B52 RELATION BETWEEN THE AREA OF A PLANE 
and AK can be drawn, cutting off triangles ADC and AEC, the one 
less, and the other greater, than ABC, but neither of them differing 
from the triangle ABC by an area so great as a given area; while at 
the same time the difference between the sum of the angles of the 
triangle ABC and the sum of the angles of either of the triangles, 
ACD, ACE, is less than any given angle. 
If the hypothesis be made that the angles of a plane triangle are 
together (see Cor. Prop. II.) equal to two right angles, the problem 
can be effected by the methods which Euclid describes. 
We onlyneed, therefore, to show how it can be performed on the 
hypothesis that the angles of a plane triangle are not equal to two 
right angles. Bisect CG in F; and jom AF. The triangles ABC 
and ACF have a comon side AC. Therefore (Cor. 4) the area of the 
one will (on the hypothesis on which we are now proceeding) be less 
than, equal to, or greater than, the area of the other, according as 
the sum of the angles of the former is greater than, equal to, or less 
than, the sum of the angles of the latter. Now we can find the sum 
of the angles of each by construction. Therefore we can tell whether 
the triangle ACF is less than, equal to, or greater than, the triangle 
ABC. Should the triangle ACF be greater than the triangle ABC, 
we may repeat the construction ; bisecting CF, and drawing a line 
from A to the point of section. By repeating this construction suf- 
ficiently often, the base (such as CD) of the triangle (such as ACD) 
ultimately obtained will become less than any assignable line; and 
hence the area of the triangle will become (Prop. III.) less than any 
assignable area, and consequently less than the triangle ABC. Let 
ACD, the triangle obtained by bisecting CE, and joining AD, be less 
than the triangle ABC ; the triangle AEC bemg greater than ABC. 
Bisect DE in the point ¢; and join A¢. Find, as above, whether the 
‘triangle AC¢ is less or greater than the triangle ABC, or equal to it. 
Should it be greater, the triangle ABC lies between the limits, ACD 
cand ACé; but should it be less, the triangle ABC lies between the 
limits AC¢ and ACE. And soon. Ultimately we obtain two limits, 
which we may suppose to be represented by the triangles ACD and 
ACE, between which the triangle ABC lies, the base DE of the tri- 
-angle ADE, which is the difference of the limits, beg made as small 
as we please. Therefore (Prop. III.) the area of the triangle ADE 
becomes-ultimately indefinitely small; so that each of the triangles 
ACD and ACE becomes indefinitely near in area to the triangle ABC. 
