90 Doctrine of Parallels. 
to describe that species of surface to be one which can contain 
all lines that can pass through a given point to cut a given straight 
line indefinitely produced ; but how is it to be ascertained, that 
any two points not in that given line being taken in that surface, 
the line joining them shall cut the lines between, that lie in and 
define the surface ? 
The undeniable existence of these defects, in addition to the 
vexed one of parallels, gives dignity and value to an attempt of © 
Mr. T. Perroner Txomeson, of Cambridge University, in Eng- 
land, in his ‘“ Geometry without axioms,” to deduce the element- 
ary properties of the straight line and plane from the sphere alone. 
The same author has discussed with much sagacity, in his ap- 
pendix, the methods and devices,—about thirty in all,—which, 
during the historical period of the science, have been proposed 
to obviate or evade the difficulty relating to parallelism ; and he 
has exposed, with a clearness seemingly incontestible, fallacies in 
each. Mr. 'T'sompson has also propounded, in his text, a series 
of propositions which purport to compose a rigorous weet of the 
entire doctrine; and this, not improbably, with justice ; but the 
prolixity of the process is sufficient, in some instances, to deter one, 
for the moment, from pursuing it through the somewhat intricate 
figures. 
The author who has made the most persevering and repeated 
endeavors to complete the doctrine of parallels—supplanting cer- 
tain of his earlier methods by new ones, and pertinaciously main- 
taining to the last the rigorous character of others—is the cele- 
brated Lecenpre. The conception of the method first given in _ 
the twelfth edition of his Geometry is very elegant ; as also its 
execution in all those steps which are rigorous, and by which it 
is in fact demonstrated, that if a triangle be given, there may be 
constructed another triangle having its three angles equal, in their 
sum, to the sum of the three of the original triangle, and two of 
its angles less than any assigned angle. Hence it was inferred 
that by the repeated bisection of one of the angles at the base, 
the whole triangle may be considered as coinciding with its base, 
and the exterior angle as less than any angle assigned. The 
‘weakness of the conclusion is found in this,—that, since the sides 
_ containing the continually bisected angle increase pers passu with 
he diminution of the angle, the apex of the triangle may, for 
t that appears, be of any assignable length, and may be com- 
it, therefore, to subtend an assignable angle. 
