C S. AK as EC is to CG; and KA to AH as GC is to CF; then o, Book XIby equality, as BA is to AH, fo is CE to CF. Compleat the n parallelogram BH, and folid ÁL: Then, because the three plain b 12. 6angles, containing the folid angle at A, are equal to the three plain © 22. 5. angles containing the folid angle at C, and the fides about the equal angles proportional, the parallelogram KB is fimilar to the parallelogram GE. For the fame reason, KH is fimilar to GF, and HB to FE; therefore the three parallelograms of the folid AL are fimilar to the three parallelograms of the folid CD; but these three parallelograms are equal and fimilar to the three oppofite ones d; therefore the folid AL is fimilar to the folid CD: Which was to be done. d. , d 24. e def. D IF a folid parallelopipedon be cut by a plain passing through the diagonals of two oppofite plains, that folid will be bisected by the plain. b 240 If the folid parallelopipedon AB be cut by the plain GAEF, pafling through the diagonals GF, AE, of two oppofite plains, then the folid AB is bifected by the plain GAEF. For, because the triangles CGF, GBF, are equal, and likewife the triangles ADE, AEHa, and the parallelograms AC, BE, for they are a 34. I. oppofite, and likewife GH equal to CE; the prifm contained by the two triangles CGF, ADE, and the three parallelograms GE, AC, CE, is equal to the prifm contained by the triangles GFB, AEH, and the three parallelograms GE, BE, AB c ̧c def. Io. Wherefore, &c. S PROP. XXIX. THEOR. OLID parallelepipedons, conftitute upon the fame base, having the fame altitude, and whofse insistent right lines are in the same right line, are equal to one another. Let the folid parallelopipedons CM, BF, be conftitute upon the fame bafe AB, having the fame altitude, and whofe infiftent right lines AF, AG, LM, LN, CD, CE, BH, BK, are in the fame right lines FN, DK; then the folid CM is equal to the folid CN. For, becaufe CH, CK, are parallelograms, DH, EK, are each equal to CB; therefore DH is equal to a 34. I. EK. Take EH from, or add to both, then there will remain HK b 8. I. C 24. Book XI. HK equal to DE, and the triangle DEC equal to HKBb, and In the parallelogram DG to HN; but the parallelogram CF is equal to BM, and CG to BN, for they are oppofite; therefore the prifm contained by the two triangles AFG, DEC, and the three parallelograms CF, DG, CG, is equal to the prism contained by the two triangles LMN, HKB, and the three paralled def. 1o. lograms BM, HN, BN d; add, or take away the folid whose base is the parallelogram AB, oppofite to the parallelogram GEHM; then the folid ČM is equal to the folid CN. Wherefore, &c. a 24. b 29. S PROP. XXX. THE OR. OLID parallelopipedons, conftitute upon the fame base, having the fame altitude, and whofe infiftent right lines are not in the fame right line, are equal to one another. Let there be folid parallelopipedons CM, CN, having equal altitudes, ftanding on the fame base AB, and whofe infiftent right lines AF, AG, LM, LN, CD, CE, BH, BK, are not in the fame right lines; then the folid CM will be equal to the folid CN. For, produce NK, DH, till they meet in R; and draw GE, FM, meeting in X; likewife produce GE, FM, to the points O, P; join AX, LO, CP, BR; then the folid CM, whose base is the parallelogram ACBL, oppofite to the equal parallelogram FDHM 2, is equal to the folid CO ', whose base is the fame parallelogram AB, oppofite to the equal parallelogram XR; for they stand upon the fame base AB, and the infiftent lines AF, AX, LM, LO, CD, CP, BH, BR, are in the fame right lines FO, DR; but the folid CO is equal to the folid CN, for they have the same base AB, oppofite to the parallelograms XR, GK, each equal to AB, and their infiftent right lines