Vitruvius, in his Architecture, reports, that Hiero, king of Sicily, having employed an artist to make a crown of pure gold, which was designed to be dedicated to the gods, suspected that the goldsmith had stolen part of the gold, and substituted silver in its place: being desirous of discovering the cheat, he proposed the question to Archimedes, desiring to know if he could, by his art, discover whether any other metal were mixed with the gold. This celebrated mathematician being soon afterwards bathing himself, observed, that as he entered the bath, the water ascended, and flowed out of it; and as he came out of it, the water descended in like manner: from which he inferred that if a mass of pure gold, silver, or any other metal, were thrown into a vessel of water, the water would ascend in proportion to the bulk of the metal. Being intensely occupied with the invention, he leaped out of the bath, and ran naked through the streets, crying, "I have found it, I have found it!"

The way in which he applied this circumstance to the solution of the question proposed was this: he procured two masses, the one of pure gold, and the other of pure silver, each equal in weight to the crown, and consequently of unequal magnitudes; then immersing the three bodies separately in a vessel of water, and collecting the quantity of water expelled by each, he was presently enabled to detect the fraud, it being obvious, that if the crown expelled more water than the mass of gold, it must be mixed with silver or some baser metal. Suppose, for instance, in order to apply it to the question, that each of the three masses weighed eighteen pounds; and that the mass of gold displaced one pound of water, that of silver a pound and a half, and the crown one pound and a quarter only: then, since the mass of silver displaced half a pound of water more than the same weight or gold, and the crown a quarter of a pound more than the gold, it appears, from the rule of proportion, that half a pound is to eighteen pounds, as a quarter is to nine pounds; which was, therefore, the quantity of silver mixed in the crown.

Since the time of Archimedes, several other methods have been devised for solving this problem; but the most natural and easy is, that of weighing the crown both in air and water, and observing the difference.