[Illustration 69]

[Illustration 70]

The square is principally used in preliminary plotting:  it is the determining figure in many of the palaces of the Italian Renaissance; the Arc de Triomphe, in Paris is a modern example of its use (Illustrations 59, 60).  The circle is often employed in conjunction with the square and the triangle.  In Thomas Jefferson’s Rotunda for the University of Virginia, a single great circle was the determining figure, as his original pen sketch of the building shows (Illustration 61).  Some of the best Roman triumphal arches submit themselves to a circular synopsis, and a system of double intersecting circles has been applied, with interesting results, to facades as widely different as those of the Parthenon and the Farnese Palace in Rome, though it would be fatuous to claim that these figures determined the proportions of the facades.

By far the most important figure in architectural proportion, considered from the standpoint of geometry, is the equilateral triangle.  It would seem that the eye has an especial fondness for this figure, just as the ear has for certain related sounds.  Indeed it might not be too fanciful to assert that the common chord of any key (the tonic with its third and fifth) is the musical equivalent of the equilateral triangle.  It is scarcely necessary to dwell upon the properties and unique perfection of this figure.  Of all regular polygons it is the simplest:  its three equal sides subtend equal angles, each of 60 degrees; it trisects the circumference of a circle; it is the graphic symbol of the number three, and hence of every threefold thing; doubled, its generating arcs form the vesica piscis, of so frequent occurrence in early Christian art; two symmetrically intersecting equilateral triangles yield the figure known as “Solomon’s Seal,” or the “Shield of David,” to which mystic properties have always been ascribed.

It may be stated as a general rule that whenever three important points in any architectural composition coincide (approximately or exactly) with the three extremities of an equilateral triangle, it makes for beauty of proportion.  An ancient and notable example 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