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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 slim
volume of privately printed verse it was my first book. I worked hard
on it. Fifteen years elapsed between its beginning and completion;
it was twice published serially--written, rewritten and
tre-written--before it reached its ultimate incarnation in book form.

Confronted now with the opportunity to revise the text again, I find
myself in the position of a surgeon who feels that the operation he
is called upon to perform may perhaps harm more than it can help.
Prudence therefore prevails over my passion for dissection: warned by
eminent examples, I fear that any injection of my more mature and less
cocksure 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 few
verbal changes, and whatever reservations I have about it shall
be stated in this preface. These are not many nor important: _The
Beautiful Necessity_ contains nothing that I need repudiate or care to
contradict.

Its thesis, briefly stated, is that art in all its manifestations is
an expression of the cosmic life, and that its symbols constitute a
language by means of which this life is published and represented. Art
is at all times subject to the _Beautiful Necessity_ of proclaiming
the _world order_.

In attempting to develop this thesis it was not necessary (nor as
I now think, desirable) to link it up in so definite a manner with
theosophy. The individual consciousness is colored by the particular
medium through which it receives truth, and for me that medium was
theosophy. 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 to
imply a loss of faith in the fundamental tenets of theosophy, and such
an implication would not be true.

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

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

In tracing everything back to the logarithmic spiral (which embodies
the principle of extreme and mean ratios) I consider that Mr. Hambidge
has made one of those generalizations which reorganizes the old
knowledge and organizes the new. It would be only natural if in his
immersion in his idea he overworks it, but Mr. Hambidge is a man of
such intellectual integrity and thoroughness of method that he may be
trusted not to warp the facts to fit his theories. The truth of the
matter is that the entire field of research into the mathematics of
Beauty is of such richness that wherever a man plants his metaphysical
spade 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 to
conflict, it is less likely that one or the other of us is mistaken
than 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 still
remain the greatest justification and confirmation of my fundamental
contention--that art is an expression of the _world order_ and
is therefore orderly, organic; subject to mathematical law, and
susceptible of mathematical analysis.

CLAUDE BRAGDON

Rochester, N.Y.

April, 1922

THE ART OF ARCHITECTURE

One of the advantages of a thorough assimilation of what may be called
the theosophic idea is that it can be applied with advantage to every
department of knowledge and of human activity: like the key to a
cryptogram it renders clear and simple that which before seemed
intricate and obscure. Let us apply this key to the subject of art,
and to the art of architecture in particular, and see if by so
doing we may not learn more of art than we knew before, and more of
theosophy too.

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

In such a spectrum of the arts each one occupies a definite place, and
all together form a series of which music and architecture are the two
extremes. That such is their relative position may be demonstrated in
various ways. The theosophic explanation involving the familiar idea
of the "pairs of opposites" would be something as follows. According
to the Hindu-Aryan theory, Brahma, that the world might be born, fell
asunder into man and wife--became in other words _name and form_[A]
The two universal aspects of name and form are what philosophers call
the two "modes of consciousness," one of time, and the other of space.
These are the two gates through which ideas enter phenomenal life; the
two 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 of
one or the other of them, and may be classified accordingly. In such a
classification music is seen to be allied to time, and architecture to
space, because music is successive in its mode of manifestation, and
in time alone everything would occur successively, one thing following
another; while architecture, on the other hand, impresses itself upon
the beholder all at once, and in space alone all things would exist
simultaneously. Music, which is in time alone, without any relation to
space; and architecture, which is in space alone, without any relation
to 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 others
the elements of both time and space enter in varying proportion,
either actually or by implication. Poetry and the drama are allied to
music inasmuch as the ideas and images of which they are made up are
presented successively, yet these images are for the most part
forms of space. Sculpture on the other hand is clearly allied to
architecture, and so to space, but the element of _action_, suspended
though it be, affiliates it with the opposite or time pole. Painting
occupies a middle position, since in it space instead of being
actual has become ideal--three dimensions being expressed through
the mediumship of two--and time enters into it more largely than into
sculpture by reason of the greater ease with which complicated action
can be indicated: a picture being nearly always time arrested in
midcourse as it were--a moment transfixed.

In order to form a just conception of the relation between music and
architecture it is necessary that the two should be conceived of not
as standing at opposite ends of a series represented by a straight
line, but rather in juxtaposition, as in the ancient Egyptian symbol
of a serpent holding its tail in its mouth, the head in this case
corresponding to music, and the tail to architecture; in other words,
though in one sense they are the most-widely separated of the arts, in
another 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 of
the other arts, convertible into the other, by reason of the
correspondence subsisting between intervals of time and intervals of
space. A perception of this may have inspired the famous saying
that architecture is _frozen music_, a poetical statement of a
philosophical truth, since that which in music is expressed by means
of harmonious intervals of time and pitch, successively, after the
manner of time, may be translated into corresponding intervals of
architectural void and solid, height and width.

In another sense music and architecture are allied. They alone of
all the arts are purely creative, since in them is presented, not
a likeness of some known idea, but _a thing-in-itself_ brought to
a distinct and complete expression of its nature. Neither a musical
composition nor a work of architecture depends for its effectiveness
upon resemblances to natural sounds in the one case, or to natural
forms in the other. Of none of the other arts is this to such a degree
true: they are not so much creative as re-creative, for in them all
the artist takes his subject ready made from nature and presents it
anew according to the dictates of his genius.

The characteristic differences between music and architecture are the
same as those which subsist between time and space. Now time and space
are such abstract ideas that they can be dealt with best through
their corresponding correlatives in the natural world, for it is a
fundamental theosophic tenet that nature everywhere abounds in such
correspondences; that nature, in its myriad forms, is indeed the
concrete presentment of abstract unities. The energy which everywhere
animates form is a type of time within space; the mind working in
and through the body is another expression of the same thing.
Correspondingly, music is dynamic, subjective, mental, of one
dimension; while architecture is static, objective, physical, of three
dimensions; sustaining the same relation to music and the other
arts as does the human body to the various organs which compose, and
consciousnesses which animate it (it being the reservatory of these
organs and the vehicle of these consciousnesses); and a work of
architecture in like manner may and sometimes does include all of the
other arts within itself. Sculpture accentuates and enriches, painting
adorns, works of literature are stored within it, poetry and the drama
awake its echoes, while music thrills to its uttermost recesses, like
the very spirit of life tingling through the body's fibres.

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

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

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

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

This architecture, anterior to the Christian era, may be broadly
divided into three great periods, during which it was successively
practiced by three peoples: the Egyptians, the Greeks and the Romans.
Then intervened the Dark Ages, and a new art arose, the Gothic, which
was a flowering out in stone of the spirit of Christianity. This was
in turn succeeded by the Renaissance, the impulse of which remains
to-day unexhausted. In each of these architectures the peculiar genius
of a people and of a period attained to a beautiful, complete and
coherent utterance, and notwithstanding the considerable intervals
of time which sometimes separated them they succeeded one another
logically and inevitably, and each was related to the one which
preceded 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 birth
and training for their high office, tried by the severest ordeals
and bound by the most solemn oaths. The priests were honored and
privileged above all other men, and spent their lives dwelling
apart from the multitude in vast and magnificent temples, dedicating
themselves to the study and practice of religion, philosophy, science
and art--subjects then intimately related, not widely separated as
they are now. These men were the architects of ancient Egypt: theirs
the minds which directed the hands that built those time-defying
monuments.

The rites that the priests practiced centered about what are known
as the Lesser and the Greater Mysteries. These consisted of
representations by means of symbol and allegory, under conditions
and amid surroundings the most awe-inspiring, of those great truths
concerning man's nature, origin and destiny of which the priests--in
reality a brotherhood of initiates and their pupils--were the
custodians. These ceremonies were made the occasion for the initiation
of neophytes into the order, and the advancement of the already
initiated into its successive degrees. For the practice of such rites,
and others designed to impress not the elect but the multitude, the
great temples of Egypt were constructed. Everything about them was
calculated to induce a deep seriousness of mind, and to inspire
feelings of awe, dread and even terror, so as to test the candidate's
fortitude of soul to the utmost.

