We imagine ourselves in the centre of a sphere which we allow to expand
uniformly on all sides. Whilst the inner wall of this sphere withdraws
from us into ever greater distances, it grows flatter and flatter
until, on reaching infinite distance, it turns into a plane. We thus
find ourselves surrounded everywhere by a surface which, in the strict
mathematical sense, is a plane, and is yet one and the same surface on
all sides. This leads us to the conception of the plane at infinity as
a self-contained entity although it expands infinitely in all
directions.

This property of a plane at infinity, however, is really a property of
any plane. To realize this, we must widen our conception of infinity by
freeing it from a certain one-sidedness still connected with it. This
we do by transferring ourselves into the infinite plane and envisaging,
not the plane from the point, but the point from the plane. This
operation, however, implies something which is not obvious to a mind
accustomed to the ordinary ways of mathematical reasoning. It therefore
requires special explanation.

In the sense of Euclidean geometry, a plane is the sum-total of
innumerable single points. To take up a position in a plane, therefore,
means to imagine oneself at one point of the plane, with the latter
extending around in all directions to infinity. Hence the journey from
any point in space to a plane is along a straight line from one point
to another. In the case of the plane being at infinity, it would be a
journey along a radius of the infinitely large sphere from its centre
to a point at its circumference.

In projective geometry the operation is of a different character. Just