
At certain periods of the world's history, periods of mystical enlightenment, men have been wont to use the human figure, the soul's temple, as a sort of archetype for sacred edifices (Illustration 41). The colossi, with calm inscrutable faces, which flank the entrance to Egyptian temples; the great bronze Buddha of Japan, with its dreaming eyes; the little known colossal figures of India and China—all these belong scarcely less to the domain of architecture than of sculpture. The relation above referred to however is a matter more subtle and occult than mere obvious imitation on a large scale, being based upon some correspondence of parts, or similarity of proportions, or both. The correspondence between the innermost sanctuary or shrine of a temple and the heart of a man, and between the gates of that temple and the organs of sense is sufficiently obvious, and a relation once established, the idea is susceptible of almost infinite development. That the ancients proportioned their temples from the human figure is no new idea, nor is it at all surprising. The sculpture of the Egyptians and the Greeks reveals the fact that they studied the body abstractly, in its exterior presentment. It is clear that the rules of its proportions must have been established for sculpture, and it is not unreasonable to suppose that they became canonical in architecture also. Vitruvius and Alberti both lay stress on the fact that all sacred buildings should be founded on the proportions of the human body.
In France, during the Middle Ages, a Gothic cathedral became, at the hands of the secret masonic guilds, a glorified symbol of the body of Christ. To practicalminded students of architectural history, familiar with the slow and halting evolution of a Gothic cathedral from a Roman basilica, such an idea may seem to be only the maunderings of a mystical imagination, a theory evolved from the inner consciousness, entitled to no more consideration than the familiar fallacy that vaulted nave of a Gothic church was an attempt to imitate the green aisles of a forest. It should be remembered however that the habit of the thought of that time was mystical, as that of our own age is utilitarian and scientific; and the chosen language of mysticism is always an elaborate and involved symbolism. What could be more natural than that a building devoted to the worship of a crucified Savior should be made a symbol, not of the cross only, but of the body crucified?
The vesica piscis (a figure formed by the developing arcs of two equilateral triangles having a common side) which in so many cases seems to have determined the main proportion of a cathedral plan—the interior length and width across the transepts—appears as an aureole around the figure of Christ in early representations, a fact which certainly points to a relation between the two (Illustrations 42, 43). A curious little book, The Rosicrucians, by Hargrave Jennings, contains an interesting diagram which well illustrates this conception of the symbolism of a cathedral. A copy of it is here given. The apse is seen to correspond to the head of Christ, the north transept to his right hand, the south transept to the left hand, the nave to the body, and the north and south towers to the right and left feet respectively (Illustration 44).
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 aweinspiring 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 clusters, and the circumference of each shaft of such a cluster 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 balustrade occurred it was proportioned not with relation to the height of the wall or the column below, as in classic architecture, but with relation to a man's stature.
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 balustrade 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.
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 circumferences of a circle having its center in the navel (Illustrations 45, 46).
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 Caesar, 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 twothirds 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 twothirds 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 allpervasive 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 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
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 basreliefs 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 IledeFrance—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 cooerdinating 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 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 6770). 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 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.
Number is the within of all things—the "first form of Brahman." It is the measure of time and space; it lurks in the heartbeat 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 selfmoving 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 swiftglancing 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 fivepointed 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 rightangled 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).
Every building is a symbol of some number or group of numbers, and other things being equal the more perfect the numbers involved the more beautiful will be the building (Illustrations 7682). The numbers 5 and 7—those which occur oftenest—are the most satisfactory because being of small quantity, they are easily grasped by the eye, and being odd, they yield a center or axis, so necessary in every architectural composition. Next in value are the lowest multiples of these numbers and the least common multiples of any two of them, because the eye, with a little assistance, is able to resolve them into their constituent factors. It is part of the art of architecture to render such assistance, for the eye counts always, consciously or unconsciously, and when it is confronted with a number of units greater than it can readily resolve, it is refreshed and rested if these units are so grouped and arranged that they reveal themselves as factors of some higher quantity.
