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Vibrating Strings and Eighteenth-Century Mechanics

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Abstract

In the late 1740s an acrimonious dispute broke out between Jean le Rond d’Alembert and Leonhard Euler over the solution of the two-dimensional wave equation. The issues were the nature of legitimate mathematical functions, the definition of continuity, and differentiability. A description of this overheated exchange illustrates the relations between mathematics, metaphysics and experimental philosophy during the eighteenth century.1

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References

  1. For details of the mathematical issues involved and the opinions of Euler and d’Alembert see, Ivor Grattan-Guinness, From the Calculus to Set Theory (London: Duckworth, 1980), chap. 1

    MATH  Google Scholar 

  2. Jerome R. Ravetz, “Vibrating Strings and Arbitrary Functions,” in The Logic of Personal Knowledge, (Glencoe II.: Free Press, 1961), 71–88

    Google Scholar 

  3. and Clifford Truesdell, “The Rational Mechanics of Flexible or Elastic Bodies, 1638–1788,” in Euler, Opera Omnia, series 2, vol. 11, part 2, (Bern: 1957), 240–262, although Truesdell is too partial to Euler. Henk Bos also noted this partiality

    Google Scholar 

  4. in H. J. M. Bos, “Mathematics and Rational Mechanics,” in Ferment of Knowledge: Studies in the Historiography of the Eighteenth Century, G. S. Rousseau and Roy Porter, eds. (Cambridge: Cambridge University press, 1980), 327–355. For d’Alembert

    Google Scholar 

  5. see Thomas Hankins, Jean d’Alembert: Science and the Enlightenment (Oxford: The Clarendon Press, 1970).

    MATH  Google Scholar 

  6. Michel Paty, d’Alembert et son temps: Éléments de biographie (Strasbourg: Université Louis Pasteur, 1977) characterized Alembert’s previous work in mechanics more as “a branch of mathematics than as an experimental science,” in reference to d’Alembert, Traité de Mécanique (Paris, 1743).See also Hankins, “Introduction,” to Traité.

    Google Scholar 

  7. See also, G. F. Wheeler and W. P. Crummet, “The Vibrating String Controversy,” Amer. J. Phys. 55 (1987): 33–37.

    Article  MathSciNet  ADS  Google Scholar 

  8. Taylor’s condition was that the acceleration of the oscillating body was proportional to its distance from its equilibrium position, i.e., simple harmonic motion. Jean le Rond d’Alembert, “Recherches sur la courbe que forme une corde tendue mise en vibration,” Mém. Acad. Sci. Berlin 3 (1747) [1749]: 214–219, p. 214, and “Suite des recherches sur la courbe que forme une corde tendue, mise en vibration,” same journal, 220–249.

    Google Scholar 

  9. See S. S. Demidov, “Création et développement de la théorie des équations différentielles aux dérivées partielles dans les travaux de J. d’Alembert,” Rev. Hist. Sci. 35 (1982): 3–42

    MathSciNet  Google Scholar 

  10. and Steven B. Engelmann, “D’Alembert et les équations au dérivées partielles,” Dix-huit. siècle 16 (1984): 7–203. Notation remained remarkably fluid until the late nineteenth century.

    Google Scholar 

  11. See Florian Cajori, A History of Mathematical Notations, (Chicago: Open Court Press, 1929.)

    MATH  Google Scholar 

  12. D’Alembert obtained this solution through a change of coordinates, not the method of separation of variables. D’Alembert called this “the method of multipliers.” For discussions, see Demidov, “Création,” J. R. Ravetz, “Vibrating Strings,” and Grattan-Guinness, From Calculus. D’Alembert was not the first to refer to families of curves rather than a generalized functional solution to a partial differential equation. See Steven B. Engelmann, Families of Curves and the Origins of Partial Differentiation (Amsterdam: North-Holland, 1984).

