Archimedes' principle is named after Archimedes of Syracuse, the first to discover this law. When he did, he ran naked out of his bathtub.<ref name=acottLaw>Template:Cite journal.</ref> Archimedes' principle relates buoyancy to displacement.<ref>Displacement here means the water pushed aside by the bottle (or other object).</ref>
Archimedes' treatise, On floating bodies, proposition five states:
For more general objects, floating and sunken, and in gases as well as liquids (i.e. a fluid), Archimedes' principle may be stated thus in terms of forces:
For a sunken object, the volume of displaced fluid is the volume of the object, and for a floating object on a liquid, the weight of the displaced liquid is the weight of the object.
Briefly: Buoyancy = weight of displaced fluid.
To use the principle to tell the difference between gold and another substance, the concept of mass (in practice, weight) must be added.
Apply this formula to a suitably pure piece of gold:
That gives you the density of the gold sample. The only unknown is the density of the (gold) object; the density of water is 1.
Repeat for the experimental object (non-gold), when you get a different (and usually lesser) density.
Now you can tell what is gold and what is not, and collect your reward from the King of Syracuse. That is why Archimedes shouted "Eureka!"
We do not actually know if Archimedes used exactly this method. An alternative is to use a scale. On one side put the object to be tested (e.g. the crown). On the other side put gold of equal weight. Immerse the scales. The gold will go down, and the crown up (if it is not gold). That is because, being less dense than gold, it occupies a larger volume and receives more buoyancy.