Final answer:
Archimedes used density to determine the purity of a gold crown by applying Archimedes' principle; substances have unique densities, which can be measured by weighing the object both in and out of water.
Step-by-step explanation:
Archimedes used density to discover if a crown was made of pure gold because substances have their own unique densities. Density is the mass per unit volume of a substance. Hence, pure gold has a distinctly high density compared to other metals. If gold is alloyed with a less dense metal, the crown's average density would decrease. By applying Archimedes' principle, which states that the buoyant force on a submerged object is equal to the weight of the fluid it displaces, Archimedes could determine if the density (and therefore the purity) of the crown was the same as that of pure gold. This involves weighing the crown in air and then in water to calculate its density.
Archimedes' principle also implies that an object's apparent weight loss when submerged in a fluid is equal to the fluid's weight that has been displaced. Using this principle, if a crown purported to be made of gold does not displace an amount of water whose weight is equal to that expected from an object of pure gold's density, it indicates that the crown is not made of pure gold.