To assess the purity of Hiero’s golden crown/wreath (suspected goldsmith of replacing some gold with silver) measure density (density = purity)
Archimedes’ Principle of Buoyancy:
Vitruvius, On Architecture, c. 15BC, nearly 200 years after Archimedes
THE STORY in Marcus Vitruvius Pollio (aka, Vitruvius): de Architectura (On Architecture), Book IX – “Introduction” 9-12
9. Though Archimedes discovered many curious matters which evince great intelligence, that which I am about to mention is the most extraordinary. Hiero, when he obtained the regal power in Syracuse, having, on the fortunate turn of his affairs, decreed a votive crown of gold to be placed in a certain temple to the immortal gods, commanded it to be made of great value, and assigned an appropriate weight of gold to the manufacturer. He, in due time, presented the work to the king, beautifully wrought, and the weight appeared to correspond with that of the gold which had been assigned for it.
10. But a report having been circulated, that some of the gold had been abstracted, and that the deficiency thus caused had been supplied with silver, Hiero was indignant at the fraud, and, unacquainted with the method by which the theft might be detected, requested Archimedes would undertake to give it his attention. Charged with this commission, he by chance went to a bath, and being in the vessel, perceived that, as his body became immersed, the water ran out of the vessel. Whence, catching at the method to be adopted for the solution of the proposition, he immediately followed it up, leapt out of the vessel in joy, and, returning home naked [tunic only], cried out with a loud voice that he had found that of which he was in search, for he continued exclaiming, in Greek, εὑρηκα, (I have found it out) [Eureka!].
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11. After this, he is said to have taken two masses, each of a weight equal to that of the crown, one of them of gold and the other of silver. Having prepared them, he filled a large vase with water up to the brim, wherein he placed the mass of silver, which caused as much water to run out as was equal to the bulk thereof. The mass being then taken out, he poured in by measure as much water as was required to fill the vase once more to the brim. By these means he found out what quantity of water was equal to a certain weight of silver.
12. He then placed the mass of gold in the vessel, and, on taking it out, found that the water which ran over was lessened, because, as the magnitude of the gold mass was smaller than that containing the same weight of silver. After again filling the vase by measure, he put the crown itself in, and discovered that more water ran over then than with the mass of gold that was equal to it in weight; and thus, from the superfluous quantity of water carried over the brim by the immersion of the crown, more than that displaced by the mass, he found, by calculation, the quantity of silver mixed with the gold, and made manifest the fraud of the manufacturer.
The Short Version: the local tyrant contracts the ancient Greek polymath Archimedes to detect fraud in the manufacture of a golden crown. Said tyrant, name of Hiero, suspects his goldsmith of leaving out some measure of gold and replacing it with silver in a wreath dedicated to the gods. Archimedes accepts the challenge and, during a subsequent trip to the public baths, realizes that the more his body sinks into the water, the more water is displaced--making the displaced water an exact measure of his volume. Because gold weighs more than silver, he reasons that a crown mixed with silver would have to be bulkier to reach the same weight as one composed only of gold; therefore it would displace more water than its pure gold counterpart. Realizing he has hit upon a solution, the young Greek math whiz leaps out of the bath and rushes home naked crying "Eureka!" Or, translated: "I've found it! I've found it!"
(Biello, David. “Fact or Fiction?: Archimedes Coined the Term "Eureka!" in the Bath.” Scientific American 8 Dec. 2006 .)
Whenever we have a mental breakthrough, whenever we find the answer to a problem we’ve been working on, whenever we finally get it – whatever “it” has been that we’ve been toiling over, then we exclaim, “Eureka!” (now, as for the running-down-the-street-naked part of the story, well, we tend to overlook that).