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 so the Triangle D E C is + equal † 8. ** to the Triangle HKB, and the Parallelogram D G equal to the Parallelogram HN; for the same Reason the Triangle AF G is equal to the Triangle L MN. Now the Parallelogram CF I is equal to the Paralle- 1 24 of tbis. logram BM, and the Parallelogram C G to the Parallogram B N, for they are 'opposite. Therefore the Prism 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. 30.cf angles L MN, N B K, and the three Parallelogramstbis. BM,HN, B N. Let the common Sotid, whose Balo is the Parallelogram AB, opposite to the Parallelogram GEHM, be added, then the whole folid Parallelopipedon C M is equal to the whole solid Parallelopipedon B F. Therefore, folid Parallelopipedons, being constituted upon the same Base, and having the same Altitude, and whose insistent Lines are in the same Right Lines, are equal to one another; which was to be demonstrated. PROPOSITION XXX. THEOREM, Lines, are equal to one another. having equal Altitudes, and standing on the same For, lee NK, D H, be produced, and G E, FM, be drawn, mecting each other in the Points R, X: Let alfo F M, G E, be produced to the Points O, P, and join AX, LO, CP, BR. The Solid CM, whose Base is the Parallelogram ACB L, being opposite to the Parallelogram DU M, is * equal to the Solid CO, * 29ef this, whose whose Base is the Parallelogram A'C BL, being oppofite to XPRO, for they stand upon the same Bale ACBL; and the insistent Lines A F, AX, LM, LO, CD, CP, BH, BR, are in the same Right Lines FO, DR: But the Solid CO, whose Base is the Pa rallelogram ACBL, being opposite to X PRO, is • 29 of tbis. * equal to the Solid CN, whose Base is the Paralle Jogram ACBL, being opposite to GEKN; for they stand upon the same Base A CBL, and their infittest Lines AG, A X, CE, CP, LN, LO, BK, BR, are in the same Right Lines GP, NR: Wherefore the Solid C M shall be equal to the Solid C N. Therefore, folid Parallelopipedons, being constituted upon the same Baje, and having the same Altitude, whose infiftent Lines are not placed in the same Right Lines, are equal to one antotber; which was to be demonstrated. PROPOSITION XXXİ. THE OR EM. equal Bases, and having the same Altitude, are tuted upon the equal Bases A B, CD, and having the fame Altitude. I say, the Solid A E is equal to the Solid CF. First, Let H K, BE, A G; L M, OP, DF,CE, RS, be at Right Angles to the Bases A B, CD; Jet the Angle ALB not be equal to the Angle CRD, and produce CR to T, so that RT be equal to AL; then make the Angle TRY, at the Point R, in the Right Line RT, equal * to the Angle ALB; make R Y equal to L B; draw X Y thro' the Point Y, * parallel to RT I, 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 R X shall be equal and similar to the Parallelogram HL. And again, because A L is equal to R T, and L M o RS, and they contain equal Angles, the Parallelogram RY hall thall be equal and similar to the Parallelogram AM. For the fame Reason the Parallelogram LE is equal and similar to the Parallelogram SY; therefore three Parallelograms of the Solid A E are equal and similar to three Parallelograms of the Solid Y Y; and so the three opposite ones of one Solid are + also equal and † 24 of bu. similar to the three opposite ones of the other: Therefore the whole folid Parallelopipedon AE is equal to the whole folid Parallelopipedon Y Y. Let DR, XY, be produced, and meet each other in the Point 12, and let TQ be drawn through T* parallel to D1, and pro- • 31. ” duce TQ, O D, till they meet in V, and compleat the Solids 124,RI: Then the Solid 422, whose Base is the Parallelogram R¥, and 2r is that opposite to it, is 1 1 29 of the equal to the Solid Y Y, whofe Bafe is the Parallelogram Ry, and Y 0, is that opposite to it, for they stand upon the fame Base R ¥, have the same Altitude, and their infiftent Lines R 12, RY, TQ, TX, SZ, SN, YT, 40, are in the fame Right Lines 12 X, ZQ; but the Solid Y is equal to the Solid A E; and so A E is equal to the Solid Y 1. Again, because the Parallelogram R YXT is equal to the Parallelogram 12 T, for it stands on the same Bare RT, and between the fame Parallels RT, 12 X; and : the Parallelogram RYXT is equal to the Parallelogram CD, because it is also equal to A B; the Parallelogram 82 T is equal to