It is highly likely that when he was a young man, Archimedes studied with the successors of Euclid in Alexandria. He gained a reputation in his own time which few other mathematicians of this period achieved because of his inventions. He invented the compound pulley, and many other devices, some which were used to wage war. Yet Archimedes believed that pure mathematics was the only worthy pursuit.

The achievements of Archimedes are quite outstanding. He is considered by most historians of mathematics as one of the greatest mathematicians of all time. He perfected a methods of integration which allowed him to find areas, volumes, surface areas, and centers of mass of many geometrical objects. His work on integration gave birth to the calculus perfection much later by Kepler, Cavalieri, Fermat, Leibniz and Newton.

Archimedes was able to apply the method of exhaustion, which is the early form of integration, to obtain a whole range of important results. Archimedes also gave an accurate approximation to pi, studied spirals, and showed that he could approximate square roots accurately. He invented a system for expressing large numbers. His most famous theorem gives the weight of a body immersed in a liquid, called Archimedes' principle. There are also lost works on semi-regular polyhedra, balances and levers, and mirrors.

The story of Archimedes' death is famous. He was killed during the capture of Syracuse by the Romans. Here is Plutarch's version:

Archimedes was, as fate would have it, intent upon working out some problem by a diagram, and having fixed his mind alike and his eyes upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming up to him, commanded him to follow to Marcellus; which he declining to do before he had worked out his problem to a demonstration, the soldier, enraged, drew his sword and ran him through.Archimedes considered his most significant accomplishments were those concerning a cylinder circumscribing a sphere, and he asked for a representation of this together with his result on the ratio of the two, to be inscribed on his tomb.

It is perhaps surprising that the mathematical works of Archimedes were relatively little known immediately after his death. Only after Eutocius brought out editions of some of Archimedes works, with commentaries, in the sixth century were the remarkable treatises to become more widely known.