Doctrine of Parallels. 93 
tained either by both or by neither of two such precisely similar 
spaces on opposite sides of the line. This is an assertion intelli- 
gibly true, without the requirement of any subtleties or even any 
conjectures as to the proper choice of sides in the alternative; but 
our argument, as will presently be seen, allows the objector to be 
dogmatical as to the one side or the other, or to stand in hesitancy 
between the two. But, if any one should deem it an additional 
satisfaction to conceive, with the utmost precision, what is inten- 
ded by the expression “a line belonging to or contained by an an- 
gular space,” I may, without involving the merits of the argument 
in any specific definition of which it and its fundamental princi- 
ple are independent, explain my individual conception of the in- 
terpretation, which, moreover, I suppose to coincide with that in- 
terpretation and idea that would spontaneously suggest itself to a 
mind imbued with even no more than the most elementary geo- 
metrical conceptions. Every geometrical magnitude is a definite 
extension. A sphere, for example, which is given in dimensions 
and fixed in position, occupies, throughout its entire extension, 
place or position,—that is to say, throughout the whole, poinis 
may be taken,—and those belong to or are contained by the sphere. 
The same is true of points of an angular space, and, by conse- 
quence, of a line of the same. The idea, like the idea of distance, 
is simple and plain, and, like that, capable of being referred to and 
recognized but scarcely of being made plainer by definition. 
If, however, any one should suppose, that a line which belongs 
to or is contained by an angular space must have a portion of that 
space on both sides of it, the reader can judge, after becoming 
possessed of the argument, whether, even in the sense of such a 
definition, he can deny my principle, as above stated, or subvert 
its conclusive application. Yet such a definition does, in eflect, 
deny that a magnitude occupies place up to its extreme bounda- 
ries. A better definition would be, that the contained line must 
be between the bounding lines of the angular space ; but this would 
compel the definer to adopt the negation of our alternative,—so as 
to begin, not like our argument, “ the line does or does not belong,” 
&c., but, “the line does not belong,” &c. 
One caution’ only is requisite to bespeak a due appreciation of 
the chain of proof I am about to offer. Lest the reader might, in 
the outset, enter upon it with a presumption that from such a ~ S 
truism as that above stated, no conclusion, and none, especially, 
