1843.] Treatment of Geometry as a branch of Analysis. 1 19 



12. We are now in a condition to discuss the geometrical properties 

 implicated inequations (a) and (/3). The first set can be presented in 

 a more convenient form by eliminating cos B and cos C from the first 

 by the help of the second and third. 



It then becomes 



a 2 = i2 ^ c ? _2iccosA "1 



symmetrically b® = at +c ? — 2 a c cos B > . . . . (y) 

 and c2 = «2 -f b% — 2 a b cos C ) 



On these two sets of equations, /3 dependent on the sinal and y 

 on the cosinal forms of the functions, the entire geometry of triangles 

 can be raised with little more difficulty than is experienced in the 

 deduction of a corollary. 



Taking the first equation of ^3, it can be changed to the proportion 

 b : c — sin B : sin C* Hence \{ b — c ; sin B = sin C. It will 

 follow that B = C (I. 5) for the ambiguity B = ir — C cannot take 

 place, since two angles of a triangle cannot both be obtuse. Similarly, 

 if B = C ; b = c (I. 6). If b 7 c, sin B 7 sin C, and therefore B must 

 be greater than C (I. 18). The converse evidently follows (I. 19). 



Again, by composition the proportion becomes 



b + c : c = sin B + sin C : sin C. 

 and compounding this with a proportion derived from the 2d of /3, 

 b -+- c : a = sin B + sin C : sin A. 



Suppose another triangle A'B'C on the same base a inclosed within 



ABC, so that B 7 B' and C 7 C then also A' 7 A. This triangle 



will also have 



V + c' : a = sin B' + sin C : sin A'. 



Compounding b+c: 6'+c'=(sin B + sin C) sin A': (sin B' -J- sin C) sin A. 



But the second antecedent is entirely greater than its consequent ; 



... Hc7&' + c' (I. 21) 



If A' fall on the base, b' + c' will equal a 



.-. b + c7a (1.20) 



13. Consulting the first of y. As A increases while acute, cos A 

 decreases, hence a less amount is taken from b® + c 2 and a conse- 

 quently increases. When A becomes obtuse, cos A is negative, and the 

 third term therefore additive ; now also then increase of A adds more 

 to b 2 -\- c 2 and therefore to a 2 - Always therefore if b and c remain 



* This notation is German, and very expressive, proportion being the equality of 

 ratios. 



