Augustin Louis Cauchy

Augustin Louis Cauchy
Augustin Louis Cauchy
Born	August 21 1789(1789-08-21) Dijon, France
Died	23 May 1857 Paris, France
Residence	France
Nationality	French
Field	calculus
Institutions	École Centrale du Panthéon École Nationale des Ponts et Chaussées École polytechnique
Alma mater	École Nationale des Ponts et Chaussées
Known for	Cauchy integral theorem
Religious stance	Catholic

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.

Biography

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. He did this because he found it impossible to take the necessary oaths to the new government as he remained loyal to the House of Bourbon. 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. (Note: At that time, Turin was the capital of the Kingdom of Sardinia, which unified Italy later in 1871. Now Turin is just a city in northern Italy.)

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. Subequently, Cauchy lived in the France ruled by the emperor Napoleon III until his death in 1857.

Cauchy married Aloise de Bure in 1818. She was a close relative of a publisher who published most of Cauchy's works. 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.

Cauchy had two daughters.

Work

The genius of Cauchy was illustrated in his simple solution of the problem of Apollonius, that is, to describe a circle touching three given circles, which he discovered in 1805, his generalization of Euler's formula 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).

He wrote numerous treatises and made 789 contributions to scientific journals. These writings covered notable topics including the theory of series (where he developed with perspicuous skill the notion of convergency), the theory of numbers and complex quantities, the theory of groups and substitutions, and 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. He also contributed significant research in mechanics, 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.

Other significant contributions include being the first to prove the Fermat polygonal number theorem. He created the residue theorem and used it to derive a whole host of most interesting series and integral formulas and was the first to define complex numbers as pairs of real numbers. He also discovered many of the basic formulas in the theory of q-series. His collected works, Œuvres complètes d'Augustin Cauchy, have been published in 27 volumes.

Although generally rigorous, he was way ahead of the rest of his field at the time, and thus one of his theorems was exposed to a "counter-example" by Abel, later fixed by the inclusion of uniform continuity.

In a paper published in 1855, two years before his death, he discussed some theorems, one of which is similar to the "Argument Principle" in many modern textbooks on complex analysis. In modern control theory textbooks, the Cauchy argument principle is quite frequently used to derive the Nyquist stability criterion, which can be used to predict the stability of negative feedback amplifier and negative feedback control systems. Thus Cauchy's work has strong impact on both pure mathematics and practical engineering.

Politics and religious beliefs

Augustin Louis Cauchy grew up in the house of a staunch royalist. This made his father flee with the family to Arcueil during the French Revolution. Their life there was apparently hard and Cauchy spoke of living on rice, bread, and crackers during the period. In any event he inherited his father's staunch royalism and hence refused to take oaths to any government after the overthrow of Charles X.

He was an equally staunch Catholic and a member of the Society of Saint Vincent de Paul. He also had links to the Society of Jesus and defended them at the Academy when it was politically unwise to do so. His zeal for his faith may have led to his caring for Charles Hermite during his illness and leading Hermite to become a faithful Catholic. It also inspired Cauchy to plea on behalf of the Irish during the Potato Famine.

His royalism and religious zeal also made him contentious, which caused difficulties with his colleagues. He felt that he was mistreated for his beliefs, but his opponents felt he intentionally provoked people by berating them over religious matters or by defending the Jesuits after they had been suppressed. Niels Henrik Abel called him a "bigoted Catholic" and added he was "mad and there is nothing that can be done about him," but at the same time praised him as a mathematician. Cauchy's views were widely unpopular among mathematicians and when Guglielmo Libri Carucci dalla Sommaja was made chair in mathematics before him he, and many others, felt his views were the cause. When Libri was accused of stealing books he was replaced by Joseph Liouville which caused a rift between him and Cauchy. Another dispute concerned Jean Marie Constant Duhamel and a claim on inelastic shocks. Cauchy was later shown, by Jean-Victor Poncelet, that he was in the wrong. Despite that Cauchy refused to concede this and nursed a bitterness on the whole issue.

His daughter indicated his last moments brought him a certain calm and that his final words were "Jesus, Mary, and Joseph."

(For corroboration of claims here see the link to MacTutor History of Mathematics archive for his and Hermite's biographies)

See also

• Cauchy integral theorem
• Cauchy's integral formula
• Cauchy-Schwarz inequality
• Cauchy's theorem (group theory)
• Cauchy's theorem (geometry)
• 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)
• Cauchy functional equation
• Cauchy-Peano theorem
• Cauchy argument principle
• Nyquist stability criterion

References

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External links

• John J. O'Connor and Edmund F. Robertson. Augustin Louis Cauchy at the MacTutor archive
• Cauchy criterion for convergence
• Œuvres complètes d'Augustin Cauchy Académie des sciences (France). Ministère de l'éducation nationale.
• MacTutor biography

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