PROP. VIII. THEOR. Solid parallelepipeds which have the same altitude, are to one another as their bases, Let AB, CD be solid parallelepipeds of the same altitude : they are to one another as their bases ; that is, as the base AE to the base CF, so is the solid AB to the solid CD. To the straight line FG apply the parallelogram FH equal (Cor, 45. 1.) to AE, so that the angle FGH be equal to the angle LCG; B K and complete the solid parallelepiped GK upon the base FH, one of whose insisting lines is FD, whereby the solids CD, GK must be of the same altitude. Therefore the solid AB is equal (7. 3, Sup.) to the solid GK, because they are upon equal bases AE, FH, and are of the same altitude : and because the solid parallelepiped CK is cut by the plane DG which is parallel to its opposite planes, the base HF is (3. 3. Sup.) to the base FC, as the soli HD to the solid DC: But the base HF is equal to the base AE, and the solid GK to the solid AB: therefore, as the base AE to the base CF, so is the solid AB ta the solid CD, Wherefore solid parallelepipeds, &c. Q. E. D. : Cor. 1. From this it is manifest, that prisms upon triangular bases, and of the same altitude, are to one another as their bases. Let the prisms BNM, DPG, the bases of which are the triangles AEM, CFG, have the same altitude ; complete the parallelograms AE, CF, and the solid parallelepipeds AB, CD, in the first of which let AN, and in the other let CP be one of the insisting lines. And because the solid parallelepipeds AB, CD have the same altitude, they are to one another as the base AE is to the base CF; wherefore the prisms, which are their halves (4.3. Sup:) are to one another, as the base AE to the base CF; that is, as the triangle AEM to the triangle CFG, Cor. 2. Also a prism and a parallelepiped, which have the same al. titude, are to one another as their bases; that is, the prism BNM is to the parallelepiped CD as the triangle AEM to the parallelogram LG. For by the last Cor. the prism BNM is to the prism DPG as the triangle AME to the triangle CGF, and therefore the prism BNM is to twice the prism DPG as the triangle AME to twice the triangle CGF (4. 5.); that is, the prism BNM is to the parallelepiped CD as the triangle AME to the parallelogram LG. PROP. IX. THEOR. Solid parallelepipeds are to one another in the ratio that is compounded of the ratios of the areas of their bases, and of their altitudes. Let AF and GO be two solid parallelepipeds, of which the bases are the parallelograms AC and GK, and the altitudes, the perpendicu. lars let fall on the planes of these bases from any point in the opposite planes EF and MO; the solid AF is to the solid GO in a ratio compounded of the ratios of the base AC to the base GK, and of the per. pendicular on AC, to the perpendicular on GK. Case 1. When the insisting lines are perpendicular to the bases AC and GK, or when the solids are upright. In GM, one of the insisting lines of the solid GO, take GQ equal to AE, one of the insisting lines of the solid AF, and through Q let a plane pass paralleļ to the plane GK, meeting the other insisting lines 2 H B of the solid GO in the points R, S and T. It is evident that GS is a solid parallelepiped (def. 5. 3. Sup.), and that it has the same altitude with AF, viz. GQ or AE. Now the solid AF is to the solid GO in a ratio compounded of the ratios of the solid AF to the solid GS (def. 10. 5.), and of the solid GS to the solid GO; but the ratio of the solid AF to the solid GS, is the same with that of the base AC to the base GK (8. 3. Sup.), because their altitudes AE and GQ are equal; and the ratio of the solid GS to the solid GO, is the same with that of GQ to GM (3. 2. Sup.); therefore, the ratio wbich is compounded of the ratios of the solid AF to the solid GS, and of the solid GS to the solid GO, is the same with the ratio which is compounded of the ratios of the base AC to the base GK, and of the altitude AE to the altitude GM (F. 5.). But the ratio of the solid AF to the solid GO, is that which is compounded of the ratios of AF to GS, and of GS to GO; therefore, the ratio of the solid AF to the solid GO is compounded of the ratios of the base AC to the base GK, and of the altitude AE to the altitude GM. Case 2. When the insisting lines are not perpendicular to the bases. Let the parallelograms AC and GK be the bases as before, and let AE and GM be the altitudes of two parallelepipeds Y and Z on these bases. Then, if the upright parallelepipeds AF and GO be constituts. ed on the bases AC and GK, with the altitudes AE and GM, they will be equal to the parallelepipeds Y and Z (7. 3. Sup.). Now, the solids AF and GO, by the first case, are in the ratio compounded of the ratios of the bases AC and GK, and of the altitudes AE and GM ; therefore also the solids Y and Z have to one another a ratio that is compounded of the same ratios. Therefore, &c. Q. E. D. Cor. 1. Hence, two straight lines may be found having the same ratio with the two parallelepipeds AF and GO. TO AB, one of the sides of the parallelogram AC, apply the parallelogram BV equal to GK, having an angle equal to the angle BAD (44. 1.); and as AE to GM, so let AV be to AX (12. 6.), then AD is to AX as the solid AF to the solid GO. For the ratio of AD to AX is compounded of the ratios (def. 10. 5.) of AD to AV, and of AV to AX ; but the ratio of AD to AV is the same with that of the parallelogram AC to the parallelogram BV (1. 