- Caritat , M.-J.-A.-N. 1771 . Marquis de Condorcet, ‘Eloge de M. Fontaine’ . Hist. Acad. Roy. Sci. , : 105 – 130 . publ. 1774
- Taton , René . 1970–1978 . “ Fontaine (Fontaine des Bertins), Alexis ” . In Dictionary of scientific biography Edited by: Gillispie , Charles C. Vol. 5 , 54 – 55 . New York 15 vols.
- Voltaire to Condorcet, 24 The complete works of Voltaire Besterman Theodore Geneva and Oxfordshire 1773 December 85–134 241 241 in Correspondence and related documents, vol. 50 1968–1976. vol. 40 (of this series)
- Voltaire to d'Alembert, 15 The complete works of Voltaire, vols. 85–134: Correspondence and related documents Besterman Theodore 1773 December 223 223 1968–1976
- Fontaine , Alexis . 1770 . Traité de calcul différentiel et intégral Paris ‘Table des mémoires contenus dans ce volume’. I have cited the second edition of Fontaine's complete works, the first edition having been published in Paris in 1764. The two editions are virtually identical; only the title Mémoires données `a l'Académie royale des Sciences non imprimés dans leurs temps was changed in 1770. All subsequent references to the complete works will be to the second edition.
- See Henry Charles Sur quelques billets inédits de Lagrange Bulletino di bibliografia e di storia delle scienze matematiche e fisiche 1886 19 129 135 (p. 129); and Gustave Eneström, ‘Lettres inédites de Joseph-Louis Lagrange `a Leonard Euler publiées par B. Boncompagni’, ibid., 12 (1879), 828–838.
- Many years after his initial exposition of his method, Fontaine referred to this commutativity of the two differential operations as ‘a theorem found in the calculations in the solution of the problem of the tautochrones that I gave in 1734’. Addition `a la méthode pour la solution des probl`emes de maximis et minimis Mém. Acad. Roy. Sci. 1767 588 613 publ. 1770 (p. 597, note 2). By 1767, however, Lagrange had introduced independent variations d and δ, and ‘… in taking Euler's treatise “de infinitis curvis ejusdem generis”, published in Tome VII of the Commentaries of the St. Petersburg Academy [the volume for the years 1734–1735, published in 1740], as a guide, he obtains immediately the identity: δd(y)-dδF(y), and in general δdmF(y)=dm δF(y)’ (quoted from G. Eneström (footnote 6), 832).
- For example, see Lagrange Joseph-Louis Essai d'une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies Miscellanea Taurinensis 1760–1761 2 173 195 (Oeuvres de Lagrange, vol. 1, 335–362, where on p. 336 Lagrange introduces two independent variations d and δ and explains that ‘… δZ will express a difference in Z which will not be the same as dZ, but which will however be formed by the same rules, so that by having an arbitrary equation dZ=mdx one will have δZ=mδx as well’).
- Fontaine . 1770 . Traité de calcul différentiel et intégral 1 – 5 . Paris ‘Nouvelle méthode pour la solution des probl`emes de maximis et minimis’.
- See L'Académie Royale des Sciences, registre de proc`es-verbaux 1732 May 170r 170r 7 also Hist. Acad. Roy. Sci., (1732: publ. 1735), 71.
- Fontaine . 1770 . Traité de calcul différentiel et intégral 3 – 3 . Paris The diagram above appears on ‘Page 22, Plate 1’, located at the end of the treatise. Interestingly enough, this is exactly the same as the solution to be found in Woodhouse's chapter devoted to Lagrange's first memoir on the calculus of d- and δ-variations (see his A treatise on isoperimetrical problems and the calculus of variations (1810, Cambridge, Eng.); reprinted in facsimile as A history of the calculus of variations in the eighteenth century (1964, New York), 85–86). Woodhouse apparently intended this as an illustration of the Lagrangean method, and in particular to show how the brachistochrone in a void could be derived in a manner that, according to Woodhouse, was ‘less peculiar and geometrical’ than earlier derivations, while Fontaine is nowhere mentioned in the course of the argument. Woodhouse did include Fontaine's Royal Academy of Sciences Mémoire of 1767 on maxima and minima (see footnote 7) in his ‘List of foreign authors that treat of the subject of the present work’, but there is no mention of the complete works of 1764 in which Fontaine's first memoir on maxima and minima is to be found. Had Fontaine had an opportunity to examine Volume 2 of the Turin Academy memoirs, which contains Lagrange's first memoir on the calculus of d- and δ-variations, prior to the publication of his own collected works in 1764? Lagrange's patron d'Alembert received a copy of the second volume of the Turin Academy memoirs by 1762 (see the letter from d'Alembert to Lagrange, dated 15 November 1762, in Oeuvres de Lagrange, vol. 13, 7–8), but Lagrange's memoir on calculus of variations contained therein was never a topic of discussion between d'Alembert and Lagrange until March of 1768 (see d'Alembert to Lagrange, 20 November 1768, ibid., 121). But this is merely suggestive; it is not possible at the moment to say with certainty that Fontaine, who was certainly not on the same terms with Lagrange that d'Alembert was, had not seen Lagrange's memoir prior to the publication of his own complete works in 1764.
