respect to the similar A BCD and bed so as to make the equal diedrals A-BC-D and a-bc-d. The same may be said of any other adjacent faces. In the same manner we may consider each of the corresponding polygons forming faces of the similar polyedra; and find that the ratios between the several similar polygons will be the same as the ratio of the squares of any homologous lines; and the sums of all of them will have the same ratio. There are, as already seen, surfaces entirely curved, as a sphere, and surfaces partly curved and partly plane. In any case the similar surfaces may be considered as the limits toward which similar inscribed or circum scribed polyedra approach as the number of faces is increased. Lines may be chosen that shall be corresponding lines in the two similar surfaces under consideration and which shall be corresponding lines in the approaching polyedra. The surfaces of the approaching similar polyedra are to each other as the squares of the homologous lines which may be so chosen as to remain unaltered during the approach. S2 S2 Sa L2 L2 These surfaces represented by S1, S1, S2, S2, S3, 83, etc., which are similar at each step and continually approach the curved or mixed surfaces, always have the same ratio The limits toward which they approach will have the same ratio (§ 96). Hence the theorem. NOTE. - In the above article is established the second of the three great principles of geometry as stated in § 69, Note. Exercises. 1. From the formula for the surface of a sphere, show that the surfaces of two spheres are to each other as the squares of their radii. 2. Show that the ratio of the areas of similar zones on different spheres equals the ratio of the squares on the diameters of the spheres. 3. Making use of a spherical blackboard, show how to find approximately the distance from New York to Queenstown. 4. With the same materials find approximately the distance from San Francisco to Yokohama, and show how near to the Aleutian Islands the shortest arc will pass. CHAPTER XIII. VOLUMES. 144. Definitions. 1. As has already been stated, a volume is an enclosed and limited portion of space. The figure may be real or imaginary. 2. In $57 the area of a rectangle is represented by ab, the product of two ad The figure is called a rectangular parallelopiped. It is also a right prism with rectangular base. If the volume of one rectangular parallelopiped be represented by V, and the volume of another by v, the three edges meeting at a vertex in the one being a, b, and c, and in the other, x, y, and z, we have, THEOREM. The volumes of two rectangular parallelopipeds are to each other as the products of the three edges meeting at a vertex. Exercises. 1. Show that the theorem might be stated thus: The volumes of two rectangular parallelopipeds are to each other as the products of their bases and altitudes. 2. If the bases happen to be equal, they are to each other as their altitudes. 3. If their altitudes happen to be equal, they are to each other as their bases. 4. Show that if the figures are similar, their volumes will be to each other as the cubes on any of the corresponding lines. 145. THEOREM. An oblique prism is equivalent to a right prism, the base of which is a right section of the oblique prism, and the altitude of which is equal to an edge of the oblique prism. Let A-I represent the oblique prism, and K-N a right section, and QS another right section at a distance from K-N equal to an edge, AF, of truncated prism A-N as moved in the direction of the edges. When it shall have been moved the distance AF, the oblique polygon ABCDE, forming its lower base, will coincide with FGHIJ, forming its upper base, the right section KMNOP will coincide with the right section QRSTU, and the lateral faces will coincide, so |