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Augustin Louis Cauchy



Augustin Louis Cauchy (đôi khi tên họ được viết Cô-si) là một nhà toán học người Pháp sinh ngày 21 tháng 8 năm 1789 tại Paris và mất ngày 23 tháng 5 năm 1857 cũng tại Paris. Ông vào học Trường Bách Khoa Pháp (École Polytechnique) lúc 16 tuổi. Năm 1813, ông từ bỏ nghề kỹ sư để chuyên lo về toán học. Ông dạy toán ở Trường Bách Khoa và thành hội viên Hàn Lâm Viện Khoa Học Pháp.

Công trình lớn nhất của ông là lý thuyết hàm số với ẩn số tạp. Ông cũng đóng góp rất nhiều trong lãnh vực toán tích phân và toán vi phân. Ông đã đặt ra những tiêu chuẩn Cauchy để nghiên cứu về sự hội tụ của các dãy trong toán học.

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CAUCHY Augustin (1789-1857)

Ce mathématicien réputé est un littéraire au départ...


Sa réputation de savant s'installe




Ce mathématicien réputé est un littéraire au départ... puis il se tourne vers les sciences pour préparer l'entrée à Polytechnique et ensuite aux Ponts où il est admis premier en 1807.

Très vite, il va se consacrer essentiellement à ses recherches mathématiques : il rédige des mémoires très remarqués sur les polygones et les polyèdres ; en 1815, il démontre l'un des deux théorèmes de Fermat ; il multiplie les publications, espérant être élu à l'Académie, il est finalement nommé membre de l'Institut en 1816 (par une ordonnance royale qui exclut pour des raisons politiques Lazare Carnot et Gaspard Monge). Sa fidélité monarchiste a été payante...


Mais, à la révolution de 1830, Cauchy refuse de prêter serment à Louis-Philippe et s'exile : il perd ses titres de professeur à Polytechnique, à la faculté des sciences, et d'ingénieur des ponts et chaussées.

Il va traverser une période d'isolement et ce n'est qu'en 1849, sous le régime républicain, qu'il retrouvera ses fonctions.

Son oeuvre scientifique est considérable : plus de 800 articles sur les sujets mathématiques et physiques les plus variés. Il a été un fondateur, bouleversant certaines parties de la physique mathématique. Il est à l'origine de l'analyse moderne : on lui doit notamment la théorie des équations différentielles et la théorie mécanique de l'élasticité.

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Biographies of Mathematicians - Cauchy
His Life
Cauchy was born on August 21, 1789 in Paris, France.

Austin Louis Cauchy was born on August 21, 1789 in Paris, France. He was the oldest of six children, born to a Catholic Lawyer, classical scholar, police officer, and strong supporter of the king. With his family, Cauchy's father retreated to the country to escape the gruesome aftermath of the Revolution. For the first twenty years of Cauchy's life he was undernourished, frail, and weak. His father educated him until the age of 13. When Cauchy was young Laplace and Lagrange took part in his mathematical education. Lagrange advised his father to allow his son to have a good knowledge of languages before pursuing mathematics. Between 1802 and 1804 Cauchy attended Ecole Centrale du Pantheon where he studied the classical languages. In 1805 Cauchy took the entrance exam for the Ecole Polytechnique. He was examined by Biot and placed second. There he took courses by Lacroix, de Prony, and Hachette while his analysis teacher was Ampere. After graduating in 1807 Cauchy entered Ecole des Ponts et Chaussees, a engineering school. While he was there he was assigned the Ourcq Canal project under Pierre Girard.

Cauchy took his first job in 1810 at Cherbourg. Here he worked on port facilities for Napoleon's English invasion fleet. While taking a heavy workload Cauchy pursued his mathematical research. In 1811 he proved that the angles of a complex polyhedron are determined by its faces. On this topic Cauchy wrote his first paper and later submitted another paper about polygons and polyhedra. In 1812 Cauchy investigated symmetric functions and submitted a memoir later published in the Ecole Polytechnique in 1815. A memoir on definite integrals was published in 1814, this became the basis of his theory of complex functions.

