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The citizens of ancient Syracuse would have recognized the man who is said to have bustled past them naked and dripping and shouting, "Eureka!" (I have found it!). It was Archimedes, celebrated mathematician, scientist, inventor, and confidant of the king.

That Archimedes seemed oblivious to his own nakedness and to onlookers' bemused stares was perhaps only mildly scandalous, given his reputation for eccentricity. To residents of this long-ago Sicilian city-state, Archimedes had always occupied an enchanted middle ground: one foot planted squarely in the world of men, the other dancing to some private muse of nature.

Did Archimedes really run stark naked through the streets of Syracuse? And, if he did, what bolt of inspiration sparked his unclothed euphoria? Archimedes' streak is among the oldest running accounts of the dazzle of scientific genius, vying with Plato's story of Archimedes' predecessor Thales absentmindedly falling down a well while pondering the stars.

Archimedes' naked run first appears, paradoxically, in the work On architecture by the first-century B.C. Roman designer-builder Vitruvius. (Here, too, is the description that inspired Leonardo da Vinci's famous drawing of the Vitruvian Man.) Vitruvius was a great admirer of classical Greek geometers, especially Archimedes. The great sage's notorious dash appears in Vitruvius' book right after instructions on how to use the Pythagorean theorem to compute the optimum rise of a staircase. Where Vitruvius got this story, almost two centuries after Archimedes' death, he doesn't say. But there it is - as stark as Archimedes supposedly was that memorable day - the Latin word nudus.

Archimedes was naked and wet, Vitruvius tells us, because only moments earlier, he had leaped from his bath, elated at his flash of insight into a puzzling problem. The Syracusan king, Hieron II, had given the royal metal-smith a specific weight of gold to be fashioned into a splendid wreathlike crown. Now the king suspected that the completed crown, destined to adorn the statue of a deity, had been cut with less valuable silver and that the smith had pocketed the unused gold. Hieron directed Archimedes to establish the crown's makeup without sampling or defacing it in any way.

Archimedes knew that gold is more dense than silver. So if a certain weight of silver had been substituted for the same weight of gold, the crown would occupy a larger space than an identical one of pure gold. But how does one measure the volume of an irregular crown?

Stepping into his brimful bath, as legend has it, Archimedes noticed water splashing over the rim. The more of him that was immersed, the more water overflowed. Eureka! The mundane had become momentous; to find the crown's volume, all Archimedes had to do was immerse the crown in a vessel full of water and measure the spillage.

The trouble with Vitruvius's scenario is twofold. First, Archimedes' epiphany about volume measurement seems rather humdrum compared to his other Eureka-worthy insights; it's hard to imagine him exulting in the streets about it. Second, as Archimedes' latter-day admirer Galileo pointed out in 1586, to prove such metallurgical fakery in the manner Vitruvius describes would have required a degree of measurement precision simply unavailable in ancient times. The difference in the volume of overflow produced by immersing a pure gold crown versus an adulterated one would have been too minute to discern. Galileo could not believe that the vaunted Archimedes would have posed such a flawed technique. "[T]his seems, so to say, a crude thing," mused the twenty-two-year-old Galileo, "far from scientific precision; and it will seem even more so to those who have read and understood the very subtle inventions of this divine man in his own writings; from which one most clearly realizes how inferior all other minds are to Archimedes'..." {Fermi, L. and Bernardini, G. Galileo and the Scientific Revolution. Basic Books, Inc., New York, 1961; www.math.nyu.edu/~crorres/Archimedes/Crown/bilancetta.html} The overflow method, Galileo concluded, was a misstatement by Vitruvius, who evidently had ignored the practicalities of the problem.

More likely, as Galileo proposed in his first-ever scientific pamphlet, La Bilancetta (The Little Balance), Archimedes used a hydrostatic version of a beam balance, a device that resembles the proverbial scales of justice. From one arm of the beam, Archimedes suspended the suspect crown, and from the other arm an equivalent weight of pure gold. The two objects being of equal weight and situated equal distances from the central pivot, the beam would have been stable in any orientation. A nudge from Archimedes' finger would have brought it to its proper starting orientation - perfectly horizontal.

