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Archimedes' father was an astronomer and he himself inherited an interest in the subject. He was educated in Alexandria and spent most of the rest of his life in his birthplace, Syracuse, under the patronage of King Hieron. Archimedes was without question the greatest mathematician and scientist that classical Greek civilization produced and is usually considered to be one of the greatest mathematicians of all time. He was held in very high regard even by his contemporaries, and Karl Friedrich Gauss thought that only Isaac Newton was Archimedes' equal as a mathematician. Archimedes was as much an applied mathematician as a pure mathematician. He was very much interested in putting his theoretical discoveries to practical use and is known to have been skilled in making his own equipment and carrying out his own experiments. It is no exaggeration to describe Archimedes as the creator of the science of mechanics. Naturally before his time many isolated facts had been discovered, but it was only with him that mechanics became a unified body of theory capable of yielding new and unexpected practical applications.

Archimedes was able to find methods for determining the center of gravity of a variety of bodies. He also gave the first general theory of levers, and organized a practical demonstration to show how, with a suitable series of levers, a very small force is capable of moving a very large weight. He amazed his contemporaries by arranging for the king of Syracuse to move a large ship simply by pressing a small lever. In connection with his work on levers Archimedes made one of his famous statements, “Give me a firm place to stand on and I will move the Earth.” Archimedes also had a practical interest in optics, although no writings of his on the subject have come down to us. He put all this newfound theoretical knowledge to deadly effect when Syracuse was besieged by the Romans, by designing and building a variety of war machines. Among these were enormous mirrors to focus the Sun's rays and set fire to the Roman ships, and a variety of catapults.

Archimedes also successfully applied his scientific discoveries in hydrostatics. He designed all sorts of pumps, and the Archimedean water-screw is still widely used. But his most famous practical success was in solving a problem presented to him by King Hieron. Hieron wished to know whether a newly made crown, which was supposed to be of pure gold was, as he suspected, partly silver. Archimedes solved the problem by grasping the concept of relative density. By immersing successively the crown itself and pieces of gold and silver of equal weight in full containers of water and observing the amount of water each displaced, Archimedes was able to show that the crown was indeed not made of pure gold. One of the famous stories associated with Archimedes tells how this occured to him when he was getting into his bath and observed how the more of his body was immersed the more water overflowed from the bath. He saw instantly how to solve his problem, leaped from the bath, and rushed through the streets, stark naked, shouting “Eureka!” (I have found it).

Archimedes' work in applied mathematics and science ensured his great contemporary fame, but some of his greatest work was probably in his more esoteric researches in pure mathematics. Like all Greek mathematicians, his interest was primarily concentrated on geometry. Arithmetic was greatly hampered by a very cumbersome system of notation. Although Archimedes himself invented a much improved system for notation of very large numbers, algebra had yet to be invented, in Europe at least. Archimedes' most profound achievement was to perfect the ‘method of exhaustion’ for calculating the areas and volumes of curved figures. The method involves successively approximating the figure concerned by inscribed and circumscribed polygons. This method essentially used the concept of limit – a concept that took some time for later European mathematicians to grasp. Archimedes used this method to determine an approximate value for π, which was not to be improved on for many centuries.

Archimedes was put to death by a Roman soldier when the Romans, under general Marcellus, finally successfully besieged Syracuse. The killing was against the orders of Marcellus who respected Archimedes and wished for him to be protected. Archimedes was apparently drawing mathematical symbols in the sand when killed.