XVI. On the mode of constructing the Three Conic XVII. On the analogous properties of the Normal, in all the Conic Sections CHAP. VII. 93 102 On the method of finding the Dimensions of Conic Sections whose Latera-recta are given, and of describing such as shall pass through certain given points. XVIII. On the method of finding the dimensions of Conic XIX. On the method of describing Conic Sections which CHAP. VIII. 110 116 On the Quadrature of the Conic Sections. XX. On the relation which obtains between the areas of Conic Sections of the same kind, having the same vertex and axis; and on the Quadrature of the Parabola, Ellipse, and Hyperbola, 123 XXI. On the Quadrature of the PARABOLA, according to the method of the Ancients 132 ERRATA. Page 17. In Art. 24, of this page, Cor. should have immediately pre 123. In this Figure, the lines CQ, CP, drawn from the centre of the Hyperbola, should not have cut the Curve. CONIC SECTIONS. CHAP. I. INTRODUCTION. A CONE is a solid figure formed by the revolution of a right-angled triangle about one of its sides (Euc. B. XI. Def. 18.). From the manner in which this solid is generated, it is evident that if it be cut by a plane parallel to its base, the intersection of the plane with its surface will be a circle; and if it be cut by a plane passing through its vertex, the intersection will be a triangle. If the plane by which the cone is cut be not parallel to the base, or does not pass through the vertex, then the line traced out upon its surface will be one of those curves more particularly distinguished by the name of CONIC SECTIONS, the properties B properties of which are to be made the subject of the following Treatise. I. (1.) Let BEFGp be a cone, and let it be cut by a plane BEnG perpendicular to its base and passing through its vertex; then the section BEG will be a triangle. Next, let it be cut by a plane pДon at right angles to the plane BEn G, and parallel to a plane touching the side BE of the cone; then the curve line pPA00, which is formed by the intersection of this latter plane with the surface of B D G tion CPDO of this plane with the surface of the cone will be a circle. Since the plane BEn G divides the cone into two equal parts, CD (the common intersection of the planes BEnG, CPDON) will be a diameter of that circle; and for the same reason EG will be a diameter of the circle Ep Go F. Let ANn be the common intersection of |