J. L. Lagrange's changing approach to the foundations of the calculus of variations

citation_journal_title=Journal de l'Ecole polytechnique; citation_title=Mémoire sur les conditions d'intégrabilité des fonctions différentielles; citation_author=Joseph Bertrand; citation_volume=17; citation_publication_date=1841; citation_pages=249-275; citation_id=CR1 citation_journal_title=Archive for History of Exact Sciences; citation_title=Differentials, Higher-Order Differentials and the Derivative in the Leibnizian Calculus; citation_author=Henk Bos; citation_volume=14; citation_publication_date=1974; citation_pages=1-90; citation_id=CR2 citation_journal_title=Histoire de l'Académie Royale des Sciences avec les Mémoires de Mathématique et de Physique; citation_title=Recherches générales sur le calcul intégral; citation_author=Alexis Clairaut; citation_volume=1739; citation_publication_date=1739; citation_pages=425-436; citation_id=CR3 Condorcet, J. M. Marquis de [1765]: Du Calcul Intégral (Paris, 1765). Courant, R. & Hilbert, D. [1953]: Methods of Mathematical Physics Volume 1 (Interscience, 1953). Euler, Leonhard Leonhardi Euleri Opera Omnia Series I Opera Mathematica (Various cities, 1911–1956) Volumes 13 22 24 25. Series IVa Commercium Epistolicum (Basel, 1980) Volume 5. This volume of Series IVa is edited by A. P. Juškevič & R. Taton and contains selected French translations of Latin correspondence as well as valuable detailed notes of information. citation_journal_title=Commentarii academiae scientiarum Petropolitanae; citation_title=De Infinitis Curvis Eiusdem Genesis Seu Methodus Inveniendi Aequationes Pro Infinitis Curvis Eiusdem Generis; citation_author=Leonhard Euler; citation_volume=7; citation_publication_date=1734; citation_pages=174-189; citation_id=CR7 Euler, Leonhard [1744]: Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes Sive Solutio Problematis Isoperimetrici Latissimo Sensu Accepti (Lausanne, 1744) = Opera S.I 24 (1952), ed., Constantin Carathéodory citation_journal_title=Opera S.IVa; citation_title=Letter to Lagrange 6 September 1755; citation_author=Leonhard Euler; citation_volume=5; citation_publication_date=1755; citation_pages=375-478; citation_id=CR9 citation_journal_title=Opera S.IVa; citation_title=Letter to Lagrange 24 April 1756; citation_author=Leonhard Euler; citation_volume=5; citation_publication_date=1756; citation_pages=386-390; citation_id=CR10 citation_journal_title=Novi commentarii academiae scientiarum Petropolitanae; citation_title=Elementa Calculi Variationum; citation_author=Leonhard Euler; citation_volume=10; citation_publication_date=1764; citation_pages=51-93; citation_id=CR11 citation_journal_title=Novi ... Petropolitanae; citation_title=Analytica Explicatio Methodi Maximorum et Minimorum; citation_author=Leonhard Euler; citation_volume=10; citation_publication_date=1764; citation_pages=94-134; citation_id=CR12 Euler, Leonhard [1770]: “Appendix de Calculo Variationum”, Institutiones Calculi Integralis, Volume 3 (1770) = Opera A.I 13 (1914) pp. 369–469. citation_journal_title=Novi ... Petropolitanae; citation_title=Methodus Nova et Facilis Calculum Variationum Tractandi; citation_author=Leonhard Euler; citation_volume=16; citation_publication_date=1771; citation_pages=35-70; citation_id=CR14 citation_journal_title=Archive for History of Exact Sciences; citation_title=J. L. Lagrange's Early Contributions to the Principles and Methods of Mechanics; citation_author=Craig Fraser; citation_volume=28; citation_publication_date=1983; citation_pages=197-241; citation_id=CR15 Goldstein, Herbert [1950]: Classical Mechanics (Addison-Wesley, 1950). Goldstine, Herman H. [1980]: A History of the Calculus of Variations from the 17th through the 19th Century (Springer-Verlag, 1980). Grabiner, Judith V. [1981]: The Origins of Cauchy's Rigorous Calculus (M.I.T. Press, 1981). Huke, Aline [1930]: “An Historical and Critical Study of the Fundamental Lemma in the Calculus of Variations”, Contributions to the Calculus of Variations (University of Chicago Press, 1931) pp. 45–160. Jordan, Camille [1896]: Cours d'Analyse de l'Ecole Polytechnique, Tome Troisième Calcul Intégral (Paris, 1896). citation_journal_title=Historia Mathematica; citation_title=The History of Differential Forms from Clairaut to Poincaré; citation_author=Victor J. Katz; citation_volume=8; citation_publication_date=1981; citation_pages=161-188; citation_id=CR21 Kline, Morris [1972]: “The Calculus of Variations in the Eighteenth Century”, Mathematical Thought from Ancient to Modern Times (Oxford, 1972) pp. 573–591. Lagrange, Joseph Louis Œuvres de Lagrange J. A. Serret & G. Darboux, eds., 14 volumes (Paris, 1867–1892). The page references which I give in this article are, with one exception, to the edition of the particular work in question which appears in