TRIANGLE AND THE SUM OF THE ANGLES. 353 
At the same time (Prop. III.) the sum of the angles of the triangle 
ADE becomes indefinitely near to two right angles. Let 8 be the 
sum of the angles of the triangle ABC; S,, the sum of the angles of 
the triangle ACD ; S,, the sum of those of the triangle ACE; and 3, 
the difference betwixt two right angles and the sum of the angles of 
the triangle ADE. Then 6 is equal to the difference betwixt S, and 
S,5 so that, since 6 ultimately becomes indefinitely smal!, the differ- 
ence betwixt S, and S, ultimately becomes indefinitely small. And 
(Cor. 4) S is termediate betwixt S, and S,. Therefore ultimately 
its difference from either of them becomes indefinitely small. 
Proposition V. 
@ gifsa line LD (Fig. 12) be drawn from L to any pomt D in the base 
of a triangle LBC; and if A represent the area, and S the sum of 
FIGl2 
E Caiienm an Mec 
the angles, of the triangle LBD; and a represent the area, and s the 
sum of the angles, of the triangle LDC; then, reasoning on the hy- 
pothesis that the angles of a plane triangle are (see Cor. Prop. I1.) un- 
equal to two right angles, we can prove that A:a=2—S:2—s; a 
right angle being taken as the unit of measure. 
For, by taking FD sufficiently small, the triangle LFD can be made 
(Prop. III.) smaller than any given space; the sum of its angles also 
falling short of two right angles by an angle less than any given 
angle. Having cut off a small triangle LFD from LBD, we can next 
(Cor. 5, Prop. IV.) draw lines LG, LG,, LG,, &c., (only the first of 
these lines is expressed in the figure), in such a manner that the tri- 
angle LGF shall differ from the triangle LFD 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 LFD by an angle less than any 
Vou, V. 2B 
