case of
the Venetian Ducal Palace, the numbers involved are too great for
counting, but other and different arithmetical truths are portrayed;
for example, the multiplication of the first arcade by 2 in
the second, and this by 3 in the cusped arches, and by 4 in the
quatrefoils immediately above.

[Illustration 84: NUMERATION IN GROUPS EXPRESSED ARCHITECTURALLY]

[Illustration 85: ARCHITECTURAL ORNAMENT CONSIDERED AS THE
OBJECTIFICATION OF NUMBER. MULTIPLICATION IN GROUPS OF FIVE; TWO;
THREE; ALTERNATION OF THREE AND SEVEN]

[Illustration 86]

Seven is proverbially the perfect number. It is of a quantity
sufficiently complex to stimulate the eye to resolve it, and yet so
simple that it can be analyzed at a glance; as a center with two equal
sides, it is possessed of symmetry, and as the sum of an odd and even
number (3 and 4) it has vitality and variety. All these properties a
work of architecture can variously reveal (Illustration 77). Fifteen,
also, is a number of great perfection. It is possible to arrange the
first 9 numbers in the form of a "magic" square so that the sum of
each line, read vertically, horizontally or diagonally, will be 15.
Thus:

4 9 2 = 15
3 5 7 = 15
8 1 6 = 15
-- -- --
15 15 15

Its beauty is portrayed geometrically in the accompanying figure which
expresses it, being 15 triangles in three groups of 5 (Illustration
86). Few arrangements of openings in a facade better satisfy the eye
than three superimposed groups of five (Illustrations 76-80). May not
one source of this satisfaction dwell in the intrinsic beauty of the
number 15?

In conclusion, it is perhaps well that the reader be again reminded
that these are the by-ways, and not the highways of architecture: that
the highest beauty comes always, not from beautiful numbers, nor
from likenesses to Nature's eternal patterns of the world, but from
utility, fitness, economy, and the perfect adaptation of means to
ends. But along with this truth there goes another: that in every
excellent work of architecture, in addition to its obvious and
individual beauty, there dwells an esoteric and universal beauty,
following as it does the archetypal pattern laid down by the Great
Architect for the building of that temple which is the world wherein
we dwell.

VII

FROZEN MUSIC

In the series of essays of which this is the final one, the author has
undertaken to enforce the