the Œuvres. (Thus, for example, [1806, 404] refers to page 404 of Œuvres 10.) The one exception is the Méchanique Analitique, for which I have included page numbers to the original edition of 1788 as well as to the edition in the Œuvres. See also the entry below under Méchanique Analitique. citation_journal_title=Œuvres; citation_title=Letter to Euler 28 June 1754; citation_author=Joseph Louis Lagrange; citation_volume=14; citation_publication_date=1754; citation_pages=135-138; citation_id=CR24 citation_journal_title=Œuvres; citation_title=Letter to Euler 12 August 1755; citation_author=Joseph Louis Lagrange; citation_volume=14; citation_publication_date=1755; citation_pages=138-144; citation_id=CR25 citation_journal_title=Œuvres; citation_title=Letter to Euler 20 November 1755; citation_author=Joseph Louis Lagrange; citation_volume=14; citation_publication_date=1755; citation_pages=146-151; citation_id=CR26 citation_journal_title=Euler Opera S. IVa; citation_title=Letter to Euler 5 October 1756; citation_author=Joseph Louis Lagrange; citation_volume=5; citation_publication_date=1756; citation_pages=396-411; citation_id=CR27 Lagrange, Joseph Louis [1760a]: “Essai d'une nouvelle méthode pour déterminer les maxima et les minima des formules indéfinies”, Miscellanea Taurinensia 2 (1762) = Œuvres 1 (1867) pp. 335–362. Lagrange, Joseph Louis [1760b]: “Application de la méthode exposée dans le mémoire précédent à la solution de différentes problèmes de dynamique”, Miscellanea Taurinensia 2 (1762) = Œuvres 1 pp. 365–468. Lagrange, Joseph Louis [1764]: “Recherches sur la libration de la Lune dans lesquelles on tâche de résoudre la Question proposée par l'Académie Royale des Sciences, pour le Prix de l'année 1764”, Prix de l'Académie Royale des Sciences de Paris tome IX 1780 (1777) = Œuvres 6 (1878) pp. 5–61. Lagrange, Joseph Louis [1780]: “Théorie de la libration de la lune, et des autres phénomènes qui dependent de la figure non sphérique de cette planète”, Nouveaux Memoires de l'Académie royale des Sciences de Berlin 1780 = Œuvres 5 (1870) pp. 5–122. citation_journal_title=Œuvres; citation_title=Letter to Laplace 15 September 1782; citation_author=Joseph Louis Lagrange; citation_volume=14; citation_publication_date=1782; citation_pages=115-117; citation_id=CR32 Lagrange, Joseph Louis [1788]: Méchanique Analitique (Paris, 1788). The second edition of this work appeared in two volumes as the Mécanique Analytique, the first volume in 1811 and the second (posthumously) in 1815. The edition in the Œuvres 11 and 12 (1888) is based on the second edition. Although the second edition differs in many ways from the first, not only in details of presentation but also in scope, the passages of concern in this article remain unchanged except in notation. Lagrange, Joseph Louis [1797]: Théorie des fonctions analytiques contenant les principles du calcul différentiel dégagés de toute consideration d'infiniment petits ou d'évanouissans de limites ou de fluxions et réduits à l'analyse algébrique des quantités finies, Journal de l'Ecole Polytechnique 3 (1797). Second edition published in 1813 = Œuvres 9 (1881) My references to numbered articles in the Théorie follow the edition of 1797. Lagrange, Joseph Louis [1806]: Lećons sur le calcul des fonctions (Second edition, 1806) = Œuvres 10 (1884). The lessons on the calculus of variations did not appear in the first edition, published in 1801. Mach, Ernst [1883]: The Science of Mechanics: A Critical and Historical Account of its Development (Open Court, 1960), The first edition of Mach's book appeared in German in 1883. Ovaert, Jean-Louis [1976]: “La Thèse de Lagrange et la Transformation de l'Analyse”, Philosophie et Calcul de l'Infini, Christian Houzel et al. (Paris, 1976) pp. 157–200. citation_journal_title=Mémoires de l'Académie Royale des Sciences de l'Institut de France Tome; citation_title=Mémoires sur le Calcul des Variations; citation_author=Siméon-Denis Poisson; citation_volume=XII; citation_publication_date=1831; citation_pages=223-331; citation_id=CR38 citation_journal_title=Proceedings of the American Philosophical Society; citation_title=Lagrange's personality (1736–1813); citation_author=George Sarton; citation_volume=88; citation_publication_date=1944; citation_pages=457-496; citation_id=CR39 Struik, D. J. [1969]: A Source Book in Mathematics, 1200–1800 (Harvard Press, 1969) pp. 399–413. Selected translations from Euler [1744] and Lagrange [1760a] into English. Todhunter, Isaac [1861]: A History of the progress of the calculus of variations during the nineteenth century (London, 1861). Winter, Eduard & Maria [1957]: Die Registres der Berliner Akademie der Wissenschaften 1746–1766 (Berlin, 1957). Woodhouse, Robert [1810]: A Treatise on Isoperimetrical Problems and the Calculus of Variations (Cambridge, 1810).