- Bernoulli , Jakob . 1697 . Solution problematum fraternorum … . Acta eruditorum , May : 211 – 217 . especially p. 212 (quotation) and p. 205, ‘Table IV’ (diagram) (Opera omnia, vol. 2, 768–778 (pp. 769–770)). I have quoted the English translation appearing in Woodhouse (footnote 11), 4; the expression in brackets is Woodhouse's.
- Fontaine . 1770 . Traité de calcul différentiel et intégral 1 – 1 . Paris
- Fontaine , Alexis . 1734 . Sur les courbes tautochrones . Mém. Acad. Roy. Sci. , : 371 – 379 . publ. 1736
- As indicated in Addition `a la méthode pour la solution des probl`emes de maximis et minimis Mém. Acad. Roy. Sci. 1767 588 613 publ. 1770
- As indicated in Lagrange Joseph-Louis Essai d'une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies Miscellanea Taurinensis 1760–1761 2 173 195
- ‘Differential coefficients’ as distinguished from ‘differentials’, and Euler's efforts to replace the latter by the former as the cornerstone of the infinitesimal calculus, are discussed at great length 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 esp. pp. 24–25. We know that the algebra of a single differential operator is subject to certain indeterminacies, quite apart from the matter of the validity of the formalism. The .-operator is not a function that uniquely associates a quantity [xdot] with x, but rather just specifies an ‘order of magnitude’ (namely, a first-order, infinitesimally small quantity with respect to x). So, without additional considerations that go beyond the formal algebra, formally obtained ‘differential’ equations are not entirely meaningful, because they contain non-uniquely specified quantities. Some particular first-order differential must ordinarily be chosen as constant (which turns out from a modern function-theoretic standpoint to correspond to choosing an independent variable), otherwise the ‘differential’ equation is not completely interpreted. By the same token, the particular choice of the constant first-order differential is crucial to interpreting the equation. Comparing Fontaine's definitions here with those of Euler as described by Bos (ibid., 72), we see that Fontaine's way of defining differential coefficients is equivalent to the choice of x as an independent variable. Yet he did not explicitly stipulate the equivalent in terms of differentials—namely, that [xdot] be regarded as constant. Bos explains (ibid., 24–25) that the indeterminacy ‘does not influence the computational techniques or the interpretation of first-order differential equations … although the first-order differentials themselves are indeterminate, the relations between them are determined. Also the summation [that is, integration] of differentials is not affected by this indeterminacy …’. Perhaps this is why Fontaine could neglect the stipulation of a constant, first-order .-differential in his algebra of .- and d-operations; first-order .-operations alone are involved, as we shall see in a moment, so that at no point does the need to make such a stipulation ever arise. (The same is true in Fontaine's derivation in sub-section 3.1 of the brachistochrone in a void, culminating in (3.1.9.).) On the other hand, as Bos points out himself (ibid., 11 (note 19)), the usual concept of ‘differential’ breaks down in the calculus of several independent variables, while part of my contention will be that the calculus of several variables is precisely what is at issue in Fontaine's case.
- As indicated in Lagrange Joseph-Louis Essai d'une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies Miscellanea Taurinensis 1760–1761 2 173 195
- Fontaine Sur les courbes tautochrones Mém. Acad. Roy. Sci. 1734 378 378 publ. 1736 (‘Exemple IV’), 371 (diagram).
