354 RELATION BETWEEN THE AREA OF A PLANE 
given angle; and that the triangle LGG, shall differ from the triangle 
LGF by a space less than any given space, the sum of its angles at the 
same time differing from the sum of the angles of the triangle LGF 
by.an angle less than any given angle; and so on, till the whole of 
the triangle LBD has been exhausted, except a remainder LBE, which 
is less than the triangle to which it is adjacent. Proceed next to 
divide the triangle LDC into triangles LDT, LTH, Sc., related to the 
triangle LFD and to one another in the same manner as the triangles 
LFG, LGG,, &c.; the remainder LMC being finally left over, less 
than the triangle to which it is adjacent. Then, since any two adja- 
cent triangles in the series, 
TAD Gr DAG Gee SOC 2 0). af Sieca) aceite. aloha ee ol 
which together constitute the triangle LDH, may be made as nearly 
equal as we please, we can make every one of them as nearly equal to 
the first as we please. And, from a similar consideration, it appears 
that we can at the same time make the sum of the angles of any tri- 
angle in the=series as nearly equal as we please to the sum of the 
angles of the first. In like manner we can make every one of the 
triangles in the series, 
EOD AUR TAN ES cree HER GEA aT He Me EAM acu ((2)) 
which together constitute the triangle LDM, as nearly equal to LDF 
as we please ; the sum of the angles of each being at the same time 
made as nearly equal as we please to the sum of the angles of the tri- 
angle LDF. Let there be N terms in the series (1), and 2 im the 
series (2). Then 
ED e=tN times sD one Ue gene 
Q being a quantity which we may arrange to have as small as we 
please. In like manner, 
TN times, UE Diagn ete cone (4) 
q bemg a quantity which we may arrange to have as small as we 
please. Again, if S, be the sum of the anglesYof the triangle LFD, 
S, @/, the sum of the angles of the triangle LFG, S, wih, the 
sum of the angles of the triangle LG@G,, and so on, and S, the sum 
of the angles of the triangle LED, we have 
8S, = NS,—2(N—-1) wh, wh, & &e. 
PZ Ooh IN Q'S) eemin tad Mo eshore ot 105) 