He was appointed assistant professor of analysis at the Ecole Polytechnique in 1815. The Grand Prix of the French Academy of Science was awarded to him in 1816. Cauchy submitted a paper to the Institute solving one of Fermat's claims on polygonal numbers.

In 1817 Cauchy filled in for Biot at the College de France, where he lectured on the methods of integration. These methods he discovered, but did not publish. Cauchy was the first to make a rigorous study of the conditions of convergence of infinite series in addition to his definition of an integral. Cours d'analyse, text on developing basic theorems of calculus as precisely as possible, was designed for the students at Ecole Polytechnique by Cauchy. He began his study of the calculus of residues in 1826 in Sur un nouveau genre de calcul analogue au calcul infinetesimal while in 1829 in Lecons sur le Calcul Differential he defined for the first time a complex function of a complex variable.

Cauchy was disliked by many of the other scientists. Abel wrote of him after his visit to the Institute in 1826, "Cauchy is mad and there is nothing that can be done about him, although, right now, he is the only one who knows how mathematics should be done." When Cauchy refused to swear an oath of allegiance to the new regime, because of political events in France, he lost all his positions there.

In Turin during 1831 he accepted an offer from the King of Piedmont for a chair of theoretical physics. He taught in Turin and Menbrea attended his courses, he commented, "very confused, skipping suddenly from one idea to another, from one formula to the next, with no attempt to give a connection between them. His presentations were obscure clouds, illuminated from time to time by flashes of pure genius ... of the thirty who enrolled with me, I was the only one to see through it."

In 1833 Cauchy moved from Turin to Prague in order to tutor the grandson of Charles X. As this quote shows he was not very successful, "When questioned by Cauchy on a problem in descriptive geometry, the prince was confused and hesitant ... As with mathematics, the prince showed very little interest ... Cauchy became annoyed and screamed and yelled. The queen sometimes said to him, 'too loud, not so loud'."

There were discussions in Prague with Bolzano about how much Cauchy's definition of continuity is due to Bolanzo. Cauchy's definition was formed before Bolzano's seems to be more convincing, though.

After moving back to Paris in 1838 Cauchy regained his position at the Academy; however, he did not get his teaching positions back because he refused to take the oath. Later in 1839 the position at the Bureau Des Longitudes was open. Although Cauchy was elected, because he did not take the oath he was not allowed to attend any meetings or receive a salary. The mathematics chair at the college de France became vacant in 1843. Cauchy should have easily won on account of his mathematical abilities, but due to religious and political views he was not chosen. During the time after that Cauchy's mathematical output was less than before.

He did important work on applications to mathematical physics, mathematical astronomy, and differential equations. His four volume text Exercises d'analyse et de physique mathemtique was published between 1840 and 1847. Cauchy still did not change in his views and continued to give his colleagues problems. Cauchy stole many of his ideas from his colleagues, they referred to him as "cochon", which is French for pig. In the last few years of his life he had a dispute with Duhamel regarding a result on the inelastic shocks. Cauchy claimed to be the first to give the results in 1832, but Poncelet referred to his own work on the subject in 1826. Even though Cauchy was proved wrong he would never admit it. Cauchy died at about four a.m. on May 27, 1857. His last words were "Men pass away, but their deeds abide". Many terms in mathematics bare his name, the Cauchy integral theorem, in the theory of complex functions, the Cauchy-Kovalevskaya existence theorem for the solution of partial differential equations, the Cauchy-Riemann equations, and the Cauchy sequences. A book of his collected works entitled Oeuvres completes d'Augustin Cauchy (1882-1870) was published in 27 volumes.

Cauchy died on May 23, 1857 in Sceaux, France.

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Encyclopedia of World Biography on Augustin Louis Cauchy

The French mathematician Augustin Louis Cauchy (1789-1857) provided the foundation for the modern period of rigor in analysis. He launched the theory of functions of a complex variable and was its authoritative pioneer developer.

