in Plane Trigonometry. 169 



By applying both formulae, the same result will be obtained in 

 two different ways, a thing always desirable, and better than a 

 verification, obtained by a mere repetition of the calculation. 



3. These formulae (which are not given as new) may be 

 briefly demonstrated as follows : 



Because sin A : sin C : : a : c, 

 and sin B : sin C : : b : c; 

 Therefore sin A + sin B : sin C : : a + b : c; 

 also sin A sin B : sin C : : a b : c. 

 But because sin A + sin B = 2 sin ^ (A + B) cos (A B) , 

 and sin A sin B = 2 cos ^ (A + B) sin ^ (A B) , 

 and sin C = sin (A + B) = 2 sin (A + B) cos ^ (A + B) , 

 Therefore, sin A + sin B : sin C : : cos i (A B) : cos ^ (A + B) ; 

 in like manner, sin A sin B : sin C : : sin (A B) : sin 4 (A + B) ; 

 Hence, cos (A B) : cos $(A + B) : : a + b : c; 

 Also, sin HA B) : sin | (A + B) : : a b : c. 



4. An example will serve to shew the distinction between the 

 common method and that proposed in this paper. 



The sides of a triangle being a, b, c ; 

 and the opposite angles A, B, C . 



(side a = 169584 feet ) . . , _, . 



There are given \ side b = 119613 I to find the Slde C in two 



1 angle C = 60 43' 36" j wa y s ' 



a + b 289197 5-4611938 



a b 49971 4-6987180 



tan i (A + B) 59 38' 12" 10-2322235 



14-9309415 



tan HA B) 16 26 0-23 9 -4697477 



A = 76 4 12-23 

 B = 43 12 11-77 



VOL. X- P. I. 



