The most famous shout in science is a verb conjugation. Eureka is not a noun or an exclamation in Greek. It is simply "I have found," the first person singular perfect of heuriskein, "to find."
The word owes its fame to Archimedes of Syracuse, who lived from about 287 to 212 BCE. King Hiero II suspected that a goldsmith had mixed cheaper metal into a gold crown. Archimedes had to find out without damaging the crown.
The story says the answer came to him in the bath. As he got in, the water rose. He saw that a submerged object pushes aside its own volume of water. That meant he could measure the crown's volume, and so its density. He leapt out and ran naked through the streets of Syracuse, shouting "Eureka!"
How much of that happened is unknown. The earliest account comes from the Roman architect Vitruvius, in De architectura. He wrote around two centuries after the event supposedly took place. Plutarch retold it in shorter form about a century after that.
Historians have doubts about the method as well as the run. The difference in displaced water between a pure crown and an alloyed one would have been tiny. Measuring it accurately was probably beyond the tools of the time. Galileo later suggested Archimedes more likely used a hydrostatic balance, weighing the crown against gold in water. So the discovery is plausible, but the bathtub scene is a story, not a record.
English borrowed the word around 1600, dropping the Greek h along the way. German, Danish and Finnish kept the h. By then it had become a general cry of discovery, not a reference to crowns.
California adopted "Eureka" as its state motto in 1963. It points to the 1848 gold find near Sutter's Mill, a different kind of discovery in the ground.
The same Greek verb gave English heuristic, first recorded in 1821. A heuristic is a rule of thumb for finding an answer. It is, in a sense, a method for reaching eureka.