    Google Scholar 

  13. Leonhard Euler, “Sur la vibration des cordes,” Mém. Acad. Sci. Berlin 4 (1748) [1750]: 69–85, trans, From Leonhard Euler Nova Acta Eruditorum (1749): 512–527, reprinted in Euler, Leonhard Euler Opera Omnia series 2, vol. 10: 63–78, p. 64 and p. 72 respectively.

    Google Scholar 

  14. See Louise Ahrndt and Robert William Gollard, “Euler’s Troublesome Series: An Early Example of the Use of Trigonometric Series,” Hist. Math. 20 (1993): 54–62

    Article  MATH  Google Scholar 

  15. and Victor Katz, “The Calculus of Trigonometric Functions,” Hist. Math. 14 (1987): 311–324.

    Article  MATH  Google Scholar 

  16. See Euler, “Sur la vibration.” D’Alembert’s reply to Euler’s initial critique is in d’Alembert, “Addition au mémoire sur la courbe que forme une corde tendue, mise en vibration,” Mém Acad. Sci. Berlin 6 (1750): 355–360, where he in turn critiqued Euler’s first paper. He reiterated his definition of the necessary geometric continuity for the curves

    Google Scholar 

  17. in d’Alembert, “Recherches sur les vibrations des cordes sonores,” in Opuscules Mathématiques (1761), vol. 1, 1–73. Euler’s rejoinder appeared

    Google Scholar 

  18. as Euler, “De motu fili fîexilis, corpusculis quotcumque onusti,” Novi Comm. Acad. Sci. Petropolitanae 9 (1762–63): 215–245.

    Google Scholar 

  19. See Euler to Müller, April 1763, in Paul Heinrich von Fuss, comp. Correspondance mathématique et physique de quelques célèbres géomètres du xviii e siècle (New York; Johnson Reprint of 1843 edition, 1968), vol. 2, p. 215. For Euler on d’Alembert, see vol. 2, p. 71. For his style, see also his letters to Müller on candidates for the St. Petersburg Academy, and his letters to the chaplain to Frederick, Prince of Wales in vol. 3.

    Google Scholar 

  20. D’Alembert’s later mathematical work appeared in nine volumes, d’Alembert, Opuscules Mathématiques, (Paris, 1761–1780). See Demidov, “Création,” for d’Alembert and partial differential equations. Euler’s work on partial differential equations culminated

    Google Scholar 

  21. in Euler, Institutiones Calculi Integralis, (St. Petersburg, 1770).

    Google Scholar 

  22. See Demidov, “The Study of Partial Differential Equations of the First Order in the Eighteenth and Nineteenth Century,” Arch. Hist. Exact Sci. 27 (1982): 325–350. A short account of the history of partial differential equations is in

    Article  MathSciNet  Google Scholar 

  23. J. Lützen, “Partial Differential Equations,” in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Grattan-Guinness, ed. (London: Routledge, 1994), 2 vols., vol. 1: 452–469.

    Google Scholar 

  24. Daniel Bernoulli, “Theoremata de oscillationibus corporum filo flexili connexorum et catenae verticaliter suspensae,” Comm. Acad. Sci. Imp. Petropolitanae, 6 (1732–33): 108–122, and, “Demonstrationes theorematum suorum de oscillationibus corporum filo flexili connexorum et catenae verticaliter suspensae,” same journal, 7 (1740): 162–173.

    Google Scholar 

  25. Euler, “De oscillationibus fill flexilis quotcumque pondusculis onusti,” Comm. Acad. Sci. Imp. Petropolitanae, 8 (1736): 30–47. Details of Euler’s derivation are in Truesdell, “Rational Mechanics,” 162–165

    Google Scholar 

  26. and H. F. K. Burckhardt, “Entwicklungen nach oscillirenden Functionen und Integration der differential Gleichungen der mathematischen Physik,” Jahresber. dtsch. Math. Verein, 10 pt. 2 (1901–1908).

