# What does “Eureka” mean? Displacement of liquids

...the total volume occupied by the submerged solid and liquid together is exactly equal to the volume of the solid plus the volume of the liquid.

Legend has it that the Greek philosopher Archimedes was tasked with figuring out whether an allegedly gold crown was adulterated with cheaper metals. He knew that gold is denser than the baser metals, so he could tell if the crown was really gold if he could measure its density. Measuring its weight was no problem, but how to measure the volume of such a complicated shape?

If a solid is submerged in a liquid, the liquid cannot occupy the same space as the solid, so it is forced out of the way. At the same time it flows, so it fills in all the holes, crevices, nooks and crannies, leaving no gaps. And since it's incompressible, the volume occupied by the liquid itself doesn't change. Therefore the total volume occupied by the submerged solid and liquid together is exactly equal to the volume of the solid plus the volume of the liquid. So if you measure the apparent “change” in the volume of the liquid, that's just the volume of the submerged solid object. Since measuring the volume of a liquid is pretty easy, the problem is solved: you now have an accurate way to measure the volume of any solid object, no matter how complicated its shape.

According to the story, Archimedes arrived at this insight while bathing, and was so excited that he jumped from the bath yelling “Eureka!” (“I found it!”) and ran home naked through the streets of Syracuse. Dubious story, but sound physics.

—Roger Tobin