How did the scientist Archimedes distinguish between real and fake crowns?

There is a little story about the discovery of this law, you might as well read it.

There is a story about the discovery of the principle of buoyancy: According to legend, King Hernon of Syracuse asked craftsmen to make him a pure gold crown. But after it was finished, the king suspected that the gold crown made by the craftsman was not entirely gold, but it was indeed as heavy as the pure gold that was originally given to the goldsmith. Did the craftsmen embezzle gold? Wanting to test the authenticity without destroying the crown, this problem not only stumped the king, but also made the ministers look at each other. On the advice of a minister, the king invited Archimedes to examine it. At first, Archimedes was thinking hard but could do nothing. One day, he was taking a bath at home. When he sat in the tub, he saw the water overflowing and felt his body being gently lifted. He suddenly realized that he could determine the specific gravity of the gold crown by measuring the displacement of a solid in water. He jumped out of the bathtub excitedly and ran out without even putting on his clothes, shouting "Eureka! Eureka!" (Eureka, meaning "I found it". Greek: ερηκα)

After further experiments, he came to the palace. He put the crown and the same weight of pure gold in two basins filled with water. He compared the water overflowing from the two basins and found that the crown was in the basin. More water overflowed than the other basin. This shows that the volume of the crown is larger than that of pure gold of the same weight, and the density is different, which proves that the crown is mixed with silver.

The significance of this experiment is far greater than finding out that the goldsmith deceived the king. Archimedes discovered the law of buoyancy (Archimedes' principle): the buoyancy gained by an object in a liquid is equal to the buoyancy he obtained. Displace the weight of the liquid. Until modern times, people are still using this principle to calculate the specific gravity of objects and determine the load capacity of ships.