In return for the success of his policies, the king commissioned a large golden crown to be made and placed in a shrine as an offering to the gods. To the contractor, he weighed out the exact amount of gold to be used in the crown. When the crown was finished, it exactly fulfilled the king's weight requirement, and was accordingly set in place on the altar. Later, however, a rumor surfaced that the some of the gold had been removed and replaced with an equal weight of silver. The king became upset at this disrespect for his authority, but lacking a method of proving the fraud, was left without recourse. In an effort to remedy the situation, he asked Archimedes to investigate the matter in hopes of finding a solution.

While the problem rested upon his mind, Archimedes went to take a bath. As he was sitting down in the bath, he noticed that the amount of water that overflowed the side was equal to the amount of his body he submerged. Realizing that he could use this to determine the actual volume of the crown and, comparing it with the volume of an equal mass of pure gold, thereby solving the problem, Archimedes leapt from the tub, so great was his joy, and ran naked through the streets shouting," Heuraka! Heuraka!"

Plutarch, in his "Marcellus," tells how Archimedes was persuaded to put his knowledge to practical use in the construction of engines for use in the defense of the city of Syracuse. Although Archimedes refused to write upon the subject because of a crudeness associated with utilitarianism, and for that matter, any earthly topic. Yet, Plutarch discusses a letter Archimedes wrote to his kinsman, King Heiro. In the letter, he claims that with a given force, he can move any object, no matter how heavy. He even claimed to be able to move this world(hence, "Give me a lever and a place to stand, and I can move the world.") The King challenged him on this statement, so he had a ship dragged up onto the beach by many men, and then loaded it with a full of cargo and crew. Archimedes, apparently through the use of pulleys, was then able to drag the ship over the land as easily as if it were pulled through water, to the amazement of all who were watching.