Multiplying the proportions (1) and (2), term by term, we have, ABC ~ {SH : DEF x {Sh :: AB® : DEP. That is, S-ABC : S-DEF :: AB: : DE'. PROPOSITION XXXII. THEOREM. In two similar polyedrons : 1. The surfaces are to each other as the squares of their homologous edges. 2. The volumes are to each other as the cubes of these edges. First. — The areas of similar polygons being proportional to the squares of their homologous sides, the homologous faces of the two polyedrons form a series of equal ratios; hence, the sums of these faces, that is, the surfaces of the two polyedrons, are to each other as the squares of these same sides or edges. Secondly. —Similar tetraedrons being proportional to the cubes of their homologous edges, the tetraedrons of which the two polyedrons are composed form a series of equal ratios; hence the sums of the antecedents and of the consequents, that is, the volumes of the two polyedrons, are to each other as the cubes of these same edges. CoR. —Two similar pyramids are to each other as the cubes of their homologous edges, as also two similar prisms. EXERCISES ON BOOK VI. THEOREMS. I Show that two tetraedrons are equal, First.—When they have an equal diedral contained by two plane faces equal each to each, and similarly situated. Second. -When they have an equal face adjacent to three diedrals, equal cach to each, and similarly situated. Third.—When they have three faces equal each to each, and similarly situated. Fourth.-When they have one edge equal, and the plane angles of three faces equal, and similarly situated. Fifth. --When they have an edge and five diedrals equal each to each, and similarly situated. 2. Two triangular prisms are equal when they have their lateral faces equal and arranged in the same manner. 3. The volume of a triangular prism is equal to the product of one of its lateral faces by half the distance from this face to the opposite edge. 4. Every plane which contains the line joining the middle points of two opposite faces of a parallelopipedon will divide that parallelopipedon into two equal parts. 5. Show that the formula (a + b)' = a + 3aRb + zab' + 3 is verified by the geometrical construction when a and b are two parts of a given line. 6. Two tetraedrons which have a solid angle equal, are to each other as the product of the three edges which meet in the vertices of these equal solid angles. 7. The plane bisector of a diedral angle of a tetraedron divides the opposite edge into two parts proportional to the faces adjacent to this diedral. 8. If a plane be drawn, containing one edge of a tetraedron and the middle point of the opposite edge, it will divide the tetraedron into two equivalent tetraedrons. 9. Every plane which passes through the middle points of two opposite edges of a tetraedron, divides this body into two equivalent parts. 10. The six planes drawn perpendicular to the six edges of a tetraedron at their middle points, meet in a common point which is equidistant from the four vertices of the tetraedron. II. The six plane bisectors of the diedral angles of a tetraedron meet in a common point equidistant from the four faces of the tetraedron. 12. The four straight lines which join the vertices of a tetraedroni with the intersections of the medians of the opposite faces meet in a common point which divides each line of junction in the ratio of 3 to 1. (That is, the part of the line towards the vertex will be to the part towards the face as 3 to 1.) 13. The four perpendiculars erected to the faces of a tetraedron at the centres of their circumscribed circles meet in a common point. 14. The three straight lines which join the middle points of the opposite edges of a tetraedron meet in a common point which bisects these lines. 15. The different points of intersection mentioned in Exercises 9 to 14 inclusive, all become one and the same point when the tetraedron is regular. 16. When a tetraedron has one of its triedrals trirectangular, then the square of the area of the face opposite to this right triedral is equal to the sum of the squares of the other three faces. 17. The distance of the centre of a parallelopipedon from any plane is equal to one-eighth of the sum of the distances of its eight vertices from the same plane. 18. If a point is at a constant distance from the centre of a parallelopipedon, the sum of the squares of its distances from the vertices is constant. 19. The altitude of a regular tetraedron is equal to the sum of the perpendiculars let fall from any point taken in the interior of the tetraedron on its four faces. 