The avenue of approach to an Egyptian temple was flanked on both
sides, sometimes for a mile or more, with great stone sphinxes--that
emblem of man's dual nature, the god emerging from the beast. The
entrance was through a single high doorway between two towering
pylons, presenting a vast surface sculptured and painted over with
many strange and enigmatic figures, and flanked by aspiring obelisks
and seated colossi with faces austere and calm. The large court thus
entered was surrounded by high walls and colonnades, but was open
to the sky. Opposite the first doorway was another, admitting to a
somewhat smaller enclosure, a forest of enormous carved and painted
columns supporting a roof through the apertures of which sunshine
gleamed or dim light filtered down. Beyond this in turn were other
courts and apartments culminating in some inmost sacred sanctuary.

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

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

In the architecture of Greece the note of dread and mystery yields to
one of pure joyousness and freedom. The terrors of childhood have been
outgrown, and man revels in the indulgence of his unjaded appetites
and in the exercise of his awakened reasoning faculties. In Greek art
is preserved that evanescent beauty of youth which, coming but once
and continuing but for a short interval in every human life, is yet
that for which all antecedent states seem a preparation, and of
which all subsequent ones are in some sort an effect. Greece typifies
adolescence, the love age, and so throughout the centuries humanity
has turned to the contemplation of her, just as a man all his life
long secretly cherishes the memory of his first love.

An impassioned sense of beauty and an enlightened reason characterize
the productions of Greek architecture during its best period. The
perfection then attained was possible only in a nation whereof the
citizens were themselves critics and amateurs of art, one wherein
the artist was honored and his work appreciated in all its beauty
and subtlety. The Greek architect was less bound by tradition and
precedent than was the Egyptian, and he worked unhampered by any
restrictions save such as, like the laws of harmony in music, helped
rather than hindered his genius to express itself--restrictions
founded on sound reason, the value of which had been proved by
experience.

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

[Illustration 1]

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

Just as every detail of a Greek temple was adjusted to its position
and expressed its office, so the building itself was made to fit
its site and to show forth its purpose, forming with the surrounding
buildings a unit of a larger whole. The Athenian Acropolis is an
illustration of this: it is an irregular fortified hill, bearing
diverse monuments in various styles, at unequal levels and at
different angles with one another, yet the whole arrangement seems as
organic and inevitable as the disposition of the features of a face.
The Acropolis is an example of the ideal architectural republic
wherein each individual contributes to the welfare of all, and at the
same time enjoys the utmost personal liberty (Illustration 1).

Very different is the spirit bodied forth in the architecture of
Imperial Rome. The iron hand of its sovereignty encased within the
silken glove of its luxury finds its prototype in buildings which were
stupendous crude brute masses of brick and concrete, hidden within a
covering of rich marbles and mosaics, wrought in beautiful but often
meaningless forms by clever degenerate Greeks. The genius of Rome
finds its most characteristic expression, not in temples to the high
gods, but rather in those vast and complicated structures--basilicas,
amphitheatres, baths--built for the amusement and purely temporal
needs of the people.

If Egypt typifies the childhood of the race and Greece its beautiful
youth, Republican Rome represents its strong manhood--a soldier filled
with the lust of war and the love of glory--and Imperial Rome its
degeneracy: that soldier become conqueror, decked out in plundered
finery and sunk in sensuality, tolerant of all who minister to his
pleasures but terrible to all who interfere with them.

The fall of Rome marked the end of the ancient Pagan world. Above
its ruin Christian civilization in the course of time arose. Gothic
architecture is an expression of the Christian spirit; in it is
manifest the reaction from licentiousness to asceticism. Man's
spiritual nature, awakening in a body worn and weakened by
debaucheries, longs ardently and tries vainly to escape. Of some such
mood a Gothic cathedral is the expression: its vaulting, marvelously
supported upon slender shafts by reason of a nicely adjusted
equilibrium of forces; its restless, upward-reaching pinnacles and
spires; its ornament, intricate and enigmatic--all these suggest the
over-strained organism of an ascetic; while its vast shadowy interior
lit by marvelously traceried and jeweled windows, which hold the eyes
in 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 enforces
the theosophical teaching that everything of man's creating is made in
his own image. Architecture mirrors the life of the individual and of
the race, which is the life of the individual written large in time
and space. The terrors of childhood; the keen interests and appetites
of youth; the strong stern joy of conflict which comes with manhood;
the lust, the greed, the cruelty of a materialized old age--all these
serve but as a preparation for the life of the spirit, in which the
man becomes again as a little child, going over the whole round, but
on a higher arc of the spiral.

The final, or fourth state being only in some sort a repetition of
the first, it would be reasonable to look for a certain correspondence
between Egyptian and Gothic architecture, and such a correspondence
there is, though it is more easily divined than demonstrated. In
both there is the same deeply religious spirit; both convey, in some
obscure yet potent manner, a sense of the soul being near the surface
of life. There is the same love of mystery and of symbolism; and in
both may be observed the tendency to create strange composite figures
to typify transcendental ideas, the sphinx seeming a blood-brother to
the gargoyle. The conditions under which each architecture flourished
were not dissimilar, for each was formulated and controlled by small
well-organized bodies of sincerely religious and highly enlightened
men--the priesthood in the one case, the masonic guilds in the
other--working together toward the consummation of great undertakings
amid a populace for the most part oblivious of the profound and subtle
meanings of which their work was full. In Mediæval Europe, as in
ancient Egypt, fragments of the Ancient Wisdom--transmitted in the
symbols and secrets of the cathedral builders--determined much of
Gothic architecture.

The architecture of the Renaissance period, which succeeded the
Gothic, corresponds again, in the spirit which animates it, to Greek
architecture, which succeeded the Egyptian, for the Renaissance as
the name implies was nothing other than an attempt to revive Classical
antiquity. Scholars writing in what they conceived to be a Classical
style, sculptors modeling Pagan deities, and architects building
according to their understanding of Vitruvian methods succeeded
in producing works like, yet different from the originals they
followed--different because, animated by a spirit unknown to the
ancients, they embodied a new ideal.

In all the productions of the early Renaissance, "that first
transcendent springtide of the modern world," there is the evanescent
grace 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 artist
sought to attain to a certain abstract perfection of type; to build
a temple which should combine all the excellencies of every similar
temple, to carve a figure, impersonal in the highest sense, which
should embody every beauty. The artist of the Renaissance on the other
hand delighted not so much in the type as in the variation from it.
Preoccupied with the unique mystery of the individual soul--a sense of
which was Christianity's gift to Christendom--he endeavored to portray
that wherein a particular person is unique and singular. Acutely
conscious also of his own individuality, instead of effacing it he
made his work the vehicle and expression of that individuality. The
history of Renaissance architecture, as Symonds has pointed out,
is the history of a few eminent individuals, each one moulding and
modifying the style in a manner peculiar to himself alone. In the
hands of Brunelleschi it was stern and powerful; Bramante made
it chaste, elegant and graceful; Palladio made it formal, cold,
symmetrical; while with Sansovino and Sammichele it became sumptuous
and bombastic.

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

The final part of this second or modern architectural cycle lies still
in the future. It is not unreasonable to believe that the movement
toward mysticism, of which modern theosophy is a phase and the
spiritualization of science an episode, will flower out into an
architecture which will be in some sort a reincarnation of and a
return 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 European
architecture it is possible to trace a gradual growth or unfolding
as of a plant. It is a fact fairly well established that the Greeks
derived their architecture and ornament from Egypt; the Romans in
turn borrowed from the Greeks; while a Gothic cathedral is a lineal
descendant from a Roman basilica.

[Illustration 2]

[Illustration 3]

The Egyptians in their constructions did little more than to place
enormous stones on end, and pile one huge block upon another. They
used many columns placed close together: the spaces which they spanned
were inconsiderable. The upright or supporting member may be said to
have been in Egyptian architecture the predominant one. A vertical
line therefore may be taken as the simplest and most abstract symbol
of Egyptian architecture (Illustration 2). It remained for the Greeks
fully to develop the lintel. In their architecture the vertical
member, or column, existed solely for the sake of the horizontal
member, or lintel; it rarely stood alone as in the case of an Egyptian
obelisk. The columns of the Greek temples were reduced to those
proportions most consistent with strength and beauty, and the
intercolumnations were relatively greater than in Egyptian examples.
It may truly be said that Greek architecture exhibits the perfect
equality and equipoise of vertical and horizontal elements and these
only, no other factor entering in. Its graphic symbol would therefore
be composed of a vertical and a horizontal line (Illustration 3). The
Romans, while retaining the column and lintel of the Greeks, deprived
them of their structural significance and subordinated them to the
semicircular 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 and
vault are all retained in changed form, but that which more than
anything else differentiates Gothic architecture from any style which
preceded it is the introduction of the principle of an equilibrium of
forces, of a state of balance rather than a state of rest, arrived at
by the opposition of one thrust with another contrary to it. This fact
can be indicated graphically by two opposing inclined lines, and
these united to the preceding symbol yield an accurate abstract of the
elements of Gothic architecture (Illustration 5).

