PROP. XXXII. THEOR. Solid parallelopipeds which have the same altitude, are to See N. one another as their bases. Let AB, CD be solid parallelopipeds of the same altitude: they shall be to one another as their bases; that is, as the base AE to the base CF, so shall the solid AB be to the solid CD. 31. 11. To the straight line. FG apply the parallelogram FH equal* to AE, so that the angle FGH may be equal Cor. 45.1. to the angle LCG; and upon the base FH complete the solid parallelopiped GK, one of whose insisting lines is FD, whereby the solids CD, GK must be of the same altitude: therefore the solid AB is equal* 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 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. Wherefore, solid parallelopipeds, &c. Q. E. D. O P L COR. From this it is manifest, that prisms upon triangular bases, of the same altitude, are to one another as their bases. Let the prisms, the bases of which are the triangles AEM, CFG, and NBO, PDQ the triangles opposite to them, have the same altitude: they shall be to one another as their bases. Complete the parallelograms AE, CF, and the solid parallelopipeds AB, CD, in the first of which let MO, and in the other let GQ 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, are to one another, as the base AE to the base CF; that is, as the triangle AEM to the triangle CFG. 25. 11. 28.11. *24. 11. * C. 11. * 1. 6. 1.6. * 1. 6. 25.11. * 25. 11. 25. 11. PROP. XXXIII. THEOR. Similar solid parallelopipeds are one to another in the triplicate ratio of their homologous sides. Let AB, CD be similar solid parallelopipeds, and the side AE homologous to the side CF: the solid AB shall have to the solid CD the triplicate ratio of that' which AE has to CF. B X HP R G M Produce AE, GE, HE, and in these produced take EK equal to CF, EL equal to FN, and EM equal to FR; and complete the parallelogram KL, and the solid KO. Because KE, EL are equal to CF, FN each to each, and the angle KEL equal to the angle CFN, because it is equal to the angle AEG, which is equal to CFN, by reason that the solids AB, CD are similar; therefore the parallelogram KL is similar and equal to the parallelogram CN. For the same reason the parallelogram MK is similar and equal to CR, and also OE to FD. Therefore three parallelograms of the solid KO are equal and similar to three parallelograms of the solid CD: and the three opposite ones in each solid are equal and similar to these: therefore the solid KO is equal* and similar to the solid CD. Complete the parallelogram GK; and upon the bases GK, KL, complete the solids EX, LP, so that EH be an insisting straight line in each of them, whereby they must be of the same altitude with the solid AB. And because the solids AB, CD are similar, and, by permutation, as AE is to CF, so is EG to FN, and so is EH to FR: but FC is equal to EK, and FN to EL, and FR to EM; therefore, as AE to EK, so is EG to EL, and so is HE to EM: but as AE to EK, so* is the parallelogram AG to the parallelogram GK; and as GE to EL, so is* GK to KL; and as HE to EM, so* is PE to KM: therefore as the parallelogram AG to the parallelogram GK, so is GK to KL, and PE to KM: but as AG to GK, so is* the solid AB to the solid EX; and as GK to KL, so* is the solid EX to the solid PL; and as PE to KM, so* is the solid PL to the solid KO: and therefore as the solid AB to the solid EX, so is EX to PL, and PL to KO: but if four magnitudes be continued proportionals, the first is said to have to the fourth the triplicate + ratio of +11 Def. 5. which it has to the second; therefore the solid AB has to the solid KO, the triplicate ratio of that which AB has to EX: but as AB is to EX, so is the parallelogram AG to the parallelogram GK, and the straight line AE to the straight line EK; wherefore the solid AB has to the solid KO, the triplicate ratio of that which AE has to EK: but the solid KO is equal to the solid CD, and the straight line EK is equal to the straight line CF; therefore the solid AB has to the solid CD the triplicate ratio of that which the side AE has to the homologous side CF. Therefore, similar solid parallelopipeds, &c. Q. E. D. COR. From this it is manifest, that, if four straight lines be continual proportionals, as the first is to the fourth, so is the solid parallelopiped described from the first to the similar solid similarly described from the second; because the first straight line has to the fourth the triplicate ratio of that which it has to the second. PROP. D. THEOR. Solid parallelopipeds which are contained by parallelo- See N. grams equiangular to one another, each to each, that is, of which the solid angles are equal, cach to each, have to one another the ratio which is the same with the ratio compounded of the ratios of their sides. Let AB, CD be