(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 segment of a paraboloid of revolution floating with its base upwards or downwards (but completely above or completely below the surface) for a number of cases differing (1) according to the relation between the length of the axis of the paraboloid and the principal parameter of the generating parabola, and (2) according to the specific gravity of the solid in relation to the fluid; where the position of rest and stability is such that the axis of the solid is not vertical, the angle at which it is inclined to the vertical is fully determined.
The idea of specific gravity appears all through, though this actual term is not used. Archimedes speaks of