The question asked how Archimedes can make a more sensitive measurement on whether or not the crown is part silver.
The answer I was looking for is that Archimedes balances the crown with some pure gold, then submerges scale, crown, and gold all at once under water (or mercury, say). If the crown and gold have different densities, the buoyant force on them from the water will be different, and the scale will no longer balance.
One reason this procedure is nice is that no measuring is calculating is required. You only need to say whether or not the balance is even.
Let’s take a look at why this procedure is more sensitive than Archimedes’ original one, in which he looks at the volume of water displaced by the crown and pure gold.
Silver weighs ten times as much as water, and gold 20 times as much. (I looked this up online, since I didn’t have a scale on hand to weigh my large stashes of gold and silver.)
Next, let’s say that Archimedes’ balance is sensitive to one part in one thousand, so that the pure gold he weighs out has the same mass as the crown to this accuracy.
The improved test, where the scale goes underwater, will detect a difference in the gold and silver as long as the difference in the buoyant forces between them makes up one thousandth part their weight.
Let’s say the crown is made of up a fraction silver and gold, and that its total mass is one crownmass. Then the total buoyant force on it is . The buoyant force on the pure gold is just . The difference must be more than for us to detect it. That gives
The total gold and silver in the crown add to one, so . Using this
So under these assumptions, Archimedes can measure as little as 2% silver in the crown with the improved method.
When only 2% of the crown is silver, the additional volume displaced by the crown is quite small. The silver takes up twice as much space as the gold, so the added volume is 2% of the total volume, compared to pure gold. Also remember that there is a 1/1000 part error in the masses of the gold and crown, but this is small enough compared to the 2% that it can be ignored.
2% is a very small signal for this method. If you simply dunked the gold in the bucket, the water level might rise 1cm. Then you’d be looking for the water level to rise 1.02cm when adding the crown, which is not reasonable to detect with your eye.
You might try to be more clever, for example, by measuring in a vase with a very thin neck. Then, if you had a door in the bottom of the vase, you could load the gold there and fill the vase until the water came up to the bottom of the thin neck. Then drain the water, set it aside, and trade the gold for the crown. Put the water back in, and see if the height of the water in the thin neck vase has raised visibly. This would magnify the height change. However, it might be tough to keep track of all the water in this procedure.
Ultimately, I think the underwater weighing would be more effective, assuming the water doesn’t do anything funny to the bearing in the scale, or otherwise affect its ability to balance things.
Also, note that if Archimedes were to conduct the experiment with mercury rather than water, the difference between the two methods would become more dramatic. The under-mercury weighing is more effective, since mercury is very dense, and the difference between silver and gold is far more dramatic. The volume-difference method would be the same, because the volumes of the two masses are unchanged. Analogously, if Archimedes had been trying to distinguish lighter metals, the underwater weighing method would again be more effective.