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THE BEAUTIFUL
NECESSITY

THE BEAUTIFUL
NECESSITY Seven Essays on
Theosophy and Architecture

by
Claude Bragdon

ROCHESTER, N. Y.
THE MANAS PRESS
1 9 io

Copyright KflO
h
Claude Bragdon

"Let us build altars to the Beautiful Necessity" — Emerson

CONTENTS i
The Theosophic View of the Art of Architecture 9-21
II
Unity and Polarity . . . . . .22-32
III
Changeless Change . . . . . .33-49
IV
The Bodily Temple . . . . . 5° -59
V
Latent Geometry . . . . . .60-75
VI
The Arithmetic of Beauty . . . . .76-84
VII
Frozen Music  85-92
Conclusion ........ 93
lo/jf/i^ 15" 12 9 1

THE THEOSOPHIC VIEW OF THE
ART OF ARCHITECTURE
ONE of the many 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 was in
tricate and obscure. Let us apply this key to the subject of art, and to the
art of architecture in particular, and let us 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 on the
physical plane our only avenue of knowledge is sense perception, a more
exact expression of the theosophic idea would be: Everything is the ex
pression 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 so 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 rain
bow-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 ex
tremes. That such is their relative position may be demonstrated in various [9l

io THE BEAUTIFUL NECESSITY i
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.* 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, if we were only keen enough to per
ceive 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 manifesta
tion, and in time alone everything would occur successively, one thing follow
ing 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
proportions, either actually or by implication. Poetry and the drama are
allied to music insomuch 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 ex
pressed 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 mid-
course, 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
*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. Schopen
hauer says : "Polarity, or the sundering of a force into two quantitatively 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."

i THE ART' OF ARCHITECTURE n
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, con
vertible 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 state
ment 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 expres
sion 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 best understood through their cor
responding correlatives in the natural world, for it is a fundamental theo
sophic 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 informs matter is a type of time
within space ; the mind working in and through the body is another expres
sion of the same thing. Accordingly, 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

12 THE BEAUTIFUL NECESSITY i
accentuates and enriches, painting adorns it, 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 readily apparent.
Music is interior, abstract, subjective, speaking directly to the soul in a
simple and universal language whose meaning is made personal and particu
lar in the breast of each listener : "Music alone of all 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 require
ments. Music is pure beauty, the voice of the unfettered and perpetually
evanishing soul of things; architecture is that soul imprisoned in a form,
become subject to the law of casualty, 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 in 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 implied, for physical re-birth is but a single mani
festation of that universal law of alternation of state, of animation of vehicles,
and progression through successive 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, carry
ing along and bringing into manifestation in each higher arc of the spiral the
experience accumulated in all preceding states, and at the same time unfold
ing that power of the Self peculiar to the plane in which it happens to be
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

i THE ART OF ARCHITECTURE 13
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 what is popularly
known as the 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 suc
ceeded 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 which 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,

14 THE BEAUTIFUL NECESSITY i
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 either
side, 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 awfulness 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 agony
and toil 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 fleshly 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 typified 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

i THE ART OF ARCHITECTURE 15
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 perfec
tion 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 har
mony 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 appro
priate 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.
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 func
tion, 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

i6

THE BEAUTIFUL NECESSITY

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 i).
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 expres
sion, not in temples to the high gods, but rather in those vast and complicated

i THE ART OF ARCHITECTURE 17
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 degen
eracy: 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 vault
ing, 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 organ
ism 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

18 THE BEAUTIFUL NECESSITY i
strange, composite figures to typify transcendental ideas, the sphinx seem
ing a blood brother to the gargoyle. The conditions under which each archi
tecture flourished were not dissimilar, for each was formulated and con
trolled 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 towards 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 Mediaeval Europe, as in
ancient Egypt, fragments of the Secret Doctrine — 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 that 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 Renais
sance, 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

THE ART OF ARCHITECTURE

19

it formal, cold, symmetrical; while with Sansovino and Sammichele it
became sumptuous and bombastic.
As the Renaissance ripened to its decay, 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 towards
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 ancient
verities. In studying these salient 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.

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

20

THE BEAUTIFUL NECESSITY

been in Egyptian architecture the predominant one. A vertical line, there
fore, 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 semi-circular 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

i THE ART OF ARCHITECTURE 21
inclined lines, and these united to the preceding symbol yield an accurate
abstract of the elements of Gothic architecture (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 depart
ment 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.

