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Stories of Science:
Archimedes and the bathtub

By ItsNotMagicItsScience
Image Source: http://en.wikipedia.org/wiki/File:Domenico-Fetti_Archimedes_1620.jpg

Science is peppered with stories of discovery, with some of the biggest breakthroughs coming from seemingly everyday occurrences. One of the oldest and well-known tale revolves around Archimedes' legendary “Eureka!” moment when he realised the principle of buoyancy while taking a bath.

But did the event actually happen, or was it an ancient story that got misinterpreted as the years went by? Let's have a closer look into this discovery story.

The story
Archimedes's tale takes place some 2,200 years ago when King Hieron II of Syracuse in Sicily gave a jeweller a bar of gold and ordered him to make it into a crown. The king, however, suspected that the jeweller had substituted some of the gold for cheaper metal like silver, while pocketing the leftover gold.

The king had no way of proving his suspicions, so he asked Archimedes – a Greek  mathematician, engineer, inventor, and astronomer – to find a definitive answer. Archimedes had spent a long time trying to figure out the answer, which came to him when he noticed how water would splash out of his bath tub the moment he stepped into it, and the more he stepped into the tub, even more water got displaced.

At the time, Archimedes had known that gold was denser than silver, so if a certain weight of silver had been substituted for the same weight of gold, the crown would occupy a larger space than an identical one of pure gold.

So to find the crown’s volume, all Archimedes had to do was essentially immerse the crown and exact measurement of pure gold in a tub filled with water to the brim, measure the spillage, and compare the volume of spillages – if the jeweller had indeed made a crown of pure gold the volume should be the same.

Archimedes was said to be so thrilled with this discovery that he immediately hopped out of the bath and ran onto the streets naked shouting 'Eureka!' 'Eureka!'.

And in case you were wondering, the jeweller was indeed cheating the king.

What probably happened
Archimedes's discovery was told by Vitruvius, a Roman architect, writer and engineer (smart people back then seemed to be doing everything) in a book written two centuries after Archimedes had died. Where Vitruvius got his sources from, he didn't say, but he did write about a scene where Archimedes was running out naked and wet, after he had leaped from his bath following that discovery.

Whether or not Archimedes used such a simple comparative method of displacement to prove the crown's impurity was up for debate as early as 1586. Galileo Galilei, a big fan of Archimedes, pointed out that the king's problem needed a method with more precise measurements that was not available during that time; Galileo thought that the difference in the volume of overflow produced by immersing a pure gold crown versus an impure one would be too small to differentiate.

“This seems, so to say, a crude thing,” wrote Galileo, “far from scientific precision; and it will seem even more so to those who have read and understood the very subtle inventions of this divine man in his own writings; from which one most clearly realises how inferior all other minds are to Archimedes’…”

Instead, Galileo proposed a method in which the crown is balanced on a scale against pure gold in air, and then the scale is submerged with crown and gold in water to see if they still balance.

In Galileo's plan, if the crown was made of pure gold, the buoyant forces on the crown and the gold bars would be the same and the balance would remain horizontal. This would happen because Archimedes's principle states that the same weight of the same substance must occupy the same volume, whatever the shape.

If the crown was impure, it would have a slightly larger volume than one of pure gold – remember, silver takes up more space than the equivalent weight of gold. Immersed in water, a larger-volume crown would be buoyed more strongly than the matching gold bar; this would cause the balance to tip, with the crown side higher than the side containing the pure gold bar.

What Archimedes's discovery meant
Archimedes' soak in the tub gave rise to Archimedes Principle, which states that when a body is immersed in water, it experiences a kind of force we call buoyancy. This force is equal to the weight of the water displaced by the body.

Buoyancy explains why somethings float, and others don't. A ball of steel, for instance, will sink because it's unable to displace water that equals its weight.  But steel of the same weight but shaped as a bowl will float because the weight gets distributed over a larger area and the steel displaces water equal to its weight.



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