    Google Scholar 

  27. Bernoulli to Euler, September 1736, in Fuss, Correspondance, vol. 2 p. 434. See also Bernoulli to Euler, March 1739, Fuss, vol. 2, p. 456. In his November 1740 letter to Euler Bernoulli discussed the fundamental frequencies of pipes of different lengths, Fuss, vol. 2, p. 465. O. B. Sheynin also argues that Euler treated physical problems as purely mathematical ones. See Sheynin, “Euler’s Treatment of Observations,” Arch. Hist. Exact Sci. 9 (1972): 45–56.

    Article  MathSciNet  MATH  Google Scholar 

  28. Bernoulli, “Commentationes de oscillationibus compositis praesertim iis quae fiunt in corporibus ex filo flexili suspensis,” Comm. Acad. Sci. Imp. Petropolitanae, 12 (1740): 97–108.

    Google Scholar 

  29. Bernoulli to Euler, 28 January, 1741 in Fuss, Correspondance, vol. 2 p. 469. Euler published his solution as an appendix in Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes (Geneva: 1744), “Additamentum I De curvis elasticis.” For details of Euler’s work see Truesdell, “Rational Mechanics,” 192–219. Truesdell sees the exchange purely in terms of the mathematics involved and characterizes Bernoulli as “whining” about Euler’s mathematical success while slacking off himself.

    Google Scholar 

  30. The delayed papers were, Bernoulli, “De Vibrationibus et Sono Laminarum Elasticarum, Commentationes Physico-Geometricae,” Comm. Acad. Sci. Imp. Petropolitanae, 13 (1741) [1751]: 105–120, and “De Sonis Multifariis quos Laminae Elasticae Diversmode edunt disquisitiones Mechanico-Geometricae Experimentis Acusticis Illustratae et Confirmatae,” same journal, 167–196. While both Bernoulli and Euler continued to publish in the St. Petersburg journals, Euler was always careful to keep in close touch with the remaining members of the Academy. Bernoulli’s comments on the vibrating strings problem are in Bernoulli to Johann III Bernoulli (probably), 7 October 1753, in Fuss Correspondance, vol. 2. See also Truesdell, “Rational Mechanics,” p. 254.

    Google Scholar 

  31. The series appeared in Bernoulli to Johann II Bernoulli (probably), undated, Fuss, Correspondance, vol. 2. 653–655. Fuss dated this letter between 1754 and 1766. Truesdell narrowed the dates to between 1754 and 1755. See Truesdell, “Rational Mechanics,” p. 257. Truesdell argues that this letter was meant for Euler. Bernoulli’s series appeared in Bernoulli, “Réflexions et éclaircissemens sur les nouvelles vibrations des cordes exposées dans les mémoires de l’Académie de 1747 et 1748,” Nouveaux Mém. Acad. Sci. Berlin 9 (1753) [1755]: 147–172. In a second paper Bernoulli applied the principle of superposition. Bernoulli, “Sur le mélange de plusieurs espèces de vibrations simples isochrones, qui peuvent coexister dans un même système de corps,” same journal: 173–195.

    Google Scholar 

  32. Euler’s critique of Bernoulli and further remarks on discontinuous functions appear in Euler, “Remarques sur les mémoires précédentes de Mr. Bernoulli,” Nouveaux Mém. Acad. Sci. Berlin 9 (1753) [1755]: 196–222. The papers to which Euler refers are Bernoulli, “Réflexions,” and “Sur le mélange.”

    Google Scholar 

  33. Bernoulli, “Lettre de M. Daniel Bernoulli à M. Clairaut au sujet des nouvelles découvertes faites sur les vibrations des cordes tendues ”J. des Sçavans (March 1758): 157–166, p. 157. See also,Bernoulli, “Mémoire sur les vibrations des cordes d’une épaisseur inégale,” Mém. Acad. Sci. Berlin 21 (1765): 281–306, p. 283.

