VI.29 Augustin-Louis Cauchy
b. Paris, 1789; d. Sceaux, France, 1857
Real and complex analysis; mechanics; number theory;
equations and algebra
Trained as a roads and bridges engineer at the École Polytechnique (hereafter, “EP”) and the École des Ponts et Chaussées (1805-10), Cauchy passed his career as an academic at the EP and the Paris Faculté des Sciences of the Université de France until 1830, when he left France with the deposed royal family after the revolution of that year. He returned only in 1838, and later taught in the Paris Faculté.
Of Cauchy’s many contributions to pure and applied mathematics, the best known are in mathematical analysis. In the foundations of real variables, he replaced all previous approaches to the theory with one that (in more developed forms) has now become standard: (i) lay down an explicit theory of limits; (ii) formulate definitions carefully, and in general terms; (iii) define the derivative of a function as the limiting value of the difference quotient, its integral as the limiting value of a sequence of partition sums, its continuity in terms of the joint passage to limits of any sequence of its argument and of its corresponding values, and the sum of a convergent infinite series as the limiting value of its partial sums. A key ingredient in all this was the idea that (iv) limits may not exist: their existence has to be justified carefully. Similarly, (v) the existence of solutions to differential equations has to be proved, not just assumed.
This approach brought a new level of rigor to analysis; for example, for the first time THE FUNDAMENTAL THEOREM OF CALCULUS [I.3 §5.5] was a genuine theorem, governed by conditions on the function. However, this emphasis on limits made the theory hard for beginners: it was not liked by staff or students at the EP, where he taught it in this form between 1816 and 1830 and published it extensively, especially in his Cours d’Analyse (1821) and his Résumé (1823) of the calculus. Its rise to standard educational practice was very gradual, both in France and elsewhere.
Another major innovation of Cauchy dates from 1814, when he began to create complex-variable analysis. Initially the integrand was a complex function but the limits of integration were real; however, from 1825 on they too became complex, and in this form he found many theorems on the residues of functions over closed domains of various shapes. Unusually for him, his progress was fitful, and he cast the theory in terms of the complex plane only in the mid 1840s. He also studied the general theory of complex functions, including their expansion in power series of various kinds.
Cauchy’s main single achievement in applied mathematics lies in linear elasticity theory, where in the 1820s he used stress-strain models to analyze the behavior of various kinds of surfaces and solids; later he adapted it to study aspects of (aetherian) optics. In the 1810s he studied deep-body fluid dynamics, where he found Fourier-integral solutions. In this and several other areas he was in some competition with Fourier and, especially, Poisson, regarding both the quality of the theory and the chronology of its development.
Cauchy’s other contributions lie in basic mechanics (derived from the EP teaching); singular and general solutions of differential equations; the theory of equations, especially methods that helped in the rise of group theory; algebraic number theory; perturbation theory in celestial mechanics; and an astounding paper of 1829 on quadratic forms, which could have launched the spectral theory of matrices had its author recognized its significance!
Further Reading
Belhoste, B. 1991. Augustin-Louis Cauchy. A Biography. New York: Springer.
Cauchy, A. L. 1882-1974. Oeuvres Complètes, twelve volumes in the first series and fifteen in the second. Paris: Gauthier-Villars.