THE BEAUTIFUL NECESSITY

Seven Essays on Theosophy and Architecture

by CLAUDE BRAGDON, F. A. I. A. 

MCMXXII

 "Let us build altars to the Beautiful Necessity" --EMERSON

By the Same Author: Episodes From An Unwritten History The Golden Person In The Heart Architecture And Democracy A Primer Of Higher Space Four Dimensional Vistas Projective Ornament Oracle

CONTENTS

 I THE ART OF ARCHITECTURE

 II UNITY AND POLARITY

 III CHANGELESS CHANGE

 IV THE BODILY TEMPLE

 V LATENT GEOMETRY

 VI THE ARITHMETIC OF BEAUTY

 VII FROZEN MUSIC

 CONCLUSION

PREFACE TO THE SECOND EDITION

_The Beautiful Necessity_ was first published in 1910. Save for a slimvolume of privately printed verse it was my first book. I worked hardon it. Fifteen years elapsed between its beginning and completion;it was twice published serially--written, rewritten andtre-written--before it reached its ultimate incarnation in book form. 

Confronted now with the opportunity to revise the text again, I findmyself in the position of a surgeon who feels that the operation heis called upon to perform may perhaps harm more than it can help. Prudence therefore prevails over my passion for dissection: warned byeminent examples, I fear that any injection of my more mature and lesscocksure consciousness into this book might impair its unity--that I"never could recapture the first fine careless rapture. "

The text stands therefore as originally published save for a fewverbal changes, and whatever reservations I have about it shallbe stated in this preface. These are not many nor important: _TheBeautiful Necessity_ contains nothing that I need repudiate or care tocontradict. 

Its thesis, briefly stated, is that art in all its manifestations isan expression of the cosmic life, and that its symbols constitute alanguage by means of which this life is published and represented. Artis at all times subject to the _Beautiful Necessity_ of proclaimingthe _world order_. 

In attempting to develop this thesis it was not necessary (nor asI now think, desirable) to link it up in so definite a manner withtheosophy. The individual consciousness is colored by the particularmedium through which it receives truth, and for me that medium wastheosophy. Though the book might gain a more unprejudiced hearing, and from a larger audience, by the removal of the theosophic"color-screen, " it shall remain, for its removal now might seem toimply a loss of faith in the fundamental tenets of theosophy, and suchan implication would not be true. 

The ideas in regard to time and space are those commonly currentin the world until the advent of the Theory of Relativity. To ageneration brought up on Einstein and Ouspensky they are bound toappear "lower dimensional. " Merely to state this fact is to dealwith it to the extent it needs to be dealt with. The integrity of myargument is not impaired by these new views. 

The one important influence that has operated to modify my opinionsconcerning the mathematical basis of the arts of space has been thediscoveries of Mr. Jay Hambidge with regard to the practice of theGreeks in these matters, as exemplified in their temples and theirceramics, and named by him _Dynamic Symmetry_. 

In tracing everything back to the logarithmic spiral (which embodiesthe principle of extreme and mean ratios) I consider that Mr. Hambidgehas made one of those generalizations which reorganizes the oldknowledge and organizes the new. It would be only natural if in hisimmersion in his idea he overworks it, but Mr. Hambidge is a man ofsuch intellectual integrity and thoroughness of method that he may betrusted not to warp the facts to fit his theories. The truth of thematter is that the entire field of research into the mathematics ofBeauty is of such richness that wherever a man plants his metaphysicalspade he is sure to come upon "pay dirt. " _The_ _Beautiful Necessity_represents the result of my own prospecting; _Dynamic Symmetry_represents the result of his. If at any point our findings appear toconflict, it is less likely that one or the other of us is mistakenthan that each is right from his own point of view. Be that as it may, I should be the last man in the world to differ from Mr. Hambidge, for if he convicted me of every conceivable error his work would stillremain the greatest justification and confirmation of my fundamentalcontention--that art is an expression of the _world order_ andis therefore orderly, organic; subject to mathematical law, andsusceptible of mathematical analysis. 

CLAUDE BRAGDON

Rochester, N. Y. 

April, 1922

I

THE ART OF ARCHITECTURE

One of the advantages of a thorough assimilation of what may be calledthe theosophic idea is that it can be applied with advantage to everydepartment of knowledge and of human activity: like the key to acryptogram it renders clear and simple that which before seemedintricate and obscure. Let us apply this key to the subject of art, and to the art of architecture in particular, and see if by sodoing we may not learn more of art than we knew before, and more oftheosophy too. 

The theosophic idea is that everything is an expression of theSelf--or whatever other name one may choose to give to that immanentunknown reality which forever hides behind all phenomenal life--butbecause, immersed as we are in materiality, our chief avenue ofknowledge is sense perception, a more exact expression of thetheosophic idea would be: Everything is the expression of the Selfin terms of sense. Art, accordingly, is the expression of the Self interms of sense. Now though the Self is _one_, sense is not one, butmanifold: and therefore there are _arts_, each addressed to someparticular faculty or group of faculties, and each expressing someparticular quality or group of qualities of the Self. The white lightof Truth is thus broken up into a rainbow-tinted spectrum of Beauty, in which the various arts are colors, each distinct, yet merging oneinto another--poetry into music; painting into decoration; decorationbecoming sculpture; sculpture--architecture, and so on. 

In such a spectrum of the arts each one occupies a definite place, andall together form a series of which music and architecture are the twoextremes. That such is their relative position may be demonstrated invarious ways. The theosophic explanation involving the familiar ideaof the "pairs of opposites" would be something as follows. Accordingto the Hindu-Aryan theory, Brahma, that the world might be born, fellasunder into man and wife--became in other words _name and form_[A]The two universal aspects of name and form are what philosophers callthe two "modes of consciousness, " one of time, and the other of space. These are the two gates through which ideas enter phenomenal life; thetwo boxes, as it were, that contain all the toys with which we play. Everything, were we only keen enough to perceive it, bears the mark ofone or the other of them, and may be classified accordingly. In such aclassification music is seen to be allied to time, and architecture tospace, because music is successive in its mode of manifestation, andin time alone everything would occur successively, one thing followinganother; while architecture, on the other hand, impresses itself uponthe beholder all at once, and in space alone all things would existsimultaneously. Music, which is in time alone, without any relation tospace; and architecture, which is in space alone, without any relationto time, are thus seen to stand at opposite ends of the art spectrum, and to be, in a sense, the only "pure" arts, because in all the othersthe elements of both time and space enter in varying proportion, either actually or by implication. Poetry and the drama are allied tomusic inasmuch as the ideas and images of which they are made up arepresented successively, yet these images are for the most partforms of space. Sculpture on the other hand is clearly allied toarchitecture, and so to space, but the element of _action_, suspendedthough it be, affiliates it with the opposite or time pole. Paintingoccupies a middle position, since in it space instead of beingactual has become ideal--three dimensions being expressed throughthe mediumship of two--and time enters into it more largely than intosculpture by reason of the greater ease with which complicated actioncan be indicated: a picture being nearly always time arrested inmidcourse as it were--a moment transfixed. 

In order to form a just conception of the relation between music andarchitecture it is necessary that the two should be conceived of notas standing at opposite ends of a series represented by a straightline, but rather in juxtaposition, as in the ancient Egyptian symbolof a serpent holding its tail in its mouth, the head in this casecorresponding to music, and the tail to architecture; in other words, though in one sense they are the most-widely separated of the arts, inanother they are the most closely related. 

Music being purely in time and architecture being purely in space, each is, in a manner and to a degree not possible with any ofthe other arts, convertible into the other, by reason of thecorrespondence subsisting between intervals of time and intervals ofspace. A perception of this may have inspired the famous sayingthat architecture is _frozen music_, a poetical statement of aphilosophical truth, since that which in music is expressed by meansof harmonious intervals of time and pitch, successively, after themanner of time, may be translated into corresponding intervals ofarchitectural void and solid, height and width. 

In another sense music and architecture are allied. They alone ofall the arts are purely creative, since in them is presented, nota likeness of some known idea, but _a thing-in-itself_ brought toa distinct and complete expression of its nature. Neither a musicalcomposition nor a work of architecture depends for its effectivenessupon resemblances to natural sounds in the one case, or to naturalforms in the other. Of none of the other arts is this to such a degreetrue: they are not so much creative as re-creative, for in them allthe artist takes his subject ready made from nature and presents itanew according to the dictates of his genius. 

The characteristic differences between music and architecture are thesame as those which subsist between time and space. Now time and spaceare such abstract ideas that they can be dealt with best throughtheir corresponding correlatives in the natural world, for it is afundamental theosophic tenet that nature everywhere abounds in suchcorrespondences; that nature, in its myriad forms, is indeed theconcrete presentment of abstract unities. The energy which everywhereanimates form is a type of time within space; the mind working inand through the body is another expression of the same thing. Correspondingly, music is dynamic, subjective, mental, of onedimension; while architecture is static, objective, physical, of threedimensions; sustaining the same relation to music and the otherarts as does the human body to the various organs which compose, andconsciousnesses which animate it (it being the reservatory of theseorgans and the vehicle of these consciousnesses); and a work ofarchitecture in like manner may and sometimes does include all of theother arts within itself. Sculpture accentuates and enriches, paintingadorns, works of literature are stored within it, poetry and the dramaawake its echoes, while music thrills to its uttermost recesses, likethe very spirit of life tingling through the body's fibres. 

Such being the relation between them, the difference in the nature ofthe ideas bodied forth in music and in architecture becomes apparent. Music is interior, abstract, subjective, speaking directly to the soulin a simple and universal language whose meaning is made personal andparticular in the breast of each listener: "Music alone of all thearts, " says Balzac, "has power to make us live within ourselves. "A work of architecture is the exact opposite of this: existingprincipally and primarily for the uses of the body, it is like thebody a concrete organism, attaining to esthetic expression only inthe reconciliation and fulfilment of many conflicting practicalrequirements. Music is pure beauty, the voice of the unfetteredand perpetually vanishing soul of things; architecture is that soulimprisoned in a form, become subject to the law of causality, beatenupon by the elements, at war with gravity, the slave of man. One isthe Ariel of the arts; the other, Caliban. 

Coming now to the consideration of architecture in its historicalrather than its philosophical aspect, it will be shown how certaintheosophical concepts are applicable here. Of these none is morefamiliar and none more fundamental than the idea of reincarnation. Byreincarnation more than mere physical re-birth is meant, for physicalre-birth is but a single manifestation of that universal law ofalternation of state, of animation of vehicles, and progressionthrough related planes, in accordance with which all things move, and as it were make music--each cycle complete, yet part of a largercycle, the incarnate monad passing through correlated changes, carrying along and bringing into manifestation in each successive arcof the spiral the experience accumulated in all preceding states, and at the same time unfolding that power of the Self peculiar to theplane in which it is momentarily manifesting. 

This law finds exemplification in the history of architecture in theorderly flow of the building impulse from one nation and one countryto a different nation and a different country: its new vehicle ofmanifestation; also in the continuity and increasing complexity ofthe development of that impulse in manifestation; each "incarnation"summarizing all those which have gone before, and adding some newfactor peculiar to itself alone; each being a growth, a life, withperiods corresponding to childhood, youth, maturity and decadence;each also typifying in its entirety some single one of theselife-periods, and revealing some special aspect or power of the Self. 

For the sake of clearness and brevity the consideration of only oneof several architectural evolutions will be attempted: that which, arising in the north of Africa, spread to southern Europe, thence tothe northwest of Europe and to England--the architecture, in short, ofthe so-called civilized world. 

This architecture, anterior to the Christian era, may be broadlydivided into three great periods, during which it was successivelypracticed by three peoples: the Egyptians, the Greeks and the Romans. Then intervened the Dark Ages, and a new art arose, the Gothic, whichwas a flowering out in stone of the spirit of Christianity. This wasin turn succeeded by the Renaissance, the impulse of which remainsto-day unexhausted. In each of these architectures the peculiar geniusof a people and of a period attained to a beautiful, complete andcoherent utterance, and notwithstanding the considerable intervalsof time which sometimes separated them they succeeded one anotherlogically and inevitably, and each was related to the one whichpreceded and which followed it in a particular and intimate manner. 

The power and wisdom of ancient Egypt was vested in its priesthood, which was composed of individuals exceptionally qualified by birthand training for their high office, tried by the severest ordealsand bound by the most solemn oaths. The priests were honored andprivileged above all other men, and spent their lives dwellingapart from the multitude in vast and magnificent temples, dedicatingthemselves to the study and practice of religion, philosophy, scienceand art--subjects then intimately related, not widely separated asthey are now. These men were the architects of ancient Egypt: theirsthe minds which directed the hands that built those time-defyingmonuments. 

The rites that the priests practiced centered about what are knownas the Lesser and the Greater Mysteries. These consisted ofrepresentations by means of symbol and allegory, under conditionsand amid surroundings the most awe-inspiring, of those great truthsconcerning man's nature, origin and destiny of which the priests--inreality a brotherhood of initiates and their pupils--were thecustodians. These ceremonies were made the occasion for the initiationof neophytes into the order, and the advancement of the alreadyinitiated into its successive degrees. For the practice of such rites, and others designed to impress not the elect but the multitude, thegreat temples of Egypt were constructed. Everything about them wascalculated to induce a deep seriousness of mind, and to inspirefeelings of awe, dread and even terror, so as to test the candidate'sfortitude of soul to the utmost. 

