displaces a certain volume of water, V say, (2) that a weight W of gold displaces a certain other volume of water, V1 say, and (3) that a weight W of silver displaces a volume V2.
From (2) it follows, by proportion, that a weight w1 of gold will displace w1/W · V1 of the fluid, and from (3) it follows that a weight w2 of silver displaces w2/W · V2 of the fluid.
Hence V = w1/W · V1 + w2/W · V2;
therefore WV = w1V1 + w2V2,
that is, (w1 + w2) V = w1V1 + w2V2,
so that w1/w2 = (V2 − V) / (V − V1),
which gives the required ratio of the weights of gold and silver contained in the crown.
The last two propositions of Book I. investigate the case of a segment of a sphere floating in a fluid when the base of the segment is