24 of this, is equal to EK. Let E H, which is common, be taken away, then the Remainder DE will be equal to the Remainder HK, and fo the Triangle DE Cis + equal † 8. ». to the Triangle HKB, and the Parallelogram DG equal to the Parallelogram HN; for the fame Reason the Triangle A F G is equal to the Triangle L M N. Now the Parallelogram C F is equal to the Parallelogram B M, and the Parallelogram CG to the Parallogram BN, for they are 'oppofite. Therefore the Prifm contained under the two Triangles AF G, DEC, and the three Parallelograms C F, DG, CG, is equal to the Prism contained under the two Tri- *Def. 30.of* angles L MN, BK, and the three Parallelograms BM, HN, BN. Let the common Solid, whose Base is the Parallelogram A B, oppofite to the Parallelogram GEHM, be added, then the whole folid Parallelopipedon CM is equal to the whole folid Parallelopipedon BF. Therefore, folid Parallelopipedons, 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; which was to be demonftrated. PROPOSITION XXX. Solid Parallelopipedons, being conftituted upon the LET there be folid Parallelopipedons CM, CN, this. For, let NK, DH, be produced, and G E, F M, be drawn, meeting each other in the Points R, X: Let alfo FM, GE, be produced to the Points O, P, and join AX, LO, CP, BR. The Solid CM, whofe Bafe is the Parallelogram ACB L, being oppofite to the Parallelogram FDHM, is equal to the Solid CO, 29 of this whofe whose Base is the Parallelogram A ̊C BL, being oppofite to X PR O, for they ftand upon the fame Base ACBL; and the infiftent Lines A F, A X, LM, LO, CD, CP, BH, BR, are in the fame Right Lines FO, DR: But the Solid C O, whofe Bafe is the Parallelogram AC BL, being opposite to X P R O, is 29 of this. * equal to the Solid CN, whofe Bafe is the ParalleJogram AC BL, being oppofite to GEKN; for they stand upon the fame Bafe AC BL, and their infiftent Lines AG, A X, CE, CP, LN, LO, BK, BR, are in the fame Right Lines G P, NR: Wherefore the Solid C M fhall be equal to the Solid C N. Therefore, folid Parallelopipedons, being conftituted upon the fame Baje, and having the fame Altitude, whofe infiftent Lines are not placed in the fame Right Lines, are equal to onë another; which was to be demonstrated. PROPOSITION XXXI. THEOREM. Solid Parallelopipedons, being conftituted upon equal Bafes, and having the fame Altitude, are equal to one another. LETAE, CF, be folid Parallelopipedons, conftituted upon the equal Bafes A B, CD, and having the fame Altitude. I fay, the Solid A E is equal to the Solid C F. Firft, Let HK, BE, AG, LM, OP, DF, CE, R S, be at Right Angles to the Bases A B, CD; let the Angle ALB not be equal to the Angle CRD, and produce C R to T, fo that R T be equal to AL, then make the Angle TRY, at the Point R, in the Right Line R T, equal to the Angle AL B; make RY equal to LB; draw X Y thro' the Point Y, * parallel to RT, and compleat the Parallelogram R X, and the Solid Y. Therefore, because the two Sides TR, RY, are equal to the two Sides AL, LB, and they contain equal Angles; the Parallelogram RX fhall be equal and fimilar to the Parallelogram H L. And again, because A L is equal to R T, and L M to RS, and they contain equal Angles, the Parallelogram R fhall be equal and fimilar to the Parallelogram A M. For the fame Reafon the Parallelogram LE is equal and fimilar to the Parallelogram SY; therefore three Parallelograms of the Solid A E are equal and fimilar to three Parallelograms of the Solid Y Y; and fo the three oppofite ones of one Solid are + also equal and +24 of this. fimilar to the three oppofite ones of the other: Therefore the whole folid Parallelopipedon AE is equal to the whole folid Parallelopipedon Y. Let D R, XY, be produced, and meet each other in the Point 2, and let TQ be drawn through T* parallel to D, and pro- * 31. duce TQ,OD, till they meet in V, and compleat the Solids, RI: Then the Solid Y, whofe Base is the Parallelogram RY, and r is that oppofite to it, is ‡ ‡-29 of tim equal to the Solid YY, whofe Bafe is the Parallelogram RY, and Yo, is that oppofite to it, for they stand upon the fame Base RY, have the fame Altitude, and their infiftent Lines R 2, RY, TQ, TX, SZ, SN, г, Y, are in the fame Right Lines NX, Zo; but the Solid Y Y is equal to the Solid AE; and fo AE is equal to the Solid Y. Again, because the Parallelogram R Y XT is equal to the Parallelogram T, for it ftands on the fame Bafe RT, and between the fame Parallels R T, 2 X; and: the Parallelogram R Y XT is equal to the Parallelogram CD, because it is alfo equal to A B; the Parallelogram T is equal to the