At this point, Archimedes might have immersed the suspended objects into vessels of water. There were two possible outcomes. If the crown was indeed made of pure gold, the buoyant forces on it and on the matching gold slug would be identical and the balance would remain horizontal. (Why? The same weight of the same substance must occupy the same volume, whatever it shape; in that case, Archimedes tells us, both the crown and the gold slug are buoyed identically by the water.) The other possible outcome is for an adulterated crown, which would have had a slightly larger volume than one of pure gold - silver takes up more space than the equivalent weight of gold. Immersed in water, a larger-volume crown would be buoyed more strongly than the matching gold slug; the beam of the balance would tip, the crown side higher than the slug side. Evidently, the balance did tip, proving the artisan's guilt.

In his paper, Galileo lays out the presumed design of Archimedes' hydrostatic balance, down to wrapping the arms of the apparatus with thin wire to create a hyper-precise measurement scale. "Since the wires are very fine, as is needed for precision, it is not possible to count visually, because the eye is dazzled by such small spaces. To count them easily, therefore, take a most sharp stiletto and pass it slowly over said wires." {Hoddeson, Lillian H. "How Did Archimedes Solve King Hiero's Crown Problem? - An Unanswered Question." The Physics Teacher 10 (1972): 17.} Galileo doesn't tell us whether he put his envisioned Archimedean balance to the test. But a modern analysis reveals that such a device applied to Hieron's crown could reasonably have detected as little as 6.5% silver intermixed with gold. If Galileo was right, the larcenous goldsmith stood little chance against the detecting skill of Archimedes.

The rousing vision of a precise hydrostatic balance, rather than the trivial overflow to measure volume, is more likely to have sent Archimedes flying from his bath. But was he truly so bedazzled that he neglected to put on his clothes? Genius often has its eccentricities, a certain disconnection from reality among them. Isaac Newton is known to have inadvertently ambled out of his house while deep in thought.

Whenever Archimedes is pictured in his watery moment of inspiration, he is typically alone in a tub in what appears to be his home. But the Greeks of antiquity, like the Romans after them, bathed in a public facility with attendants to receive and store patrons' garments. If the legend is true, Archimedes would have been at the public baths when he had his Eureka! moment and might have rushed home to pursue the idea, perhaps neglecting to retrieve his clothes. Even then, history's premier naked scientist might not have been completely bare. According to classicist Lydia Lake, "naked" in antiquity - Greek gumnos, Roman nudus - had a dual meaning: either stripped of all clothing or else lacking an outer garment, such as a toga or scarf-like wrap (chlamys). Archimedes might have sprinted through the streets of Syracuse in his underwear - what we would today recognize as a tunic. (Some folklorists speculate that Archimedes' horse-borne successor, Lady Godiva, was likewise not naked, merely stripped of her marks of rank.) Ancient farmers routinely worked the fields in their undergarments; when the "naked" Cincinnatus was called before the Roman senate to become dictator, he dressed by draping a toga over his work tunic. Warriors sans armor were also said to be naked.

In the end, it doesn't much matter whether Archimedes ran through ancient streets naked or in his underwear. Or whether he ran through the streets at all. Nor does it matter whether Archimedes' brain swam with images of overflowing tubs or hydrostatic balances. The Eureka! story - fact or fable - is the touchstone of creative epiphany. It represents the culmination of Archimedean effort: that still mysterious cognitive path that scientists tread toward solutions, fundamental truths, and new ideas. Blazing this path is, in Cicero's words, the "sweetest sustenance of souls," and every scientific explorer is driven onward by its sublime energy. Refracted through the long-ago tale of a naked man in the thrall of an idea is the timeless image of the scientist, patiently applying unbiased eye, hand, and mind toward the explication of nature. And every once in a while, if the experiment is apt, if the circumstances are right, if the will bears up against repeated failure - Eureka!

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My math teacher talked about this, and this was very informing and easy to understand!

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