The Syracusan Mathematician Who Ran Naked Through the Streets in About 250 BCE Shouting Eureka After Working Out the Principle of Displacement in His Bath

Archimedes of Syracuse (c. 287 – 212 BCE) was the senior mathematician of the late Hellenistic Mediterranean. His mathematical and engineering work for the Syracusan royal court included the screw pump, the compound pulley, the proof that the volume of a sphere is two-thirds the volume of its circumscribed cylinder (which he asked to have engraved on his tombstone), and the defence of Syracuse against the Roman siege of 214-212 BCE.

The Eureka story is the most familiar of his achievements. It is recorded by the Roman architect-writer Vitruvius in the preface to Book 9 of De architectura, written about 25 BCE — approximately two centuries after Archimedes’ lifetime.

What Vitruvius says

King Hiero II of Syracuse had commissioned a votive gold crown for a temple. He suspected the goldsmith of having fraudulently substituted some of the gold with silver. He asked Archimedes to determine whether the crown was pure gold, without damaging the crown itself.

Archimedes worked on the problem for several weeks without progress. The conventional method — weighing the crown against an equal mass of gold — could not detect partial substitution if the goldsmith had simply removed some gold and replaced it with the equivalent weight of silver.

The answer came in the public bath. As Archimedes lowered himself into the bath he noticed the water rising in proportion to the volume of his body. The realisation was that a submerged body displaces a volume of water equal to its own volume — and that the volume of an irregularly-shaped object could therefore be measured by submerging it in a calibrated container.

The volume-and-weight measurement would distinguish gold from silver. Gold’s density (19.3 g/cm³) is almost twice silver’s (10.5 g/cm³). A crown that was pure gold would have a smaller volume per unit weight than a crown of mixed gold and silver. The same weight of metal would displace less water if pure gold than if alloyed.

Archimedes leapt out of the bath and ran home through the Syracuse streets shouting Heúrēka (“I have found it”), forgetting to dress.

What he found

Vitruvius’s narrative ends with the discovery and does not record the analytical result. The Hiero crown story is, in Vitruvius’s telling, an anecdote about the moment of insight rather than a forensic report. Later medieval and Renaissance reconstructions usually add the conclusion that the crown was indeed fraudulent and that the goldsmith was executed. The original story does not say so.

Whether it happened

Galileo argued in 1586 that the displacement method as Vitruvius describes it would not have produced sufficient precision to detect a small silver substitution in a crown of typical Hellenistic size. Galileo proposed an alternative reconstruction using a hydrostatic balance — weighing the crown in air and then while submerged — which would have given Archimedes the same answer with much higher precision and which is described in Archimedes’ own surviving work On Floating Bodies.

The modern reconstruction is that the displacement insight was real — the principle is correct, and it appears throughout Archimedes’ later work — but that the running-naked-through-Syracuse story is a Vitruvian literary embellishment. The mathematical principle survived. The cry of Heúrēka did too.