There is an old story about a king and his crown. the king hired a goldsmith to make him a crown a pure gold. after the king received the crown, he started to worry that the goldsmith might have cheated him. he thought maybe the goldsmith didn't use pure gold, maybe he used a mix of metals to save money. the king asked a man named archimedes to figure out if the crown was pure gold or not. archimedes reasoned that the density of a lump of gold and a crown of pure gold would be the same. how could he prove if that crown was pure gold without melting or otherwise destroying the crown?