34. I.

PROPOSITION XXVIII.

THEOREM.

If a folid Parallelepipedon be cut by a Plane paffing thro' the Diagonals of two oppofite Planes, that Solid will be bifected by the Plane.

LET

ET the folid Parallelepipedon A B be cut by the Plane CDEF, paffing thro' the Diagonals C F, DE, of two oppofite Planes. I fay, the Solid A B is bifected by the Plane CDEF.

For, because the Triangle C G F is equal to the Triangle CBF, and the Triangle ADE to the Triangle +24 of tbis. DEH, and the Parallelogram CA to + the Parallelogram BE, for it is oppofite to it; and the Parallelogram G E to the Parallelogram CH; the Prifm contained by the two Triangles CGF, A DE, and the three Parallelograms GE, AC, CE, is equal to the Prifm contained under the two Triangles CF B, DEH, and the three Parallelograms CH, BE, CE; for ‡ they are contained under Planes equal in Number and Magnitude. Therefore, the whole Solid A B is bifected by the Plane CDEF; which was to be demonftrated.

Def. 10. of this.

* 34. 1.

PROPOSITION XXIX.

THEOREM.

Solid Parallelepipedons, being conftituted upon the fame Bafe, and having the fame Altitude, and whofe infiftent Lines are in the fame Right Lines, are equal to one another.

LET the folid Parallelepipedons CM, BF, be

conftituted upon the fame Bafe A B, with the fame Altitude, whofe infiftent Lines A F, A G, LM, LN, CD, CE, BH, BK, are in the fame Right Lines FN, DK. I fay, the Solid CM is equal to the Solid B F.

For, because CH, CK, are both Parallelograms, CB fhall be * equal to DH, or E K; wherefore DH

P

is equal to E K. Let E H, which is common, be taken away, then the Remainder DE will be equal to the Remainder H K, and fo the Triangle DE C is + equal + 8. . to the Triangle HK B, and the Parallelogram DG equal to the Parallelogram HN; for the fame Reafon the Triangle A F G is equal to the Triangle L M N. Now the Parallelogram CF is equal to the Paralle-‡ 24 of thi logram BM, and the Parallelogram C G to the Parallelogram BN, for they are oppofite. Therefore the Prifin contained under the two Triangles AFG, DEC, and the three Parallelograms CF, DG,CG, is equal to the Prism contained under the two Tri-* Def. on angles L MN, HB K, and the three Parallelograms of tbit. BM, HN, BN. Let the common Solid, whofe Base is the Parallelogram A B, oppofite to the Parallelogram GEHM, be added, then the whole folid Parallelepipedon CM is equal to the whole folid Parallelepipedon B F. Therefore, folid Parallelepipedons, being conftitued upon the fame Bafe, and having the fame Altitude, and whofe infiftent Lines are in the fame Right Lines, are equal to one another; which was to be demonftrated.

PROPOSITION, XXX.

THE ORE M.

Solid Parallelepipedons, being conftituted upon the
fame Bafe, and having the fame Altitude, whofe
infiftent Lines are not placed in the fame Right
Lines, are equal to one another.

LET there be folid Parallelepipedons CM, CN,
having equal Altitudes, and ftanding on the fame
Bafe A B, and whofe infiftent Lines A F, AG, LM,
LN, CD, CE, BH, BK, are not in the fame Right
Lines. I fay, the Solid C M is equal to the Solid
CN.

For let NK, DH, be produced, and G E, FM, be drawn, meeting each other in the Points R, X: Let. alfo FM, GE, be produced to the Points O, P, and join A X, LO, CP, BR. The Solid C M, whofe Base is the Parallelogram A CBL, being oppofite to the Parallelogram F DH M, is equal to the Solid CO, 29 of this whofe

6

whofe Bafe is the Parallelogram A CBL, being oppofite to XPR O, for they ftand upon the fame Bafe ACBL; and the infiftent Lines AF, AX, LM, LO, CD, CP, BH, BR, are in the fame right Lines FO, DR: But the Solid C O, whofe Bafe is the Parallelogram A CBL, being oppofite to X PRO, is *29 of this.* equal to the Solid CN, whofe Bafe is the Parallelogram A CBL, being oppofite to GEKN; for they ftand upon the fame Bafe A CB L, and their infiftent Lines AG, A X, CE, CP, LN, LO, BK, BR, are in the fame Right Lines GP, NR: Wherefore the Solid C M fhall be equal to the Solid CN. Therefore, folid Parallelepipedons, being conftituted upon the fame Bafe, and having the fame Altitude, whofe infiftent Lines are not placed in the fame Right Lines, are equal to one another; which was to be demonstrated.

PROPOSITION XXXI.

THEOREM.

Solid Parallelepipedons, being conflituted upon equal Bafes, and having the fame Altitude, are equal to one another.

LETAE, CF, be folid Parallelepipedons, conftituted upon the equal Bafes A B, CD, and having the fame Altitude. I fay, the Solid A E is equal to the Solid CF.

*

Firft, Let HK, BE, AG, LM, OP, DF, C#, R S, be at Right Angles to the Bafes A B, CD; let the Angle A LB not be equal to the Angle CR D, and produce CR to T, fo that R T be equal to AL; then make the Angle TRY, at the Point R, in the Right Line RT, equal to the Angle ALB; make RY equal to LB; draw X Y, thro' the Point Y, parallel to RT, and compleat the Parallelogram RX, and the Solid Y. Therefore, because the two Sides TR, RY, are equal to the two Sides AL, L B, and they contain equal Angles; the Parallelogram R X fhall be equal and fimilar to the Parallelogram H L. And again, because, AL is equal to RT, and L M to RS, and they contain equal Angles, the Parallelogram R v shall