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 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.