parallel Lines in Geometry. 165 
independently of the theory of parallels, that the three angles of 
a triangle are together equal to two right angles. This is ac- 
complished by proving indirectly that the angles of a triangle 
can neither be less nor greater than two right angles. And, 
when we reflect that the whole difficulty is occasioned by the im- 
perfect nature of the definition of a straight line, we are led to 
. suspect that it is necessary to employ the indirect mode of rea- 
*. 
soning. One objection may be miade to Legendre’s demonstra- 
tion; for we are required to admit that, through a point situated 
within a rectilineal angle, at least one straight line may be drawn 
that shall meet both the sides of the angle; a hypothesis, which, 
although it be very probable, is yet in some degree uncertain 
and precarious. Thinking to gratify the lovers of speculative 
geometry, I shall now add a demonstration of the same proposi- 
tion, which requires no new principles, and is liable to no ob- 
jection excepting the length that always attends indirect investi- 
gations. 
Prop. I. Fig. 5 (Plate III.) 
To construct a triangle that shall have the sum of its angles 
equal to the sum of the angles of a given triangle, and one of 
its angles equal to, or less than, half any proposed angle of the 
given triangle. 
Let ABC be the given triangle, and ABC one of its angles : 
bisect the side AC, opposite to ABC, in E; join BE, and, 
having produced it, cut off EF equal to BE; join CF: the 
sum of the angles of the triangle BFC will be equal to the 
sum of the augles of the triangle A BC; and one of the angles: 
vee or BFC will be equal to, or less than, half the angle 
BC. ' 
The construction being the same as in the 16th of the first 
book of Euclid, it may be proved, as in that proposition, that the 
two triangles AEB and CEF are equal in all respects. Where- 
fore, the angle BAE being equal to ECF, the whole angle 
BCF is equal to the two angles BAE and BCE; and, the 
angle ABE being equal to EFC, the whole angle ABC is 
equal to the two angles CBE and EFC. Consequently the 
three angles BCF, CBE, and EFC are equal to the three 
angles BAC, ACB, and ABC. Again, if BC be equal to 
CF, the angles EBC and EFC will be equal to one another, 
_ and to the half of ABC; but, if BC and CF be unequal, the 
angles E BC and EFC will likewise be unequal, and one of them 
will be less than the half of ABC. 
This proposition may be considered as a corollary to the 16th 
of the first book of Euclid. 
Prop. 
