V

LATENT GEOMETRY

[Illustration 51: THE HEXAGRAM AND EQUILATERAL TRIANGLE IN NATURE]

It is a well known fact that in the microscopically minute of nature, units everywhere tend to arrange themselves with relation to certain simple geometrical solids, among which are the tetrahedron, the cube, and the sphere. This process gives rise to harmony, which may be defined as the relation between parts and unity, the simplicity latent in the infinitely complex, the potential complexity of that which is simple. Proceeding to things visible and tangible, this indwelling harmony, rhythm, proportion, which has its basis in geometry and number, is seen to exist in crystals, flower forms, leaf groups, and the like, where it is obvious; and in the more highly organized world of the animal kingdom also; though here the geometry is latent rather than patent, eluding though not quite defying analysis, and thus augmenting beauty, which like a woman is alluring in proportion as she eludes (Illustrations 51, 52, 53).

[Illustration 52: PROPORTIONS OF THE HORSE]

[Illustration 53]

By the true artist, in the crystal mirror of whose mind the universal harmony is focused and reflected, this secret of the cause and source of rhythm—that it dwells in a correlation of parts based on an ultimate simplicity—is instinctively apprehended. A knowledge of it formed part of the equipment of the painters who made glorious the golden noon of pictorial art in Italy during the Renaissance. The problem which preoccupied them was, as Symonds says of Leonardo, "to submit the freest play of form to simple figures of geometry in grouping." Alberti held that the painter should above all things have mastered geometry, and it is known that the study of perspective and kindred subjects was widespread and popular.

[Illustration 54]

The first painter who deliberately rather than instinctively based his compositions on geometrical principles seems to have been Fra Bartolommeo, in his Last Judgment, in the church of St. Maria Nuova, in Florence. Symonds says of this picture, "Simple figures—the pyramid and triangle, upright, inverted, and interwoven like the rhymes of a sonnet—form the basis of the composition. This system was adhered to by the Fratre in all his subsequent works" (Illustration 54). Raphael, with that power of assimilation which distinguishes him among men of genius, learned from Fra Bartolommeo this method of disposing figures and combining them in masses with almost mathematical precision. It would have been indeed surprising if Leonardo da Vinci, in whom the artist and the man of science were so wonderfully united, had not been greatly preoccupied with the mathematics of the art of painting. His Madonna of the Rocks, and Virgin on the Lap of Saint Anne, in the Louvre, exhibit the very perfection of pyramidal composition. It is however in his masterpiece, The Last Supper, that he combines geometrical symmetry and precision with perfect naturalness and freedom in the grouping of individually interesting and dramatic figures. Michael Angelo, Andrea del Sarto, and the great Venetians, in whose work the art of painting may be said to have culminated, recognized and obeyed those mathematical laws of composition known to their immediate predecessors, and the decadence of the art in the ensuing period may be traced not alone to the false sentiment and affectation of the times, but also in the abandonment by the artists of those obscurely geometrical arrangements and groupings which in the works of the greatest masters so satisfy the eye and haunt the memory of the beholder (Illustrations 55, 56).

[Illustration 55: THE EMPLOYMENT OF THE EQUILATERAL TRIANGLE IN
RENAISSANCE PAINTING]

[Illustration 56: GEOMETRICAL BASIS OF THE SISTINE CEILING PAINTINGS]

[Illustration 57: ASSYRIAN; GREEK]

[Illustration 58: THE GEOMETRICAL BASIS OF THE PLAN IN ARCHITECTURAL
DESIGN]

[Illustration 59]

Sculpture, even more than painting, is based on geometry. The colossi of Egypt, the bas-reliefs of Assyria, the figured pediments and metopes of the temples of Greece, the carved tombs of Revenna, the Della Robbia lunettes, the sculptured tympani of Gothic church portals, all alike lend themselves in greater or less degree to a geometrical synopsis (Illustration 57). Whenever sculpture suffered divorce from architecture the geometrical element became less prominent, doubtless because of all the arts architecture is the most clearly and closely related to geometry. Indeed, it may be said that architecture is geometry made visible, in the same sense that music is number made audible. A building is an aggregation of the commonest geometrical forms: parallelograms, prisms, pyramids and cones—the cylinder appearing in the column, and the hemisphere in the dome. The plans likewise of the world's famous buildings reduced to their simplest expression are discovered to resolve themselves into a few simple geometrical figures. (Illustration 58). This is the "bed rock" of all excellent design.

[Illustration 60: EGYPTIAN; GREEK; ROMAN; MEDIÆVAL]

[Illustration 61: JEFFERSON'S PEN SKETCH FOR THE ROTUNDA OF THE
UNIVERSITY OF VIRGINIA]

[Illustration 62: APPLICATION OF THE EQUILATERAL TRIANGLE TO THE
ERECHTHEUM AT ATHENS]

[Illustration 63]

[Illustration 64: THE EQUILATERAL TRIANGLE IN ROMAN ARCHITECTURE]

But architecture is geometrical in another and a higher sense than this. Emerson says: "The pleasure a palace or a temple gives the eye is that an order and a method has been communicated to stones, so that they speak and geometrize, become tender or sublime with expression." All truly great and beautiful works of architecture from the Egyptian pyramids to the cathedrals of Ile-de-France—are harmoniously proportioned, their principal and subsidiary masses being related, sometimes obviously, more often obscurely, to certain symmetrical figures of geometry, which though invisible to the sight and not consciously present in the mind of the beholder, yet perform the important function of coördinating the entire fabric into one easily remembered whole. Upon some such principle is surely founded what Symonds calls "that severe and lofty art of composition which seeks the highest beauty of design in architectural harmony supreme, above the melodies of gracefulness of detail."

[Illustration 65: THE EQUILATERAL TRIANGLE IN ITALIAN ARCHITECTURE]

[Illustration 66: THE HEXAGRAM IN GOTHIC ARCHITECTURE]

There is abundant evidence in support of the theory that the builders of antiquity, the masonic guilds of the Middle Ages, and the architects of the Italian Renaissance, knew and followed certain rules, but though this theory be denied or even disproved—if after all these men obtained their results unconsciously—their creations so lend themselves to a geometrical analysis that the claim for the existence of certain canons of proportion, based on geometry, remains unimpeached.

[Illustration 67]

[Illustration 68]

The plane figures principally employed in determining architectural proportion are the circle, the equilateral triangle, and the square—which also yields the right angled isosceles triangle. It will be noted that these are the two dimensional correlatives of the sphere, the tetrahedron and the cube, mentioned as being among the determining forms in molecular structure. The question naturally arises, why the circle, the equilateral triangle and the square? Because, aside from the fact that they are of all plane figures the most elementary, they are intimately related to the body of man, as has been shown (Illustration 45), and the body of man is as it were the architectural archetype. But this simply removes the inquiry to a different field, it is not an answer. Why is the body of man so constructed and related? This leads us, as does every question, to the threshold of a mystery upon which theosophy alone is able to throw light. Any extended elucidation would be out of place here: it is sufficient to remind the reader that the circle is the symbol of the universe; the equilateral triangle, of the higher trinity (atma, buddhi, manas); and the square, of the lower quaternary of man's sevenfold nature.

[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 façades 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 façades.

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 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 formulæ 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.