Chapter 5

The figure has been variously analyzed with a view to establishing numerical ratios between its parts (Illustrations 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 Cæsar, formulated the important proportions of the statues of classical 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 (Illustrations 45, 47).

Such analyses 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 assist the architectural designer to arrive at beauty of proportion by methods of his own, not perhaps in the shape of rigid formulæ, 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

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).

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.

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).

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.

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."

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.

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.

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.

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.

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.

Number is the within of all things--the "first form of Brahman." It is the measure of time and space; it lurks in the heart-beat and is blazoned upon the starred canopy of night. Substance, in a state of vibration, in other words conditioned by number, ceaselessly undergoes the myriad transmutations which produce phenomenal life. Elements separate and combine chemically according to numerical ratios: "Moon, plant, gas, crystal, are concrete geometry and number." By the Pythagoreans and by the ancient Egyptians sex was attributed to numbers, odd numbers being conceived of as masculine or generating, and even numbers as feminine or parturitive, on account of their infinite divisibility. Harmonious combinations were those involving the marriage of a masculine and a feminine--an odd and an even--number.

Numbers progress from unity to infinity, and return again to unity as the soul, defined by Pythagoras as a self-moving number, goes forth from, and returns to God. These two acts, one of projection and the other of recall; these two forces, centrifugal and centripetal, are symbolized in the operations of addition and subtraction. Within them is embraced the whole of computation; but because every number, every aggregation of units, is also a new unit capable of being added or subtracted, there are also the operations of multiplication and division, which consists in one case of the addition of several equal numbers together, and in the other, of the subtraction of several equal numbers from a greater until that is exhausted. In order to think correctly it is necessary to consider the whole of numeration, computation, and all mathematical processes whatsoever as _the division of the unit_ into its component parts and the establishment of relations between these parts.

The progression and retrogression of numbers in groups expressed by the multiplication table gives rise to what may be termed "numerical conjunctions." These are analogous to astronomical conjunctions: the planets, revolving around the sun at different rates of speed, and in widely separated orbits, at certain times come into line with one another and with the sun. They are then said to be in conjunction. Similarly, numbers, advancing toward infinity singly and in groups (expressed by the multiplication table), at certain stages of their progression come into relation with one another. For example, an important conjunction occurs in 12, for of a series of twos it is the sixth, of threes the fourth, of fours the third, and of sixes the second. It stands to 8 in the ratio of 3:2, and to 9, of 4:3. It is related to 7 through being the product of 3 and 4, of which numbers 7 is the sum. The numbers 11 and 13 are not conjunctive; 14 is so in the series of twos, and sevens; 15 is so in the series of fives and threes. The next conjunction after 12, of 3 and 4 and their first multiples, is in 24, and the next following is in 36, which numbers are respectively the two and three of a series of twelves, each end being but a new beginning.

It will be seen that this discovery of numerical conjunctions consists merely of resolving numbers into their prime factors, and that a conjunctive number is a common multiple; but by naming it so, to dismiss the entire subject as known and exhausted, is to miss a sense of the wonder, beauty and rhythm of it all: a mental impression analogous to that made upon the eye by the swift-glancing balls of a juggler, the evolutions of drilling troops, or the intricate figures of a dance; for these things are number concrete and animate in time and space.

The truths of number are of all truths the most interior, abstract and difficult of apprehension, and since knowledge becomes clear and definite to the extent that it can be made to enter the mind through the channels of physical sense, it is well to accustom oneself to conceiving of number graphically, by means of geometrical symbols (Illustration 72), rather than in terms of the familiar arabic notation which though admirable for purposes of computation, is of too condensed and arbitrary a character to reveal the properties of individual numbers. To state, for example, that 4 is the first square, and 8 the first cube, conveys but a vague idea to most persons, but if 4 be represented as a square enclosing four smaller squares, and 8 as a cube containing eight smaller cubes, the idea is apprehended immediately and without effort. The number 3 is of course the triangle; the irregular and vital beauty of the number 5 appears clearly in the heptalpha, or five-pointed star; the faultless symmetry of 6, its relation to 3 and 2, and its regular division of the circle, are portrayed in the familiar hexagram known as the Shield of David. Seven, when represented as a compact group of circles reveals itself as a number of singular beauty and perfection, worthy of the important place accorded to it in all mystical philosophy (Illustration 73). It is a curious fact that when asked to think of any number less than 10, most persons will choose 7.

Every form of art, though primarily a vehicle for the expression and transmission of particular ideas and emotions, has subsidiary offices, just as a musical tone has harmonics which render it more sweet. Painting reveals the nature of color; music, of sound--in wood, in brass, and in stretched strings; architecture shows forth the qualities of light, and the strength and beauty of materials. All of the arts, and particularly music and architecture, portray in different manners and degrees the truths of number. Architecture does this in two ways: esoterically as it were in the form of harmonic proportions; and exoterically in the form of symbols which represent numbers and groups of numbers. The fact that a series of threes and a series of fours mutually conjoin in 12, finds an architectural expression in the Tuscan, the Doric, and the Ionic orders according to Vignole, for in them all the stylobate is four parts, the entablature 3, and the intermediate column 12 (Illustration 74). The affinity between 4 and 7, revealed in the fact that they express (very nearly) the ratio between the base and the altitude of the right-angled triangle which forms half of an equilateral, and the musical interval of the diminished seventh, is architecturally suggested in the Palazzo Giraud, which is four stories in height with seven openings in each story (Illustration 75).