Doctrine of Parallels. 97 
not intersect at the same time. A threefold distinction must, 
therefore, actually subsist, namely, lines that intersect BC, lines 
that do not intersect it, and a peculiar line limiting the two, or di- 
viding between the spaces that can contain them ; and which, 
with reference to the line BC, may not improperly be said to 
touch it in C. 
The same may, in like manner, be proved respecting any recti- 
linear figures whatever. But if a figure be curvilinear, it may be 
circumscribed by a rectilinear figure whose boundary shall touch 
the curvilinear at any given point; or in other words, may coin-, 
cide with the curve at that point. Then the curvilinear figure at 
the point of coincidence, is separated, equally with the rectilinear, 
from the spaces without; by the right line which contains the one 
given point, which point, therefore, belongs to neither the space 
within nor the space without, but to the boundary. Therefore 
any point whatever of the curve does not belong to the curvili- 
near space ; which consequently is, as in the former case, a divid- 
ing line. : 
Lastly, if a solid be cut by a plane, there is a sectional figure 
which is constituted by all that is common to the plane and the. 
solid, and no more. But the boundary of the section has been. 
proved not to belong to its superficial extension and therefore does. 
not belong to the solid. But it is the section of the solid’s-bound- 
ary by the plane which constitutes the boundary of the figure, 
and therefore the boundary of the solid does not, at this line, be- 
long to the solid; and the same may be proved at any point what- 
ever in the surface of the solid. 
ScHotrum.—Besides the evidence which the principle and man- 
ner of reasoning employed above carry in themselves, there are 
two incidental symptoms or indexes of genuineness. One is that 
the resulting truth in relation to a parallel, is made to depend, as at 
ought to depend, upon a property of the infinite straight line in. 
distinction from the finite. Thus, in comparing the proposition 
and the corollary, we find that the line AC in the latter cannot 
- exist under the conditions of the former, because beyond every 
such line others may be drawn intersecting the indefinitely pro- 
~ duced line. 'The other is that the truth of the corollary is dedu- 
cible, at least in a partial ease, by a method quite independent of 
its own argument. For if, in fig. @ of the proposition, the angu- 
lar space DAC, instead of being defined as that which can con- 
Szcoxp Series, Vol. 1, No. 1.—Jan. 1846. 13 