[Illustration 4]

[Illustration 5]

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

[Footnote A: The quaint Oriental imagery here employed should not
blind the reader to the precise scientific accuracy of the idea of
which this imagery is the vehicle. Schopenhauer says: "Polarity, or
the sundering of a force into two quantitively different and opposed
activities striving after re-union,... is a fundamental type of almost
all the phenomena of nature, from the magnet and the crystal to man
himself."]

II

UNITY AND POLARITY

Theosophy, both as a philosophy, or system of thought, which discovers
correlations between things apparently unrelated, and as a life,
or system of training whereby it is possible to gain the power to
perceive and use these correlations for worthy ends, is of great value
to the creative artist, whose success depends on the extent to which
he works organically, conforming to the cosmic pattern, proceeding
rationally and rhythmically to some predetermined end. It is of value
no less to the layman, the critic, the art amateur--to anyone in
fact who would come to an accurate and intimate understanding and
appreciation of every variety of esthetic endeavor. For the benefit of
such I shall try to trace some of those correlations which theosophy
affirms, and indicate their bearing upon art, and upon the art of
architecture in particular.

One of the things which theosophy teaches is that those transcendent
glimpses of a divine order and harmony throughout the universe
vouchsafed the poet and the mystic in their moments of vision are not
the paradoxes--the paronomasia as it were--of an intoxicated state of
consciousness, but glimpses of reality. We are all of us participators
in a world of concrete music, geometry and number--a world of
sounds, odors, forms, motions, colors, so mathematically related and
coordinated 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 of
art, for art is idealized creation: nature carried to a higher power
by reason of its passage through a human consciousness. Thought and
emotion tend to crystallize into forms of beauty as inevitably as does
the frost on a window pane. Art therefore in one of its aspects is the
weaving of a pattern, the communication of an order and a method to
the material or medium employed. Although no masterpiece was ever
created by the conscious following to set rules, for the true artist
works unconsciously, instinctively, as the bird sings or as the bee
builds its honey-cell, yet an analysis of any masterpiece reveals the
fact that its author (like the bird and the bee) has "followed the
rules without knowing them."

Helmholtz says, "No doubt is now entertained that beauty is subject
to laws and rules dependent on the nature of human intelligence. The
difficulty consists in the fact that these laws and rules, on whose
fulfilment beauty depends, are not consciously present in the mind of
the artist who creates the work, or of the observer who contemplates
it." Nevertheless they are discoverable, and can be formulated,
after a fashion. We have only to read aright the lessons everywhere
portrayed in the vast picture-books of nature and of art.

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

The second law, not contradicting but supplementing the first is the
law of _Polarity_, i.e., duality. All things have sex, are either
masculine or feminine. This too is the reflection on a lower plane
of one of those transcendental truths taught by the Ancient Wisdom,
namely that the Logos, in his voluntarily circumscribing his infinite
life in order that he may manifest, encloses himself within his
limiting veil, _maya_, and that his life appears as spirit (male), and
his _maya_ as matter (female), the two being never disjoined during
manifestation. The two terms of this polarity are endlessly repeated
throughout nature: in sun and moon, day and night, fire and water, man
and woman--and so on. A close inter-relation is always seen to subsist
between corresponding members of such pairs of opposites: sun, day,
fire, man express and embody the primal and active aspect of the
manifesting deity; moon, night, water, woman, its secondary and
passive aspect. Moreover, each implies or brings to mind the others
of its class: man, like the sun, is lord of day; he is like fire, a
devastating 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 the
constant and characteristic distinction between the two terms is a
thing far beyond mere contrast.

In music they are the major and minor modes: the typical, or
representative chords of the dominant seventh, and of the tonic (the
two chords into which Schopenhauer says all music can be resolved): a
partial dissonance, and a consonance: a chord of suspense, and a chord
of satisfaction. In speech the two are vowel, and consonant sounds:
the type of the first being _a_, a sound of suspense, made with the
mouth open; and of the second _m_, a sound of satisfaction, made by
closing the mouth; their combination forms the sacred syllable Om
(_Aum_). In painting they are warm colors, and cold: the pole of
the first being in red, the color of fire, which excites; and of the
second in blue, the color of water, which calms; in the Arts of design
they are lines straight (like fire), and flowing (like water); masses
light (like the day), and dark (like night). In architecture they are
the column, or vertical member, which resists the force of gravity;
and the lintel, or horizontal member, which succumbs to it; they are
vertical 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 sex
quality, and to make this easier some sort of classification and
analysis must be attempted. Those things which are allied to and
partake of the nature of _time_ are masculine, and those which are
allied to and partake of the nature of _space_ are feminine: as
motion, and matter; mind, and body; etc. The English words "masculine"
and "feminine" are too intimately associated with the idea of physical
sex properly to designate the terms of this polarity. In Japanese
philosophy and art (derived from the Chinese) the two are called _In_
and _Yo_ (In, feminine; Yo, masculine); and these little words, being
free from the limitations of their English correlatives, will be found
convenient, 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, and
color composition of Japanese prints and kakemonos to that exhibited
in the vastly more pretentious easel pictures of modern Occidental
artists--a superiority now generally acknowledged by connoisseurs--is
largely 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 the
vegetable kingdom. The trunk of a tree is Yo, its foliage, In; and
in each stem and leaf the two are repeated. A calla, consisting of
a single straight and rigid spadix embraced by a soft and tenderly
curved spathe, affords an almost perfect expression of the
characteristic differences between Yo and In and their reciprocal
relation to each other. The two are not often combined in such
simplicity and perfection in a single form. The straight, vertical
reeds which so often grow in still, shallow water, find their
complement in the curved lily-pads which lie horizontally on its
surface. Trees such as pine and hemlock, which are excurrent--those in
which the branches start successively (i.e., after the manner of time)
from a straight and vertical central stem--are Yo; trees such as
the elm and willow, which are deliquescent--those in which the trunk
dissolves as it were simultaneously (after the manner of space) into
its branches--are In. All tree forms lie in or between these two
extremes, and leaves are susceptible of a similar classification. It
will 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 elements
arranged with relation to axial lines; the second, by elements
arranged with relation to focal points (Illustrations 6,7).

The student should train himself to recognize In and Yo in all
their Protean presentments throughout nature--in the cloud upon the
mountain, the wave against the cliff, in the tracery of trees against
the sky--that he may the more readily recognize them in his chosen
art, whatever that art may be. If it happens to be painting, he will
endeavor to discern this law of duality in the composition of every
masterpiece, recognizing an instinctive obedience to it in that
favorite device of the great Renaissance masters of making an
architectural setting for their groups of figures, and he will
delight to trace the law in all its ramifications of contrast between
complementaries in line, color, and mass (Illustration 8).

