Archimedes

of his body, he thought of applying the same idea to the case of the crown and measuring the volumes of water displaced respectively (1) by the crown itself, (2) by the same weight of pure gold, and (3) by the same weight of pure silver. This gives an easy means of solution. Suppose that the weight of the crown is W, and that it contains weights w1 and w2, of gold and silver respectively. Now experiment shows (1) that the crown itself 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 (1) entirely above and (2) entirely below the surface of the fluid; and it is shown that the segment will in either case be in equilibrium in the position in which the axis is vertical, the equilibrium being in the first case stable.

Book II. is a geometrical tour de force. Here, by the methods of pure geometry, Archimedes investigates the positions of rest and stability of a right