Augustin Louis Cauchy was born in Paris on Aug. 21, 1789, 38 days after the fall of the Bastille. His father, Louis François, was a parliamentary lawyer, lieutenant of police, and ardent royalist. Sensing the political wind, he moved the family to his country cottage at Arcueil, where they lived for nearly 11 years. Here young Cauchy received a strict religious education from his mother and an elementary classical education from his father, who wrote his own textbooks in verse.

By 1800 the political situation had stabilized and the family moved back to Paris. At the age of 16 Cauchy entered the École Polytechnique, at that time the best school in the world for a budding mathematician. Originally designed to produce military engineers for the Revolutionary armies of France, the school developed as a revolutionary (in method) educational institution. Teaching was linked with research as the nation's finest mathematicians created pure mathematics in discussion with their students and showed them how mathematical theory and practice nourished one another at the very edge of invention.

As Lagrange and Laplace had predicted, Cauchy was a brilliant academic success. In the realm of personal relationships he was not so successful. The generally anticlerical polytechnicians simply could not believe that a brilliant student as aggressively pious and evangelically Catholic as Cauchy could exist. His imperturbability on the matter progressively amused, bewildered, irritated, and infuriated them. It was a pattern of responses that was to become typical in his social relationships. Many years later, after Cauchy had become the most influential mathematician in the world, the naive young genius Abel would conclude that Cauchy was insane. How else could a man of science be so bigoted in religious matters?

From Engineer to Mathematician

From the Polytechnique, Cauchy passed to the École des Ponts et Chaussées, where he studied engineering for 3 years. Upon graduation in 1810, he was sent to Cherbourg as a military engineer. But he could not stay away from pure mathematics. In his spare time he began to review all mathematics, "clearing up obscurities" and inventing new methods for the "simplification of proofs and the discovery of new propositions." He displayed the power and originality of these methods in a series of papers that impressed even the sophisticated mathematical community of Paris. Among these researches were two on polyhedrons, one on symmetric functions, and one on determinants. In the last paper Cauchy reorganized all that was then known about the subject and gave the word "determinant" its modern meaning. All this spare-time work had two results: it broke Cauchy's health, and he abandoned engineering to devote his life to mathematics.

If the mathematical community had been impressed by Cauchy the hobbyist mathematician, it was dazzled by Cauchy the full-time professional. In 1815 he proved a Fermat conjecture on polygonal (figurate) numbers that had defeated some of the world's best mathematicians. In the following year he demonstrated his versatility by winning the grand prize of the Académie des Sciences with a mathematical treatment of wave propagation on the surface of a fluid. Meanwhile, he had obtained his first teaching position, at the Polytechnique. He was appointed professor there in 1816, and before long he was also lecturing at the Collége de France and the Sorbonne.

At the age of 27 Cauchy was elected to the Académie des Sciences-an unusual honor for so young a man. In his case, there were some who insisted that there was nothing honorable about it. The chair which Cauchy filled had belonged to Gaspard Monge, the father of descriptive geometry, first director of the École Polytechnique, and loyal follower of Napoleon I. The restored Bourbon regime demanded that Monge be expelled from the academy. The academicians complied and elected Cauchy in his place. Cauchy, as rigidly ultraroyalist in politics as he was ultra-Catholic in religion, could never see anything improper about the procedure.

In 1818, securely established as the outstanding mathematician of France, Cauchy married Aloise de Bure. They had two daughters.

Prolific Decade

Cauchy worked as if he expected his worth to be measured by the sheer weight of his publications. His ideas, touching upon nearly every branch of mathematics, pure and applied, seemed to materialize as fast as he could write them down. There were occasions when he would produce two full-length papers in one week.

One of Cauchy's major interests in these years was the attempt to repair the logical foundations of analysis in such a way that this branch of mathematics would have "all the rigor required in geometry." This was a problem of long standing.

In his devastating criticism of the Newton-Leibniz calculus, Bishop Berkeley had suggested that the faulty reasoning of the calculus led to correct results because of compensating errors. Maclaurin and Lagrange accepted the criticism and both made heroic efforts to construct a logical justification for the methods of the differential calculus. Neither succeeded.
Cauchy did not quite succeed either. But he took a great step in the right direction when he made the concept of limit the basis for the whole development. His definition of continuity and the derivative in terms of limit was quite modern. But to say that Cauchy" gave the first genuinely mathematical definition of limit, and it has never required modification" is quite wrong.