solid parallelopipeds, of which AB is contained by the parallelograms AE, AF, AG which are equiangular, each to each, to the parallelograms CH, CK, CL, which contain the solid CD. The ratio which the solid AB has to the solid CD, shall be the same with that which is compounded of the ratios of the sides AM to DL, AN to DK, and AO to DH. Produce MA, NA, OA to P, Q, R, so that AP be equal to DL, AQ to DK, and AR to DH; and complete the solid parallelopiped AX contained by the parallelograms AS, AT, AV similar and equal to CH, CK, CL, each to each. Therefore the solid AX is equal to the solid CD. Complete likewise the solid C. 11. AY, the base of which is AS, and AO one of its insisting straight lines. Take any straight line a, and as MA to AP, so make a to b; and as NA to AQ, so † 12, 6, make b to c; and as AO to AR, so c to d. Then, be * P *32.11. * 25. 11. cause the parallelogram AE is equiangular to AS, AE is to AS, as the straight line a to c, as is demonstrated in the 23d Prop. Book VI.: and the solids AB, AY, being betwixt the parallel planes BOY, EAS, are of the same altitude: therefore the solid AB is to the solid AY, as the base AE to the base AS; that is, as the straight line a is to c. And the solid AY is to the solid AX, as the base OQ is to the base QR; that is, as the straight line OA to AR; that is, as the straight line c to the straight line d. D is to the solid AY, as a is ex æquali, the solid AB is H But to the solid AX, or CD which is equal to it, as the straight line a is to d. * Def. A.5. the ratio of a to d is said to be compounded* of the ratios of a to b, b to c, and c to d, which are the same with the ratios of the sides MA to AP, NA to AQ, and OA to AR, each to each: and the sides AP, AQ, AR are equal to the sides DL, DK, DH, each to each: therefore the solid AB has to the solid CD the ratio which is the same with that which is compounded of the ratios of the sides AM to DL, AN to DK, and AO to DH. Q. E. D. See N. PROP. XXXIV. THEOR. The bases and altitudes of equal solid parallelopipeds, are reciprocally proportional: and if the bases and altitudes be reciprocally proportional, the solid parallelopipeds are equal. Let AB, CD be two solid parallelopipeds: and first, let the insisting straight lines AG, EF, LB, HK; CM, NX, OD, PR, be at right angles to the bases. If the solid AB be equal to the solid CD, their bases shall be reciprocally proportional to their altitudes; that is, as the base EH is to the base NP, so shall CM be to AG. F H P If the base EH be equal to the `K BR D base NP, then because the solid AB is likewise equal to the solid CD, CM shall be equal to AG: because if the bases EH, NP be equal, X but the altitudes AG, CM be not 'equal, neither shall the solid AB be equal to the solid CD: but the solids are equal, by the hypothesis; therefore the altitude CM is not unequal to the altitude AG; that is, they are equal. Wherefore, as the base EH to the base NP, so is CM to AG. R D K B M Next, let the bases EH, NP not be equal, but EH greater than the other: then since the solid AB is equal to the solid CD, CM is therefore greater than AG: for if it be not, neither also in this case would the solids AB, CD be equal, which, by the hypothesis, are equal. Make then CT equal to AG, and complete the solid parallelopiped CV, H L P A E 32. 11. 25. 11. of which the base is NP, and altitude CT. Because the solid AB is equal to the solid CD, therefore the solid AB is to the solid CV, as the solid CD to the *7.5. solid CV: but as the solid AB to the solid CV, so * is the base EH to the base NP; for the solids AB, CV are of the same altitude: and as the solid CD to CV, SO is the base MP to the base PT, and so is the straight line MC* to CT; and CT is equal to AG: *1.6. therefore as the base EH to the base NP, so is MC to AG. Wherefore the bases of the solid parallelopipeds AB, CD are reciprocally proportional to their altitudes. Let now the bases of the solid parallelopipeds AB, CD be reciprocally proportional to their altitudes, viz. as the base EH is to the base NP, so let CM be to AG: the solid AB shall be equal to the solid CD. K BR D F H P A E C * A.5. If the base EH be equal to the base NP, then since EH is to NP as the altitude of the solid CD is to the altitude of the solid AB, therefore the altitude of CD is equal* to the altitude of AB: but solid parallelopipeds upon equal bases, and of the same altitude, are equal to one another; therefore the solid AB is * 31. 11. equal to the solid CD. * But let the bases EH, NP be unequal, and let EH be the greater of the two. Therefore, since, as the base EH to the base NP, so is CM the altitude of the solid CD to AG the altitude of AB, CM is greater than AG. Therefore, as before, take CT equal to AG, and complete the solid CV. And because the base EH |