The Culture of Research Mathematics in 1860s Prussia: Adolph Mayer and the Theory of the Second Variation in the Calculus of Variations

  • Craig Fraser
Conference paper
Part of the Proceedings of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques book series (PCSHPM)


The paper examines the intellectual culture of higher mathematics in Prussia and more broadly in Germany in the middle of the nineteenth century. There was at this time a strong ethos of pure mathematics in which the subject was pursued more for its intrinsic interest than for its utility or practical applications. This outlook was reflective of the prominence of neohumanism in German culture of the period. In the case of mathematics, it led to a higher degree of logical sophistication in the elaboration of theories. The work of Adolph Mayer in the calculus of variations at Prussia’s University of Königsberg is presented as a case study that illustrates the outlook and underlying values of German higher mathematics in the second half of the century. The self-consciously theoretical character of this mathematics distinguished it in a qualitative way from the style and mentality of the Enlightenment masters of analysis a century earlier. Our study provides evidence for the basic historicity of the development of analysis from 1750 to 1870.


Carl Jacobi Adolph Mayer Königsberg school Neohumanism Pure mathematics Calculus of variations Second variation Sufficient conditions 


  1. Ben-David, Joseph. 1971. The Scientist’s Role in Society A Comparative Study. Prentice-Hall. Englewood Cliffs, New Jersey.Google Scholar
  2. Bolza, Oscar. 1909. Vorlesungen über Variationsrechnung. Teubner. Leipzig and Berlin.Google Scholar
  3. Cantor, Moritz. 1889.“Richelot, Friedrich Julius,” in Allgemeine Deutsche Biographie, V. 28, 432. Duncker & Humblot, Leipzig.Google Scholar
  4. Clebsch, Rudolf Alfred. 1858a. “Ueber die Reduktion der zweiten Variation auf ihre einfachste Form.” Journal für die reine und angewandte Mathematik 55, 254-273.MathSciNetCrossRefGoogle Scholar
  5. Clebsch, Rudolf Alfred. 1858b. “Ueber diejenigen Probleme der Variationsrechnung, welche nur eine unäbhangige Variable enthalten.” Journal für die reine und angewandte Mathematik 55, 335-355.CrossRefGoogle Scholar
  6. Deakin, Michael A. B. 1981. “The development of the Laplace transform, 1737-1937: I. Euler to Spitzer, 1737-1880.” Archive for History of Exact Sciences 25 (1981), pp. 343-390.MathSciNetCrossRefGoogle Scholar
  7. Delaunay, Charles. 1841. “Thèse sur la distinction des maxima et des minima dans les questions qui dépendent de la méthode des variations.” Journal de Mathématiques pures et appliqées 6, 209-237.Google Scholar
  8. Ferraro, Giovanni. 2008. The Rise and Development of the Theory of Series up to the Early 1820s. In the series Sources and Studies in the History of Mathematics and Physical Sciences. Springer Science and Business Media, New York.Google Scholar
  9. Forman, Paul. 1971. “Weimar Culture, Causality and Quantum Theory, 1918-1927: Adaptation by German physicists and Mathematicians to a Hostile Intellectual Environment,” Historical Studies in the Physical Sciences 3 (1971), 1-115.CrossRefGoogle Scholar
  10. Fraser, Craig. 1994. “Calculus of variations.” In Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Ed. I. Grattan-Guinness (Routledge, 1994), V.1, pp. 342-350.Google Scholar
  11. Fraser, Craig. 1996. “Jacobi’s Result (1837) in the Calculus of Variations and its Reformulation by Otto Hesse (1857). A study in the changing interpretation of mathematical theorems,” in Jahnke et Knoche (1996), pp. 149-172.Google Scholar
  12. Fraser, Craig. 2003. “The calculus of Variations: A Historical Survey,” in Jahnke (2003), 355-384.Google Scholar
  13. Fraser, Craig. 2009. “Sufficient Conditions, Fields and the Calculus of Variations,” Historia Mathematica 36 (2009), 420-427.CrossRefGoogle Scholar
