SPHERE, a regular solid'figure, every point of whose surface is equally distant from its center; and whose outline is traced by a circle revolving round its diameter. All sections of a sphere by a plane are necessarily circles, and all sections by planes passing through the center, or by planes cutting the sphere at equal distances from the center, are equal. The former sections are called great, and the latter small, circles. Small cir cles may vary In size between a mere point and a great circle, approaching either limit as nearly as we please. The surface of a sphere is equal to that of four of its great cir cles, or (taking .c for the radius of the sphere) to 4:r.e: and its volume to that of a cone whose altitude is twice that of the sphere, or 4x, and whose base is a great circle of the 4x 47r.
sphere, the formula for it being X or - The most remarkable geometrical 3 property of the sphere is the relation which its surface and volume bear to those of the " circumscribing cylinder, i.e., a cylinder length and diameter of each end are
each equal to the diameter of the sphere, and in which, therefore, the sphere will be exactly contained. The concave surface of such a cylinder is exactly equal to the sur face of the sphere; and not only so, but if a section parallel to the base of the cylinder be made through both cylinder and sphere, the curved surfaces of the portions cut off are equal, whether such portion be cut off from one end or be intercepted between two parallel sections; it follows from this that the curved surface of any section of a sphere with parallel ends is equal to the product.of the circumference of a great circle of sphere by the height or thickness of the section, and that the curved surfaces of all sec tions of a sphere are proportional to the thickness of such sections, The volume of the sphere, also, is equal to two-thirds of that of the circumscribing cylinder.