Dem.—Let AB be the diameter of the semicircle which describes the sphere. Take two points, E, F, indefinitely near each other in the semicircle. Join EF, and produce to meet the tangent CD parallel to AB in N. Draw EI, FK parallel to PQ. Produce EI to meet AB in G. Let O be the centre. Join OE. Now we have FE : KI :: EN : IN [VI. ii.]; but EN : IN :: OE : EG, because the triangles ENI and OEG are similar. Hence FE : KI :: OE : EG; but OE = IG. Hence EF : IK :: IG : EG; and IG : EG :: circumference of circle described by the point I : circumference of circle described by the point E. Hence the rectangle contained by EF, and circumference of circle described by E is equal to the rectangle contained by IK, and circumference of circle described by I—that is, the portion of the spherical surface described by EF is equal to the portion of the cylindrical surface described by IK. Hence it is evident, if planes be drawn perpendicular to the diameter AB—that the portions of cylindrical and spherical surface between any two of them are equal. Hence the whole spherical surface is equal to the cylindrical surface described by CD. Or thus: Conceive the whole surface of the sphere divided into an indefinitely great number of equal parts, then it is evident that each of these may be regarded as the base of a pyramid having the centre of the sphere as a common vertex. Therefore the volume of the sphere is equal to the whole area of the surface multiplied by one-third of the radius. Hence if S denote the surface, we have S × = [vi., Cor. 6]; therefore S = 4πr2. That is, the area of the surface of a sphere is equal four times the area of one of its great circles. Exercises. 1. The convex surface of a cone is equal to half the rectangle contained by the circumference of the base and the slant height. 2. The convex surface of a right cylinder is equal to the rectangle contained by the circumference of the base and the altitude. 3. If P be a point in the base ABC of a triangular pyramid O–ABC, and if parallels to the edges OA, OB, OC, through P, meet the faces in the points a, b, c, the sum of the ratios 4. The volume of the frustum of a cone, made by a plane parallel to the base, is equal to the sum of the three cones whose bases are the two ends of the frustum, and the circle whose diameter is a mean proportional between the end diameters, and whose common altitude is equal to one-third of the altitude of the frustum. 5. If a point P be joined to the angular points A, B, C, D of a tetrahedron, and the joining lines, produced if necessary, meet the opposite faces in a, b, c, d, the sum of the ratios 6. The surface of a sphere is equal to the rectangle by its diameter, and the circumference of a great circle. 7. The surface of a sphere is two thirds of the whole surface of its circumscribed cylinder.