Therefore AC passes through O and is a diameter of the lower base. Hence A0=BP. $170 8 158 Also AO was proved parallel to BP. Hence the figure ABPO is a parallelogram. $126 $117 Q. E. D. 776. COR. I. The axis of a circular cylinder passes through the centres of all sections parallel to its base. 777. COR. II. A right circular cylinder may be generated by the revolution of a rectangle about one of its sides as an axis. 778. Defs. For this reason a right circular cylinder is also called a cylinder of revolution. The radius of the base of a cylinder of revolution is called the radius of the cylinder. 9. Def.-A plane is tangent to a cylinder when it rough an element and meets its surface nowhere 780. A plane passing through a tangent to the base of a cylinder and the element drawn at the point of contact is tangent to the cylinder. GIVEN the cylinder ST, the tangent AD to its base, and the element AB drawn through the point of contact. TO PROVE that the plane CM, passing through AD and AB, is tangent to the cylinder. If the plane should meet the surface of the cylinder in any point X, not in AB, draw the element SY passing through X. Then SY would lie in the plane CM. § 526 II, IV Therefore AD would meet the curve AT in two points, A and Y. This cannot be, since AD is tangent to the base. Hence the plane CM does not meet the surface of the cylinder except in AB. It is therefore tangent to the cylinder. 8 779 Q. E. D. 781. COR. I. Through a given element, one and only one plane tangent to the cylinder can be drawn. 782. COR. II. If a plane is tangent to a cylinder, its intersection with the plane of the base is tangent to the base. 783. COR. III. The intersection of two planes tangent to a cylinder is parallel to the elements. 784. Exercise.-Show how to draw a plane through a given point tangent to a cylinder. THE CONE 785. 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 given fixed point. Thus, if the straight line OB passes through the point 0 and moves so as continually to intersect the curve CD, the surface generated - CBD is a conical surface. 786. Defs.-The moving line is called the generatrix; the fixed curve the directrix; the fixed point the vertex. Any straight line in the surface, as OA, representing one position of the generatrix, is called an element of the surface. 787. Remark.-If the generatrix is of indefinite length, as BOA, the conical surface consists of two symmetrical parts, each of indefinite extent, lying on opposite sides of the vertex, as O-CBD and O-GAF. The directrix may be any curve whatever. But for the student who has not studied the appendix the proofs are rigorous only when the directrix is considered to be the circumference of a circle. 788. Defs.-A cone is a solid bounded by a closed conical surface and a plane. The conical surface is called the lateral surface and the section made by the plane the base of the cone. The vertex of the conical surface is called the vertex of the cone, and the elements of the conical surface are also called elements of the cone. PROPOSITION V. THEOREM 789. Every section of a cone made by a plane passing through its vertex and cutting its base is a triangle. GIVEN the cone O-CABD, whose base is cut in the line AB by a plane passed through 0. TO PROVE the section made by this plane is a triangle. The intersection AB is a straight line. $528 We therefore need only to prove that the intersections. OA and OB are straight. Draw straight lines from 0 to A and B. These straight lines lie in both the cutting plane and the conical surface. $$ 524, 785 Therefore they form the intersections of this plane and the conical surface. Hence the section made by the plane OAB is a triangle. Q. E. D. 790. Defs.-A cone whose base is a circle is called a circular cone. The straight line joining the vertex of a circular cone to the centre of its base is the axis of the cone. |