It is used to determine the density of a body experimentally; for if W is the weight of a body weighed in a balance in air (strictly _in vacuo_), and if W´ is the weight required to balance when the body is suspended in water, then the upward thrust of the liquid or weight of liquid displaced is W - W´, so that the _specific gravity_ (S.G.), defined as the ratio of the weight of a body to the weight of an equal volume of water, is W/(W - W´). As stated first by Archimedes, the principle asserts the obvious fact that a body displaces its own volume of water; and he utilized it in the problem of the determination of the adulteration of the crown of Hiero. He weighed out a lump of gold and of silver of the same weight as the crown; and, immersing the three in succession in water, he found they spilt over measures of water in the ratio 1/14 : 4/77 : 2/21 or 33 : 24 : 44; thence it follows that the gold : silver alloy of the crown was as 11 : 9 by weight. 13. _Theorem._--The resultant vertical thrust on any portion of a curved surface exposed to the pressure of a fluid at rest under gravity is the weight of fluid cut out by vertical lines drawn round the boundary of the curved surface. _Theorem._--The resultant horizontal thrust in any direction is obtained by drawing parallel horizontal lines round the boundary, and intersecting a plane perpendicular to their direction in a plane curve; and then investigating the thrust on this plane area, which will be the same as on the curved surface. The proof of these theorems proceeds as before, employing the normality principle; they are required, for instance, in the determination of the liquid thrust on any portion of the bottom of a ship. In casting a thin hollow object like a bell, it will be seen that the resultant upward thrust on the mould may be many times greater than the weight of metal; many a curious experiment has been devised to illustrate this property and classed as a hydrostatic paradox (Boyle, _Hydrostatical Paradoxes_, 1666). [Illustration: FIG. 2.] Consider, for instance, the operation of casting a hemispherical bell, in fig. 2. As the molten metal is run in, the upward thrust on the outside mould, when the level has reached PP´, is the weight of metal in the volume generated by the revolution of APQ; and this, by a theorem of Archimedes, has the same volume as the cone ORR´, or 1/3 [pi]y³, where y is the depth of metal, the horizontal sections being equal so long as y is less than the radius of the outside hemisphere. Afterwards, when the metal has risen above B, to the level KK´, the additional thrust is the weight of the cylinder of diameter KK´ and height BH. The upward thrust is the same, however thin the metal may be in the interspace between the outer mould and the core inside; and this was formerly considered paradoxical.