BOOK XI, PROP. XXX. THEOR. OLID parallelopipeds upon the fame base, and of the fame altitude, the infifting ftraight lines of which are not terminated in the fame ftraight lines, are equal to one another. Let the parallelopipeds CM, CN be upon the fame base AB, and of the fame altitude; but their infifting straight lines AF, AG, LM, LN, CD, CE, BH, BK not terminated in the fame ftraight lines; the folids CM, CN are equal. Produce FD, MH, and NG, KE, and let them meet one another in the points O, P, Q, R; and join AO, LP, BQ, CR: and because the plane LBHM is parallel to the oppofite plane ACDF, and that the plane LBHM is that in which are the parallels LB, MPHQ, in which alfo is the figure BLPQ; and the plane ACDF is that in which are the parallels AC, FODR, in which also is the figure CAOR; therefore the figures BLPQ, CAOR are in parallel planes: In like manner, it may be fhewn, that the figures ALPO, CBQR are in parallel planes and the planes ACBL, ORQP are parallel; therefore the folid CP is a parallelopiped: But the folid CM, of which the bafe is ACBL, a 29. 11. to which FDHM is the oppofite parallelogram is equal to the folid CP, of which the base is the parallelogram ACBL, to which *The infifting ftraight lines are the fides of the parallelograms betwixt the base and the opposite plane. 199 which OROP is the one oppofite, becaufe they are upon the Book XI. fame bafe, and their infifting straight lines AF, AO, CD, CR; W LM, LP, BH, BQ are in the fame ftraight lines FR, MQ and a the folid CP is equal to the folid CN; for they are upon the a 29. 11. fame base ACBL, and their infisting straight lines AO, AG, LP, LN; CR, CE, BQ, BK are in the same straight lines ON, RK; therefore the folid CM is equal to the folid CN. Wherefore, folid parallelopipeds, &c. Q. E. D. SOLI PROP. XXXI. THEOR. OLID parallelopipeds which are upon equal bases, Let the folid parallelopipeds AE, CF be upon equal bafes AB, CD, and be of the fame altitude; the folid AE is equal to the folid CF. to the IS. I. b 13. 11. First, Let the infifting straight lines be at right angles to the bafes AB, CD, and let the bafes be equiangular, having the angle ALB equal to CLD, and let them be placed in the fame plane, and fo as that CL, LB be in a ftraight line; therefore AL, LD are in a straight line 2; and the straight line LM, which is a 3. Cor. at right angles to the plane in which the bases are in the point L, is common b two folids AE, CF; let the other infifting lines of the folids be AG, HK, BE; DF, OP, CN. Produce OD, HB, and let them meet in Q, and complete the parallelo F R M E G K B piped LR, of which the C L bafe is the parallelogram LQ, and LM is one of H its infifting lines: Therefore, because the parallelogram AB is equal to CD, as AB to LQ, fo is CD to LQ: and because the c7. 5. parallelopiped AR is cut by the plane LMEB, which is parallel to the oppofite planes AK, DR; as the bafe AB to the base LQ. d d fo is the folid AE to the folid LR: and because the parallelo- d 25. 11. piped CR is cut by the plane LMFD parallel to the oppofite planes CP, BR; as the bafe CD to the base LQ, fo is the folid CF to the folid LR: But as AB to LQ, fo is CD to LQ, as before was proved; therefore as the folid AE to the folid LR, fo is the folid CF to the folid LR; and therefore the folid AE is equal to the folid CF. e f e 11. 5. f 9.5. Next, BOOK XI. Next, Let the infifting lines be at right angles to the bafes, but the bafes AB, CD not be equiangular. At the point L in g23. 1. the straight line LB, make the angle BLS equal to CLD, and let LS meet HA in S, and complete the parallelogram BS, and the parallelopiped SE, of which the bafe is SB, and one of the infifting lines is LM: Then, the folid SE is equal to the folid h 29. 11. AE, because they are upon the fame bafe LE, and of the fame altitude, and their infisting lines LS, LA, BT, BH; MV, M t k 35. I. L V SA k MG, EX, EK are in the fame ftraight lines SH, VK: and becaufe the parallelogram AB is equal to SB, for they are upon the fame base LB, and between the same parallels LB, AT; and that the bafe AB is equal to the bafe CD; therefore SB is equal to CD; and the angle SLB is equal to the angle CLD: therefore, by the first case, the solid SE is equal to the folid CF : but the folid SE is equal to the folid AE, as was demonstrated; therefore the folid AE is equal to the folid CF. But, if the infisting lines AG, HK, BE, LM; CN, RS, DF, OP be not at right angles to the bafes AB, CD: From the MX; 111.11. points G, K, E, M; N, S, F, P draw1 GQ, KT, EV, NY, SZ, FI, PU perpendiculars 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, IU, UY : Then, because QG, KT are at right angles to the fame plane, m 6. 