Archimedes of Syracuse (c. 287–212 BC) is the closest the ancient world came to a one-man mathematics department, and he did something astonishing: he invented the core idea of integration roughly nineteen centuries before Newton and Leibniz got the credit.

His method of exhaustion found the area of a curved shape by filling it with more and more straight-edged pieces and seeing what the total crept towards — which is exactly the limit of a Riemann sum, two thousand years early.

Squeezing π out of polygons

He pinned down \pi by trapping a circle between polygons with more and more sides — eventually a 96-gon inside and a 96-gon outside — and proved

He also showed a sphere is exactly two-thirds of its surrounding cylinder in volume — a result he was so proud of that he asked for a sphere-in-a-cylinder to be carved on his tomb.

Eureka, levers, and a fatal devotion to circles

The legends are as good as the maths. "Eureka!" — leaping from his bath on realising that a floating body displaces its weight in water. "Give me a place to stand and I will move the Earth" — on the power of the lever. His war machines reportedly held off the Roman siege of Syracuse for years. When the city finally fell, a Roman soldier found him drawing diagrams in the sand; his supposed last words — "do not disturb my circles" — got him run through. He was, to the end, more interested in geometry than in self-preservation.

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The full story (with far fewer jokes) is on Wikipedia: Archimedes — Wikipedia.