One of the greatest mathematicians of all time, Archimedes was born and lived in Syracuse, Sicily. His interest in maths probably came from his father, an astronomer, and it is also likely that he studied with the successors of Euclid in Alexandria.
His contemporaries paid most attention to his successes as an engineer. He produced a variety of war machines for the Syracuse King to help him fend off invasion from Roman forces.
He is best known for the world’s most famous bath during which he realised that the amount of water which overflowed from the tub was proportional to the amount of his body that was submerged.
“Eureka” (I have found it) he yelled as he ran, naked, down the street. He is also credited with developing systems of pulleys and levers, the Archimedes screw used to remove water from the hold of a ship, a planetarium and many other remarkable mechanical feats.
Archimedes himself believed that pure maths was the only worthy pursuit. He was most proud of finding the volume of a sphere and he also looked at the relationship between spheres and cylinders.
Inscribed on his gravestone was the 2:3 (volume) ratio he discovered between a sphere and the smallest cylinder that encloses it.
He also developed the most accurate measure of pi, found a new way to approximate square roots and devised his own number system which could utilise larger numbers than the unwieldy Greek system.
From the perspective of history, his greatest idea was probably the invention of integral calculus using a theory of integration.
He also anticipated the invention of differential calculus. None of this fundamental work survived, and none of it was further developed. They were independently re-discovered by Leibniz and Newton much later.
He died an untimely death, apparently working on a mathematical problem, at the hands of one of the Roman soldiers he had worked so hard to keep out of Syracuse.