How did Archimedes determine the adulteration of gold by his principle?

First, he took pure gold equal to the weight of the crown to be checked for adulteration. Then it is confirmed that the weight of the crown and the weight of the gold it takes are equal.
Now he immersed the crown in a water container completely filled with water and determined the amount of water removed by immersing the crown. In the same way, the pure gold taken equal to the weight of the crown was immersed in water and the amount of water removed was determined.

If the amount of water removed by dipping the crown and the gold is equal, then we can say that the volume of the crown and the volume of pure gold are equal.

Earlier we took the mass of pure gold and crown to be equal. And now the volume has become equal.

The density of any object is equal to the ratio of its mass to its volume. Now if two objects have equal mass and volume then their density must also be equal. In the same way, crown and pure gold have the same mass and volume, so their density is equal. That means the crown and pure gold have equaled their elemental density or found their elemental similarity. That means the crown is made of pure gold.

And if there was no similarity in volume, we could say with certainty that the density or composition of the pure gold and the crown are not the same. That means the crown is not pure gold. This is how Archimedes diagnosed the adulteration of gold in the crown.