TRIANGLE AND THE SUM OF THE ANGLES, 345 
sum of the angles of the triangle zty is equal to two right angles. 
Therefore the sum of the angles of the triangle 2zy is equal to two 
right angles. . 
Cor.—Kither every triangle has the sum of its angles equal to two 
right angles, or no triangle has the sum of its angles so great (see 
Prop. I.) as two right angles. 
Provosirion III, 
If the base CD of a triangle ACD (Fig. 6) be diminished indefi- 
nitely according to any law, while neither of the other sides becomes 
greater than a given line AB, the area of the triangle ACD becomes 
ultimately less than any finite , 
space L (Fig 5); and the sum | 
of its angles does not ulti- 
mately differ from two right 
angles by any finite angle. 
For, within the area L take 
a pomt F. Then, by choosing | 
a radius sufficiently small, we © 
ean describe, with F as a 
centre, a circle lying wholly 
within L, and therefore less 
than L. Draw a diameter EG, 
with a radius HF perpendicu- 
lar to it. Join EH; and from 
any point M in EH let fall MN perpendicular on EF. By bisecting NF, 
and again bisecting the parts obtained, and so on, we can diyide NF 
into ” equal parts; where m may be taken greater than any number 
that can be named, Let NF be so divided into the a equal parts, 
