direction. Projective geometry is able to state that a point moving
along this line in one direction will eventually return from the other.
To see this, we imagine two straight lines a and b intersecting at P.
One of these lines is fixed (a); the other (b) rotates uniformly about
C. Fig. 7 indicates the rotation of b by showing it in a number of
positions with the respective positions of its point of intersection
with a (P1, P2. . .). We observe this point moving along a, as a result
of the rotation of b, until, when both lines are parallel, it reaches
infinity. As a result of the continued rotation of b, however, P does
not remain in infinity, but returns along a from the other side. We
find here two forms of movement linked together - the rotational
movement of a line (b) on a point (C), and the progressive movement of
a point (P) along a line (a). The first movement is continuous, and
observable throughout within finite space. Therefore the second
movement must be continuous as well, even though it partly escapes our
observation. Hence, when P disappears into infinity on one side of our
own point of observation, it is at the same time in infinity on the
other side. In order words, an unlimited straight line has only one
point at infinity.

It is clear that, in order to become familiar with this aspect of
geometry, one must grow together in inward activity with the happening
which is contained in the above description. What we therefore intend
by giving such a description is to provide an opportunity for a
particular mental exercise, just as when we introduced Goethe's botany
by describing a number of successive leaf-formations. Here, as much as
there, it is the act of 're-creating' that matters.

The following exercise will help us towards further clarity concerning
the nature of geometrical infinity.