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

With reference to architecture, it is true, generally speaking,
that architectural forms have been developed through necessity, the
function seeking and finding its appropriate form. For example, the
buttress of a Gothic cathedral was developed by the necessity of
resisting the thrust of the interior vaulting without encroaching upon
the nave; the main lines of a buttress conform to the direction of the
thrust, and the pinnacle with which it terminates is a logical shape
for the masonry necessary to hold the top in position (Illustration
9). Research along these lines is interesting and fruitful of result,
but there remains a certain number of architectural forms whose origin
cannot be explained in any such manner. The secret of their undying
charm lies in the fact that in them In and Yo stand symbolized and
contrasted. They no longer obey a law of utility, but an abstract
law of beauty, for in becoming sexually expressive as it were, the
construction itself is sometimes weakened or falsified. The familiar
classic console or modillion is an example: although in general
contour it is well adapted to its function as a supporting bracket,
embedded in, and projecting from a wall, yet the scroll-like ornament
with which its sides are embellished gives it the appearance of
not entering the wall at all, but of being stuck against it in some
miraculous manner. This defect in functional expressiveness is
more than compensated for by the perfection with which feminine
and masculine characteristics are expressed and contrasted in the
exquisite double spiral, opposed to the straight lines of the moulding
which it subtends (Illustration 10). Again, by fluting the shaft of a
column its area of cross-section is diminished but the appearance of
strength is enhanced because its masculine character--as a supporting
member 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 they
are architecture epitomized as it were. A building consists of a wall
upholding a roof: support and weight. The type of the first is the
column, which may be conceived of as a condensed section of wall; and
of the second, the lintel, which may be conceived of as a condensed
section of roof. The column, being vertical, is Yo; the lintel, being
horizontal, is In. To mark an entablature with horizontal lines in the
form of mouldings, and the columns with vertical lines in the form
of flutes, as is done in all the "classic orders," is a gain
in 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 the
capital, which is In. The capital is itself twofold, consisting of a
curved member and an angular member. These two appear in their utmost
simplicity in the _echinus_ (In) and the _abacus_ (Yo) of a
Greek Doric cap. The former was adorned with painted leaf forms,
characteristically feminine, and the latter with the angular fret
and meander (Illustration 12). The Ionic capital, belonging to a more
feminine style, exhibits the abacus subordinated to that beautiful
cushion-shaped member with its two spirally marked volutes. This,
though a less rational and expressive form for its particular office
than is the echinus of the Doric cap, is a far more perfect symbol of
the feminine element in nature. There is an essential identity between
the Ionic cap and the classic console before referred to--although
superficially the two do not resemble each other--for a straight line
and a double spiral are elements common to both (Illustration 10).
The Corinthian capital consists of an ordered mass of delicately
sculptured leaf and scroll forms sustaining an abacus which though
relatively masculine is yet more curved and feminine than that of any
other style. In the caulicole of a Corinthian cap In and Yo are again
contrasted. In the unique and exquisite capital from the Tower of the
Winds at Athens, the two are well suggested in the simple, erect, and
pointed leaf forms of the upper part, contrasted with the complex,
deliquescent, rounded ones from which they spring. The essential
identity of principle subsisting between this cap and the Renaissance
baluster by San Gallo is easily seen (Illustration 13).

[Illustration 11]

[Illustration 12]

This law of sex-expressiveness is of such universality that it can
be made the basis of an analysis of the architectural ornament of any
style or period. It is more than mere opposition and contrast. The egg
and tongue motif, which has persisted throughout so many centuries and
survived so many styles, exhibits an alternation of forms resembling
phallic emblems. Yo and In are well suggested in the channeled
triglyphs and the sculptured metopes of a Doric frieze, in the
straight and vertical mullions and the flowing tracery of Gothic
windows, in the banded torus, the bead and reel, and other familiar
ornamented mouldings (Illustrations 14, 15, 16).

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

[Illustration 13: CAPITAL FROM THE TOWER OF THE WINDS, ATHENS;
CORINTHIAN CAP FROM HADRIAN BUILDINGS, ATHENS; ROSETTE FROM TEMPLE OF
MARS, 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 two
towers, man and woman--that Adam and that Eve for whose redemption
according to current teaching Christ suffered and was crucified. The
north or right-hand tower ("the man's side") was called the sacred
male pillar, Jachin; and the south, or left-hand tower ("the woman's
side"), the sacred female pillar, Boaz, from the two columns flanking
the gate to Solomon's Temple--itself an allegory to the bodily temple.
In only a few of the French cathedrals is this distinction clearly
and consistently maintained, and of these Tours forms perhaps the most
remarkable example, for in its flamboyant façade, over and above the
difference in actual breadth and apparent sturdiness of the two towers
(the south being the more slender and delicate), there is a clearly
marked distinction in the character of the ornamentation, that of the
north tower being more salient, angular, radial--more masculine in
point of fact (Illustration 17). In Notre Dame, the cathedral of
Paris, as in the cathedral of Tours, the north tower is perceptibly
broader than the south. The only other important difference appears to
be in the angular label-mould above the north entrance: whatever may
have been its original function or significance, it serves to define
the tower sexually, so to speak, as effectively as does the beard on
a man's face. In Amiens the north tower is taller than the south, and
more massive in its upper stages. The only traceable indication of
sex in the ornamentation occurs in the spandrels at the sides of the
entrance 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 Middle
Ages, just as anciently in Egypt and again in Greece--in fact wherever
and whenever the Secret Doctrine was known--sex was attributed to
numbers, odd numbers being conceived of as masculine, and even, as
feminine. Two, the first feminine number, thus became a symbol of
femininity, accepted as such so universally at the time the cathedrals
were 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. VATICAN
MUSEUM; FRIEZE IN THE EMPIRE STYLE BY PERCIER AND FONTAINE.
FRIEZE FROM THE TEMPLE OF VESTA AT TIVOLI. (ROMAN); ROMAN DORIC
FRIEZE--VIGNOLE]

[Illustration 16]

[Illustration 17]

The vital, organic quality so conspicuous in the best Gothic
architecture has been attributed to the fact that necessity determined
its characteristic forms. Professor Goodyear has demonstrated that
it may be due also in part to certain subtle vertical leans and
horizontal bends; and to nicely calculated variations from strict
uniformity, which find their analogue in nature, where structure is
seldom rigidly geometrical. The author hazards the theory that still
another reason why a Gothic cathedral seems so living a thing is
because it abounds in contrasts between what, for lack of more
descriptive adjectives, he is forced to call masculine and feminine
forms.

[Illustration 18]

[Illustration 19]

Ruskin says, in _Stones of Venice_, "All good Gothic is nothing more
than 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 vaults
beneath, to the crown-like points that enrich the smallest niche of
its doorway, one law and one expression will be found in all. The
modes of support and of decoration are infinitely various, but the
real character of the building, in all good Gothic, depends on the
single lines of the gable over the pointed arch endlessly rearranged
and repeated." These two, an angular and a curved form, like the
everywhere recurring column and lintel of classic architecture,
are but presentments of Yo and In (Illustration 18). Every Gothic
traceried window, with straight and vertical mullions in the
rectangle, losing themselves in the intricate foliations of the arch,
celebrates the marriage of this ever diverse pair. The circle and the
triangle are the In and Yo of Gothic tracery, its Eve and Adam, as
it were, for from their union springs that progeny of trefoil,
quatrefoil, cinquefoil, of shapes flowing like water, and shapes
darting like flame, which makes such visible music to the entranced
ear.

[Illustration 20: SAN GIMIGNANO S. JACOPO.]

By seeking to discover In and Yo in their myriad manifestations, by
learning to discriminate between them, and by attempting to express
their characteristic qualities in new forms of beauty--from
the disposition of a façade to the shaping of a moulding--the
architectural designer will charge his work with that esoteric
significance, that excess of beauty, by which architecture rises
to the dignity of a "fine" art (Illustrations 19, 20). In so doing,
however, he should never forget, and the layman also should ever
remember, that the supreme architectural excellence is fitness,
appropriateness, the perfect adaptation of means to ends, and the
adequate expression of both means and ends. These two aims, the one
abstract and universal, the other concrete and individual, can always
be combined, just as in every human countenance are combined a type,
which is universal, and a character, which is individual.

CHANGELESS CHANGE

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

The preceding essay was devoted for the most part to that "inevitable
duality" which finds concrete expression in countless pairs of
opposites, 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 of
fulfilment; in speech by vowel and consonant sounds, epitomized in _a_
and in _m_; in painting by warm colors and cold, epitomized in red
and blue; in architecture by the vertical column and the horizontal
lintel, by void and solid--and so on.

TRINITY

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

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

It would be supererogatory to dwell at any length on this "trinity
of manifestation" as the concrete expression of that unmanifest and
mystical trinity, that _three-in-one_ which under various names occurs
in every world-religion, where, defying definition, it was wont
to find expression symbolically in some combination of vertical,
horizontal and curved lines. The anstated cross of the Egyptians
is such a symbol, the Buddhist wheel, and the fylfot or swastika
inscribed within a circle, also those numerous Christian symbols
combining the circle and the cross. Such ideographs have spelled
profound meaning to the thinkers of past ages. We of to-day are not
given to discovering anything wonderful in three strokes of a pen,
but every artist in the weaving of his pattern must needs employ these
mystic symbols in one form or another, and if he employs them with
a full sense of their hidden meaning his work will be apt to gain
in originality and beauty--for originality is a new and personal
perception of beauty, and beauty is the name we give to truth we
cannot understand.

