EXERCISE. How many edges has each of the five regular polyhedrons? COR. The number of face angles in any polyhedron is twice as great as the number of edges; for each face has the same number of sides and angles, and two sides form but one edge. Hence the number of face angles in a polyhedron is even; the number of triangular, pentagonal, heptagonal, &c., faces, must be even; and the number of solid angles formed by 3, 5, 7, &c., plane angles, must be even. XI.* Theorem. The sum of all the face angles of any polyhedron is equal to four right angles, taken as many times as the polyhedron has vertices, less two. TO BE PROVED. The sum of the face angles S = 4R (v-2). PROOF. Let A1, A2, N1, N2, N3, then A1 = . be the sums of the angles, and the number of sides, of the faces respectively; 2Rn1 4R, A2 = 2Rn2 4R, A3 = 2Rn3 - 4R Adding these equations, member to member, we have A1 + A2 + A3 + . . . = S = 2R (n1 + N2 + (I., 35, Cor. 1). BOOK VII. THE THREE ROUND BODIES. THE CYLINDER. I. DEF. 1. The locus of the parallel lines drawn from all the points in the circumference of a circle to a plane parallel to that circle is a cylindrical surface; * and the solid enclosed is a cylinder. DEF. 2. Any one of the parallel lines is called an element; as, Aa, Bb. m с COR. 1. All the elements, Aɑ, Bb, . are equal (V., 22). A M /B COR. 2. The locus of all the intersec tions of the elements and the plane is the circumference of a circle equal to the given circle. For from the centre, M, draw Mm || Aa; then Mm Aa, and AMma is a parallelogram: hence MA = ma. For the same reason, the dis = tance from m to the extremity, b, of any element, Bb, equals the radius MB: hence abc is a circle drawn from m as a centre, with a radius equal to AM. DEF. 3. The two circles are called the bases of the cylinder; the line Mm, joining their centres, the axis; the perpendicular distance between the bases, the altitude; and the locus of the elements, the lateral surface. COR. 3. Every section of a cylinder made by a plane paral * A cylindrical surface in its most extended sense is the locus of the parallel lines drawn through all the points of any curve. lel to the bases is a circle equal to the bases; for it is the base of a new cylinder cut off by this plane. M m COR. 4. Every section of a cylinder made by a plane passing through an element is a parallelogram; for a plane passing through an element, Aa, will cut the circumference of the base in a second point, B. Through B draw Bb || Aa; then Bb is in the plane AaB, and, by the definition, is an element of the cylinder : Aa, and AabB is a parallelogram (I., 39). DEF. 4. If the plane AabB be turned about Aa, the intersection Bb will at length fall upon Aɑ. In this position, the plane has but one line common with the cylinder, and is called a tangent plane. hence Bb B DEF. 5. A right cylinder is one whose elements are perpendicular to its base. It may be generated by the revolution of a rectangle about one of its sides. This side is the axis of the cylinder. The opposite side generates the curved surface; and the other two sides, the bases. SCHOLIUM. A cylinder may be considered a prism of an infinite number of sides (IV., 10). THE CONE. II. DEF. 1. The locus of all the lines drawn through a point, and the circumference of a circle not in the same plane, is a conical surface; and the solid enclosed A is a cone. DEF. 2. The lines are called the elements of the cone; the point, the vertex; the circle, the base; the perpendicular distance of the vertex from the base, the altitude; the line drawn from the vertex to the middle of the base, the axis; and the locus of the elements, the lateral surface. M DEF. 3. A right cone is a cone whose axis is perpendicular to its base. It may be generated by the revolution of a right-angled triangle, ADM, about one of the perpendicular sides, MD. SCHOLIUM. A cone may be considered a pyramid of an infinite number of sides (IV., 10). III. A B Theorem. Every section of a cone made by a plane passing through the vertex is a triangle. For let a plane pass through the vertex, M, cutting the base in AB. Join MA and MB; then MA and MB are elements of the cone, and lie in the plane, since they have each two points in common with the plane hence they are the intersection of the plane and lateral surface; and MAB is a triangle. B M C DEF. If the plane MAB be turned about MA, the intersection, MB, will fall upon MA. In this position, the plane has but one line, MA, in common with the cone; and is called a tangent plane. EXERCISE. Prove that the triangular section passing through the axis of an oblique cone, and perpendicular to the base, has the same altitude as the cone, also the greatest and least elements for its sides.. IV. Theorem. Every section of a cone made by a plane parallel to the base is a circle. HYPOTH. In the cone M-ABC, the plane section, abc, is parallel to the base ABC. TO BE PROVED. abc is a circle. A PROOF. Let MD be the axis, cutting abc in d. B M planes intersecting ABC in AD, BD, . . . and abe in ad, bd, . ... Then ad | AD, bd. || BD (V., 19), and AD MD BD MD = bd Md (III., 21): hence is a circle described from d as a centre with a radius ad. COR. The axis of a cone passes through the centre of all the circular sections parallel to the base. DEF. A truncated cone is the portion of a cone included between the base and a plane cutting the cone. When the cutting plane is parallel to the base, the truncated cone is called a frustum of a cone. The altitude of a frustum is the perpendicular distance between its bases. THE SPHERE.† V. DEF. The locus of all the points that are equally distant from a given point is a spherical surface; and the solid enclosed A by that surface is a sphere. The given point is B the centre of the sphere; the line drawn from the centre to the surface, the radius; and the line drawn through the centre, and terminated both ways, the diameter. A sphere may be generated by the revolution C of a semicircle, ABC, about its diameter, AC. †The recitation-room should be furnished with a spherical blackboard, on which the student should draw the diagrams of spherical surfaces. |