music; it is in the pulses of the blood no less than in the starred
curtain of the sky. It is a necessary concomitant alike of the sharp
bargain, the chemical experiment, and the fine frenzy of the poet.
Music is number made audible; architecture 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 æsthetic--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.