Cauchy defines "limit" as follows: "When the values successively assigned to the same variable indefinitely approach a fixed value, so as to end by differing from it as little as desired, this fixed value is called the limit of all the others."

As a rough description of the limit idea, Cauchy's "definition" may have merit. But it is verbal, intuitive, crammed with undefined terms, and therefore absolutely nonmathematical in the modern sense. Strangely enough, Cauchy did give a precise mathematical definition of convergent series, and he went on to establish criteria for convergence. It is said that Laplace, after hearing Cauchy's first lectures on series, rushed home in a panic, barred his door, and laboriously tested all the series in his masterpiece, the Mécanique céleste, using Cauchy's criteria. This story, perhaps apocryphal, nevertheless indicates how Cauchy's methods began to set new standards of rigor in analysis.

Between 1825 and 1831 Cauchy published a series of papers which created a new branch of analysis, the theory of functions of a complex variable. It is the principal mathematical tool used in vast domains of physics.

A Matter of Principle

The Revolution of 1830 sent Charles X into exile. The new king, Louis Philippe, demanded oaths of allegiance from the professors of France. Cauchy refused. He had already sworn his oath to Charles. Stripped of all his positions, he exiled himself to Switzerland, leaving his family in Paris.

In 1831 Cauchy was appointed professor of mathematical physics at Turin. Two years later Charles summoned him to Prague to tutor Henri, his 13-year-old grandson. Cauchy, ever the faithful legitimist, agreed to supervise the education of the future pretender. His family joined him in Prague in 1834. Playing Aristotle to Henri's Alexander consumed most of Cauchy's waking hours and sharply curtailed his mathematical output. It never ceased entirely, however. Among the important papers of this period were a long memoir on the dispersion of light, and the first existence proofs for the solution to a system of differential equations.

In 1838 Cauchy and family returned to Paris. Charles had baroneted him, but the title was no help in getting a position, since Baron Cauchy still refused to take the oath. At last, after the Revolution of 1848, the oath was abolished, and Cauchy resumed his old professorship at the Polytechnique. Louis Napoleon reinstituted the oath in 1852, but Cauchy was specifically exempted.

Meanwhile Cauchy's rate of publication reached and even surpassed previous limits. Of special merit in the more than 500 papers that appeared after 1838 were treatises on the mechanics of continuous media, the first rigorous proof of Taylor's theorem, a remarkably modern representation of complex numbers in terms of polynomial congruences, and a collection of papers on the theory of substitutions.

Cauchy's Influence on Mathematics

If the worth of a mathematician were to be measured by the number of times his name appeared in modern college textbooks, Cauchy might be ranked as the greatest of them all. His long-standing influence and fame are due in part to the fact that he swamped the competition with the published word. He was the first mathematician to realize that the greatest material engine of mathematical progress was the printing press. He knew that the entire mathematical community, from professor to arithmetic teacher, took its cue from published papers and textbooks. He literally imprinted his ideas upon a generation.

This practice of rapid publication, together with Cauchy's rather flowery style, had its dangers. Abel, for one, had difficulty in understanding some of Cauchy's papers. "His works are excellent, but he writes in a very confusing manner." But Cauchy's style of writing was the least of the offenses he committed against Abel in particular and mathematics in general. The 15-year delay in the publication of Abel's masterpiece--from 1826 to 1841--was largely due to Cauchy's cavalier treatment of it. Abel died in 1829, the same year in which Cauchy contributed to the suppression of young Galois's epochmaking discoveries. Galois died in 1832. It was this contemptuous attitude toward younger mathematicians, together with his religious and political bigotry, that made Cauchy unpopular with many of his colleagues. After all, it was difficult to overlook the fact that Galois had been a radical republican.

Cauchy died on May 23, 1857, after a short illness. His last words were, "Men die but their works endure."

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Augustin Louis Cauchy from Encyclopedia of World Biography. ©2005-2006 Thomson Gale, a part of the Thomson Corporation. All rights reserved.