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Augustin Louis Cauchy (21 August 1789 – 23 May 1857; pronounced ) was a French mathematician, who is one of the most prominent mathematicians of the first half of the nineteenth century. He started the project of formulating and proving the theorems of infinitesimal 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, through his perspicuous and rigorous methods Cauchy exercised a great influence over his contemporaries and successors. His writings cover the entire range of mathematics and mathematical physics.

Cauchy was a prolific writer, he wrote approximately eight hundred research articles and five complete textbooks. He was a devout Roman Catholic, strict (Bourbon) royalist, and a close associate of the Jesuit order.

Biography
Cauchy's father (Louis-François Cauchy) was a high official in the Parisian Police of the Old Régime. He lost his position because of the French Revolution (July 14, 1789) that broke out one month before Augustin-Louis was born. The Cauchy family survived the revolution and the following Reign of Terror (1794) by escaping to Arcueil, where Cauchy received his first education, from his father. After the death of Robespierre (1794), it was safe for the family to return to Paris. There Louis-François Cauchy found himself a new bureaucratic job, and quickly moved up the ranks. When Napoleon Bonaparte came to power (1799), Louis-François Cauchy was further promoted, and became Secretary-General of the Senate, working directly under Laplace (who is now better known for his work on mathematical physics). The famous mathematician Lagrange was also no stranger in the Cauchy family.

On Lagrange's advice, Augustin-Louis was enrolled in the École Centrale du Panthéon, the best secondary school of Paris at that time, in the fall of 1802. Most of the curriculum consisted of classical languages; the young and ambitious Cauchy, being a brilliant student, won many prizes in Latin and Humanities. In spite of these successes, Augustin-Louis chose an engineering career, and prepared himself for the entrance examination to the École Polytechnique.

In 1805 he placed second out of 293 applicants on this exam; naturally, he was admitted. One of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education. The school functioned under military discipline, which caused the young and pious Cauchy some problems in adapting. Nevertheless, he finished the Polytechnique in 1807, at the age of 18, and went on to the École des Ponts et Chaussées (School for Bridges and Highways). He graduated in civil engineering, with the highest honors.

Cauchy entered l'École Centrale du Panthéon in 1802, proceeded to the École Polytechnique in 1805, and to l'École Nationale des Ponts et Chaussées in 1807, afterwards adopting 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 after Charles X of France of the House of Bourbon was ousted. He did this because he found it impossible to take the necessary oaths to the new government because 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 (at that time, Turin was the capital of the Kingdom of Sardinia, which unified Italy later in 1871.)

Cauchy married Aloise de Bure in 1818. She was a close relative of the 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: Marie Françoise Alicia (1819) and Marie Mathilde (1823).

Work
The genius of Cauchy was illustrated in his simple solution of the problem of Apollonius&mdash;describing a circle touching three given circles&mdash;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).

Other significant contributions include being the first to prove the Fermat polygonal number theorem. Cauchy created the residue theorem, used it to derive a whole host of 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, are published in 27 volumes.

In a paper published in 1855, two years before Cauchy's 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 a 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 Lois-François Cauchy spoke of living on rice, bread, and crackers during the period. A paragraph from an undated letter from Louis-François to his mother in Rouen, cited by C A Valson in |La Vie et les Travaux du baron Cauchy (Volume 1, Pg 13) says:We never had more than a half pound of bread — and sometimes not even that. This we supplement with little supply of hard crackers and rice that we are allotted. Otherwise, we are getting along quite well, which is the important thing and goes to show that human beings can get by with little. I should tell you that for my children's pap I still have a bit of fine flour, made from wheat that I grew on my own land. I had three bushels, and I also have a few pounds of potato starch. It is as white as snow and very good, too, especially for very young children. It, too, was grown on my own land (9). '' 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 plead 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."

Works by A. Cauchy

 * Oeuvres complètes d'Augustin Cauchy publiées sous la direction scientifique de l'Académie des sciences et sous les auspices de M. le ministre de l'Instruction publique (27 volumes) (Paris : Gauthier-Villars et fils, 1882-1974)
 * Analyse algèbrique (Imprimerie Royale, 1821)
 * Nouveaux exercices de mathématiques (Paris : Gauthier-Villars, 1895)
 * Exercices d'analyse et de physique mathematique (Volume 1)
 * Exercices d'analyse et de physique mathematique (Volume 2)
 * Exercices d'analyse et de physique mathematique (Volume 3)
 * Exercices d'analyse et de physique mathematique (Volume 4) (Paris: Bachelier, 1840-1847)
 * Cours d'analyse de l'École royale polytechnique