32.11. +1.6. 25. 11. † 9.5. 11. RD K B MV P CN is to the base NP, as CM to AG, Second general case. Let the insisting straight lines FE, BL, GA, KH; XN, DO, MC, RP not be at angles to the bases of the solids. In this case, likewise, if the solids AB, CD be equal, their bases shall 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. 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. R D Because the solid AB is equal to the solid CD, and * 29. or 30. that the solid AB is equal to the solid BT, for they are upon the same base FK, and of the same altitude; and that the solid 29. or 30. CD is equal* to the solid DZ, being upon the same base XR, and of the same altitude; therefore the solid BT is equal to the 11. H S P 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 shall be 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 altitudes 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: K B H P s 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 to the solid BA, 29. or 30. 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. Therefore, the bases, &c. Q. E. D. 11. PROP. XXXV. THEOR. If, 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 sides, each 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 shall be equal to the angle MDN. * D B K E N L M 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 straight lines AB, AC, DE, DF, draw perpendiculars KB, KC, NE, NF; and join HB, BC, ME, EF. Because HK is perpendicular to the plane BAC, the plane HBK which passes through HK is at right angles to the plane BAC; and AB is drawn in the plane BAC at right angles to the common section BK of the two planes; therefore AB is perpendicular* to the plane HBK, and makes right angles* with every straight line meeting it in that plane: but BH meets it in that plane; therefore ABH is a right angle: for the same reason DEM is a right angle, and is therefore equal to the angle ABH: and the angle HAB is equal + to the angle MDE: therefore in the two triangles HAB, MDE there are two angles in one equal to two angles in the other, each to each, and one side equal to one side, opposite to one of the equal angles in each, viz. HA equal to DM; therefore the remaining sides are equal*, each to each: wherefore AB is equal to DE. In the same manner, if HC and MF be joined, it may be demonstrated that AC is equal to DF: therefore, since AB is equal to DE, BA and AC are equal to ED and DF each to each; and the angle BAC is equalt to the angle EDF: wherefore the base BC is equal* to the base EF, and the remaining angles to the remaining angles: therefore the angle ABC is equal to fore in the two triangles BCK, EFN, there are two angles in one equal to two angles in the other, each to each, and one side equal to one side adjacent to the equal angles in each, viz. BC equal to EF; therefore the other sides are equal to the other sides; BK then is equal to EN: and AB is equal to DE; wherefore AB, BK are equal to DE, EN, each to each; and they contain right angles; wherefore the base AK is equal to the base DN. And since AH is equal to DM, the square of AH is equal to the square of DM: but the squares of AK, KH are equal to the square* of AH, *47.1. because AKH is a right angle; and the squares of DN, NM are equal to the square of DM, for DNM is a right angle: wherefore the squares of AK, KH are equal to the squares of DN, NM: and of these the square of AK is equal to the square of DN; therefore the remaining square of KH is equal to the remaining square of NM; and the straight line KH to the straight line NM: and because HA, AK, are equal to MD, DN, each to each, and the base HK to the base MN, as has been proved; therefore the angle HAK is equal to the angle MDN. Therefore, if from the 8. 1. vertices, &c. Q. E. D. * COR. From this it is manifest, that if from the vertices of two equal plane angles, there be elevated two equal straight lines containing equal angles with the sides of the angles, each to each; the perpendiculars drawn froni the extremities of the equal straight lines to the planes of the first angles are equal to one another. Another Demonstration of the Corollary. Let the plane angles BAC, EDF be equal to one another, and let AH, DM, be two equal straight lines above the planes of the angles, containing equal angles with BA, AC, ED, DF, each to each, viz. the angle 13. 11. See N. 26. 11. HAB equal to MDE, and HAC equal to the angle Because the solid angle at A is contained by the three plane angles BAC, BAH, HAC, which are, each to each, equal to the three plane angles EDF, EDM, MDF, containing the solid angle at D; the solid angles at A and D are equal, and therefore coincide with one another; to wit, if the plane angle BAC be applied to the plane angle EDF, the straight line AH coincides with DM, as was shewn in Prop. B. of this book: and because AH is equal to DM, the point H coincides with the point M: wherefore HK, which is perpendicular to the plane BAC, coincides with MN* which is perpendicular to the plane EDF, because these planes coincide with one another. Therefore HK is equal to MN. Q. E. D. PROP. XXXVI. THEOR. If three straight lines be proportionals, the solid parallelopiped described from all three, as its sides, is equal to the equilateral parallelopiped described from the mean proportional, one of the solid angles of which is contained by three plane angles equal, each to each, to the three plane angles containing one of the solid angles of the other figure. Let A, B, C be three proportionals, viz. A to B, as B to C: the solid described from A, B, C shall be equal to the equilateral solid described from B, equiangular to the other. Take a solid angle D contained by three plane angles EDF, FDG, GDE; and make each of the straight lines ED, DF, DG equal to B, and complete the solid parallelopiped DH: make LK equal to A, and at the point K in the straight line LK, make* a solid angle contained by the three plane angles LKM, MKN, NKL, equal to the angles EDF, FÖG, GDE, each to each; and make KN equal to B, and KM equal to C; and complete the solid parallelopiped KO. And because, as A is to B, so is B to C, and that A is equal to LK, and B to each of the straight lines DE, |