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Eureka (Greek "I have found it") is an exclamation used as an interjection to celebrate a discovery. It is most famously attributed to Archimedes; he reportedly uttered the word when, while bathing, he suddenly understood that the volume of an irregular object could be calculated by finding the volume of water displaced when the object was submerged in water. After making this discovery, he is said to have leapt out of his bathtub and run through the streets of Syracuse naked. Archimedes' insight led to the solution of a problem that had been asked of him by Hiero of Syracuse, on how to assess the purity of an irregular golden crown: by dividing the object's weight by its volume, one could calculate its density, an important indicator of purity. This story first appeared in written form in Vitruvius' books of architecture, two centuries after it supposedly took place.^[1] 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.^[2]

Another famous mathematician, Carl Friedrich Gauss, echoed Archimedes when in 1796 he wrote in his notebook, "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.^[3]

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.^[4]

The city of Eureka, California, founded in 1850, uses the California State Seal as its official seal. 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.^[5] Many places, works of culture, and other objects have since been named "Eureka"; see Eureka 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.

Hēurēka is the 1st person singular perfect indicative active of the Greek verb heuriskein, (Greek εὑρίσκειν) meaning "to find"; it means "I have found it", or more accurately, "I am in a state of having found it". The English version of the word is pronounced IPA: /juːˈriːkə/; in ancient Greek ηὕρηκα (later εὕρηκα) would have been pronounced IPA: [ˈheʷreːka] in both former and later forms, while the modern Greek pronunciation is [ˈevrika].