- Hence the methods of Jakob Bernoulli, Johann I Bernoulli, and Brook Taylor, applied to the brachistochrone is a resistant medium, had been subject to certain inherent limitations: ‘… they do not extend to cases, in which the differential function expressing the maximum should depend on a quantity, not given except under the form of a differential equation, and that not integrable; for instance, they will not solve the case of the curve of the quickest descent, in a resisting medium, the descending body being solicited by any forces whatever’ Woodhouse Traité de calcul différentiel et intégral Paris 1770 30 30
- Archives Bernoulli Letter from Johann I Bernoulli to Maupertuis Universitätsbibliothek Basel March 1734 2r 2r 9 No. 33 in ms. LIa 662
- Letter from Euler to Clairaut Leonhardi Euleri opera omnia Taton René Youschkevitch A.P. Basel 1742 January–February 5 110 117 published in ser. 4A 1980 (pp. 113–114). I am indebted to Mr. Taton for having made copies of these letters, transcribed from the originals located in Leningrad, available to me in advance of publication. In 1764 Euler repeated his sentiments concerning Fontaine's solution to the tautochrone of 1734 in Leonhard Euler, ‘Dilucidationes de tautochronis in medio resistente’, Novi comment. Acad. Sci. Petrop., 10 (1764: publ. 1766), 156–178 (Opera omnia, ser. 2, vol. 6, 189–208 (p. 192)).
- d'Alembert , Jean LeRond . 1751–1765 . “ Tautochrone ” . In Encyclopédie, ou dictionnaire raisonné … Edited by: d'Alembert and Diderot . Vol. 15 , 946 – 946 . Paris and Neufchastel 17 vols
- Euler Leonhard Principia motus fluidorum Novi comment. Acad. Sci. Petrop. 1756–1757 6 271 311 publ. 1761 (Opera omnia, ser. 2, vol. 12, 133–168, especially p. 137 (including note 1); and p. lxiii (for Clifford Truesdell's English translation of the passage)). Truesdell dates this treatise by Euler to the year 1752. The evolution away from equations among differentials and toward equations among differential coefficients, as this occurred in Euler's work, is discussed at great length in Bos (footnote 17), who makes it clear how this was an essential step toward the removal of the ambiguities inherent in higher-order ‘differential’ equations—ambiguities that resulted from holding different choices of first-order differentials constant in the process of applying differential operators to obtain higher-order equations.
- As indicated in Lagrange Joseph-Louis Essai d'une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies Miscellanea Taurinensis 1760–1761 2 173 195
- As indicated in Lagrange Joseph-Louis Essai d'une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies Miscellanea Taurinensis 1760–1761 2 173 195
- I am grateful for certain criticisms from Professor Hans Freudenthal which enabled me to identify and to correct one rather gross technical error in an earlier version of this argument. This theorem appeared explicitly in the work of Fontaine and in that of his colleague Clairaut, as opposed to the implicit form that it took in Fontaine's work of 1734, during a dispute between the two around 1738–1741 over the most efficacious means for integrating (or at least reducing to quadratures) inhomogeneous, first-order ordinary differential equations of first degree and total (or ‘Pfaffian’) differential equations in three variables (in this regard, see, for example Clairaut Alexis-Claude Recherches générales sur le calcul intégral Mém. Acad. Roy. Sci. 1739 425 436 publ. 1741 especially p. 428), which are problems that I have chosen not to take up at this time. Euler, too, apparently arrived at this theorem in his own fashion during the 1730s (see, for example, his ‘De infinitis curvis ejusdem generis. Seu methodus inveniendi aequationes pro infinitis curvis ejusdem generis’, Comment. Acad. Sci. Petrop., 7 (1734–1735: publ. 1740), 174–189 (Opera omnia, ser. 1, vol. 22, 36–56 (pp. 38–39)); but Euler's work in this area was unknown in France until late 1740. A sufficient condition for (1.2.2) is that both mixed, second-order partial differential coefficients be continuous as functions of two variables, but this subtlety was only introduced when derivatives replaced differential coefficients in the course of the nineteenth-century program to provide foundations for the infinitesimal calculus in terms of theories of functions and limits.
