Draw SO perpendicular to the base also be perpendicular to the base abcde. plane at the point o then will SO be to So, as SA is to Sa (P. III.), or as AB is to ab; hence, 180 So: AB: ab. ABCDE; it will Let it pierce that But the bases being similar polygons, we have (B. IV., P. XXVII.), base ABCDE : base abcde AB2 :: ab2. B Multiplying these proportions, term by term, we have, ᎪᏴ base ABCDE × SO : base abcde × So :: AB3: ab. But, base ABCDE × SO Pyramid S-ABCDE, and the volume of the pyramid is equal to the volume of the base abcde So is equal to S-abcde; hence, pyramid S-ABCDE pyramid S-abcde :: AB3· ab'; which was to be proved. Cor. Similar pyramids are to each other as the cubes of their altitudes, or as the cubes of any other homologous lines. If we denote the volume of any prism by V, its base by B, and its altitude by II, we shall have (P. XIV.), If we denote the volume of any pyramid by V, its base by B, and its altitude by II, we have (P. XVII.), If we denote the volume of the frustum of any pyramid by V, its lower base by B, its upper base by b, and its altitude by II, we shall have (P. XVIII., C.), V = {(B + b + √B × b) × II. (3.) REGULAR POLYEDRONS. A REGULAR POLYEDRON is one whose faces are all equal regular polygons, and whose. polyedral angles are equal, each to each. There are five regular polyedrons, namely: 1. The TETRAEDRON, or regular pyramid-a polyedron bounded by four equal equilateral triangles. 2. The IIEXAEDRON, or cube-a polyedron bounded by six equal squares. 3. The OCTAEDRON-a polyedron bounded by eight equal equilateral triangles. 4. The DODECAEDRON-a polyedron bounded by twelve equal and regular pentagons. 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. 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. reason In the Dodecaedron, the regular pentagons are grouped about the polyedral angles in sets of three, and for the same as before, they cannot be grouped in any greater number, so as to form a salient polyedral angle. Furthermore, no other regular polygons can be grouped so as to form a salient polyedral angle; therefore, Only five regular polyedrons can be formed. 14 R 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 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 K C The line DC, which generates the convex surface, 18, 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 equa to either base: hence, any section of a cylinder by a plan 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 straight 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. |