continuum of two dimensions, and space an infinite continuum of
three dimensions. What is here meant by the number of dimensions,
I think I may assume to be known.
Now we take an example of a two-dimensional continuum which is
finite, but unbounded. We imagine the surface of a large globe and
a quantity of small paper discs, all of the same size. We place
one of the discs anywhere on the surface of the globe. If we move
the disc about, anywhere we like, on the surface of the globe,
we do not come upon a limit or boundary anywhere on the journey.
Therefore we say that the spherical surface of the globe is an
unbounded continuum. Moreover, the spherical surface is a finite
continuum. For if we stick the paper discs on the globe, so that
no disc overlaps another, the surface of the globe will finally
become so full that there is no room for another disc. This simply
means that the spherical surface of the globe is finite in relation
to the paper discs. Further, the spherical surface is a non-Euclidean
continuum of two dimensions, that is to say, the laws of disposition
for the rigid figures lying in it do not agree with those of the
Euclidean plane. This can be shown in the following way. Place
a paper disc on the spherical surface, and around it in a circle
place six more discs, each of which is to be surrounded in turn
by six discs, and so on. If this construction is made on a plane
surface, we have an uninterrupted disposition in which there are
six discs touching every disc except those which lie on the outside.
[Figure 1: Discs maximally packed on a plane]
On the spherical surface the construction also seems to promise
success at the outset, and the smaller the radius of the disc
in proportion to that of the sphere, the more promising it seems.
But as the construction progresses it becomes more and more patent
that the disposition of the discs in the manner indicated, without
interruption, is not possible, as it should be possible by Euclidean
geometry of the the plane surface. In this way creatures which
cannot leave the spherical surface, and cannot even peep out from
the spherical surface into three-dimensional space, might discover,
merely by experimenting with discs, that their two-dimensional
"space" is not Euclidean, but spherical space.
From the latest results of the theory of relativity it is probable
that our three-dimensional space is also approximately spherical,
that is, that the laws of disposition of rigid bodies in it are
not given by Euclidean geometry, but approximately by spherical
geometry, if only we consider parts of space which are sufficiently
great. Now this is the place where the reader's imagination boggles.
"Nobody can imagine this thing," he cries indignantly. "It can be
said, but cannot be thought. I can represent to myself a spherical
surface well enough, but nothing analogous to it in three dimensions."
[Figure 2: A circle projected from a sphere onto a plane]
We must try to surmount this barrier in the mind, and the patient
reader will see that it is by no means a particularly difficult
task. For this purpose we will first give our attention once more to
the geometry of two-dimensional spherical surfaces. In the adjoining
figure let _K_ be the spherical surface, touched at _S_ by a plane,
_E_, which, for facility of presentation, is shown in the drawing as
a bounded surface. Let _L_ be a disc on the spherical surface. Now