The king of Syracuse had sent a lump of gold to the goldsmith to have a new crown made from it.
Although the finished crown weighed exactly as much as the lump of gold had, the king suspected that he had been ripped of by the smith. Believing that he had kept some of the gold and made up the weight with copper and silver, the king asked Archimedes if he could prove his theory.
Archimedes reasoned that a block of gold would weigh more than a block of silver or copper of the same size but thought that the king would probably not be pleased if he melted the crown to test it.
It is believed that, one day in the public bath house, Archimedes jumped into a tub full of water making it overflow. This gave him the idea of how to solve the problem.
He was so excited that, without thinking to get dressed, he ran through the streets of the town yelling 'Eurika' which means 'I have found it'.
He had figured that the crown would displace its own volume of water and so would a lump of gold of the same weight.
During his experiments he found that the water displaced was equal to the apparent loss of weight of a suspended object.
The crown and the gold weighed the same in the air but Archimedes found that, in water, the crown weighed less (displaced more water) showing that it's volume was larger.
This showed that the crown was made of an alloy rather than the pure gold given by the king and that the goldsmith had cheated.