[Terr_]

> That is not how a balance scale works. The two sides balance, because the weigher makes them balance.

I think you're confusing the tool-calibration step with the actual measuring. As long as a scale is built symmetrically, an empty scale will read as balanced in any environment. Both sides are made from the same materials with the same densities and displacements etc.

Once you apply the dissimilar samples (such as a known 1kg mass of styrofoam versus a known 1kg mass of steel) it will cease to read as perfectly even, because the air-buoyancy of the samples will be different. Perhaps not enough to see easily, but it's there.

____

It may help to consider that for this experiment we do not actually need to see an "equal" weight-measurement from any kind of scale. What we're actually trying to check is that the readout doesn't change when swapping the surroundings from air to water, ex:

    diff_air = weight(real_gold, air) - weight(suspicious_crown, air)
    diff_water = weight(real_gold, water) - weight(suspicious_crown, water)
    assert(diff_air == diff_water, "Error, density mismatch detected")

A good counterfeiter will ensure diff_air==0, but that's just them trying to cheat a much-simpler "very similar mass" test, and it isn't a prerequisite for this "same density" test.

[Lanzaa]

> ... consider that for this experiment we do not actually need to see an "equal" weight-measurement from any kind of scale.

Ah, you seem to be assuming that the mass of the crown is known and an equivalent mass of gold can be produced.

> What we're actually trying to check is that the readout doesn't change ...

I'm not sure what you mean by readout. If one side is A and the other B the balance only has three readings, A>B or A=B or A<B.

Typically I would expect the crown to be put on one side of a balance. Then the weigher would search for the amount of gold that makes the balance, well, balance (A=B). In effect the weigher chooses diff_air==0.

> A good counterfeiter will ensure diff_air==0

I don't see how the counterfeiter has any influence on diff_air, since the mass is not known beforehand.

Your equations seem correct.

[Terr_]

> Ah, you seem to be assuming that the mass of the crown is known and an equivalent mass of gold can be produced.

The first is effectively true because you can weigh the crown in air, which for goldish-density crownlike objects is so relatively thin that you'll get similar results to doing the comparisons in vacuum.

The second assumption of a reference-sample was explicitly stated earlier in the thread, inside the Wikipedia quote.

> I don't see how the counterfeiter has any influence on diff_air, since the mass is not known beforehand.

The crooked crown-maker already knows (A) how much true-gold mass their customer is expecting them to deliver and (B) they have many opportunities to mass-measure and adjust the profitably-adulterated not-quite-pure-gold object they are creating.

[hddqsb]

You should use "/" instead of "-" in your equations. If you have two objects of the same density but with different sizes, the ratio of their (apparent) weights will be the same in air and in water, but the difference will be different!