  14. Friedman, Michael and Nordmann, Alfred (Eds.). 2006. The Kantian Legacy in Nineteenth-century Science. MIT Press. Cambridge, MA.Google Scholar
  15. Fries, Jakob Friedrich. 1822. Die mathematische Naturphilosophie. C. F. Winter. Heidelberg.Google Scholar
  16. Gerstell, Marguerite. 1975. “Prussian Education and Mathematics,” American Mathematical Monthly 82, pp. 240-245.MathSciNetCrossRefGoogle Scholar
  17. Goldstine, Herman H. 1980. A History of the Calculus of Variations from the 17th through the 19th Century. Springer-Verlag. New York and Berlin.Google Scholar
  18. Gregory, Frederick. 1983. “Neo-Kantian Foundations of Geometry in the German Romantic Period.” Historia Mathematica 10, 184-201.MathSciNetCrossRefGoogle Scholar
  19. Hacking, Ian. 2014. Why Is There Philosophy of Mathematics At All? Cambridge University Press. Cambridge, UK.Google Scholar
  20. Hesse, Ludwig Otto 1857. “Über die Criterien des Maximums und Minimums der einfachen Integrale.” Journal für die reine und angewandte Mathematik 54, 227-273.CrossRefGoogle Scholar
  21. Husserl, Edmund 1882. Beiträge zur Theorie von Variationsrechnung. Unpublished dissertation University of Vienna. In 1983 a French translation of Husserl’s dissertation was published under the title Contributions à la théorie du calcul des variations. Trans. Mlle Devouard. Edited by J. Vauthier. No. 65 of Queen’s Papers in Pure and Applied Mathematics (Eds. A. J. Coleman et al). Kingston, Ontario, Canada.Google Scholar
  22. Jacobi, Carl Gustav 1837. “Zur Theorie der Variations-Rechnung und der Differential-Gleichungen.” Journal für die reine und angewandte Mathematik 17, 68-82.MathSciNetCrossRefGoogle Scholar
  23. Jahnke, H. Niels. 2003. A History of Analysis. American Mathematical Society.Google Scholar
  24. Jahnke, H. Niels and Michael Otte (Eds.). 1981. Epistemological and Social Problems of the Sciences in the Early Nineteenth Century. D. Reidel Publishing Company. Dordrecht, Holland; Hingham, MA.Google Scholar
  25. Jahnke, H. Niels and Norbert Knoche (Eds.) 1996. History of Mathematics and Education, Volume 11 of the series “Studien zur Wissenschafts-, Sozial und Bildungsgeschichte der Mathematik”. Vandenhoeck. Göttingen.Google Scholar
  26. Jordan, Camille. 1896. Cours d’analyse de l’École polytechnique V. 3: “Calcul intégral. Équations différentielles.” Paris.Google Scholar
  27. Klein, Felix. 1926. Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert Teil 1. Berlin.Google Scholar
  28. Knorr Cetina, Karin. 1999. Epistemic Cultures: How Sciences Make Knowledge. Harvard University Press. Cambridge, Massachusetts.Google Scholar
  29. Lagrange, Joseph L. 1797. Théorie des fonctions analytiques. Paris. The second edition appeared in 1813 and is reprinted as Oeuvres 9.Google Scholar
  30. Lagrange, Joseph L. 1801. Leçons sur le calcul des fonctions. Paris. Reissued in 1804 in Journal de l’École Polytechnique, 12 cahier, tome 5.Google Scholar
  31. Lagrange, Joseph L. 1806. Leçons sur le calcul des fonctions. Nouvelle edition. Paris. This edition includes additions and “un traité complèt du calcul des variations.” Reprinted as Oeuvres 10.Google Scholar
  32. Lagrange, Joseph L. 1867-1892. Oeuvres de Lagrange. 14 volumes. Paris: Gauthier-Villars.Google Scholar
  33. Liebmann, Heinrich. 1908. “Adolf Mayer †,” in Jahresbericht der Deutschen Mathematiker-Vereinigung. V. 17, 355-362.Google Scholar
  34. Lindelöf, Lorenz and François Moigno. 1861. Leçons de calcul différentiel de calcul intégral. Tome quatrième. - Calcul des variations. Authored by Lindelöf and revised in collaboration with Moigno. Paris.Google Scholar
  35. Mainardi, Gaspare. 1852. “Sul Calculo dell variazioni.” Annali di scienze mathematiche e fisiche 3 (1852), 149-192.Google Scholar