    Google Scholar 

  34. H. Floris Cohen, Quantifying Music: The Science of Music in the First Stage of the Scientific Revolution (Boston MA: Reidel, 1984) notes that Euler’s theory of consonance and those of other mathematicians was “largely abstracted from physical and physiological considerations and went back to operations with number,” p. 237. There was no physical theory of consonance until the work of Hermann von Helmholtz in the nineteenth cen- tury.

    Google Scholar 

  35. See also Albert Cohen, Music in the French Royal Academy of Sciences (Princeton NJ: Princeton University Press, 1981)

    Google Scholar 

  36. and Jamie C. Kassler, “The ‘Science’ of Music to 1830,” Arch. Int. Hist. Sci. 30(1990): 111–135.

    Google Scholar 

  37. Historians have traced mathematicians’ understandings of functions in Jerome R. Ravetz, “Vibrating Strings,” Grattan-Guinness, Foundation, A. P. Iushevich, “The Concept of the Function,” Arch. Hist. Exact Sci. 16 (1976): 37–85

    Google Scholar 

  38. Pierre Dugac, “Des fonctions comme expressions analytiques aux fonctions représentables analytiquement,” in Mathematical Perspectives: Essays on Mathematics and Its Historical Development Joseph Dauben, ed. (New York: Academic Press, 1981), 13–36.

    Google Scholar 

  39. During the eighteenth century there was no general proof for the convergence of infinite series of sines or cosines. Mathematicians sought functional equivalents to such series. See Grattan-Guinness, Foundations, Victor Katz, “The Calculus of Trigonometric Functions,” Hist. Math. 14 (1987): 311–324, and L. A. and R. W. Golland, “Euler’s Troublesome Series.”

    Article  MATH  Google Scholar 

  40. Joseph Louis Lagrange, “Recherches sur la nature, et la propagation du son,” Misc. Taurin, 1 (1754): 1–112, and “Nouvelles recherches sur la nature et la propagation du son,” same journal, 2 (1760–1761): 11–172. Reprinted in Oeuvres de Lagrange (Paris: Gauthier-Villars, 1867), 14 vols., vol. 1, 39–148 and 151–316, respectively.

    Google Scholar 

  41. For the role of elasticity in the development of the calculus see, Clifford Truesdell, “The Influence of Elasticity on Analysis,” Bull Amer. Math. Soc. 9 (1983): 293–310. This is also implicit in discussions by historians of eighteenth-century mathematics; see, Grattan-Guinness, From Calculus, and Foundations.

    Article  MathSciNet  MATH  Google Scholar 

  42. Brook Taylor, “De motu nervi tensi,” Phil. Trans. R. Soc. London 28 (1713): 26–32.

    Google Scholar 

  43. See L. Feigenbaum, “Brook Taylor and the method of Increments,” Arch. Hist. Exact Sci. 34 (1985): 1–140.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  44. See, John T. Cannon and Siglia Dostrovsky, The Evolution of Dynamics: Vibration Theory from 1687 to 1742 (New York: Springer-Verlag, 1981).

    Book  MATH  Google Scholar 

  45. Johann I Bernoulli, “Theoremata selecta pro conservatione virium vivarum demonstranda et expérimenta confirmanda excerpta ex epistolis datis ad filium Danielem,” Comm. Acad. Sci. Imp. Petropolitanae, 2 (1727): 200–207. Proofs appeared in the following volume in Bernoulli, “Meditationes de chordis vibrantibus, cum pondusculis aequali intervallo a se invicem dissitis, ubi mimirum ex principio virium vivarum quaeritur numerus vibrantionum chordae pro una oscillatione penduli datae longitudinis,” same journal, 3 (1728): 13–28.

    Google Scholar 

  46. The history of the calculus up to the work of d’Alembert is in Florian Cajori, “The Early History of Partial Differential Equations,” Amer. Math. Monthly 35 (1928): 459- 467.