The avenue of approach to an Egyptian temple was flanked on bothsides, sometimes for a mile or more, with great stone sphinxes--thatemblem of man's dual nature, the god emerging from the beast. Theentrance was through a single high doorway between two toweringpylons, presenting a vast surface sculptured and painted over withmany strange and enigmatic figures, and flanked by aspiring obelisksand seated colossi with faces austere and calm. The large court thusentered was surrounded by high walls and colonnades, but was opento the sky. Opposite the first doorway was another, admitting to asomewhat smaller enclosure, a forest of enormous carved and paintedcolumns supporting a roof through the apertures of which sunshinegleamed or dim light filtered down. Beyond this in turn were othercourts and apartments culminating in some inmost sacred sanctuary. 

Not alone in their temples, but in their tombs and pyramids andall the sculptured monuments of the Egyptians, there is the sameinsistence upon the sublimity, mystery and awefulness of life, which they seem to have felt so profoundly. But more than this, theconscious thought of the masters who conceived them, the buildings ofEgypt give utterance also to the toil and suffering of the thousandsof slaves and captives which hewed the stones out of the heart of therock, dragged them long distances and placed them one upon another, sothat these buildings oppress while they inspire, for there is in themno freedom, no spontaneity, no individuality, but everywhere the feltpresence of an iron conventionality, of a stern immutable law. 

In Egyptian architecture is symbolized the condition of the human soulawakened from its long sleep in nature, and become conscious at onceof its divine source and of the leaden burden of its fleshy envelope. Egypt is humanity new-born, bound still with an umbilical cord tonature, and strong not so much with its own strength as with thestrength of its mother. This idea is aptly symbolized in thosegigantic colossi flanking the entrance to some rock-cut temple, whichthough entire are yet part of the living cliff out of which they werefashioned. 

In the architecture of Greece the note of dread and mystery yields toone of pure joyousness and freedom. The terrors of childhood have beenoutgrown, and man revels in the indulgence of his unjaded appetitesand in the exercise of his awakened reasoning faculties. In Greek artis preserved that evanescent beauty of youth which, coming but onceand continuing but for a short interval in every human life, is yetthat for which all antecedent states seem a preparation, and ofwhich all subsequent ones are in some sort an effect. Greece typifiesadolescence, the love age, and so throughout the centuries humanityhas turned to the contemplation of her, just as a man all his lifelong secretly cherishes the memory of his first love. 

An impassioned sense of beauty and an enlightened reason characterizethe productions of Greek architecture during its best period. Theperfection then attained was possible only in a nation whereof thecitizens were themselves critics and amateurs of art, one whereinthe artist was honored and his work appreciated in all its beautyand subtlety. The Greek architect was less bound by tradition andprecedent than was the Egyptian, and he worked unhampered by anyrestrictions save such as, like the laws of harmony in music, helpedrather than hindered his genius to express itself--restrictionsfounded on sound reason, the value of which had been proved byexperience. 

The Doric order was employed for all large temples, since it possessedin fullest measure the qualities of simplicity and dignity, theattributes appropriate to greatness. Quite properly also its formulaswere more fixed than those of any other style. The Ionic order, the feminine of which the Doric may be considered the correspondingmasculine, was employed for smaller temples; like a woman it was moresupple and adaptable than the Doric, its proportions were more slenderand graceful, its lines more flowing, and its ornament more delicateand profuse. A freer and more elaborate style than either of these, infinitely various, seeming to obey no law save that of beauty, wasused sometimes for small monuments and temples, such as the Tower ofthe Winds, and the monument of Lysicrates at Athens. 

[Illustration 1]

Because the Greek architect was at liberty to improve upon the work ofhis predecessors if he could, no temple was just like any other, and they form an ascending scale of excellence, culminating in theAcropolis group. Every detail was considered not only with relationto its position and function, but in regard to its intrinsic beauty aswell, so that the merest fragment, detached from the building of whichit formed a part, is found worthy of being treasured in our museumsfor its own sake. 

Just as every detail of a Greek temple was adjusted to its positionand expressed its office, so the building itself was made to fitits site and to show forth its purpose, forming with the surroundingbuildings a unit of a larger whole. The Athenian Acropolis is anillustration of this: it is an irregular fortified hill, bearingdiverse monuments in various styles, at unequal levels and atdifferent angles with one another, yet the whole arrangement seems asorganic and inevitable as the disposition of the features of a face. The Acropolis is an example of the ideal architectural republicwherein each individual contributes to the welfare of all, and at thesame time enjoys the utmost personal liberty (Illustration 1). 

Very different is the spirit bodied forth in the architecture ofImperial Rome. The iron hand of its sovereignty encased within thesilken glove of its luxury finds its prototype in buildings which werestupendous crude brute masses of brick and concrete, hidden within acovering of rich marbles and mosaics, wrought in beautiful but oftenmeaningless forms by clever degenerate Greeks. The genius of Romefinds its most characteristic expression, not in temples to the highgods, but rather in those vast and complicated structures--basilicas, amphitheatres, baths--built for the amusement and purely temporalneeds of the people. 

If Egypt typifies the childhood of the race and Greece its beautifulyouth, Republican Rome represents its strong manhood--a soldier filledwith the lust of war and the love of glory--and Imperial Rome itsdegeneracy: that soldier become conqueror, decked out in plunderedfinery and sunk in sensuality, tolerant of all who minister to hispleasures but terrible to all who interfere with them. 

The fall of Rome marked the end of the ancient Pagan world. Aboveits ruin Christian civilization in the course of time arose. Gothicarchitecture is an expression of the Christian spirit; in it ismanifest the reaction from licentiousness to asceticism. Man'sspiritual nature, awakening in a body worn and weakened bydebaucheries, longs ardently and tries vainly to escape. Of some suchmood a Gothic cathedral is the expression: its vaulting, marvelouslysupported upon slender shafts by reason of a nicely adjustedequilibrium of forces; its restless, upward-reaching pinnacles andspires; its ornament, intricate and enigmatic--all these suggest theover-strained organism of an ascetic; while its vast shadowy interiorlit by marvelously traceried and jeweled windows, which hold the eyesin a hypnotic thrall, is like his soul: filled with world sadness, dead to the bright brief joys of sense, seeing only heavenly visions, knowing none but mystic raptures. 

Thus it is that the history of architecture illustrates and enforcesthe theosophical teaching that everything of man's creating is made inhis own image. Architecture mirrors the life of the individual and ofthe race, which is the life of the individual written large in timeand space. The terrors of childhood; the keen interests and appetitesof youth; the strong stern joy of conflict which comes with manhood;the lust, the greed, the cruelty of a materialized old age--all theseserve but as a preparation for the life of the spirit, in which theman becomes again as a little child, going over the whole round, buton a higher arc of the spiral. 

The final, or fourth state being only in some sort a repetition ofthe first, it would be reasonable to look for a certain correspondencebetween Egyptian and Gothic architecture, and such a correspondencethere is, though it is more easily divined than demonstrated. Inboth there is the same deeply religious spirit; both convey, in someobscure yet potent manner, a sense of the soul being near the surfaceof life. There is the same love of mystery and of symbolism; and inboth may be observed the tendency to create strange composite figuresto typify transcendental ideas, the sphinx seeming a blood-brother tothe gargoyle. The conditions under which each architecture flourishedwere not dissimilar, for each was formulated and controlled by smallwell-organized bodies of sincerely religious and highly enlightenedmen--the priesthood in the one case, the masonic guilds in theother--working together toward the consummation of great undertakingsamid a populace for the most part oblivious of the profound and subtlemeanings of which their work was full. In Mediæval Europe, as inancient Egypt, fragments of the Ancient Wisdom--transmitted in thesymbols and secrets of the cathedral builders--determined much ofGothic architecture. 

The architecture of the Renaissance period, which succeeded theGothic, corresponds again, in the spirit which animates it, to Greekarchitecture, which succeeded the Egyptian, for the Renaissance asthe name implies was nothing other than an attempt to revive Classicalantiquity. Scholars writing in what they conceived to be a Classicalstyle, sculptors modeling Pagan deities, and architects buildingaccording to their understanding of Vitruvian methods succeededin producing works like, yet different from the originals theyfollowed--different because, animated by a spirit unknown to theancients, they embodied a new ideal. 

In all the productions of the early Renaissance, "that firsttranscendent springtide of the modern world, " there is the evanescentgrace and beauty of youth which was seen to have pervaded Greek art, but it is a grace and beauty of a different sort. The Greek artistsought to attain to a certain abstract perfection of type; to builda temple which should combine all the excellencies of every similartemple, to carve a figure, impersonal in the highest sense, whichshould embody every beauty. The artist of the Renaissance on the otherhand delighted not so much in the type as in the variation from it. Preoccupied with the unique mystery of the individual soul--a sense ofwhich was Christianity's gift to Christendom--he endeavored to portraythat wherein a particular person is unique and singular. Acutelyconscious also of his own individuality, instead of effacing it hemade his work the vehicle and expression of that individuality. Thehistory of Renaissance architecture, as Symonds has pointed out, is the history of a few eminent individuals, each one moulding andmodifying the style in a manner peculiar to himself alone. In thehands of Brunelleschi it was stern and powerful; Bramante madeit chaste, elegant and graceful; Palladio made it formal, cold, symmetrical; while with Sansovino and Sammichele it became sumptuousand bombastic. 

As the Renaissance ripened to decay its architecture assumed more andmore the characteristics which distinguished that of Rome during thedecadence. In both there is the same lack of simplicity and sincerity, the same profusion of debased and meaningless ornament, and there isan increasing disposition to conceal and falsify the construction bysurface decoration. 

The final part of this second or modern architectural cycle lies stillin the future. It is not unreasonable to believe that the movementtoward mysticism, of which modern theosophy is a phase and thespiritualization of science an episode, will flower out into anarchitecture which will be in some sort a reincarnation of and areturn to the Gothic spirit, employing new materials, new methods, and developing new forms to show forth the spirit of the modern world, without violating ancient verities. 

In studying these crucial periods in the history of Europeanarchitecture it is possible to trace a gradual growth or unfoldingas of a plant. It is a fact fairly well established that the Greeksderived their architecture and ornament from Egypt; the Romans inturn borrowed from the Greeks; while a Gothic cathedral is a linealdescendant from a Roman basilica. 

[Illustration 2]

[Illustration 3]

The Egyptians in their constructions did little more than to placeenormous stones on end, and pile one huge block upon another. Theyused many columns placed close together: the spaces which they spannedwere inconsiderable. The upright or supporting member may be said tohave been in Egyptian architecture the predominant one. A verticalline therefore may be taken as the simplest and most abstract symbolof Egyptian architecture (Illustration 2). It remained for the Greeksfully to develop the lintel. In their architecture the verticalmember, or column, existed solely for the sake of the horizontalmember, or lintel; it rarely stood alone as in the case of an Egyptianobelisk. The columns of the Greek temples were reduced to thoseproportions most consistent with strength and beauty, and theintercolumnations were relatively greater than in Egyptian examples. It may truly be said that Greek architecture exhibits the perfectequality and equipoise of vertical and horizontal elements and theseonly, no other factor entering in. Its graphic symbol would thereforebe composed of a vertical and a horizontal line (Illustration 3). TheRomans, while retaining the column and lintel of the Greeks, deprivedthem of their structural significance and subordinated them to thesemicircular arch and the semi-cylindrical and hemispherical vault, the truly characteristic and determining forms of Roman architecture. Our symbol grows therefore by the addition of the arc of a circle(Illustration 4). In Gothic architecture column, lintel, arch andvault are all retained in changed form, but that which more thananything else differentiates Gothic architecture from any style whichpreceded it is the introduction of the principle of an equilibrium offorces, of a state of balance rather than a state of rest, arrived atby the opposition of one thrust with another contrary to it. This factcan be indicated graphically by two opposing inclined lines, andthese united to the preceding symbol yield an accurate abstract of theelements of Gothic architecture (Illustration 5). 

[Illustration 4]

[Illustration 5]

All this is but an unusual application of a familiar theosophicteaching, namely, that it is the method of nature on every plane andin every department not to omit anything that has gone before, but tostore it up and carry it along and bring it into manifestation later. Nature everywhere proceeds like the jingle of _The House that JackBuilt_: she repeats each time all she has learned, and adds anotherline for subsequent repetition. 

[Footnote A: The quaint Oriental imagery here employed should notblind the reader to the precise scientific accuracy of the idea ofwhich this imagery is the vehicle. Schopenhauer says: "Polarity, orthe sundering of a force into two quantitively different and opposedactivities striving after re-union, ... Is a fundamental type of almostall the phenomena of nature, from the magnet and the crystal to manhimself. "]

II

UNITY AND POLARITY

Theosophy, both as a philosophy, or system of thought, which discoverscorrelations between things apparently unrelated, and as a life, or system of training whereby it is possible to gain the power toperceive and use these correlations for worthy ends, is of great valueto the creative artist, whose success depends on the extent to whichhe works organically, conforming to the cosmic pattern, proceedingrationally and rhythmically to some predetermined end. It is of valueno less to the layman, the critic, the art amateur--to anyone infact who would come to an accurate and intimate understanding andappreciation of every variety of esthetic endeavor. For the benefit ofsuch I shall try to trace some of those correlations which theosophyaffirms, and indicate their bearing upon art, and upon the art ofarchitecture in particular. 

