8 



MATHEMATICS. SPHERICAL GEOMETRY. 



[PROP. vi. ir. 



Fig. 7. 



'/. \ V im-l.wi-.l liy the rc 

 / V. V X, X/ is callod a 

 spherical triangle. 



It will be observed, that 

 the great circle in a sphrri- 

 cal triangle is analogous to 

 the straight line in the case 

 of a plane triangle j but 

 is this difference to 

 be observed, that two 

 straight lines, when pro- 

 duced, never meet, whereas 

 two great circles, when 

 produced, always meet, viz., in a point distanced from 

 the ..tlu-r by a whole semicircle. To consider the result 

 of this circumstance, we will suppose the section made 

 by the plane of the paper (AB4) Fig. 7, to be one great 

 circle, and the two others to be A C<ic and B Cbc, O the 

 centre of the sphere, AO./, B OA, COc, are diameters. 

 It will be seen that A BC is a spherical triangle, as in 

 the case of figure 0. But the three circles, in addition 

 to ABC, make seven other spherical triangles, riz., 

 alC, B C, bC A, on the one 

 hemisphere, ABCu, and AB<?, 

 ubc, bcA., Bca, on the hemis- 

 phere Aeia. 



Also, it is plain that the 

 triangles on the one hemis- 

 phere are equal to those on 

 the other, each to each ; thus, 

 the triangle A B C is equal to 

 the triangle ale. For AC', 

 1 and Cue are each halves of the 

 same circle, and . . are equal 

 to each other ; take away the 

 common part C, and we have 

 left A C = oc. Similarly A B = ab, and B C = be. 

 Again, the angle of the triangle acb, which is the angle 

 between the planes, is equal to the angle ACB, which 

 is also the angle between the planes ; similarly the other 

 angles are equal, and they are described on the surface 

 of the same sphere ; if, therefore, the triangles were 

 KU]"-I imposed, they would coincide, and are . . equal. 

 N.B. The student will do well to consider very care- 

 fully the above observations : he must also take notice 

 of the assumption, that triangles taken off the surface 

 of the same sphere will coincide, provided their sides 

 and angles are equal. This is merely assuming that 

 the curvature of the same sphere is the same at all 

 parts ; which is obviously true, as the following con- 

 sideration will assure us : Suppose we have two 

 spheres of equal radii suppose these centres to coincide 

 then, since every point in each sphere is equally 

 distant from their common centre, their surfaces coin- 

 tide, and will continue to coincide however we may 

 move either of them, provided their centres continue 

 to coincide. 



It is plain that the side A B measures the angle AOB. 

 Hence the side of a spherical triangle is spoken of as an 

 nnu'le, riz., the plane angle it subtends at the centre of 

 the sphere. 



PROPOSITION VL 



Any two tides of a spherical triangle are together greater 

 than the third, and the three sides of the triangle are 

 tuyether leu than four r'ujht angles. 



For (Planes, see Prop. 20), if a solid angle be con- 

 tained by three plane angles, any two are greater than 

 a third ; but (Fig. 7) the solid angle at O is contained 

 by A O B, B O C, C O A, . . any two of these are greater 

 than a third ; and hence any two of the three sides of 

 ABC (which sides measure these angles respectively) 

 must be greater than the third. 



Again (Planes, Prop. 21), the throe angles, AOB, 

 BO C, C O A, are together less than four right angles ; 

 and . . the three sides, A B, B C, C A, which measures 

 these angles must be less than four right angles. Q.E.I). 



l)Er. IX. A lune is the portion of the surface of a 

 sphere inclosed by the arcs of two great circles. 



Tli us (Fig. 7), A Bo is a lime. 



DBF. X. The angle of a lune is the angle between 

 the two great circles which Ixmnd it. 

 Thus (Fig. 7), B Ao is the angle of the lune. 



PROPOSITION VII. 



On equal spheres, if the angles of dm Imtes are equal, the 

 lunes themselves ait equal. 



Let ACBD, AEBF, be two lunes described on 

 equal and coincident spheres, having the angle CAD 

 = angle KA F, these lines shall be equal ; for suppose 

 the one sphere to revolve till the circle A E B coincides 

 with A C B, then because an- 

 gle C A U = angle E A F, we 

 must liave AFB coinciding 

 with A D B. The lunes, 

 therefore, coincide and are 

 equal. 



JS.B. In the above demon- 

 stration we have assumed 

 that the lunes have the 

 same extremities, A B : we 

 are obviously entitled to do 

 this, since by shifting the 

 spheres these extremities can 

 be brought to coincide. Also, 

 the proposition is plainly true when the lunos are on 

 the same sphere. Also, it is manifest that the greater 

 luue has the greater angle, and vice versa, 



PROPOSITION VIII. 



In the same or equal spheres, lunes are to each other in the 

 ratio of their angles. 



For let I and m be two lunes, the angles of which are 

 a and 6 ; let A be any multiple of o, and L the lune cor- 

 responding to A ; then it is plain that L is the same 

 multiple of I that A is of a. Similarly let B be any 

 multiple of 6, and let M be the corresponding lune ; then 

 it is plain that M is the same multiple of m that B is of 

 6. We have then four magnitudes, I, m, a, b ; ami of 

 the first and third we have taken any equimultiples, L 

 and A ; and of the second and fourth wo have taken any 

 equimultiples, M and B. Now, by last proposition, if 

 L > M, A is > B ; if equal, equal ; if less, less .'. (Dcf. 

 V., p. 671), I : m : : a : b. Q. E. D. 



COR. It is plain that the area of half a hemisphere is 

 a lune whose angle is a right angle, .'. if X be the area 

 of a sphere, and if A be any lune whose angle is B, 



^ 

 A : -j : : B : one right angle. 



.*. A : X : : B : four right angles. 



Or area of lune : area of sphere : : angle of lune : four 

 right angles. 



DKF. XI. The spherical excess of a spherical triangle 

 is the excess of the siun of its three angles over two right 

 angles. 



It will be seen by the next proposition that the sum 

 of the three angles of a spherical triangle are really 

 greater than two right angles the excess of the angles 

 above two right angles is clearly due to the spliericity of 

 the triangle ; hence the term "spherical excess." 



PROPOSITION IX. 



To prove that the angles of a spherical trifinglr nr,- /..,/,//,, r 

 greater than two right angles, and that the area of a 

 spherical triangle hat to the area of half tlw sphere on 

 which it is described tlie same ratio that the spherical 

 excess has to two right angles. 



For (Fig. 7) let AB C be the triangle, then the lunes 

 corresponding to each of the angles an' II i A correspond- 

 ing to A, B6A corresponding to B, C'ibc corresponding 

 to C. For the sake of brevity, call these luues re- 

 spectively LH L.,, I. 



'Iheii by Corol. to Prop. VIII., 



LI : area of sphere : : A : 4 right angles. 



or, L, : area of hemisphere I : A : 2 right ang.es. 