    Article  MathSciNet  MATH  Google Scholar 

  47. John L. Greenberg, “Mathematical Physics in Eighteenth-Century France,” Isis, 11 (1986): 59–78 discusses the early hostility then acceptance of the calculus in Paris

    Article  Google Scholar 

  48. and in Greenberg, “Alexis Fontaine’s Integration of Ordinary Differential Equations and the Origins of the Calculus of Several Variables,” Ann. Sci. 39 (1982): 1–36 explores the development of the calculus from Johann I Bernoulli and Maupertuis to Alexis-Claude Clairaut. S. S. Demidov examines the history of partial differential equations

    Article  MATH  Google Scholar 

  49. in Demidov, “La naissance de la théorie des équations différentielles aux dérivées partielles,” Proceedings of the xiv International Congress of the History of Science 2 (1974): 111–113

    Google Scholar 

  50. and in Demidov, “The Study of partial differential Equations of the first Order in the Eighteenth and Nineteenth Centuries,” Arch. Hist. Exact Sc. 26 (1982): 325–350. For the overall development of the calculus see

    Article  MathSciNet  MATH  Google Scholar 

  51. Carl B. Boy er A History of the Calculus (New York: Dover, 1959).

    Google Scholar 

  52. See d’Alembert, “Recherches sur la courbe,” “Suite des Recherches.” For how these papers fît into the development of the calculus see Demidov, “Création et développement de la théorie des équations différentielles aux dérivées partielles dans les travaux de J. d’Alembert,” Rev. Hist. Sci. 35 (1982): 3–42.

    MathSciNet  Google Scholar 

  53. For details of Lagrange’s goals, see Grattan-Guinness, Foundations. For Lagrange’s continuing interest in the foundations of the calculus see Judith Grabiner, The Origins of Cauchy’s Rigorous Calculus (Cambridge MA.: MIT Press, 1981).

    MATH  Google Scholar 

  54. See also Craig Fraser, “Joseph Louis Lagrange’s algebraic Vision of the Calculus,” Hist. Math. 14 (1984): 38–53, “The Calculus as Algebraic Analysis,” Arch. Hist. Exact Sci. 39 (1989): 317–335, and “Lagrange’s Analytical Mathematics,” Studies Hist. Phil. Sci. 21 (1990): 243–256.

    Article  MathSciNet  ADS  Google Scholar 

  55. Lagrange’s early interest in discontinuous functions also emerged in his correspondence to Euler. See Lagrange, Oeuvres vol. 14, Correspondance, letters to Euler of August 1758 and October 1759. After Lagrange’s paper on vibrating strings Euler’s work on discontinuous functions appeared in Euler, “De motu vibratorio fili flexilis, corpusculis quotcunque onusti,” Novi Comm. Acad. Sci. Petropolitanae, 9 (1762–1763): 215–245, “Ëclairissemens sur le mouvement des cordes vibrantes,” Misc. Taurin, 3 (1762–1763): 27–59, “Sur le mouvement d’une corde, qui au commencement n’a été ébranlée que dans une partie,” Mém. Acad. Sci. Berlin, 21 (1765): 307–334, “De chordis vibrationibus disquisito ulterior,” Novi Comm. Acad. Sci. Petropolitanae, 17 (1772): 381—409, “Consideratio motus plane singularis qui in filo flexili locum nature potest,” Novi Acta Acad. Sci. Petropolitanae, 2 (1784): 103–120. See also, Lagrange, “Recherches sur la nature et la propagation du son.”

    Google Scholar 

  56. Lagrange was already known to Euler through correspondence. In 1754 they began corresponding on new methods of calculating maxima and minima as well as other issues in the calculus. See Craig Fraser, “J. L. Lagrange’s Early Contributions to the Principles and Methods of Mechanics,” Arch. Hist. Exact Sci. 28–29 (1983–84): 197–242.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  57. See also, Jean Itard, “Lagrange, Joseph Louis,” in Dict. Sci. Bio. Charles C. Gillispie, ed. (New York; Scribners, 1971), 14 vols., vol. 7, 559–573.