One of the things which theosophy teaches is that those transcendentglimpses of a divine order and harmony throughout the universevouchsafed the poet and the mystic in their moments of vision are notthe paradoxes--the paronomasia as it were--of an intoxicated state ofconsciousness, but glimpses of reality. We are all of us participatorsin a world of concrete music, geometry and number--a world ofsounds, odors, forms, motions, colors, so mathematically related andcoordinated that our pigmy bodies, equally with the farthest star, vibrate to the music of the spheres. There is a _Beautiful Necessity_which rules the world, which is a law of nature and equally a law ofart, for art is idealized creation: nature carried to a higher powerby reason of its passage through a human consciousness. Thought andemotion tend to crystallize into forms of beauty as inevitably as doesthe frost on a window pane. Art therefore in one of its aspects is theweaving of a pattern, the communication of an order and a method tothe material or medium employed. Although no masterpiece was evercreated by the conscious following to set rules, for the true artistworks unconsciously, instinctively, as the bird sings or as the beebuilds its honey-cell, yet an analysis of any masterpiece reveals thefact that its author (like the bird and the bee) has "followed therules without knowing them. "

Helmholtz says, "No doubt is now entertained that beauty is subjectto laws and rules dependent on the nature of human intelligence. Thedifficulty consists in the fact that these laws and rules, on whosefulfilment beauty depends, are not consciously present in the mind ofthe artist who creates the work, or of the observer who contemplatesit. " Nevertheless they are discoverable, and can be formulated, after a fashion. We have only to read aright the lessons everywhereportrayed in the vast picture-books of nature and of art. 

The first truth therein published is the law of _Unity_--oneness; forthere is one Self, one Life, which, myriad in manifestation, is yetin essence ever _one_. Atom and universe, man and the world--each isa unit, an organic and coherent whole. The application of this law toart is so obvious as to be almost unnecessary of elucidation, for tosay that a work of art must possess unity, must seem to proceed from asingle impulse and be the embodiment of one dominant idea, is to statea truism. In a work of architecture the coördination of its variousparts with one another is almost the measure of its success. Weremember any masterpiece--the cathedral of Paris no less than thepyramids of Egypt--by the singleness of its appeal; complex it may be, but it is a coordinated complexity; variety it may possess, but it isa variety in an all-embracing unity. 

The second law, not contradicting but supplementing the first is thelaw of _Polarity_, i. E. , duality. All things have sex, are eithermasculine or feminine. This too is the reflection on a lower planeof one of those transcendental truths taught by the Ancient Wisdom, namely that the Logos, in his voluntarily circumscribing his infinitelife in order that he may manifest, encloses himself within hislimiting veil, _maya_, and that his life appears as spirit (male), andhis _maya_ as matter (female), the two being never disjoined duringmanifestation. The two terms of this polarity are endlessly repeatedthroughout nature: in sun and moon, day and night, fire and water, manand woman--and so on. A close inter-relation is always seen to subsistbetween corresponding members of such pairs of opposites: sun, day, fire, man express and embody the primal and active aspect of themanifesting deity; moon, night, water, woman, its secondary andpassive aspect. Moreover, each implies or brings to mind the othersof its class: man, like the sun, is lord of day; he is like fire, adevastating force; woman is subject to the lunar rhythm; like water, she is soft, sinuous, fecund. 

The part which this polarity plays in the arts is important, and theconstant and characteristic distinction between the two terms is athing far beyond mere contrast. 

In music they are the major and minor modes: the typical, orrepresentative chords of the dominant seventh, and of the tonic (thetwo chords into which Schopenhauer says all music can be resolved): apartial dissonance, and a consonance: a chord of suspense, and a chordof satisfaction. In speech the two are vowel, and consonant sounds:the type of the first being _a_, a sound of suspense, made with themouth open; and of the second _m_, a sound of satisfaction, made byclosing the mouth; their combination forms the sacred syllable Om(_Aum_). In painting they are warm colors, and cold: the pole ofthe first being in red, the color of fire, which excites; and of thesecond in blue, the color of water, which calms; in the Arts of designthey are lines straight (like fire), and flowing (like water); masseslight (like the day), and dark (like night). In architecture they arethe column, or vertical member, which resists the force of gravity;and the lintel, or horizontal member, which succumbs to it; they arevertical lines, which are aspiring, effortful; and horizontal lines, which are restful to the eye and mind. 

It is desirable to have an instant and keen realization of this sexquality, and to make this easier some sort of classification andanalysis must be attempted. Those things which are allied to andpartake of the nature of _time_ are masculine, and those which areallied to and partake of the nature of _space_ are feminine: asmotion, and matter; mind, and body; etc. The English words "masculine"and "feminine" are too intimately associated with the idea of physicalsex properly to designate the terms of this polarity. In Japanesephilosophy and art (derived from the Chinese) the two are called _In_and _Yo_ (In, feminine; Yo, masculine); and these little words, beingfree from the limitations of their English correlatives, will be foundconvenient, Yo to designate that which is simple, direct, primary, active, positive; and In, that which is complex, indirect, derivative, passive, negative. Things hard, straight, fixed, vertical, are Yo;things soft, curved, horizontal, fluctuating, are In--and so on. 

[Illustration 6: WILD CHERRY; MAPLE LEAF]

[Illustration 7: CALLAS IN YO]

In passing it may be said that the superiority of the line, mass, andcolor composition of Japanese prints and kakemonos to that exhibitedin the vastly more pretentious easel pictures of modern Occidentalartists--a superiority now generally acknowledged by connoisseurs--islargely due to the conscious following, on the part of the Japanese, of this principle of sex-complementaries. 

Nowhere are In and Yo more simply and adequately imaged than in thevegetable kingdom. The trunk of a tree is Yo, its foliage, In; andin each stem and leaf the two are repeated. A calla, consisting ofa single straight and rigid spadix embraced by a soft and tenderlycurved spathe, affords an almost perfect expression of thecharacteristic differences between Yo and In and their reciprocalrelation to each other. The two are not often combined in suchsimplicity and perfection in a single form. The straight, verticalreeds which so often grow in still, shallow water, find theircomplement in the curved lily-pads which lie horizontally on itssurface. Trees such as pine and hemlock, which are excurrent--those inwhich the branches start successively (i. E. , after the manner of time)from a straight and vertical central stem--are Yo; trees such asthe elm and willow, which are deliquescent--those in which the trunkdissolves as it were simultaneously (after the manner of space) intoits branches--are In. All tree forms lie in or between these twoextremes, and leaves are susceptible of a similar classification. Itwill be seen to be a classification according to time and space, for the characteristic of time is _succession_, and of space, _simultaneousness_: the first is expressed symbolically by elementsarranged with relation to axial lines; the second, by elementsarranged with relation to focal points (Illustrations 6, 7). 

The student should train himself to recognize In and Yo in alltheir Protean presentments throughout nature--in the cloud upon themountain, the wave against the cliff, in the tracery of trees againstthe sky--that he may the more readily recognize them in his chosenart, whatever that art may be. If it happens to be painting, he willendeavor to discern this law of duality in the composition of everymasterpiece, recognizing an instinctive obedience to it in thatfavorite device of the great Renaissance masters of making anarchitectural setting for their groups of figures, and he willdelight to trace the law in all its ramifications of contrast betweencomplementaries in line, color, and mass (Illustration 8). 

[Illustration 8: THE LAW OF POLARITY CLEOPATRA MELTING THE PEARL. BYTIEPOLO]

With reference to architecture, it is true, generally speaking, that architectural forms have been developed through necessity, thefunction seeking and finding its appropriate form. For example, thebuttress of a Gothic cathedral was developed by the necessity ofresisting the thrust of the interior vaulting without encroaching uponthe nave; the main lines of a buttress conform to the direction of thethrust, and the pinnacle with which it terminates is a logical shapefor the masonry necessary to hold the top in position (Illustration9). Research along these lines is interesting and fruitful of result, but there remains a certain number of architectural forms whose origincannot be explained in any such manner. The secret of their undyingcharm lies in the fact that in them In and Yo stand symbolized andcontrasted. They no longer obey a law of utility, but an abstractlaw of beauty, for in becoming sexually expressive as it were, theconstruction itself is sometimes weakened or falsified. The familiarclassic console or modillion is an example: although in generalcontour it is well adapted to its function as a supporting bracket, embedded in, and projecting from a wall, yet the scroll-like ornamentwith which its sides are embellished gives it the appearance ofnot entering the wall at all, but of being stuck against it in somemiraculous manner. This defect in functional expressiveness ismore than compensated for by the perfection with which feminineand masculine characteristics are expressed and contrasted in theexquisite double spiral, opposed to the straight lines of the mouldingwhich it subtends (Illustration 10). Again, by fluting the shaft of acolumn its area of cross-section is diminished but the appearance ofstrength is enhanced because its masculine character--as a supportingmember resisting the force of gravity--is emphasized. 

[Illustration 9: CROSS SECTION OF BUTTRESS. ]

The importance of the so-called "orders" lies in the fact that theyare architecture epitomized as it were. A building consists of a wallupholding a roof: support and weight. The type of the first is thecolumn, which may be conceived of as a condensed section of wall; andof the second, the lintel, which may be conceived of as a condensedsection of roof. The column, being vertical, is Yo; the lintel, beinghorizontal, is In. To mark an entablature with horizontal lines in theform of mouldings, and the columns with vertical lines in the formof flutes, as is done in all the "classic orders, " is a gainin functional and sex expressiveness, and consequently in art(Illustration 11). 

[Illustration 10: CORINTHIAN MODILLION; CLASSIC CONSOLE; IONIC CAP]

The column is again divided into the shaft, which is Yo; and thecapital, which is In. The capital is itself twofold, consisting of acurved member and an angular member. These two appear in their utmostsimplicity in the _echinus_ (In) and the _abacus_ (Yo) of aGreek Doric cap. The former was adorned with painted leaf forms, characteristically feminine, and the latter with the angular fretand meander (Illustration 12). The Ionic capital, belonging to a morefeminine style, exhibits the abacus subordinated to that beautifulcushion-shaped member with its two spirally marked volutes. This, though a less rational and expressive form for its particular officethan is the echinus of the Doric cap, is a far more perfect symbol ofthe feminine element in nature. There is an essential identity betweenthe Ionic cap and the classic console before referred to--althoughsuperficially the two do not resemble each other--for a straight lineand a double spiral are elements common to both (Illustration 10). The Corinthian capital consists of an ordered mass of delicatelysculptured leaf and scroll forms sustaining an abacus which thoughrelatively masculine is yet more curved and feminine than that of anyother style. In the caulicole of a Corinthian cap In and Yo are againcontrasted. In the unique and exquisite capital from the Tower of theWinds at Athens, the two are well suggested in the simple, erect, andpointed leaf forms of the upper part, contrasted with the complex, deliquescent, rounded ones from which they spring. The essentialidentity of principle subsisting between this cap and the Renaissancebaluster by San Gallo is easily seen (Illustration 13). 

[Illustration 11]

[Illustration 12]

This law of sex-expressiveness is of such universality that it canbe made the basis of an analysis of the architectural ornament of anystyle or period. It is more than mere opposition and contrast. The eggand tongue motif, which has persisted throughout so many centuries andsurvived so many styles, exhibits an alternation of forms resemblingphallic emblems. Yo and In are well suggested in the channeledtriglyphs and the sculptured metopes of a Doric frieze, in thestraight and vertical mullions and the flowing tracery of Gothicwindows, in the banded torus, the bead and reel, and other familiarornamented mouldings (Illustrations 14, 15, 16). 

There are indications that at some time during the developmentof Gothic architecture in France, this sex-distinction became arecognized principle, moulding and modifying the design of a cathedralin much the same way that sex modifies bodily structure. The masonicguilds of the Middle Ages were custodians of the esoteric--which isthe theosophic--side of the Christian faith, and every student ofesotericism knows how fundamental and how far-reaching is this idea ofsex. 

[Illustration 13: CAPITAL FROM THE TOWER OF THE WINDS, ATHENS;CORINTHIAN CAP FROM HADRIAN BUILDINGS, ATHENS; ROSETTE FROM TEMPLE OFMARS, ROME; CAULICULUS OF CORINTHIAN CAP; BULUSTER BY SAN GALLO]

[Illustration 14: EGG AND TONGUE; BEAD AND REEL; BANDED TORUS]

The entire cathedral symbolized the crucified body of Christ; its twotowers, man and woman--that Adam and that Eve for whose redemptionaccording to current teaching Christ suffered and was crucified. Thenorth or right-hand tower ("the man's side") was called the sacredmale pillar, Jachin; and the south, or left-hand tower ("the woman'sside"), the sacred female pillar, Boaz, from the two columns flankingthe gate to Solomon's Temple--itself an allegory to the bodily temple. In only a few of the French cathedrals is this distinction clearlyand consistently maintained, and of these Tours forms perhaps the mostremarkable example, for in its flamboyant façade, over and above thedifference in actual breadth and apparent sturdiness of the two towers(the south being the more slender and delicate), there is a clearlymarked distinction in the character of the ornamentation, that of thenorth tower being more salient, angular, radial--more masculine inpoint of fact (Illustration 17). In Notre Dame, the cathedral ofParis, as in the cathedral of Tours, the north tower is perceptiblybroader than the south. The only other important difference appears tobe in the angular label-mould above the north entrance: whatever mayhave been its original function or significance, it serves to definethe tower sexually, so to speak, as effectively as does the beard ona man's face. In Amiens the north tower is taller than the south, andmore massive in its upper stages. The only traceable indication ofsex in the ornamentation occurs in the spandrels at the sides of theentrance arches: those of the north tower containing single circles, and those of the south tower containing two in one. This difference, small as it may seem, is significant, for in Europe during the MiddleAges, just as anciently in Egypt and again in Greece--in fact whereverand whenever the Secret Doctrine was known--sex was attributed tonumbers, odd numbers being conceived of as masculine, and even, asfeminine. Two, the first feminine number, thus became a symbol offemininity, accepted as such so universally at the time the cathedralswere built, that two strokes of a bell announced the death of a woman, three, the death of a man. 

