Interpretation of For mules in Spherical Trigonometry, 21 



are those which alone are recognised by writers on trigo- 

 nometry. With regard to the others, they are evidently 

 the remaining arcs of the great circles of which a\ b\ c 

 are parts, and they are therefore each greater than a semi- 

 circle. In fact, if the triangles be described, whose sides are 

 a', h\ c\ and whose angular points are A, B, C, so that A 

 and B are joined by c\ A and C by U, and B and C by a' ; 

 then 2'K—c is the other arc joining A and B; 2^ — 8' the 

 other arc joining A and C, &c. These larger arcs, being the 

 continuations of a', b\ c\ will evidently form with each other 

 angles which are vertically opposite, and therefore equal, to 

 those made by «', b\ c'. They will therefore answer to the 

 conditions of the problem, since they are arcs of great circles 

 making with each other angles equal to the given angles. 



5. It ought to be remarked that a'\ U\ c" do not form on 

 the surface of the sphere, a triangle in the ordinary sense 

 of the term : they do not form a space nsihiclithey hound isoith- 

 out intersectiiig it. The truth is, however, that in both plane 

 and spherical trigonometry, so far as the computation of sides 

 and angles alone is concerned, we have nothing to do with the 

 surface of the triangle, or with any surface whatever; the 

 lengths of lines and their relative positions being the sole sub- 

 ject of consideration. Thus, when the three sides of a sphe- 

 rical triangle are given, to find the angles, the problem is 

 simply this: Given three points on the surface of a sphere^ to 



find the angles made by the great circles passing through them. 

 So likewise, when the three angles of a spherical triangle are 

 given, to find the sides, the problem, without reference to a 

 triangle, is simply, to fnd three points on the surface of the 

 sphere^ such that the arcs joining them may make with each other 

 angles equal to given angles : and it is easy to see that other 

 problems may be expressed in a similar way ; and that the 

 same views, as well as some others in this paper, may also be 

 extended to plane triangles. 



6. The problem in which two sides of a spherical triangle, 

 and the contained angle are given, to compute the remaining 

 angles and the third side, may also be viewed in a similar 

 light. Thus, the third side is found by means of the for- 

 mula 



cos a — cos A sin 6 sin c + cos b cos c ; 



and it is plain, that a will have two values of the forms a' and 

 (fl"=)2 7r— a'. The smaller is that which is recognised by 

 the writers on trigonometry as the third side ; while the other 

 is the remaining arc of the great circle of which a' is one part. 

 This will be illustrated by taking on a common globe a point 



M 



