εὕρηκα!
“εὕρηκα!” (“Eureka!”) cried A, leaping from his bath and running naked through the streets of S to the royal palace. He had noticed the water level rise when he got into the bath, and realised that this gave a way of measuring the volume of a complex shape.
That solved a little problem he had been working on:
H, the king of S, had given a quantity of gold to a jeweller to be made into a crown. H suspected that he had been cheated; the crown weighed the right amount, but was thought to be a mixture of gold and (cheaper) silver. He asked A, the leading scientist of the era, to solve the problem for him.
Gold and silver have different densities, so if the density of the crown could be measured the jeweller could be caught (or exonerated).
Density is weight divided by volume. Weight is easy, but how to calculate the volume of a shape as complex as a crown?
After his bath and his eureka moment he knew how; he filled a vessel with water, dropped in the crown and measured the volume of the water that flowed out. A bit of arithmetic and the density was not that of gold. Crooked jeweller duly caught!
This is the story we all know, but there are a number of problems with it. Mainly, and sadly, there is little evidence that any of it happened.
A, H and S certainly existed, H may well have had a crown, but the rest is very doubtful.
The idea that a volume could be measured in this way may have originated with A, but may already have been understood. It is not likely that the measurements could have been done sufficiently accurately at the time to distinguish a minor admixture of silver with gold.
A’s Principle
However there is a way that it could have been done that follows not from the above but from something that A certainly did discover: A’s Principle. This says that the buoyancy experienced by an object immersed in a fluid is equal to the weight of the fluid that it displaces.
Buoyancy is an upwards force. It is what enables items to float. If the buoyancy of a fully immersed object is greater than its weight the upwards force wins and the object rises to the surface. This happens if the density of the object is less than that of the fluid.
If the density of an object is greater than that of the fluid, its weight is greater than the buoyancy and it sinks.
Examples
A piece of wood in water floats because wood is less dense than water. If pushed down it will displace water the weight of which is more than its weight. It will rise out of the water until the amount of water displaced weighs the same as the wood.
A brick dropped into water sinks. It is denser than water and so displaces water that weighs less than its weight. The buoyancy it experiences is less than its weight, so it goes down.
A steel boat can float even though steel is denser than water because it is shaped so that it displaces more than its weight of water. The boat has air inside it that makes its total density less than that of water.
Suppose a light plastic dinghy and a person sitting in it together weigh 100kg. How much water is displaced when it is floated in water?
We can take the density of water to be 1 (1 gram per millilitre, 1 kilogram per litre, 1 tonne per cubic metre), so 100 litres of water must be displaced.
This sounds like a lot, but if we suppose that the footprint of the boat, it’s area when seen from above, is about 1 square metre then it need only push down into the water 10 centimetres to displace this much water. The 10cm depth that it pushes down is called its draft, by the way.
H’s crown
Gold crowns sink in water, of course, but they still experience buoyancy. So this gives a way of solving the bent jeweller problem using a simple balance. What we do is balance the crown against a solid lump of pure gold:
Now immerse the whole thing in water. If the crown is made of a mixture of gold and silver its density will be different from the pure gold, so its volume will be different, so its buoyancy will be different, and the balance will tip. If the crown is made of pure gold it will experience the same buoyancy as the lump and the balance will stay level.
Did A do this? We don’t know. He was certainly capable of thinking of it, and his very own principle is what makes it work.
The cache
Oh yes, the cache.
You know who A is. Count the letters in his name (his name as we English speakers spell it).
You may know what city S is, count its letters.
You may not know who H was, look him up and count his letters.
Let’s suppose that the crown is a mixture of silver and gold. Silver is less dense than gold, so when the balance above is immersed in water, which way will it tip? If it tips so that the crown is lower B=3, if it tips so that the gold is lower B=2.
A narrowboat (canal boat) is 21 metres long and 2 metres wide.
Assume its footprint is a rectangle (ie it is not pointy at the front or round at the back but squared off – not very good boat design but it makes the sum easier).
Assume it has vertical sides and a flat bottom (close to correct for narrowboats).
If it weighs 29.4 tonnes, what is its draft in decimetres? This is D.
You should have five smallish whole numbers; A, S, H, B, D
The cache is at N52 34.abc W001 07.def
Where
a = B + D - S
b = H + B - D
c = S + B - D
d = S + B - A
e = A + H - D
f = A + H – S
There is extra information about the actual cache and you can check your answer at:
Congratulations to:
Nadiagoescaching on FTF
hal-an-tow on STF
And the prize for the most amusing log so far goes to...
chilli.monster
Well done to all the solvers and finders!