Book XI. a C. II. a Produce MA, NA, OA to P, Q, R, fo that AP be equal un 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 to the folid CD. Complete likewise the folid AY, the bafe 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, so make b to c; and as AO to AR, foc to d: Then, because the parallelogram AE is equiangular to AS, AE is to AS, as the ftraight line a to c, as is demonftrated in the 23. Prop. Book 6. and the folids AB, AY, being betwixt the parallel planes BOY, EAS, are of the fame altitude. Therefore the folid AB b 32. II. is to the folid AY, as the bafe AE to the bafe AS; that is, as the ftraight line a is to c. And the folid AY is to the folid 25, 11. AX, as the bafe OQ is to the bafe QR; that is, as the ftraight line OA to AR; that is, as the ftraight line c to the straight line d. And because the folid AB is to the folid 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 faid to def. A. 5. be compounded of the ratios of a to b, b to c, and c to d, which are the fame 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. QE. D. PROP PROP. XXXIV. THEOR. Book XI. THE bases and altitudes of equal folid parallelepi- See N. peds, are reciprocally proportional; and if the bafes and altitudes be reeiprocally proportional, the folid parallelepids are equal. Let AB, CD be equal folid parallelepipeds; their bafes are reciprocally proportional to their altitudes; that is, as the base EH is to the bafe 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 bafes. As the base EH to the bafe NP, fo is peither fhall 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 bafe EH to the base NP, fo is CM to AG. Next, Let the bafes EH, NP not be equal, but EH greater than the other: Since then the folid AB is equal to the folid CD, CM is therefore CT. Because the folid A E С N AB is equal to the folid CD, therefore the folid AB is to the a 7. 5. b 32. 11. C25. II. Book XI. folid CV, as the folid CD to the folid CV. But as the fo~ lid AB to the folid CV, fob is the bafe EH to the base 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 ftraight line MC to CT; and CT is equal to AG. Therefore, as the bafe EH to the base NP, so is MC to AG. Wherefore the bases of the folid parallelepipeds AB, CD are reciprocally proportional to their altitudes. d I. 6. Let now the bases of the folid parallelepipeds AB, CD be reciprocally proportional to their altitudes; viz. as the bafe 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 K B R D G M X + A. 5. NP, as the altitude of the folid CD is to the altitude of the folid AB, therefore the altitude of CD is equal to the altitude of AB. But folid parallelepipeds upon equal bafes, and of the f 31. 11. folid altitude, are equal f 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 base NP, fo is CM the alti e tude of the folid CD to Σ R D X K B V lid CV. And, because the bafe EH is to the H EH is to the bafe NP, as MC to CT. But as the base EH is to NP, fob is the folid AB to the folid CV; for the folids AB, CV are of the fame altitude; and as MC to CT, fo is the bafe MP to the bafe PT, C 25. 11. PT, and the folid CD to the folid c CV: And therefore as the Book XI. folid AB to the folid CV, fo is the folid CD to the folid CV; that is, each of the folids AB, CD has the fame ratio to the' folid CV; and therefore the folid AB is equal to the folid CD. Second general cafe. Let the infifting straight lines FE, BL, GA, KH; XN, DO, MC, RP not be at right angles to the bafes of the folids; and from the points F, B, K, G; X, D, R, M draw perpendiculars to the planes in which are the bafes EH, NP, meeting thofe planes in the points S, Y, V, T; Q, I, U, Z; and complete the folids FV, XU, which are parallelepipeds, as was proved in the laft part of prop. 31. of this book. In this cafe likewife, if the folids AB, CD be equal, their bases are reciprocally proportional to their altitudes, viz. the base EH to the bafe NP, as the altitude of the folid CD to the altitude of the folid AB. Because the folid AB is equal to the folid CD, and that the folid BT is equal g to the g 29.or30. folid BA, for they are upon the fame bafe FK, and of the 11. fame altitude; and that the folid DC is equal to the folid DZ, being upon the fame bafe XR, and of the fame altitude; therefore the folid BI is equal to the folid DZ: But the bafes are reciprocally proportional to the altitudes of equal folid parallelepipeds of which the infifting ftraight lines are at right angles to their bafes, as before was proved: Therefore as the bafe FK to the bafe XR, fo is the altitude of the folid DZ to the altitude of the folid BT: And the bafe FK is equal to the bafe EH, and the bafe XR to the bafe NP: Wherefore, as the bafe EH to the base NP, fo is the altitude of the folid DZ to the altitude of the folid BT: But the altitudes of the folids DZ, DC, as alfo of the folids BT, BA are the fame. Therefore, as the bafe EH to the bafe NP, fo is the altitude of the the folid paralleude Book XI. folid CD to the altitude of the folid AB; that is, the bases of AB, CD are reciprocally proportional to their altitudes. Next, Let the bafes of the folids AB, CD be reciprocally proportional to their altitudes, viz. the bafe EH to the bafe NP, as the altitude of the folid CD to the altitude of the fo lid AB; the folid AB is equal to the folid CD The fame conftruction being made; because, as the bafe EH to the base NP, fo is the altitude of the folid CD to the altitude of the folid AB; and that the base EH is equal to the base FK; and NP to XR; therefore the bafe FK is to the bafe XR, as the altitude of the folid CD to the altitude of AB: But the alti as alfo of CD and XR, fo is the alti tudes of the folids AB, BT are the fame, DZ; therefore, as the bafe FK to the bafe tude of the folid DZ to the altitude of the folid BT: Wherefore the bafes of the folids BT, DZ are reciprocally proportional to their altitudes; and their infifting straight lines are at g29. or 30. right angles to the bafes; wherefore, as was before proved, the folid BT is equal to the folid DZ: But BT is equal g to the folid BA, and DZ to the folid DC, because they are upon the fame bafes, and of the fame altitude. Therefore the folid AB is equal to the folid CD. Q. E. D. PROP. |