Augustin-Louis Cauchy (1789–1857) is the man who walked into calculus, looked at a century of brilliant-but-hand-wavy arguments, and said: prove it. Newton and Leibniz had built the machine and it plainly worked; Cauchy was the one who finally pinned down why — giving rigorous definitions of the limit, continuity, the derivative and the convergence of a series.

Almost everything in a modern analysis course wears his name somewhere: the Cauchy sequence, the Cauchy–Schwarz inequality, the root test, and (over in complex analysis) the Cauchy integral theorem.

An industrial quantity of theorems

Cauchy was relentlessly prolific — he published around 789 papers, on so many topics that the French Academy reportedly had to cap how long any single article could be just to slow him down. When a Cauchy sequence's terms get arbitrarily close to each other, the sequence is "trying" to converge:

That one idea — convergence you can check without already knowing the limit — is the hinge the whole theory of the real numbers turns on.

The referee from hell

He was also famously prickly: a devout, stubborn royalist who exiled himself rather than swear an oath to a government he disliked. And he was a terrifying referee. Two of the most brilliant young mathematicians who ever lived — Niels Abel and Évariste Galois — sent him their revolutionary work, and Cauchy reportedly mislaid or sat on it. Both died tragically young before getting their due. Genius, yes; gentle mentor, not so much.

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The full story (with far fewer jokes) is on Wikipedia: Augustin-Louis Cauchy — Wikipedia.