We have also (Theo. VI. Cor., Bk. IV.), ABC:AGC::AB: A G or D Е. But the similar triangles A B C, D E F give AB:DE::AC:DF'; hence (Theo. IX. Bk. II.), ABC:AGC::AGC: DEF'; that is, the base AGC is a mean proportional between the bases ABC, DEF of the frustum. Secondly. Let G H I K L-MNOPQ be the frustum The bases of the two pyramids may be regarded as situated in the same plane, in which case the plane MNOPQ produced will form in the triangular pyramid a section, DE F', at the same distance above the common plane of the bases; and therefore the section DEF will be to the section MNOPQ as the base ABC is to the base G H I K L (Theo. XVIII. Cor. 1); and since the bases are equivalent, the sections will be so likewise. Hence, the pyramids MNO P Q - T, DEF-S, having the same altitude and equivalent bases, are equivalent. For the same reason, the entire pyramids GHI KL-T, ABC-S are equivalent; consequently, the frustums GHIKL-MNOPQ,AB C - D E F, are equivalent. But the frustum ABC-DEF has been shown to be equivalent to the sum of three pyramids having for their common altitude the altitude of the frustum, and whose bases are the two bases of the frustum, and a mean proportional between them. Hence the proposition is true of the frustum of any pyramid. THEOREM XXIV. 321. Similar pyramids are to each other as the cubes of their homologous edges. homologous polyedral angles at the vertices are equal (274); hence the smaller pyramid may be so applied to the larger, that the polyedral angle S shall be common to both. In that case, the bases ABC, DEF will be parallel; for, since the homologous faces are similar, the angle SDE is equal to SAB, and SEF to SBC; hence the plane ABC is parallel to the plane DEF (Theo. VI.). Then let S O be drawn from the vertex S perpendicular to the plane ABC, and let P be the point where this perpendicular meets the plane DEF. From what has already been shown (Theo. XVIII.), we shall have SO:SP::SA:SD::AB:DE; and consequently, +SO:+SP : : A B : D E. But the bases ABC, D E F are similar; hence (Theo. XIX. Bk. IV.), Multiplying together the corresponding terms of these two proportions, we have 3 ABCX+SO:DEFצSP: : A B3 : D E3. Now, A B C X SO represents the volume of the pyramid ABC-S, and DEFX SP that of the pyramid DEF-S (Theo. XXII.); hence two similar pyramids are to each other as the cubes of their homologous edges. EXERCISES FOR ORIGINAL THOUGHT, ON REVIEW. 322. 1. A straight line cannot be partly in a plane, and partly out of it. 2. If two planes cut each other, their common section is a straight line. 3. If two planes are perpendicular to the same straight line they are parallel to each other. 4. In every parallelopipedon the opposite faces are equal and parallel. 5. The diagonals of every parallelo pipedon bisect each other. 6. The volume of a triangular prism is equal to the product of the area of either of its rectangular sides as a base multiplied by half its alti tude on that base. E A A D H F G 0 D B C 7. All prisms of equal bases and altitudes are equal in volume, whatever be the figure of their bases. BOOK VI. THE THREE ROUND BODIES. DEFINITIONS 323. A Cylinder is a volume which may be described by the revolution of a rectangle turning about one of its sides, which remains immovable; as the solid described by the rectangle ABCD revolving about its side A B. 324. The Bases of the cylinder are the circles described by the sides, AC, BD, of the revolving rectangle, which are adjacent to the immovable side, A B. A C B D The Axis of the cylinder is the straight line joining the centers of its two bases; as the immovable line A B. The Convex Surface of the cylinder is described by the side CD of the rectangle, opposite to the axis A B. 325. A Cone is a solid which may be described by the revolution of a right-angled triangle turning about one of its perpendicular sides, which remains immovable; as the solid described by the right-angled triangle ABC revolving about its perpendicular side A B. The Base of the cone is the circle described by the revolution of the side BC, which is perpendicular to the immovable side. A B C The Convex Surface of a cone is described by the hypothenuse, A C, of the revolving triangle. The Vertex of the cone is the point A, where the hypothenuse meets the immovable side. The Axis of the cone is the straight line joining the vertex to the center of the base; as the line A B. The Altitude of a cone is a line drawn from the vertex perpendicular to the base; and is the same as the axis, A B. The Slant Hight, or Side, of a cone, is a straight line drawn from the vertex to the circumference of the base; as the lins A C. 326. The Frustum of a cone is the part of a cone included between the base and a plane parallel to the base; as the solid CD- F. FA E C D B The Axis, or Altitude, of the frustum, is the perpendicular line AB included between the two bases; and the Slant Hight, or Side, is that portion of the slant hight of the cone which lies between the bases; as FC. 327. Similar Cylinders, or Cones, are those whose axes are to each other as the radii, or diameters, of their bases. 328. A Sphere is a solid, or volume, bounded by a curved surface, all points of which are equally distant from a point within, called the center. The sphere may be conceived to be formed by the revolution of a semicircle, DAE, about its diameter, DE, which remains fixed. 329. The Radius of a sphere is a straight line drawn from the center to any point in the surface, as the line CB. A D C B E The Diameter, or Axis, of a sphere is a line passing through the center, and terminated both ways by the surface, as the line D E. |