    Google Scholar 

  58. John Greenberg, The Problem of the Earth’s Shape from Newton to Clairaut: The Rise of Mathematical Science in Eighteenth-Century Paris and the Fall of “Normal” Science (New York: Cambridge University Press, 1995).

    MATH  Google Scholar 

  59. SeeMaupertuis, “Sur la figure de la terre et sur les moyens que l’astronomie, et la géographie fournissent pour la déterminer,” Mém. Acad. Sci. Paris (1733): 153–164,“Sur la figure de la terre,” same journal, (1735): 98–105, and Examen désintéressée des différents ouvrages qui ont été faits pour déterminer la figure de la terre (Oldenberg, 1738).

    Google Scholar 

  60. Maupertuis, Discours sur les différentes figures des astres (Paris, 1732), Maupertuis, “De figure quas Fluida rotata induere possant,” Phil Trans. R. Soc. London (1733): 240–256, and “Loi du repos des corps,” Mém. Hist. Acad. Sci. (1740): 170–176. For a detailed discussion of Maupertuis’ mathematics

    Google Scholar 

  61. see Greenberg, “Mathematical Physics in Eighteenth-Century France,” Isis, 77 (1986): 59–78, and Shape of the Earth, chap. 5, and chap. 7, 243–258.

    Article  MathSciNet  MATH  Google Scholar 

  62. Clairaut, Théorie de la figure de la terre, tirée des principes de l’hydrostatique (Paris, 1743).

    Google Scholar 

  63. See also,Greenberg, “Breaking a “Vicious Circle.” Unscrambling A.-C. Clairaut’s Iterative Method of 1743,” Hist. Math.. 15 (1988): 228–239 and Shape of the Earth, chaps. 6 and 9. Greenberg considers Clairaut’s place in the development of the calculus in chaps. 7 and 8. It is well to remember that the reduction of the data was not done by Clairaut.

    Article  MATH  Google Scholar 

  64. Philip Chandler, “Clairaut’s Critique of Newtonian Attraction: Some Insights into his Philosophy of Science” Ann. Sci. 32 (1975): 369–378.

    Article  MathSciNet  MATH  Google Scholar 

  65. This begins early. Newton’s fluxions emerged from his mechanics. Questions of mechanics exercised Johann I Bernoulli and other early developers of the calculus. Of the mathematicians mentioned in this chapter see,Euler, Mechanica sive motus scientia analytica exposita (St. Petersburg, 1736), 2 vols. The calculus of variation began similarly in the consideration of mechanical problems by Newton, Leibniz, Brook Taylor, Johann I Bernoulli then Daniel Bernoulli amongst others. As a branch of the calculus Euler systematized it in

    Google Scholar 

  66. Euler, Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes (St. Petersburg, 1744). For a brief account see Craig Fraser, “Calculus of Variations,” Encyclopedia of Hist. Philos. Math. vol. 1, 342–350.

    Google Scholar 

  67. See also H. H. Goldstine, A History of the Calculus of Variations (New York: Springer Verlag, 1980). For the technical developments in mechanics

    MATH  Google Scholar 

  68. see, A. T. Grigorian, “On the Development of the Variational Principles of Mechanics,” Arch. Int. Hist. Sci. 18 (1965): 23–35.