II
UNITY AND POLARITY
THEOSOPHY, both as a doctrine, or system of thought which dis
covers correlations between things apparently unrelated, and as a life,
or system of training whereby it is possible to gain the power to per
ceive 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 lay
man, the critic, the art amateur — to anyone, in fact, who would come to an
accurate and intimate understanding and appreciation of every variety of
aesthetic 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, that is, of sounds, odors, forms,
motions, colors, so mathematically related and co-ordinated 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
[22]

n UNITY AND POLARITY 23
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 of set rules, for the true artist works uncon
sciously, 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 lesson of the Good Law everywhere portrayed in the vast
picture-book 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 co
ordination 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 co-ordinated 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 might
manifest, incloses himself within his limiting veil, Maya, and that His Life
appears as Spirit (male) and his Maya as Matter (female), the two being
never disjointed 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

24 THE BEAUTIFUL NECESSITY n
discerned to subsist between corresponding members of such pairs of oppo
sites : sun, day, fire, man, express and embody the primal and active aspect of
the manifesting deity; moon, night, water, woman, its secondary and pas
sive. Moreover, each in a sense implies, or brings to mind, the others of its
class: man, like the sun is lord of day, a direct and devastating force like
fire;' 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 the minor modes : the typical, or
representative chords of the dominant seventh and of the tonic (the two
chords into which Schopenhauer affirms 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. 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 supporting
member, which resists the force of gravity, and the horizontal member, or
lintel, which succumbs to it ; they are vertical lines, which are aspiring, effort
ful, 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 a 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 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 com
plex, indirect, derivative, passive, negative. Things hard, straight, fixed,

II

UNITY AND POLARITY

25

^xJjVO

VO

IN i

IN-

It^

f ]

™~\S
tvo
WILD GHEMIY
MAPLE. LEAF
CALLAjU
vertical are Yo; things
soft, curved, horizontal,
fluctuating are In, and
so on.
In passing it may be
said that the superiority of
the line, mass, and color
composition of Japanese
prints and kakemonos to
6 7 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 con
scious 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 comple
ment in the curved lily-pads which lie horizontally on its surface. Trees
such as the 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, simultane
ously (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 simultaneous-
ness; 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 art 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
26

THE BEAUTIFUL NECESSITY

ii

THE LAW OF POLARITY

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).
With reference to architecture, as a general proposition it is true 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 en
croaching upon the nave; the main
lines of a buttress conform to the
direction of the thrust, and the pin
nacle with which it terminates is a
logical shape for the masonry neces
sary to hold the top in position (Il
lustration 9). Research along these
lines is very 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 sexu
ally expressive, as it were, the con
struction itself is sometimes weak
ened or falsified. The familiar
classic console or modillion is an a
example: although in general con
tour 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

CLEOPATRA MELTING THE
PEARL. £>Y TIEPOLO

II

UNITY AND POLARITY

27

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 expressive
ness 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 mascu
line character — as a supporting member resisting the force
of gravity — is emphasized.
The importance of the so-called "orders" lies in the
9 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 con
densed 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 so-called Classic Orders, is a
gain in functional and sex expressiveness, and
consequently in art (Illustration 11).
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 (Il
lustration 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

/^^^U^PWo

-IN

COEJNTHIAN MODILUON
^ — ~?=Z CLASSIC
\*±W<0 CONSOLE

28

THE BEAUTIFUL NECESSITY

n

nature. There is an essential identity between
the Ionic cap and the Classic console before
referred to, although superficially the two do not
resemble one another, as a straight line and a
double spiral are elements common to both (Il
lustration io). 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, con
trasted with the complex, deliquescent, rounded ones from which they spring.
The essential identity of principle subsisting between this cap and the Renais
sance baluster by San Gallo is apparent (Illustration 13).
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 channel triglyphs and the sculptured
metopes of a Doric frieze, in the straight and vertical mullions and the
flowing tracery of Gothic

-YO

-IN

windows, in the banded
torus, the bead and reel,
and other familiar orna
mented mouldings (Il
lustrations 14, 15, 16).
There are indications
that at some time dur
ing 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

GRjBOAN DORIC CAP

lONIG GAP

II

UNITY AND POLARITY

29

bodily structure. The ma
sonic guilds of the Middle
Ages were custodians of the
esoteric — which is the the
osophic — side of the Chris
tian faith, and every student
of the Secret Doctrine knows
how fundamental and far-
reaching is this idea of sex.
The entire cathedral sym
bolized the crucified body of
Christ; its two towers, man
and woman — that Adam and
Eve, for whose redemption,
according to popular belief,
Christ suffered and was cru
cified. The north, or right
hand tower ("the man's
side") was called the sacred
male pillar, Jachin; and the

Yo>

IN-t-

CAPITAL- FROM
THE/ TOWER. OF
THE. WIND*. ATHENS.

ROSETTE. FROM
TEMPLE. OF MARS,
EOME

THE?

CORINTHIAN CAP
FROM HADRIAN
BUILDINGS. ATHENS.

j*YO

CAUUCULOJII
OT CORINTHIAN CAP

BALUSTER) J3Y SAN GALLO.

13

EGGANDTONGUE.
Y3 IN

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 of 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
facade, 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 (Illus
tration 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

BEAD AND K£M,

EANDED TOEUS

3°

THE BEAUTIFUL NECESSITY

n

VO IN YO IN

i n| n n In n n it n n n n
wU
FRIEZE OF TH5 FARNESE PALACE

YO

¦y'.'V.t.1^

ROMAN CONSOLE.
VATICAN MUSEUM

FRIEZE IN THL EMPIRE STYLE
BY PERCIER AND FONTAINE.