There is a raison d'etre for string courses other than to mark the position of a floor on the interior of a building, and for quoins and pilasters other than to indicate the presence of a transverse wall. These sometimes serve the useful purpose of so subdividing a facade that the eye estimates the number of its openings without conscious effort and consequent fatigue (Illustration 82). The tracery of Gothic windows forms perhaps the highest and finest architectural expression of number (Illustration 83). Just as thirst makes water more sweet, so does Gothic tracery confuse the eye with its complexity only the more greatly to gratify the sight by revealing the inherent simplicity in which this complexity has its root. Sometimes, as in the case of the Venetian Ducal Palace, the numbers involved are too great for counting, but other and different arithmetical truths are portrayed; for example, the multiplication of the first arcade by 2 in the second, and this by 3 in the cusped arches, and by 4 in the quatrefoils immediately above.
Seven is proverbially the perfect number. It is of a quantity sufficiently complex to stimulate the eye to resolve it, and yet so simple that it can be analyzed at a glance; as a center with two equal sides, it is possessed of symmetry, and as the sum of an odd and even number (3 and 4) it has vitality and variety. All these properties a work of architecture can variously reveal (Illustration 77). Fifteen, also, is a number of great perfection. It is possible to arrange the first 9 numbers in the form of a "magic" square so that the sum of each line, read vertically, horizontally or diagonally, will be 15. Thus:
4 9 2 = 15 3 5 7 = 15 8 1 6 = 15 — — — 15 15 15
Its beauty is portrayed geometrically in the accompanying figure which expresses it, being 15 triangles in three groups of 5 (Illustration 86). Few arrangements of openings in a facade better satisfy the eye than three superimposed groups of five (Illustrations 7680). May not one source of this satisfaction dwell in the intrinsic beauty of the number 15?
In conclusion, it is perhaps well that the reader be again reminded that these are the byways, and not the highways of architecture: that the highest beauty comes always, not from beautiful numbers, nor from likenesses to Nature's eternal patterns of the world, but from utility, fitness, economy, and the perfect adaptation of means to ends. But along with this truth there goes another: that in every excellent work of architecture, in addition to its obvious and individual beauty, there dwells an esoteric and universal beauty, following as it does the archetypal pattern laid down by the Great Architect for the building of that temple which is the world wherein we dwell.
VII
FROZEN MUSIC
In the series of essays of which this is the final one, the author has undertaken to enforce the truth that evolution on any plane and on any scale proceeds according to certain laws which are in reality only ramifications of one ubiquitous and ever operative law; that this law registers itself in the thing evolved, leaving stamped thereon as it were fossil footprints by means of which it may be known. In the arts the creative spirit of man is at its freest and finest, and nowhere among the arts is it so free and so fine as in music. In music accordingly the universal law of becoming finds instant, direct and perfect selfexpression; music voices the inner nature of the willtolive in all its moods and moments; in it form, content, means and end are perfectly fused. It is this fact which gives validity to the before quoted saying that all of the arts "aspire toward the condition of music." All aspire to express the law, but music, being least encumbered by the leaden burden of materiality, expresses it most easily and adequately. This being so there is nothing unreasonable in attempting to apply the known facts of musical harmony and rhythm to any other art, and since these essays concern themselves primarily with architecture, the final aspect in which that art will be presented here is as "frozen music"—ponderable form governed by musical law.
Music depends primarily upon the equal and regular division of time into beats, and of these beats into measures. Over this soundless and invisible warp is woven an infinitely various melodic pattern, made up of tones of different pitch and duration arithmetically related and combined according to the laws of harmony. Architecture, correspondingly, implies the rhythmical division of space, and obedience to laws numerical and geometrical. A certain identity therefore exists between simple harmony in music, and simple proportion in architecture. By translating the consonant toneintervals into number, the common denominator, as it were, of both arts, it is possible to give these intervals a spatial, and hence an architectural, expression. Such expression, considered as proportion only and divorced from ornament, will prove pleasing to the eye in the same way that its correlative is pleasing to the ear, because in either case it is not alone the special organ of sense which is gratified, but the inner Self, in which all senses are one. Containing within itself the mystery of number, it thrills responsive to every audible or visible presentment of that mystery.
