is number made visible;
nature geometrizes not alone in her crystals, but in her most
intricate arabesques.

If number be indeed the universal solvent of all forms, sounds,
motions, may we not make of it the basis of a new aesthetic--a loom on
which to weave patterns the like of which the world has never seen? To
attempt such a thing--to base art on mathematics--argues (some one
is sure to say) an entire misconception of the nature and function of
art. "Art is a fountain of spontaneous emotion"--what, therefore,
can it have in common with the proverbially driest, least spontaneous
preoccupation of the human mind? But the above definition concludes
with the assertion that this emotion reaches the soul "through various
channels." The transit can be effected only through some sensuous
element, some language (in the largest sense), and into this the
element of number and form must inevitably enter--mathematics is
"there" and cannot be thought or argued away.

[Illustration: PLATE XI. IMAGINARY COMPOSITION: THE PORTAL]

But to make mathematics, and not the emotion which it expresses, the
important thing, is not this to fall into the time-worn heresy of
art for art's sake, that is, art for form's sake--art for the sake of
mathematics? To this objection there is an answer, and as this answer
contains the crux of the whole matter, embraces the proposition by
which this thesis must stand or fall, it must be full and clear.

What is it, in the last analysis, that all art which is not
purely personal and episodical strives to express? Is it not the
_world-order_?--the very thing that religion, philosophy, science,
strive according to their different natures and methods to express?
The perception of the world-order by the artist arouses an emotion to
which he can give vent only in terms of number; but number is itself
the most abstract expression of the world order. The form and content
of art are therefore not different, but the same. A deep sense of this
probably inspired Pater's famous saying that all art aspires toward
the condition of music; for music, from its very nature, is the
world-order uttered in terms of number, in a sense and to a degree not
attained by any other art.

This is not mere verbal juggling. We have suffered so long from an
art-phase which exalts the personal, as opposed to the cosmic, that
we have lost sight of the fact that the great arts of antiquity,
preceding the Renaissance, insisted on the cos