King
Heiron had just had a new crown forged.
The crown was to be of pure gold, but Heiron worried that the goldsmith
had mixed cheaper metals in (and kept the rest of the gold for himself). Heiron asked Archimedes to figure out
whether or not the crown was made of real gold. How he did so was up to him. There was only one rule: the crown must
not be harmed.
The
need to preserve the crown intact ruled out melting down the crown. Archimedes puzzled for many days over
the way to figure out the crown’s composition. The crown, in the shape of a laurel
leaf, gave no clue about its metallurgy by appearance alone. The crown couldn’t be melted to measure
its volume, cut in half, or in any way dissected. Archimedes was worried that the problem
was insoluble, and that his failure would not only provoke the ire of the king
but damage the scholar’s reputation.
A bath provided the answer. Upon lowering himself into the water, Archimedes noticed that water spilled over the sides of the tub. The volume of his body was displacing water. The same could be done with the crown; once the volume was determined, it could be calculated with weight in order to determine density. If a block of pure gold of equal weight displaced the same amount of water, the crown was legitimate. If it did not, a goldsmith had likely just risked and lost his neck. Archimedes was so moved by his find that he jumped out of the tub and ran through the streets shouting, “Eureka!” (“I have found it!”). He was so excited he hadn’t even stopped to dress.