The cathedral builders excelled all others in the artfulness with which they established and maintained a relation between their architecture and the stature of a man. This is perhaps one reason why the French and English cathedrals, even those of moderate dimensions are more truly impressive than even the largest of the great Renaissance structures, such as St. Peter's in Rome. A gigantic order furnishes no true measure for the eye: its vastness is revealed only by the accident of some human presence which forms a basis of comparison. That architecture is not necessarily the most awe-inspiring which gives the impression of having been built by giants for the abode of pigmies; like the other arts, architecture is highest when it is most human. The mediaeval builders, true to this dictum, employed stones of a size proportionate to the strength of a man working without unusual mechanical aids; the great piers and columns, built up of many such stones, were commonly subdivided into cl.u.s.ters, and the circ.u.mference of each shaft of such a cl.u.s.ter approximated the girth of a man; by this device the moulding of the base and the foliation of the caps were easily kept in scale. Wherever a bal.u.s.trade occurred it was proportioned not with relation to the height of the wall or the column below, as in cla.s.sic architecture, but with relation to a man's stature.

[Ill.u.s.tration 48: FIGURE DIVIDED ACCORDING TO THE EGYPTIAN CANON]

It may be stated as a general rule that every work of architecture, of whatever style, should have somewhere about it something fixed and enduring to relate it to the human figure, if it be only a flight of steps in which each one is the measure of a stride. In the Farnese, the Riccardi, the Strozzi, and many another Italian palace, the stone seat about the base gives scale to the building because the beholder knows instinctively that the height of such a seat must have some relation to the length of a man's leg. In the Pitti palace the bal.u.s.trade which crowns each story answers a similar purpose: it stands in no intimate relation to the gigantic arches below, but is of a height convenient for lounging elbows. The door to Giotto's campanile reveals the true size of the tower as nothing else could, because it is so evidently related to the human figure and not to the great windows higher up in the shaft.

[Ill.u.s.tration 49: THE MEDIaeVAL METHOD OF DRAWING THE FIGURE]

The geometrical plane figures which play the most important part in architectural proportion are the square, the circle and the triangle; and the human figure is intimately related to these elementary forms.

If a man stand with heels together, and arms outstretched horizontally in opposite directions, he will be inscribed, as it were, within a square; and his arms will mark, with fair accuracy, the base of an inverted equilateral triangle, the apex of which will touch the ground at his feet. If the arms be extended upward at an angle, and the legs correspondingly separated, the extremities will touch the circ.u.mferences of a circle having its center in the navel (Ill.u.s.trations 45, 46).

[Ill.u.s.tration 50]

The figure has been variously a.n.a.lyzed with a view to establishing numerical ratios between its parts (Ill.u.s.trations 47, 48, 49). Some of these are so simple and easily remembered that they have obtained a certain popular currency; such as that the length of the hand equals the length of the face; that the span of the horizontally extended arms equals the height; and the well known rule that twice around the wrist is once around the neck, and twice around the neck is once around the waist. The Roman architect Vitruvius, writing in the age of Augustus Caesar, formulated the important proportions of the statues of cla.s.sical antiquity, and except that he makes the head smaller than the normal (as it should be in heroic statuary), the ratios which he gives are those to which the ideally perfect male figure should conform. Among the ancients the foot was probably the standard of all large measurements, being a more determinate length than that of the head or face, and the height was six lengths of the foot. If the head be taken as a unit, the ratio becomes 1:8, and if the face--1:10.

Doctor Rimmer, in his _Art Anatomy_, divides the figure into four parts, three of which are equal, and correspond to the lengths of the leg, the thigh and the trunk; while the fourth part, which is two-thirds of one of these thirds, extends from the sternum to the crown of the head. One excellence of such a division aside from its simplicity, consists in the fact that it may be applied to the face as well. The lowest of the three major divisions extends from the tip of the chin to the base of the nose, the next coincides with the height of the nose (its top being level with the eyebrows), and the last with the height of the forehead, while the remaining two-thirds of one of these thirds represents the horizontal projection from the beginning of the hair on the forehead to the crown of the head. The middle of the three larger divisions locates the ears, which are the same height as the nose (Ill.u.s.trations 45, 47).

Such a.n.a.lyses of the figure, however conducted, reveals an all-pervasive harmony of parts, between which definite numerical relations are traceable, and an apprehension of these should a.s.sist the architectural designer to arrive at beauty of proportion by methods of his own, not perhaps in the shape of rigid formulae, but present in the consciousness as a restraining influence, acting and reacting upon the mind with a conscious intention toward rhythm and harmony. By means of such exercises, he will approach nearer to an understanding of that great mystery, the beauty and significance of numbers, of which mystery music, architecture, and the human figure are equally presentments--considered, that is, from the standpoint of the occultist.

V

LATENT GEOMETRY

[Ill.u.s.tration 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 a.n.a.lysis, and thus augmenting beauty, which like a woman is alluring in proportion as she eludes (Ill.u.s.trations 51, 52, 53).

[Ill.u.s.tration 52: PROPORTIONS OF THE HORSE]

[Ill.u.s.tration 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.

[Ill.u.s.tration 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" (Ill.u.s.tration 54). Raphael, with that power of a.s.similation which distinguishes him among men of genius, learned from Fra Bartolommeo this method of disposing figures and combining them in ma.s.ses 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 (Ill.u.s.trations 55, 56).

[Ill.u.s.tration 55: THE EMPLOYMENT OF THE EQUILATERAL TRIANGLE IN RENAISSANCE PAINTING]

[Ill.u.s.tration 56: GEOMETRICAL BASIS OF THE SISTINE CEILING PAINTINGS]

[Ill.u.s.tration 57: a.s.sYRIAN; GREEK]

[Ill.u.s.tration 58: THE GEOMETRICAL BASIS OF THE PLAN IN ARCHITECTURAL DESIGN]

[Ill.u.s.tration 59]

Sculpture, even more than painting, is based on geometry. The colossi of Egypt, the bas-reliefs of a.s.syria, 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 (Ill.u.s.tration 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. (Ill.u.s.tration 58). This is the "bed rock"

of all excellent design.

[Ill.u.s.tration 60: EGYPTIAN; GREEK; ROMAN; MEDIaeVAL]

[Ill.u.s.tration 61: JEFFERSON'S PEN SKETCH FOR THE ROTUNDA OF THE UNIVERSITY OF VIRGINIA]

[Ill.u.s.tration 62: APPLICATION OF THE EQUILATERAL TRIANGLE TO THE ERECHTHEUM AT ATHENS]

[Ill.u.s.tration 63]

[Ill.u.s.tration 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 princ.i.p.al and subsidiary ma.s.ses 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 coordinating 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."

[Ill.u.s.tration 65: THE EQUILATERAL TRIANGLE IN ITALIAN ARCHITECTURE]

[Ill.u.s.tration 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 a.n.a.lysis that the claim for the existence of certain canons of proportion, based on geometry, remains unimpeached.

[Ill.u.s.tration 67]

[Ill.u.s.tration 68]

The plane figures princ.i.p.ally 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 (Ill.u.s.tration 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.

[Ill.u.s.tration 69]

[Ill.u.s.tration 70]

The square is princ.i.p.ally 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 (Ill.u.s.trations 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 (Ill.u.s.tration 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 a.s.sert 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 circ.u.mference 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 (Ill.u.s.tration 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 (Ill.u.s.trations 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 ill.u.s.tration 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 (Ill.u.s.tration 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.

[Ill.u.s.tration 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 a.n.a.lysis 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