Who they were
He was born at Syracuse around 287 BC, the son of an astronomer named Phidias, in a Greek city that was then the largest in the western Mediterranean. He appears to have studied at Alexandria and to have corresponded with its mathematicians for the rest of his life, sending them results ahead of the proofs. He was killed in 212 BC when the Romans took Syracuse, against the orders of the general besieging it; a century and a half later Cicero, serving in Sicily, found his neglected tomb by the sphere and cylinder carved on it.
What they did
He advanced geometry, approximating pi and establishing the relation of a sphere to its enclosing cylinder, and he founded hydrostatics with the principle of buoyancy. He also devised machines and war engines that helped defend Syracuse against Rome. The famous “Eureka” bath story is a later anecdote rather than established fact.
Legacy
His methods anticipated integral calculus by nearly two thousand years. His name survives in Archimedes’ principle and the Archimedes screw, and he remains a symbol of scientific genius.