In architecture, this trinity of vertical, horizontal and curved
lines finds admirable illustration in the application of columns and
entablature to an arch and impost construction, so common in Roman and
Renaissance work. This is a redundancy, and finds no justification in
reason, 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 transition
between the columns and the entablature, and completes the trinity
of vertical, horizontal and curved lines (Illustration 21). In the
entrances to many of the Gothic cathedrals and churches the same
elements are better because more logically disposed. Here the
horizontal lintel and its vertical supports are not decorative merely,
but really perform their proper functions, while the arch, too, has
a raison d'être in that it serves to relieve the lintel of the
superincumbent weight of masonry. The same arrangement sometimes
occurs in classic architecture also, as when an opening spanned by a
single arch is subdivided by means of an order (Illustration 22).

Three is pre-eminently the number of architecture, because it is the
number of space, which for us is three-dimensional, and of all the
arts architecture is most concerned with the expression of spatial
relations. The division of a composition into three related parts is
so universal that it would seem to be the result of an instinctive
action of the human mind. The twin pylons of an Egyptian temple with
its entrance between, for a third division, has its correspondence in
the two towers of a Gothic cathedral and the intervening screen
wall of the nave. In the palaces of the Renaissance a threefold
division--vertically by means of quoins or pilasters, and horizontally
by means of cornices or string courses--was common, as was also the
division into a principal and two subordinate masses (Illustration
23).

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

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

CONSONANCE

"As is the small, so is the great" is a perpetually recurring phrase
in the literature of theosophy, and naturally so, for it is a succinct
statement of a fundamental and far-reaching truth. The scientist
recognizes it now and then and here and there, but the occultist
trusts it always and utterly. To him the microcosm and the macrocosm
are one and the same in essence, and the forth-going impulse
which calls a universe into being and the indrawing impulse which
extinguishes it again, each lasting millions of years, are echoed and
repeated in the inflow and outflow of the breath through the nostrils,
in nutrition and excretion, in daily activity and nightly rest, in
that 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 throughout
its parts. Each leaf on a tree is itself a tree in miniature, each
blossom a modified leaf; every vertebrate animal is a complicated
system of spines; the ripple is the wave of a larger wave, and that
larger wave is a part of the ebbing and flowing tide. In music this
law 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 sense
in the repetition of a major theme in the minor, or in the treble and
again in the bass, with modifications perhaps of time and key. In
the art of painting the law is exemplified in the repetition with
variation of certain colors and combinations of lines in different
parts of the same picture, so disposed as to lead the eye to some
focal point. Every painter knows that any important color in his
picture must be echoed, as it were, in different places, for harmony
of the whole.

[Illustration 24]

In the drama the repetition of a speech or of an entire scene, but
under circumstances which give it a different meaning, is often most
effective, as when Gratiano, in the trial scene of _The Merchant
of Venice_ taunts Shylock with his own words, "A Daniel come to
judgment!" 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 proper
person and no longer as the boy Ganymede.

These recurrences, these inner consonances, these repetitions with
variations are common in architecture also. The channeled triglyphs of
a 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 are
but elaborated and embellished dentils. Each pinnacle of a Gothic
cathedral is a little tower with its spire. As Ruskin has pointed out,
the great vault of the cathedral nave, together with the pointed roof
above it, is repeated in the entrance arch with its gable, and the
same two elements appear in every statue-enshrining niche of the
doorway. In classic architecture, as has been shown, instead of the
arch and gable, the column and entablature everywhere recur under
different forms. The minor domes which flank the great dome of the
cathedral of Florence enhance and reinforce the latter, and prepare
the eye for a climax which would otherwise be too abrupt. The central
pavilion of the Château Maintenon, with its two turrets, echoes the
entire façade with its two towers. Like the overture to an opera, it
introduces 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 obscurely
in the form of recurring numerical ratios, identical geometrical
determining figures, parallel diagonals and the like, which will be
discussed in a subsequent essay. It has also to do with style and
scale, the adherence to substantially one method of construction and
manner of ornament, just as in music the key, or chosen series of
notes, may not be departed from except through proper modulations, or
in 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 different
figures. 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_--not
identity, but difference. It shows itself for the most part as a
perceptible and piquant variation between individual units belonging
to the same class, type, or species.

No two trees put forth their branches in just the same manner, and no
two leaves from the same tree exactly correspond; no two persons
look alike, though they have similiar members and features; even the
markings 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 some
occult law, some theosophical truth; and this law of Diversity in
Monotony is the presentment of the truth that identity does not
exclude difference. The law is binding, yet the will is free: all men
are brothers united by the ties of brotherhood, yet each is unique, a
free agent, and never so free as when most bound by the Good Law. This
truth nature beautifully proclaims, and art also. In architecture it
is admirably exemplified in the metopes of the Parthenon frieze: seen
at a distance these must have presented a scarcely distinguishable
texture of sunlit marble and cool shadow, yet in reality each is
a separate work of art. So with the capitals of the columns of the
wonderful sea-arcade of the Venetian Ducal palace: alike in general
contour they differ widely in detail, and unfold a Bible story.
In Gothic cathedrals, in Romanesque monastery cloisters, a teeming
variety of invention is hidden beneath apparent uniformity. The
gargoyles of Notre Dame make similiar silhouettes against the sky,
but seen near at hand what a menagerie of monsters! The same spirit of
controlled individuality, of liberty subservient to the law of all, is
exemplified in the bases of the columns of the temple of Apollo near
Miletus--each one a separate masterpiece of various ornamentation
adorning an established architectural form (Illustration 28).

[Illustration 28]

[Illustration 29]

The builders of the early Italian churches, instinctively obeying this
law of Diversity in Monotony, varied the size of the arches in the
same arcade (Illustration 29), and that this was an effect of art and
not of accident or carelessness Ruskin long ago discovered, and the
Brooklyn Institute surveys have amply confirmed his view. Although
by these means the builders of that day produced effects of deceptive
perspective, of subtle concord and contrast, their sheer hatred of
monotony and meaningless repetition may have led them to diversify
their arcades in the manner described, for a rigidly equal and regular
division lacks interest and vitality.

BALANCE

If one were to establish an axial plane vertically through the center
of 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 about
balanced each other. Similarly, in all our bodily movements, for every
change of equilibrium there occurs an opposition and adjustment of
members of such a nature that an axial plane through the center of
gravity would divide the body into two substantially equal masses,
as in the case of the tree. This physical plane law of Balance
shows itself for the most part on the human plane as the law of
Compensation, whereby, to the vision of the occultist, all accounts
are "squared," so to speak. It is in effect the law of Justice, aptly
symbolized by the scales.

The law of Balance finds abundant illustration in art: in music by
the opposition, the answering, of one phrase by another of the same
elements and the same length, but involving a different sequence of
intervals; in painting by the disposition of masses in such a way that
they 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 of
Balance is in the symmetrical disposition of the elements, whether of
plan or of elevation, on either side of axial lines. A far more subtle
and vital illustration of the law occurs when the opposed elements do
not exactly match, but differ from each other, as in the case of the
two towers of Amiens, for example. This sort of balance may be said to
be characteristic of Gothic, as symmetry is characteristic of Classic,
architecture.

RHYTHMIC CHANGE

There is in nature a universal tendency toward refinement and
compactness of form in space, or contrariwise, toward increment
and diffusion; and this manifests itself in time as acceleration or
retardation. It is governed, in either case, by an exact mathematical
law, like the law of falling bodies. It shows itself in the widening
circles which appear when one drops a stone into still water, in the
convolutions of shells, in the branching of trees and the veining
of leaves; the diminution in the size of the pipes of an organ
illustrates it, and the spacing of the frets of a guitar. More and
more science is coming to recognize, what theosophy affirms, that the
spiral vortex, which so beautifully illustrates this law, both in its
time and its space aspects is the universal archetype, the pattern of
all that is, has been, or will be, since it is the form assumed by the
ultimate physical atom, and the ultimate physical atom is the physical
cosmos in miniature.

This Rhythmic Diminution is everywhere: it is in the eye itself, for
any series of mathematically equal units, such for example as the
columns and intercolumnations of a colonnade, become when seen
in perspective rhythmically unequal, diminishing according to the
universal law. The entasis of a Classic column is determined by this
law, the spirals of the Ionic volute, the annulets of the Parthenon
cap, obey it (Illustration 30).

