Let's begin with the story: the local tyrant contracts the ancient Greek polymath Archimedes to detect fraud in the manufacture of a golden crown. Said tyrant, name of Hiero, suspects his goldsmith of leaving out some measure of gold and replacing it with silver in a wreath dedicated to the gods. Archimedes accepts the challenge and, during a subsequent trip to the public baths, realizes that the more his body sinks into the water, the more water is displaced--making the displaced water an exact measure of his volume. Because gold weighs more than silver, he reasons that a crown mixed with silver would have to be bulkier to reach the same weight as one composed only of gold; therefore it would displace more water than its pure gold counterpart. Realizing he has hit upon a solution, the young Greek math whiz leaps out of the bath and rushes home naked crying "Eureka! Eureka!" Or, translated: "I've found it! I've found it!"
Several millennia later, the scientific world is replete with the exclamation, and many people have received inspiration in the shower. The mathematical conjectures of Henri Poincar¿, Einstein's theory of relativity, Newton getting dinged on the head with an apple and discovering gravity--all have been described as eureka moments. Edgar Allan Poe wrote a prose poem to science by that title and the prospectors of California's gold rush were so fond of the phrase that it crept into that state's motto. Even the American Association for the Advancement of Science calls its breaking scientific news site EurekAlert.
Too bad, then, that Archimedes probably never uttered the phrase in that way.
First and foremost, Archimedes himself never wrote about this episode, although he spent plenty of time detailing the laws of buoyancy and the lever (prompting him to reputedly pronounce: "Give me a place to stand and I will move the earth"), calculating the ratio of circles we know as pi, and starting along the path to the integral calculus that would not be invented for another 2,000 years, among other mathematical, engineering and physical feats.
The oldest authority for the naked-Archimedes eureka story is Vitruvius, a Roman writer, who included the tale in his introduction to his ninth book of architecture some time in the first century B.C. Because this was nearly 200 years after the event is presumed to have taken place, the story may have been improved in the telling. "Vitruvius may have gotten it wrong," says Chris Rorres, a mathematician at the University of Pennsylvania and a self-described Archimedes "groupie." "The volumetric method works in theory so it sounds right but when you actually try it you find that the real world gets in the way."
In fact, Rorres is one of a long line of scientists, including Galileo, who have read the account and thought "That can't be right." As Galileo showed in his tract La Bilancetta, or "The Little Balance," a scientist of Archimedes' stature could have achieved a far more precise result using his own law of buoyancy and an accurate scale, something far more common in the ancient world than a very precise pycnometer, which is used to measure displacement. (The surface tension of water can render the volume of a light object like a wreath unmeasurable.) "There may be some truth to it," Rorres adds. "Archimedes did measure the volume of things but the eureka moment was maybe due to his original discovery [concerning buoyance], not to sitting in the bathtub and then running through the streets of Syracuse naked."
Much like Newton's apple, the exclamation persists because of the enduring power of the story: a golden crown, a life in the balance, a naked mathematician. Archimedes was a font of both mathematical insight and smart quotes as well as the hero of some really great stories. (One credits him with the invention of the death ray--actually an array of mirrors to focus sunlight--to set fire to an invading Roman fleet.) The suspect foundations of the eureka moment take nothing away from the word's ability to uniquely and concisely convey the flash of inspiration.