With Lagrange and Gauss the nineteenth-century rigorization of analysis got under way. This work was considerably furthered and strengthened by the great French mathematician Augustin-Louis Cauchy, the most outstanding analyst of the first half of the nineteenth century.

Cauchy was born in Paris in 1789 and received his early education from his father. Later, at the Ecole Centrale du Pantheon, he excelled in ancient classical studies. In 1805 he entered the Ecole polytechnique and won the admiration of Lagrange. Two years later he enrolled at the Ecole des Ponts et Chaussees, where he prepared himself to be a civil engineer. Under the persuasion of Lagrange, he decided to give up civil engineering in favor of pure science and accepted a teaching post at the Ecole Polytechnique.

Cauchy wrote extensively and profoundly in both pure and applied mathematics, and he can probably be ranked next to Euler in volume of output. His collected works contain , in addition to several books, seven hundred and eighty-nine papers, some of which are very extensive works, and  fill twenty-four large quarto volumes. This work is of uneven quality, and consequently Cauchy (quite unlike the case of Gauss) has been criticized for overproduction and over hasty composition. A story is told in connection with Cauchy’s prodigious productivity. In 1835 the Academy of Sciences began publishing its Comptes Rendus. So rapidly did Cauchy supply this journal with articles that the Academy became alarmed over the mounting printing bill, and accordingly passed a rule, still in force today, limiting all published papers to maximum length of four pages. Cauchy had to seek other outlets for his longer papers, some of which exceeded a hundred pages.

Cauchy’s numerous contributions to advanced mathematics include researches in convergence and divergence of infinate series, real and complex function theory, differential equations, determinants, probability and mathematical physics. His name is met by the students of calculus in the so-called Cauchy root test and Cauchy ratio test for covergence or divergence of a series of positive terms and in the Cauchy product of two given series. Even in a first course in complex function theory, one encounters the Cauchy inequality, Cauchy’s integral formula, Cauchy’s integral theorem and the basic Cauchy-Reimann differential equations. Much of the treatment in our present-day college calculus texts is due to Cauchy, such as the basic concept of limit and continuity.

Cauchy’s contribution to determinant theory, starting with a large eighty-four page memoir in 1812, mark him as the most prolific contributor in this field. It was in his 1812 paper that Cauchy gave the first proof of an important and useful theorem. Incidentally, it was Cauchy who, in 1840, introduced the word “characteristic” into matrix theory.

Cauchy’s work exhibits great attention to rigor, and as such was largely responsible for inspiring other mathematicians to attempt the banishment of blind formal manipulation and of intuitive proofs  from analysis.

Cauchy was ardent partisan of the Bourbons and, after the revolution of 1830, was forced to give up his professorship at the Ecole Polytechnique and was excluded from public employment for eighteen years. Part of this time he spent in exile in Turin and Prague, and part in teaching in church schools in Paris. In 1848 he was allowed to return to a professorship at the Ecole Polytechnique without having to take the oath of allegiance to the new goverment.

It has been said that in religion he was bigoted, that he was an indefatigable worker and that he possessed a narrow conceit and often ignored the meritorious efforts of younger men. Nevertheless on the other side of the coin, it should be pointed out that in 1843 Cauchy published, in the form of an open letter, a magnificent defence of freedom of conscience and thought. This letter helped to bring home to the government the stupidity of academic repression, and Louis Philipe was ousted, one of the first acts of the succeeding Provisional Government was to abolish the detestable oath of allegiance.

Cauchy died suddenly on May 23, 1857, when he was sixty-eight years old. He had gone to the country to rest and to cure a bronchial trouble, on to smitten by a fatal fever. Just before his death he was talking with the Archbishop of Paris. His last words, addressed to the Archbishop, were “Men pass away, but their deeds abide.”