PROPOSITION XC.- THEOREM. A с L K M N D B H If two parallel planes AB, CD be cut by a third plane EH, they have the same inclination to that plane. Let the 1s EF and GH be the common sections of the plane EH, with the two planes AB and CD; and from K, any point in EF, draw in the plane EH the 1 KM at gute Ls to EF, and let it meet GH in L; draw also KN at pe Ls to EF in the plane AB; and through the 8 KM, KN, let a plane be made to pass, cutting the plane CD in the line LO. And :: EF and GH are the common sections of the plane EH, with the two parallel planes AB and CD, EF is || GH, (Prop. 89). But EF is at r Ls to the plane that passes through KM and KN, (Prop. 80); : it is at riLs to the lines KM and KN; .. GH is also at gut Ls to the same plane, (Prop. 83); and it is .. at re _s to LM, LO, which it meets in that plane .. since LM and LO are at gu Ls to LG, the common section of the two planes CD and EH, the ZOLM is the inclination of the plane CD to the plane EH, (Def. 6). For the same reason the (NKM is the inclination of the plane AB to the plane EH. But :: KN and LO are parallel, being the common sections of the parallel planes AB and CD, with a third plane; the interior LNKM is = the exterior LOLM; that is, the inclination of the plane AB to the plane EH is = the inclination of the plane CD to the same plane EH. PROPOSITION XCI.-THEOREM. If two straight lines AB, CD be cut by parallel planes GH, KL, MN, in the points A, E, B; C, F, D; they shall be cut in the same ratio : that is, AE: EB= CF: FD. Join AC, BD, AD, and let AD meet the plane KL in the point X; and join EX, XF. : the two parallel planes KL, MN, are cut by the plane EBDX, the common sections xx EX, BDare parallel, (Prop. 89). For the same reason, the two parallel planes GH, KL, are cut by the plane AXFC, the common sections AC, XF are parallel. And ::: EX is || BD, a side of the AABD, AE: EB=AX: XD, (Prop. 59). Again, :: XF is || AC, a side of the AADC, AX:XD=CF: FD... AE: EB=CF: FD, (Alg. 102). B F PROPOSITION XCII.-THEOREM. If a straight line AB be perpendicular to a plane CD, any plane CE touching it will be perpendicular to the same plane CD. For let CBG be their line of common section, and from any point G, in CG, let EG be drawn I to it in the plane CE. Also, let BF, GH be o tto CG in the plane CD. Then .: AB is + to the plane CD, ABF is a rL; and since EG is || AB, and GH is || BF, the LEGĦ is also a 74L, (Prop. 85). But the LEGH is the inclination of the plane CE to the plane CD, (Def. 6); .. the plane CE is at r*Ls to the plane CD. PROPOSITION XCIII.-THEOREM. If two planes AB, BC, cutting one another, be each of them perpendicular to a third plane ADC, their common section BD will also be perpendicular to the same plane. From D in the plane ADC, draw DE to AD, and DF + to DC. DE is to AD, the common section of the planes AB and ADC,and : the plane AB is at ni Ls to ADC, DE is at 74 Ls to the plane AB, and .. also to the | BD in that plane. For the same reason DF is at no Ls to DB. And since BD is at ht Ls to both the lines DE and DF, it is at pe Ls to the plane in which DE and DF are; that is, to the plane ADC, (Prop. 80). B T SOLID GEOMETRY. DEFINITIONS. I. A solid angle is that which is formed by more than two plane angles at the same point, but not in the same plane. res. II. Two solids bounded by planes are similar, when their solid angles are equal, and their plane figures similar, each to each. III. A parallelopiped is a solid bounded by six planes, of which the opposite ones are parallel. If the adjacent sides be perpendicular to one another, it is a rectangular parallelopiped. IV. A cube is a rectangular parallelopiped, of which the six sides are sq V. A prism is a solid of which the sides are parallelograms, and the ends are plane rectilineal figures. VI. A pyramid is a solid of which the sides are triangles, having a common vertex, and the base any plane rectilineal figure. If the base be a triangle, it is a triangular pyramid. If a square, it is a square pyramid. VII. A cylinder is a solid described by the revolution of a rectangle about one of its sides remaining fixed; which side is named the axis ; and either of the circles described by its adjacent sides the base of the cylinder. VIII. A cone is a solid described by the revolution of a right angled triangle about one of its sides remaining fixed, which is called the axis; and the circle described by the other side is the base of the cone. IX. Cones or cylinders are similar when they are described by similar figures. PROPOSITION XCIV.