The last-mentioned theorem naturally connects itself with the story of the crown made for Hieron. It was suspected that this was not wholly of gold but contained an admixture of silver, and Hieron put to Archimedes the problem of determining the proportions in which the metals were mixed. It was the discovery of the solution of this problem when in the bath that made Archimedes run home naked, shouting [Greek: eureka, eureka]. One account of the solution makes Archimedes use the proposition last quoted; but on the whole it seems more likely that the actual discovery was made by a more elementary method described by Vitruvius. Observing, as he is said to have done, that, if he stepped into the bath when it was full, a volume of water was spilt equal to the volume 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 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.