In recognition of the same principle of Rhythmic Diminution a building
is 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 attenuated
and highly ornamented orders above the simpler and sturdier, as in the
Roman Colosseum, or in the Palazzo Uguccioni, in Florence--to mention
only two examples out of a great number. In the Riccardi Palace
an effect of increasing refinement is obtained by diminishing the
boldness of the rustication of the ashlar in successive stories; in
the Farnese, by the gradual reduction of the size of the angle quoins
(Illustration 30). In an Egyptian pylon it is achieved most simply by
battering the wall; in a Gothic cathedral most elaborately by a
kind of segregation, or breaking up, analogous to that which a tree
undergoes--the strong, relatively unbroken base corresponding to
the trunk, the diminishing buttresses to the tapering limbs, and
the multitude of delicate pinnacles and crockets, to the outermost
branches and twigs, seen against the sky.

RADIATION

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

The idea here expressed by the term "radiation" is a familiar one
to all students of theosophy. The Logos radiates his life and light
throughout his universe, bringing into activity a host of entities
which become themselves radial centers; these generate still others,
and so on endlessly. This principle, like every other, patiently
publishes itself to us, unheeding, everywhere in nature, and in
all great art as well; it is a law of optics, for example, that all
straight lines having a common direction if sufficiently prolonged
appear to meet in a point, i.e., radiate from it (Illustration 31).
Leonardo da Vinci employed this principle of perspective in his Last
Supper to draw the spectator's eye to the picture's central figure,
the point of sight toward which the lines of the walls and ceiling
converge centering in the head of Christ. Puvis de Chavannes, in his
Boston Library decoration, leads the eye by a system of triangulation
to the small figure of the Genius of Enlightenment above the central
door (Illustration 32); and Ruskin, in his _Elements of Drawing_, has
shown how artfully Turner arranged some of his compositions to attract
attention 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 a
central point, are familiar examples. The distinctive characteristic
of Gothic construction, the concentration or focalization of the
weight of the vaults and arches at certain points, is another
illustration of the same principle applied to architecture,
beautifully exemplified in the semicircular apse of a cathedral, where
the lines of the plan converge to a common center, and the ribs of the
vaulting meet upon the capitals of the piers and columns, seeming
to radiate thence to still other centers in the loftier vaults which
finally 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 the
cathedrals of Paris and of Amiens, illustrate Radiation, in the one
case masculine: straight, angular, direct; in the feminine: curved,
flowing, sinuous. The same _Beautiful Necessity_ determined the
characteristics of much of the ornament of widely separated styles
and periods: the Egyptian lotus, the Greek honeysuckle, the Roman
acanthus, Gothic leaf work--to snatch at random four blossoms from
the sheaf of time. The radial principle still inherent in the debased
ornament of the late Renaissance gives that ornament a unity, a
coherence, 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 their
application to the art of architecture. The list is by no means
exhausted, but it is not the multiplicity and diversity of these laws
which is important to keep in mind, so much as their relatedness and
coördination, for they are but different aspects of the One Law, that
whereby the Logos manifests in time and space. A brief recapitulation
will serve to make this correlation plain, and at the same time fix
what 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 its
essence twofold, there is the law of _Polarity_; but this duality is
not static but dynamic, the two parts acting and reacting upon one
another to produce a third--hence the law of _Trinity_. Given this
third term, and the innumerable combinations made possible by
its relations to and reactions upon the original pair, the law
of _Multiplicity in Unity_ naturally follows, as does the law
of _Consonance_, or repetition, since the primal process of
differentiation tends to repeat itself, and the original combinations
to reappear--but to reappear in changed form, hence the law of
_Diversity in Monotony_. The law of _Balance_ is seen to be but a
modification of the law of Polarity, and since all things are waxing
and waning, there is the law whereby they wax and wane, that of
_Rhythmic Change_. _Radiation_ rediscovers and reaffirms, even in the
utmost complexity, that essential and fundamental unity from which
complexity was wrought.

Everything, beautiful or ugly, obeys and illustrates one or another of
these laws, so universal are they, so inseparably attendant upon every
kind of manifestation in time and space. It is the number of them
which finds illustration within small compass, and the aptness and
completeness of such illustration, which makes for beauty, because
beauty is the fine flower of a sort of sublime ingenuity. A work of
art is nothing if not _artful_: like an acrostic, the more different
ways it can be read--up, down, across, from right to left and from
left to right--the better it is, other things being equal. This
statement, of course, may be construed in such a way as to appear
absurd; what is meant is simply that the more a work of art is
freighted and fraught with meaning beyond meaning, the more secure its
immortality, the more powerful its appeal. For enjoyment, it is not
necessary that all these meanings should be fathomed, it is only
necessary that they should be felt.

Consider for a moment the manner in which Leonardo da Vinci's Last
Supper, an acknowledged masterpiece, conforms to everyone of the laws
of beauty enumerated above (Illustration 32). It illustrates the law
of Unity in that it movingly portrays a single significant episode in
the life of Christ. The eye is led to dwell upon the central personage
of this drama by many artful expedients: the visible part of the
figure of Christ conforms to the lines of an equilateral triangle
placed exactly in the center of the picture; the figure is separated
by a considerable space from the groups of the disciples on either
hand, 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 finds
fulfilment in the complex and flowing lines of the draped figures
contrasted with the simple parallelogram of the cloth-covered table,
and the severe architecture of the room. The law of Trinity is
exemplified in the three windows, and in the subdivision of the twelve
figures of the disciples into four groups of three figures each. The
law of Consonance appears in the repetition of the horizontal lines
of the table in the ceiling above; and in the central triangle before
referred to, continued and echoed, as it were, in the triangular
supports of the table visible underneath the cloth. The law of
Diversity in Monotony is illustrated in the varying disposition of the
heads of the figures in the four groups of three; the law of Balance
in the essential symmetry of the entire composition; the law of
Rhythmic Change in the diminishing of the wall and ceiling spaces; and
the law of Radiation in the convergence of all the perspective lines
to a single significant point.

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

It would be fatuous to contend that the prime object of a work of
architecture is to obey and illustrate these laws. The prime object of
a work of architecture is to fulfill certain definite conditions in a
practical, economical, and admirable way, and in fulfilling to express
as far as possible these conditions, making the form express the
function. The architect who is also an artist however will do this
and something beyond: working for the most part unconsciously,
harmoniously, joyously, his building will obey and illustrate natural
laws--these laws of beauty--and to the extent it does so it will be a
work of art; for art is the method of nature carried into those higher
regions of thought and feeling which man alone inhabits: regions which
it 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 the
body of man." If the body is, as he declares, a temple, it is not less
true that a temple or any work of architectural art is a larger body
which man has created for his uses, just as the individual self is
housed within its stronghold of flesh and bones. Architectural beauty
like human beauty depends upon the proper subordination of parts
to the whole, the harmonious interrelation between these parts, the
expressiveness of each of its function or functions, and when these
are many and diverse, their reconcilement one with another. This being
so, a study of the human figure with a view to analyzing the sources
of its beauty cannot fail to be profitable. Pursued intelligently,
such a study will stimulate the mind to a perception of those simple
yet subtle laws according to which nature everywhere works, and
it will educate the eye in the finest known school of proportion,
training it to distinguish minute differences, in the same way that
the hearing of good music cultivates the ear.

Those principles of natural beauty which formed the subject of the
two preceding essays are all exemplified in the ideally perfect human
figure. Though essentially a unit, there is a well marked division
into right and left--"Hands to hands, and feet to feet, in one body
grooms 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 has
two lips. Moreover, the terms of such pairs are masculine and feminine
with respect to each other, one being active and the other passive.
Owing to the great size and one-sided position of the liver, the right
half of the body is heavier than the left; the right arm is usually
longer and more muscular than the left; the right eye is slightly
higher than its fellow. In speaking and eating the lower jaw and under
lip are active and mobile with relation to the upper; in winking it is
the upper eyelid which is the more active. That "inevitable duality"
which is exhibited in the form of the body characterizes its motions
also. In the act of walking for example, a forward movement is
attained by means of a forward and a backward movement of the thighs
on the axis of the hips; this leg movement becomes twofold again below
the knee, and the feet move up and down independently on the axis of
the ankle. A similar progression is followed in raising the arm and
hand: motion is communicated first to the larger parts, through them
to the smaller and thence to the extremities, becoming more rapid and
complex as it progresses, so that all free and natural movements of
the limbs describe invisible lines of beauty in the air. Coexistent
with this pervasive duality there is a threefold division of the
figure into trunk, head and limbs: a superior trinity of head and
arms, and an inferior trinity of trunk and legs. The limbs are divided
threefold into upper-arm, forearm and hand; thigh, leg and foot. The
hand flowers out into fingers and the foot into toes, each with a
threefold articulation; and in this way is effected that transition
from unity to multiplicity, from simplicity to complexity, which
appears to be so universal throughout nature, and of which a tree is
the perfect symbol.