AG, AX, CE, CP, LN, LO, BK, BR, are in the fame right lines GP, NR; therefore the folid CM is equal to the folid CN. Wherefore, &c. S OLID parallelopipedons, conftitute upon equal bases, and having the fame altitudes, are equal. Let AE, CF, be folid parallelopipedons, conftitute upon the equal bafes AB, CD; and having the fame altitude, the folid AE is equal b conft. equal to the folid CF. First, let the folids AE, CF, have the Book XI. infiftent lines AG, HK, LM, BE, OP, DF, CG, RS, at right angles to the bafes AB, CD; and let the angle ALB be equal to the angle CRD. Produce CR to T; and make RT equal to LB; compleat the parallelogram DT, equiangular to AB or CD; and the folid KI, having its infiftent right lines at right angles to DT, and of the fame altitude with AE or CF. Then, because the right lines DF, RS, are at right angles to the plain OT, they are parallel and equal; therefore the a 6. parallelogram DS is equal and parallel to CP, TI: Therefore the folid CF is to the folid PI as the bafe OR is to the bafe DTC; c 25. but OR is equal to DT; therefore the folid CF is equal to the folid RI. But the folid RI is equal to the folid AE; therefore the folid AE is equal to the folid CF: But, if the angle ALB is not equal to CRD, at the point R, with the right line RF, make the angle TRY equal to the angle ALB; and make RY equal to AL; and compleat the parallelogram RX, and folid YW. Produce DR, VT, XY, to the points Q and a; and compleat the folid ae; then the parallelograms RX, RQ, are equal; and, because RX is equiangular to B, and the infift- d 35. 1. ent lines at right angles to the bafe RX, and of the same altitude with the folid AE, the plains in the folid AE are equal and fimilar to thefe in the folid YW; therefore the folid YW is equal to the folid AEd. For the fame reafon, the folid aW, whofe bafe is the parallelogram RW, and ae, that opposite to it, is equal to the folid YW, whofe bafe is the parallelogram RW, and Yf, that oppofite to it; for they ftand upon the fame bafe RW, have the fame altitude, and their infiftent lines Ra, RY, TX, TQ, SZ, SN, We, Wf, are in the same right lines aX, Zf; but the folid YW is equal to the folid CF; therefore the folid aW is equal to the folid CF. d def. Io. Now, let the infiftent lines ML, EB, GA, KH, NO, SD, PC, FR, not be at right angles to the bafes AB, CD, the folid AE will be equal to the folid CF. For, from the points G, K, E, M, P, F, S, N, let fall the right lines Mƒ, ET, GY, Kg, PX, FW, Na, SI, perpendicular to the plain of the bafes AB, CD, meeting them in the points f, Y, g, T, X, W, I, a;e II. and join ƒY, Yg, gT, Tƒ, Xa, XW, WI, Ia; then, becaufe GY, Kg, are at right angles to the fame plain, they are parallel f. For the fame reafon, Mf is parallel to ET. But MG is 6. 6. parallel to EK; therefore the plains MY, KT, of which the one paffes through GY, Yf, and the other through Kg, gT, which are parallel to GY, YF, and not in the fame plain with them, are parallel to one anothers, and equal and parallel to their 15. oppofite plains; therefore fE is a parallelopipedon. It may be proved in the fame manner, that aF is a parallelopipedon; but Q the Book XI. the folid GT is equal to the folid PI; for they are upon equal bases, and of the fame altitude, from what has been demonftrated; and the folid GT is equal to the folid AE; and the folid XF to the folid aS; therefore the folid AE is equal to the folid CF. Wherefore, &c. h 29. or 30. a 31. b 25. PRO P. XXXII. THEOR. SOLID parallelopipedons that have the fame altitude are to one another as their bases. Let AB, CD, be folid parallelopipedons, having the fame altitude as the base AE is to the bafe CF, fo is the folid AB to the folid CD. a For, to the right line FG, apply the