в Book XI. cause the parallelogram AB is equal to SB, for they are upon w the fame bafe LB, and between the fame parallels LB, AT; 8 35. I. and that the base the folid AE is e qual to the folid X CF; but the folid AE is equal to the folid SE, as was demonftrated; therefore the folid SE is equal to the folid CF. But, if the infifting ftraight lines AG, HK, BE, LM; CN, RS, DF, OP, be not at right angles to the bafes AB, CD; in this cafe likewife the folid AE is equal to the folid CF: From the points G, K, E, M, N, S, F, P, draw the ftraight lines h 11. II. GQ, KT, EV, MX; NY, SZ, FI, PU, perpendicular to the plane in which are the bases AB, CD; and let them meet it in the points Q, T, V, X; Y, Z, I, U, and join QT, TV, VX, XQ; YZ, ZÏ, IU, UY: Then, because GQ, KT, are at right i 6. 11. А HQ T angles to the fame plane, they are parallel to one another : And MG, EK are parallels; therefore the planes MQ, ET, of which one paffes through MG, GQ, and the other through EK, KT which are parallel to MG, GQ, and not in the fame k 15. 11. plane with them, are parallel to one another: For the fame reafon, the planes MV, GT are parallel to one another: Therefore the folid QE is a parallelepiped: In like manner, it may be proved, that the folid YF is a parallelepiped: But, from what has been demonftrated, the folid EQ is equal to the folid FY, because they are upon equal bafes MK, PS, and of the fame altitude, and have their infifting ftraight lines at right angles to to the bafes: And the folid EQ is equal to the folid AE; and Book XI. II. SOLID parallelepipeds which have the fame altitude, are see N. to one another as their bases. Let AB, CD be folid parallelepipeds of the fame altitude: They are to one another as their bafes; that is, as the base AE to the base CF, fo is the folid AB to the folid CD. a To the ftraight line FG apply the parallelogram FH equal a Cor.45.1. to AE, fo that the angle FGH be equal to the angle LCG; and complete the folid parallelepiped GK upon the base FH, one of whose infifting lines is FD, whereby the folids CD, GK must be of the fame altitude: Therefore the folid AB is equal b 31. II. το the folid GK, because they are upon equal B D K Q bafes AE, FH, and are of the fame altitude: And because the fo lid parallelepi ped CK is cut C E A M by the plane DG which is parallel to its oppofite planes, the base COR. From this it is manifeft that prifms upon triangular bafes, of the fame altitude, are to one another as their bafes. Let the prifms, the bafes of which are the triangles AEM, CFG, and NBO, PDQ the triangles oppofite to them, have the fame altitude; and complete the parallelograms AE, CF, and the folid parallelepipeds AB, CD, in the firft of which let MO, and in the other let GQ be one of the infifting lines. And because the folid parallelepipeds AB, CD have the same altitude, they are to one another as the bafe AE is to the base CF; Book XI ₫ 28. 11. CF; wherefore the prifms, which are their halves, are to one another, as the bafe AE to the bafe CF; that is, as the triangle AEM to the triangle CFG. a 24. II. b C. II. € I. 6. SIMILAR folid parallelepipeds are one to another in the triplicate ratio of their homologous fides. Let AB, CD be fimilar folid parallelepipeds, and the fide AE homologous to the fide CF: The folid AB has to the folid CD, the triplicate ratio of that which AE has to CF. Produce AE, GE, HE, and in thefe produced take EK equal to CF, EL equal to FN, and EM equal to FR; and complete the parallelogram KL, and the folid KO: Because KE, EL are equal to CF, FN, and the angle KEL equal to the angle CFN, because it is equal to the angle AEG which is equal to CFN, by reafon that the folids AB, ČD are fimilar; therefore the paral lelogram KL is fimilar and equal to the parallelogram CN: For the fame reason, the parallelogram MK is fimilar and equal to CR, and alfo OE : B X D H P R G N K C F A L M to FD There- с and CI. 6. 237 d 25. II, and as HE to EM, foc is PE to KM: Therefore as the parallelo- Book XI. gram AG to the parallelogram GK, fo is GK to KL, and PE to KM: But as AG to GK, fod is the folid AB to the folid EX; and as GK to KL, fod is the folid EX to the folid PL; and as PE to KM, fo d is the folid PL to the folid KO: And therefore as the folid AB to the folid EX, fo is EX to PL, and PL to KO: But if four magnitudes be continual proportionals, the firft is faid to have to the fourth the triplicate ratio of that which it has to the fecond: Therefore the folid AB has to the folid KO, the triplicate ratio of that which AB has to EX: But as AB is to EX, fo is the parallelogram AG to the parallelogram GK, and the ftraight line AE to the straight line EK. Wherefore the folid AB has to the folid KO, the triplicate ratio of that which AE has to EK. And the folid KO is equal to the folid CD, and the ftraight line EK is equal to the ftraight line CF. Therefore the folid AB has to the folid CD, the triplicate ratio of that which the fide AE has to the homologous fide CF, &c. Q. E. D. COR. From this it is manifeft, that, if four ftraight lines be continual proportionals, as the firft is to the fourth, fo is the folid parallelepiped defcribed from the first to the fimilar folid fimilarly defcribed from the fecond; because the first ftraight line has to the fourth the triplicate ratio of that which it has to the second. PROP. D. THE Q R. SOLID parallelepipeds contained OLID parallelepipeds contained by parallelograms See N, equiangular to one another, each to each, that is, of which the folid angles are equal, each to each, have to one another the ratio which is the fame with the ratio compounded of the ratios of their fides. Let AB, CD be folid parallelepipeds, of which AB is con tained by the parallelograms AE, AF, AG equiangular, each to each, to the parallelograms CH, CK, CL which contain the folid CD. The ratio which the folid AB has to the folid CD is the fame with that which is compounded of the ratios of the fides AM to DL, AN to DK, and AO to DH. Produce Book XI. a C. II. b 32. II. Produce MA, NA, OA to P, Q, R, fo that AP be equal to DL, AQ to DK, and AR to DH; and complete the folid parallelepiped AX contained by the parallelograms AS, AT, AV fimilar and equal to CH, CK, CL, each to each. Therefore the folid AX is equal to the folid CD. Complete likewife the folid AY, the bafe of which is AS, and of which AO is one of its infifting ftraight lines. Take any ftraight line a, and as MA to AP, fo make a to b; and as NA to AQ, fo make b to c; and as AO to AR, fo c to d: Then, because the parallelogram AE is equiangular to AS, AE is to AS, as the ftraight line a to c, as is demonftrated in the 23. Prop. Book 6. and the folids AB, AY, being betwixt the parallel planes BY, EAS, are of the fame altitude. Therefore the folid AB is the folid AY, as the base AE to the base AS; that is, as the ftraight line a is to c. And the folid AY is to the folid € 25. II. AX, as the bafe OQ is to the bafe QR; that is, as the straight line OA to AR; that is, as the ftraight line c to the straight line d. And because the folid AB is to the folid AY, as a is to c, and the folid AY to the folid AX, as c is to d; ex aequali, the folid AB is to the folid AX, or CD which is equal to it, as the straight line a is to d. But the ratio of a to d is said to d def. A. 5. be compounded of the ratios of a to b, b to c, and c to d, which are the fame with the ratios of the fides MA to AP, NA to AQ, and OA to AR, each to each. And the fides AP, AQ, AR are equal to the fides DL, DK, DH, each to each. There fore the folid AB has to the folid CD the ratio which is the fame with that which is compounded of the ratios of the fides AM to DL, AN to DK, and AO to DH. Q. E. D. PROP |
