Augustin Louis Cauchy: Encyclopedia - Augustin Louis Cauchy

Augustin Louis Cauchy

Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. He started the project of formulating and proving the theorems of calculus in a rigorous manner and was thus an early pioneer of analysis. He also gave several important theorems in complex analysis and initiated the study of permutation groups. A profound mathematician, Cauchy exercised by his perspicuous and rigorous methods a great influence over his contemporaries and successors. His writings cover the entire range of mathematics and mathematical physics.

Having received his early education from his father Louis François Cauchy (1760–1848), who held several minor public appointments and counted Lagrange and Laplace among his friends, Cauchy entered the École Centrale du Panthéon in 1802, and proceeded to the École Polytechnique in 1805, and to the École Nationale des Ponts et Chaussées in 1807. Having adopted the profession of an engineer, he left Paris for Cherbourg in 1810, but returned in 1813 on account of his health, whereupon Lagrange and Laplace persuaded him to renounce engineering and to devote himself to mathematics. He obtained an appointment at the École Polytechnique, which, however, he relinquished in 1830 on the accession of Louis-Philippe, finding it impossible to take the necessary oaths. A short sojourn at Fribourg in Switzerland was followed by his appointment in 1831 to the newly-created chair of mathematical physics at the University of Turin.

In 1833 the deposed king Charles X of France summoned Cauchy to be tutor to his grandson, the duke of Bordeaux, an appointment which enabled Cauchy to travel and thereby become acquainted with the favourable impression which his investigations had made. Charles created him a baron in return for his services. Returning to Paris in 1838, Cauchy refused a proffered chair at the Collège de France, but in 1848, the oath having been suspended, he resumed his post at the École Polytechnique, and when the oath was reinstituted after the coup d'état of 1851, Cauchy and François Arago were exempted from it.

Cauchy had two brothers: Alexandre Laurent Cauchy (1792–1857), who became a president of a division of the court of appeal in 1847, and a judge of the court of cassation in 1849; and Eugène François Cauchy (1802–1877), a publicist who also wrote several mathematical works.

The genius of Cauchy was illustrated in his simple solution of the problem of Apollonius, i.e. to describe a circle touching three given circles, which he discovered in 1805, his generalization of Euler's theorem on polyhedra in 1811, and in several other elegant problems. More important is his memoir on wave propagation, which obtained the Grand Prix of the Institut in 1816. His greatest contributions to mathematical science are enveloped in the rigorous methods which he introduced. These are mainly embodied in his three great treatises, Cours d'analyse de l'École Polytechnique (1821); Le Calcul infinitésimal (1823); Leçons sur les applications de calcul infinitésimal; La géométrie (1826–1828); and also in his Courses of mechanics (for the École Polytechnique), Higher algebra (for the Faculté des Sciences), and of Mathematical physics (for the Collège de France). His treatises and contributions to scientific journals (to the number of 789) contain investigations on the theory of series (where he developed with perspicuous skill the notion of convergency), on the theory of numbers and complex quantities, the theory of groups and substitutions, the theory of functions, differential equations and determinants. He clarified the principles of the calculus by developing them with the aid of limits and continuity, and was the first to prove Taylor's theorem rigorously, establishing his well-known form of the remainder. In mechanics, he made many researches, substituting the notion of the continuity of geometrical displacements for the principle of the continuity of matter. In optics, he developed the wave theory, and his name is associated with the simple dispersion formula. In elasticity, he originated the theory of stress, and his results are nearly as valuable as those of Simeon Poisson.

His collected works, Œuvres complètes d'Augustin Cauchy, have been published in 27 volumes.

See also

• Cauchy integral theorem
• Cauchy's integral formula
• Cauchy-Schwarz inequality
• Cauchy distribution
• Cauchy determinant
• Cauchy formula for repeated integration
• Cauchy sequence
• Cauchy-Riemann equations
• Cauchy-Frobenius lemma
• Cauchy product
• Cauchy principal value
• Cauchy-Binet formula
• Cauchy-Euler equation
• Cauchy's equation
• Cauchy problem
• Cauchy horizon
• Cauchy boundary condition
• Cauchy surface
• Cauchy-Kovalevskaya theorem
• Maclaurin-Cauchy test
• Cauchy's radical test
• Cauchy (crater)

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