  36. Mayer, Adolph. 1866. Beiträge zur Theorie der Maxima und Minima der einfachen Integrale. Habilitationsschrift. B. G. Teubner. Leipzig.Google Scholar
  37. Mayer, Adolph. 1868. “Ueber die Kriterien des Maximums und Minimums der einfachen Integrale.” Journal für die reine und angewandte Mathematik 69, 238-263.MathSciNetCrossRefGoogle Scholar
  38. Mayer, Adolph. 1886. “Begründung der Lagrange’sche Multiplicatorenmethode in der Variationsrechnung.” Mathematische Annalen V. 26, 74-82.Google Scholar
  39. Mayer, Adolph. 1904 and 1906. “Über den Hilbertschen Unäbhangigkeitsatz in der Theorie des Maximums und Minimums der einfachen Integralen.” Mathematische Annalen, V. 58, 235-248, V.62, 335-350.Google Scholar
  40. Mehrtens, Herbert, Henk Bos and Ivo Schneider (Eds.). 1981. Social History of Nineteenth Century Mathematics. Birkhäuser. Boston, Basel and Stuttgart.zbMATHGoogle Scholar
  41. Nakane, Michiyo and Fraser, Craig G. 2002. “The Early History of Hamilton-Jacobi Dynamical Theory, 1834-1837,” Centaurus 44, 1-67.MathSciNetCrossRefGoogle Scholar
  42. Pulte, Helmut. 2006. “Kant, Fries, and the Expanding Universe of Science” in Friedman and Nordmann (Eds.), 101-122.Google Scholar
  43. Pyenson, Lewis. 1983. Neohumanism and the Persistence of Pure Mathematics in Wilhelmian Germany. Memoirs of the American Philosophical Society, Vol. 150. American Philosophical Society. Philadelphia.Google Scholar
  44. Rowe, David E. 1985. “Felix Klein’s “Erlanger Antrittsrede.” A Transcription with English Translation and Commentary.” Historia Mathematica 12, 123-141.MathSciNetCrossRefGoogle Scholar
  45. Scharlau, Winfried. 1981.”The Origins of Pure Mathematics,” in Jahnke and Otte (1981), pp. 331-347.Google Scholar
  46. Schubring, Gert. 1981. “The Conception of Pure Mathematics as an Instrument in the Professionalization of Mathematics,” in Mehrtens et al. (1981), pp. 111-143.Google Scholar
  47. Schubring, Gert. 2005. Conflicts Between Generalization, Rigor, and Intuition: Number Concepts Underlying the Development of Analysis. Springer. New York.Google Scholar
  48. Scriba, Christoph J. 1973. “Jacobi, Carl Gustav Jacob”. Dictionary of Scientific Biography V. 7, Ed. Charles C. Gillispie, pp.50-55. Charles Scribner’s Sons. New York.Google Scholar
  49. Spitzer, Simon. 1854-1855. “Über die Kriterien des Grössten and Kleinsten bei den Problemen der Variationsrechnung”, Sitzungsberichte der Mathematisch-Naturwissenschaften Classe der Kaiserlichen Akademie der Wissenschaften. Part One appears in volume 13 (1854), pp. 1014-1071. Part Two appears in volume 14 (1855), pp. 41-120. Vienna.Google Scholar
  50. Thiele, Rüdiger. 1999. “Adolph Mayer 1839-1908,” in Reiner Groβ and Gerald Wieners (Eds.), Sächsische Lebensbilder Band 4, 211-227. Verlag der Sächsischen Akademie der Wissenschaften zu Leizig.Google Scholar
  51. Thiele, Rüdiger. 2007.Von der bernoullischen Brachistochrone zum Kalibrator-Konzept : ein historischer Abriss zur Entstehung der Feldtheorie in der Variationsrechnung (hinreichende Bedingungen in der Variationsrechnung). Brepols Publishers. Turnhout, Belgium.Google Scholar
  52. R. Stephen Turner. 1971. “The Growth of Professorial Research in Prussia, 1818 to 1848 — Causes and Context,” Historical Studies in the Physical Sciences V. 3, 137–182. CrossRefGoogle Scholar
  53. VonderMühll, Karl. 1908. “Zum Andenken an Adolph Mayer (1839-1908)”, in Mathematischen Annalen V. 65, 433-434Google Scholar
  54. von Escherich, Gustav Ritter. 1899. “Die zweite Variation der einfachen Integrale,” Sitzungsberichte der Österreichische Akademie der Wissenschaften, V. 108.Google Scholar
  55. Weierstrass, Karl. 1927. Vorlesungen über Variationsrechnung. Edited by Rudolf Rothe. Akademische Verlagsgesellschaft. Leipzig.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for the History and Philosophy of Science and TechnologyUniversity of TorontoTorontoCanada

Personalised recommendations