ched only through
the fourth, which is still largely unintelligible. The case is like
that of a man promised an increase of wages after he had worked a
month, who asks for his second month's pay before he is entitled to
the first.

THE SUBJECTIVITY OF SPACE

Without going deep into the doctrine of the ideality--that is, the
purely subjective reality--of space, it is easy to show that we have
arrived at our conception of a space of three dimensions by an
intellectual process. The sphere of the senses is two-dimensional:
except for the slight aid afforded by binocular vision, sight gives
us moving pictures _on a plane_, and touch contacts _surfaces_ only.
What circumstances, we may ask, have compelled our intellect to
conceive of _solid_ space? This question has been answered as follows:

"If a child contemplates his hand, he is conscious of its existence
in a double manner--in the first place by its tangibility, the
second by its image on the retina of his eye. By repeated groping
about and touching, the child knows by experience that his hand
retains the same form and extension through all the variations of
distance and position under which it is observed, notwithstanding
that the form and extension of the image on the retina constantly
change with the different position and distance of his hand in
respect to his eye. The problem is thus set to the child's
understanding: how to reconcile to his comprehension the apparently
contradictory facts of the _invariableness_ of the object together
with the _variableness_ of its appearance. This is only possible
within a space of three dimensions, in which, owing to perspective
distortions and changes, these variations of projection can be
reconciled with the constancy of the form of a body."

Thus we have come to the idea of a three-dimensional space in order
to overcome the apparent contradictoriness of facts of sensible
experience. Should we observe in three-dimensional space
contradictory facts our reason would be forced to reconcile these
contradictions, also, and if they could be reconciled by the idea of
a four-dimensional space our reason would accept this idea without
cavil. Furthermore, if from our childhood, phenomena had been of
daily occurrence requiring a space of four or more dimensions for an
explanation conformable to reason, we should feel ourselves native
to a space of four or more dimensions.

Poincare, the great French mathematician and physicist, ar