What is the story behind “Archimedes in the bathtub”?

Greek and Roman Mathematics Read more from
Chapter The History of Mathematics

One of the most famous stories of Archimedes involves royalty: When Hiero II of Syracuse, King of Sicily, wanted to determine if a crown (actually, a wreath) he had ordered was truly pure gold or alloyed with silver—in other words, whether or not the Royal Goldsmith had substituted some of the gold with silver. The king called Archimedes to solve the problem. The Greek mathematician knew that silver was less dense than gold (or silver was not as heavy as gold), but without pounding the crown into an easily weighed cubic shape, he didn’t know how to determine the relative density of the irregularly-shaped crown.

Perplexed, the mathematician did what we all do to get good ideas: he took a bath. As he entered the tub, he noticed how the water rose, which made him realize that the volume of water that fell out of the tub was equal to that of the volume in his body. Legend has it Archimedes ran naked through the streets shouting “Eureka!” (“I have found it!”). He knew that a given weight of gold represented a smaller volume than an equal weight of silver because gold is much denser than silver, so not as much is needed to displace the water. Therefore, a specific amount of gold would displace less water than an equal weight of silver.

The next day, Archimedes submerged the crown and an amount of gold equal to what was supposed to be in the crown. He found that Hiero’s crown displaced more water than an equal weight of gold, thus proving the crown was alloyed with a less dense material (the silver) and not pure gold. This eventually led to the hydrostatic principle, as it is now called, presented in Archimedes’s appropriately-named treatise, On Floating Bodies. As for the goldsmith, he was beheaded for stealing the king’s gold.


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