5. The ICOSAEDRON-a polyedron bounded by twenty equal equilateral triangles. In the Tetraedron, the triangles are grouped about the polyedral angles in sets of three, in the Octaedron they are grouped in sets of four, and in the Icosaedron they are grouped in sets of five. Now, a greater number of equilateral triangles cannot be grouped so as to form a salient polyedral angle; for, if they could, the sum of the plane angles formed by the edges would be equal to, or greater than, four right angles, which is impossible (B. VI., P. XX.). In the Hexaedron, the squares are grouped about the polyedral angles in sets of three. Now, a greater number of squares cannot be grouped so as to form a salient polyedral angle; for the same reason as before. In the Dodecaedron, the regular pentagons are grouped about the polyedral angles in sets of three, and for the same reason as before, they cannot be grouped in any greater number, so as to form a salient polyedral angle. 80 Furthermore, no other regular polygons can be grouped as to form a salient polyedral angle; therefore, Only five regular polyedrons can be formed. 14 BOOK VIII. THE CYLINDER, THE CONE, AND THE SPHERE. DEFINITIONS. 1. A CYLINDER is a volume which may be generated by a rectangle revolving about one of its sides as an axis. E P A H Thus, if the rectangle ABCD be turned about the side AB, as an axis, it will generate the cylinder FGCQ-P. The fixed line AB is called the axis of the cylinder; the curved surface generated by the side CD, opposite the axis, is called the convex surface of the cylinder; the equal circles FGCQ, and EHDP, generated by the remaining sides BC and AD, are called bases of the cylinder; and the perpendicular distance between the planes of the bases, is called the altitude of the cylinder. M The line DC, which generates the convex surface, is, in any position, called an element of the surface; the elements are all perpendicular to the planes of the bases, and any one of them is equal to the altitude of the cylinder. Any line of the generating rectangle ABCD, as IK, which is perpendicular to the axis, will generate a circle whose plane is perpendicular to the axis, and which is equal to either base: hence, any section of a cylinder by a plane perpendicular to the axis, is a circle equal to either base. Any section, FCDE, made by a plane through the axis, is a rectangle double the generating rectangle. 2. SIMILAR CYLINDERS are those which may be generated by similar rectangles revolving about homologous sides. The axes of similar cylinders are proportional to the radii of their bases (B. IV., D. 1); they are also proportional to any other homologous lines of the cylinders. 3. A prism is said to be inscribed in a cylinder, when its bases are inscribed in the bases of the cylinder. In this case, the cylinder is said to be circumscribed about the prism. The lateral edges of the inscribed prism are elements of the surface of the circumscribing cylinder. 4. A prism is said to be circum scribed about a cylinder, when its bases are circumscribed about the bases of the cylinder. In this case, the cylinder is said to be inscribed in the prism. The lines which join the corresponding points of contact in the upper and lower bases, are common to the surface of the cylinder and to the lateral faces of the prism, and they are the only lines which are common. The lateral faces of the prism are said to be tangent to the cylinder along these lines, which are then called elements of contact. 5. A CONE is a volume which may be generated by a right-angled triangle revolving about one of the sides adja cent to the right angle, as an axis. Thus, if the triangle SAB, right-angled at A, be turned about the side SA, as an axis, it will generate the cone S-CDBE F H The fixed line SA, is called the axis of the cone; the curved surface generated by the hypothenuse SB, is called the convex surface of the cone; the circle generated by the side AB, is called the base of the cone; and the point S, is called the vertex of the cone; the distance from the vertex to any point in the circumference of the base, is called the slant height of the cone; and the per-. pendicular distance from the vertex to the plane of the base, is called the altitude of the cone. A D The line SB, which generates the convex surface, is, in any position, called an element of the surface; the elements are all equal, and any one is equal to the slant height; the axis is equal to the altitude. Any line of the generating triangle SAB, as GH, which is perpendicular to the axis, generates a circle whose plane is perpendicular to the axis: hence, any section of a cone by a plane perpendicular to the axis, is a circle. Any section SBC, made by a plane through the axis, is an isosceles triangle, double the generating triangle. 6. A TRUNCATED CONE is that portion of a cone included between the base and any plane which cuts the cone. When the cutting plane is parallel to the plane of the base, the truncated cone is called a FRUSTUM OF A CONE, and the intersection of the cutting plane with the cone is called the upper base of the frustum; the base of the cone is called the lower base of the frustum. If the trapezoid HGAB, right-an gled A 7. SIMILAR CONES are those which may be generated by similar right-angled triangles revolving about homologous sides. The axes of similar cones are proportional to the radii of their bases (B. IV., D. 1); they are also proportional to any other homologous lines of the cones. 9. A pyramid is said to be circumscribed about a cone, when its base is circumscribed about the base of the cone, and when its vertex coincides with that of the cone. In this case, the cone is said to be inscribed in the pyramid. The lateral faces of the circumscribing pyramid are tangent to the surface of the inscribed cone, along lines which are called elements of contact. 10 A frustum of a pyramid is inscribed in a frustum |