Eureka

“Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.”

Archimedes

King Heiron had another problem.  Who else should he turn to but his old friend, Archimedes, who recently helped him out of a rather embarrassing situation involving the launch of the Syracusia. This time, the King believed he was a victim of fraud.  He gave his goldsmith a generous amount of gold with instructions to make him a wreath befitting of his royal personage. When King Heiron examined the finished product, he suspected that his cunning goldsmith had somehow devised a way to keep some of the gold by substituting it with an inferior metal.  He was in a quandary. The wreath weighed the same amount as the gold that had been entrusted to the goldsmith.   There was no proof of deception.

Sudden, almost miraculous, realizations happen in the oddest moments, engendered by a movement or a fleeting thought.  Archimedes came to his brilliant conclusion when he was taking a bath.  He noticed how the water level rose as he sunk deeper into the water.  Legend has it that he leaped straight out of his bath and ran unclothed through the streets to the king, shouting with enthusiastic gusto, “Eureka! Eureka!” which means, “I’ve got it! I’ve got it!”

The answer was elegant and profoundly transformational.  Archimedes first immersed a piece of gold that weighed the same as the royal wreath and noted the rise in the water level.  He then immersed the actual wreath and noted the water level was higher than the gold.  This meant that the wreath must be a greater volume than the gold, even if it was the same weight.  Mystery solved.

Whether or not there is any truth to this legend, it does give insight into Archimedes’ scientific methodology.  He used small, everyday problems as a starting point for crucial theoretical advancements.  Perhaps Eureka was the genesis for his groundbreaking work on how things float – hydrostatics.

“Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ‘Immortality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean.”

Godfrey Harold Hardy, A Mathematician’s Apology, (1940)