I THEOR. PROP. XXVIII. THE OR. Book XI. a folid parallelepiped be cut by a plane paffing through See N. the diagonals of two of the oppofite planes; it fhall be cut in two equal parts. Let AB be a folid parallelepiped, and DE, CF, the diagonals of the oppofite parallelograms AH, GB, viz. those which are drawn betwixt the equal angles in each: And because CD, FE are each of them parallel to GA, and not in the fame plane with it, CD, FF are parallel; wherefore the diagonals CF, a 9. 117 DE are in the plane in which the parallels are, and are themselves parallels: And the plane CDEF fhall cut the folid AB into two equal parts. A D B b 16. 17. F C 34. I. Hd 24. 11. E Because the triangle CGF is equal to the triangle CBF, and the triangle DAE to DHE; and that the parallelogram CA is equal and fimilar to the oppofite one BE; and the parallelogram GE to CH: Therefore the prifm contained by the two triangles CGF, DAE, and the three parallelograms CA, GE, EC, is equal to the prifm contained by the two triangles CBF, DHE, e C. 11. and the three parallelograms BE, CH, EC, because they are contained by the fame number of equal and fimilar planes, alike fituated, and none of their folid angles are contained by more than three plane angles. Therefore the folid AB is cut into two equal parts by the plane CDEF. Q. E. D. N. B The infifting ftraight lines of a parallelepiped, mentioned in the next and fome following propofitions, are the fides of the parallelograms betwixt the base and the opposite plane parallel to it.' OLID parallelepipeds upon the fame bafe, and of the se: N. fame altitude, the infifting ftraight lines of which are terminated in the fame ftraight lines in the plane oppofite to the bafe, are equal to one another. P 4 Let Book XI. Let the folid parallelepipeds AH, AK be upon the fame base w AB, and of the fame altitude, and let their infifting ftraight See the fi- lines AF, AG, LM, LN, be terminated in the fame ftraight gures below. line FN, and CD, CE, BH, BK be terminated in the fame ftraight line DK; the folid AH is equal to the folid AK. First, Let the parallelograms DG, HN, which are oppofite to the base AB, have a common fide HG: Then, because the folid AH is cut by the plane AGHC paffing through the diago nals AG, CH of the oppofite planes ALGF, CBHD, AH is cut into two equal parts by the plane AGHC: Therefore the folid AH is double of the prifm which is contained betwixt the triangles ALG, CBH: For the fame reason, because the folid AK is cut by the plane LGHB through the diagonals LG, BH of the oppofite planes ALNG, D H K. G N B A L CBKH, the folid AK is double of the fame prifm which is contained betwixt the triangles ALG, CBH. Therefore the folid AH is equal to the folid AK. But, let the parallelograms DM, EN oppofite to the bafe, have no common fide: Then, becaufe CH, CK are parallelograms, CB is equal to each of the oppofite fides DH, EK; wherefore DH is equal to EK: Add, or take away the common part HE; then DE is equal to HK: Wherefore alfo the triangle CDE is equal to the triangle BHK: And the parallelogram DG is equal to the parallelogram HN: For the fame reafon, the triangle AFG is equal to the triangle LMN, and the parallelogram CF is equal to the parallelogram BM, and C F N H K C B A L f C. II. A CG to BN; for they are oppofite. Therefore the prism which is contained by the two triangles AFG, CDE, and the three parallelograms AD, DG, GC is equal to the prifm, contain ed by the two triangles LMN, BHK, and the three parallelograms BM, MK, KL. If therefore the prifm LMNBHK be taken taken from the folid of which the bafe is the parallelogram Book XI. AB, and in which FDKN is the one oppofite to it; and if from this fame folid there be taken the prifm AFGCDE; the remaining folid, viz. the parallelepiped AH, is equal to the remaining parallelepiped AR. Therefore folid parallelepipeds, &c. Q. E. D. PROP. XXX. THEOR. SOLID parallelepipeds upon the fame bafe, and of the see N. fame altitude, the infifting ftraight lines of which are not terminated in the fame ftraight lines in the plane oppofite to the base, are equal to one another. Let the parallelepipeds CM, CN be upon the fame base AB, and of the fame altitude, but their infisting straight lines AF, AG, LM, LN, CD, CE, BH, BK not terminated in the fame ftraight lines: The folids CM, CN are equal to one another. 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 afe the parallels LB, MHPQ, in which alfo is the figure BLPQ; and the plane ACDF is that in which are the parallels AC, FDOR, in which alfo is the figure CAOR; therefore the figures BLPQ, CAOR are in parallel planes: In like manner, because the plane ALNG is parallel to the oppofite plane CBKE, and that the plane ALNG is that in which are the parallels P 3 AL, Book XI. AL, OPGN, in which alfo is the figure ALPO; and the plane CBKE is that in which are the parallels CB, RQEK, in which alfo is the figure CBQR; therefore the figures ALPO, CBQR are in parallel planes: And the planes ACBL, ORQP are parallel; therefore the folid CP is a parallelepiped: But the folid CM, of which the bafe is ACBL, to which FDHM is the a 29. 11. oppofite parallelogram, is equal to the folid CP, of which the See N. bafe is the parallelogram ACBL, to which ORQP is the one oppofite; because they are upon the fame base, and their infifting straight lines AF, AO, CD, CR; LM, LP, BH, BQ are in the fame ftraight lines FR, MQ: And the folid CP is equal to the folid CN; for they are upon the fame bafe ACBL, and their infifting ftraight lines AO, AG, LP, LN; CR, CE, BQ, BK are in the fame ftraight lines ON, RK: Therefore the folid CM is equal to the folid CN. Wherefore folid parallelepipeds, &c. Q. E. D. SOLID parallelepipeds which are upon equal bafes, and of the fame altitude, are equal to one another. Let the folid parallelepipeds AE, CF, be upon equal bafes AB, CD, and be of the fame altitude; thé folid AE is equal to the folid CF. Firft, Let the infifting ftraight lines be at right angles to the bafes AB, CD, and let the bafes be placed in the fame plane, and a 13. II. 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 bafes are, in the point L, is common to the two 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 infifting straight lines: Therefore, because the parallelogram AB is equal to CD, as the bafe AB is to the bafe LQ, fo is the base CD to the fame ¢ 7. 5. LQ: And because the folid parallelepiped AR is cut by the plane LMEB, which is parallel to the oppofite planes AK, DR; as the base AB is to the bafe LQ, fo is d the folid AE to the d 25. 11. folid LR: For the fame reafon, 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 before was proved: Therefore as the folid AE to the folid P F R N M E G X D Q B C L A S HT LR, fo is the folid CF to the folid LR; and therefore the folid с AE is equal to the folid CF. But let the folid parallelepipeds SE, CF be upon equal bafes SB, CD, and be of the fame altitude, and let their infifting 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 to the fo- f 29. 11. lid SE, of which the base is LE, and to which SX is opposite ; 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 |