- An early application in France of (1.2.1) is to be found in Bouguer Pierre Une base qui est exposée au choc d'un fluide étant donnée, trouver l'esp`ece de conoïde dont il faut la couvrir pour que l'impulsion soit la moindre qu'il est possible Mém. Acad. Roy. Sci. 1733 85 107 publ. 1735 especially pp. 96–98, where Bouguer apparently utilized it, somewhat prophetically, to solve an extremal problem. Fontaine is said to have provided a ‘proof’ of the result in general, but a rather general exposition of (1.2.1) had in fact already appeared in Nikolaus II Bernoulli, ‘Exercitatio geometrica de trajectoriis orthogonalibus, continens varias carum tum inveniendarum tum construendarums methodos, sua vel demonstratione vel analysi munitas cum praemissa discussione quarundam ejusdem problematis solutionum. Sectio II’, Acta eruditorum, supplements, 7 (1721), 303–326, especially pp. 307–308.
- When (δ/δX) In T(X) = 0 (the tautochrone), the division by (δ/δX) In T(X) in the last step (5.3.18) of my argument above is of course not permitted. In fact, it turns out that such a step would in any event be superfluous in this case; (5.3.20) just follows from the step (5.3.17) that preceded; in other words, in one less step. I should mention that Fontaine did not justify the term-by-term comparison of the differential coefficients in his own two penultimate equations (4.1) and (4.4) (or, what amounts to the same thing, present arguments that would enable us to understand why the coefficients of the corresponding quadratic equation (5.3.19) in y should all vanish in the cases that he examined). He did mention, however, that such a term-by-term comparison was not valid in general—namely, in the presence of hypotheses of resultant forces of forms more general than the ones that he considered in his memoir (see Fontaine Sur les courbes tautochrones Mém. Acad. Roy. Sci. 1734 378 378 (‘Premi`ere remarque’)
- According to Fontaine's biographer and one-time student, the Marquis de Condorcet, Fontaine deliberately expressed himself cryptically at times because he enjoyed doing that. As Condorcet put it, Fontaine ‘… was sometimes even pleased to leave the nature and range of applicability of his methods deliberately obscure, which served above all to conceal their limits. He liked his equals in mathematics to try to guess what he was up to, and he candidly admitted everything when they had penetrated it’ ((footnote 1), 110). Needless to say, the ‘candid admissions’ were unfortunately all verbal; they did not take the form of new memoirs. Historians of the nineteenth century were a good deal less generous in their evaluation of the situation. For example, according to Alfred Maury, ‘Fontaine liked moreover, as if to further the debate and conflict, to envelop himself in obscure language and only present his ideas obliquely, which remain in the realm of the arcane’ L'ancienne Académie des Sciences , 2nd ed. Paris 1864 66 67
- Fontaine . 1734 . Sur les courbes tautochrones . Mém. Acad. Roy. Sci. , : 379 – 379 . (‘Seconde remarque’)
- Leibniz to Johann I Bernoulli, 2 Newton Isaac The correspondence of Isaac Newton Turnbull H.W. Scott J.F. Hall A.R. Tilling L. Cambridge, , Eng. 1716 April 6 321 321 in 7 vols. 1959–77 (note 7)
- Leibniz to Conti, 3 The correspondence of Isaac Newton Turnbull H.W. Scott J.F. Hall A.R. Tilling L. Cambridge, , Eng. 1716 April 6 323 323 7 vols. 1959–77 (note 1)
- Quoted from the abstract of a manuscript version of an article by Bernoulli Nikolaus II Acta eruditorum 1718 248 262 on ‘orthogonal trajectories’, the published version of which appeared in (Monmort to Newton, 16 March 1718, in Newton (footnote 35), vol. 6, 439 (note 7)).
- Nikolaus II Bernoulli Bouguer Pierre Une base qui est exposée au choc d'un fluide étant donnée, trouver l'esp`ece de conoïde dont il faut la couvrir pour que l'impulsion soit la moindre qu'il est possible Mém. Acad. Roy. Sci. 1733 305 308
- Euler Clairaut Alexis-Claude Recherches générales sur le calcul intégral Mém. Acad. Roy. Sci. 1739 425 436 publ. 1741 passim.
