During his life before the siege of Syracuse, Archimedes composed a body of treatises (more than 27, complete or partial, have survived) exploring problems of geometry and mathematics. His writings indicate particular interests in calculating the area and volume of geometric figures (the area under the arc of a parabola, for example) and the behavior of irregularly shaped geometric bodies (a trapezoid, for example) in liquids. Among his more stimulating and complex treatises is "The Sand Reckoner," in which he not only describes a new and effective method of calculating the diameter of the Sun but also hypothesizes how many grains of sand would be necessary to fill the known cosmos. As several commentators have observed, the apparent absurdity of Archimedes' goal is, fundamentally, little different from modern scientists' attempts to estimate the total mass of the universe. In many respects, Archimedes' most interesting treatise is "The Method," wherein he sets out his understanding of theoretical mathematical and geometrical reasoning and its practical applications.

Archimedes was executed in 212 BC, reportedly because his preoccupation with his studies--he was working out geometrical problems by drawing in the sand, the ancient equivalent of a blackboard--did not permit him to obey a Roman soldier's command. His grave marker, a sphere within a cylinder, illustrated one of his proudest accomplishments--calculating the ratio of 3:2 between a cylinder and sphere. That tomb was rediscovered, in a forgotten thicket at Syracuse, by the Roman orator and politician Cicero in 75 BC. . .

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