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

[Illustration 38]

[Illustration 39]

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

[Illustration 40]

The relation of these laws of beauty to the art of architecture has
been shown already. They are reiterated here only to show that man is
indeed the microcosm--a little world fashioned from the same elements
and in accordance with the same _Beautiful Necessity_ as is the
greater world in which he dwells. When he builds a house or temple he
builds it not literally in his own image, but according to the laws of
his own being, and there are correspondences not altogether fanciful
between the animate body of flesh and the inanimate body of stone. Do
we not all of us, consciously or unconsciously, recognize the fact
of character and physiognomy in buildings? Are they not, to our
imagination, masculine or feminine, winning or forbidding--_human_,
in point of fact--to a greater degree than anything else of man's
creating? They are this certainly to a true lover and student of
architecture. Seen from a distance the great French cathedrals appear
like crouching monsters, half beast, half human: the two towers stand
like a man and a woman, mysterious and gigantic, looking out over city
and plain. The campaniles of Italy rise above the churches and houses
like the sentinels of a sleeping camp--nor is their strangely human
aspect wholly imaginary: these giants of mountain and campagna have
eyes and brazen tongues; rising four square, story above story, with
a belfry or lookout, like a head, atop, their likeness to a man is not
infrequently 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 the
Ionic column we discern her stiffened, like Lot's wife, into a pillar,
with nothing to show her feminine but the spirals of her beautiful
hair. The columns which uphold the pediment of the Parthenon are
unmistakably masculine: the ratio of their breadth to their height is
the 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 mystical
enlightenment, men have been wont to use the human figure, the soul's
temple, as a sort of archetype for sacred edifices (Illustration 41).
The colossi, with calm inscrutable faces, which flank the entrance to
Egyptian temples; the great bronze Buddha of Japan, with its dreaming
eyes; the little known colossal figures of India and China--all these
belong scarcely less to the domain of architecture than of sculpture.
The relation above referred to however is a matter more subtle and
occult than mere obvious imitation on a large scale, being based upon
some correspondence of parts, or similarity of proportions, or both.
The correspondence between the innermost sanctuary or shrine of a
temple and the heart of a man, and between the gates of that temple
and the organs of sense is sufficiently obvious, and a relation once
established, the idea is susceptible of almost infinite development.
That the ancients proportioned their temples from the human figure
is no new idea, nor is it at all surprising. The sculpture of the
Egyptians and the Greeks reveals the fact that they studied the body
abstractly, in its exterior presentment. It is clear that the rules
of its proportions must have been established for sculpture, and it is
not unreasonable to suppose that they became canonical in architecture
also. Vitruvius and Alberti both lay stress on the fact that all
sacred buildings should be founded on the proportions of the human
body.

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

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

In France, during the Middle Ages, a Gothic cathedral became, at the
hands of the secret masonic guilds, a glorified symbol of the body
of Christ. To practical-minded students of architectural history,
familiar with the slow and halting evolution of a Gothic cathedral
from a Roman basilica, such an idea may seem to be only the
maunderings of a mystical imagination, a theory evolved from the inner
consciousness, entitled to no more consideration than the familiar
fallacy that vaulted nave of a Gothic church was an attempt to imitate
the green aisles of a forest. It should be remembered however that the
habit of the thought of that time was mystical, as that of our own age
is utilitarian and scientific; and the chosen language of mysticism is
always an elaborate and involved symbolism. What could be more natural
than that a building devoted to the worship of a crucified Savior
should be made a symbol, not of the cross only, but of the body
crucified?

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

[Illustration 46]

The _vesica piscis_ (a figure formed by the developing arcs of two
equilateral triangles having a common side) which in so many cases
seems to have determined the main proportion of a cathedral plan--the
interior length and width across the transepts--appears as an aureole
around the figure of Christ in early representations, a fact which
certainly points to a relation between the two (Illustrations 42,
43). A curious little book, _The Rosicrucians_, by Hargrave Jennings,
contains an interesting diagram which well illustrates this conception
of the symbolism of a cathedral. A copy of it is here given. The apse
is seen to correspond to the head of Christ, the north transept to his
right hand, the south transept to the left hand, the nave to the body,
and the north and south towers to the right and left feet respectively
(Illustration 44).

[Illustration 47]

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

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

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

The geometrical plane figures which play the most important part in
architectural proportion are the square, the circle and the triangle;
and the human figure is intimately related to these elementary forms.
If a man stand with heels together, and arms outstretched horizontally
in opposite directions, he will be inscribed, as it were, within a
square; and his arms will mark, with fair accuracy, the base of an
inverted equilateral triangle, the apex of which will touch the ground
at his feet. If the arms be extended upward at an angle, and the
legs correspondingly separated, the extremities will touch
the 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 establishing
numerical ratios between its parts (Illustrations 47, 48, 49). Some
of these are so simple and easily remembered that they have obtained
a certain popular currency; such as that the length of the hand equals
the length of the face; that the span of the horizontally extended
arms equals the height; and the well known rule that twice around
the wrist is once around the neck, and twice around the neck is once
around the waist. The Roman architect Vitruvius, writing in the age of
Augustus Cæsar, formulated the important proportions of the statues
of classical antiquity, and except that he makes the head smaller than
the normal (as it should be in heroic statuary), the ratios which
he gives are those to which the ideally perfect male figure should
conform. Among the ancients the foot was probably the standard of all
large measurements, being a more determinate length than that of the
head or face, and the height was six lengths of the foot. If the head
be taken as a unit, the ratio becomes 1:8, and if the face--1:10.

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

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

V

LATENT GEOMETRY

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

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

[Illustration 52: PROPORTIONS OF THE HORSE]

[Illustration 53]

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

[Illustration 54]

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

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

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

[Illustration 57: ASSYRIAN; GREEK]

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

[Illustration 59]

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

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

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

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

[Illustration 63]

[Illustration 64: THE EQUILATERAL TRIANGLE IN ROMAN ARCHITECTURE]

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

[Illustration 65: THE EQUILATERAL TRIANGLE IN ITALIAN ARCHITECTURE]

[Illustration 66: THE HEXAGRAM IN GOTHIC ARCHITECTURE]

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

[Illustration 67]

[Illustration 68]

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

[Illustration 69]

[Illustration 70]

The square is principally used in preliminary plotting: it is the
determining figure in many of the palaces of the Italian Renaissance;
the Arc de Triomphe, in Paris is a modern example of its use
(Illustrations 59, 60). The circle is often employed in conjunction
with the square and the triangle. In Thomas Jefferson's Rotunda for
the University of Virginia, a single great circle was the determining
figure, as his original pen sketch of the building shows (Illustration
61). Some of the best Roman triumphal arches submit themselves to a
circular synopsis, and a system of double intersecting circles has
been applied, with interesting results, to façades as widely different
as those of the Parthenon and the Farnese Palace in Rome, though
it would be fatuous to claim that these figures determined the
proportions of the façades.

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

It may be stated as a general rule that whenever three important
points in any architectural composition coincide (approximately or
exactly) with the three extremities of an equilateral triangle, it
makes for beauty of proportion. An ancient and notable example occurs
in the pyramids of Egypt, the sides of which, in their original
condition, are believed to have been equilateral triangles. It is a
demonstrable fact that certain geometrical intersections yield the
important proportions of Greek architecture. The perfect little
Erechtheum would seem to have been proportioned by means of the
equilateral triangle and the angle of 60 degrees, both in general and
in detail (Illustration 62). The same angle, erected from the central
axis of a column at the point where it intersects the architrave,
determines both the projection of the cornice and the height of
the architrave in many of the finest Greek and Roman temples
(Illustrations 67-70). The equilateral triangle used in conjunction
with the circle and the square was employed by the Romans in
determining the proportions of triumphal arches, basilicas and
baths. That the same figure was a factor in the designing of Gothic
cathedrals is sufficiently indicated in the accompanying facsimile
reproductions of an illustration from the Como Vitruvius, published in
Milan in 1521, which shows a vertical section of the Milan cathedral
and the system of equilateral triangles which determined its various
parts (Illustration 71). The _vesica piscis_ was often used to
establish the two main internal dimensions of the cathedral plan: the
greatest diameter of the figure corresponding with the width across
the transepts, the upper apex marking the limit of the apse, and the
lower, the termination of the nave. Such a proportion is seen to be
both subtle and simple, and possesses the advantage of being easily
laid out. The architects of the Italian Renaissance doubtless
inherited certain of the Roman canons of architectural proportion,
for they seem very generally to have recognized them as an essential
principle of design.

