"Eureka" comes from the Ancient Greek word εὕρηκα heúrēka, meaning "I have found (it)", which is the first person singular perfect indicative active of the verb heuriskō "I find". The reconstructed Ancient Greek pronunciation is [hěu̯rɛːka], while the Modern Greek pronunciation is [ˈevrika].
The accent of the English word is on the second syllable, following Latin accent rules, which require that a penult (next-to-last syllable) must be accented if it has a long vowel. In the Greek pronunciation, the first syllable has a high pitch accent, because the Ancient Greek rules of accent do not force accent to the penult unless the ultima (last syllable) has a long vowel. The long vowels in the first two syllables would sound like a double stress to English ears (as in the phrase Maltese cat).
The exclamation 'Eureka!' is famously attributed to the ancient Greek scholar Archimedes. He reportedly proclaimed "Eureka!" when he stepped into a bath and noticed that the water level rose—he suddenly understood that the volume of water displaced must be equal to the volume of the part of his body he had submerged. (This relation is not what is known as Archimedes' principle—that deals with the upthrust experienced by a body immersed in a fluid.) He then realized that the volume of irregular objects could be measured with precision, a previously intractable problem. He is said to have been so eager to share his discovery that he leapt out of his bathtub and ran through the streets of Syracuse naked.
Archimedes' insight led to the solution of a problem posed by Hiero of Syracuse, on how to assess the purity of an irregular golden votive crown; he had given his goldsmith the pure gold to be used, and correctly suspected he had been cheated, by the goldsmith removing gold and adding the same weight of silver. Equipment for weighing objects already existed, and now that Archimedes could also measure volume, their ratio would give the object's density, an important indicator of purity.
This story first appeared in written form in Vitruvius's books of architecture, two centuries after it supposedly took place. Some scholars have doubted the accuracy of this tale, saying among other things that the method would have required precise measurements that would have been difficult to make at the time. Galileo Galilei himself weighed in on the controversy, suggesting a design for a hydrostatic balance that could be used to compare the dry weight of an object with the weight of the same object submerged in water. For the problem posed to Archimedes, though, there is a simple method which requires no precision equipment: balance the crown against pure gold in air, and then submerge the scale with crown and gold in water to see if they still balance.
Names and mottos 
The expression is also quoted as the state motto of California, referring to the momentous discovery of gold near Sutter's Mill in 1848. The California State Seal has included the word "eureka" since its original design by Robert S. Garnett in 1849; the official text from that time describing the seal states that this word's meaning applies "either to the principle involved in the admission of the State or the success of the miner at work". In 1957, the state legislature attempted to make "In God We Trust" the state motto, but this attempt did not succeed, and "Eureka" became the official motto in 1963.
The city of Eureka, California, founded in 1850, uses the California State Seal as its official seal. Eureka is a considerable distance from Sutter's Mill, but was the jumping off point of a smaller gold rush in Trinity County, California in 1850. It is the largest of at least eleven remaining US cities and towns named for the exclamation, "eureka!". As a result of the extensive use of the exclamation dating from 1849, there were nearly 40 locales so named by the 1880s in a nation that had none in the 1840s. Many places, works of culture, and other objects have since been named "Eureka"; see Eureka (disambiguation) for a list.
"Eureka" was also associated with a gold rush in Ballarat, Victoria, Australia. The Eureka Stockade was a revolt in 1854 by gold miners against unjust mining license fees and a brutal administration supervising the miners. The rebellion demonstrated the refusal of the workers to be dominated by unfair government and laws. The Eureka Stockade has often been referred to as the "birth of democracy" in Australia.
Another mathematician, Carl Friedrich Gauss, echoed Archimedes when in 1796 he wrote in his diary, "ΕΥΡΗΚΑ! num = Δ + Δ + Δ", referring to his discovery that any positive integer could be expressed as the sum of at most three triangular numbers. This result is now known as Gauss' Eureka theorem and is a special case of what later became known as the Fermat polygonal number theorem.
See also 
- εὑρίσκω. Liddell, Henry George; Scott, Robert; A Greek–English Lexicon at Perseus Project
- "IGCSE Physics Notes: Using Archimedes Principle to Find the Density of an Object". A Star Maths & Physics Tutors. Retrieved 2012-06-06.
- Tom Clegg (2001-04-08). "Eureka!". Retrieved 2012-06-06.
- Vitruvius on Architecture, IX:Introduction:9‑12, translated into English and in the original Latin.
- The first Eureka moment, Science 305: 1219, August 2004. Fact or Fiction?: Archimedes Coined the Term "Eureka!" in the Bath, Scientific American, December 2006.
- Rorres, Chris. "The Golden Crown: Galileo's Balance". Drexel University. Retrieved 2009-03-24.
- Tipler, Paul A.; Mosca, Gene (2003), Physics for Scientists and Engineers (5th ed.), Macmillan, p. 403, ISBN 9780716783398.
- Official state law defining the motto. Accessed February 26, 2007.
- California Place Names, by Erwin Gudde, p. 105
- West, Barbara A. (2010). A Brief History of Australia. Infobase Publishing. pp. 66–67. ISBN 9780816078851.
- Bell, Eric Temple (1956). "Gauss, the Prince of Mathematicians". In Newman, James R. The World of Mathematics I. Simon & Schuster. pp. 295–339. Dover reprint, 2000, ISBN 0-486-41150-8.
- Ono, Ken; Robins, Sinai; Wahl, Patrick T. (1995). "On the representation of integers as sums of triangular numbers". Aequationes Mathematicae 50 (1–2): 73–94. doi:10.1007/BF01831114. MR 1336863.