Augustin-Louis Cauchy

born: Aug 21, 1789 in Paris
died: May 23, 1857 outside Paris

Modern mathematics is indebted to Cauchy for two of its major interests, each of which marks a sharp break with the mathematics of the 18th century. The first was the introduction of rigor into mathematical analysis. The second thing of fundamental importance was on the opposite side — the combinatorial.
(E.T. Bell)

Brilliant. Prolific. Pious. Stubborn. Based calculus on the concept of limit. Cauchy’s root and ratio tests for convergence of series, Cauchy’s inequality, Cauchy’s integral formula and theorem, Cauchy product, Cauchy-Riemann differential equations. His ideas developed into the theory of finite groups.

Cauchy was the oldest of six children of a Catholic lawyer, classical scholar, police officer and supporter of the king. In order to escape the bloody aftermath of the Revolution, his father retreated with his family to his country home in Arcueil where the family subsisted for a number of years primarily on whatever they could raise. As a consequence, Cauchy was undernourished and frail until he was 20. All of the schools were closed following the French Revolution so he was educated at home by his father and did not enter school until he was 13. He immediately started to win the academic prizes — first prize in Greek, Latin, Latin verse and even the national prize in humanities. At 16 he entered the Polytechnique; at 18 he attended a school for civil engineers; and at 21 he had a degree in civil engineering. His first assignment was to be a military engineer for Napoleon and to help build defenses at Cherbourg. He took four books with him — a physics book by Laplace, an analysis book by Lagrange, a religious book and a collection of Vergil’s Latin writings — hardly the usual collection for a young engineer. Cauchy worked hard and even found time to tutor the other engineers, to help with the local school examinations and to do mathematical research. In a letter to his mother, he said, “I get up at four and am busy from morning to night. Work doesn’t tire me; on the contrary it strengthens me and I am in perfect health.”

At 24, Cauchy returned to Paris. His mathematical researches at Cherbourg had already attracted the attention of France’s top mathematicians. At 26, he proved a conjecture of Fermat which had stumped Euler and Gauss: “Every positive integer is the sum of 3 triangular numbers, 4 square numbers, 5 pentagonal numbers, … “. He lectured on analysis at the Polytechnique and became a professor at the Sorbonne. By 27, he was among the best living mathematicians and had started work on functions of a complex variable. All 300 pages of this research was published 11 years later. During his incredible mathematical rise he also found time to marry, apparently very happily. He remained married for 40 years and had 2 daughters whom he educated at home in the way he had been educated by his father.

Encouraged by several colleagues, Cauchy finally published his analysis lectures, and they helped set the modern standard of rigor found in textbooks. Bell says, “Even today Cauchy’s definitions of limit and continuity, and much of what he wrote on the convergence of infinite series in this course of lectures, will be found in any carefully written book on the calculus.”

He refused to take a loyalty oath to the government which took power in 1830 and went into exile to take a position as professor of mathematical physics at Turin in Italy. He returned to Paris in 1838, but still refused to take a loyalty oath, so all positions at the national schools were closed to him. He got around the regulations a bit and also managed to survive by teaching in religious schools. The government was overthrown again in 1848, and the oaths were dropped until Napoleon III took charge in 1852. Napoleon reinstated the loyalty oath, but Cauchy’s stand was well known, and he was France’s leading mathematician, so word was quietly passed that he need not take an oath. He spent the rest of his life teaching at the Sorbonne in Paris.

At 67, he developed bronchial trouble and went to the country to recover. However, there he caught a fever and died. His last words were, “Men pass away but their deeds abide.”

He published a total of 789 papers. When the French Academy of Sciences started to publish a weekly journal, Comptes rendus, Cauchy swamped it with so many long papers that a rule was adopted limiting papers to a maximum of 4 pages. His works on complex functions and number theory were each over 300 pages long.

Cauchy was very pious all of his life — a trait which many of his contemporaries thought he overdid. Abel, who was himself the son of a minister and an outstanding mathematician, wrote home, “Cauchy is a bigoted Catholic — a strange thing for a man of science.” But Abel also thought Cauchy’s lectures should be read by anyone who appreciated mathematical rigor.

“The methods he introduced, his whole program of inaugurating the first period of modern rigor, and his almost unequaled inventiveness have made a mark on mathematics that is, so far as we can now see, destined to be visible for many years to come.”

Condensed from Men of Mathematics by E. T. Bell (1937, Simon and Schuster). DTH

Last Updated September 21, 2022