[Illustration 71]

Nevertheless, when all is said, it is easy to exaggerate the
importance of this matter of geometrical proportion. The designer
who seeks the ultimate secret of architectural harmony in mathematics
rather than in the trained eye, is following the wrong road to
success. A happy inspiration is worth all the formulæ in the world--if
it be really happy, the artist will probably find that he has
"followed the rules without knowing them." Even while formulating
concepts of art, the author must reiterate Schopenhauer's dictum
that the _concept_ is unfruitful in art. The mathematical analysis
of spatial beauty is an interesting study, and a useful one to the
artist; but it can never take the place of the creative faculty, it
can only supplement, restrain, direct it. The study of proportion is
to the architect what the study of harmony is to a musician--it helps
his genius adequately to express itself.

VI

THE ARITHMETIC OF BEAUTY

Although architecture is based primarily upon geometry, it is possible
to express all spatial relations numerically: for arithmetic, not
geometry, is the universal science of quantity. The relation of masses
one to another--of voids to solids, and of heights and lengths to
widths--forms ratios; and when such ratios are simple and harmonious,
architecture may be said, in Walter Pater's famous phrase, to
"aspire towards the condition of music." The trained eye, and not an
arithmetical formula, determines what is, and what is not, beautiful
proportion. Nevertheless the fact that the eye instinctively rejects
certain proportions as unpleasing, and accepts others as satisfactory,
is an indication of the existence of laws of space, based upon number,
not unlike those which govern musical harmony. The secret of the deep
reasonableness of such selection by the senses lies hidden in the very
nature of number itself, for number is the invisible thread on which
the worlds are strung--the universe abstractly symbolized.

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

[Illustration 72: A GRAPHIC SYSTEM OF NOTATION]

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

[Illustration 73]

[Illustration 74]

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

[Illustration 75]

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

[Illustration 76]

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

[Illustration 77]

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

[Illustration 78]

[Illustration 79]

[Illustration 80]

[Illustration 81]

Every building is a symbol of some number or group of numbers, and
other things being equal the more perfect the numbers involved the
more beautiful will be the building (Illustrations 76-82). The numbers
5 and 7--those which occur oftenest--are the most satisfactory because
being of small quantity, they are easily grasped by the eye, and being
odd, they yield a center or axis, so necessary in every architectural
composition. Next in value are the lowest multiples of these numbers
and the least common multiples of any two of them, because the
eye, with a little assistance, is able to resolve them into their
constituent factors. It is part of the art of architecture to
render such assistance, for the eye counts always, consciously or
unconsciously, and when it is confronted with a number of units
greater than it can readily resolve, it is refreshed and rested if
these units are so grouped and arranged that they reveal themselves as
factors of some higher quantity.

[Illustration 82]

[Illustration 83: A NUMERICAL ANALYSIS OF GOTHIC TRACERY]

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

[Illustration 84: NUMERATION IN GROUPS EXPRESSED ARCHITECTURALLY]

[Illustration 85: ARCHITECTURAL ORNAMENT CONSIDERED AS THE
OBJECTIFICATION 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 quantity
sufficiently complex to stimulate the eye to resolve it, and yet so
simple that it can be analyzed at a glance; as a center with two equal
sides, it is possessed of symmetry, and as the sum of an odd and even
number (3 and 4) it has vitality and variety. All these properties a
work of architecture can variously reveal (Illustration 77). Fifteen,
also, is a number of great perfection. It is possible to arrange the
first 9 numbers in the form of a "magic" square so that the sum of
each line, read vertically, horizontally or diagonally, will be 15.
Thus:

     4    9    2   =  15
     3    5    7   =  15
     8    1    6   =  15
    --   --   --
    15   15   15

Its beauty is portrayed geometrically in the accompanying figure which
expresses it, being 15 triangles in three groups of 5 (Illustration
86). Few arrangements of openings in a façade better satisfy the eye
than three superimposed groups of five (Illustrations 76-80). May not
one source of this satisfaction dwell in the intrinsic beauty of the
number 15?

In conclusion, it is perhaps well that the reader be again reminded
that these are the by-ways, and not the highways of architecture: that
the highest beauty comes always, not from beautiful numbers, nor
from likenesses to Nature's eternal patterns of the world, but from
utility, fitness, economy, and the perfect adaptation of means to
ends. But along with this truth there goes another: that in every
excellent work of architecture, in addition to its obvious and
individual beauty, there dwells an esoteric and universal beauty,
following as it does the archetypal pattern laid down by the Great
Architect for the building of that temple which is the world wherein
we dwell.

VII

FROZEN MUSIC

In the series of essays of which this is the final one, the author has
undertaken to enforce the truth that evolution on any plane and on
any scale proceeds according to certain laws which are in reality only
ramifications of one ubiquitous and ever operative law; that this law
registers itself in the thing evolved, leaving stamped thereon as it
were fossil footprints by means of which it may be known. In the arts
the creative spirit of man is at its freest and finest, and nowhere
among the arts is it so free and so fine as in music. In music
accordingly the universal law of becoming finds instant, direct
and perfect 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 gives
validity to the before quoted saying that all of the arts "aspire
toward the condition of music." All aspire to express the law, but
music, being least encumbered by the leaden burden of materiality,
expresses it most easily and adequately. This being so there is
nothing unreasonable in attempting to apply the known facts of musical
harmony and rhythm to any other art, and since these essays concern
themselves primarily with architecture, the final aspect in which
that art will be presented here is as "frozen music"--ponderable form
governed by musical law.

Music depends primarily upon the equal and regular division of time
into beats, and of these beats into measures. Over this soundless and
invisible warp is woven an infinitely various melodic pattern, made
up of tones of different pitch and duration arithmetically related
and combined according to the laws of harmony. Architecture,
correspondingly, implies the rhythmical division of space, and
obedience to laws numerical and geometrical. A certain identity
therefore exists between simple harmony in music, and simple
proportion in architecture. By translating the consonant
tone-intervals into number, the common denominator, as it were, of
both arts, it is possible to give these intervals a spatial, and
hence an architectural, expression. Such expression, considered as
proportion only and divorced from ornament, will prove pleasing to
the eye in the same way that its correlative is pleasing to the ear,
because in either case it is not alone the special organ of sense
which is gratified, but the inner Self, in which all senses are one.
Containing within itself the mystery of number, it thrills responsive
to every audible or visible presentment of that mystery.

[Illustration 87]

If a vibrating string yielding a certain musical note be stopped in
its center, that is, divided by half, it will then sound the octave
of that note. The numerical ratio which expresses the interval of
the octave is therefore 1:2. If one-third instead of one-half of the
string be stopped, and the remaining two-thirds struck, it will yield
the musical fifth of the original note, which thus corresponds to the
ratio 2:3. The length represented by 3:4 yields the fourth; 4:5 the
major third; and 5:6 the minor third. These comprise the principal
consonant intervals within the range of one octave. The ratios of
inverted intervals, so called, are found by doubling the smaller
number of the original interval as given above: 2:3, the fifth, gives
3:4, the fourth; 4:5, the major third, gives 5:8, the minor sixth;
5:6, the minor third, gives 6:10, or 3:5, the major sixth.

[Illustration 88: ARCHITECTURE AS HARMONY]

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

A natural deduction from these facts is that relations of
architectural length and breadth, height and width, to be "musical"
should be capable of being expressed by ratios of quantitively small
numbers, preferably an odd number and an even. Although generally
speaking the simpler the numerical ratio the more perfect the
consonance, yet the intervals of the fifth and major third (2:3 and
4:5), are considered to be more pleasing than the octave (1:2), which
is too obviously a repetition of the original note. From this it is
reasonable to assume (and the assumption is borne out by experience),
that proportions, the numerical ratios of which the eye resolves too
readily, become at last wearisome. The relation should be felt rather
than fathomed. There should be a perception of identity, and also
of difference. As in music, where dissonances are introduced to give
value to consonances which follow them, so in architecture simple
ratios should be employed in connection with those more complex.

[Illustration 89]

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