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IF

PROP. XXVIII. THEOR.

Book XI.

F a folid parallelepiped be cut by a plane paffing See N. thro' the diagonals of two of the opposite planes; it thall 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 berw xt the equal angles in each. and becaufe 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 themfeives parallels . and the plane CDEF fall cut the folid AB into two equal parts.

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B

b. 16. 11.

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A

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Because the triangle CGF is equal to the triangle CBF, and the triangle DAE to DrE; and that the parallelogram CA is equal and fimilar to the

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pofite 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 equale e. C. 11. to the 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 ftraight lines of a parallelepiped, mentioned in the next and fome following Propofitions, are the fides of the parallelograms betwixt the bafe and the oppofite plane ' parallel to it.'

SOLID

PROP. XXIX. THEOR.

OLID parallelepipeds upon the fame bafe, and of the see 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.

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Book XI.

Let the folid parallelepipeds AH, AK be upon the fame bafe AB, and of the fame altitude, and let their infiffing ftraight lines See the fi- AF, AG. LM, LN; CD, CE, BH, BK be terminated in the fame gures below. ftraight lines FN, DK. the folid AH is equal to the folid AK.

First, Let the parallelograms DG, HN which are opposite to the bafe AB have a common fide HG. then because the folid AH is cut by the plane AGHC paffing thro' the diagonals AG, CH of the oppofite planes ALGF, CBHD, AH is cut into two equal 2. 18. 11. parts by the plane AGHC.

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therefore the folid AH is double
of the prifm which is contained
by the triangles ALG, CBH. for
the fame reafon, becaufe the fo- C
lid AK is cut by the plane LGHB
thro' the diagonals LG, BH of

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the oppofite planes ALNG, CBKH, the folid AK is double of the fame priẩm which is contained by the triangles ALG, CBH. Therefore the folid AH is equal to the folid AK.

But let the parallelograms DM, EN oppofite to the base have no common fide. then becaufe CH, CK are parallelograms, CB is b. 34. 1. 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 c. 38. 1. 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 e. 24. 11. the triangle LMN, and the parallelogram CF is equal to the paral

d. 36. I.

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f. C. 11.

lelogram BM, and CG to BN; for they are oppofite. Therefore the prifm which is contained by the two triangles AFG, CDE, and the three parallelograms AD, DG, GC is equal f to the prifm contained by the two triangles LMN, BHK, and the three parallelograms BM, MK, KL. If therefore the prifm LMN, BHK be

taken from the folid of which the bafe is the parallelogram AB, Book XI. and in which FDKN is the one oppofite to it; and if from this fame fold there be taken the prism AFG, CDE; the remaining folid, 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 bafe, and of See N. the fame altitude, the infifting ftraight lines of which are not terminated in the fame ftraight 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 lives 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 be caufe the plane LBHM is parallel to the oppofite plane ACDF,

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and that the plane LBHM is that in which are 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

Book XI. are the parallels 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, OROP are parallel; therefore the folid CP is a parallelepiped. but the folid CM of which the bafe is ACBL, to which FDHM is the oppofite

a. 29. 11. parallelogram, is equal to the folid CP of which the bafe is the

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See N.

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parallelogram ACBL, to which ORQP is the one oppofite; because they are upon the fame bafe, and their infifting straight lines AF, AO, CD, CR; LM, LP, BH, BQ are in the fame straight lines FR, MQ, and the folid CP is equal to the folid CN, for they are upon the fame bafe ACBL, and their infifting straight 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.

SOLI

PROP. XXXI. THEOR.

OLID 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 bafes 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 the fides CL, LB be in a straight line; therefore the ftraight Book XI. 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 a. 13. 11. other infifting lines of the folids be AG, HK, BE; DF, OP, CN. and firft, 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. 14 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 base c. 7. 5. 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 bafe AB is to the bafe LQ, fo is the folid d. 25. AE to the folid LR. for the fame reafon, becaufe the folid parallelepiped CR is cut by the plane LMFD which is parallel to the

oppofite planes CP, BR; as the bafe CD to the bafe LQ, so is the folid CF to the folid LR. but as the bafe AB to the bafe LQ, fo the bafe

CD to the bafe LQ, as before was prov

ed. therefore as the

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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.

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But let the folid parallelepipeds SE, CF be upon equal bafes SB, CD, and be of the fame altitude, and let their infifting straight 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 folid SE, of which the bafe is LE, f. 17. and to which SX is oppofite; for they are upon the fame bafe LE, and of the fame altitude, and their infifting straight lines, viz. LA, LS, BH, BT; MG, MV, EK, EX are in the fame straight P

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