[Illustration 15: FRIEZE OF THE FARNESE PALACE; ROMAN CONSOLE. VATICANMUSEUM; FRIEZE IN THE EMPIRE STYLE BY PERCIER AND FONTAINE. FRIEZE FROM THE TEMPLE OF VESTA AT TIVOLI. (ROMAN); ROMAN DORICFRIEZE--VIGNOLE]

[Illustration 16]

[Illustration 17]

The vital, organic quality so conspicuous in the best Gothicarchitecture has been attributed to the fact that necessity determinedits characteristic forms. Professor Goodyear has demonstrated thatit may be due also in part to certain subtle vertical leans andhorizontal bends; and to nicely calculated variations from strictuniformity, which find their analogue in nature, where structure isseldom rigidly geometrical. The author hazards the theory that stillanother reason why a Gothic cathedral seems so living a thing isbecause it abounds in contrasts between what, for lack of moredescriptive adjectives, he is forced to call masculine and feminineforms. 

[Illustration 18]

[Illustration 19]

Ruskin says, in _Stones of Venice_, "All good Gothic is nothing morethan the development, in various ways, and on every conceivable scale, of the group formed by the pointed arch for the bearing line below, and the gable for the protecting line above, and from the huge, gray, shaly slope of the cathedral roof, with its elastic pointed vaultsbeneath, to the crown-like points that enrich the smallest niche ofits doorway, one law and one expression will be found in all. Themodes of support and of decoration are infinitely various, but thereal character of the building, in all good Gothic, depends on thesingle lines of the gable over the pointed arch endlessly rearrangedand repeated. " These two, an angular and a curved form, like theeverywhere recurring column and lintel of classic architecture, are but presentments of Yo and In (Illustration 18). Every Gothictraceried window, with straight and vertical mullions in therectangle, losing themselves in the intricate foliations of the arch, celebrates the marriage of this ever diverse pair. The circle and thetriangle are the In and Yo of Gothic tracery, its Eve and Adam, asit were, for from their union springs that progeny of trefoil, quatrefoil, cinquefoil, of shapes flowing like water, and shapesdarting like flame, which makes such visible music to the entrancedear. 

[Illustration 20: SAN GIMIGNANO S. JACOPO. ]

By seeking to discover In and Yo in their myriad manifestations, bylearning to discriminate between them, and by attempting to expresstheir characteristic qualities in new forms of beauty--fromthe disposition of a façade to the shaping of a moulding--thearchitectural designer will charge his work with that esotericsignificance, that excess of beauty, by which architecture risesto the dignity of a "fine" art (Illustrations 19, 20). In so doing, however, he should never forget, and the layman also should everremember, that the supreme architectural excellence is fitness, appropriateness, the perfect adaptation of means to ends, and theadequate expression of both means and ends. These two aims, the oneabstract and universal, the other concrete and individual, can alwaysbe combined, just as in every human countenance are combined a type, which is universal, and a character, which is individual. 

III

CHANGELESS CHANGE

TRINITY, CONSONANCE, DIVERSITY IN MONOTONY, BALANCE, RHYTHMIC CHANGE, RADIATION

The preceding essay was devoted for the most part to that "inevitableduality" which finds concrete expression in countless pairs ofopposites, such as day and night, fire and water, man and woman;in the art of music by two chords, one of suspense and the other offulfilment; in speech by vowel and consonant sounds, epitomized in _a_and in _m_; in painting by warm colors and cold, epitomized in redand blue; in architecture by the vertical column and the horizontallintel, by void and solid--and so on. 

TRINITY

This concept should now be modified by another, namely, that in everyduality a third is latent; that two implies three, for each sex soto speak is in process of becoming the other, and this alternationengenders and is accomplished by means of a third term or neuter, which is like neither of the original two but partakes of the natureof them both, just as a child may resemble both its parents. Twilightcomes between day and night; earth is the child of fire and water;in music, besides the chord of longing and striving, and the chord ofrest and satisfaction (the dominant seventh and the tonic), there isa third or resolving chord in which the two are reconciled. In thesacred syllable Om (_Aum_), which epitomizes all speech, the _u_ soundeffects a transition between the _a_ sound and the _m_; among theso-called primary colors yellow comes between red and blue; and inarchitecture the arch, which is both weight and support, which isneither vertical nor horizontal, may be considered the neuter of thegroup of which the column and the lintel are respectively masculineand feminine. "These are the three, " says Mr. Louis Sullivan, "theonly three letters from which has been expanded the architectural art, as a great and superb language wherewith man has expressed, throughthe generations, the changing drift of his thoughts. "

[Illustration 21: THE LAW OF TRINITY. A ROMAN IONIC ARCADE, BYVIGNOLE. --THE COLUMN, THE ENTABLATURE, AND THE ARCH CORRESPOND TO LINESVERTICAL, HORIZONTAL AND CURVED. ]

It would be supererogatory to dwell at any length on this "trinityof manifestation" as the concrete expression of that unmanifest andmystical trinity, that _three-in-one_ which under various names occursin every world-religion, where, defying definition, it was wontto find expression symbolically in some combination of vertical, horizontal and curved lines. The anstated cross of the Egyptiansis such a symbol, the Buddhist wheel, and the fylfot or swastikainscribed within a circle, also those numerous Christian symbolscombining the circle and the cross. Such ideographs have spelledprofound meaning to the thinkers of past ages. We of to-day are notgiven to discovering anything wonderful in three strokes of a pen, but every artist in the weaving of his pattern must needs employ thesemystic symbols in one form or another, and if he employs them witha full sense of their hidden meaning his work will be apt to gainin originality and beauty--for originality is a new and personalperception of beauty, and beauty is the name we give to truth wecannot understand. 

In architecture, this trinity of vertical, horizontal and curvedlines finds admirable illustration in the application of columns andentablature to an arch and impost construction, so common in Roman andRenaissance work. This is a redundancy, and finds no justification inreason, because the weight is sustained by the arch, and the "order"is an appendage merely; yet the combination, illogical as it is, satisfies the sense of beauty because the arch effects a transitionbetween the columns and the entablature, and completes the trinityof vertical, horizontal and curved lines (Illustration 21). In theentrances to many of the Gothic cathedrals and churches the sameelements are better because more logically disposed. Here thehorizontal lintel and its vertical supports are not decorative merely, but really perform their proper functions, while the arch, too, hasa raison d'être in that it serves to relieve the lintel of thesuperincumbent weight of masonry. The same arrangement sometimesoccurs in classic architecture also, as when an opening spanned by asingle arch is subdivided by means of an order (Illustration 22). 

Three is pre-eminently the number of architecture, because it is thenumber of space, which for us is three-dimensional, and of all thearts architecture is most concerned with the expression of spatialrelations. The division of a composition into three related parts isso universal that it would seem to be the result of an instinctiveaction of the human mind. The twin pylons of an Egyptian temple withits entrance between, for a third division, has its correspondence inthe two towers of a Gothic cathedral and the intervening screenwall of the nave. In the palaces of the Renaissance a threefolddivision--vertically by means of quoins or pilasters, and horizontallyby means of cornices or string courses--was common, as was also thedivision into a principal and two subordinate masses (Illustration23). 

[Illustration 22: THE LAW OF TRINITY. THE TRINITY OF HORIZONTAL VERTICALAND CURVED LINES. ]

The architectural "orders" are divided threefold into pedestal orstylobate, column and entablature; and each of these is again dividedthreefold: the first into plinth, die and cornice; the second intobase, shaft and capital; the third into architrave, frieze andcornice. In many cases these again lend themselves to a threefoldsubdivision. A more detailed analysis of the capitals already shown tobe twofold reveals a third member: in the Greek Doric this consistsof the annulets immediately below the abacus; in the other orders, thenecking which divides the shaft from the cap. 

CONSONANCE

"As is the small, so is the great" is a perpetually recurring phrasein the literature of theosophy, and naturally so, for it is a succinctstatement of a fundamental and far-reaching truth. The scientistrecognizes it now and then and here and there, but the occultisttrusts it always and utterly. To him the microcosm and the macrocosmare one and the same in essence, and the forth-going impulsewhich calls a universe into being and the indrawing impulse whichextinguishes it again, each lasting millions of years, are echoed andrepeated in the inflow and outflow of the breath through the nostrils, in nutrition and excretion, in daily activity and nightly rest, inthat longer day which we name a lifetime, and that longer rest in_Devachan_--and so on until time itself is transcended. 

[Illustration 23]

In the same way, in nature, a thing is echoed and repeated throughoutits parts. Each leaf on a tree is itself a tree in miniature, eachblossom a modified leaf; every vertebrate animal is a complicatedsystem of spines; the ripple is the wave of a larger wave, and thatlarger wave is a part of the ebbing and flowing tide. In music thislaw is illustrated in the return of the tonic to itself in the octave, and its partial return in the dominant; also in a more extended sensein the repetition of a major theme in the minor, or in the treble andagain in the bass, with modifications perhaps of time and key. Inthe art of painting the law is exemplified in the repetition withvariation of certain colors and combinations of lines in differentparts of the same picture, so disposed as to lead the eye to somefocal point. Every painter knows that any important color in hispicture must be echoed, as it were, in different places, for harmonyof the whole. 

[Illustration 24]

In the drama the repetition of a speech or of an entire scene, butunder circumstances which give it a different meaning, is often mosteffective, as when Gratiano, in the trial scene of _The Merchantof Venice_ taunts Shylock with his own words, "A Daniel come tojudgment!" or, as when in one of the later scenes of _As You Like It_an earlier scene is repeated, but with Rosalind speaking in her properperson and no longer as the boy Ganymede. 

These recurrences, these inner consonances, these repetitions withvariations are common in architecture also. The channeled triglyphs ofa Greek Doric frieze echo the fluted columns below (Illustration 24). The balustrade which crowns a colonnade is a repetition, in some sort, of the colonnade itself. The modillions of a Corinthian cornice arebut elaborated and embellished dentils. Each pinnacle of a Gothiccathedral is a little tower with its spire. As Ruskin has pointed out, the great vault of the cathedral nave, together with the pointed roofabove it, is repeated in the entrance arch with its gable, and thesame two elements appear in every statue-enshrining niche of thedoorway. In classic architecture, as has been shown, instead of thearch and gable, the column and entablature everywhere recur underdifferent forms. The minor domes which flank the great dome of thecathedral of Florence enhance and reinforce the latter, and preparethe eye for a climax which would otherwise be too abrupt. The centralpavilion of the Château Maintenon, with its two turrets, echoes theentire façade with its two towers. Like the overture to an opera, itintroduces themes which find a more extended development elsewhere(Illustration 26). 

[Illustration 25]

[Illustration 26]

[Illustration 27]

This law of Consonance is operative in architecture more obscurelyin the form of recurring numerical ratios, identical geometricaldetermining figures, parallel diagonals and the like, which will bediscussed in a subsequent essay. It has also to do with style andscale, the adherence to substantially one method of construction andmanner of ornament, just as in music the key, or chosen series ofnotes, may not be departed from except through proper modulations, orin a specific manner. 

Thus it is seen that in a work of art, as in a piece of tapestry, the same thread runs through the web, but goes to make up differentfigures. The idea is deeply theosophic: one life, many manifestations;hence, inevitably, echoes, resemblances--_Consonance_. 

DIVERSITY IN MONOTONY

Another principle of natural beauty, closely allied to the foregoing, its complement as it were, is that of _Diversity in Monotony_--notidentity, but difference. It shows itself for the most part as aperceptible and piquant variation between individual units belongingto the same class, type, or species. 

No two trees put forth their branches in just the same manner, and notwo leaves from the same tree exactly correspond; no two personslook alike, though they have similiar members and features; even themarkings on the skin of the thumb are different in every human hand. Browning says, "As like as a hand to another hand! Whoever said that foolish thing, Could not have studied to understand--"

Now every principle of natural beauty is but the presentment of someoccult law, some theosophical truth; and this law of Diversity inMonotony is the presentment of the truth that identity does notexclude difference. The law is binding, yet the will is free: all menare brothers united by the ties of brotherhood, yet each is unique, afree agent, and never so free as when most bound by the Good Law. Thistruth nature beautifully proclaims, and art also. In architecture itis admirably exemplified in the metopes of the Parthenon frieze: seenat a distance these must have presented a scarcely distinguishabletexture of sunlit marble and cool shadow, yet in reality each isa separate work of art. So with the capitals of the columns of thewonderful sea-arcade of the Venetian Ducal palace: alike in generalcontour they differ widely in detail, and unfold a Bible story. In Gothic cathedrals, in Romanesque monastery cloisters, a teemingvariety of invention is hidden beneath apparent uniformity. Thegargoyles of Notre Dame make similiar silhouettes against the sky, but seen near at hand what a menagerie of monsters! The same spirit ofcontrolled individuality, of liberty subservient to the law of all, isexemplified in the bases of the columns of the temple of Apollo nearMiletus--each one a separate masterpiece of various ornamentationadorning an established architectural form (Illustration 28). 

