TRIANGLE AND THE SUM OF THE ANGLES. 355 
where, since we may arrange to have h,, ho, &c., as small as we 
please, we may understand that / is a quantity which we can arrange 
to have as small as we please. In lke manner, if S, be the sum of 
the angles of the triangle LDM, we can get 
De Fil CAre Or en Rly alle atAiodys> dees waits crauaifsaihtl 4, Secu y)) 
& bemg a quantity which we can arrange to have as small as we 
please. Hence, from (5) and (6), we can order our construction so as 
to make the ratio, 2—S, : 2—S., as nearly equal as we please to the 
ratio, N : 2; the same means by which this is secured having the 
effect of rendering [see (3) and (4)] the ratio, LED : LMD, as nearly 
equal as we please to the ratio, N: Hence we can order our con- 
struction so as to make the two ratios, 
LED : LMD, 
and, 2—S, : 2—S,, 
as nearly equal as we please. This is accomplished by the means 
above described, whatever be the length of the lme FD. It may 
therefore be still accomplished, though FD be taken indefinitely 
small. But as FD is indefinitely diminished, the area of the triangle 
LFD, and therefore that of the triangle LBE is (Prop. III.) indefi- 
nitely diminished. Hence, as FD is indefinitely diminished, the ratio 
of the triangles LED and LBD ultimately becomes indefinitely near 
to a ratio of equality ; the ratio of the triangles LDM and LCM aiso, 
becoming, under the same circumstances, indefinitely near to a ratio, 
of equality. Consequently, by taking FD small enough, the ratio,, 
LBD: LCD, or, A:a, becomes indefinitely near to the ratio, 
LED: LMD. In like manner it can be proved, that, as FD becomes 
indefinitely small, the ratio, 2—S, : 2—S,, approximates indefinitely 
to the ratio, 2—S:2—s. Therefore the ratio, A: a, cannot differ 
by any finite amount from the ratio, 2—S: 2—s. That is, 
Propostrion VI. 
If BGC and HCF (Fig. 13) be any two plane triangles, S being 
the sum of the angles of the former, and s the sum of the angles of 
the latter; then, reasoning on the hypothesis that the angles of a 
