sion in the Tuscan, the Doric, and the Ionic orders according to
Vignole, for in them all the stylobate is four parts, the entablature
3, and the intermediate column 12 (Illustration 74). The affinity
between 4 and 7, revealed in the fact that they express (very nearly)
the ratio between the base and the altitude of the right-angled
triangle which forms half of an equilateral, and the musical interval
of the diminished seventh, is architecturally suggested in the Palazzo
Giraud, which is four stories in height with seven openings in each
story (Illustration 75).

[Illustration 78]

[Illustration 79]

[Illustration 80]

[Illustration 81]

Every building is a symbol of some number or group of numbers, and
other things being equal the more perfect the numbers involved the
more beautiful will be the building (Illustrations 76-82). The numbers
5 and 7--those which occur oftenest--are the most satisfactory because
being of small quantity, they are easily grasped by the eye, and being
odd, they yield a center or axis, so necessary in every architectural
composition. Next in value are the lowest multiples of these numbers
and the least common multiples of any two of them, because the
eye, with a little assistance, is able to resolve them into their
constituent factors. It is part of the art of architecture to
render such assistance, for the eye counts always, consciously or
unconsciously, and when it is confronted with a number of units
greater than it can readily resolve, it is refreshed and rested if
these units are so grouped and arranged that they reveal themselves as
factors of some higher quantity.

[Illustration 82]

[Illustration 83: A NUMERICAL ANALYSIS OF GOTHIC TRACERY]

There is a raison d'etre for string courses other than to mark the
position of a floor on the interior of a building, and for quoins and
pilasters other than to indicate the presence of a transverse wall.
These sometimes serve the useful purpose of so subdividing a facade
that the eye estimates the number of its openings without conscious
effort and consequent fatigue (Illustration 82). The tracery of Gothic
windows forms perhaps the highest and finest architectural expression
of number (Illustration 83). Just as thirst makes water more sweet, so
does Gothic tracery confuse the eye with its complexity only the more
greatly to gratify the sight by revealing the inherent simplicity in
which this complexity has its root. Sometimes, as in the