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Did Archimedes shout “Eureka”?

7 min read

Nobody knows, and there is reason to doubt it. The only detailed source is the Roman architect Vitruvius, in Book IX of De Architectura, written around 30 to 15 BC, nearly two centuries after Archimedes died in 212 BC. He says Archimedes, asked by King Hiero II to test a golden wreath, noticed water overflowing his bath, ran home naked shouting “Eureka” (“I have found it”). Archimedes' own surviving works never mention the crown.

The claim, and the verdict

The claim: King Hiero of Syracuse suspected his goldsmith of mixing silver into a golden crown. Archimedes, stepping into a bath, saw the water rise, realised he could measure the crown's volume this way, and ran through the streets naked shouting “Eureka”.

The verdict: a legend with a real scientist at its centre. Archimedes (c. 287 to 212 BC) lived in Syracuse and worked for Hiero II, and he did found the science of floating bodies. But the bath story first appears in writing about 190 years after his death, in a Roman book about architecture, with no source named. Archimedes' own treatise on floating bodies does not mention it. And the method Vitruvius describes would have been hard to make work with a real wreath. Most historians treat the story as an anecdote that may contain a memory of a real problem, but not as a record of what happened.

What Vitruvius actually wrote

Marcus Vitruvius Pollio was a Roman military engineer and architect who dedicated his ten books On Architecture to Augustus, probably between about 30 and 15 BC. In the preface to Book IX he praises great discoveries and tells the story of Archimedes and the crown.

Hiero, he says, had vowed a golden crown (a votive wreath) to the gods for a temple. He weighed out the gold for the contractor, and the finished wreath weighed the same. But a rumour said some gold had been kept back and silver added. Hiero asked Archimedes to find out without damaging the wreath. Archimedes went to the baths, noticed that as he lowered himself into the tub the water ran over, and saw how to solve it. Overjoyed, he jumped out and ran home naked, shouting in Greek “Eureka, eureka”.

Vitruvius then gives the method: take a lump of gold and a lump of silver, each the same weight as the wreath. Put each in turn into a vessel filled to the brim and measure how much water overflows. Silver, being less dense, displaces more water than the same weight of gold. The wreath displaced more water than the gold lump, so it contained silver, and the theft was proved.

What the evidence shows: a dated source table

DateSourceWhat it says
c. 287 to 212 BCArchimedes' life in SyracuseWorks for King Hiero II; killed by a Roman soldier when Syracuse falls in 212 BC.
3rd century BCArchimedes, On Floating Bodies, Book IProves that a body in a fluid is buoyed up by a force equal to the weight of fluid it displaces. No crown, no bath.
c. 30 to 15 BCVitruvius, De Architectura, Book IX, prefaceThe first and fullest telling of the crown, the bath and “Eureka”.
c. AD 100Plutarch, essay against the Epicureans (Moralia)Mentions in passing that Archimedes leapt from his bath shouting “I have found it”, as an example of the joy of discovery.
c. 1586Galileo Galilei, La Bilancetta (The Little Balance)Argues that Vitruvius' overflow method is too crude, and proposes a hydrostatic balance instead.
1906Johan Ludvig Heiberg identifies the Archimedes PalimpsestThe only surviving Greek text of On Floating Bodies. Still no crown story.

The biggest problem is practical. Gold is about 19.3 times as dense as water and silver about 10.5. A wreath of one kilogram with 30 per cent silver would displace only about 13 cubic centimetres more water than a kilogram of pure gold: less than a tablespoon. Measuring that by catching overflow from a brim-full vessel, with drips and surface tension, would be very hard.

Galileo, as a young man in 1586, thought Archimedes must have used a better method based on his own principle. Hang the wreath and an equal weight of pure gold from a balance, then lower both into water. If the wreath contains silver, it has more volume, receives more buoyant force, and the balance tips. That works with ordinary equipment and is exactly the kind of reasoning in On Floating Bodies. Many historians of science think, if Archimedes ever tested a wreath, this is closer to how he would have done it.

How the story spread

Vitruvius was copied throughout the Middle Ages and printed in the Renaissance, and his book was read by every architect and engineer. The Eureka story suited an age that admired sudden insight. By the eighteenth century “Eureka” was an ordinary English word for a moment of discovery. It became the motto of the state of California, linked to the gold rush of 1849, and it has named countless products since.

The image of the naked scientist running through the streets also made the story memorable, and it fits other legends about Archimedes: that he was so absorbed in a geometry problem when the Romans took Syracuse that he ignored the soldier who killed him, and that he boasted he could move the Earth with a lever and a place to stand. Those, too, come from writers centuries later, Plutarch among them, and should be treated as stories.

What Archimedes actually wrote

Archimedes' surviving books are rigorous, proof by proof, in the style of Euclid. In On Floating Bodies he states what we now call Archimedes' principle: a solid body immersed in a fluid is lighter by the weight of the fluid it displaces. In On the Equilibrium of Planes he sets out the law of the lever, starting from postulates like this one:

“Equal weights at equal distances are in equilibrium and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance.”
Archimedes, On the Equilibrium of Planes, Book I, Postulate 1 (trans. T. L. Heath, 1897)

In Measurement of a Circle he proved that pi lies between 3 10/71 and 3 1/7. In The Method, preserved only in the palimpsest, he explained to Eratosthenes that he often found results first by imagining figures balanced on a lever, and only then proved them by strict geometry. That is a real record of how he discovered things, in his own words, and it is less dramatic than a bath.

You can ask him about the lever, the sphere and the cylinder, and the crown story by talking with Archimedes in Yuga, by voice or text. It is an AI recreation built from his works and the ancient record, not Archimedes, and it can be wrong, so cite his treatises and Vitruvius directly.

How to check an ancient anecdote

  1. Find the earliest author who tells it and how long after the event he wrote. Here, about 190 years.
  2. Ask whether that author names a source. Vitruvius does not.
  3. Compare with the person's own writings. Archimedes never mentions the crown.
  4. Test whether the story works in practice. The overflow method is doubtful; the balance method works.

More: questions to ask Archimedes, Pi Day and Archimedes, famous mathematicians in history and how to check if a quote is real.

Questions people ask

What does “Eureka” mean?
It is Greek for “I have found it”, from the verb heuriskein, to find. In Vitruvius' story Archimedes shouts it twice as he runs home from the baths. In English it has long been used for a moment of sudden discovery, and it is the state motto of California.
Where does the Eureka story come from?
From Vitruvius, a Roman architect, in the preface to Book IX of his ten books On Architecture, written around 30 to 15 BC and dedicated to Augustus. That is roughly 190 years after Archimedes died in 212 BC. Plutarch, around AD 100, mentions the shout briefly. Archimedes' own writings do not mention it.
How did Archimedes test the golden crown?
According to Vitruvius, by comparing how much water overflowed when he dipped the crown, an equal weight of gold, and an equal weight of silver into a full vessel. Galileo argued in 1586 that this would be too imprecise, and that Archimedes more likely weighed the crown against pure gold while both hung in water, using his own principle of buoyancy.
Was Archimedes' principle discovered in a bath?
We do not know. Archimedes proved the principle in On Floating Bodies, a careful mathematical treatise, without describing how he arrived at it. The bath story is a later anecdote. Whatever started the idea, the principle itself, that a body in a fluid is buoyed up by the weight of fluid it displaces, is fully his.

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