182L] geometrically the Cases of Spherical Triangles. 26Js 



the arcs B A, B C, fig. 4, equal to the given sides, and make 

 B O L equal to the given angle ; let fall C E perpendicular 

 to O B ; on O L take O M = C E, draw M N parallel to it, and 

 on C E produced take E P = O N. Join O A, let fall P D per- 

 pendicular to it; take NK = P D, and join M K; then 

 M K N is equal to the required angle A. 



Case 3. — When two sides B C, C A, and an angle A opposite 

 one of them are given. 



First, to find the third side A B. 



With the chord of 60° as radius, and from any centre O, fig. 5, 

 describe a circle, set off the arcs C B, C A, equal to the given, 

 sides, and make COL equal tathe given angle, draw A I per- 

 pendicular to O C, take O M = A I, and draw M IS perpendi- 

 cular to O C ; from M apply M F = chord B C, meeting C O 

 produced, if necessary, in F ; make I P = O N, and from P as a 

 centre, with the distance F N, intersect the circumference of the 

 circle in H and H'' : the arc C H, or C H', is equal to the third 

 side required. 



Secondly, to find the angle B opposite the other given side. 



With the chord of 60° describe a circle as before, take the 

 arcs C B, C A, fig. 6, equal to the given sides, and make COL 

 equal to the given angle ; dravv A I, B K, perpendicular to O C, 

 take O M = A I, and from M apply M H = B K, meeting C O^ 

 produced, if necessary, in H : the angle M H C, or its supple- 

 ment M H C^ is equal to the required angle B. 



Thirdly, to find the angle C included by the given sides. 



Find the third side A B by the first part of this case, and the 

 -angle C from the three sides by Case 1. 



This third Case it is well known frequently admits of two solu- 

 tions, and in those instances in which the data lead to a double 

 solution, these constructions will accordingly give two values for 

 €ach of the required parts. In the first part of the Case, if H' 

 either falls upon C, or on the same side of C with B, there will 

 be only one value of the side A B, viz. C H. In the second 

 part of the case, if either of the angles M H C, M H C^, taken as 

 the value of B make i (A + B) of the same affection with 

 -i- (B C -f C A) then either of these angles may be that required^ 

 but if only one of the values of B is in accordance with the 

 theorem, ^ (A + B) hke -i- (B C + C A), this value is that 

 required. 



In the third part of the Case, the angle C will have one or two 

 values according as the side A B has one or two values. 



Without giving the demonstration of these constructions with, 

 all the rigour of the Euclidean geometry, the mathematical 

 reader will, it is presumed, be convinced of their accuracy from, 

 the following general explanation. Suppose ocb a, fig. 7, to be 

 a spherical tetraedron whose radius, o «, is equal to the radius o€ 

 the scale of chords made use of in the preceding constructions^ 

 and whose convex surface, a c b, forms the triangle proposed to 



