AE to the solid LR: for the same reason because the solid parallelopiped CR is cut by the plane LMFD, which is parallel to the opposite planes CP, BR; as the base CD to the base LQ; so is the solid CF to the solid LR, but as the base AB to the base LQ, so the base CD to the base LQ. as has been proved: therefore as the solid AE to the solid LR, so is the solid CF to the solid LR; and therefore the solid AE is equal (9. 5.) to the solid CF.. But let the solid parallelopipeds, SE, CF be upon equal bases SB, Cn and be of the same altitude, and let their insisting straight lines be at right angles to the bases; and place the bases SB, CD in the same plane, so that CL, LB be in a straight line; and let the angles SLB, CLD, be unequal; the solid SE is also in this case equal to the solid CF. Produce DL, TS until they meet in A, and from B draw BH parallel to DA; and let HB, OD produced meet in Q, and complete the solids AE, LR: therefore the solid AE, of which the base is the parallelogram LE, and AK the plane opposite to it, is equal (5. 3. Sup.) to the solid SE, of which the base is LE, and SX the plane opposite; for they are upon the same base LE, and of the same altitude, and their insisting straight lines, viz. LA, LS, BH, BT; MG, MU, EK, EX, are in the same straight lines AT, GX: and because the parallelogram AB is equal (35. 1.) to SB, for they are upon the same base LB, and between the same parallels LB, AT; and because the base SB is equal to the base CD; therefore the base AB is equal to the base CD: but the angle ALB is equal to the angle CLD: therefore, by the first case, the solid AE is equal to the solid CF; but the solid AE is equal to the solid SE, as was demonstrated: therefore the solid SE is equal to the solid CF. Case 2. If the insisting straight lines AG, HK, BE, LM; CN, RS, DF, OP, be not at right angles to the bases AB, CD; in this case likewise the solid AE is equal to the solid CF. Because solid parallelopipeds on the same base, and of the same altitude, are equal (6. 3. Sup.), if two solid parallelopipeds be constituted on the bases AB and CD of the same altitude with the solids AE and CF, and with their insisting lines perpendicular to their bases, they will be equal to the solids AE and CF; and, by the first case of this proposition, they will be equal to one another; wherefore, the solids AE and CF are also equal. PROP. VIII. THEOR. Solid parallelopipeds which have the same altitude, are to one another as their bases. Let AB, CD be solid parallelopipeds 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. Prop. 45. 1.) to AE, so that the angle FGH be equal to the angle LCG; and complete the solid parallelopiped GK upon the base, FH, one of whose in sisting 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 parallelopiped 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 solid 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 to the solid CD. 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 paral lelopipeds 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 parallelopipeds 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 parallelopiped, which have the same altitude, are to one another as their bases; that is, the prism BNM is to the parallelopiped 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 parallelopiped CD as the triangle AME to the parallelogram LG. PROP. IX. THEOR. Solid parallelopipeds 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 parallelopipeds, of which the bases are the parallelograms AC and GK, and the altitudes, the perpendiculars 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 perpendicular 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 parallel to the plane GK, meeting the other insisting lines of the solid GO in the points R, S and T. It is evident that GS is a solid parallelopiped (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 which 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 compound. ed 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 parallelopipeds Y and Z on these bases. Then, if the upright parallelopipeds AF and GO be constituted on the bases AC and GK, with the altitudes AE and GM, they will be equal to the parallelopipeds 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. COR. 1. Hence, two straight lines may be found having the same ratin with the two parallelopipeds 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 (Prop. 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 parallelopipeds, and if to AB, to the perpendicular from A upon DC, and to the altitude of the parallelopiped 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 parallelopiped GO, the numbers L, l, m, n, be proportional; the solid AF is to the solid GO as LX MXN to xmxn. For it may be proved, as in the 7th of the 1st of the Sup. that LX MX N is to lxmxn in the ratio compounded of the ratio of LXM to lxm, and of the ratio of N to n. Now the ratio of LXM to lxm 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 parallelopipeds, by hypothesis, therefore, the ratio of LX MXN to lx mxn is compounded of the ratio of the areas of the bases, and of the ratio of the altitudes of the parallelopipeds AF and GO; and the ratio of the parallelopipeds themselves is shewn, in this proposition, to be compounded of the same ratios; therefore it is the same with that of the product LX MXN to the product xmxn. COR. 3. Hence all prisms are to one another in the ratio compounded of the ratios of their bases, and of their altitudes. For every prism is equal to a parallelopiped of the same altitude with it, and of an equal base (2. Cor. 8. 3. Sup.). PROP. X. THEOR. Solid parallelopipeds, which have their bases and altitudes reciprocally proportional, are equal; and parallelopipeds which are equal, have their bases and altitudes reciprocally proportional. Let AG and KQ be two solid parallelopipeds, 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 be to 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 S 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 compounded of the ratios of AE to KO, 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 sclid 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. 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 parallelopipeds are to one another in the triplicate ratio of their homologous sides. Let AG, KQ be two similar parallelopipeds, of which AB and KL are two homologous sides; the ratio of the solid AG to the solid KQ is triplicate of the ratio of AB to KL. Because the solids are similar, the parallelograms AF, KP 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 |