Kernel
Archimedes is antiquity's only figure who combines first-rate mathematician, first-rate physicist, and first-rate engineer in one person. The law of the lever, the calculation of π to within a hair, the hydrostatic principle that bears his name, the Archimedean screw still in use today — and a death at Roman hands in besieged Syracuse that became a permanent parable about war and learning.
Contribution
Method of exhaustion (a forerunner of integral calculus, 1,900 years before Newton). The Method, a long-lost text recovered from a 13th-century palimpsest in 1906, shows Archimedes using mechanical reasoning to anticipate integration. The siege engines, mirror-arrays (probably mythologized), and screw pumps that engineered ancient warfare and irrigation.
Lineage
Studied at Alexandria, returned to Syracuse, corresponded with Eratosthenes. His texts travel west via Roman compilers, then to Arabic mathematicians, then back to Renaissance Italy where they decisively influence Galileo's mechanics.
Civilization-scale significance
The figure to whom Newton was implicitly compared by every European mathematician until calculus rendered the comparison moot. Even now, what Archimedes was trying to do in The Method reads as modern.