an interesting study, and a useful one to the
artist; but it can never take the place of the creative faculty, it
can only supplement, restrain, direct it. The study of proportion is
to the architect what the study of harmony is to a musician--it helps
his genius adequately to express itself.

VI

THE ARITHMETIC OF BEAUTY

Although architecture is based primarily upon geometry, it is possible
to express all spatial relations numerically: for arithmetic, not
geometry, is the universal science of quantity. The relation of masses
one to another--of voids to solids, and of heights and lengths to
widths--forms ratios; and when such ratios are simple and harmonious,
architecture may be said, in Walter Pater's famous phrase, to
"aspire towards the condition of music." The trained eye, and not an
arithmetical formula, determines what is, and what is not, beautiful
proportion. Nevertheless the fact that the eye instinctively rejects
certain proportions as unpleasing, and accepts others as satisfactory,
is an indication of the existence of laws of space, based upon number,
not unlike those which govern musical harmony. The secret of the deep
reasonableness of such selection by the senses lies hidden in the very
nature of number itself, for number is the invisible thread on which
the worlds are strung--the universe abstractly symbolized.

Number is the within of all things--the "first form of Brahman." It
is the measure of time and space; it lurks in the heart-beat and is
blazoned upon the starred canopy of night. Substance, in a state of
vibration, in other words conditioned by number, ceaselessly undergoes
the myriad transmutations which produce phenomenal life. Elements
separate and combine chemically according to numerical ratios:
"Moon, plant, gas, crystal, are concrete geometry and number." By
the Pythagoreans and by the ancient Egyptians sex was attributed to
numbers, odd numbers being conceived of as masculine or generating,
and even numbers as feminine or parturitive, on account of their
infinite divisibility. Harmonious combinations were those
involving the marriage of a masculine and a feminine--an odd and an
even--number.

[Illustration 72: A GRAPHIC SYSTEM OF NOTATION]

Numbers progress from unity to infinity, and return again to unity as
the soul, defined by Pythagoras as a self-moving number, goes forth
from, and returns to God. These two acts, one of projection and the
other of recall; these two fo