Book XI. another as the bafe AE is to the bafe CF; wherefore the prifms, which are their halves, are to one another as the base AE to the base CF; that is, as the triangle AEM to the triangle CFG. d. 28. II. IMILAR folid parallelepipeds are one to another in the triplicate ratio of their homologous fides. Let AB, CD be fimilar folid parallelepipeds, and the fide AE homologous to the fide CF. the folid AB has to the solid 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 EM 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 fimilar; therefore the parallelogram KL is fimilar and equal to the parallelogram CN. for the fame reason, the parallelogram MK is fimilar and equal to CR, and alfo OE to FD. therefore three parallelograms of the folid KO are B.X D of the folid CD. and G the three oppofite ones in each folid are a. 24. 11. equal a and fimilar to thefe. therefore b. C. II. c. I. 6. the folid KO is equal b and fimilar to the the parallelogram GK, and complete the folids EX, LP upon the bases GK, KL, so that EH be an infisting straight line in each of them, whereby they must be of the fame altitude with the folid AB. and because the folids AB, CD are fimilar, and by permutation, as AE is to CF, fo is EG to FN, and fo 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 fo is HE to EM. but as AE to EK, fo is the parallelogram AG to the parallelogram GK; and c. I. 6. d. 25. 11. as GE to EL, fo 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, fod is the folid AB to the folid EX; and as GK to KL, fod is the folid EX to the folid PL; and as PE to KM, fod is the folid PL to the folid KO. and therefore as the folid 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 folid KO, the triplicate ratio of that which AB has to EX. but as AB is to EX, fo is the parallelogram AG to the parallelogram GK, and the straight line AE to the straight line EK. wherefore the folid 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 ftraight line EK is equal to the straight line CF. Therefore the folid AB has to the folid CD, the triplicate ratio of that which the fide AE has to the homologous fide CF. Q. E. D. COR. From this it is manifeft, that if four straight lines be continual proportionals, as the firft is to the fourth, fo is the folid parallelepiped described from the first to the similar solid fimilarly described from the fecond; because the firft ftraight line has to the fourth, the triplicate ratio of that which it has to the fecond. SOLU PROP. D. THEOR. OLID parallelepipeds contained by parallelograms See N. equiangular to one another, each to each, that is, of which the folid angles are equal, each to each; have to one another the ratio which is the fame 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 fame with that which is compounded of the ratios of the fides AM to DL, AN to DK, and AO to DH. Book XI. a. C. II. Produce MA, NA, OA to P, Q, R, so that AP be equal to DL, AQ to DK, and AR to DH; and complete the folid parallelepiped AX contained by the parallelograms AS, at, av fimilar and equal to CH, CK, CL, each to each. therefore the folid AX is equal a to the folid CD. complete likewise the solid AY the base of which is AS, and of which AO is one of its infifting straight lines. Take any ftraight line a, and as MA to AP, fo make a to b; and as NA to AQ, 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 23. Prop. Book 6. and the solids AB, AY, being betwixt the parallel planes BOY, EAS, are of the fame altitude, b. 32. 11. therefore the folid AB is to the folid AY, as the base AE to the base AS; that is, as the ftraight line a is to c. and the folid AY c. 25. 11. is to the folid 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 ftraight line d. and because the folid AB is to the solid AY, as a is to c, and the folid AY to the folid AX, as c is to d; ex aequali, the folid AB is to the folid AX, or CD which is equal to it, as the 'ftraight line a is to d. but the ratio of a to d is said to be compounded d of the ratios of a to b, b to c, and c to d, which are the same with the ratios of the fides MA to AP, NA to AQ, and OA to AR, each to each. and the fides AP, AQ, AR are equal to the fides DL, DK, DH, each to each. Therefore the folid AB has to the folid CD the ratio which is the fame with that which is compounded of the ratios of the fides AM to DL, AN to DK, and AO to DH. Q. E. D. d. Def.A.5. THE PROP. XXXIV. THEOR. Book XI. HE bafes and altitudes of equal folid parallelepi- See N. peds are reciprocally proportional; and if the bafes and altitudes be reciprocally proportional, the folid parallelepipeds are equal. Let AB, CD be equal folid parallelepipeds; their bases are reciprocally proportional to their altitudes; that is, as the base EH is to the base NP, fo is the altitude of the folid CD to the altitude of the folid AB. First, Let the infifting ftraight lines AG, EF, LB, HK, CM, NX, OD, PR be at right angles to the bases. as the base EH to equal, neither shall the folid AB be equal to the folid CD. but the folids are equal, by the hypothefis. therefore the altitude CM is not unequal to the altitude AG; that is, they are equal. wherefore as the base EH to the bafe NP, fo is CM to AG. Next, Let the bafes EH, NP not be equal, but EH greater than the other. fince then the folid AB is equal to the folid CD, CM and altitude CT. Because the folid AB is equal to the folid CD. a. 7. 5. b. 32. II. Book XI. therefore the folid AB is to the folid CV, as a the folid CD to the folid CV. but as the folid AB to the folid CV, fob is the base EH to the bafe NP; for the folids AB, CV are of the fame altitude; and as the folid CD to CV, fo is the bafe MP to the base PT, and fod is the straight line MC to CT; and CT is equal to AG. therefore as the bafe EH to the bafe NP, fo is MC to AG. wherefore the bases of the folid parallelepipeds AB, CD are reciprocally proportional to their altitudes. C. 25. II. d. 1. 6. Let now the bases of the folid parallelepipeds AB, CD be reciprocally proportional to their altitudes; viz. as the base EH to the bafe NP, fo the altitude of the folid CD to the altitude of the folid AB; the folid AB is equal to the folid CD. let the in K BR D e. A. 5. e therefore the altitude of CD is equal to the altitude of AB. but folid parallelepipeds upon equal bases, and of the fame altitude are f. 31. 11. equal to one another; therefore the folid AB is equal to the folid CD. But let the bafes EH, NP be unequal, and let EH be the greater of the two. therefore, fince as the bafe EH to the bafe NP, fo is CM the altitude of the folid CD to AG the altitude of AB, CM is greater e than AG. a gain, take CT equal to AG, RD and complete, as before, the folid CV. and, becaufe K B M X H the base EH is to the bafe C G T |