GEORGE BRUCE HALSTED — NEW ELEMENTS OE GEOMETRY. 
9 
? (*-A)=X+*. 
47T 
| (Jb)=y+j. 
J| ■{— C)=Z. 
47T 
R=X+Y+Z+^+A. 
Hence we g*et 
A+B+C=*+^(a-*--. y+*) (1). 
It would still remain now to demonstrate, in the hypothesis that 
the sum of the angles of a triangle S<>, that A=x-\-y — zfor every 
r or at least for r— oo; but to undertake this labor would be in 
vain. On the contrary, for the condition S< r, we find always 
A <A+y— as we will immediately see. 
When CC ' —r increases, the points B', B", B'" withdraw from 
the point B in the direction AB, while the lines CB', CB", CB' ' ' ap- 
proach to a certain limit CD (Fig-. 5), which in the new theory I 
have designated as a 'parallel to AB, and which makes with CA, 
CB angles ACD=?r — A— «, BCD=B— /?, such that « and /? are any 
positive quantities. 
Moreover, by making- 
CB ' —r sufficiently great, 
the angles B'CD, B"CD,C 
B'”CD become as small as 
we choose. Calling-, then, P 
the surface of the triangle 
ACB', we will have for r 
very great (Fig-. 4) without 
appreciable error 
J 
;r~(" — A— a)R= A (*-A)R+P-tf. 
47T 47T 
Rence .v-P (-A./.R (2). 
A 7T 
Putting- for the moment the angle C'CD = M (Fig-. 5) the already 
given equation — B)R=2^(Y+_y) may be written 
A( x _b)R=A-MR+P— A+y— z (3) 
4 7T 4 7T ( 
In this substituting- the value 
M=A+C+a 
we have an equation which combined with equation (2) conducts 
anew to equation (1) and so verifies it. 
If we make 
