i But the solid al is to the solid AR as Ar is to AW (VIII. 11.); whence, also, ac is to eg as AP is to AW (V. 11.), or AC to Eg as Eo to Aw. Again, let ac be to Eg as Eo is to Aw; then will AR be equal to EY. For, since Al is to ex as Ac to ÉG (VIII. 10.), and AC to EG as Eo to Aw (by Hyp.), AL will be to Ey as EO to AW (V. 11.) But eo, or Ap, is to aw as AL is to AR (VIII. 11.); therefore Al will be to Ey as al is to AR (V. 11.) And since the antecedents are equal, the confequents will also be equal ; whence the solid Ar is equal to the solid EY, as was to be shewn. Cor. The same proportion will hold of prisms in general; these being equal to rectangular parallelepipedons of equal bases and altitudes. PRO P. XIII. THEOREM. Similar rectangular parallelepipedons are to each other as the cubes of their like fides. Let AF, kp be two fimilar rectangular parallelepipedons, whose like fiđes are AB, KL; then will af be to KP as the cube of AB is to the cube of kl. For let AT, KW be two cubes standing on AX, K2, the squares of the fides AB, KL. Then fince parallelepipedons on the fame base are to each other as their altitudes (VIII. 11.), AF will be to An as Ah to AV, or AB; and Ko to ks as KR TO KY, or KL. But the planes ABEH, KLOR being similar (VIII. Def. 2.), AH, will be to AB as KR is to KL (VI. Def. 1.); whence AF is to An as KP to ks (V. 11.); or ar to KP as An to Ks (V. 15.) Again, since parallelepipedons of the fame altitude are to each other as their bases (VIII. 10.), at will be to, An as Ax to Ac; and kw to ks as Kz to KM. And because ax, or the square of "AB, is to Ac, as K2, or the square of kl, is to KM (VI. 17.); AT will be to An as kw is to Ks (V. 11.); or at to kw as an to Ks (V. 15.) But AF has been shewn to be to KP as an is to KS; therefore, also, AF is to KP as Ar to kw (V. 11.) Q. Ę. D. Cor. I. Similar rectangular parallelepipedons are to each other as the cubes of their altitudes; these being considered as like sides of the folids. COR. 2. Every prism being equal to a parallelepipedon of an equal base and altitude (VIII. 9. Cor.), all fimilar prisms will be to each other as the cubes of their altitudes, or like sides. PROP. PROP. XIV. THEOREM. If a pyramid be cut by a plane parallel to its base, the section will be to the base as the squares of their distances from the vertex. C B Let EDABC be a pyramid, and mo a section parallel to the base Ac; then will mo be to AC as the squares of their distances from the vertex. For draw Es perpendicular to the plane of the base AC (VII. 9.); and join Ds and pr. Then, since mp, mn are parallel to AD, AB (VII. 12), the angle pmn will be equal to the angle DAB (VII. 7.); and pm will be to DA as Em to EA, or as mn to AB (VI. 3.) For a like reason each of the angles in the section mo are equal to their corresponding angles in the base AC, and the sides about them are proportional ; whence mo is similar to ac (VI. Def. 1.) And because pm is parallel to DA, and pr to Ds (VII. 12.), pm will be to da as Ep to ED, or as Er to Es (VI. 3.) The lines pm, DA, Er and es being, therefore, proportional, the square of pm will be to the square of DA, as the square of Er is to the square of Es (VI. 19. Cor.) But the square of pm is to the square of DA as mo is to AC (VI. 17.); whence the square of Er is to the square of Es as mo is to ac (V. 11.) Q. E. D, Cor. If a pyramid be cut by a plane parallel to its base, the section will be fimilar to the base. PRO P. XV. THEOREM. Pyramids of equal bases and altitudes are equal to each other. Let EDABC, LKFGH be any two pyramids, of which the base ac is equal to the base Fh, and the altitude Es to the altitude LP; then will EDABC be equal to LKFGH. For make er equal to Lo; and draw the sections mn, vw, parallel to the bases AC, FH. Then, by the last proposition, the square of Er is to the square of Es as mn is to AC; and the square of Lo to the square of LP as vw is to FH. And fince the square of Er is equal to the square of Lo (Const. and II. 2.); and the square of Es to the square of of LP (Hyp. and II. 2.), mn will be to Ac as vw is to FH (V.9.) But ac is equal to FH, by hypothesis; whence mn is, allo, equal to vw (V. 10.) And, in the fame manner, it may be fhewn, that any other sections, at equal distances froin the vertices, are equal to each other. Since, therefore, every section in the pyramid EDABC is equal to its corresponding section in the pyramid LKFGH, the pyramids themfelves, which are composed of those fections, must also be equal. Q. E. D. PRO P. XVI. THEOREM. Every pyramid of a triangular base, is the third part of a prism of the fame base and altitude. Let Dabc be a pyramid, and FDABE a prism, standing upon the fame base ABC, and having the fame altitude; then will DABC be a third of FDABE. For in the planes of the three sides of the prism, draw the diagonals DB, DC and ce: |