and S', and let the surface of the prism be P. Hence P is greater than S, and therefore the perimeter of the base of P is greater than that of S', or than that of the given cylinder ; that is, the perimeter of a polygon inscribed in a circle is greater than the circle itself, which is absurd. 2o. Let S be less than S'; by reasoning precisely similar to that used in Prop. II. we may prove, that if this were the case, the perimeter of a polygon circumscribed round a circle is less than that of the circle itself. Hence it follows that S and S' are equal. PROPOSITION IV. (190) The cylindrical surface of a cylinder is equal to the rectangle under its altitude and the cir cumference of its base. For it is equal to a prism with an isoperimetrical base, (189). (191) Cor. 1.-A cylindrical surface is represented numerically by the product of its altitude into the circumference of its base. (192) Cor. 2.-Let a and r represent the altitude and radius, as in (182). Then S = 27 r a. (193) Cor. 3.-Since the circumferences of circles are as their diameters, cylindrical surfaces are as the rectangles under their altitudes, and the diameters of their bases. (194) Cor. 4.-Cylindrical surfaces with equal bases are as their altitudes, and with equal altitudes are as the circumferences of their bases. (195) Cor. 5.—The surfaces of similar cylinders are as the squares of their diameters or altitudes. (196) Cor. 6.-In equal cylindrical surfaces the diameters and altitudes are reciprocally proportional, and vice versa. (197) DEF.—A cone is a solid produced by the revolution of the hypotenuse of a right angled triangle about one of the sides as a fixed axis. Let PB be the hypotenuse, and P A the fixed axis. The side A B moving in a plane at right angles to P A describes in that plane a circle BMN. This circle is called the base of the cone. A convex surface, called the conical surface, is described by the motion of the hypotenuse; and as this line, in all its successive positions, must be entirely in the conical surface, it follows that every right line drawn from P to a point in the circumference of the base must be entirely in the conical surface. В M The point P is called the verter, and every right line drawn from that point to the circumference of the base is called a side of the cone. (198) Der.—The line P A is called the axis of the cone. (199) Cor. 1.-It is evident, therefore, that if any plane pass through the vertex and intersect the conical surfaces, the intersections will be two sides of the cone; and as the intersection with the base will be a chord of the circle, the whole intersection will be a triangle MPN. (200) Cor. 2.-What has been observed of the base A B of the generating triangle, is also applicable to any perpendicular G H to the axis terminated in the hypotenuse of the generating triangle. Such a line GH moves in a plane perpendicular to PA, and its extremity H describes a circle. Hence it follows, that every section of a cone by a plane parallel to the base is a circle, the centre of which is in the axis, and the radius GH of which is to that A B of the base, as their distances from the vertex, that is, as PG: PA. PROPOSITION V. (201) ' If a plane PA B be drawn through the axes of a cone intersecting the conical surface in the side PB, and through P B another plane CBPD be drawn at right angles to the former, this plane will lie entirely outside the conical surface, except in the line P B, in which it meets it. For let any other plane PAH be drawn through the axis PA and intersecting the plane CBPD. In the right angled triangle AB H the hypotenuse AH is greater than the side A B or the radius of the base, and therefore the point H lies outside the base. The same may be proved of every section parallel to the base, and therefore it follows that the plane CB P D meets the conical surface only in PB, E lying elsewhere wholly outside it. (202) Dec.-Such a line is called a tangent plane to the conical surface, which it touches in the line PB. (203) Cor. 1.-Hence all tangent planes pass through the vertex, and the lines of contact are sides of the cone. B H the cone. (204) If the circumference of the base be divided at three or more points, and lines be drawn from these to the vertex P, as also lines joining the points of division so as to form a polygon inscribed in the base, these lines will form the edges of a pyramid whose base is the polygon inscribed in the base, and whose sides are the lines drawn to the vertex, and all of which are sides of Such a pyramid is said to be inscribed in the cone. (205) If several tangent planes be drawn to the same cone intersecting each other, and also the plane of the base of the cone produced, their intersections with the plane of the base will be tangents to the base itself, and the planes may be so disposed that these tangents shall form a polygon circumscribing the base. This polygon will be the base of a pyramid whose lateral faces are the tangent planes. Such a pyramid is said to circumscribe the cone. (206) It is evident that both the volume and surface of the cone are greater than those of any inscribed, and less than those of any circumscribed pyramid. This observation is applicable to the surfaces whether the bases be parts of them or not. (207) It is also evident, that the number of sides of the bases of the inscribed or circumscribed pyramid may be increased until the difference between its volume or surface and that of the cone shall be less than any given magnitude. PROPOSITION VI. (208) If a cone and pyramid have equal bases and equal altitudes they will have equal volumes. This proposition is proved in exactly the same manner as (180). In fact, the same words may be used here, changing cylinder into cone, and prism into pyramid. (209) Cor. 1.