If a vibrating string yielding a certain musical note be stopped in its center, that is, divided by half, it will then sound the octave of that note. The numerical ratio which expresses the interval of the octave is therefore 1:2. If onethird instead of onehalf of the string be stopped, and the remaining twothirds struck, it will yield the musical fifth of the original note, which thus corresponds to the ratio 2:3. The length represented by 3:4 yields the fourth; 4:5 the major third; and 5:6 the minor third. These comprise the principal consonant intervals within the range of one octave. The ratios of inverted intervals, so called, are found by doubling the smaller number of the original interval as given above: 2:3, the fifth, gives 3:4, the fourth; 4:5, the major third, gives 5:8, the minor sixth; 5:6, the minor third, gives 6:10, or 3:5, the major sixth.
Of these various consonant intervals the octave, fifth, and major third are the most important, in the sense of being the most perfect, and they are expressed by numbers of the smallest quantity, an odd number and an even. It will be noted that all the intervals above given are expressed by the numbers 1, 2, 3, 4, 5 and 6, except the minor sixth (5:8), and this is the most imperfect of all consonant intervals. The subminor seventh, expressed by the ratio 4:7 though included among the dissonances, forms, according to Helmholtz, a more perfect consonance with the tonic than does the minor sixth.
A natural deduction from these facts is that relations of architectural length and breadth, height and width, to be "musical" should be capable of being expressed by ratios of quantitively small numbers, preferably an odd number and an even. Although generally speaking the simpler the numerical ratio the more perfect the consonance, yet the intervals of the fifth and major third (2:3 and 4:5), are considered to be more pleasing than the octave (1:2), which is too obviously a repetition of the original note. From this it is reasonable to assume (and the assumption is borne out by experience), that proportions, the numerical ratios of which the eye resolves too readily, become at last wearisome. The relation should be felt rather than fathomed. There should be a perception of identity, and also of difference. As in music, where dissonances are introduced to give value to consonances which follow them, so in architecture simple ratios should be employed in connection with those more complex.
Harmonics are those tones which sound with, and reinforce any musical note when it is sounded. The distinguishable harmonics of the tonic yield the ratios 1:2, 2:3, 3:4, 4:5, and 4:7. A note and its harmonics form a natural chord. They may be compared to the widening circles which appear in still water when a stone is dropped into it, for when a musical sound disturbs the quietude of that pool of silence which we call the air, it ripples into overtones, which becoming fainter and fainter, die away into silence. It would seem reasonable to assume that the combination of numbers which express these overtones, if translated into terms of space, would yield proportions agreeable to the eye, and such is the fact, as the accompanying examples sufficiently indicate (Illustrations 8790).
The interval of the subminor seventh (4:7), used in this way, in connection with the simpler intervals of the octave (1:2), and the fifth (2:3), is particularly pleasing because it is neither too obvious nor too subtle. This ratio of 4:7 is important for the reason that it expresses the angle of sixty degrees, that is, the numbers 4 and 7 represent (very nearly) the ratio between onehalf the base and the altitude of an equilateral triangle: also because they form part of the numerical series 1, 4, 7, 10, etc. Both are "mystic" numbers, and in Gothic architecture particularly, proportions were frequently determined by numbers to which a mystic meaning was attached. According to Gwilt, the Gothic chapels of Windsor and Oxford are divided longitudinally by four, and transversely by seven equal parts. The arcade above the roses in the facade of the cathedral of Tours shows seven principal units across the front of the nave, and four in each of the towers.
A distinguishing characteristic of the series of ratios which represent the consonant intervals within the compass of an octave is that it advances by the addition of 1 to both terms: 1:2, 2:3, 3:4, 4:5, and 5:6. Such a series always approaches unity, just as, represented graphically by means of parallelograms, it tends toward a square. Alberti in his book presents a design for a tower showing his idea for its general proportions. It consists of six stories, in a sequence of orders. The lowest story is a perfect cube and each of the other stories is 1112ths of the story below, diminishing practically in the proportion of 8, 7, 6, 5, 4, 3, allowing in each case for the amount hidden by the projection of the cornice below; each order being accurate as regards column, entablature, etc. It is of interest to compare this with Ruskin's idea in his Seven Lamps, where he takes the case of a plant called Alisma Plantago, in which the various branches diminish in the proportion of 7, 6, 5, 4, 3, respectively, and so carry out the same idea; on which Ruskin observes that diminution in a building should be after the manner of Nature.