parallelogram FH, equal to the parallelogram AE; upon the bafe FH, compleat the folid GK, of the fame altitude as CD; then the folid AB is equal to the folid GK ; but the folid CK is cut by the plain DG, parallel to the oppofite plain; therefore the folid CD is to the folid GK, as CF is to FH; that is, AB is to CD as AE is to CF. Wherefore, &c. à 24. PROP. XXXIII. THEOR. SIMILAR Solid parallelopipedons are to one another in the triplicate ratio of their homologous fides. Let AB, CD be fimilar folid parallelopipedons, and let the fide AE be homologous to the fide CF; then the folid AB has to the folid CD a triplicate ratio of that which the fide AE has to CF. For, produce AE, GE, HE, to K, L, M; make EK equal to CF, EL to FN, and EM to FR; and the angle KEL is equal to CFN; for AEG is equal to CFN; and compleat the parallelogram KL, and the folid KO; then the parallelogram KL is fimilar and equal to the parallelogram CN. For the fame reason, the parallelogram KM is equal and fimilar to the parallelogram CR, and OE to FD; therefore the whole folid KO is equal and fimilar to the folid CD. Likewife, compleat the parallelogram HL, and folids EX, LP, upon the bases GK, KL, having the fame altitude as AB, for EH is an infiftent line to both; but the folid OK is proved fimilar to CD; and AB is given given fimilar to CD; therefore AB is fimilar to OK b. Then, Book XI. because AB is fimilar to CD, and the bafe AG to CN, as AE and 12. 5. C I. G. is to CF, fo is EG to FN; and fo is EH to FR ; and, because b 21. 6. FC is equal to EK, and FN to EL, and FR to EM, as AE is to EK, fo is EG to EL; and fo is the parallelogram AG to GK ; but, as GE is to EL, fo is the parallelogram GK to KL ; and, as HE is to EM, fo is the parallelogram PE to KM ; therefore, as AG is to GK, fo is GK to KL; and fo is PE to KM; but, as AG is to GK, fo is the folid AB to the folid EX d; for the d 25. plain GH is parallel to the oppofite plains; and, as GK is to KL, fo is the folid EX to PLd; and, as PE is to KM, fo is the folid PL to the folid KO. Then, because the four folids AB, EX, P, KO, are proportionals, AB has to KO a triplicate proportion of what AB has to EX ; but, as AB e def, 11. is to EX, fo is the parallelogram AG to GKd; and fo is the right line AE to the right line EK; therefore the folid AB has to the folid KO a triplicate ratio of what AE has to EK; but the folid KO is equal and fimilar to the folid CD; and the right line EK equal to CF: Therefore the solid AB has to the folid CD a triplicate ratio of what the homologous fide AE has to the homologous fide CF. Wherefore, &c. COR. Hence, if four right lines be proportional, as the first is to the fourth, so is a folid parallelopipedon described on the first, to a fimilar one described on the second, PRO P. XXXIV. THEOR. THE bases and altitudes of equal folid parallelopipedons are reciprocally proportional; and parallelopipedons, whose bafes and altitudes are reciprocally proportional, are equal. Let AB, CD, be equal folid parallelopipedons; then the base EH is to the base NP, as the altitude of the folid CD is to the altitude of the folid AB. First, let the infiftent right lines AG, EF, LB. HK, CM, NX, OD, PR, be at right angles to their bases; then, as the bafe EH is to the bafe NP, fo is CM to AG. For, if the base EH is equal to the bafe NP; then the altitudes CM, AG, are equal. For, if the bafes EH, NP, are equal, but the altitudes not equal, then the folids are not equal ; but the folid AB is a 31, put equal to the folid CD; therefore the altitudes CM, AG, are equal: Therefore, as the bafe EH is to the bafe NP, fo is CM to AG. Now, let the bafes be unequal, and let EH be the 32. greater, then the altitude CM will be greater than the altitude AG; |