Most who thought about nature in antiquity were content with narratives of cause and effect; Archimedes of Syracuse (c. 287–212 BCE) instead took a physical phenomenon and turned it into a chain of proof as rigorous as Euclid's geometry. In his treatise "On the Equilibrium of Planes" he derived the law of the lever — weights balancing in inverse proportion to their arms — from a handful of axioms, among the earliest known cases of a law of nature being demonstrated mathematically.

In "On Floating Bodies" he showed that a body immersed in a fluid is buoyed up by a force equal to the weight of the fluid it displaces. The tale Vitruvius relays two centuries later — solving whether a crown was pure gold in the bath and running out shouting "Eureka" — is most likely a legend; but the principle behind it, measuring density by displacement, is a genuine contribution and still bears his name.

In mathematics he went further still. By inscribing and circumscribing 96-sided polygons on a circle he showed that π lies between 3 + 10/71 and 3 + 1/7 — the first bounding of a constant squeezed from above and below. He found the area of a parabolic segment and the 2:3 ratio between the volume of a sphere and its cylinder by cutting figures into infinitesimal parts and summing them — the intuition of integral calculus, sensed two thousand years early. His own favourite result, the sphere-and-cylinder theorem, he asked to have carved on his tomb.

Archimedes was killed in 212 BCE during Rome's siege of Syracuse, cut down by a soldier in the captured city. Part of his work could only be recovered in 1906, from a faint parchment overwritten with a prayer book — the Archimedes Palimpsest. His method is the seed of the idea that nature is written in quantitative law; from that seed Galileo and Newton would grow.