Abstract
Sophie Germain set out to derive the mathematical theory to describe the complex phenomena manifested on Chladni’s vibrating plates. To do that, Germain sought to obtain a clear understanding of the theories advanced by Euler, the Bernoullis, d’Alembert, and Lagrange, and she tried to extend and improve their analysis. This was a daunting task. Her predecessors had worked for many years to formulate the mathematical foundation for elasticity that was in place in 1809.
That among all curves of the same length which not only pass through the points A and B, but are also tangent to given straight lines at these points, that curve be determined in which the value of \( \mathop \smallint \nolimits_{A}^{B} \frac{ds}{{R^{2} }} \) be a minimum.
—EULER, 1744
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Notes
- 1.
Bernoulli I (1705).
- 2.
Calinger (1996).
- 3.
Euler (1736).
- 4.
Euler (1744).
- 5.
Goldstine (1980), p. 67.
- 6.
Correspondence from Daniel Bernoulli to Euler. Quoted statement (in Latin) is at the end of the letter dated 20 October 1742. Letter XXVI in Fuss (1843), and online at The Euler Archive.
- 7.
Euler (1744). An English translation of the first selections of this book, along with some brief commentary, is found in Struik (1969) pp. 399–406.
- 8.
Straub (1990).
- 9.
Timoshenko (1983).
- 10.
Ibid., p. 34.
- 11.
Euler (1744) Additamentum 1, De curvis elasticis, pp. 245–310. Equation appears in p. 285.
- 12.
Euler (1757).
- 13.
Euler (1862).
- 14.
Young (1807). See especially Lecture 13, On Passive Strength and Friction, pp. 109–113; Definition of squeeze–stretch ratio, p. 105.
- 15.
D’Alembert’s principle is an alternative form of Newton’s second law of motion. This principle reduces a problem in dynamics to a problem of statics.
- 16.
Euler (1740).
- 17.
D’Alembert (1747).
- 18.
Euler (1748–1749).
- 19.
Wheeler and Crummet (1987).
- 20.
Straub (1970–1990).
- 21.
Daniel Bernoulli to Euler. Letter LVII dated 7 October 1753 in Fuss (1843) and online at the Euler Archive.
- 22.
Euler (1766).
- 23.
Jacques II Bernoulli married a granddaughter of Euler in St Petersburg. He died in 1789 by drowning in the Neva River.
- 24.
Bernoulli II (1789).
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Musielak, D. (2020). Euler and the Bernoullis. In: Sophie Germain. Springer Biographies. Springer, Cham. https://doi.org/10.1007/978-3-030-38375-6_5
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