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 solid parallelopiped AX contained by the parallelograms AS, AF, AV similar and equal to CH, CK, CL, each to each. Therefore the solid AX is equal (C. 11.) to the solid CD. Complete likewise the solid AY, the base of which is AS, and of which AO is one of its insisting straight lines. Take any straight line a, and as MA to AP, so make a to b, and as NA to AQ, so make b to c; and as AO to AR, so 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 23d prop. book 6: and the solids AB, AY, being betwixt the parallel planes BOY, EAS, are of the same altitude. Therefore the solid AB is to the solid AY, as (32. 11.) the base AE to the base AS; that is, as the straight line a is to c. And the solid AY, is to the solid AX, as (25. 11.) 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 because the solid AB is to the solid AY, as a is to c, and the solid AY to the solid AX as c is to d; ex æquali, the solid 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 said to be compounded (def. A. 5.) of the ratios of a to b, b to c, and c to d, which are the same with 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 solid AB has to the solid 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 AO to DH. Q. E. D. PROP. XXXIV. THEOR. THE bases and altitudes of equal solid parallelopipeds, are reciprocally proportional: and if the bases and altitudes be reciprocally proportional, the solid parallelopipeds are equal.* Let AB, CD be equal solid parallelopipeds; their bases are reciprocally proportional to their altitudes; that is, as the base EH is to the base NP, so is the altitude of the solid CD to the altitude of the solid AB. As the First, let the insisting straight lines, AG, EF, LB, HK; CM, NX, OD, PR be at right angles to the bases. base EH to the base NP, so is K CM to AG. If the base EH be equal, but the altitudes AG, CM be not equal, neither A B R D G F M X LP shall the solid AB be equal to the solid CD. But the solids are equal, by the hypothesis. Therefore the altitude CM is not unequal to the altitude AG; that is, they are equal. Wherefore as the base EH to the base NP, so is CM to AG. K R D X B M F G Next, let the bases EH, NP not be equal; but EH greater than the other since then the solid AB is equal to the solid CD, CM is therefore greater than AG: for if it be not, neither also, in this case, would the solids AB, CD be equal, which, by the hypothesis, are equal. Make then CT equal to AG, and complete the solid parallelo- H piped CV of which the base. is NP, and altitude CT. Because the solid AB is equal to the solid CD, therefore the solid as (7. 5.) the solid CD to the solid CV. • See Note. AB is to the solid CV, to the solid CV, so (32. 11.) is the base EH to the base NP; for the solids AB, CV are of the same altitude; and as the solid CD to CV, so (25. 11.) is the base MP to the base PT, and so (1.6.) is the straight line MC to CT; and CT is equal to AG. Therefore, as the base EH to the base NP, so is MC to AG. Wherefore, the bases of the solid parallelopipeds AB, CD are reciprocally proportional to their altitudes. H K B R D G F M X Let now the bases of the solid parallelopipeds AB, CD be reciprocally proportional to their altitudes; viz. as the base EH to the base NP, so the altitude of the solid CD to the altitude of the solid AB; the solid AB is equal to the solid CD. Let the insisting lines. be, as before, at right angles to the bases. Then, if the base EH be equal to the base NP, since EH is to NP, as the altitude of the solid CD is to the altitude of the solid AB, therefore the altitude of CD is equal (A. 5.) to the altitude of AB. But solid parallelopipeds upon equal bases, and of the same altitude, are equal (31. 11.) to one another: therefore the solid AB is equal to the solid CD. P A E C N But let the bases EH, NP be unequal, and let EH be the greater of the two. Therefore, since as the base EH to the base NP, so is CM the altitude of the solid CD to AG the altitude of AB, CM is greater (A. 5.) than AG. Again, take CT equal to AG, and complete, as before, the solid CV. K R D B M X base EH is to the base NP, as MC to CT. But as the base EH is to NP, so (32. 11.) is the solid AB to the solid CV; for the solids AB, CV are of the same altitude; and as MC to CT, so is the base MP to the base PT, and the solid GD to the solid (25. 11.) CV: and therefore as the solid AB to the solid CV, so is the solid CD to the solid CV; that is, each of the solids AB, CD has the same ratio to the solid CV; and therefore the solid AB is equal to the solid CD. Second general case. Let the insisting straight lines FE, BL, GA, KH; XN, DO, MC, RP not be at right angles to the bases of the solids; and from the points F, B, K, G; X, D, R, M draw perpendiculars to the planes in which are the bases EH, NP meeting those planes in the points S, Y, V, T; Q, I, U, Z; and complete the solids FV, XU, which are parallelopipeds, as was proved in the last part of Prop. 31. of this Book. In this case likewise, if the solids AB, CD be equal, their bases are reciprocally proportional to their altitudes, viz. the base EH to the base NP, as the altitude of the solid CD to the altitude of the solid AB. Because the solid AB is equal to the solid CD, and that the solid BT is equal (29. or 30. 11.) to the solid BA, for they are upon the same base FK, and of the : same altitude; and that the solid DC is equal (29. or 30. 11.) to the solid DZ, being upon the same base XR, and of the same altitude; therefore the solid BT is equal to the solid DZ but the bases are reciprocally proportional to the altitudes of equal solid parallelopipeds of which the insisting straight lines are at right angles to their bases, as before was proved. Therefore as the base FK to the base XR, so is the altitude of the solid DZ to the altitude of the solid BT: and the base FK is equal to the base EH, and the base XR to the base NP. Wherefore, as the base EH to the base NP, so is the altitude of the solid DZ to the altitude of the solid BT: but the altitudes of the solids DZ, DC, as also of the solids BT, BA are the same. Therefore as the base EH to the base NP, so is the altitude of the solid CD to the altitude of the solid AB; that is, the bases of the solid parallelopipeds AB, CD are reciprocally proportional to their altitudes, Next, let the bases of the solids AB, CD be reciprocally proportional to their altitudes, viz. the base EH to the base NP, as the altitude of the solid CD to the altitude of the solid AB; the solid AB is equal to the solid CD: the same construction being made: because as the base EH to the base NP, so is the altitude of the solid CD to the altitude of the solid AB; and that the base EH is equal to the base FK; and NP to XR; therefore the base FK is to the base XR, as the altitude of the solid CD to the altitude of AB. But the alti tudes of the solids AB, BT are the same, as also of CD and DZ; therefore as the base FK to the base XR, so is the altitude of the solid DZ to the altitude of the solid BT: wherefore the bases of the solids BT, DZ are reciprocally proportional to their altitudes; and their insisting straight lines are at right angles to the bases; wherefore, as was before proved, the solid BT is equal to the solid DZ: but BT is equal (29. or 30. 11.) to the solid BA, and DZ to the solid DC, because they are upon the same bases, and of the same altitude. Therefore the solid AB is equal to the solid CD. Q. E. D. 1 |