Archimedes was the greatest philosophy/alchemist/astronomer in the city. The king was tormented with anxiety over the purity of his gold. He summoned Archimedes and implored him to use his knowledge of metals to devise a method of testing the gold. Archimedes thought long and hard, he wandered the city streets and became covered with dust. At home he refused his meals, paced the rooms, annoyed the slaves and generally drove his wife to distraction. They went to bed, but archimedes could not sleep, he would get up and scribble at his desk and mutter to himself, and his poor wife could not sleep a wink either. On the third evening she could bear it no longer. 'Archimedes!' she cried 'for my sake, take a bath!'
So archimedes got into his hot relaxing bath, saw the water level rise, and the rest is classical.