fore IK: BC:: LM: EF. But BC is equal to EF, by hypothesis; therefore IK is also equal to LM. In the same manner, it is shown that any other sections, at equal distance from the vertex, are equal to each other. Since, then, every section in the cone is equal to the corresponding section in the pyramids, and the heights are equal, the solids ABC, DEF, composed of those sections, must be equal also. Q. E. D. PROP. XI. Every triangular prism may be divided into three equal triangular pyramids of the same base and altitude with the prism. Let ABCDEF be a prism. In the planes of the three sides of the prism, draw the diagonals BF, BD, CD. B Then the two planes BDF, BCD divide the whole prism into the three pyramids BDEF, DABC, DBCF; which are proved to be all equal to one another as follows: E A C Since the opposite ends of the prism are equal to each other, the pyramid whose base is ABC and vertex D, is equal to the pyramid whose. base is DEF and vertex B (Prop. 10), being pyramids of equal base and altitude. But the latter pyramid, whose base is DEF and vertex B, may be considered as having BEF for its base and D for its vertex, and this is equal to the third pyramid, whose base is BCF and vertex D, being pyramids of the same altitude (since they have the same vertex and their bases are in the same plane) and equal bases BEF, BCF (th. 19). Consequently, all the three pyramids which compose the prism are equal to each other, and each pyramid is the third part of the prism, or the prism is triple of the pyramid. Q. E. D. Corol. 1. Any triangular pyramid is the third part of a triangular prism of the same base and altitude (this follows from the last Prop., 10). Corol. 2. Every pyramid, whatever its figure may be, is the third part of a prism of the same base and altitude. This follows from Props. 3 and 10, or may be proved by dividing the given prism into triangular prisms, and the given pyramid into triangular pyramids, all having a common altitude. Corol. 3. Any right cone is the third part of a cylinder, or of a prism, of equal base and altitude; since it has been proved that a cylinder is equal to a prism, and a cone equal to a pyramid, of equal base and altitude. Corol. 4. The measure of a pyramid or cone will be the product of its base by the third of its altitude. (See note to Prop. 6.) Scholium. Whatever has been demonstrated of the proportionality of prisms or cylinders holds equally true of pyramids or cones, the former being always triple the latter when they have the same base and altitude; viz., that similar pyramids or cones are as the cubes of their like linear sides, or diameters, or altitudes, &c. The tangent plane to a cone is analogous to that of a cylinder. The contact is a right-lined element. Every tangent plane to a cone passes through the vertex. The surface of a cone developes into the sector of a circle, and is measured by the circumference of the base multiplied by half the apophthegm. A pyramid is inscribed in a cone when the base of the pyramid is inscribed in that of the cone, and they have the same vertex. A cone is a pyramid of an infinite number of triangular faces. EXERCISES. 1. Prove that the four diagonals of a parallelopipedon meet in the same point. 2. That the square of each diagonal of a rectangular parallelopipedon is equal to the sum of the squares of its three edges. 3. Construct a parallelopipedon upon three lines perpendicular to each other as edges. 4. Prove that the two lines joining the points of the opposite faces of a parallelopipedon, in which the diagonals of those faces intersect, bisect each other at the point where the diagonals of the solid meet. 5. Prove that two polyhedrons which have the same vertices are identical. 6. That two prisms are equal when they have three faces forming a polyhedral angle of the one equal to the same in the other, and arranged in the same order. 7. That two right prisms are equal when they have equal bases and altitudes. 8. That two tetrahedrons are equal, 1°. When they have a diedral angle equal, comprehended between equal faces, arranged in the same manner in both; 2°. When they have one trihedral angle, comprehended by three equal faces in each, and arranged in the same order. 3°. Corol. When they have their edges all equal and arranged in the same order. 4°. When they have two faces and two adjacent diedrals equal. 