11. they are parallels: and MG, EK are parallels; therefore the n 15. 11. planes MQ, ET paffing through them are parallel to one another: For the fame reafon, the planes MV, GT are parallel; therefore therefore the folid QE is a parallelopiped. In like manner, it Book XI. may be proved, that the folid YF is a parallelopiped: 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 lines at right angles to the bafes and the folid EQ is equal to the folid AE; and the o 29. or folid FY to the folid CF, because they are upon the fame bases, and of the fame altitude: Therefore the folid AE is equal to the folid CF. Wherefore folid parallelopipeds, &c. Q. E. D. PROP. XXXII, THEOR. OLID parallelopipeds which have the fame alti- SOL Let AB, CD be folid parallelopipeds of the fame altitude; they are to one another as their bases; that is, as the base AE to the base CF, so is the folid AB to the solid CD. Produce CG to H; and to FG apply the parallelogram FH a 30. 11. equal to AE, having FGH for one of its angles; and complete aCor.45.1. b the parallelopiped GK upon the bafe FH, and having FD for one of its infifting lines; therefore the folids CD, GK are of the fame altitude; therefore the folid AB is equal to the folid GK, b 31. 11. because they are upon equal bases AE, FH, and are of the fame altitude and because the folid CK is cut by the plane DG, which is parallel to its oppofite planes, the bafe HF is to the bafe FC, as the folid HD to the folid DC; But the base HF c 25. 11. is equal to the base AE, and the folid GK to the folid AB; therefore, as the bafe AE to the base CF, fo is the folid AB to the folid CD. Wherefore, &c. Q. E. D. COR. 1. From this it is manifeft, that prisms upon triangular bafes, of the fame altitude, are to one another as their bafes. Let the prifms AEM-NBO and CFG-PDQ have the fame altitude; and complete the parallelograms AE, CF, and the pa#allelopipeds AB, CD, of which the infifting lines are MO, GQ: Cc and BOOK XI. and because the parallelopipeds have the fame altitude, AB is to ✅CD, as the base AE to CF; wherefore the prisms, which are d 28. 11. their halves, are to one another, as AE to CF; that is, as the triangle AEM to CFG. COR. 2. Alfo, any prifms of the fame altitude are to one another as their bafes: For prifms upon any other figures may be divided into prifms having triangular bafes: And each triangle is to each triangle, as the prifm upon the first to the prifm eCor.24.5. upon the other; therefore the whole prifm is to the whole as the base of the first to the base of the other. a 3. Cor. 15. 1. C 32. II. ST PROP. XXXIII. THEOR. IMILAR folid parallelopipeds are one to another in the triplicate ratio of their homologous fides. Let AB, CD be fimilar parallelopipeds, and the fide AE homologous to the fide CE: The folid AB has to the folid CD, the triplicate ratio of that which AE has to CE. B X H P G Let the bafes AG, CN be in the fame plane, and the folids AB, CD on different fides of it, and let AE, EC be in a straight line; therefore, because the angles AEG, CEN are equal, GE, EN are in a straight line 2: And if the plane ED be produced beyond GN, it fhall coincide with the plane GH, because these planes have the fame inclination to the common plane; therefore, because the angle GEH is equal to REN, RE is in the fame ftraight line with EH: complete the parallelogram GC and the pa. rallelopipeds EX, NP upon the bafes GC, CN, fo that EH be an infifting line in each of them, whereby they must be of the fame altitude with the folid AB: and because the folids AB, CD are fimilar, as AE to EG, fo is CE to b A E N R D EN; and, by alternation, as AE to EC, fo is GE to EN: For the fame reason, as AE to EC, fo is HE to ER: But as AE to b 1. 6. EC, fob is the parallelogram AG to GC, and the folid AB to the folid EX, because they have the fame altitude; and as GE to EN, fo is the base GC to the base CN, and the folid EX to the folid NP: and as HE to ER, fob is the base HC to the bafe CR, and the folid NP to the folid CD: But as AE to EC, fo is GE to EN, and HE to ER; therefore, as the folid AB to 11. 5. the folid EX, fodis EX to NP, and NP to CD: But if four magnitudes |