Final answer:
Archimedes discovered his principle while trying to determine if a crown made for King Hiero was pure gold, by comparing the water displacement caused by the crown to that caused by the same weight of pure gold.
Step-by-step explanation:
Archimedes' principle was discovered as he tried to solve a practical problem for King Hiero of Syracuse. The king suspected that a crown made for him was not pure gold and gave Archimedes the task of verifying its purity without damaging it. The discovery came when Archimedes noticed the water displacement caused by his body when he entered a bath, leading him to realize that the volume of displaced water was equal to the volume of the object submerged, thus related to its density.
He understood that by measuring the volume of water displaced by the crown and comparing it to the displacement of the same weight of pure gold, he could determine if the crown was made of pure gold or a less dense alloy. Archimedes thus found that the buoyant force on an object is equal to the weight of the fluid it displaces, a principle that could be used to determine the density and therefore the purity of the crown. Upon making this discovery, there is a famous tale of him shouting 'Eureka!' (meaning 'I found it!') and running through the streets of Syracuse, overwhelmed with joy.