PROPOSITION XXV. THEOREM 632. The bases of a cylinder are equal. Hyp. ABE and DCG are the bases of the cylinder BG. To prove Base ABE = base DCG. Proof. A, B, E, are any three points in lower base of cylinder, AD, BC, and EG are elements. Draw AE, AB, BE, DG, DC, and CG. .. A ABE = A DGC. Place the lower base on the upper base so that the equal A coincide. Then the bases will coincide, because the points A, B, and E are any three points in the perimeter of the lower base, and therefore every point in the perimeter of the lower base will fall in the perimeter of the upper base. :: the bases are equal. Q.E.D. 633. Cor. 1. Any two parallel sections cutting all the elements of a cylinder are equal. Hint. — (What is the solid included between these sections ?) 634. CoR. 2. Any section of a cylinder parallel to the base is equal to the base. PROPOSITION XXVI. THEOREM 635. The lateral area of a circular cylinder is equal to the product of the perimeter of a right section of the cylinder by an element. Hyp. S is the lateral area, P the perimeter of a right section, and E an element of the cylinder AK; S' is the lateral area, P' the perimeter of a section of a prism with a regular polygon as base, inscribed in cylinder AK. To prove S= P x E. Proof. The edge of the inscribed prism coincides with an element of the cylinder. (Why?) :: S' = P'x E. (547) If the number of faces of the inscribed prism be increased, S' will approach S as a limit, and P' will approach P as a limit. But S'= P x E. .:S=PX E. (213) 636. Cor. 1. The lateral area of a cylinder of revolution is the product of the circumference of its base by its altitude. 637. Cor. 2. If S denote the lateral area, T the total area, H the altitude, and R the radius of a cylinder of revolution, S=2 RH, and PROPOSITION XXVII. THEOREM 638. The volume of a circular cylinder is equal to the product of its base by its altitude. Hyp. V the volume, B the base, and H the altitude, of cylinder AK; V' the volume, B' the regular polygon forming the base of a prism inscribed in AK. To prove V=B x H. 639. COR. For a cylinder of revolution with radius of base V= R x H. Ex. 1129. Two cylinders of revolution have equal altitudes and their radii are respectively 3 and 4. Find a third cylinder of revolution of the same altitude and equivalent to the sum of the two given cylinders. PROPOSITION XXVIII. THEOREM 640. The lateral areas, or the total areas, of similar cylinders of revolution are to each other as the squares of their radii, or as the squares of their altitudes; and their volumes are to each other as the cubes of their radii, or as the cubes of their altitudes. Hyp. S, S', are the lateral areas; T, T', the total areas, V, V', the volumes, R, R', the radii, and H, H', the altitudes of two similar cylinders of revolution. CONES 641. DEF. A conical surface is a surface generated by a moving straight line which continually intersects a given fixed curve and constantly passes through a fixed point not in the same plane with the curve. 642. DEF. The generatrix of the surface is the moving straight line; the directrix is the given curve; and the vertex is the fixed point. 644. DEF. The upper and lower nappes are the portions of the conical surface formed above and below the vertex when the length of the generatrix is unlimited. 645. DEF. A cone is a solid bounded by a conical surface and a plane cutting all its elements. 646. DEF. The lateral area of the cone is the conical surface; the base of the cone is the plane surface; the vertex of the cone is the vertex of the conical surface; and the elements of the cone are the elements of the conical surface. |