FOURTH BOOK. DEFINITIONS. 1. Let there be a circle, and a fixed point, without its plane, and let a straight line, always passing through that point, and indefinitely extended both ways, revolve about the circle in its circumference ; the two surfaces thus described are each of them named a conic surface, the fixed point its vertex, the circle its base, and the straight line passing through the vertex and the centre its axis (first figure to proposition 1). 2. A cone is a solid bounded by a circle and a conic surface. The fixed point is called the vertex, and the circle the base of the cone. Also, the straight line passing through the fixed point, and the centre of the circle, is named the axis of the cone. 3. Let there be a circle, and any straight line intersecting its plane at the centre, and let another straight line, always parallel to the former, revolve about the circle in its circumference; the surface thus described is named a cylindric surface, of which the circle is the base, and the straight line through its centre the axis (second figure to proposition 1). 4. A cylinder is a solid bounded by two equal and parallel circles, and a cylindric surface. Either of the circles is named the base, and the straight line joining their centres the axis of the cylinder. 5. A cone or cylinder is termed right or oblique, according as the axis is perpendicular or oblique to the base. 6. The section of a cone or cylinder is termed parallel or oblique, according as its plane is parallel or inclined to the base. 7. When a cone or cylinder is cut by a plane touching the axis, and perpendicular to the base, and by another plane perpendicular to the former, in such a manner, that the common section of the two planes make angles with the common sections of the first plane and the surface, alternately equal to those which the common section of the first plane and the base makes with the same lines on the same side, the section of the second plane is called a subcontrary section (figures to Prop. 2). COROLLARIES FROM THE DEFINITIONS. Cor. 1.-A straight line joining the vertex, and any point in a conic surface, lies wholly in that surface; its continuation one way is in the same, and its continuation the other way in the opposite surface; also, every such line meets the circumference of the base. COR. 2.-Any plane touching the axis of a conic surface, cuts that surface in two straight lines, as AD, AF (fig. to Prop. 1). CoR. 3.- If a cone be cut by a plane touching the axis, or by a plane through the vertex, and any two points in the circumference of the base, the section is a tri angle, as AFD (first figure to Prop. 1). Cor. 4.-If a plane, touching a tangent to the base, pass through the vertex, it touches the conic surface in the straight line joining the vertex and the point of contact, every point in the plane, except in that straight line, being without the surface. Cor. 5.-A straight line, drawn from any point in a cy lindric surface, parallel to the axis, lies wholly in that surface. Cor. 6.—Any plane touching the axis of a cylindric surface, cuts that surface in two parallel straight lines. PROPOSITION I. If a conic or cylindric surface be cut by a plane parallel to the base, their line of common section is the circumference of a circle, having its centre in the axis. 1. Let ABDCA be a conic surface, of which BCD is the base, and AE the axis, and let it be cut by the plane GOL, parallel to BCD, the line of common section GKHL is the circumference of a circle having its centre in AE. For, let any two planes, ABC, AFD, touching the axis AE, cut the surfaces in the straight lines AB, AC, AF, IN AD (Cor. 2 to Def.); the base BCD, in the diameters BEC, DEF; and the parallel section GOL, in the straight lines GOH, KOL. Then BC is parallel to GH, and FD to KL. Hence, by similar triangles, ED:OK (= AE: AO) = EC:0H. But ED = EC, therefore OK = OH. Consequently all straight lines drawn from the point 0, where the axis meets the parallel plane GOL, to terminate in the line of common section GKHL, are equal to one another, and GKHL is the circumference of a circle, of which O is be the centre. 