6.) or GK; and the ratio of AV to AX is the same with that of AE to GM; therefore the ratio of AD to AX is compounded of the ratios of AC to GK, and of AE to GM (E. 5.). But the ratio of the solid AF to the solid GO is compounded of the same ratios ; therefore, as AD to AX, so is the solid AF to the solid GO. Cor. 2. If AF and GO are two parallelepipeds, and if to AB, to the perpendicular from A upon DC, and to the altitude of the parallellepiped AF, the numbers L, M, N be proportional: and if to AB, to GH, to the perpendicular from G on LK, and to the altitude of the parallelepided GO, the numbers L, l, m, n be proportional; the solid AF is to The solid GO as LXMXN to lXmXn. For it may be proved, as in the 7th of the 1st of the Sup. that LX BIX N is to lXmXn in the ratio compounded of the ratio of LXMI to lXm, and of the ratio of N to n. Now the ratio of LXM to lton is that of the area of the parallelogram AC to that of the parallelogram GK; and the ratio of N to n is the ratio of the altitudes of the parallelepipeds, by hypothesis, therefore, the ratio of LXMXN to i Xnxn is compounded of the ratio of the areas of the bases, and of the ratio of the altitudes of the parallelepipeds AF and GO; and the ratio of the parallelepipeds themselves is shewn, in this proposition, to be compounded of the same ratios; therefore it is the same with that of the product L XMXN to the product l xm Xn. Cor. 3. Hence all prisms are to one another in the ratio compounded of the ratios of their bases, and of their altitudes. prism is equal to a parallelepiped of the same altitude with it, and of an equal base (2. Cor. 8. 3. Sup.). For every PROP. X. THEOR. Solid parallelepipeds, which have their bases and altitudes reciprocally proportional, are equal ; and parallelepipeds which are equal, have their bases and altitudes reciprocally proportional. Let AG and KQ be two solid parallelepipeds, of which the bases are AC and KM, and the altitudes AE and KO, and let AC be to KM as KO to AE ; the solids AG and KQ are equal. As the base AC to the base KM, so let the straight line KO belto the straight line S. Then, since AC is to KM as KO to S, and also by hypothesis, AC to KM as Ko to AE, KO has the same 'ratio to $ that it has to AE (11. 5.) ; wherefore AE is equal to S (9. 5.). But the solid AG is to the solid KQ, in the ratio compounded of the ratios of AE to KO, and of AC to KM (9. 3. Sup.), that is, in the ratio com. pounded of the ratios of AE to Ão, and of KO to S. And the ratio of AE to S is also compounded of the same ratios (def. 10. 5.); therefore, the solid AG has to the solid KQ the same ratio that AE has to S. " But AE was proved to be equal to S, therefore AG is equal to KQ Again, if the solids AG and KQ be equal, the base AC is to the base KM as the altitude KO to the altitude AE. Take S, so that AC may be to KM as KO to S, and it will be shewn, as was done above, that the solid AG is to the solid KQ as AE to S; now, the solid AG is, by hypothesis, equal to the solid KQ; therefore, AE is equal to S; but, by construction, AC is to KM, as KO is to S; therefore, AC is to KM as Ko to AE. Therefore, Q. E. D. Cor. In the same manner. It may be demonstrated, that equal prisms have their bases and altitudes reciprocally proportional, and conversely. PROP. XI. - THEOR. Similar solid parallelepipeds are to one another in the triplicate ratio of their homologous sides. Let AG, KQ be two similar parallelepipeds, of which AB and KL are two homologous sides ; the ratio of the solid AG to the solid KQ its triplicate of the ratio of AB to KL. Because the solids are similar, the parallelograms AF, KB are similar (def. 2. 3 Sup.), as also the parallelograms AH, KR; therefore, the ratios of AB to KL, of AE to KO, and of AD to KN are all equal (def. 1. 6.). But the ratio of the solid AG to the solid KQ is compounded of the ratios of AC to KM, and of AE to KO. Now, the ratio of AC to KM, because they are equiangular parallelograms, is compounded (23. 6.) of the ratios of AB to KL, and of AD to KN. Wherefore, the ratio of AG to KQ is compounded of the three ratios of AB to KL, AD to KN, and AE to KO.; and these three ratios have already been proved to be equal; therefore, the ratio that is compounded of them, viz. the ratio of the solid AG to the solid KQ, is triplicate of any of them (def. 12. 5.) ; it is therefore triplicate of the ratio of AB to KL. Therefore, similar solid parallelepipeds, &c. Q. E. D. Cor. 1. If as AB to KL, so KL to m, and as KL to m, so is-m to ng then AB is to n as the solid AG to the solid KQ. For the ratio of AB to nis triplicate of the ratio of AB to KL (def. 12. 5.), and is therefore equal to that of the solid AG to the solid KQ. Cor. 2. As cubes are similar solids, therefore the cube on AB is to the cube on KL in the triplicate ratio of AB to KL, that is in the same ratio with the solid AG, to the solid KQ. Similar solid parallelepipeds are therefore to one another as the cubes on their homologous sides. Cor. 3. In the same manner it is proved, that similar prisms are to one another in the triplicate ratio, or in the ratio of the cubes of their homologous sides. PROP. XII. THEOR. If two triangular pyramids, which have equal bases and altitudes, be cut by planes that are parallel to the bases, and at equal distances from them, the sections are equal to one another. Let ABCD and EFGH be two pyramids, having equal bases BDC and FGH, and equal altitudes, viz. the perpendiculars AQ, and ES, |