Abstract
Up until this time Mademoiselle Germain had not published a thing. Then there occurred a remarkable event that established her as an author. A German scholar, Chladni, came to Paris to repeat his intriguing experiments on the vibration of elastic plates. They caused a sensation. Napoleon, before whom they were performed, took a lively interest in them, regretted that they had not been subject to calculation, and, in order to facilitate this, proposed that the Institute offer a prix extraordinaire. Yet geometers were completely discouraged by a word from Lagrange, who said that the solution of this problem would require a new kind of analysis. Despite the imposing authority of this geometer of Turin, Mademoiselle Germain never despaired of success. She studied these phenomena in a thousand ways, applied analysis to them, and submitted a memoir in competition wherein she gave an equation for elastic surfaces.1
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Chapter Five. the One Entry
Libri, G.: 1832, ‘Notice sur Mlle. Sophie Germain’, in Lherbette, (ed.), Considérations générales sur l’état des sciences et des lettres, Lachevardière, Paris, 1833, p. 13.
Lagrange, J. L.: 1788, Mécanique analytique. Chez la Veuve Desaint, Paris.
Euler, L.: 1779, ‘Investigatio Motuum quibus laminae et virgae elasticae contremiscunt’ Acta Acad. Sci., Petrop. 1, pp. 103–161. Leonhard Euler (1707–1783), a Swiss mathematician, is recognized, like Gauss, as one of the finest minds to have worked in both pure and applied mathematics. Euler wrote most of his memoirs in Latin — a language in which Sophie Germain had no formal training.
Lagrange, J. L.: 1788, op. cit., ‘Avertissement de la première édition.’
Germain, S.: 1821, Recherches sur la théorie des surfaces élastiques, Mme. Ve. Courcier, Paris, 1821, (Avertissement). It is unlikely that Sophie Germain witnessed Chladni’s experiments in the company of Laplace and Napoleon. More probably, Chladni performed before small, informal gatherings of members of the First Class. Lagrange could have invited Sophie Germain to one of these.
Euler, L.: 1779, op. cit.
Ibid., p. 264.
Ibid., p. 265.
A manuscript containing Sophie Germain’s analysis of Euler’s investigation of the vibration of a beam supported at an interior point is found in the Bibliothèque Nationale, MS. Fr. 9115 f. 155–194. It carries the title ‘Remarques sur le mémoire d’Euler: Investigatio motuum quibus laminae et virgae elasticae constremiscunt’, in Acta Acad. Petrop Ann. (1779) p. 103 et seq.’
Legendre is referring to the pagination of Euler, L.: 1779, op. cit.
Bibliothèque Nationale, MS. Fr. 9118. Also in Stupuy, op. cit., p. 287. Stupuy consistently erred in the designation of the trigonometric function cotangent. He wrote ‘cos’ where Legendre wrote ‘cot’.
In her ‘Remarques sur le mémoire d’Euler…’, Sophie Germain deduced that in order to obtain solutions which may be interpreted in terms of Euler’s Case V, the two equations \( {\text{tang}}(1 - \lambda )\omega = \frac{{{e^{{(1 - \lambda )\omega }}} - {e^{{ - (1 - \lambda )\omega }}}}}{{{e^{{(1 - \lambda )\omega }}} - {e^{{ - (1 - \lambda )\omega }}}}} \) and \( {\text{tang}}\lambda \omega = \frac{{{e^{{\lambda \omega }}} - {e^{{ - \lambda \omega }}}}}{{{e^{{\lambda \omega }}} - {e^{{ - \lambda \omega }}}}} \) must be simultaneously satisfied by ω. For most values of λ, She showed that this is impossible.
Bibliothèque Nationale, MS. Fr. 9118. Also in Stupuy, op. cit., p. 291. Legendre missed the point that Sophie Germain had made with respect to solutions of the second kind. She is correct, Legendre in error.
Ibid., p. 295.
Sophie Germain’s 1811 entry is held at the Archives de l’Académie des Sciences. A draft is contained in MS. 2381, Bibliothèque de l’Institut.
Bibliothèque Nationale, MS. Fr. 9118. Also in Stupuy, op. cit., p. 298.
Ibid., p. 299.
Bibliothèque Nationale, MS. Fr. 9114. Also in Stupuy, op. cit.
Bibliothèque Nationale, MS. Fr. 9118. Also in Stupuy, op. cit., p. 300. Legendre urged Sophie Germain to consult Lagrange’s analysis of the membrane. He was, in effect, telling her how Lagrange had obtained the differential equation for the plate starting from her hypothesis (that the elastic force is proportional to (l/r)+(l/r’)), then applying the techniques of the variational calculus. Lagrange’s adoption of Sophie Germain’s expression for the elastic force and derivation, of what has since become a familiar equation governing the bending of plates, borders on serendipity. (See note 25, ch. 6).
Sophie Germain, 1811 entry, op. cit.
One sees that I take V((1/r) + (l/r’)) for the moment of this force. It would take too long to discuss here those considerations that led me to choose this function of osculatory radii. But it is easy to see that even if the expression for elastic moment ought to contain other functions of these radii than this one I have adopted (such as products or powers of these same quantities), the results, when applied to this problem, would not change since (as is clear in what follows), (l/r)·(l/r’). for example, may always be neglected with respect to ((1/r) + (1/r’)). Ibid.
As we shall see, in Chapter 7 below, this note was made public in the course of a polemic involving Poisson and Navier. See Navier. 1828, ‘Remarques sur l’article de M. Poisson’, Annal, de chimie 39, p. 149. The original is held at the Bibliothèque Nationale, MS. Fr. 9118. How it ended up among Sophie Germain’s papers is unclear.
The actual letter sent to Legendre is missing, perhaps destroyed. Sophie Germain’s draft is found in the Bibliothèque Nationale, MS. Fr. (Nouv. Acq.) 5166.
Institut de France. I re Classe. Travaux divers, vol. 2, no. 24, (1811–1816), (Public Session of 6 January, 1812).
An interesting aspect of the variational approach to the plate-bending problem is that if one assumes, as Sophie Germain did, that the elastic force is proportional to V((l/r) + (1/r’)) one obtains, as Langrange did, the correct equation governing the motion of the interior points, but some of the equations governing the behavior at the plate’s edges, obtained in this way, are in error. (See note 25, Chapter 6). To Kirchoff (1827–1888) we owe the resolution of these difficulties. Kirchoff, G. R.: 1850, ‘Ueber das Gleichgewicht und die Bewegung einer elastischen Scheibe’, Grelles Journal 40.
Bibliothèque Nationale, MS. Fr. (Nouv. Acq.) 5166.
…“sum of the moments of elastic forces that act throughout the plate.” Sophie Germain, 1813 Entry, Archives de l’Acad. des Sciences, p. 2.
Bibliothèque Nationale, MS. Fr. 9118. Also in Stupuy, op. cit., p. 303.
Institut de France. I re Classe. Travaux divers, op. cit. (Public Session of 3 January 1814).
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© 1980 D. Reidel Publishing Company, Dordrecht, Holland
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Bucciarelli, L.L., Dworsky, N. (1980). The One Entry. In: Sophie Germain. Studies in the History of Modern Science, vol 6. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9051-7_5
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