What is the story of Archimedes’ inspiration in the bathtub when he yelled Eureka?

Better yet, we can give you an account from the first century B.C., written by Marcus Vitruvius Pollio, a Roman architect, engineer, and writer:

In the case of Archimedes, although he made many wonderful discoveries of diverse kinds, yet of them all, the following, which I shall relate, seems to have been the result of a boundless ingenuity.

Hiero, after gaining the royal power in Syracuse, resolved, as a consequence of his successful exploits, to place in a certain temple a golden crown, which he had vowed to the immortal gods.

He contracted for its making at a fixed price, and weighed out a precise amount of gold to the contractor. At the appointed time the latter delivered to the king’s satisfaction an exquisitely finished piece of handiwork, and it appeared that in weight the crown corresponded precisely to what the gold had weighed.

But afterwards a charge was made that gold had been abstracted and an equivalent weight of silver had been added in the manufacture of the crown. Hiero, thinking it an outrage that he had been tricked, and yet not knowing how to detect the theft, requested Archimedes to consider the matter.

The latter, while the case was still on his mind, happened to go to the bath, and on getting into a tub, observed that the more his body sank into it the more water ran out over the tub.

As this pointed out the way to explain the case in question, he jumped out of the tub and rushed home naked, crying with a loud voice that he had found what he was seeking; for as he ran he shouted repeatedly in Greek,


Taking this as the beginning of his discovery, it is said that he made two masses of the same weight as the crown, one of gold and the other of silver. After making them, he filled a large vessel with water to the very brim, and dropped the mass of silver into it.

As much water ran out as was equal in bulk to that of the silver sunk in the vessel. Then, taking out the mass, he poured back the lost quantity of water, using a pint measure, until it was level with the brim as it had been before. Thus he found the weight of silver corresponding to a definite quantity of water.

After this experiment, he like-wise dropped the mass of gold into the full vessel and, on taking it out and measuring as before, found that not so much water was lost, but a smaller quantity: namely, as much less as a mass of gold lacks in bulk compared to a mass of silver of the same weight.

Finally, filling the vessel again and dropping the crown itself into the same quantity of water, he found that more water ran over the crown than for the mass of gold of the same weight.

Hence, reasoning from the fact that more water was lost in the case of the crown than in that of the mass, he detected the mixing of silver with the gold, and made the theft of the contractor perfectly clear.

It’s a great story, written two centuries after Archimedes’ death (he was killed by a Roman soldier during the sacking of Syracuse).

It might even be true, although Chris Rorres, a mathematics professor at Drexel University, gives us some reason to doubt. “Suppose the dishonest goldsmith replaced 30 percent (300 grams) of the gold in the wreath by silver,” he writes.

“The difference in the level of water displaced by the wreath and the gold is 0.41 millimeters. This is much too small a difference to accurately measure the overflow from, considering the possible sources of error due to surface tension, water clinging to the gold upon removal, air bubbles being trapped in the lacy wreath, and so forth.”