Archimedes of Syracuse is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time,^[42] Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems.^[43] These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.^[44]

Archimedes' other mathematical achievements include deriving an approximation of pi, defining and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, working on statics and hydrostatics. Archimedes' achievements in this area include a proof of the law of the lever,^[45] the widespread use of the concept of center of gravity,^[46] and the enunciation of the law of buoyancy or Archimedes' principle.

Archimedes died during the siege of Syracuse, when he was killed by a Roman soldier despite orders that he should not be harmed.