2. Let MBDCN be a cylindric surface, of which BCD is the base, and AE the axis, and let it be cut by the plane GOL, parallel to BCD, the line of common section GKHL is the circumference of a mi circle, having its centre in AE. For, let any two planes, ABC, AFD, ak touching the axis AĒ, cut the surface in the straight lines MB, NC; LF, KD (Cor. 6 to Def.), the base BCD, in the diameters BEC, DEF, and the parallel section GOL, in the straight lines GOH, KOL. Then BC is parallel to GH, and se FD to KL. But NC, KD, are each parallel to AE. Therefore OD, OC, are parallelograms, and OK = ED = EC = OH. Whence GKHL is the circumference of a circle, of which O is the centre. Cor. 1.-If a conic surface be cut by a plane, parallel to the base, the solid betwixt that plane and the vertex is a cone. Cor. 2.-If a cylindric surface be cut by two planes pa rallel to the base, the solid betwixt them, and the solids between each of them and the base, are cylinders. Cor. 3.-If a cone or cylinder be cut by a plane parallel to the base, the section is a circle. COR. 4.-Any plane touching the axis of a cone or cy linder, cuts every parallel section in its diameter. HA PROPOSITION II. Every subcontrary section of an oblique cone or cylinder is a circle. Let the cone or cylinder MBCN be cut by the plane MC, perpendicular to the base, touching the axis, and meeting the surface in the straight lines MB, M/ N NC, and the base in the diameter BC. Let it be cut by another plane DFG, perpendicular to the former, so that their B{_----line of common sec ------- BA tion DF may form, with one of the lines MB, NC, the angle NFD, equal to the angle MBC, which BC forms with the other on the same side. The latter section DGF, which is called a subcontrary section, is a circle, and DF its diameter. For, let the parallel section HGI pass through any point K in DF. Because HGI is parallel to the base, it is perpendicular to MC, and therefore GK (So. Ge. I. 18) its line of common section, with DGF, is perpendicular to the same. Thus, GK is at right angles to DF and HI, and since HI is a diameter of the parallel section (IV.1, Cor. 4), the rectangle HK · KI = GK2. But the triangles HDK, IKF, being similar (for the angles at K are equal, and the angle NFD = MBC = DHK), DK :HK = KI:KF, and the rectangle DK · KF=HK KI. Consequently the rectangle DK · KF = GK, and the section DGF a circle, having DF for a diameter. Cor.—Every subcontrary section of an oblique cylinder is equal to its base. For the triangles HDK, IKF, are isosceles. PROPOSITION III. Every section of a cone or cylinder, by a plane meeting the conic or cylindric surface on every side, that is neither a parallel nor a subcontrary section, is an ellipse. 19 Let EHFG be a section of a cone or cylinder MNP, by a plane meeting the surface on every side, but neither a parallel nor a subcontrary section, EHFG is an ellipse. For, let C be the middle of EF, and K any other point in it'; Ne P N ---through C and K let planes pass parallel to the base cutting MNP in AB, OR, and EHFG in HG and LD. Also, let the plane MNP cut the base in a diameter NP perpendicular to the line of common section of the plane of the base and of the section EHG. Then HG is parallel to that line of common section (So. Ge. I. 14), and AB to NP; therefore AB is also perpendicular to HG (So. Ge. I. 9). For a similar reason, OR is perpendicular to LD. Now (IV. 1), the sections AHBG, OLRD, are circles, of which AB, OR, are diameters, and therefore AC · CB = CG”, and OK · KR = DK2. By similar triangles EC: EK = AC:OK. CF:KF=CB: KR. EC2: EK · KF= CG2: DK2. Consequently EHFG is an ellipse, of which EF and GH are two conjugate diameters. PROPOSITION IV. If a cone be cut by a plane parallel to another, touching the conic surface, the section is a parabola. Through any point B, in the circumference of the base BGC, let the tangent DE be drawn, the plane ADE, touching that line, and the vertex A, touches the conic surface in the straight line AB (Cor. 4 to Def.). Let the cone be cut by the plane VGN, parallel to ADE, and meeting the base in GN ; the section GVN is a parabola. For, let ABC, the plane of AB and the axis, intersect GVN, in the line VK, and the base in the diameter |