Abhandlungen zur Geschichte der Mathematik.

On an allusion in Aristotle to a construction for parallels. 159
from the given straight line is negligible). But how are we to draw,
through the point, a straight line in the plane of the meridian? It is
practically equivalent (says PHILOPONUS, as I read him) to assuming another very distant point in the meridian plane and joining the given point
to it (the assumed point must be at a very great distance, because otherwise the distance of the given point from the given straight line would
not be negligible in comparison with its distance from the assumed point).
But again, how can we join a point to another point so distant that no
ruler will reach to it? The objector will not grant us the existence of
a point in the meridian plane which can be used to draw a straight line
to. He will, in fact, assert (and rightly) that we cannot really direct a
straight line to the assumed distant point except by drawing it, without
more ado, parallel to the given straight line. And herein is the petitio principii.
In modern mathematical language we may put the matter thus. Assuming that two straight lines whose intersection is at an infinite distance
are parallel, we are to imagine an infinitely distant point on the given
straight line, and we are to draw another straight line from the given
point to the infinitely distant point. We cannot in practice do this, and
the infinitely distant point is of no use to us; our only method is to
draw a parallel to begin with, in order, as it were, to locate the infinitely distant point. The objector will rightly say that the infinitely
distant point cannot be admitted at all except as the very point in which
a parallel will intersect the given straight line; and the petitio principii is obvious.
If the method of drawing parallels condemned by ARISTOTLE was
substantially that above described, the idea underlying it would be curiously similar to that which suggested to the editors of certain English
text-books of elementary geometry (e. g. J. M. WILSON) the direction
theory of parallels. According to this theory different straight lines may
have either the same or different directions, and parallels are then defined
as straight lines which are not parts of the same straight line but have
the same direction. But these editors give us no definition or notion
of direction except with reference to straight lines which meet, and then
they straightway proceed to use the term with reference to straight lines
which do not meet, though they can attach no geometrical meaning to
the same direction as applied to the latter class of lines. The logical
fallacy could not be better exposed than it is by C. L. DODGSON in
EUCLID and his modern rivals, the fact being that the whole idea of
the same direction as applied to non-coincident straight lines is derived