- Nicole François Méthode générale pour déterminer la nature des courbes qui coupent une infinité d'autres courbes données de position, en faisant toujours un angle constant Mém. Acad. Roy. Sci. 1715 54 61 publ. 1717 (‘Exemple III’); his ‘Solution d'un probl`eme proposé par Mr. Bernoulli, professeur de mathématique `a Basle’, ‘L'Académie Royale des Sciences, registre de proc`es-verbaux’, 9 April 1718, 97v–101r; and his ‘Solution nouvelle d'un probl`eme proposé aux géom`etres Anglais par feu M. Leibniz, peu de temps avant sa mort’, Mém. Acad. Roy. Sci., (1725: publ. 1727), 130–153.
- Nicole François Usage des suites pour la résolution de plusieurs probl`emes de la méthode inverse des tangentes Mém. Acad. Roy. Sci. 1737 50bis 85bis publ. 1740 (p. 59bis)
- Nicole , François . 1737 . Usage des suites pour la résolution de plusieurs probl`emes de la méthode inverse des tangentes . Mém. Acad. Roy. Sci. , : 73 – 73 . (diagram follows p. 85). The constant on the left hand side of (6.2.1) is written as a square root in order to preserve dimensional homogeneity, in accordance with traditional geometrical algebra.
- Nicole , François . 1737 . Usage des suites pour la résolution de plusieurs probl`emes de la méthode inverse des tangentes . Mém. Acad. Roy. Sci. , : 70 – 70 . (diagram follows p. 85).
- Concerning the treatise De scientia infiniti that Leibniz had hoped to write but never achieved, see Costabel Pierre De scientia infiniti Leibniz 1646–1716. Aspects de l'homme et l'oeuvre Paris 1968 105 117 in especially pp. 114–117
- Fontaine , Alexis . July 1737 . Mani`ere d'éviter les suites dans certains probl`emes, dont la solution semble d'abord en dépendre’, ‘L'Académie Royale des Sciences, registre de proc`es-verbaux July , 139r – 140v . 6
- Fontaine Alexis ’Mani`ere d'éviter les suites dans certains probl`emes, dont la solution semble d'abord en dépendre’, ‘L'Académie Royale des Sciences, registre de proc`es-verbaux’ July 1737 139v 139v 6 (including diagram)
- Fontaine Fontaine Alexis Mani`ere d'éviter les suites dans certains probl`emes, dont la solution semble d'abord en dépendre’, ‘L'Académie Royale des Sciences, registre de proc`es-verbaux 1737 July 139v 139v 6 The ‘Plumatif’ of the ‘Registre de proc`es-verbaux’, housed in the Archives of the Royal Academy of Sciences, is a handwritten copy of the handwritten original—not the original proceedings. This means that Fontaine's memoir as it appears in the ‘Plumatif’ is a handwritten document once or twice removed from Fontaine's manuscript. This is an important consideration, because the symbols that appear in the ‘Plumatif’ (for example, for ‘d’) need not have been Fontaine's own. Now the notations are uncertain, so that the matter is serious. I have not been able to locate Fontaine's manuscript, which would have settled the matter. The paper was never published.
- Fontaine Alexis Mani`ere d'éviter les suites dans certains probl`emes, dont la solution semble d'abord en dépendre’, ‘L'Académie Royale des Sciences, registre de proc`es-verbaux July 1737 139v 139v 6 Fontaine says: ‘… nous aurons dF = fdx + ((nF – fx)/p)dp, ou il ne restera plus qu'`a mettre pour p et dp leur valeur [tirée de l'equation `a la courbe SMm, pour avoir l'equation `a la courbe AMB]’. I should point out that Euler also solved what amounts to Nicole's ‘fourth problem’, using exactly the same argument as Fontaine, in his Mechanica sive motus scientia analytice exposita (2 vols., 1736, St. Petersburg), vol. 2 (Opera omnia, ser. 2, vol. 2, 43–45 (‘Propositio 14’)). Meanwhile, there is considerable evidence that French mathematicians were not yet familiar with Euler's Mechanica in 1737. A short review of this work did appear nevertheless in the Journal de Trévoux for the year 1737—and during the month of July no less! (see Journal de Trévoux (1737), 1318–1321 (or reprinted in facsimile as Journal de Trévoux, 37 (1968, Geneva), 335–336)). Full length, detailed reviews only appeared for the first time in the Journal de Trévoux for 1740.