    MATH  Google Scholar 

  69. Both Lagrange and Euler did both. Their work discussed here led them into new channels of mathematics which they then explored as mathematics, not as outcomes of the problems of mechanics. See Fraser, “J. L. Lagrange’s Changing Approach to the Foundations of the Calculus of Variations,” Arch. Hist. Exact Sci. 32 (1985): 151–191. Lagrange, Mécanique Analytique (Paris: 1788) was a text on the the calculus and partial differential equations pertinent to mechanics.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  70. See also S. B. Engelmann, “Lagrange’s Early Contributions to the Theory of First Order Partial Differential Equations,”Hist. Math. 7 (1980): 7–23. Daniel Bernoulli’s and Euler’s work on the oscillations of lamina led Euler to a general framework for the solution of n-th order differential equations.

    Article  Google Scholar 

  71. See Demidov, “On the History of the Theory of Linear Differential Equations,” Arch. Hist. Exact Sci. 28 (1983): 369–387. See also Demidov, “The Study of Partial Differential Equations of the First Order.”

    Article  MathSciNet  MATH  Google Scholar 

  72. For d’Alembert on the vis viva controversy, see Hankins, “Eighteenth Century Attempts to solve the vis viva Controversy,” Isis, 56 (1965): 281–297, and d’Alembert.

    Article  Google Scholar 

  73. See also David Papineau, “The Vis Viva Controversy,” Stud. Hist. Phil. Sci. 8 (1977): 111–142, and J. Morton Briggs, “d’Alembert: Philosophy and Mechanics in the Eighteenth Century,” University of Colorado Stud. (1964): 38–56. For Euler’s metaphysics

    Article  Google Scholar 

  74. see Jean Dhombres, “Les présupposes d’Euler,” Rev. Hist. Sci. 40 (1987): 179–202

    MathSciNet  MATH  Google Scholar 

  75. and Stephen Gaukroger, “Euler’s Concept of Force,” Brit. J. Hist. Sci. 15 (1982): 132–154.

    Article  MathSciNet  MATH  Google Scholar 

  76. For the principle of Least Action see Pierre Brunet, Etude historique sur le principe de la moindre action (Paris: Hermann et Cie, 1938), p. 6

    MATH  Google Scholar 

  77. Philip Jourdain, “The Nature and Validity of the Principle of Least Action,” Monist, 23 (1913): 277–293.

    Google Scholar 

  78. See also A. Kneser, Das Prinzip der Kleinsten Wirkung von Leibniz bis zur Gegenwart (Leipzig: Teubner, 1928).

    MATH  Google Scholar 

  79. For Leonard Euler see Truesdell, “The Rational Mechanics,” and, “Euler’s Contribution to the Theory of Ships and Mechanics,” Centaums, 26 (1982): 323–335, O. B. Sheynin, “Euler’s Treatment of Observations.” For an account of Willem Jakob van ‘sGravesande’s experiments

    Article  MathSciNet  ADS  Google Scholar 

  80. see Pierre Brunet, Les physiciens hollandais et la méthode expérimentale en France au xviiie siècle, (Paris: Albert Blanchard, 1926)

    Google Scholar 

  81. and Edward G. Ruestow, Physics at Seventeenth and Eighteenth Century Leiden: Philosophy and the New Science in the University (The Hague: Nijhoff, 1973).

    Book  Google Scholar 

  82. On Desaguliers and his imitators, see Larry Stewart, The Rise of Public Science: Rhetoric, Technology, and Natural Philosophy in Newtonian Britain, 1660–1750 (New York: Cambridge University Press, 1992).

    Google Scholar 

  83. See, George Atwood Treatise on the Rectilinear Motion and Rotation of Bodies (London: 1784). On Atwood’s machines and his teaching

    Google Scholar 

  84. see Simon Schaff er, “Machine Philosophy: Demonstration Devices in Georgian Mechanics,” Osiris, 9 (1993): 157–182, 159–163.

