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. H H' -R- 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. 643. DEF. An element of a conical surface is the generatrix in any position. 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. 647. DEF. A circular cone is a cone whose base is a circle; the axis of a circular cone is the straight line joining the vertex and the center of the base. 648. DEF. A right circular cone is one whose axis is perpendicular to the base; an oblique circular cone is one whose axis is not perpendicular to the base. NOTE. In this book only circular cones are treated of. 649. DEF. A cone of revolution is a right circular cone, for this latter may be generated by a right triangle revolving about one of its arms as an axis. 650. DEF. The altitude of a cone is the perpendicular distance from the vertex to the plane of the base. 651. DEF. Similar cones of revolution are cones generated by the revolution of similar right triangles about homologous arms. 652. DEF. A tangent line to a cone is a line which touches the cone in one point only and does not intersect it. 653. DEF. A tangent plane to a cone is a plane which contains one element of the cone and but one, and does not intersect the cone. 654. DEF. A pyramid is inscribed in a cone when its base is inscribed in the base of the cone and its vertex coincides with the vertex of the cone. 655. A pyramid is circumscribed about a cone when its base is circumscribed about the base of the cone and its vertex coincides with the vertex of the cone. 656. DEF. A frustum of a cone is the portion included between its base and a plane parallel to the base. The lower base of the frustum is the base of the cone, and the upper base is the section made by the plane. PROPOSITION XXIX. THEOREM 657. Every section of a cone made by a plane passing through its vertex is a triangle. Hyp. ABC is a section of the cone made by plane passing through vertex A. To prove Proof. ABC a triangle. BC is a straight line. (Why ?) AB and AC are elements of the conical surface. .. AB and AC are straight lines. AB and AC are in the cutting plane. (Why?) .. they are intersections of the plane and the conical surface. .. ABC, the section, is a triangle. (Why?) Q.E.D. 658. COR. Every section of a right cone made by a plane passing through its vertex is an isosceles triangle. |