- As indicated in Bouguer Pierre Une base qui est exposée au choc d'un fluide étant donnée, trouver l'esp`ece de conoïde dont il faut la couvrir pour que l'impulsion soit la moindre qu'il est possible Mém. Acad. Roy. Sci. 1733 85 107 publ. 1735
- Bernoulli Johann I Méthode pour trouver les tautochrones, dans des milieux résistants, comme le quarré des vitesses Mém. Acad. Roy. Sci. 1730 78 101 publ. 1732 (p. 80, ‘Scholie’, for the reference to Nikolaus II Bernoulli's memoir) (Opera omnia, vol. 3, 173–197 (pp. 175–176)).
- One should probably allow for the possibility that Fontaine obtained some insights to the solution of Nicole's problems from Nikolaus II Bernoulli's memoir. It is not even hard to imagine how such a hypothetical case of ‘plagiarism’ could have easily gone unnoticed in France in 1737. The deaths of Johann I Bernoulli's correspondents Pierre-Rémond de Montmort (1719) and Pierre Varignon (1722) ushered in a period of little correspondence between Johann I Bernoulli and the French. With the onset around 1730 of his correspondence with Maupertuis, member of a new generation of French mathematicians, Bernoulli began to pick up where he had left off with the French some ten years earlier. Meanwhile Fontaine, who belonged to a still younger generation of French mathematicians, was far more adept than his Royal Academy overseer Maupertuis and would have had a sharper eye for the frontiers of mathematical research buried away in that rather all-purpose, foreign journal Acta eruditorum. Moreover, the Acta may have been relatively scarce in France at this time for economic reasons. For example, in reply to a request from the Chevalier de Louville in 1720 for a copy of Acta eruditorum for 1719, J.-N. Delisle wrote: ‘Mr. Montaland [one of the leading bookvendors in Paris] no longer imports books from Germany, because of too great a difference in the rate of exchange’ (see the letter from Delisle to the Chevalier de Louville, 8 Correspondance Paris Observatory 1720 August I 2r 2r No. 194 in ms. B1,1 J.-N. Delisle's When Louville made the same request in 1721, Delisle replied: ‘I still haven't been able to find Acta eruditorum for 1719. Mrs. Montaland and Caveliers have no more of them, and Mr. Varignon did not buy it, having found it to be too expensive …’ (Delisle to the Chevalier de Louville, 26 January 1721, No. 6 in ms. B1,2 (J.-N. Delisle's Correspondance, Tome II, Paris Observatory), 1r). Indications such as these concerning Acta eruditorum, together with the fact that Clairaut betrayed a total ignorance of Nikolaus II Bernoulli's memoir in 1740 (see the letters from Clairaut to Euler, 17 September 1740, and Euler to Clairaut, 19 October 1740 (footnote 23), 68–78), suggest that one ought to question seriously the extent of French familiarity with the younger Bernoulli's memoir prior to 1740.
- Lagrange Joseph-Louis Nouvelles réflexions sur les tautochrones Nouv. Mém. Acad. Sci. Belles-Lettres Berlin 1770 97 122 publ. 17720(pp. 97–98) (Oeuvres de Lagrange, vol. 3, 157–186 (p. 158)).
- Bolza , Oskar . 1904 . Lectures on the calculus of variations 19 – 19 . Chicago 1961, New York
- Let us note that a local minimum principle is not what is required in some problems of this type—for example, the path of quickest descent is not in general the local path of quickest descent (see Woodhouse Traité de calcul différentiel et intégral Paris 1770 6 7 When it is not so, the path of quickest descent is determined (in Fontaine's notation) by dFL(⋅/v)=0, where FL(⋅/v) extends over a finite interval of time, rather than by an equation like (3.1.3).
- Clifford Truesdell has made the distinction between ‘true minimum principles’ on one hand, and ‘variational principles’ or formal expressions in variations on the other. Truesdell intimates that by means of formal rearrangements—for example, sufficiently general ‘variational principles’—one can arrive at would-be generalizations of just about any collection of mechanical laws or principles (see, for example, his Whence the law of moment of momentum? Essays in the history of mechanics New York 1968 239 271 in his especially p. 242 (note 4)). The question is: what do such ‘generalizations’ achieve? Truesdell intimates that such generalizations are purely formal, and consequently artificial. I am inclined to agree, and the study of Fontaine's work in particular sheds further light on this, if from a different angle. Namely, if we set aside true minimum principles, which are a fundamental part of certain problems (like the isoperimetricals), then to say that the range of problems considered by Fontaine falls within the scope of the ‘fluxio-differential method’ has no more depth than an assertion, say, to the effect that each of a group of problems is a problem in the calculus of several variables! Needless to say, the same is true of certain ‘generalizations’ obtained via the ‘variational principles’ of a ‘calculus of variations’.
