PROP. XXVIII. THEOR. Book XI. IF F a folid parallelepiped be cut by a plane paffing thro' See N. the diagonals of two of the oppofite planes; it fhall be cut into two equal parts. Let AB be a folid parallelepiped, and DE, CF the diagonals of the oppofite parallelograms AH, GB, viz. thofe 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, EF are parallel; wherefore the diagonals CF, DE are in the plane in a. 9. 11. which the parallels are, and are them felves parallels . and the plane CDEF C B b. 16. 11. F C. 34. I. D H E d. 24. II. Becaufe 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 e. C. 11. prifm contained by the two triangles CBF, DHE, 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 straight lines of a parallelepiped, mentioned in the next and fome following Propofitions, are the fides of the ⚫ parallelograms betwixt the base and the oppofite plane parallel ' to it.' PROP. XXIX. THEOR. SOLID parallelepipeds upon the fame bafe, and of the See N. fame altitude, the infifting straight lines of which are terminated in the fame ftraight lines in the plane oppofite to the bafe, are equal to one another. Let Book XI. Let the folid parallèlepipeds AH, AK be upon the fame base AB, and of the fame altitude, and let their infifting straight lines AF, AG, See the fi- LM, LN; CD, CE, BH, BK be terminated in the fame straight gures be-lines FN, DK. the folid AH is equal to the folid AK., low. First, Let the parallelograms DG, HN which are opposite to the base AB have a common fide HG. then because the solid AH is cut by the plane AGHC paffing through the diagonals AG, CH of the a. 28. 11. oppofite planes ALGF, CBHD, AH is cut into two equal parts by the plane AGHC. therefore the D H folid AH is double of the prifm which is contained by the tri F K N of the oppofite planes ALNG, CBKH, the folid AK is double of the fame prifm which is contained by the triangles ALG, CBH. Therefore the folid AH is equal to the folid AK. then DE is to the tri But let the parallelograms DM, EN opposite to the base have no common fide. then because CH, CK are parallelograms, CB is eb. 34. 1. qual b to each of the oppofite fides DH, EK; wherefore DH is équal to EK. add, or take away the common part HE; e. 38. 1. equal to HK. wherefore alfo the triangle CDE is equal d. 36. 1. angle BHK. and the parallelogram DG is equal to the parallelo gram HN. for the fame reason, the triangle AFG is equal to the . 24. 11. triangle LMN, and the parallelogram CF is equal to the paralleló gram BM, and CG to BN; for they are oppofite. Therfore the prism which is contained by the two triangles AFG, CDE, and the 1.C.. three parallelograms AD, DG, GC is equal f to the prifm con tained by the two triangles LMN, BHK, and the three parallelograms BM, MK, KL. If therefore the prifm LMN, BHK be taken from from the folid of which the base is the parallelogram AB, and in Book XI. which FDKN is the one oppofite to it; and if from this fame folid there be taken the prism AFG, CDE; the remaining solid, viz. the parallelepiped AH, is equal to the remaining parallelepiped AK. Therefore folid parallelepipeds, &c. Q. E. D. PROP. XXX. THEOR. SOLID parallelepipeds upon the fame base, and of see N. the fame altitude, the infifting ftraight lines of which are not terminated in the same straight lines in the plane oppofite to the bafe, are equal to one another. Let the parallelepipeds CM, CN be upon the fame base AB, and of the fame altitude, but their infifting ftraight lines AF, AG, LM, LN, CD, CE, BH, BK not terminated in the fame straight 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 are the parallels LB, MHPQ, in which also is the figure BLPQ; and the plane ACDF is that in which are the parallels AC, FDOR, in which also 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 Book XI. are the parallels AL, OPGN, in which alfo is the figure ALPO; anɗ the plane CBKE is that in which are the parallels CB, RQEK, in which also 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 base is ACBL, to which FDHM is the oppofite paralle29. 11. logram, is equal to the folid CP of which the base is the parallelo See N. a gram ACBL, to which OROP is the one oppofite; because they are upon the fame bafe, and their infifting ftraight 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 base ACB L, 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 solid parallelepipeds, &c. Q. E. D. PROP. XXXI. THEOR. SOLID parallelepipeds which are upon equal bases, and of the fame altitude, are equal to one another. Let the folid parallelepipeds AE, CF, be upon equal bases AB,* CD, and be of the fame altitude; the folid AE is equal to the folid CF. First, Let the infifting straight lines be at right angles to the bafes AB, CD, and let the bafes be placed in the fame plane, and fo as that that the fides CL, LB be in a straight line; therefore the ftraight Book Xİ. line LM which is at right angles to the plane in which the bafes are, a in the point L, is common to the two folids AE, CF; let the a. 13. 11. other infisting lines of the folids be AG, HK, BE; DF, OP, c. 7. 5. 7.5. CN. and first, let the angle ALB be equal to the angle CLD; then AL, LD are in a straight line b. produce OD, HB, and let them b. 14. 1. 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 cqual to CD, as the bafe AB is to the bafe LQ, fo is the bafe CD to the fame 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 base LQ, fo is the folid AE to the folid d. 25. 11. LR. for the fame reason, because the folid parallelepiped CR is cut by the plane LMFD which is parallel to the oppofite planes CP, LR, fo is the folid CF to the folid LR; and therefore the folid 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 ftraight 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 folid SE of which the bafe is LE, and to which SX is f. 29. 11. 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 arc in the fame ftraight lines AT, GX. and |