I teach physics and when we talk about density it's fun to tell the story of Archimedes and the crown. For the uninitiated the story goes that the king was gifted a crown that was claimed to be made of solid gold. The king wanted to know if it was actually solid gold, or fake (perhaps mixed with silver). The king tasked Archimedes with finding a way to tell without damaging the crown. Archimedes thought long and hard about the problem, and then one day in the bath though of the following solution:
You place the crown on one side of a balance scale, and have gold of known purity placed on the other side until the balance balances. Now the mass of the crown equals the mass of the gold. The question is about their volume. So you get a bucket of water, note the height of the water. Place the crown in the water and note how high up the water rises. Then remove the crown and place the identical mass of gold in the water. If the water rises to the same height, then the crown is pure gold. If, however, the crown is mixed with a lighter element like silver, then the crown will be 'fluffier' than a pure gold crown and push the water up higher.
The story goes that he jumped out of the bath tub running naked through the streets yelling "Eureka!"
Students love this story and naturally ask the question "So was the crown real or fake?" and I cant find an answer. Whenever I google this I just get retellings of the story I wrote above that ends with an explanation of the physics. I cant find the actual story. Is this story a made up parable like Newton getting hit with the apple? Or did this really happen? If so what was the outcome?
Thanks.
That great story is told by Marcus Vitruvius, in the ninth book of "De architectura", chapter III. The story is not long, and it is very curious. Sadly, we don't know what happened to the goldsmith, but he was found guilty.
Let me give you the text, as it may come in very handy:
Taking this as the beginning of his discovery, it is said that he made two masses of the same weight as the crown, one of gold and the other of silver. After making them, he filled a large vessel with water to the very brim, and dropped the mass of silver into it. As much water ran out as was equal in bulk to that of the silver sunk in the vessel. Then, taking out the mass, he poured back the lost quantity of water, using a pint measure, until it was level with the brim as it had been before. Thus he found the weight of silver corresponding to a definite quantity of water.
After this experiment, he likewise dropped the mass of gold into the full vessel and, on taking it out and measuring as before, found that not so much water was lost, but a smaller quantity: namely, as much less as a mass of gold lacks in bulk compared to a mass of silver of the same weight. Finally, filling the vessel again and dropping the crown itself into the same quantity of water, he found that more water ran over for the crown than for the mass of gold of the same weight. Hence, reasoning from the fact that more water was lost in the case of the crown than in that of the mass, he detected the mixing of silver with the gold, and made the theft of the contractor perfectly clear.
I have to warn you, of course, that the text comes from Vitruvius, a Roman author from the 1st century BCE, who was born around 80 BCE and died in 15 BCE. This puts the story some 200 years after the alleged facts, so details may not be completely precise. Hiero II of Syracuse was tyrant of that city between 265 and 215 BCE, and his name is mentioned by Vitruvius, so that gives a chronological frame of reference. However, being that the details provided by Vitruvius in his books are coherent and can be corroborated by other evidence, we shall accept his account as true.
As for what would befall the fraudster goldsmith, it is not mentioned in the story told by Vitruvius, and in the end it does not really matter. The account of Vitruvius is not there to illustrate us on the harshness or lightness of the laws of Syracuse, but to show a scientific principle, how Archimedes came to it, and one possible real life application.