Why did Archimedes decide that the goldsmith that made the kings crown should be excecuted?

During Archimedes' time in Syracuse, King Hieron, the king of Syracuse and a close friend of Archimedes, commissioned the construction of a beautiful golden crown. When the goldsmith returned the crown, the king suspected him of stealing some of the gold and replacing it in the crown with silver. The king wanted to make sure that he got all of his gold back, so he asked Archimedes to discover the true contents of the crown. If Archimedes could determine the crown's true contents, he could tell the king whether or not the goldsmith gave him all of his gold back. Archimedes pondered this problem for some time, and one fateful morning in the baths of Syracuse he found the answer. Archimedes was stepping into the tub when he noticed that this action caused some of the water in the tub to overflow. He suddenly realized that the volume of the water that was being displaced equaled the volume of his body in the tub. He then ran through the streets of Syracuse naked yelling "Eureka," which means "I found it." He believed that he could measure the volumes of all irregular solids by placing them in water and noting the amount of water displaced. He solved the king's problem by placing equal masses of gold and silver in water. The silver had a greater volume because it displaced more water. That meant that the silver was less dense than the gold because it had the same weight but a larger volume. Archimedes then compared the lump of gold to the crown. The two had the same mass, but when Archimedes measured their volumes, the crown had a larger volume. Therefore, the crown was less dense and not made of pure gold. The king thus knew that the goldsmith had replaced some of the gold in the crown with silver, and he had the goldsmith executed.