occurs
in the pyramids of Egypt, the sides of which, in their original
condition, are believed to have been equilateral triangles. It is a
demonstrable fact that certain geometrical intersections yield the
important proportions of Greek architecture. The perfect little
Erechtheum would seem to have been proportioned by means of the
equilateral triangle and the angle of 60 degrees, both in general and
in detail (Illustration 62). The same angle, erected from the central
axis of a column at the point where it intersects the architrave,
determines both the projection of the cornice and the height of
the architrave in many of the finest Greek and Roman temples
(Illustrations 67-70). The equilateral triangle used in conjunction
with the circle and the square was employed by the Romans in
determining the proportions of triumphal arches, basilicas and
baths. That the same figure was a factor in the designing of Gothic
cathedrals is sufficiently indicated in the accompanying facsimile
reproductions of an illustration from the Como Vitruvius, published in
Milan in 1521, which shows a vertical section of the Milan cathedral
and the system of equilateral triangles which determined its various
parts (Illustration 71). The _vesica piscis_ was often used to
establish the two main internal dimensions of the cathedral plan: the
greatest diameter of the figure corresponding with the width across
the transepts, the upper apex marking the limit of the apse, and the
lower, the termination of the nave. Such a proportion is seen to be
both subtle and simple, and possesses the advantage of being easily
laid out. The architects of the Italian Renaissance doubtless
inherited certain of the Roman canons of architectural proportion,
for they seem very generally to have recognized them as an essential
principle of design.

[Illustration 71]

Nevertheless, when all is said, it is easy to exaggerate the
importance of this matter of geometrical proportion. The designer
who seeks the ultimate secret of architectural harmony in mathematics
rather than in the trained eye, is following the wrong road to
success. A happy inspiration is worth all the formulae in the world--if
it be really happy, the artist will probably find that he has
"followed the rules without knowing them." Even while formulating
concepts of art, the author must reiterate Schopenhauer's dictum
that the _concept_ is unfruitful in art. The mathematical analysis
of spatial beauty is