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 the solid being lighter or heavier than the fluid or equally heavy with it, or when a ratio has to be expressed, he speaks of a solid the weight of which (for an equal volume) has a certain ratio to that of the fluid.
The _editio princeps_ of the works of Archimedes with the commentaries of Eutocius was brought out by Hervagius (Herwagen) at Basel in 1544. D.
Rivault (Paris, 1615) gave the enunciations in Greek and the proofs in Latin somewhat retouched. The _Arenarius_ (_Sandreckoner_) and the _Dimensio circuli_ with Eutocius's commentary were edited with Latin translation and notes by Wallis in 1678 (Oxford). Torelli's monumental edition (Oxford, 1792) of the Greek text of the complete works and of the commentaries of Eutocius, with a new Latin translation, remained the standard text until recent years; it is now superseded by the definitive text with Latin translation of the complete works, Eutocius's commentaries, the fragments, scholia, etc., edited by Heiberg in three volumes (Teubner, Leipzig, first edition, 1880-1; second edition, including the newly discovered _Method_, etc., 1910-15).
Of translations the following may be mentioned. The Aldine edition of 1558, 4to, contains the Latin translation by Commandinus of the _Measurement of a Circle_, _On Spirals_, _Quadrature of the Parabola_, _On Conoids and Spheroids_, _The Sandreckoner_. Isaac Barrow's version was contained in _Opera Archimedis_, _Apollonii Pergoei conicorum libri_, _Theodosii Sphoerica_, _methodo novo illustrata et demonstrata_ (London, 1675). The first French version of the works was by Peyrard in two volumes (second edition, 1808). A valuable German translation, with notes, by E. Nizze, was published at Stralsund in 1824. There is a complete edition in modern notation by T. L. Heath (_The Works of Archimedes_, Cambridge, 1897, supplemented by _The Method of Archimedes_, Cambridge, 1912).