Augustin-Louis Cauchy was, in many respects, the father of modern mathematical analysis - an idea I discussed in some detail when I noded my homework under Augustin-Louis Cauchy and the Rigorisation of Analysis. However, it is not this alone for which he is famous, and I've been commissioned by the powers that be to furnish a potted biography that's more comprehensive than that essay.

Augustin-Louis Cauchy was born on August 21, 1789, in Paris, at the dawn of the French Revolution. His father, Louis-François Cauchy, was a city police chief, and before the child was more than a few weeks old, the family had had to flee to more rural areas to escape retribution from the revolutionary mob. The astronomer Bailly was among those directing the Storming of the Bastille shortly afterwards, and the Cauchys were well out of it. In 1794, when the Reign of Terror was over, they returned to Paris, and Cauchy Sr got a new job. Napoleon Bonaparte's arrival in the Brumaire coup of 1799 further advanced the family's fortunes, and young Augustin-Louis was offered academic advancement by Count Pierre Laplace, and in 1805 he entered the École Polytechnique. (Literally 'Polytechnic School - actually one of the principal universities.)

After graduation, Cauchy went to work on the Ourcq Canal project, but returned to the École as a teacher when political changes in 1815 left many vacancies. While there, he made himself unpopular with staff and students alike, but succeeded in producing his crucial textbook Cours d'Analyse de l'École Royale Polytechnique: Première Partie: Analyse Algébrique (Analysis Course of the Royal École Polytechnique: First Part: Algebraic Analysis), which lays down the theory of limits as now understood, and represented a tremendous advance from the infinitesimal theory which had prevailed beforehand.

Cauchy did not make friends easily. One of his students, the great Niels Henrik Abel, said:
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.
Abel died, at home in Norway, in 1829, some three years after making this assertion. Cauchy had still not returned Abel's work to him, and when the work finally did get marked, the comments Cauchy made reflected neither his own talents nor those of Abel. Shortly afterwards, Cauchy's conservatism and awkwardness exiled him from his native France. In July 1830 the Orleanist claimant Louis Philippe became king, and the Bourbon loyalist Cauchy moved to Switzerland, teaching for a time at the Académie Helvétique, before moving on to Turin. At about this time Cauchy was considered to have failed to swear an oath of allegiance to the new French régime, and lost all standing in Paris. Critical opinion of the courses he taught in Turin was poor, and he seemed to be suffering a depression or mania of some kind. In 1833 he headed on to Prague, where he acted as tutor to the grandson of the emperor Charles X. Here also, Cauchy's disjointed teaching style and short temper failed to win him friends.

Whilst in Prague, Cauchy met with Bernard Bolzano, a Czech priest who was working on the question of continuity. Their encounter is not closely documented, but it seems clear that both learned from it. In 1838 Cauchy returned to Paris, but although he regained hism post at the École and a new one at the Bureau des Longitudes, his continued failure to swear allegiance meant that he could not teach or earn money in these positions. He subsequently failed (in 1843) to gain the mathematics chair at the Collège de France, due to the unpopularity of his ultra-Catholic, pro-Jesuit views and his reactionary political stance. During this time, although his work was less intense than whilst at the École, he produced important results on differential equations, mathematical physics and astronomy.

The downfall of Louis Philippe in 1848 allowed Cauchy to regain his full standing in the mathematical community, but he never did achieve the post at the Collège, this going to Liouville in 1850 following the flight of Libri, who had defeated Liouville and Cauchy but was suspected (correctly) o theft. Cauchy died in an appropriately pious and Catholic manner, calling on Jesus, Mary and Joseph before he passed on, on May 23, 1857.

His many achievements are summed up in Professor Bruno Belhoste's 'Cauchy 1789-1857', translated into English as Augustin-Louis Cauchy: A Biography, like this:
Cauchy's creative genius found broad expression not only in his work on the foundations of real and complex analysis, areas to which his name is inextricably linked, but also in many other fields. Specifically, in this connection, we should mention his major contributions to the development of mathematical physics and to theoretical mechanics... his two theories of elasticity and his investigations on the theory of light, research which required that he develop whole new mathematical techniques such as Fourier transforms, diagonalisation of matrices, and the calculus of residues.
Cauchy discoveries include:

Sources: My own essay cited above, and its sources listed there; Belhoste's book; and assorted websites, in order to remind me what Professor Belhoste's excellent biography had to say. I snagged the node-list from pax music's node Augustin Cauchy.