PROPOSITION XXXVI. THEOREM. 663. Any section of a cone parallel to the base is to the base as the square of the altitude of the part above the section is to the square of the altitude of the cone. Α Then Let B denote the base of the cone, H its altitude, b a section of the cone parallel to the base, and Let B' denote the base of an inscribed pyramid, b' the base of the pyramid formed in the section of the cone. | B :P :: Ho : ho, $ 566 (any section of a pyramid Il to its base is to the base as the square of the I from the vertex to the plane of the section is to the square of the altitude of the pyramid). Now let the number of lateral faces of the inscribed pyramid be indefinitely increased, the new edges continually bisecting the arcs in the base of the cone. Then B' and b' approach B and b respectively as their limits. But however great the number of lateral faces of the pyramid, B': 6 :: H2 : h2. •. B:6:: Ho :ào, $ 199 Q. E. D. PROPOSITION XXXVII. THEOREM. 664. The volume of any cone is equal to the product of one-third of its base by its altitude. § 574 Let V denote the volume, B the base, and H the al titude of the cone. We are to prove V = { B X H. Let the volume of an inscribed pyramid A-C D E F G be denoted by V', and its base by B'. H will also be the altitude of this pyramid. V = { B' X H, Now, let the number of lateral faces of the inscribed pyramid be indefinitely increased, the new edges continually bisecting the arcs in the base of the cone. Then V' approaches to V as its limit, and B' to B as its limit. Vi= Į B' X H. 8 199 Q. E. D. 665. COROLLARY 1. If the cone be a cone of revolution, and R be the radius of the base, then B=k? (§ 381); .:. V=ļa Ro H. 666. Cor. 2. Similar cones of revolution are to each other as the cubes of their altitudes, or as the cubes of the radii of their bases. For, let R and R' be the radii of two similar cones of revolution, H and H' their altitudes, V and V' their volumes. Since the generating triangles are similar, we have H:H'::R: R'. PROPOSITION XXXVIII. THEOREM. 667. A frustum of any cone is equivalent to the sum of three cones whose common altitude is the altitude of the frustum and whose bases are the lower base, the upper base, and a mean proportional between the bases of the frustum. Let V denote the volume of the frustum, B its lower base, b its upper base, and H its altitude. Let V' denote the volume of an inscribed frustum of a pyramid, B' its lower base, b' its upper base. Its altitude will also be H. Then, Vi= H (B' + V + V B' X 6'), $ 578 (a frustum of any pyramid is to the sum of threc pyramids whose common altitude is the altitude of the frustum, and whose bases are the lower base, the upper base, and a mean proportional between the bases of the frustum). Now, let the number of lateral faces of the inscribed frustum be indefinitely increased, the new edges continually bisecting the arcs in the bases of the frustum of the cone. But however great the number of lateral faces of the frustum of the pyramid, V= H (B' + %! + VB' X 6'. Q. E. D. 668. COROLLARY. If the frustum be that of a cone of revolution, and R and r be the radii of its bases, we have B = 1 R”, and b = a pa, VBXb=Rr. and BOOK VIII. THE SPHERE. ON SECTIONS AND TANGENTS. 669. DEF. A Sphere is a solid bounded by a surface all points of which are equally distant from a point within called the centre. A sphere may be generated by the revolution of a semicircle about its diameter as an axis. 670. DEF. A Radius of a sphere is the distance from its centre to any point in the surface. All the radii of a sphere are equal. 671. DEF. A Diameter of a sphere is any straight line passing through the centre and having its extremities in the surface of the sphere. All the diameters of a sphere are equal, since each is equal to twice the radius. 672. DEF. A Section of a sphere is a plane figure whose boundary is the intersection of its plane with the surface of the sphere. 673. Def. A line or plane is Tangent to a sphere when it has one, and only one, point in common with the surface of the sphere. 674. Def. Two spheres are tangent to each other when their surfaces have one, and only one, point in common. 675. DEF. A polyhedron is circumscribed about a sphere when all of its faces are tangent to the sphere. In this case the sphere is inscribed in the polyhedron. 676. Def. A polyhedron is inscribed in a sphere when all of its vertices are in the surface of the sphere. In this case the sphere is circumscribed about the polyhedron. 677. DEF. A Cylinder or cone is circumscribed about a sphere when its bases and cylindrical surface, or its base and conical surface, are tangent to the sphere. In this case the sphere is inscribed in the cylinder or cone. PROPOSITION I. THEOREM. 678. Every section of a sphere made by a plane is a circle. Let the section A B C be a plane section of a sphere whose centre is 0. We are to prove section A B C a circle. From the centre 0 draw ODI to the section, and draw the radii 0 A, O B, O C, to different points in the boundary of the section. In the rt. A ODA, O D B and 0 DC, (being radii of the sphere). .. the rt. A ODA, O D B and 0 DC are equal, $ 109 (two rt. A are equal when they have a side and hypotenuse of the one equal respectively to a side and hypotenuse of the other). .:. DA, D B and D C are equal, (being homologous sides of equal A). Q. E. D. 679. COROLLARY 1. The line joining the centres of a sphere and a circle of a sphere is perpendicular to the circle. 680. COR. II. If R, r and p, respectively, denote the radius of a sphere, the radius of a circle of a sphere, and the perpendicular from the centre of the sphere to the circle, then r = V RP – p? Therefore all circles of a sphere equally distant from the centre are equal, and of two circles unequally distant from the centre of the sphere the more remote is the smaller. Again, if p=0, then r = R, and the centre of the sphere and the centre of the circle coincide ; such a section is the greatest possible circle of the sphere. |