9. That two pyramids are equal when they have their bases and two other faces forming a trihedral angle, with the base equal in each. 10. Polyhedrons may be divided into tetrahedrons by planes passing diagonally through the edges. 11. Polyhedrons are equal when composed of the same number of equal tetrahedrons. 12. Prove that polyhedrons are equal when their faces and diedral angles are equal, and disposed in the same order. 13. Prove that similar polyhedrons are composed of the same number of similar tetrahedrons. 14. That polyhedrons are similar when their faces are all equal, each to each, and equally inclined. 15. That polyhedrons are similar when they have a face in each similar, and their homologous vertices out of this face are determined by tetrahedrons having a triangular face in the homologous face.* 16. That two pyramids are similar when they have their edges parallel. 17. That two regular polyhedrons of the same kind are similar. 18. That a plane passed through two edges of a parallelopiped, diagonally opposed to each other, divides the parallelopiped into two symmetrical triangular prisms equal in volume. * This is the definition of similar polyhedrons given by Legendre. 19. That two prisms of the same base are proportional to their altitudes. 20. That two regular pyramids are equal when their base and an edge of the one are equal to the same in the other. 21. That similar pyramids are to each other as the cubes of their homologous sides. 22. That similar polyhedrons are as the cubes of their homologous sides. 23. That the surfaces of similar polyhedrons are as the squares of their homologous sides. 24. Cut a pyramid by a plane, parallel to the base, in such a way that the section shall be to the base in the ratio of two given lines. 25. Also, so that the convex surface of the superior portion shall be one third that of the whole pyramid. 26. Show how to construct a pyramid when the base and two adjacent triangular faces are given. 27. A prism with the same data. 28. A parallelopipedon with given base, and edge meeting it. 29. With given edges to construct the faces of a tetrahedron. 30. With three edges forming a trihedral angle, and their angles, to construct the faces of a triangular prism. 31. The same for a pentagonal prism, three faces, forming a trihedral angle, being given. 32. Show how to construct a cylinder similar to a given cylinder, and whose base shall be to that of the given in the ratio of two given lines. 33. Show in what the frustum of a cone develops, and what is the measure of its surface. 34. Prove that every plane parallel to the axes of a cylinder cuts its surface in two lines parallel to the axis. 35. That if a circle and a line tangent to it revolve about a common axis passing through the center of the circle, the curve of contact of the cone generated by the line, and the sphere generated by the circle, will be a circle whose plane is perpendicular to the axis. 36. That similar cones and cylinders are proportional, their surfaces to the squares, and their volumes to the cubes of their altitudes, elements, diameters of bases, or any homologous lines. PROP. XII. The frustum of a pyramid is composed of three pyramids having for a common altitude the altitude of the frustum, and for bases the upper and lower base of the frustum and a mean proportional between them. a Let a plane be passed through the three points a, C, b; it cuts off a triangular pyramid having acb for a base, and C in the plane of the lower base of the frustum for a vertex. This is evidently the first pyramid of the enunciation. Again: suppose a plane be passed through the three points A, C, b; it cuts off a pyramid having ABC for base and b for vertex, the second pyramid of the enunciation. There remains the pyramid CAba, which is equal to the pyramid CADb (bD being drawn parallel to aA), having the same vertex C and an equal base, since the diagonal Ab bisects the parallelogram ADba. This last pyramid being considered as having its vertex at b and its base ADC, has the altitude of the frustum, and it remains to show that the triangle ADC, which is its base, is a mean proportional between the triangles ABC and abc. For this purpose, let us observe that A D B (1) ▲ ABC: ▲ ADC:AB: AD because they have a common vertex, and, therefore, are to each other as their bases, which are in the same straight line. Also, that ▲ ADC: ▲ abc: AC: ac = (2) because the angle A the angle a (corol. 1, Prop. 16, Geom. of Planes), and (th. 60, cor. 3) ▲ ADC: abc:: AD × AC: ab × ac, and AD ab. Also, that AB: ab:: AC: ac = (3) |