[Illustration 28]

[Illustration 29]

The builders of the early Italian churches, instinctively obeying thislaw of Diversity in Monotony, varied the size of the arches in thesame arcade (Illustration 29), and that this was an effect of art andnot of accident or carelessness Ruskin long ago discovered, and theBrooklyn Institute surveys have amply confirmed his view. Althoughby these means the builders of that day produced effects of deceptiveperspective, of subtle concord and contrast, their sheer hatred ofmonotony and meaningless repetition may have led them to diversifytheir arcades in the manner described, for a rigidly equal and regulardivision lacks interest and vitality. 

BALANCE

If one were to establish an axial plane vertically through the centerof a tree, in most cases it would be found that the masses of foliage, however irregularly shaped on either side of such an axis, just aboutbalanced each other. Similarly, in all our bodily movements, for everychange of equilibrium there occurs an opposition and adjustment ofmembers of such a nature that an axial plane through the center ofgravity would divide the body into two substantially equal masses, as in the case of the tree. This physical plane law of Balanceshows itself for the most part on the human plane as the law ofCompensation, whereby, to the vision of the occultist, all accountsare "squared, " so to speak. It is in effect the law of Justice, aptlysymbolized by the scales. 

The law of Balance finds abundant illustration in art: in music bythe opposition, the answering, of one phrase by another of the sameelements and the same length, but involving a different sequence ofintervals; in painting by the disposition of masses in such a way thatthey about equalize one another, so that there is no sense of "strain"in the composition. 

In architecture the common and obvious recognition of the law ofBalance is in the symmetrical disposition of the elements, whether ofplan or of elevation, on either side of axial lines. A far more subtleand vital illustration of the law occurs when the opposed elements donot exactly match, but differ from each other, as in the case of thetwo towers of Amiens, for example. This sort of balance may be said tobe characteristic of Gothic, as symmetry is characteristic of Classic, architecture. 

RHYTHMIC CHANGE

There is in nature a universal tendency toward refinement andcompactness of form in space, or contrariwise, toward incrementand diffusion; and this manifests itself in time as acceleration orretardation. It is governed, in either case, by an exact mathematicallaw, like the law of falling bodies. It shows itself in the wideningcircles which appear when one drops a stone into still water, in theconvolutions of shells, in the branching of trees and the veiningof leaves; the diminution in the size of the pipes of an organillustrates it, and the spacing of the frets of a guitar. More andmore science is coming to recognize, what theosophy affirms, that thespiral vortex, which so beautifully illustrates this law, both in itstime and its space aspects is the universal archetype, the pattern ofall that is, has been, or will be, since it is the form assumed by theultimate physical atom, and the ultimate physical atom is the physicalcosmos in miniature. 

This Rhythmic Diminution is everywhere: it is in the eye itself, forany series of mathematically equal units, such for example as thecolumns and intercolumnations of a colonnade, become when seenin perspective rhythmically unequal, diminishing according to theuniversal law. The entasis of a Classic column is determined by thislaw, the spirals of the Ionic volute, the annulets of the Parthenoncap, obey it (Illustration 30). 

In recognition of the same principle of Rhythmic Diminution a buildingis often made to grow, or appear to grow lighter, more intricate, finer, from the ground upward, an end attained by various devices, one of the most common being the employment of the more attenuatedand highly ornamented orders above the simpler and sturdier, as in theRoman Colosseum, or in the Palazzo Uguccioni, in Florence--to mentiononly two examples out of a great number. In the Riccardi Palacean effect of increasing refinement is obtained by diminishing theboldness of the rustication of the ashlar in successive stories; inthe Farnese, by the gradual reduction of the size of the angle quoins(Illustration 30). In an Egyptian pylon it is achieved most simply bybattering the wall; in a Gothic cathedral most elaborately by akind of segregation, or breaking up, analogous to that which a treeundergoes--the strong, relatively unbroken base corresponding tothe trunk, the diminishing buttresses to the tapering limbs, andthe multitude of delicate pinnacles and crockets, to the outermostbranches and twigs, seen against the sky. 

RADIATION

The final principle of natural beauty to which the author would callattention is the law of _Radiation_, which is in a manner a returnto the first, the law of _Unity_. The various parts of any organismradiate from, or otherwise refer back to common centers, or foci, and these to centers of their own. The law is represented in itssimplicity in the star-fish, in its complexity in the body of man; atree springs from a seed, the solar system centers in the sun. 

The idea here expressed by the term "radiation" is a familiar oneto all students of theosophy. The Logos radiates his life and lightthroughout his universe, bringing into activity a host of entitieswhich become themselves radial centers; these generate still others, and so on endlessly. This principle, like every other, patientlypublishes itself to us, unheeding, everywhere in nature, and inall great art as well; it is a law of optics, for example, that allstraight lines having a common direction if sufficiently prolongedappear to meet in a point, i. E. , radiate from it (Illustration 31). Leonardo da Vinci employed this principle of perspective in his LastSupper to draw the spectator's eye to the picture's central figure, the point of sight toward which the lines of the walls and ceilingconverge centering in the head of Christ. Puvis de Chavannes, in hisBoston Library decoration, leads the eye by a system of triangulationto the small figure of the Genius of Enlightenment above the centraldoor (Illustration 32); and Ruskin, in his _Elements of Drawing_, hasshown how artfully Turner arranged some of his compositions to attractattention to a focal point. 

This law of Radiation enters largely into architecture. The Colosseum, based upon the ellipse, a figure generated from two points or foci, and the Pantheon, based upon the circle, a figure generated from acentral point, are familiar examples. The distinctive characteristicof Gothic construction, the concentration or focalization of theweight of the vaults and arches at certain points, is anotherillustration of the same principle applied to architecture, beautifully exemplified in the semicircular apse of a cathedral, wherethe lines of the plan converge to a common center, and the ribs of thevaulting meet upon the capitals of the piers and columns, seemingto radiate thence to still other centers in the loftier vaults whichfinally meet in a center common to all. 

[Illustration 30]

[Illustration 31]

[Illustration 32]

The tracery of the great roses, high up in the façades of thecathedrals of Paris and of Amiens, illustrate Radiation, in the onecase masculine: straight, angular, direct; in the feminine: curved, flowing, sinuous. The same _Beautiful Necessity_ determined thecharacteristics of much of the ornament of widely separated stylesand periods: the Egyptian lotus, the Greek honeysuckle, the Romanacanthus, Gothic leaf work--to snatch at random four blossoms fromthe sheaf of time. The radial principle still inherent in the debasedornament of the late Renaissance gives that ornament a unity, acoherence, and a kind of beauty all its own (Illustration 35). 

[Illustration 33]

[Illustration 34]

Such are a few of the more obvious laws of natural beauty and theirapplication to the art of architecture. The list is by no meansexhausted, but it is not the multiplicity and diversity of these lawswhich is important to keep in mind, so much as their relatedness andcoördination, for they are but different aspects of the One Law, thatwhereby the Logos manifests in time and space. A brief recapitulationwill serve to make this correlation plain, and at the same time fixwhat has been written more firmly in the reader's mind. 

[Illustration 35]

[Illustration 36]

First comes the law of _Unity_; then, since every unit is in itsessence twofold, there is the law of _Polarity_; but this duality isnot static but dynamic, the two parts acting and reacting upon oneanother to produce a third--hence the law of _Trinity_. Given thisthird term, and the innumerable combinations made possible byits relations to and reactions upon the original pair, the lawof _Multiplicity in Unity_ naturally follows, as does the lawof _Consonance_, or repetition, since the primal process ofdifferentiation tends to repeat itself, and the original combinationsto reappear--but to reappear in changed form, hence the law of_Diversity in Monotony_. The law of _Balance_ is seen to be but amodification of the law of Polarity, and since all things are waxingand waning, there is the law whereby they wax and wane, that of_Rhythmic Change_. _Radiation_ rediscovers and reaffirms, even in theutmost complexity, that essential and fundamental unity from whichcomplexity was wrought. 

Everything, beautiful or ugly, obeys and illustrates one or another ofthese laws, so universal are they, so inseparably attendant upon everykind of manifestation in time and space. It is the number of themwhich finds illustration within small compass, and the aptness andcompleteness of such illustration, which makes for beauty, becausebeauty is the fine flower of a sort of sublime ingenuity. A work ofart is nothing if not _artful_: like an acrostic, the more differentways it can be read--up, down, across, from right to left and fromleft to right--the better it is, other things being equal. Thisstatement, of course, may be construed in such a way as to appearabsurd; what is meant is simply that the more a work of art isfreighted and fraught with meaning beyond meaning, the more secure itsimmortality, the more powerful its appeal. For enjoyment, it is notnecessary that all these meanings should be fathomed, it is onlynecessary that they should be felt. 

Consider for a moment the manner in which Leonardo da Vinci's LastSupper, an acknowledged masterpiece, conforms to everyone of the lawsof beauty enumerated above (Illustration 32). It illustrates the lawof Unity in that it movingly portrays a single significant episode inthe life of Christ. The eye is led to dwell upon the central personageof this drama by many artful expedients: the visible part of thefigure of Christ conforms to the lines of an equilateral triangleplaced exactly in the center of the picture; the figure is separatedby a considerable space from the groups of the disciples on eitherhand, and stands relieved against the largest parallelogram of light, and the vanishing point of the perspective is in the head of Christ, at the apex, therefore, of the triangle. The law of Polarity findsfulfilment in the complex and flowing lines of the draped figurescontrasted with the simple parallelogram of the cloth-covered table, and the severe architecture of the room. The law of Trinity isexemplified in the three windows, and in the subdivision of the twelvefigures of the disciples into four groups of three figures each. Thelaw of Consonance appears in the repetition of the horizontal linesof the table in the ceiling above; and in the central triangle beforereferred to, continued and echoed, as it were, in the triangularsupports of the table visible underneath the cloth. The law ofDiversity in Monotony is illustrated in the varying disposition of theheads of the figures in the four groups of three; the law of Balancein the essential symmetry of the entire composition; the law ofRhythmic Change in the diminishing of the wall and ceiling spaces; andthe law of Radiation in the convergence of all the perspective linesto a single significant point. 

To illustrate further the universality of these laws, consider nowtheir application to a single work of architecture: the Taj Mahal, oneof the most beautiful buildings of the world (Illustration 36). It isa unit, but twofold, for it consists of a curved part and an angularpart, roughly figured as an inverted cup upon a cube; each of these(seen in parallel perspective, at the end of the principal vista) isthreefold, for there are two sides and a central parallelogram, andtwo lesser domes flank the great dome. The composition is rich inconsonances, for the side arches echo the central one, the subordinatedomes the great dome, and the lanterns of the outstanding minaretsrepeat the principal motif. Diversity in Monotony appears abundantlyin the ornament, which is intricate and infinitely various; the law ofBalance is everywhere operative in the symmetry of the entire design. Rhythmic Change appears in the tapering of the minarets, the outlinesof the domes and their mass relations to one another; and finally, the whole effect is of radiation from a central point, of elementsdisposed on radial lines. 

It would be fatuous to contend that the prime object of a work ofarchitecture is to obey and illustrate these laws. The prime object ofa work of architecture is to fulfill certain definite conditions in apractical, economical, and admirable way, and in fulfilling to expressas far as possible these conditions, making the form express thefunction. The architect who is also an artist however will do thisand something beyond: working for the most part unconsciously, harmoniously, joyously, his building will obey and illustrate naturallaws--these laws of beauty--and to the extent it does so it will be awork of art; for art is the method of nature carried into those higherregions of thought and feeling which man alone inhabits: regions whichit is one of the purposes of theosophy to explore. 

IV

THE BODILY TEMPLE

Carlyle says: "There is but one temple in the world, and that is thebody of man. " If the body is, as he declares, a temple, it is not lesstrue that a temple or any work of architectural art is a larger bodywhich man has created for his uses, just as the individual self ishoused within its stronghold of flesh and bones. Architectural beautylike human beauty depends upon the proper subordination of partsto the whole, the harmonious interrelation between these parts, theexpressiveness of each of its function or functions, and when theseare many and diverse, their reconcilement one with another. This beingso, a study of the human figure with a view to analyzing the sourcesof its beauty cannot fail to be profitable. Pursued intelligently, such a study will stimulate the mind to a perception of those simpleyet subtle laws according to which nature everywhere works, andit will educate the eye in the finest known school of proportion, training it to distinguish minute differences, in the same way thatthe hearing of good music cultivates the ear. 

