TRIANGLE AND THE SUM OF THE ANGLES. 349 
We can go on thus indefi- A 
nitely, forming a series of 
pairs of equal triangles 
KMN and PNF, Trh and 
FLA, &c., to which there 
is no limit; and, if S, be FIGs 
the sum of the angles of g z = 3 
the first triangle in the n™ yA 
pair, and s, the sum of the 
angles of the second tri- 
angle in the x™ pair, 
ie —Sy = S—-s. 
But, as the series of tri- 
angles, FPN, FLA, &c., is he 
indefinitely increased in 
y Pasi ne 
number, by a continued 
repetition of the construc- 
tion above described, the 
base (such as AL) of the 
triangle ultimately ob- 
tained becomes indefinitely 
small. For 
BC = CD + DE 
=CD+NP + MN 
= CD + 2NP +/AL+ TA, 
and so on, without limit; so that, if the base (such as AL) of the 
triangle (such as FLA) ultimately obtained did not become indefinitely 
small, the finite ine BC would be greater than the sum of an indefi- 
nite number of lines, none of which was less than a given finite line : 
which is impossible. Since therefore the base (such as hL) of the 
triangle (such as FAL) ultimately obtained must become indefinitely 
small, the sum of the angles of the triangle (such as FL‘) ultimately 
obtained cannot (Prop. III.) differ by any finite angle from two right 
angles. That is, S, does not continue, as n is indefinitely increased, 