Parallelogram CD, and DT is fome other Parallelogram: Therefore, as the Bafe CD is to the Bale D T, fo is T to TD. And because the folid Parallelopipedon CI is cut by the Plane R F, being parallel to two oppofite Planes; it fhall be, as the Bafe CD is to the Bafe D T, fo is 25 of this. the Solid CF to the Solid R I. For the fame Reafon, because the folid Parallelopipedon I is cut by the Plane R, parallel to two oppofite Planes; as the Bafe T is to the Bafe DT, fo fhall* the Solid n be to the Solid R I. But as the Base CD is to the Base DT, fo is the Bafe T to TD: Therefore, as the Solid CF is to the Solid RI, fo is the Solid N ¥ to the Solid R I. And fince each of the Solids C F, NY, has the fame Proportion to the Solid R I, the Solid C F is equal to the Solid : But the Solid has been proved equal to the Solid AE; there fore + 9.5. fore the Solid A. E fhall be † equal to the Solid CF: But, now let the infiftent Lines A G, H K, BE, LM, CN, OP, DF, R S, not be at Right Angles to the Bases. A B, C D. I say, again, that the Solid A E is equal to the Solid C F. Let there be drawn from the Points K, E, G, M, P, F, N, S, to the Plane wherein are the Bafes A B, CD, the Perpendiculars KE, ET, GY, MO, SI, FY, NO, PX, meeting the Plane in the Points ET, Y, 0, I, Y, 1, X; and join 3 T, Y O, Z Y, TO, X Y, X Q, Q 1, ¥ I; then the Solid K is equal to the Solid PI, for they ftand on equal Bases K M, PS, have the fame Altitude, and the infiftent Lines are at Right Angles to the Bafes. But the Solid K. O, is equal to the Solid A E, 29oftbis, and the Solid PI to the Solid C F, fince they ftand upon the fame Bafe, have the fame Altitude, and their infiftent Lines are in the fame Right Line: Therefore the Solid AE fhall be equal to the Solid CF. Wherefore, folid Parallelopipedons, being conflituted upon equal Bafes, and having the fame Altitude, are equal to one another; which was to be demonstrated. 1 t PROPOSITION XXXII. PROBLEM. Sold Parallelopipedons, that have the fame Altitude are to each other as their Bases. LET AB, CD, be folid Parallelopipedons, that have the fame Altitude. I fay, they are to one another as their Bafes; that is, as the Bafe AE is to the Base C F, fo is the Solid A B to the Solid CD. For, apply a Parallelogram F H to the Right Line F G, equal to the Parallelogram A E; and compleat the folid Parallelopipedon G K upon the Bafe FH, having the fame Altitude as CD has: Then the 31 of this. Solid A B is equal to the Solid G K, for they stand upon equal Bafes A E, F H, and have the fame Altitude; and fo, because the folid Parallelopipedon C K is cut by the Plane D G, parallel to two oppofite +25 of this. Planes, it fhall be t, as the Bafe HF is to the Base FC, fo is the Solid HD to the Solid DC: But the Bafe Bafe FH is equal to the Bafe A E, and the Solid A B to the Solid G K. Therefore, as the Bafe A E is to the Bafe CF, fo is the Solid A B to the Solid C D. Wherefore, folid Parallelopipedons, that have the fame Altitude, are to each other as their Bases; which was to be demonftrated. PROPOSITION XXXIII. THEOREM. Similar folid Parallelepipedons are to one another in the triplicate Proportion of their homologous Sides. L ETA B, CD, be fimilar folid Parallelopipedons, and let the Side A E be homologous to the Side CF. I fay, the Solid A B, to the Solid C D, hath a Proportion, triplicate of that, which the Side A E has to the Side CF. For, produce A E, GE, HE, to EK, EL, EM; and make E K equal to CF, and E L to F N, and EM to FR; and let the Parallelogram K L, and likewife the Solid KO, be compleated: Then, because the two Sides K E, E L, are equal to the two Sides C F, FN; and the Angle K E L equal to the Angle CFN (fince the Angle AEG is equal to the Angle CFN, because of the Similarity of the Solids. A B, CD) the Parallelogram KL fhall be fimilar and equal to the Parallelogram CN. For the fame Reafon, the Parallelogram K M is equal and fimilar to the Parallelogram CR, and the Parallelogram OE to DF; therefore three Parallelograms of the Solid K O are equal and fimilar to three Parallelograms of the Solid CD: But those three Parallelograms are equal and fimilar to the 24 of bit three oppofite Parallelograms; therefore the whole Solid K O is equal and fimilar to the whole Solid C D. Let the Parallelogram G K be compleated, as alfo the Solids EX, LP, upon the Bafes GK, KL, having the fame Altitude as A B: And fince, because of the Similarity of the Solids AB and CD, it is, as AE is to CF, fe is EG to FN; and fo EH to FR; and FC is equal EK, and FN to EL, and FR to EM; it shall be, as Q * AE |