Those principles of natural beauty which formed the subject of thetwo preceding essays are all exemplified in the ideally perfect humanfigure. Though essentially a unit, there is a well marked divisioninto right and left--"Hands to hands, and feet to feet, in one bodygrooms and brides. " There are two arms, two legs, two ears, two eyes, and two lids to each eye; the nose has two nostrils, the mouth hastwo lips. Moreover, the terms of such pairs are masculine and femininewith respect to each other, one being active and the other passive. Owing to the great size and one-sided position of the liver, the righthalf of the body is heavier than the left; the right arm is usuallylonger and more muscular than the left; the right eye is slightlyhigher than its fellow. In speaking and eating the lower jaw and underlip are active and mobile with relation to the upper; in winking it isthe upper eyelid which is the more active. That "inevitable duality"which is exhibited in the form of the body characterizes its motionsalso. In the act of walking for example, a forward movement isattained by means of a forward and a backward movement of the thighson the axis of the hips; this leg movement becomes twofold again belowthe knee, and the feet move up and down independently on the axis ofthe ankle. A similar progression is followed in raising the arm andhand: motion is communicated first to the larger parts, through themto the smaller and thence to the extremities, becoming more rapid andcomplex as it progresses, so that all free and natural movements ofthe limbs describe invisible lines of beauty in the air. Coexistentwith this pervasive duality there is a threefold division of thefigure into trunk, head and limbs: a superior trinity of head andarms, and an inferior trinity of trunk and legs. The limbs are dividedthreefold into upper-arm, forearm and hand; thigh, leg and foot. Thehand flowers out into fingers and the foot into toes, each with athreefold articulation; and in this way is effected that transitionfrom unity to multiplicity, from simplicity to complexity, whichappears to be so universal throughout nature, and of which a tree isthe perfect symbol. 

[Illustration 37: THE LAW OF RHYTHMIC DIMINUTION ILLUSTRATED IN THETAPERING BODY, LIMBS, FINGERS & TOES. ]

[Illustration 38]

[Illustration 39]

The body is rich in veiled repetitions, echoes, _consonances_. Thehead and arms are in a sense a refinement upon the trunk and legs, there being a clearly traceable correspondence between their variousparts. The hand is the body in little--_"Your soft hand is a woman ofitself"_--the palm, the trunk; the four fingers, the four limbs; andthe thumb, the head;-each finger is a little arm, each finger tip alittle palm. The lips are the lids of the mouth, the lids are thelips of the eyes--and so on. The law of _Rhythmic Diminution_ isillustrated in the tapering of the entire body and of the limbs, inthe graduated sizes and lengths of the palm and the toes, and inthe successively decreasing length of the palm and the joints of thefingers, so that in closing the hand the fingers describe naturalspirals (Illustrations 37, 38). Finally, the limbs radiate as it werefrom the trunk, the fingers from a point in the wrist, the toes froma point in the ankle. The ribs radiate from the spinal column like theveins of a leaf from its midrib (Illustration 39). 

[Illustration 40]

The relation of these laws of beauty to the art of architecture hasbeen shown already. They are reiterated here only to show that man isindeed the microcosm--a little world fashioned from the same elementsand in accordance with the same _Beautiful Necessity_ as is thegreater world in which he dwells. When he builds a house or temple hebuilds it not literally in his own image, but according to the laws ofhis own being, and there are correspondences not altogether fancifulbetween the animate body of flesh and the inanimate body of stone. Dowe not all of us, consciously or unconsciously, recognize the factof character and physiognomy in buildings? Are they not, to ourimagination, masculine or feminine, winning or forbidding--_human_, in point of fact--to a greater degree than anything else of man'screating? They are this certainly to a true lover and student ofarchitecture. Seen from a distance the great French cathedrals appearlike crouching monsters, half beast, half human: the two towers standlike a man and a woman, mysterious and gigantic, looking out over cityand plain. The campaniles of Italy rise above the churches and houseslike the sentinels of a sleeping camp--nor is their strangely humanaspect wholly imaginary: these giants of mountain and campagna haveeyes and brazen tongues; rising four square, story above story, witha belfry or lookout, like a head, atop, their likeness to a man is notinfrequently enhanced by a certain identity of proportion--of ratio, that is, of height to width: Giotto's beautiful tower is an example. The caryatid is a supporting member in the form of a woman; in theIonic column we discern her stiffened, like Lot's wife, into a pillar, with nothing to show her feminine but the spirals of her beautifulhair. The columns which uphold the pediment of the Parthenon areunmistakably masculine: the ratio of their breadth to their height isthe ratio of the breadth to the height of a man (Illustration 40). 

[Illustration 41: THE BODY THE ARCHETYPE OF SACRED EDIFICES. ]

[Illustration 42: THE VESICA PISCIS AND THE PLAN OF CHARTRES. ]

At certain periods of the world's history, periods of mysticalenlightenment, men have been wont to use the human figure, the soul'stemple, as a sort of archetype for sacred edifices (Illustration 41). The colossi, with calm inscrutable faces, which flank the entrance toEgyptian temples; the great bronze Buddha of Japan, with its dreamingeyes; the little known colossal figures of India and China--all thesebelong scarcely less to the domain of architecture than of sculpture. The relation above referred to however is a matter more subtle andoccult than mere obvious imitation on a large scale, being based uponsome correspondence of parts, or similarity of proportions, or both. The correspondence between the innermost sanctuary or shrine of atemple and the heart of a man, and between the gates of that templeand the organs of sense is sufficiently obvious, and a relation onceestablished, the idea is susceptible of almost infinite development. That the ancients proportioned their temples from the human figureis no new idea, nor is it at all surprising. The sculpture of theEgyptians and the Greeks reveals the fact that they studied the bodyabstractly, in its exterior presentment. It is clear that the rulesof its proportions must have been established for sculpture, and it isnot unreasonable to suppose that they became canonical in architecturealso. Vitruvius and Alberti both lay stress on the fact that allsacred buildings should be founded on the proportions of the humanbody. 

[Illustration 43: A GOTHIC CATHEDRAL THE SYMBOL OF THE BODY OF JESUSCHRIST]

[Illustration 44: THE SYMBOLISM OF A GOTHIC CATHEDRAL FROM THEROSICRUCIANS: HARGRAVE JENNINGS]

In France, during the Middle Ages, a Gothic cathedral became, at thehands of the secret masonic guilds, a glorified symbol of the bodyof Christ. To practical-minded students of architectural history, familiar with the slow and halting evolution of a Gothic cathedralfrom a Roman basilica, such an idea may seem to be only themaunderings of a mystical imagination, a theory evolved from the innerconsciousness, entitled to no more consideration than the familiarfallacy that vaulted nave of a Gothic church was an attempt to imitatethe green aisles of a forest. It should be remembered however that thehabit of the thought of that time was mystical, as that of our own ageis utilitarian and scientific; and the chosen language of mysticism isalways an elaborate and involved symbolism. What could be more naturalthan that a building devoted to the worship of a crucified Saviorshould be made a symbol, not of the cross only, but of the bodycrucified?

[Illustration 45: THE GEOMETRICAL BASIS OF THE HUMAN FIGURE]

[Illustration 46]

The _vesica piscis_ (a figure formed by the developing arcs of twoequilateral triangles having a common side) which in so many casesseems to have determined the main proportion of a cathedral plan--theinterior length and width across the transepts--appears as an aureolearound the figure of Christ in early representations, a fact whichcertainly 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 conceptionof the symbolism of a cathedral. A copy of it is here given. The apseis seen to correspond to the head of Christ, the north transept to hisright 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). 

[Illustration 47]

The cathedral builders excelled all others in the artfulness withwhich they established and maintained a relation between theirarchitecture and the stature of a man. This is perhaps one reason whythe French and English cathedrals, even those of moderate dimensionsare more truly impressive than even the largest of the greatRenaissance structures, such as St. Peter's in Rome. A gigantic orderfurnishes no true measure for the eye: its vastness is revealedonly by the accident of some human presence which forms a basisof comparison. That architecture is not necessarily the mostawe-inspiring which gives the impression of having been built bygiants for the abode of pigmies; like the other arts, architecture ishighest when it is most human. The mediæval builders, true to thisdictum, employed stones of a size proportionate to the strength ofa man working without unusual mechanical aids; the great piers andcolumns, built up of many such stones, were commonly subdividedinto clusters, and the circumference of each shaft of such a clusterapproximated the girth of a man; by this device the moulding of thebase and the foliation of the caps were easily kept in scale. Wherevera balustrade occurred it was proportioned not with relation to theheight of the wall or the column below, as in classic architecture, but with relation to a man's stature. 

[Illustration 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 andenduring to relate it to the human figure, if it be only a flight ofsteps in which each one is the measure of a stride. In the Farnese, the Riccardi, the Strozzi, and many another Italian palace, the stoneseat about the base gives scale to the building because the beholderknows instinctively that the height of such a seat must have somerelation to the length of a man's leg. In the Pitti palace thebalustrade which crowns each story answers a similar purpose: itstands in no intimate relation to the gigantic arches below, but isof a height convenient for lounging elbows. The door to Giotto'scampanile reveals the true size of the tower as nothing else could, because it is so evidently related to the human figure and not to thegreat windows higher up in the shaft. 

[Illustration 49: THE MEDIÆVAL METHOD OF DRAWING THE FIGURE]

The geometrical plane figures which play the most important part inarchitectural 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 horizontallyin opposite directions, he will be inscribed, as it were, within asquare; and his arms will mark, with fair accuracy, the base of aninverted equilateral triangle, the apex of which will touch the groundat his feet. If the arms be extended upward at an angle, and thelegs correspondingly separated, the extremities will touchthe circumferences of a circle having its center in the navel(Illustrations 45, 46). 

[Illustration 50]

The figure has been variously analyzed with a view to establishingnumerical ratios between its parts (Illustrations 47, 48, 49). Someof these are so simple and easily remembered that they have obtaineda certain popular currency; such as that the length of the hand equalsthe length of the face; that the span of the horizontally extendedarms equals the height; and the well known rule that twice aroundthe wrist is once around the neck, and twice around the neck is oncearound the waist. The Roman architect Vitruvius, writing in the age ofAugustus Cæsar, formulated the important proportions of the statuesof classical antiquity, and except that he makes the head smaller thanthe normal (as it should be in heroic statuary), the ratios whichhe gives are those to which the ideally perfect male figure shouldconform. Among the ancients the foot was probably the standard of alllarge measurements, being a more determinate length than that of thehead or face, and the height was six lengths of the foot. If the headbe 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 fourparts, three of which are equal, and correspond to the lengths ofthe leg, the thigh and the trunk; while the fourth part, which istwo-thirds of one of these thirds, extends from the sternum to thecrown of the head. One excellence of such a division aside from itssimplicity, consists in the fact that it may be applied to the face aswell. The lowest of the three major divisions extends from the tip ofthe chin to the base of the nose, the next coincides with the heightof the nose (its top being level with the eyebrows), and the last withthe height of the forehead, while the remaining two-thirds of one ofthese thirds represents the horizontal projection from the beginningof the hair on the forehead to the crown of the head. The middle ofthe three larger divisions locates the ears, which are the same heightas the nose (Illustrations 45, 47). 

Such analyses of the figure, however conducted, reveals anall-pervasive harmony of parts, between which definite numericalrelations are traceable, and an apprehension of these should assistthe architectural designer to arrive at beauty of proportion bymethods of his own, not perhaps in the shape of rigid formulæ, butpresent in the consciousness as a restraining influence, acting andreacting upon the mind with a conscious intention toward rhythm andharmony. By means of such exercises, he will approach nearer to anunderstanding of that great mystery, the beauty and significance ofnumbers, of which mystery music, architecture, and the human figureare equally presentments--considered, that is, from the standpoint ofthe occultist. 

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 certainsimple geometrical solids, among which are the tetrahedron, the cube, and the sphere. This process gives rise to harmony, which may bedefined as the relation between parts and unity, the simplicity latentin the infinitely complex, the potential complexity of that which issimple. Proceeding to things visible and tangible, this indwellingharmony, rhythm, proportion, which has its basis in geometry andnumber, is seen to exist in crystals, flower forms, leaf groups, andthe like, where it is obvious; and in the more highly organized worldof the animal kingdom also; though here the geometry is latent ratherthan patent, eluding though not quite defying analysis, and thusaugmenting beauty, which like a woman is alluring in proportion as sheeludes (Illustrations 51, 52, 53). 

[Illustration 52: PROPORTIONS OF THE HORSE]

[Illustration 53]

By the true artist, in the crystal mirror of whose mind the universalharmony is focused and reflected, this secret of the cause and sourceof rhythm--that it dwells in a correlation of parts based on anultimate simplicity--is instinctively apprehended. A knowledge of itformed part of the equipment of the painters who made glorious thegolden noon of pictorial art in Italy during the Renaissance. Theproblem which preoccupied them was, as Symonds says of Leonardo, "to submit the freest play of form to simple figures of geometry ingrouping. " Alberti held that the painter should above all things havemastered geometry, and it is known that the study of perspective andkindred subjects was widespread and popular. 

[Illustration 54]

The first painter who deliberately rather than instinctively basedhis compositions on geometrical principles seems to have been FraBartolommeo, in his Last Judgment, in the church of St. Maria Nuova, in Florence. Symonds says of this picture, "Simple figures--thepyramid and triangle, upright, inverted, and interwoven like therhymes of a sonnet--form the basis of the composition. This system wasadhered to by the Fratre in all his subsequent works" (Illustration54). Raphael, with that power of assimilation which distinguisheshim among men of genius, learned from Fra Bartolommeo this methodof disposing figures and combining them in masses with almostmathematical precision. It would have been indeed surprising ifLeonardo da Vinci, in whom the artist and the man of science wereso wonderfully united, had not been greatly preoccupied with themathematics of the art of painting. His Madonna of the Rocks, andVirgin on the Lap of Saint Anne, in the Louvre, exhibit the veryperfection of pyramidal composition. It is however in his masterpiece, The Last Supper, that he combines geometrical symmetry and precisionwith perfect naturalness and freedom in the grouping of individuallyinteresting and dramatic figures. Michael Angelo, Andrea del Sarto, and the great Venetians, in whose work the art of painting may be saidto have culminated, recognized and obeyed those mathematical laws ofcomposition known to their immediate predecessors, and the decadenceof the art in the ensuing period may be traced not alone to the falsesentiment and affectation of the times, but also in the abandonment bythe artists of those obscurely geometrical arrangements and groupingswhich in the works of the greatest masters so satisfy the eye andhaunt the memory of the beholder (Illustrations 55, 56). 

