taken from the folid of which the base is the parallelogram AB, Book XI. and in which FDKN is the one oppofite to it; and if from this fame folid there be taken the prifm AFG, CDE; the remaining folid, viz. the parallelepiped AH, is equal to the remaining parallelepiped AK. Therefore folid parallelepipeds, &c. Q. E, D.

SOL

PROP. XXX. THEOR.

OLID parallelepipeds upon the fame base, 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 infisting straight 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

Book XI. are the parallels AL, OPGN, in which also is the figure ALPO; and 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 bafe is ACBL, to which FDHM is the oppofite a. 29. 11. parallelogram, is equal to the folid CP of which the base is the

a

See N.

A

a

parallelogram ACBL, to which ORQP is the one oppofite; because they are upon the same base, and their insisting 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 infisting 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.

PROP. XXXI. THEOR.

OLID parallelepipeds which are upon equal bafes, and of the fame are to

SOL

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 infisting straight lines be at right angles to the bases AB, CD, and let the bases be placed in the same plane, and so 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 bases are,

in the point L, is common a 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 first, let the angle ALB be equal to the angle CLD; then AL, LD are in a ftraight 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 infisting straight lines. therefore because the parallelogram AB is equal to CD, as the base AB is to the base Q, fo is the bafe c. 7. 5. CD to the fame LQ. and because the folid parallelepiped AR is cut by the plané 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. XI. AE to the folid LR. for the fame reafon, because the folid parallelepiped CR is cut by the plane LMFD which is parallel to the

folid AE to the folid LR, fo is the folid CF to the folid LR; and

e

therefore the folid AE is equal to the folid CF.

e. 9. 5.

But let the folid parallelepipeds SE, CF be upon equal' bafes SB, CD, and be of the same altitude, and let their infisting straight lines be at right angles to the bases; 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 also 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 base is the parallelogram LE, and AK the one oppofite to it, is equal f to the folid SE, of which the bafe is LE, f. 29. 11. 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 ftraight

P

Book XI. lines AT, GX. and because the parallelogram AB is equal to SB, for they are upon the fame base LB, and between the fame parallels LB, AT; and

g. 35. I.

that the bafe SB is P

F

R

to the folid CF; but the folid AE is equal to the folid SE, as was demonstrated; therefore the folid SE is equal to the solid 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 GQ, KT, EV, MX; h. 11. 11. NY, SZ, FI, PU, perpendicular to the plane in which are the bafes AB, CD; and let them meet it in the points Q, T, V, X;

h

i. 6. II.

A HQ T

Y, Z, I, U, and join QT, TV, VX, XQ; YZ, ZI, IU, UY. then because GQ, KT, are at right 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, k. 15. 11. and not in the fame plane with them, are parallel to one another. for the fame reason, the planes MV, GT are parallel to one another. therefore 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

1. 29.or 30.

have their infisting straight lines at right angles to the bafes. and Book XI. the folid EQ is equal to the folid AE; and the folid FY to the folid CF; because they are upon the fame bafes and of the fame altitude. therefore the folid AE is equal to the folid CF. Wherefore folid parallelepipeds, &c. Q. E. D.

II.

LID parallelepipeds which have the fame altitude, See N. are to one another as their bases.

SOLI

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 the folid AB to the folid CD.

To the straight line FG apply the parallelogram FH equal a to a. Cor.45.1 AE, fo that the angle FGH be equal to the angle LCG; and complete the folid parallelepiped GK upon the base FH, one of whose infifting lines is FD, whereby the folids CD, GK must be of the

fame altitude. therefore the folid AB is equal to the folid GK, b. 31. 11. because they are

which is parallel

the folid parallel

C

to its oppofite planes, the bafe HF is to the bafe FC, as the folid c. 25. 11. HD to the folid DC. but the bafe HF is equal to the bafe AE, and the folid GK to the folid AB. therefore as the base AE to the bafe 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 bafes, of the fame altitude, are to one another as their bases.

Let the prifms the bases of which are the triangles AEM, CFG, and NBO, PDQ the triangles oppofite to them, have the same 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 folid parallelepipeds AB, CD have the fame altitude, they are to one