Eureka!
Exactly what Archimedes did after shouting "Eureka!" and running naked through the streets is a matter of some dispute. According to Vitruvius, in his Ten Books on Architecture, Hiero of Syracuse - Hieron II - gave thanks to the gods for gaining the throne by donating a wreath of gold to be placed on the head of one of the statues in the main temple of Syracuse. He gave a certain quantity of gold to a goldsmith and commissioned him to execute the work of making the wreath and in due course the goldsmith returned with the wreath, which was weighed in Hiero's presence and found to correspond exactly with the gold the man had been given.
Later on, however, someone informed Hiero that the goldsmith had actually kept some of the gold for himself and had smelted an equal weight of silver in with the gold so that the weight remained the same, though obviously the value of the wreath was now that much less. An examination of the wreath failed to detect any obvious difference between it and so much pure gold, but the information was apparently trustworthy and Hiero charged Archimedes with the task of finding some method by which all doubts could be either confirmed or disproved.
Archimedes pondered the matter for some time until he took a bath one day and his slave had negligently filled the tub so full that as Archimedes lowered himself into the water, the tub overflowed, which drew Archimedes' attention to the fact that a body placed in water displaces an equal volume of water. Suddenly struck with the significance of this fact, Archimedes shouted "Eureka!" (I have it!), leaped out of the bath and ran home, naked, in order to confirm his intuition by experiment.
According to Vitruvius' tale, Archimedes filled a jar to the brim and then placed in it a lump of silver of the weight of the disputed crown, carefully collecting the water which overflowed and measuring it. He then refilled the jar and this time placed a lump of gold in it. Again he collected and measured the water which overflowed and, finding that they were a different quantity, he was confident enough to go and demonstrate his discovery to the king using the golden wreath instead of the silver. The difference between the amount of water which overflowed from the gold and that which overflowed from the wreath was enough to convict the dishonest goldsmith.
Unfortunately for the story, the amount of gold involved must have been 2lbs or less, as the largest known gold wreath from the period in question weighs 19oz. A couple of its leaves are missing, but even so the most it could realistically weigh would be 20oz. Even if we assume that the goldsmith put in 7oz of silver - which would be pushing his luck considerably as the gold would no longer be a rich yellow - the difference would be around a third of an ounce of water - which would require considerable accuracy in the measuring!
Some have pointed out that in Archimedes' own writings there is no reference to the story and no indication that he knew or understood the principles that underlie it. On the other hand, Archimedes devised a "Law of Buoyancy" which states that "Any object immersed in water is bouyed up by a force equal to the weight of the fluid displaced by the object." This, of course, is why iron boats float: by beating the iron out thin and forming it into a bowl shape you can make it displace more water than the iron weighs - ergo, the bowl floats. It is also the basis of working out whether the gold was mixed with silver.
The simplest way of demonstrating the solution would be to get a lump of gold of the correct weight and put it in a basin of water that was full right up to the top. Water would overflow, but once things had settled down you remove the lump, taking care not to cause any more water to overflow. Then - very, very carefully - you put the wreath into the basin. If it is of pure gold no more water will overflow. If, however, it is of silver and therefore takes up more room than the lump of gold, more water will flow out of the basin and you can send the police round to pick up the goldsmith.
That's the theory! In practice it would be almost impossible to perform the task without spilling more water: there would be air bubbles trapped in the leaves of the wreath which would make it appear to be larger than it really is. Unless Archimedes was somehow able to solve that problem, he would be responsible for falsely convicting the goldsmith - unless, of course, that is what happened! Perhaps the goldsmith was innocent after all and Archimedes was guilty of careless procedures.
Some experts therefore consider it more likely that Archimedes used his Bouyancy Principle by suspending the wreath from one arm of a balance and an equal weight of gold from the other, then lowering the whole thing into a tub of water. No need to measure the amount of water displaced; all you did was look at the balance arm. If it remained horizontal then both items were of the same material because they displaced the same amount of water and were bouyed up to exactly the same extent. If, however, one item displaced a greater amount of water it would be bouyed up to a greater extent and the balance arm would tilt.
If this was indeed the method Archimedes used, I take my hat off to him - his mind was truly devious, for the overflowing water method follows naturally from getting into an over-filled bath, but differential levels of a balance arm dipped in water do not and it is greatly to Archimedes credit that his mind was able to make the conceptual leap from the one to the other.
the bowl floats Of course, if water can get into the bowl through a hole, it fills up the bowl and the bowl no longer displaces a large amount of water. It simply displaces the same amount of water as a lump of iron of the same weight - and as iron is heavier than water, the bowl (or the boat) promptly sinks. Return
© Kendall K. Down 2009