[Illustration 55: THE EMPLOYMENT OF THE EQUILATERAL TRIANGLE INRENAISSANCE PAINTING]

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

[Illustration 57: ASSYRIAN; GREEK]

[Illustration 58: THE GEOMETRICAL BASIS OF THE PLAN IN ARCHITECTURALDESIGN]

[Illustration 59]

Sculpture, even more than painting, is based on geometry. The colossiof Egypt, the bas-reliefs of Assyria, the figured pediments andmetopes of the temples of Greece, the carved tombs of Revenna, the Della Robbia lunettes, the sculptured tympani of Gothic churchportals, all alike lend themselves in greater or less degree to ageometrical synopsis (Illustration 57). Whenever sculpture suffereddivorce from architecture the geometrical element became lessprominent, doubtless because of all the arts architecture is the mostclearly and closely related to geometry. Indeed, it may be said thatarchitecture is geometry made visible, in the same sense that musicis number made audible. A building is an aggregation of the commonestgeometrical forms: parallelograms, prisms, pyramids and cones--thecylinder appearing in the column, and the hemisphere in the dome. The plans likewise of the world's famous buildings reduced to theirsimplest expression are discovered to resolve themselves into a fewsimple 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 THEUNIVERSITY OF VIRGINIA]

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

[Illustration 63]

[Illustration 64: THE EQUILATERAL TRIANGLE IN ROMAN ARCHITECTURE]

But architecture is geometrical in another and a higher sense thanthis. Emerson says: "The pleasure a palace or a temple gives the eyeis that an order and a method has been communicated to stones, so thatthey speak and geometrize, become tender or sublime with expression. "All truly great and beautiful works of architecture from the Egyptianpyramids to the cathedrals of Ile-de-France--are harmoniouslyproportioned, their principal and subsidiary masses being related, sometimes obviously, more often obscurely, to certain symmetricalfigures of geometry, which though invisible to the sight and notconsciously present in the mind of the beholder, yet perform theimportant function of coördinating the entire fabric into one easilyremembered whole. Upon some such principle is surely founded whatSymonds calls "that severe and lofty art of composition which seeksthe highest beauty of design in architectural harmony supreme, abovethe 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 thebuilders of antiquity, the masonic guilds of the Middle Ages, andthe architects of the Italian Renaissance, knew and followed certainrules, but though this theory be denied or even disproved--if afterall these men obtained their results unconsciously--their creationsso lend themselves to a geometrical analysis that the claim for theexistence of certain canons of proportion, based on geometry, remainsunimpeached. 

[Illustration 67]

[Illustration 68]

The plane figures principally employed in determining architecturalproportion are the circle, the equilateral triangle, and thesquare--which also yields the right angled isosceles triangle. Itwill be noted that these are the two dimensional correlatives of thesphere, the tetrahedron and the cube, mentioned as being among thedetermining forms in molecular structure. The question naturallyarises, why the circle, the equilateral triangle and the square?Because, aside from the fact that they are of all plane figures themost elementary, they are intimately related to the body of man, ashas been shown (Illustration 45), and the body of man is as it werethe architectural archetype. But this simply removes the inquiry toa different field, it is not an answer. Why is the body of man soconstructed and related? This leads us, as does every question, tothe threshold of a mystery upon which theosophy alone is able to throwlight. Any extended elucidation would be out of place here: it issufficient to remind the reader that the circle is the symbol of theuniverse; the equilateral triangle, of the higher trinity (_atma, buddhi, manas_); and the square, of the lower quaternary of man'ssevenfold nature. 

[Illustration 69]

[Illustration 70]

The square is principally used in preliminary plotting: it is thedetermining 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 conjunctionwith the square and the triangle. In Thomas Jefferson's Rotunda forthe University of Virginia, a single great circle was the determiningfigure, as his original pen sketch of the building shows (Illustration61). Some of the best Roman triumphal arches submit themselves to acircular synopsis, and a system of double intersecting circles hasbeen applied, with interesting results, to façades as widely differentas those of the Parthenon and the Farnese Palace in Rome, thoughit would be fatuous to claim that these figures determined theproportions of the façades. 

By far the most important figure in architectural proportion, considered from the standpoint of geometry, is the equilateraltriangle. It would seem that the eye has an especial fondness forthis figure, just as the ear has for certain related sounds. Indeed itmight not be too fanciful to assert that the common chord of any key(the tonic with its third and fifth) is the musical equivalent ofthe equilateral triangle. It is scarcely necessary to dwell uponthe properties and unique perfection of this figure. Of all regularpolygons it is the simplest: its three equal sides subtend equalangles, each of 60 degrees; it trisects the circumference of a circle;it is the graphic symbol of the number three, and hence of everythreefold thing; doubled, its generating arcs form the _vesicapiscis_, of so frequent occurrence in early Christian art; twosymmetrically intersecting equilateral triangles yield the figureknown as "Solomon's Seal, " or the "Shield of David, " to which mysticproperties have always been ascribed. 

It may be stated as a general rule that whenever three importantpoints in any architectural composition coincide (approximately orexactly) with the three extremities of an equilateral triangle, itmakes for beauty of proportion. An ancient and notable example occursin the pyramids of Egypt, the sides of which, in their originalcondition, are believed to have been equilateral triangles. It is ademonstrable fact that certain geometrical intersections yield theimportant proportions of Greek architecture. The perfect littleErechtheum would seem to have been proportioned by means of theequilateral triangle and the angle of 60 degrees, both in general andin detail (Illustration 62). The same angle, erected from the centralaxis of a column at the point where it intersects the architrave, determines both the projection of the cornice and the height ofthe architrave in many of the finest Greek and Roman temples(Illustrations 67-70). The equilateral triangle used in conjunctionwith the circle and the square was employed by the Romans indetermining the proportions of triumphal arches, basilicas andbaths. That the same figure was a factor in the designing of Gothiccathedrals is sufficiently indicated in the accompanying facsimilereproductions of an illustration from the Como Vitruvius, published inMilan in 1521, which shows a vertical section of the Milan cathedraland the system of equilateral triangles which determined its variousparts (Illustration 71). The _vesica piscis_ was often used toestablish the two main internal dimensions of the cathedral plan: thegreatest diameter of the figure corresponding with the width acrossthe transepts, the upper apex marking the limit of the apse, and thelower, the termination of the nave. Such a proportion is seen to beboth subtle and simple, and possesses the advantage of being easilylaid out. The architects of the Italian Renaissance doubtlessinherited certain of the Roman canons of architectural proportion, for they seem very generally to have recognized them as an essentialprinciple of design. 

[Illustration 71]

Nevertheless, when all is said, it is easy to exaggerate theimportance of this matter of geometrical proportion. The designerwho seeks the ultimate secret of architectural harmony in mathematicsrather than in the trained eye, is following the wrong road tosuccess. A happy inspiration is worth all the formulæ in the world--ifit be really happy, the artist will probably find that he has"followed the rules without knowing them. " Even while formulatingconcepts of art, the author must reiterate Schopenhauer's dictumthat the _concept_ is unfruitful in art. The mathematical analysisof spatial beauty is an interesting study, and a useful one to theartist; but it can never take the place of the creative faculty, itcan only supplement, restrain, direct it. The study of proportion isto the architect what the study of harmony is to a musician--it helpshis genius adequately to express itself. 

VI

THE ARITHMETIC OF BEAUTY

Although architecture is based primarily upon geometry, it is possibleto express all spatial relations numerically: for arithmetic, notgeometry, is the universal science of quantity. The relation of massesone to another--of voids to solids, and of heights and lengths towidths--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 anarithmetical formula, determines what is, and what is not, beautifulproportion. Nevertheless the fact that the eye instinctively rejectscertain 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 deepreasonableness of such selection by the senses lies hidden in the verynature of number itself, for number is the invisible thread on whichthe worlds are strung--the universe abstractly symbolized. 

Number is the within of all things--the "first form of Brahman. " Itis the measure of time and space; it lurks in the heart-beat and isblazoned upon the starred canopy of night. Substance, in a state ofvibration, in other words conditioned by number, ceaselessly undergoesthe myriad transmutations which produce phenomenal life. Elementsseparate and combine chemically according to numerical ratios:"Moon, plant, gas, crystal, are concrete geometry and number. " Bythe Pythagoreans and by the ancient Egyptians sex was attributed tonumbers, odd numbers being conceived of as masculine or generating, and even numbers as feminine or parturitive, on account of theirinfinite divisibility. Harmonious combinations were thoseinvolving the marriage of a masculine and a feminine--an odd and aneven--number. 

[Illustration 72: A GRAPHIC SYSTEM OF NOTATION]

Numbers progress from unity to infinity, and return again to unity asthe soul, defined by Pythagoras as a self-moving number, goes forthfrom, and returns to God. These two acts, one of projection and theother of recall; these two forces, centrifugal and centripetal, aresymbolized in the operations of addition and subtraction. Within themis embraced the whole of computation; but because every number, everyaggregation of units, is also a new unit capable of being addedor subtracted, there are also the operations of multiplication anddivision, which consists in one case of the addition of several equalnumbers together, and in the other, of the subtraction of severalequal numbers from a greater until that is exhausted. In order tothink correctly it is necessary to consider the whole of numeration, computation, and all mathematical processes whatsoever as _thedivision of the unit_ into its component parts and the establishmentof relations between these parts. 

[Illustration 73]

[Illustration 74]

The progression and retrogression of numbers in groups expressed bythe multiplication table gives rise to what may be termed "numericalconjunctions. " These are analogous to astronomical conjunctions: theplanets, revolving around the sun at different rates of speed, andin widely separated orbits, at certain times come into line with oneanother 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 theirprogression come into relation with one another. For example, animportant conjunction occurs in 12, for of a series of twos it isthe sixth, of threes the fourth, of fours the third, and of sixes thesecond. It stands to 8 in the ratio of 3:2, and to 9, of 4:3. It isrelated to 7 through being the product of 3 and 4, of which numbers 7is the sum. The numbers 11 and 13 are not conjunctive; 14 is so inthe series of twos, and sevens; 15 is so in the series of fives andthrees. The next conjunction after 12, of 3 and 4 and their firstmultiples, is in 24, and the next following is in 36, which numbersare respectively the two and three of a series of twelves, each endbeing but a new beginning. 

[Illustration 75]

It will be seen that this discovery of numerical conjunctions consistsmerely of resolving numbers into their prime factors, and that aconjunctive number is a common multiple; but by naming it so, todismiss the entire subject as known and exhausted, is to miss asense of the wonder, beauty and rhythm of it all: a mental impressionanalogous to that made upon the eye by the swift-glancing balls of ajuggler, the evolutions of drilling troops, or the intricate figuresof a dance; for these things are number concrete and animate in timeand space. 

[Illustration 76]

The truths of number are of all truths the most interior, abstractand difficult of apprehension, and since knowledge becomes clear anddefinite to the extent that it can be made to enter the mind throughthe channels of physical sense, it is well to accustom oneself toconceiving of number graphically, by means of geometrical symbols(Illustration 72), rather than in terms of the familiar arabicnotation which though admirable for purposes of computation, is oftoo condensed and arbitrary a character to reveal the properties ofindividual 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 if4 be represented as a square enclosing four smaller squares, and 8as a cube containing eight smaller cubes, the idea is apprehendedimmediately and without effort. The number 3 is of course thetriangle; the irregular and vital beauty of the number 5 appearsclearly in the heptalpha, or five-pointed star; the faultless symmetryof 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 itselfas a number of singular beauty and perfection, worthy of the importantplace accorded to it in all mystical philosophy (Illustration 73). Itis a curious fact that when asked to think of any number less than 10, most persons will choose 7. 

[Illustration 77]

Every form of art, though primarily a vehicle for the expression andtransmission 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 thequalities of light, and the strength and beauty of materials. Allof the arts, and particularly music and architecture, portray indifferent manners and degrees the truths of number. Architecture doesthis in two ways: esoterically as it were in the form of harmonicproportions; and exoterically in the form of symbols which representnumbers and groups of numbers. The fact that a series of threes anda series of fours mutually conjoin in 12, finds an architecturalexpression in the Tuscan, the Doric, and the Ionic orders according toVignole, for in them all the stylobate is four parts, the entablature3, and the intermediate column 12 (Illustration 74). The affinitybetween 4 and 7, revealed in the fact that they express (very nearly)the ratio between the base and the altitude of the right-angledtriangle which forms half of an equilateral, and the musical intervalof the diminished seventh, is architecturally suggested in the PalazzoGiraud, which is four stories in height with seven openings in eachstory (Illustration 75). 

