Archimedes’ Eureka Moment

By Math Is Fun, January 8, 2022

Son is very interested in ancient Greek and its famous people. So I decided to give him a mini-series on Greek Mathematicians. Archimedes was the first.

Archimedes was one of the greatest scientists of ancient Greek, from the city state of Syracuse in Sicily, then a Greek colony. His achievements spanned mathematics, physics and astronomy. He also had a few famous stories.

Legend has it that the King of Syracuse had a crown made of pure Gold, which he supplied. But he suspected the dishonest Goldsmith replaced some of the Gold with silver, so he could steal the Gold. The King asked Archimedes to find a way to prove it, one way or another. The condition was that Archimedes could not damage the crown.

It was a difficult problem. The Goldsmith was smart and sneaky. The crown weighed the same as the Gold the King provided. If some Gold was replaced with silver, the Goldsmith mixed the silver well with the rest of the Gold, so there was no way to tell from the appearance.

Archimedes tried several days to find a solution to no avail. Exhausted, he decided to take a bath. When he was lying down in the bathtub, he saw the water spilled out. Suddenly, a flash of brilliance struck him. He was so excited. He ran out to the streets shouting “Eureka! Eureka! Eureka!”. Eureka is Greek and means “I have found it!”. Archimedes was so excited. But he forgot one thing: since he was taking a bath, he was not dressed!

So what was Archimedes’ solution to the King’s problem? Archimedes went to the King and performed the experiments in front of the King. He put a large bowl full of water on top of another container. Then he submerged the crown in the bowl and collected the water spilled. The volume of the spilled water equaled the volume of the crown. This was a clever way to measure the volume of the crown, even if its shape was irregular.

As the next step, Archimedes asked the King to measure some Gold of the same weight as the crown. He repeated the same water spill experiment and measured the volume of the Gold. If the crown was made of pure gold, the volume of the crown should be equal to the volume of the Gold of the same weight. It turned out the volume of the crown was much larger, that meant the crown was mixed with lighter ingredients – for example Silver.

The King got his answer. The Goldsmith got his punishment. Archimedes left a famous Eureka moment in history.