AS much water ran out as was equal in bulk to that of the silver sunk in the vessel. Then, taking out the mass, he poured back the lost quantity of water, using a pint measure, until it was level with the brim as it had been before. Thus he found the weight of silver corresponding to a definite quantity of water.
After this experiment, he likewise dropped the mass of gold into the full vessel and, on taking it out and measuring as before, found that not so much water was lost, but a smaller quantity: namely, as much less as a mass of gold lacks in bulk compared to a mass of silver of the same weight.* Finally, filling the vessel again and dropping the crown itself into the same quantity of water, he found that more water ran over for the crown than for the mass of gold of the same weight. Hence, reasoning from the fact that more water was lost in the case of the crown than in that of the mass, he detected the mixing of silver with the gold, and made the theft of the contractor perfectly clear.
abridged
* The craftsman had taken care to make his crown weigh the same as the lump of gold given to him by Hiero, thinking that this was the only test anyone could make. But Archimedes realised that gold is unusually heavy, so if something else had been mixed into it, the crown would have to be bulkier in order to register the same overall weight. But how to measure the volume (not weight) of metal in such a curiously-shaped object without melting it down? The water displacement test found this out: the crown displaced more water than a solid block of gold did, revealing that it was slightly too bulky and therefore an excess of some lighter material had been added in.