    Article  ADS  Google Scholar 

  85. Gerard L’E. Turner, “Physical Science at Oxford in the Eighteenth Century,” in The History of the University of Oxford vol. 5 The Eighteenth Century, L. S. Sutherland and L. G. Mitchell, eds. (Oxford: Oxford University Press, 1986), 659–681. On the place of experiment in general see

    Google Scholar 

  86. John Schuster and Graeme Watchins, “Natural Philosophy, Experiment, and Discourse in the Eighteenth Century,” in Experimental Inquiries: Historical, Philosophical, and Sociological Studies of Experiment, Homer Le Grand ed. (Dordrecht: Kluwer Academic, 1990)

    Google Scholar 

  87. See W. D. Hackmann, “The Relation between Concept and Instrument Design in Eighteenth-Century Experimental Science,” Ann. Sci. 36 (1979): 205–224.

    Article  Google Scholar 

  88. In France such experimental forms of mechanics were more likely to be met in courses designed for engineers. See, Grattan-Guinness, “Varieties of Mechanics by 1800,” Hist. Math. 17 (1990); 313–338, 321–322

    Article  MathSciNet  MATH  Google Scholar 

  89. see also, C. C. Gillispie Lazare Carnot, Savant: (Princeton NJ: Princeton University Press, 1971).

    MATH  Google Scholar 

  90. On Holland, see H. A. M. Snelders, “Professors, Amateurs and Learned Societies: The Organization of the Natural Sciences,” in The Dutch Republic in the Eighteenth Century, M. Jacob and W. W. Mijnhardt, eds. (Ithaca NY: Cornell University Press, 1992), 308–328. For public lecturing in Britain see, British Journal for the History of Science March 1995 issue on this topic, Larry Stewart, The Rise of Public Science

    Google Scholar 

  91. and Jan Golinski, Science as Public Culture: Chemistry and Enlightenment in Britain, 1760–1820 (New York: Cambridge University Press, 1992). On London see

    Google Scholar 

  92. A. Q. Morton, “Lectures on Natural Philosophy, 1750–1765: S. C. T. Demainbray (1710–1782) and the ‘Inattention’ of his Countrymen,” Brit. J. Hist. Sci. 23 (1990): 411–434 and the economic hazards of a career in public lecturing and on the patrons of such lecture courses

    Article  Google Scholar 

  93. John R. Milburne, “The London Evening Courses of Benjamin Morton and James Ferguson, Eighteenth Century Lecturers on Experimental Philosophy,” Ann. Sci. 40 (1983): 437–455, and “James Ferguson’s Lecture Tour of the English Midlands in 1771,” same journal, (1985): 597–415

    Article  Google Scholar 

  94. and Colin Russell, Science and Social Change in Britain and Europe, 1700–1900 (New York: St. Martin’s Press, 1983).

    Google Scholar 

  95. Wilhelm Jacob van ‘sGravesande, Mathematical Elements of Natural Philosophy Confirmed by Experiment, 4th ed. (London: 1731) 2 vols. For ‘sGravesande and physics as experiment, see Pierre Brunet, Les physiciens hollandais.

    Google Scholar 

  96. Leonhard Euler, Letters on Different Subjects in Natural Philosophy, addressed to a German Princess 2 vols. (New York: Arno reprint of 1833 edition). This was not a popularization as depicted

    Google Scholar 

  97. in Walter D. Wetzels, “Popularization of the New Physics: Euler’s Letters;” Stud. Voltaire Eighteenth Cent. 264(1989): 796–800. Their metaphysical and “antique” character is discussed in

    Google Scholar 

  98. John Heilbron, Electricity in the Seventeenth and Eighteenth Centuries: A Study of Early Modern Physics (Berkeley, CA: University of California Press, 1979), p. 72–73.

    Google Scholar 

  99. See also Casper Hakfoort, Optics in the Age of Euler: Conceptions of the Nature of Light (Cambridge: Cambridge University Press, 1995)

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Garber, E. (1999). Vibrating Strings and Eighteenth-Century Mechanics. In: The Language of Physics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1766-4_2

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  • DOI: https://doi.org/10.1007/978-1-4612-1766-4_2

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