the Parallelogram CD, and DŤ is some other Parallelogram: Therefore, as the Base CD is to the Bale D T, fo is 2 T to TD. And because the solid Parallelopipedon C I is cut by the Plane RF, being parallel to two opposite Planes ; it Thall be as the Base CD is to the Base DT, so is * 25 of this the Solid CF to the Solid RI. For the fame Reafon, because the folid Parallelopipedon 21 is cut by the Plane Ry, parallel to two oppofite Planes ; as the Base 12 T is to the Base DT, lo shall the Solid 124 be to the Solid R I. But as the Bare CD is to the Base D T, ļo is the Base 12 T to TD: Therefore, as the Solid CF is to the Solid R I, so is the Solid 12 y to the Solid RI. And since each of the Solids CF, 224, has the same Proportion to the Solid® R I, the Solid CF is equal to the Solid 24: But the Solid 24 has been proved equal to the Solid AE; there fore 1 fore the Solid A E shall be † equal to the Solid C Fs But, now let the infiftent Lines A G, H K, BE, LM, CN, OP, DF, RS, not be at Right Angles to the Bafes A B CD.. I say, again, that the Solid A E is equal to the Solid CF. 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, E T, GY, MO, SI, FY, N 12, PX, meeting, the Plane in the Points. T, Y, 0, I, 4, 82, X; and join 2 T, Y0, 2Y, T9, XY, X12, 22 I, ¥I; then the Solid K ® is equal to the Solid Pl, for they ftand on equal Bases KM, PS, have the same Altitude, and the infiftent Lines are at Right Angles to the Bases. But the Solid K Q, is equal to the Solid A E, 1 29 oftbis, and the Solid P I to the Solid CF, since they ftand upon the same Rase, have the fame Altitude, and their infiftent Lines are in the same Right Line: Therefore the Solid AE shall be equal to the Solid CF. Wherefore, folid Parallelepipedons, being constituted upon equal Bases, and having the fame Altitude, are equal to one another; which was to be demonstrated. PROPOSITION XXXII. . PROBLEM. tude, are to each other as their Bases. LET AB, CD, be folid Parallelopipedons, that have the same Altitude. I say, they are to one another as their Bases; that is, as the Base A E is to the Bale CF, fo is the Solid A B to the Solid CD. For, apply a Parallelogram F H to the Right Line FG, equal to the Parallelogram AE, and compleat the folid Parallelopipedon G K upon the Base FH, having the fame Altitude as CD has : Then the 31 of tbis. Solid A B is * equal to the Solid GK, for they stand upon equal Bases A E, F H, and have the fame Altitude; and fo, because the folid Parallelopipedon CK is cut by the Plane DG; parallel to two opposite 25 of sbis. Planes, it shall be t, as the BaleHF is to the Base FC, so is the Solid HD to the Solid DC. But the Bafé Base F H is equal to the Base A E, and the Solid AB to the Solid G K. Therefore, as the Basc A E is to the Base CF, so is the Solid A B to che Solid CD. Wherefore, solid Parallelopipedons, that have the same Altitude, are to each other as their Bases; which was to be demonstrated. PROPOSITION XXXIII. THEOREM. in the triplicate Proportion of their bomologous L and let the Side A E be homologous to the Side CF. I say, the Solid A B, to the Solid CD, hath a Proportion, triplicate of that, which the Side A E has to the Side CF. For, produce A E, G E, 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 likewise the Solid KO, be compleated : Then, because the two Sides KE, EL, are equal to the two Sides CF, FN; and the Angle K E L equal to the Angle CFN (since she Angle AEG is equal to the Angle CFN, because of tbe Similarity of the Solids. A B, C D) the Parallelogram K L Thall be similar and equal to the Pa. rallelogram CN. For the fame Reason, the Parallelogram K M is equal and similar to the Parallelogram CR, and the Parallelogram O E to DF; therefore three Parallelograms of the Solid K O are equal and similar to three Parallelograms of the Solid CD: But those three Parallelograms are * equal and similar to the 24 of ibig three oppofite Parallelograms; therefore the whole So. lid K O is equal and funilar to the whole Solid CD. Let the Parallelogram G K be compleated, as also the Solids EX, LP, upon the Bases GK, KL, having the fame Altitude as A B: And since, because of the Similarity of the Solids AB and CD, it is, as AE is to CF, is EG to FN; and so EH O FR ; and FC is equal EK, and FN to EL, and FR to EM; it Thall be, as . AE |