20. First. The volume of a truncated triangular prism is equivalent to three pyramids, having the base of the prism for a common base, and the three points in which the three edges pierce the inclined cutting plane as vertices. Second.—It is also equivalent to its base multiplied by one-third of the distance of this base to the point of intersection of the medians of the inclined upper base. Third. It is also equal to its right section multiplied by one-third of the sum of its three edges. 21. The volume of a truncated parallelopipedon is equal to the product of its right section by the arithmetical mean of its four lateral edges. If planes be drawn through the vertices of a tetraedron parallel to the opposite faces, the tetraedron formed by these planes is not similar to the first. 22. 23. of the volumes of two similar polyedrons are proportional to the cubes of their homologous faces. PROBLEMS. 1. Find a point in the interior of a tetraedron, such that being joined to the four vertices, the tetraedron is divided into four equivalent tetraedrons. 2. Draw a plane parallel to the base of a pyramid cutting off a small pyramid' which shall be of the given pyramid - 4 64 3. In a tetraedron, S-ABC, through E, the middle point of the edş? SB, let the plane DEF be passed parallel to the base ABC; the plane EGH, parallel to the face ASC, and the plane EDH; the pyramid, S-ABC, is thus divided into two equivalent triangular prisms, and into two tetraedrons of the same base and altitude. Divide these two pyramids in the same manner, and deduce the volume of the pyramid as the limit of the sum of the series of successive prisms thus obtained. 4. Cut a cube by a plane so as to make the intersection a regular hexagon. 5. Compute the altitude of a prism, knowing the volume, v, and the base, b; the same of a pyramid. 6. Given in a frustum of a pyramid the lower base, B, the height, h, and the volume, v. Compute the other base. 7. Compute the surface and volume of a regular tetraedron, the edge being given. 8. Find the entire surface of a regular pyramid, the slant height being 12 feet, and each side of the hexagonal base being 3 feet. 9. Find the convex surface of the frustum of a pentagonal regular pyramid whose slant height is 40 feet, each side of the lower base 8 feet, and each side of the upper 5 feet. 10. Find the surface and volume of a frustum of a pyramid whose bases are squares, each side of the lower base being 12 feet, each side of the upper base 6 feet, and the height 4 feet. 11. Find the surface and volume of a block of marble in the shape of a rectangular parallelopipedon whose three dimensions are 4 feet 6 inches, 2 feet 3 inches, 3 feet 9 inches. 12. Find the whole surface of a triangular prism whose altitude is 15 feet, and its bases equilateral triangles, whose sides are 4 feet. 13. Find the volume of a regular hexagonal pyramid whose slant height is 18 feet, and each side of the base 5 feet. 14. Find the volume of the frustum of a pyramid, given the altitude 12 feet, and the bases regular dodecagons, the radii of whose circumscribing circles are 3.6 and .8 feet respectively. 15. Compute the three dimensions of a rectangular parallelopipedon, knowing that they are proportional to the numbers J, $, and }, the volume of the parallelopipedon being 2 cubic yards. 16. Compute the volume of a rectangular parallelopipedon, of which the surface is 5 square yards, and the three dimensions proportional to the numbers 4, 6, 9. 17. The height of a pyramid is 4.5 meters, and its base is a square whose side is 1.2 meters. Compute the corresponding dimensions of a similar pyramid, the volume of which is 7.29 cubic meters. 18. The base of a regular pyramid is a hexagon, each of whose sides is 3 feet in length, and its convex surface is ten times the area of its base. Find its height. 19. Find the volume of a right truncated triangular prism whose base is an equilateral triangle, the side of which is 6 feet, and the three lateral edges of which are 10, 12, and 15 feet. (See Exercises, Theorem 20.) 20. The greatest pyramid of Egypt is 150 yards high, and its base is a square whose side is 250 yards. Find its volume and its convex surface. 21. The edge, SA, of a pyramid, S-ABCD, being four feet six inches long, find the parts into which it is divided by a plane parallel to the base which divides the convex surface, first, into two equivalent parts; second, into two parts proportional to the numbers 3 and 5. 22. A regular pyramid has a hexagon for its base whose side is 15 feet, and its faces make with the base an angle equal to two-thirds of a right angle. Find the volume. 23. A right prism has for its base a regular hexagon. Find its altitude, knowing that its volume is 3 cubic feet and its convex surface 12 square feet. |