-THEOREM. case, If a solid angle A be contained by three plane angles BAC, CAD, DAB, any two of these are together greater than the third. If all the Ls be =, or if the two greater be =, the proposition is evident. In any other let BAC be the greatest Z, and let BAE be cut off from it, = DAB. Through any B point E in AE, let the | BEC be drawn in the plane of the <BAC to meet its sides in B and C. Make AD=EA, and join BD, DC. - the As BAD, BAE have AD=AE and AB common, and the included Ls BAE, BAD=, the base BE is = BD, (Prop. 5). But the two sides BD, DC are 7 BE, EC, .. DC is Ž EC; and since the side AE is = AD, and AC common to the two As ACD, ACE, but the base DC 7 EC, the CAD is EAC, (Prop. 14). Consequently the sum of the L8 BAD, CAD is = the ŹBAC. B PROPOSITION XCV.—THEOREM. All the plane angles which form any solid angle, are together less than four right angles. Let there be a solid L at the point A contained by the plane Ls BAC, CAD, DAE, E EAB, these Ls, taken together, are four god Ls. For, through any point B in AB, let a plane be extended to meet the sides of the solid / in the lines of common section BC, CD, DE, EB, thereby forming the pyramid BCDE-A; and from any point within the rectilineal figure BCDE, which is the base of the pyramid, let the s OB, OC, OD, OE, be drawn to the angular points of the figure, which will thereby be divided into as many A8 as the pyramid has sides. Then :: each of the solid Ls at the base of the pyramid is contained by three plane Ls, any two of them are together 7 the third, (Prop. 94.) Thus, ABE, ABC, are together 7 EBC. .. the Ls at the bases of the As which have their vertex at A, are together 7 the Ls at the bases of the As which have their vertex at 0. But all the Ls of the former are together = all the Ls of the latter. .. the remaining Ls at A are together < the remaining Ls at 0, that is, <four right angles, (Prop. 1, cor. 3.) PROPOSITION XCVI.—THEOREM. If two solid angles be each of them contained by three plane angles, and have these angles equal, each to each, and alike situated, the two solid angles are equal. Let there be a solid angle at A contained by the three plane L: BAC, CAD, DAB, and a solid angle at E, contained by the three plane Ls FEG, GEH, HEF, = the former, each to each, and alike situated, these solid angles are equal. For let the 18 AB, AC, AD, EF, EG, EH, be all equal, and let their extremities be joined by the lines BC, CD, DB, FG, GH, HF; and thus there are formed two isosceles pyramids, BCD-A, and FGH-E. Upon the bases BCD, FGH, let the Is AK, AA. B EL, fall from the vertices A, E. Then : the ro Ld A$ AKB, AKC, AKD, have equal hypotenuses, and the side AK common, their other sides, KB, KC,KD, (Prop.39,cor. 2), are also equal; ::. K is the centre of the circle that circumscribes the A BDC. In like manner, L is the centre of the circle that circumscribes the AFHG. But these As are equal in every respect. For the sides BC, FG, are equal, .: they are the bases of equal and similar AS BAC, EFG; and for the same reason, CD=GH, and BD=HF. If, .. the pyramid BCD-A be applied to the pyramid FGH-E, their bases BCD and FGH will coincide. Also the point K will fall on L, •: in the plane of the circle about FGH, no other point than the centre is equally distant from the three points in the circumference, the perpendicular KA will coincide with LE, (Prop. 86, cor. 2), and the point A will coincide with E, the ro La A8 AKB, ELF, have equal hypotenuses, (Prop. 39, cor. 2), and BK=FL. But the points B, D, C, coincide with F, H, G, each with each ; :. the 1s AB, AC, AD, coincide with EF, EG, EH, and the plane L: BAC, CAD, DAB, with FEG, GEH, HEF, each with each. Consequently the solid _s themselves coincide and are equal. Cor. 1. If two solid angles, each contained by three plane angles, have their linear sides, or the planes that bound them, parallel each to each, the solid angles are equal, (Props. 84 and 89.) Cor. 2. If two solid angles, each contained by the same number of plane angles, have their linear or plane sides parallel, each to each, the solid angles are equal. For each of the solid angles may be divided into solid Zs, each contained by three plane Ls, and the parts being equal, and alike situated, the wholes are equal. PROPOSITION XCVII.-THEOREM. If two triangular prismg, ABC-DEF, and GHK-LMN, have the plane angles BAC, CAD, DAB, and HGK, KGL, LGH, and also the linear sides AB, AC, AD, and GH, GK, GL, about two of their solid angles at A and G equal, each to each, and alike situated, the prisms are equal and similar. For, since the solid Zs A, G, are each contained by three plane Ls, which are B C M TUTTI |