SPHERE, properties of the. 1. All spheres are to one another as the cubes of their diameters. 2. The surface of a sphere is equal to four times the area of one of its great circles, as is demonstrated by Archimedes in his book of the Sphere and Cylinder, lib. i. prop. 37. Hence, to find the superficies of any sphere, we have this easy rule; let the area of a great circle be multiplied by 4, and the pro duct will be the superficies : or, accord ing to Euclid, lib. VI. prop. 20. and lib. xii. prop. 2. the area of a given sphere, CEBD (fig. 3.) is equal to that of a circle, whose radius is equal to the diameter of the sphere BC. Therefore, having mea sured the circle described with the radius BC, this will give the surface of the sphere. 3. The solidity of a sphere is e9ual to the surface multiplied into one third of the radius : or, a sphere is equal to two thirds of its circumscribing cylin der, having its base equal to a great circle of the sphere. Let ABEC (fig. 4 and 5.) be the quadrant of a circle, and ABDC the circumscribed square, equal twice the triangle ADC: by the revolution of the figures about the right line AC, as an axis, a hemisphere will be generated by the quadrant, a cylinder of the same base and height of the square, and a cone by the triangle : let these three be cut any how by the plane HF, parallel to the base AB ; and the section of the cylinder will be a circle, whose radius is PH ; in the hemisphere, a circle whose radius is FE ; and in the cone, a circle of the radius FG. But EA' (=HP) = EF' + FA' :
but AF' = FG', because AC = CD ; and therefore HP' =EF' VG' ; or' the cir cle of the radius HF is equal to a circle of the radius EF, together with a circle of the radius GF : and since this is true every where, all the circles together de scribed by the respective radii HF, that is, the cylinder, are equal to all the circles described by the respective radii EF and FG, that is, to the hemisphere and cone taken together. But by Euclid. lib. xii. prop. 10. the cone generated by the tri angle DAC, is one third part of the cylin der generated by the square BC ; whence it follows, that the hemisphere generated by the rotation of the quadrant ABEC, is equal to the remaining two-thirds of the cylinder, and that the whole sphere is two-thirds of the cylinder circumscrib ed about it. Hence it follows, that a sphere is equal to a cone whose height is equal to the semi-diameter of the sphere, and its base equal to the super fices of the sphere, or to the area of four great circles of the sphere, or to that of a circle, whose radius is equal to the dia meter of the sphere.