Why Archimedes shouted EUREKA?

Archimedes, according to legend, shouted "Eureka!" after leaping out of his bathtub and running naked down the streets of Syracuse (a Greek colony in ancient Sicily). Eureka means "I found it!" in Greek, and he was yelling it because he was ecstatic at finding the solution to a problem that had been put to him by the king of Syracuse.

It seems the king was suspicious that a jeweler was cheating him about the true amount of gold he had used to make the royal crown. It was known how much a given amount of gold should weigh, but the problem was that the crown was already finished and shaped into an intricate design, and there was no good way to measure the volume of the crown without destroying it.

When Archimedes lowered himself into his full bathtub, he noticed that his body displaced an amount of water, spilling over the edge. He reasoned that an object lowered in water would displace a volume of water equal to its own, so he now had a way to accurately measure the volume of the crown. The crown was tested, and it measured out at less that the amount of gold the jeweler had charged the king for, and he was punished severely.

Archimedes was considered a scientific genius of his time, and was also credited for defending his city of Syracuse against the Romans by using cranes that overturned ships, and mirrors that concentrated sunlight on them and caused them to burn.