and fo as that the fides CL, LB be in a straight line; there- Book XI. fore the ftraight line LM, which is at right angles to the plane in which the bases are, in the point L, is common to the two a 13. 11. folids AE, CF; let the other infifting lines of the folids be AG, HK, BE; DF, OP, CN : And first, let the angle ALB be equal to the angle CLD; then AL, LD are in a straight line ". b 14. I. Produce OD, HB, and let them meet in Q, and complete the folid parallelepiped LR, the base of which is the parallelogram LQ, and of which LM is one of its infisting straight lines: Therefore, because the parallelogram AB is equal to CD, as the bafe AB is to the bafe LQ, fo is the bafe CD to the fame c 7. s. LQ: And because the folid parallelepiped AR is cut by the plane LMEB, which is parallel to the oppofite planes AK, DR; as the bafe AB is to the bafe LQ, fo is the folid AE to the d 25. 11. folid LR: For the fame reason, because the folid parallelepiped CR is cut by the plane LMFD, which is parallel to the oppofite planes CP, BR; as the base CD to the bafe LQ, fo is the folid CF to the folid LR: But as the bafe AB to the base LQ, fo the bafe CD to the bafe LQ, as P F R N M E G X K D Q B before was proved: Therefore as the C L folid AE to the fo lid LR, fo is the folid CF to the folid LR; and therefore the folid AE is equal to the folid CF. e But let the folid parallelepipeds SE, CF be upon equal bafes SB, CD, and be of the fame altitude, and let their infisting ftraight lines be at right angles to the bafes; and place the bafes SB, CD in the fame plane, fo that CL, LB be in a straight line; and let the angles SLB, CLD be unequal; the folid SE is alfo in this cafe equal to the folid 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 folids AE, LR: Therefore the folid AE, of which the bafe is the parallelogram LE, and AK the one oppofite to it, is equal f to the fo- f 29. 11. lid SE, of which the bafe is LE, and to which SX is oppofite; for they are upon the fame bafe LE, and of the fame altitude, and their infifting ftraight lines, viz. LA, LS, BH, BT; MG, MV, EK, EX are in the fame ftraight lines AT, GX: And be caufe a Book XI. caufe the parallelogram AB is equal to SB, for they are upin on the fame bafe LB, and between the fame parallels LB, AT; 8 35. I. and that the base SB is equal to the bafe CD: therefore qual to the folid C L A S HT CF; But if the infifting straight lines AG, HK, BE, LM; CN, RS, DF, OP, be not at right angles to the bafes AB, CD; in this cafe likewise the folid AE is equal to the folid CF: From the points G, K, E, M, N, S, F, P, draw the straight lines h 11. 11. 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, ZI, IÚ, ÚY: Then, because GQ, KT, are at right M E P i 6. 11. k 15. 11. angles to the fame plane, they are parallel i 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 plane with them, are parallel to one another: For the fame reafon, the planes MV, GT are parallel to one another: There fore 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. the folid FY to the folid CF; because they are upon the fame bases and of the fame altitude: Therefore the folid AE is equal 1 29. or 30. to the folid CF. Wherefore folid parallelepipeds, &c. Q.E.D. II. SOLI PROP. XXXII. THEOR. LID parallelepipeds which have the fame altitude, see N. are to one another as their bases. Let AB, CD be folid parallelepipeds of the fame altitude: They are to one another as their bases; that is, as the base AE to the bafe CF, fo is the folid AB to the folid CD. To the ftraight line FG apply the parallelogram FH equal a Cor. 45. Bi 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 whofe infifting lines is FD, whereby the folids CD, GK must be of the fame altitude: Therefore the folid AB is equalb b 31. 14. to the folid AE, FH, and G H altitude: And because the fo ped CK is cut by the plane DG which is parallel to its oppofite planes, the base HF is to the base FC, as the folid HD to the folid DC: But c 25. Is; the bafe HF is equal to the base AE, and the folid GK to the folid AB: Therefore, as the base AE to the base CF, fo is the folid AB to the folid CD. Wherefore folid parallelepipeds, &c. Q. E. D. COR. From this it is manifeft that prifms upon triangular bas fes, 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 first of which let MO, and in the other let GQ be one of the infifting lines. And because the solid parallelepipeds AB, CD have the fame altitude, they are to one another as the bafe AE is to the base CF; Book XI. 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. 28. II. a 24. Ir. b C. II. c 1. 6. ST IMILAR 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, CD are fimilar; therefore the parallelogram KL is fimilar and equal to the parallelogram CN: For the fame reafon, the parallelogram MK is fimilar and equal to CR, and alfo OE to FD: Therefore three parallelograms of the folid KO are equal and fimilar to three parallelograms of the folid CD: And the three oppofite ones in each folid are equal and fimilar to thefe : Therefore the fo a lid KO is equal b B X D H P R G N K C F A E L M and fimilar to the folid CD: Complete the parallelogram GK, and complete the folids EX, LP upon the bafes GK, KL, fo that EH be an infifting ftraight line in each of them, whereby they must be of the fame altitude with the folid AB: And becaufe the folids AB, CD are fimilar, and, by permutation, as AE is to CF, fo is EG to FN, and fo is EH to FR; and FC is equal to EK, and FN to EL, and FR to EM; therefore as AE to EK, fo is EG to EL, and fo is HE to EM: But as AE to EK, fo is the parallelogram AG to the parallelogram GK; and as GE to EL, fo is GK to KL; and and as HE to EM, fo 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 folide 1. 6. EX; and as GK to KL, fod is the folid EX to the folid PL;d 25. 11. and as PE to KM, fod 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 straight 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 straight line has to the fourth the triplicate ratio of that which it has to the fecond. SOL PROP. D. THEOR. 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 contained 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 |