . “ Le calcul intégral. Première méthode ” . In Mémoires données à l'Académie Royale des Sciences non imprimés dans leurs temps Paris in Fontaine's republished as Traité de calcul différentiel et intégral (1770, Paris), 24–83 (pp. 24–28 (‘Définitions et propositions fondamentales’), 29–37 (‘Le calcul des équations aux premières différences’)).
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• Letter from Fontaine to d'Ortous de Mairan, 22 June 1741 (Archives Bertrand (Carton I), L'Académie Royale des Sciences). See also ‘L'Académie Royale des Sciences, registre de procès-verbaux’, 1 July 1741, 221. This was just after Fontaine had sent his pli cacheté to the Royal Academy L'Académie Royale des Sciences, registre de procès-verbaux 1742 January 14 21 17 On 13 June 1741, or just a few days before the dispatch of the pli cacheté, Fontaine sent to d'Ortous de Mairan a ‘très interessante lettre sur ses travaux scientifiques et dans laquelle il revendique la découverte d'un théorème sur le calcul intégral dont Clairaut s'attribue l'honneur’ (quoted in the ‘Fichier Charavay’, 74, flo-for, No. 25237 (192) (Salle des Manuscrits, Bibliothèque Nationale (Paris)).
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• Clairaut . 1739 . Recherches générales sur le calcul intégral . Mém. Acad. Roy. Sci. , : 433 – 433 . publ. 1741)
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