I know this test may sound unrealistic but in theory it should work as long the equipment is accurate enough.
My idea is using the Archimede's principle to test the purity of the silver coins.
The alleged story is a long time ago the Greece king ordered a new crown to be made. The head designer presented to him a new crown but rumors said that some gold was allegedly taken away by the designer and he used silver to replace the gold stolen.
So the king called in Archimede's for help and at first he was perplexed but when he took a bath he saw the water coming out of the bathtub. Archimedes then got an idea.
His idea was if he placed the crown and the same weight of gold in the same amount of water and collect the water flowing out, then by weighting the water he will be able to identify if the crown is pure gold. The crown wasn't pure gold and the head designer was executed.
So I am thinking if the right equipment is used and the condition of the coins are similar. Will it be possible to use this method to test the purity of the silver.
Only a theoretical solution unsure if it actually works.