—Hence the volume of a cone is expressed numerically by one-third of the product of the base and altitude. (210) Cor. 2.-A cone is one-third of a cylinder on the same base and in the same altitude. (211) Cor. 3.-If a be the altitude, and r the radius of the base, 17 p? a is the volume (182). (212) Cor. 4.-The volumes of cones are as the rectangle under their altitudes, and the diameters of their bases. (213) DEF.-Similar cones are those whose axes are as the radii of their bases. (214) Cor. 5.-The volumes of similar cones are as the cubes of their altitudes or diameters. (215) Cor. 6.-Cones with equal bases are as their altitudes, and with equal altitudes are as their bases. (216) COR. 7. If the volumes of cones be equal, their bases and altitudes are reciprocally proportional, and vice versá. Also if the volumes be equal, their altitudes and the squares of the diameters of their bases are reciprocally proportional, and vice versá. PROPOSITION VII. (217) The surface of a circumscribed pyramid, ex clusive of its base, is equal to half of the rectangle under the side of the cone and the perimeter of the base. For let PB (fig. Art. (201)) be the line of contact of one of the triangular faces, and let E C be the corresponding side of the base, and P A the axis of the cone. The plane D A B is perpendicular to the plane of the base, and also to the plane DBC. Hence the intersection E C of these planes is perpendicular to the plane P A B, and therefore perpendicular to PB. Hence the area of the triangle E PC is equal to half of the rectangle under the side PB of the cone, and the side E C of the polygonal base of the pyramid. The same being true for every triangular face of the pyramid, it follows that the sum of its triangular faces is equal to half the rectangle under the side of the cone, and the perimeter of the base of the pyramid. (218) CoR.—Hence the surfaces of circumscribed pyramids are as the perimeters of their bases. PROPOSITION VIII. (219) A conical surface is equal to half of the rectan gle under the side of the cone and the circumference of its base. If the conical surface be not equal to half this rectangle, let any other conical surface having the same vertex and axis and its base on the same plane be equal to it. The base of this other cone being concentrical with that of the given one, must be either contained within the base of the given cone, or must contain the base of the given cone within it. First. Suppose that it is contained within the base of the given cone. Let the surface of the given cone be S, and let the side of the given cone be s, and the circumference of its base c. cone. The surface of the lesser cone will then be į s x c. Let a polygon be circumscribed round the base of the lesser cone, so as to be contained within the base of the greater. If this polygon be the base of a pyramid circumscribing the lesser cone, its surface will be 18 x c, d being the circumference of its base, and s' the side of the lesser cone; and this surface will be greater than that of the lesser cone, and less than that of the greater; that is, $' x ' is greater than is x c, and less than S. But s' the side of the lesser cone is less than s, the side of the greater, and d the perimeter of the included polygon is less than c the circumference of the circle which includes it. Therefore į s' x d' is less than s Xc; but it was already proved to be greater than it, which is absurd. Therefore the base of the cone whose surface is equal to 1s x c is not contained within that of the given Secondly. Let it contain the base of the given cone within it. Let a polygon be circumscribed round the base of the given cone, so as to be included within the greater base. As before, let this polygon be the base of a pyramid circumscribing the given cone, and let the circumference of the base of this pyramid be c', its surface will then be is x c', and that of the greater cone which includes it, and is therefore greater than it, is j's X c. But d' the perimeter of the polygon is greater than c the circumference of the circle which it circumscribes; and therefore s x c'is greater thans x c, the contrary of which has just been proved. Hence the base of the cone whose surface is equal to jis x c can neither be within nor without that of the given cone, and therefore must coincide with it. The surface of the cone is therefore equal to half the rectangle under its side, and the circumference of its base. (220) Cor. 1.-The surface of a cone is represented numerically by half the product of its side, and the circumference of its base. (221) Cor. 2.—The surface of a cone is half that of a cylinder on the same base and with an equal side. (222) Cor. 3.-If a be the side, and r the radius of the base, the surface = Tr a. (223) Cor. 4.-Conical surfaces are as the rectangles under their sides and the diameters of their bases. (224) Cor. 5.—Conical surfaces with equal sides are as the diameters of their bases, and those with equal diameters are as their sides. (225) Cor. 6.—Similar conical surfaces are as the squares of their sides, or the diameters of their bases. (226) Cor. 7.-Equal conical surfaces reciprocate their sides and diameters. |