62 RELATIONS PERTAINING SIMPLY [BoOK I. 



tions of the second to the first, the third to the first, the second to the third. 

 Our problem, therefore, depends upon the solution of a spherical triangle, in 

 which, from one side and the adjacent angles, the other parts are to be deduced. 

 We omit, as sufficiently well known, the common precepts for this case given 

 in spherical trigonometry : another method, derived from certain equations, which 

 are sought in vain in our w y orks on trigonometry, is more conveniently employed. 

 The following are these equations, which we shall make frequent use of in future: 

 a, b, c, denote the sides of the spherical triangle, and A, B, C, the angles oppo 

 site to them respectively : 



j sini(& c) __ sin | (B C) 

 sin a cos ^ A 



-. j sin 1 (b + e) __ cosj (BC) 

 sin ^ a sin ^ A 



HI COS 



cos^a cos 



i : \ ~t - 



cos \ a sin A 



Although it is necessary, for the sake of brevity, to omit here the demonstration 

 of these propositions, any one can easily verify them in triangles of which neither 

 the sides nor the angles exceed 180. But if the idea of the spherical triangle is 

 conceived in its greatest generality, so that neither the sides nor the- angles are 

 confined within any limits whatever (which affords several remarkable advan 

 tages, but requires certain preliminary explanations), cases may exist in which it 

 is necessary to change the signs in all the preceding equations ; since the former 

 signs are evidently restored as soon as one of the angles or one of the sides is 

 increased or diminished 360, it will always be safe to retain the signs as we 

 have given them, whether the remaining parts are to be determined from a side 

 and the adjacent angles, or from an angle and the adjacent sides ; for, either 

 the values of the quantities sought, or those differing by 360 from the true val 

 ues, and, therefore, equivalent to them, will be obtained by our formulas. We 

 reserve for another occasion a fuller elucidation of this subject : because, in the 

 meantime, it will not be difficult, by a rigorous induction, that is, by a complete 

 enumeration of all the cases, to prove, that the precepts which we shall base upon 



