# Augustin Louis Cauchy

# 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 fulltime 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."

## Further Reading

There is no full-length biography of Cauchy, but E.T. Bell, *Men of Mathematics* (1937), contains a biography and a discussion of his place in the history of mathematics. An older source, David Eugene Smith, *History of Mathematics* (2 vols., 1925), gives a brief but adequate account of Cauchy's life. Herbert Westren Turnbull, *The Great Mathematicians* (1961), although it contains no biography of Cauchy, discusses him in relation to the life and work of Joseph Louis Lagrange. See also Jane Muir, *Of Men and Numbers* (1961), written in a lively and popular style and containing numerous references to Cauchy.

## Additional Sources

Belhoste, Bruno, *Augustin-Louis Cauchy: a biography,* New York: Springer-Verlag, 1991. □

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