[Illustration 78]

[Illustration 79]

[Illustration 80]

[Illustration 81]

Every building is a symbol of some number or group of numbers, andother things being equal the more perfect the numbers involved themore beautiful will be the building (Illustrations 76-82). The numbers5 and 7--those which occur oftenest--are the most satisfactory becausebeing of small quantity, they are easily grasped by the eye, and beingodd, they yield a center or axis, so necessary in every architecturalcomposition. Next in value are the lowest multiples of these numbersand the least common multiples of any two of them, because theeye, with a little assistance, is able to resolve them into theirconstituent factors. It is part of the art of architecture torender such assistance, for the eye counts always, consciously orunconsciously, and when it is confronted with a number of unitsgreater than it can readily resolve, it is refreshed and rested ifthese units are so grouped and arranged that they reveal themselves asfactors of some higher quantity. 

[Illustration 82]

[Illustration 83: A NUMERICAL ANALYSIS OF GOTHIC TRACERY]

There is a raison d'être for string courses other than to mark theposition of a floor on the interior of a building, and for quoins andpilasters other than to indicate the presence of a transverse wall. These sometimes serve the useful purpose of so subdividing a façadethat the eye estimates the number of its openings without consciouseffort and consequent fatigue (Illustration 82). The tracery of Gothicwindows forms perhaps the highest and finest architectural expressionof number (Illustration 83). Just as thirst makes water more sweet, sodoes Gothic tracery confuse the eye with its complexity only the moregreatly to gratify the sight by revealing the inherent simplicity inwhich this complexity has its root. Sometimes, as in the case ofthe Venetian Ducal Palace, the numbers involved are too great forcounting, but other and different arithmetical truths are portrayed;for example, the multiplication of the first arcade by 2 inthe second, and this by 3 in the cusped arches, and by 4 in thequatrefoils immediately above. 

[Illustration 84: NUMERATION IN GROUPS EXPRESSED ARCHITECTURALLY]

[Illustration 85: ARCHITECTURAL ORNAMENT CONSIDERED AS THEOBJECTIFICATION OF NUMBER. MULTIPLICATION IN GROUPS OF FIVE; TWO;THREE; ALTERNATION OF THREE AND SEVEN]

[Illustration 86]

Seven is proverbially the perfect number. It is of a quantitysufficiently complex to stimulate the eye to resolve it, and yet sosimple that it can be analyzed at a glance; as a center with two equalsides, it is possessed of symmetry, and as the sum of an odd and evennumber (3 and 4) it has vitality and variety. All these properties awork of architecture can variously reveal (Illustration 77). Fifteen, also, is a number of great perfection. It is possible to arrange thefirst 9 numbers in the form of a "magic" square so that the sum ofeach 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 whichexpresses it, being 15 triangles in three groups of 5 (Illustration86). Few arrangements of openings in a façade better satisfy the eyethan three superimposed groups of five (Illustrations 76-80). May notone source of this satisfaction dwell in the intrinsic beauty of thenumber 15?

In conclusion, it is perhaps well that the reader be again remindedthat these are the by-ways, and not the highways of architecture: thatthe highest beauty comes always, not from beautiful numbers, norfrom likenesses to Nature's eternal patterns of the world, but fromutility, fitness, economy, and the perfect adaptation of means toends. But along with this truth there goes another: that in everyexcellent work of architecture, in addition to its obvious andindividual beauty, there dwells an esoteric and universal beauty, following as it does the archetypal pattern laid down by the GreatArchitect for the building of that temple which is the world whereinwe dwell. 

VII

FROZEN MUSIC

In the series of essays of which this is the final one, the author hasundertaken to enforce the truth that evolution on any plane and onany scale proceeds according to certain laws which are in reality onlyramifications of one ubiquitous and ever operative law; that this lawregisters itself in the thing evolved, leaving stamped thereon as itwere fossil footprints by means of which it may be known. In the artsthe creative spirit of man is at its freest and finest, and nowhereamong the arts is it so free and so fine as in music. In musicaccordingly the universal law of becoming finds instant, directand perfect self-expression; music voices the inner nature of the_will-to-live_ in all its moods and moments; in it form, content, means and end are perfectly fused. It is this fact which givesvalidity to the before quoted saying that all of the arts "aspiretoward the condition of music. " All aspire to express the law, butmusic, being least encumbered by the leaden burden of materiality, expresses it most easily and adequately. This being so there isnothing unreasonable in attempting to apply the known facts of musicalharmony and rhythm to any other art, and since these essays concernthemselves primarily with architecture, the final aspect in whichthat art will be presented here is as "frozen music"--ponderable formgoverned by musical law. 

Music depends primarily upon the equal and regular division of timeinto beats, and of these beats into measures. Over this soundless andinvisible warp is woven an infinitely various melodic pattern, madeup of tones of different pitch and duration arithmetically relatedand combined according to the laws of harmony. Architecture, correspondingly, implies the rhythmical division of space, andobedience to laws numerical and geometrical. A certain identitytherefore exists between simple harmony in music, and simpleproportion in architecture. By translating the consonanttone-intervals into number, the common denominator, as it were, ofboth arts, it is possible to give these intervals a spatial, andhence an architectural, expression. Such expression, considered asproportion only and divorced from ornament, will prove pleasing tothe 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 sensewhich is gratified, but the inner Self, in which all senses are one. Containing within itself the mystery of number, it thrills responsiveto every audible or visible presentment of that mystery. 

[Illustration 87]

If a vibrating string yielding a certain musical note be stopped inits center, that is, divided by half, it will then sound the octaveof that note. The numerical ratio which expresses the interval ofthe octave is therefore 1:2. If one-third instead of one-half of thestring be stopped, and the remaining two-thirds struck, it will yieldthe musical fifth of the original note, which thus corresponds to theratio 2:3. The length represented by 3:4 yields the fourth; 4:5 themajor third; and 5:6 the minor third. These comprise the principalconsonant intervals within the range of one octave. The ratios ofinverted intervals, so called, are found by doubling the smallernumber of the original interval as given above: 2:3, the fifth, gives3:4, 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. 

[Illustration 88: ARCHITECTURE AS HARMONY]

Of these various consonant intervals the octave, fifth, and majorthird are the most important, in the sense of being the most perfect, and they are expressed by numbers of the smallest quantity, an oddnumber and an even. It will be noted that all the intervals abovegiven are expressed by the numbers 1, 2, 3, 4, 5 and 6, except theminor sixth (5:8), and this is the most imperfect of all consonantintervals. The sub-minor seventh, expressed by the ratio 4:7 thoughincluded among the dissonances, forms, according to Helmholtz, a moreperfect consonance with the tonic than does the minor sixth. 

A natural deduction from these facts is that relations ofarchitectural length and breadth, height and width, to be "musical"should be capable of being expressed by ratios of quantitively smallnumbers, preferably an odd number and an even. Although generallyspeaking the simpler the numerical ratio the more perfect theconsonance, yet the intervals of the fifth and major third (2:3 and4:5), are considered to be more pleasing than the octave (1:2), whichis too obviously a repetition of the original note. From this it isreasonable to assume (and the assumption is borne out by experience), that proportions, the numerical ratios of which the eye resolves tooreadily, become at last wearisome. The relation should be felt ratherthan fathomed. There should be a perception of identity, and alsoof difference. As in music, where dissonances are introduced to givevalue to consonances which follow them, so in architecture simpleratios should be employed in connection with those more complex. 

[Illustration 89]

Harmonics are those tones which sound with, and reinforce any musicalnote when it is sounded. The distinguishable harmonics of the tonicyield the ratios 1:2, 2:3, 3:4, 4:5, and 4:7. A note and its harmonicsform a natural chord. They may be compared to the widening circleswhich appear in still water when a stone is dropped into it, for whena musical sound disturbs the quietude of that pool of silence whichwe call the air, it ripples into overtones, which becoming fainterand fainter, die away into silence. It would seem reasonable to assumethat the combination of numbers which express these overtones, iftranslated into terms of space, would yield proportions agreeableto the eye, and such is the fact, as the accompanying examplessufficiently indicate (Illustrations 87-90). 

The interval of the sub-minor seventh (4:7), used in this way, inconnection with the simpler intervals of the octave (1:2), and thefifth (2:3), is particularly pleasing because it is neither tooobvious nor too subtle. This ratio of 4:7 is important for the reasonthat it expresses the angle of sixty degrees, that is, the numbers 4and 7 represent (very nearly) the ratio between one-half the base andthe altitude of an equilateral triangle: also because they form partof the numerical series 1, 4, 7, 10, etc. Both are "mystic" numbers, and in Gothic architecture particularly, proportions were frequentlydetermined by numbers to which a mystic meaning was attached. According to Gwilt, the Gothic chapels of Windsor and Oxford aredivided longitudinally by four, and transversely by seven equal parts. The arcade above the roses in the façade of the cathedral of Toursshows seven principal units across the front of the nave, and four ineach of the towers. 

A distinguishing characteristic of the series of ratios whichrepresent the consonant intervals within the compass of an octave isthat 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 asquare. Alberti in his book presents a design for a tower showing hisidea for its general proportions. It consists of six stories, in asequence of orders. The lowest story is a perfect cube and each of theother stories is 11-12ths of the story below, diminishing practicallyin the proportion of 8, 7, 6, 5, 4, 3, allowing in each case for theamount hidden by the projection of the cornice below; each order beingaccurate as regards column, entablature, etc. It is of interest tocompare this with Ruskin's idea in his _Seven Lamps_, where he takesthe case of a plant called Alisma Plantago, in which the variousbranches diminish in the proportion of 7, 6, 5, 4, 3, respectively, and so carry out the same idea; on which Ruskin observes thatdiminution in a building should be after the manner of Nature. 

[Illustration 90: ARCADE OF THE CANCELLERIA]

It would be a profitless task to formulate exact rules ofarchitectural proportion based upon the laws of musical harmony. Thetwo arts are too different from each other for that, and moreoverthe last appeal must always be to the eye, and not to a mathematicalformula, just as in music the last appeal is to the ear. Laws thereare, but they discover themselves to the artist as he proceeds, andare for the most part incommunicable. Rules and formulæ are useful andvaluable not as a substitute for inspiration, but as a guide: not aswings, but as a tail. In this connection perhaps all that is necessaryfor the architectural designer to bear in mind is that importantratios of length and breadth, height and width, to be "musical" shouldbe expressed by quantitively small numbers, and that if possible theyshould obey some simple law of numerical progression. From this basicsimplicity complexity will follow, but it will be an ordered andharmonious complexity, like that of a tree, or of a symphony. 

[Illustration 91: THE PALAZZO VERZI AT VERONA (LOWER PORTION ONLY). A COMPOSITION FOUNDED ON THE EQUAL AND REGULAR DIVISION OF SPACE, ASMUSIC IS FOUNDED ON THE EQUAL AND REGULAR DIVISION OF TIME. ]

[Illustration 92: ARCHITECTURE AS RHYTHM. A DIVISION OF SPACECORRESPONDING TO 3/4 AND 4/4 TIME. ]

In the same way that a musical composition implies the division oftime into equal and regular beats, so a work of architecture shouldhave for its basis some unit of space. This unit should be nowhere tooobvious and may be varied within certain limits, just as musical timeis retarded or accelerated. The underlying rhythm and symmetry willthus give value and distinction to such variation. Vasari tells howBrunelleschi. Bramante and Leonardo da Vinci used to work on paperruled in squares, describing it as a "truly ingenious thing, and ofgreat utility in the work of design. " By this means they developedproportions according to a definite scheme. They set to work with adivision of space analogous to the musician's division of time. The examples given herewith indicate how close a parallel may existbetween music and architecture in this matter of rhythm (Illustrations91-93). 

[Illustration 93]

It is a demonstrable fact that musical sounds weave invisible patternsin the air. Architecture, correspondingly, in one of its aspects, is geometric pattern made fixed and enduring. What could be moreessentially musical for example than the sea arcade of the VenetianDucal Palace? The sand forms traced by sound-waves on a musicallyvibrating steel plate might easily suggest architectural ornament didnot 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 theessential identity between, say, a Greek rosette and a Gothicrose-window; an arcade and an egg and dart moulding (Illustration 94). All architectural forms and arrangements which give enduring pleasureare in their essence musical. Every well composed façade makes harmonyin three dimensions; every good roof-line sings a melody against thesky. 

[Illustration 94: ARCHITECTURE AS PATTERN]

CONCLUSION

In taking leave of the reader at the end of this excursion togetheramong the by-ways of a beautiful art, the author must needs add afinal word or two touching upon the purpose and scope of these essays. Architecture (like everything else) has two aspects: it may be viewedfrom the standpoint of utility, that is, as construction; or from thestandpoint of expressiveness, that is, as decoration. No attempt hasbeen made here to deal with its first aspect, and of the second(which is again twofold), only the universal, not the particularexpressiveness has been sought. The literature of architecture is richin works dealing with the utilitarian and constructive side of theart: indeed, it may be said that to this side that literature isalmost exclusively devoted. This being so, it has seemed worth whileto 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 tohave treated the subject inductively, amassing facts and drawingconclusions, but to have done so the author would have been falseto the very principle about which the work came into being. With theacceptance of the Ancient Wisdom, the inductive method becomesno longer necessary. Facts are not useful in order to establish ahypothesis, they are used rather to elucidate a known and acceptedtruth. When theosophical ideas shall have permeated the thought ofmankind, this work, if it survives at all, will be chiefly--perhapssolely--remarkable by reason of the fact that it was among thefirst in which the attempt was made again to unify science, art andreligion, as they were unified in those ancient times and among thoseancient peoples when the Wisdom swayed the hearts and minds of men.