Cauchy attended France’s great Ecole Polytechnique from 1805 until 1807 and worked briefly as a military engineer. In 1813 he abandoned his chosen career, apparently for reasons of health, and devoted himself exclusively to mathematics. Cauchy secured an instructorship at the Polytechnique, where he rose to be professor of mechanics in 1816. During this period he undertook a thorough reorganization of the foundations of the calculus, infusing the subject, as he put it, with the same rigor that was to be found in geometry. Because of the changing political situation in 1830 Cauchy went into voluntary exile in Turin, where he obtained an appointment at the university. In 1838 he returned to Paris and resumed his teaching, although not at the Polytechnique. Cauchy was the foremost French mathematician of the nineteenth century; his 789 papers and seven books rank him second only to Euler in terms of productivity.

Cauchy’s celebrated Cours d’analyse de l’Ecole Royale Polytechnique, based on his lectures at that school, stamped elementary calculus with the character it has today. It recognizes the limit concept as the cornerstone of a firm logical explanation of continuity, convergence, the derivative and the integral. In defining "limit," he says:

When the values successivly attributed to a particular variable approach indefinitely a fixed value so as to differ from it by as little as one wishes, this latter value is called the limit of the others.

Suffice it to say, the reliance on such phrases as "as little as one wishes" denies precision to the notion. The Cours describes the derivative of y = f(x) as the limit ("when it exists") of a difference quotient

as h goes to zero. Another aspect of Cauchy’s work is a careful treatment of sequences and series. One of the basic tests for sequential convergence is a result that is today called the "Cauchy convergence criterion"; specifically, a sequence s₁, s₂, s₃, ... converges to a limit if the difference s_m - s_n can be made less than any assigned value by taking m and n sufficiently large.