ed by study, the colder and thinner becomes the air and the
fewer the contacts with the affairs of every day. The Promethean
fire of pure mathematics is perhaps the greatest of all in man's
catalogue of gifts; but it is not most itself, but least so, when,
immersed in the manifoldness of phenomenal life, it is made to serve
purely utilitarian ends.

INTUITION

Common sense, immersed in the mere business of living, knows no more
about life than a fish knows about water. The play of reason upon
phenomena dissects life, and translates it in terms of inertia. The
pure logic of mathematics ignores life and disdains its limitations,
leading away into cold, free regions of its own. Now our desire for
freedom is not to vibrate in a vacuum, but to live more abundantly.
_Intuition_ deals with life directly, and introduces us into
life's own domain: it is related to reason as flame is related to
heat. All of the great discoveries in science, all of the great
solutions in mathematics, have been the result of a _flash_ of
intuition, after long brooding in the mind. _Intuition illumines_.
Intuition is therefore the light which must guide us into that
undiscovered country conceded by mathematics, questioned by science,
denied by common sense--_The Fourth Dimension of Space_.

OUR SENSE OF SPACE

Space has been defined as "room to move about." Let us accord to
this definition the utmost liberty of interpretation. Let us
conceive of space not alone as room to move ponderable bodies in,
but as room to think, to feel, to strike out in unimaginable
directions, to overtake felicities and knowledges unguessed by
experience and preposterous to common sense. Space is not measurable:
we attribute dimensionality to space because such is the method of
the mind; and that dimensionality we attribute to space is
progressive because progression is a law of the mind. The so-called
dimensions of space are to space itself as the steps that a climber
cuts in the face of a cliff are to the cliff itself. They are not
necessary to the cliff: they are necessary only to the climber.
Dimensionality is the mind's method of mounting to the idea of the
infinity of space. When we speak of the fourth dimension, what we
mean is the fourth stage in the apprehension of that infinity. We
might as legitimately speak of a fifth dimension, but the
profitlessness of any discussion of a fifth and higher stages lies
in the fact that they can be intelligently approa