70 Reply to the criticism on Demonstration of Parallels. 
in determining the angle BAD to be such as to contain all of a 
certain class of lines, every other line is excluded from it; in 
other words, that the line AD, which is the limit of a certain 
class of lines, must itself be comprehended in that class.” Now, 
as to the assertion in the first member of this sentence, the reader 
will perceive by referring to p. 95, that what the critic has treated 
as an assumption, is, in fact, explicitly a condition of the hypothe- 
sis. For not only the application itself, in the argument, evinces 
what was meant by the concise definition of the angle BAD, as 
being “ constituted by the condition that it can contain all the 
lines drawn, &c.,”’ but the terms themselves intend that “every 
other line is excluded,’’ as completely as if that phrase had been 
adopted in the wording ;—how else could the angle be said to 
be constituted, or what is the same, defined.* If then the line 
AD is one of the class of which the angle BAD is defined just 
to contain all and “no other,” (if that last part of the expression 
is important to appear in the wording, ) it is contained by the an- 
gle, in accordance with the hypothesis, and on the reverse suppo- 
sition it is, by the hypothesis, excluded. 
I perceive, therefore, in the step objected to, no defect even of 
expression. The discerning reader cannot fail, at least, to take 
the argument, in its real force and meaning, to be as follows :— 
If you constitute BAD capable of containing all those lines and 
those only through A, which, if produced, would meet FG on 
the side towards G, you cannot constitute DAC similarly condi- 
tioned with respect to all those which would not meet. The 
ground of this assertion lies in the antecedent proof that under such 
a supposition, AD must be contained by both BAD and DAC or 
by neither, and, under the hypothesis, either side of the dilemma 
involves the absurdity that AD meets and does not meet at the 
same time. I know not what attempt can be made to escape 
from the conclusiveness of this reasoning, unless by either deny- 
ing the allowable character of the hypothesis itself, or taking the 
ground that when a line divides an angle, the being contained 
either part or not contained, cannot be predicated of the line. 
The first, or a denial of the hypothesis, is forbidden by the per- 
fection of the idea of space: the latter ground, if taken, would 
* Thus, to refer, for oe nat merely, to an instance in common parlance— 
measure were said to be constituted so as to -asheeien 231 cubic inches, 
pg toccee be just that amount and no 
