92 Doctrine of Parallels. 
exist,—and for this reason, probably, it may be that LecenpRE 
himself, in his ultimate and general memoir on the subject of par- 
allels contained in Vol. xix, of the “Mémoires de L’ Académie 
Royale,” has made no account of that method. Had the argu- 
ment, however, turned upon the comparison of the contained an- 
gular space with the exterior space in the way of ratio, instead of 
absolute excess or defect, (as it will be obvious to all familiar with 
that argument that it might have been made to do by continually 
bisecting the interior angle until a part should be found less than 
the excess of the whole exterior over the interior, ) the proof would 
have rested unapproachable by the objection named. ‘The only 
possible doubt would then have been whether the space on the 
side of the contained line opposite to the angular point is certainly 
a part of the interior angular space alone ; yet that it must be such 
is, if not perfectly axiomatic, at least so nearly axiomatic as to 
give to this simple method, so modified, in my own apprehension, 
at least, a superiority over all others, including those of LecenpRE 
himself and the suspicious although certainly simple and specious 
method of Brerrranp. 
But, not to prolong discussions and comparisons beyond the de- 
mands of my immediate object, I pass to the development of an 
attempt of my own by which I propose to complete the doctrine 
of parallels and to make the postulatum of Evcurp independent, in 
fact, (in the simple case in which one of the interior angles m 
with the third or cutting line is a right angle) of any antecedent 
proposition. 
To this end I employ a particular relation—whether positively 
or negatively assumed in hypothesis—of lines containing an angle 
to the angular space; which relation, or possible relation, if that 
qualification should be insisted on, appears not to have been here- 
tofore reflected upon, or even noticed for any purpose of demon- 
stration or investigation,—and which constitutes, therefore, if I 
err not in my application of it, a novel element of geometrical rea- 
soning ; notwithstanding that I am not able to determine what 
other than the special applications I shall make, it may be ex- 
pected to be susceptible of. 'The element or relation referred to 
___ is nothing more than the truism that, if two straight lines meet, 
either of them belongs or does not belong to—or (if that phrase- 
~ ology be preferable ) is contained or is not contained by—the angu- 
bounded by the lines, and consequently, would be con- 