It would be a profitless task to formulate exact rules of architectural proportion based upon the laws of musical harmony. The two arts are too different from each other for that, and moreover the last appeal must always be to the eye, and not to a mathematical formula, just as in music the last appeal is to the ear. Laws there are, but they discover themselves to the artist as he proceeds, and are for the most part incommunicable. Rules and formulae are useful and valuable not as a substitute for inspiration, but as a guide: not as wings, but as a tail. In this connection perhaps all that is necessary for the architectural designer to bear in mind is that important ratios of length and breadth, height and width, to be "musical" should be expressed by quantitively small numbers, and that if possible they should obey some simple law of numerical progression. From this basic simplicity complexity will follow, but it will be an ordered and harmonious complexity, like that of a tree, or of a symphony.
In the same way that a musical composition implies the division of time into equal and regular beats, so a work of architecture should have for its basis some unit of space. This unit should be nowhere too obvious and may be varied within certain limits, just as musical time is retarded or accelerated. The underlying rhythm and symmetry will thus give value and distinction to such variation. Vasari tells how Brunelleschi. Bramante and Leonardo da Vinci used to work on paper ruled in squares, describing it as a "truly ingenious thing, and of great utility in the work of design." By this means they developed proportions according to a definite scheme. They set to work with a division of space analogous to the musician's division of time. The examples given herewith indicate how close a parallel may exist between music and architecture in this matter of rhythm (Illustrations 9193).
It is a demonstrable fact that musical sounds weave invisible patterns in the air. Architecture, correspondingly, in one of its aspects, is geometric pattern made fixed and enduring. What could be more essentially musical for example than the sea arcade of the Venetian Ducal Palace? The sand forms traced by soundwaves on a musically vibrating steel plate might easily suggest architectural ornament did not the differences of scale and of material tend to confuse the mind. The architect should occupy himself with identities, not differences. If he will but bear in mind that architecture is pattern in space, just as music is pattern in time, he will come to perceive the essential identity between, say, a Greek rosette and a Gothic rosewindow; an arcade and an egg and dart moulding (Illustration 94). All architectural forms and arrangements which give enduring pleasure are in their essence musical. Every well composed facade makes harmony in three dimensions; every good roofline sings a melody against the sky.
CONCLUSION
In taking leave of the reader at the end of this excursion together among the byways of a beautiful art, the author must needs add a final word or two touching upon the purpose and scope of these essays. Architecture (like everything else) has two aspects: it may be viewed from the standpoint of utility, that is, as construction; or from the standpoint of expressiveness, that is, as decoration. No attempt has been made here to deal with its first aspect, and of the second (which is again twofold), only the universal, not the particular expressiveness has been sought. The literature of architecture is rich in works dealing with the utilitarian and constructive side of the art: indeed, it may be said that to this side that literature is almost exclusively devoted. This being so, it has seemed worth while to attempt to show the reverse of the medal, even though it be "tails" instead of "heads."
It will be noted that the inductive method has not, in these pages, been honored by a due observance. It would have been easy to have treated the subject inductively, amassing facts and drawing conclusions, but to have done so the author would have been false to the very principle about which the work came into being. With the acceptance of the Ancient Wisdom, the inductive method becomes no longer necessary. Facts are not useful in order to establish a hypothesis, they are used rather to elucidate a known and accepted truth. When theosophical ideas shall have permeated the thought of mankind, this work, if it survives at all, will be chiefly—perhaps solely—remarkable by reason of the fact that it was among the first in which the attempt was made again to unify science, art and religion, as they were unified in those ancient times and among those ancient peoples when the Wisdom swayed the hearts and minds of men.
THE END 
