120 Treatment of Geometry as a branch of Analysis. [No. 134. 



constant, the increase of A increases a, and vice versa, the increase of a 

 will increase A, (I. 24, 25). 



14. We proceed now to the general determination of triangles. 

 We might first fix the conditions necessary to determine them in in- 

 dividuality, and then in species as Euclid has done ; but it will be more 

 consonant to the spirit of analysis to obtain the most general first. 



Dividing the equations (y) by c 2 and (p) by c and writing - asm and 



b 



as n, 



we 



have 









m* = 



: 71% 



+ 1- 



■ 2n 



cos 



A 



n* = 



m 2 



+ 1 ~ 



■ 2m 



COS 



B 



1 = 



; m 2 



-f 7i 2 - 



- 2mn 



COS 



C 



1 n sin C — sin B = 0) 



> and m sin C — sin A = > 

 ) m sin B — n sin A = ) 



From these six equations, each involving three quantities, any two 

 being given, the rest will be determined. The cases will be 



First : m and n given or the ratios of the sides. Here the angles 

 are determined by their cosines, and hence no ambiguity can occur. 

 The form of the triangle is known, or its species determined (VI. 5). 



Second : m and B given or the ratios of two sides (a,c) and the in- 

 cluded angle. Still n being determined by the 2nd of the first set, the 

 rest are determined as in the former case, and no ambiguity is in- 

 volved. (VI. 6.) 



Third : A,B and therefore C given ; or the three angles. Here m 

 and n are determined by the first two of the 2nd set, and there is no 

 ambiguity. (VI. 4.) 



Fourth : m and A given or the ratio of two sides (a, c) and, an angle 

 opposite to one. In this case C is determined by the 2nd of the 2nd set : 

 the sinal function entering occasions ambiguity. If m be 7 1, a is 

 greater than c and therefore A than C, whence C cannot be obtuse and 

 there is no ambiguity ; but if m ^L 1 or a is less than c, there is no 

 way of avoiding the difficulty, unless the species of C be directly 

 given. (VI. 7) 



If now the length of one of the sides be given in addition to the 

 ratio in which it is involved, the triangle will be determined individually 

 as well as in species. This can occur in 1st, 2nd and 4th cases, which 

 produce (Euclid I. 8, 4 and 26) and (Young I. 26). There being no ratio 

 given in the third case, there is no individual triangle determined by 

 the three angles. 



