Archimedes carried out mathematical and mechanical investigations in the Hellenistic world, chiefly associated with Syracuse. Correspondence embedded in surviving works connects him with scholars including Eratosthenes and Conon in Alexandria. Details of his education and official employment are uncertain, and a lost ancient biography cannot be reconstructed from later anecdotes. His reputation rests most securely on treatises and demonstrated results, which show a programme of reasoning about geometry, equilibrium and the behaviour of bodies rather than only ingenious devices.

In On the Sphere and Cylinder he established relationships between their areas and volumes, including the proportion he reportedly wanted commemorated on his tomb. Measurement of a Circle bounded pi between close limits by comparing polygons. Other works treated spirals, conoids and the quadrature of a parabola. The method of exhaustion supported rigorous conclusions about areas and volumes. In The Method, preserved through the Archimedes Palimpsest, mechanical reasoning helped discover results subsequently justified geometrically. Calling this modern calculus would be anachronistic, although later mathematicians recognized important anticipations.

His work on plane equilibrium analysed levers and centres of gravity. On Floating Bodies examined hydrostatic balance and supplied the principle associated with buoyancy. Devices such as the water-raising screw are linked to his name, but attribution does not establish that every underlying mechanism originated with him. The gold-crown experiment and cry of Eureka come from a much later narrative by Vitruvius. They communicate the image of an inventive investigator without providing equally secure evidence for a particular event. Stories of mirrors burning Roman ships are especially doubtful.

Ancient historians connect Archimedes with machines defending Syracuse during the Roman siege of 214–212 BCE. These accounts support a practical role in military engineering, while descriptions of individual mechanisms vary. He was killed when the city fell in 212; the familiar story of a soldier interrupting his geometry was elaborated in later sources. Subsequent copying, translation and recovery of manuscripts transmitted his results unevenly across cultures and centuries. Modern study of the palimpsest recovered important lost material. His mathematical texts provide firmer evidence of achievement than the legends surrounding his inventions.