- I cannot say, however, whether the argument above helps to shed any light on the question of why the differential coefficient (and its offspring the derivative) replaced the differential as the fundamental concept of the infinitesimal calculus. In suggesting that the answer lies in part in the study of functions of more than one variable Bos Differentials, higher-order differentials and the derivative in the Leibnizian calculus Arch. hist. exact. sci. 1974–75 14 11 11 alleges that ‘The usual conceptions and techniques of differentials break down when applied to such functions, and the ensuing difficulties have to be solved by the systematic use of derivatives and partial derivatives’. On this basis one would expect the ‘fluxio-differential method’ to conceal inherent limitations, which are absent in Fontaine's approach to Nicole's ‘trajectory’ problems.
- Fontaine . July 1737 . Mani`ere d'éviter les suites dans certains probl`emes, dont la solution semble d'abord en dépendre’, ‘L'Académie Royale des Sciences, registre de proc`es-verbaux July , 24 – 37 . 6
- Bos . 1974–75 . Differentials, higher-order differentials and the derivative in the Leibnizian calculus . Arch. hist. exact. sci. , 14 : 5 – 9 . To talk of a ‘differential calculus in a single variable’ during the infancy of the infinitesimal calculus is itself a somewhat misleading modernization of the situation that existed, which I have taken the liberty of using here only for its connotative value. As Bos makes clear throughout his article, the Leibnizian differential calculus was not a ‘calculus of derivatives of functions of a single variable’, but a calculus of several non-independent variables of finite and infinitesimal magnitudes, in which infinitesimal variables of first and higher orders were obtained by successive applications of the differential treated as an operator acting on finite variables and other infinitesimal variables, respectively. One cannot emphasize enough how absolutely fundamental is Bos's work with regard to the illumination of the distinction between the Leibnizian and the modern differential calculi which in the past has been all too easily glossed over. Some of Bos's discussion reappears in his chapter 2 in I. Grattan-Guinness (ed.), From the calculus to set theory, 1630–1910. An introductory history (1980, London).
- Clairaut Alexis-Claude Sur l'intégration ou la construction des équations différentielles du premier ordre Mém. Acad. Roy. Sci. 1740 293 323 publ. 1742 especially pp. 308–310.
- Taton , René . 1951 . L'oeuvre scientifique de Gaspard Monge Paris especially pp. 160–161 for comments on the early developments which accord with this overall view.
- Clairaut's most explicit criticism of Fontaine in this regard appears in Clairaut Recherches générales sur le calcul intégral Mém. Acad. Roy. Sci. 1739 425 425 It is based on a geometric point of view that Clairaut had acquired by the time that he was fifteen years old (see his Recherches sur les courbes `a double courbure (1731, Paris), 14 (§ 30)).
- As indicated in Bouguer Pierre Une base qui est exposée au choc d'un fluide étant donnée, trouver l'esp`ece de conoïde dont il faut la couvrir pour que l'impulsion soit la moindre qu'il est possible Mém. Acad. Roy. Sci. 1733 85 107 publ. 1735
- I pose the question because Leibniz, for example, had given an illustration of ‘variation of parameters’ in a geometric context—namely, in 1692 and 1694 he determined the ‘envelope’ of a family of straight lines by differentiation with respect to the parameter of the family, followed by elimination of the parameter (see Bos Differentials, higher-order differentials and the derivative in the Leibnizian calculus Arch. hist. exact. sci. 1974–75 14 40 41 In the 1770s the ‘envelope’ of a one-parameter family of curves came to be identified with the ‘singular solution’ to the ordinary differential equation for the one-parameter family of curves—a solution not contained in the complete solution of the differential equation. The analytic side—in particular, Lagrange's method of ‘variation of constants’—has recently been discussed in S. B. Engelsman, ‘Lagrange's early contributions to the theory of first-order partial differential equations’, Historia mathematica, 7 (1980), 7–23. For the geometric side see Taton (footnote 63), chapters 4 and 6.
Annals of Science
Volume 38, 1981 - Issue 3