[spacemule]

> then immersing the apparatus in water

I'm having trouble picturing what's going on here. I'm not sure what the apparatus is. The best I can figure, it's two containers of water with the reference gold in one and the crown in the other. However, this wouldn't be any different than just weighing the two pieces of gold (which are already known to be the same mass). I tried searching for "hydrostatic balance," but nothing relevant to this came up. Since you seem to understand this, would you mind explaining (or pointing to an explanation of) Archimedes's probable solution?

[hddqsb]

Sure! (My explanation is basically equivalent to the combined explanations by @mafuy and @malcolp, but I'll post it anyway.)

The idea is to balance a scale with the crown on one side and an equal mass of gold on the other side (no water yet), and then submerge the entire scale in water. Now there will still be the same mass on each side, but there will additionally be a buoyancy force on each side proportional to its volume. So if the volume of the crown is different from the volume of the reference gold, the buoyancy forces will be different and the scale will tip.

[mafuy]

The apparatus is the scale, on which the crown and the gold were placed. They balance out when it is in air. They do not (for a fake crown), when the scale is submerged in water.

[spacemule]

Thanks for answering. I'm not even an amateur in physics, so forgive me for this elementary follow-up question. Why does it not balance in water? It would seem to me that the same weight it on each side of the scale, so the same force is pushing down on each side of the scale.