Book XI. PROP. XXXIV. THEOR. " HE bases and altitudes of equal folid parallelepipeds, See N. are reciprocally proportional; and if the bases and altitudes be reciprocally proportional, the solid parallelepipeds are equal. Let AB, CD be equal folid parallelepipeds; their bases are reci. procally proportional to their altitudes; that is, as the base EH is to the base NP, so is the altitude of the folid CD to the altitude of the solid AB. First, Let the insisting straight lines AG, EF, LB, HK ; CM, NX, OD, PR be at right angles to the bases. as the base EH to the base NP, so is CM to AG. if the base EH be equal to К. B R the base NP, then because G L P C AG, CM be not equal, neither shall the folid AB be equal to the solid CD. but the folids 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, fo is CM to AG. Next, let the bases EH, NP not be equal, but EH greater than the other. since then the solid AB is equal to the folid CD, CM is therefore greater than RD AG, for if it be not, neither also, in this case, X к E would the solids AB, CD be equal, which, by the hypothefis, are equal. Make then CT equal to H Н L AG, and complete the P folid parallelepiped CV A of which the base is NP, C N and altitude CT. Because the solid AB is cqual to the solid CD, therefore P 4 2. 7. S c. 25. II. Book XI. therefore the solid AB is to the solid CV, as a the solid CD to the solid CV. but as the solid AB to the solid CV, so b is the base EH to the base NP; for the folds AB, CV are of the same altitude; and b. 32.11. as the solid CD to CV, so is the base MP to the base PT, and fod is the straight line MC to CT; and CT is equal to AG, therefore d. 1. 6. as the base El to the base NP, fo is MC to AG. wherefore the bases of the solid parallelepipeds AB, CD are reciprocally proportional to their altitudes. Let now the bases of the folid parallelepipeds AB, CD be reciprocally proportional to their altitudes; viz. as the base EH to the base NP, fo the altitude of the folid CD to the alti K B R D tude of the folid AB; the M folid AB is equal to the so X lid CD. let the insisting lines be, as before, at right IL 0 P C N the base NP, since EH is to NP, as the altitude of the solid CD is to the altitude of the folid 6. A. S. AB, therefore the altitude of CD is equal to the altitude of AB. but solid parallelepipeds upon equal bases, and of the fame altitude f. 31. 1!. are equal f to one another; therefore the folid AB is equal to the solid CD. 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, fo is CM the altitude of the R D X G F T L P 0 A С N CT, therefore the base EH is to the base NP, as MC to CT. but as the base EH is to NP, so b is the folid AB to the folid CV; for the folids AB, CV are of the fame altitude; and as MC to CT, so is the base MP to the base PT, and the solid CD to the solid CV. and and therefore as the solid A B to the sokid CV, so is the solid CD Book XI. to the solid CV; that is, each of the folids AB, CD has the same ratio to the folid CV; and therefore the folid AB is equal to the folid 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 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 those planes in the points S, Y, V, T; Q, I, U, Z; and complete the folids FV, XU, which are parallelepipels, as was proved in the last part of Prop. 3 1 ít of this Book. In this cafe likewise, if the folids AB, CD be equal, their bases are reciprocally proportional to their altitudes, viz. the base EH to the baté NP, as the altitude of the solid CD to the altitude of the solid AB. Because the folid AB is equal to the folid CD, and that the folid BT is equal $ to the g. 29. or folid BA, for they are upon the same bate FK, and of the fame alti 30. II. tude; and that the folid DC is equal to the folid DZ, being upon the fame base XR, and of the fame altitude; therefore the folid BT is equal to the folid DZ. but the yases are reciprocally proportional to the altitudes of equal solid parallelepipeds of which the insisting straight lines are at right angles to their bafes, as before was proved therefore as the base FK to the base XR, fo is the altitude of the solid DZ to the altitude of the folid 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 folid DZ to the altitude of the folid BT. but the altitudes of the solids DZ, DC, as also of the folids BT, B A are the same. Therefore as the base EH to the base NP, so is the altitude of the Book XI. folid CD to the altitude of the folid AB; that is, the bases of the folid parallelepipeds 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 folid CD to the altitude of the folid AB; the solid AB is equal to the folid CD. the fame construction being made, because as the base EH to the base NP, fo is the altitude of the solid 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 base FK is to the base XR, as the altitude of the solid CD to the altitude of AB, but the altitudes of the solids AB, BT are the fame, as also of CD and DZ; therefore as the base FK to the base XR, fo is the altitude of the solid DZ to the altitude of the solid BT. wherefore the bases of the folids 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 folid BT is equal to the folid g. 29. OF DZ. but BT is equal & to the solid BA, and DZ to the solid DC, 30. 11. because they are upon the same bases, and of the fame altitude. Therefore the solid AB is equal to the folid CD, Q. E. D. PROP. Book XI. PROP. XXXV. THEOR. I F from the vertices of two equal plane angles there be See N. drawn two straight lines elevated above the planes in which the angles are, and containing equal angles with the sides of those angles, each to each; and if in the lines above the planes there be taken any points, and from them perpendiculars be drawn to the planes in which the first named angles are; and from the points in which they meet the planes, straight lines be drawn to the vertices of the angles first named; these straight lines shall contain equal angles with the straight lines which are above the planes of the angles. Let BAC, EDF be two equal plane angles; and from the points A, D let the straight lines AG, DM be elevated above the planes of the angles, making equal angles with their fides, cach to each; viz. the angle GAB equal to the angle MDE, and GAC to MDF; and in AG, DM let any points G, M be taken, and from them let perpendiculars GL, MN be drawn to the planes BAC, EDF meeting these planes in the points L, N; and join LA, ND. the angle GAL is equal to the angle MDN. Make AH equal to DM, and through H draw HK parallel to GL. but GL is perpendicular to the plane BAC, wherefore HK is perpendicular * to the same plane. from the points K, N, to the a. 8. 11. straight lines AB, AC, DE, DF, draw perpendiculars KB, KC, NE, NF; and join HB, BC, ME, EF. Because HK is perpendicular to the |