8. 35. 1. Book XI. lines AT, GX. and because the parall, iogram AB is equal to SB, w for they are upon the fame base LB, and between the fame paral lels LB, AT; and R M V X K e В HT But if the insisting 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 solid 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 h to the plane in which are the bases AB, CD; and let them meet it in the points Q, T, V, X; AS À H Q T C R Y Z Y, Z, I, U, and join QT, TV, VX, XQ; YZ, ZI, IU, UY. then because GQ, KT, are at right angles to the same plane, they se i. 6. II. 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 k to one another. for the same reifon, the planes MV, GT are parallel to one another. therefore the solid QE is a parallelepiped. in like manner, it may be proved, that the folid YF is a parallelepiped. but, from what has been demonstrated, the folid EQ is equal to the folid FY, because they are upon equal bases MK, PS, and of the fame altitude, and have their insisting straight lines at right angles to the bases, and Book X1.. the solid EQ is equal' to the folid AE ; and the folid FY to me the folid CF; because they are upon the fame bases and of the 1.29. or 30. fame altitude. therefore the folid AE is equal to the solid CF. Wherefore folid parallelepipeds, &c. Q. E. D. II. 0 Р PROP. XXXII. THEOR. OLID parallelepipeds which have the same altitude, see n. are to one another as their bases. Let AB, CD be folid parallelepipeds of the same 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" to a.Cor.45.6. AE, so that the angle FGH be equal to the angle LCG; and coinplete the solid parallelepiped GK upon the base FII, one of whose in fisting lines is FD, whereby the folids CD), GK must be of the fame altitude. therefore the folid AB is equal o to the folid GK, b. 31. 1:. because they are upon equal bafes B т I AE, FH, and are of the same altitude. and because L the folid parallelepiped CK is cut by the plane DG A M C G H which is parallel to its opposite planes, the base HF ise to the base FC, as the folid c. 15. 16. HD to the folid DC. but the base 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 solid parallelepipeds, &c. Q. E. D. Cor. From this it is manifest that prisms non triangular bases, of the fame altitude, are to one another as their bases. Let the prisms the bases of which are the triangles AEM, CIG, and NBO, PDO the triangles opposite to them, have the fame altitude; and complete the parallelograms AE, CF, and the folic parallelepipeds AB, CD, in the first of which let MO, and in the other let Go be one of the infifting lines, and because the folk parallelepipeds AB, CD have the fame altitude, they are to one Bouk XI. another as the base AE is to the base CF; wherefore the prisms, v which are their halves d, are to one another as the base AE to the d. 28. 11. base CF; that is, as the triangle AEM to the triangle CFG. SIMILA the triplicate ratio of their homologous fides. Let AB, CD be similar folid parallelepipeds, and the side AE homologous to the side CF. the solid AB has to the folid CD, the triplicate ratio of that which AE has to CF. Produce AE, GE, HE, and in these produced take EK equal to CF, EL equal to FN, and LM equal to FR ; and complete the parallelogram KL, and the folid KO. because KE, EL are equal to CF, FN, and the angle KEL equal to the angle CFN, because the angle AEG is equal to CFN, by reason that the folids AB, • CD are limilar; therefore the parallelogram KL is similar and equal to the parallelogram CN. for the same reason, the parallelogram B X H three parallelograms P Р R of the folid CD. and C the three opposite NI ones in each folid are K 2. 24. 11. equal · and similar C E to these. therefore M the solid KO is equal b. C. 11. b and Gmilar to the 0 folid CD. complete the parallelogram GK, and complete the folids EX, LP upon the bases GK, KL, so that EH be an insisting straight line in each of them, whereby they must be of the fame altitude with the solid AB. and because the solids AB, CD are similar, and by permutation, as AE is to CF, so is EG to FN, and so is EH to FR ; and FC is equal to EK, and FN to EL, and FR to EM; therefore as AE to EK, fo is EG to EL, and so is HE to EM. but as AE to EK, fo o is the parallelogram AG to the parallelogram GK; and as GE to EL, so is GK to KL; and as HE to EM, fo is PE Book XI. to KM. therefore as the parallelogram AG to the parallelogram GK, fo is GK to KL, and PE to KM. but as AG to GK, so d is c. 1. 6. the solid AB to the solid EX; and as GK to KL, so d is the solid d. 25. 11. EX to the solid PL ; and as PE to KM, so d is the solid PL to the folid ko. and therefore as the solid AB to the folid EX, fo is EX to PL, and PL to ko. but if four magnitudes be continual proportionals, the first is said to have to the fourth the triplicate ratio of that which it has to the fecond. therefore the folid AB has to the solid KO, the triplicate ratio of that which AB has to EX. but as AB is to EX, so is the parallelogram AG to the parallelogram GK, and the straight line AE to the straight line EK. wherefore the solid AB has to the folid KO, the triplicate ratio of that which AE has to EK, and the folid KO is equal to the folid CD, and the straight line EK is equal to the straight line CF. Therefore the solid AB has to the folii CD, the triplicate ratio of that which the side AE has to the homologous fide CF. Q. E. D. Cor. From this it is manifest, that if four straight lines be continual proportionals, as the first is to the fourth, fo is the solid parallelepiped described from the first to the similar solid similarly described from the second; because the first straight line has to the fourth, the triplicate ratio of that which it has to the second. PROP. D. THEO R. SOLI OLID parallelepipeds contained by parallelograms iecit equiangular to one another, each to each, that is, of which the solid angles are equal, each to each; have to one another the ratio which is the same with the ratio compounded of the ratios of their fides. Let AB, CD be folid parallelepipeds, of which AB is contained by the parallelograms AE, AF, AG equiangular, each to each, to the parallelograms CH, CK, CL which contain the folid CD. the ratio which the folid AB has to the folid CD is the same with that which is compounded of the ratios of the sides AM to DL, AN to DK, and AO to DH. Book XI. Produce MA, NA, OA to P, Q, R, so that AP be equal to VDL, AQ to DK, and AR to DH ; and complete the folid pa rallelepiped AX contained by the parallelograms AS, AT, AV Similar and equal to CH, CK, CL, each to each. therefore the a. C. 11. folid AX is equal to the folid CD. complete likewise the folid AY the base of which is AS, and of wbich AO is one of its in- ! filling straight lines. Take any straight line a, and as MA to AP, so make a to b; and as NA to A, fo make b to c; and as OA to AR, fo c to d. then because the parallelogram AE is equiangular to AS, AE is to AS, as the straight line a to c, as is demonstrated in the 2 3. Prop. Book 6. and the folids AB, AY, being bitwixt the parallel planes BOY, EAS, are of the same altitude. b. 32. 11 therefore the solid AB is to the solid AY, as b the base AE to the b.de AS ; that is, as the straight line a is to c. and the folid AY c. 25. 11. d. Def. a.s. is to the solid AX, as the base OQ is to the base QR ; that is, as the straight line OA to AR; that is, as the straight line c to the straight line d. and becaute the folid AB is to the folid AY, as a is tu c, and the folid AY to the folid AX, as c is to d; ex aequali, the folid AB is to the solid AX, or CD which is equal to it, as the straight line a is to d. but the ratio of a to d is faid to be compounded d of the ratios of a to b, b to c, and cio d, which are the fame with the ratios of the sides MA to AP, NA to AQ, and OA to AR, each to each. and the sides AP, AQ, AR are equal to the sides DL, DK, DH, each to each. Therefore the folid AB has to the ford CD the ratio which is the same with that which is compounded of the ratios of the sides AM to DL, AN to DK, and A to DH. Q. L. D. |