FRIEZE FROM THE
TEMPLE OF VESTA
AT TIVOLI.(ROMAN)

ROMAN DORIC FRIEZE  VIGNOLE

15
other important difference appears to be in the angular label mould above
the north entrance : whatever may have been its original function or signifi
cance, 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 indica
tion 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. 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

II

UNITY AND POLARITY

3i

YD IN

EGYPTIAN ORNAMENT
YO.tN 
GREEK, ORNAMENT
YO IN

PALAZZO GIRAUD. ROiVfi
YO IN

T

a

PALAZZO FARNESE BONE
VOIN

rails
GREEK, HONEYSUCKLE

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.
The vital, organic quality so conspicuous
in the best
Gothic archi
tecture has
been attributed
to the fact that
necessity deter
mined its char
acteristic 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.
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 be
neath, 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 decora
tion 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

16

GOTHIC/

r^

CLASSIC

THE ELEMENTS OK GOTHIC CO
AND OF GLASSIG(2> ARCHITECTURE

18

32

THE BEAUTIFUL NECESSITY

n

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, cele
brates 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 make such visible music to the entranced eye.
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
facade to the shaping of a moulding —
the architectural designer will charge
his work with that esoteric significance,
that excess of beauty, by which archi
tecture rises to the dignity of a "fine"
art (Illustrations 19, 20'j. In so doing,
however, he should never forget, and
the layman, also, should ever remember,
that the supreme architectural excel
lence is fitness, appropriateness, the
perfect adaptation of means to ends,
and the perfect 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.

YW

SAN OJMIQNANO 5. JACOPO,

20

Ill
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 fulfil
ment ; 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 archi
tecture 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 accom
plished 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
resembles 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 striv
ing 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, which epitomizes all speech, the u sound effects the
transition between the a sound and the m; among primary colors yellow
comes between red and blue; and in architecture the arch, which is both
[33]

34

THE BEAUTIFUL NECESSITY

m

THB LAW OF TRINITY

A ROMAN IONIC ARCADE. BV
VIONOLE~rH£. COLUMN. THE.
ENTABLATURE AND THE, ARCH
CORRESPOND TO LINES VERT
ICAL HORIZONTAL AND CURVED

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 Sulli
van, "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."
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 ansated cross of the
Egyptians is such a symbol, the Buddhist
wheel, and the flyflot, 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 the reason, since 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

21

Ill

CHANGELESS CHANGE

35

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'etre in that it serves
to relieve the lintel of the superincumbent weight of masonry. The same ar
rangement 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

THE LAW OF TRINITY

number of architecture, because
it is the number of our space,
which is three-dimensional, and
of all the arts architecture is
most concerned with the expres
sion of spatial relations. The
division of a composition into
three related parts is so universal
that it would seem to be the re
sult 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 corre
spondence in the two towers of
a Gothic cathedral and the inter
vening screen wall of the nave.
In the palaces of the Renaissance
a three-fold 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 (Illustra
tion 23).
The architectural "orders," so-called, 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 :

THE TRINITY OE HORIZONTAL-
^^VElcTICAL.AND CURVED LINES

22

36

THE BEAUTIFUL NECESSITY

in

THE LAW OF TRINITY: A THREEFOLD DIS
POSITION OF THE PARTS OF A BUILDING— 3

0OTHIC- NOTRE. DAME..
1  ,-£-,  3

ITALIAN RENAISSANCE.
B^LAZZO VENDRAM1N-
CALEEOl AT VENICE.

EGYPTIAN-FRONT OF TEMPLE,.

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 lit
erature of theosophy, and
naturally so, for it is a
succint statement of a
fundamental and far-
reaching truth. The scien
tist 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 im
pulse which calls a uni
verse 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, up and up and up, and forever and ever
and ever.
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 part of the ebbing
and flowing tide. In music this law is illustrated in the return of the tonic

M^

— a — r

tMUIIUra nS|H

xmmammm

:.»iuM:«j^

iDldlOj

I FRENCH RENAISSANCE- OUD6AIJ

BUArZOIiAWOUNI.FLOMNa| DE BEAUMESNIL.

Ill

CHANGELESS CHANGE

37

THE. LAW OF CONSONANCE-! REPETITION WITH VARIATION

THE. DOME OF THE
CATHEDRAL OF
FLORENCE

THE SMALL DOMES
PREPARE, THE. EYE- FOIL.
i THE GREAT DOME.

[CORINTHIAN CAPITAL AND
_1 ENTABLATURE. SHOWING
GOERESPONDENGE BETWEEN
THE/IRJ VARIOUS PARTS.

THE BEAD AND REEL
ECHO THE EGO AND
TONGUE

THE CHANNELED TRI
GLYPHS ABOVE ECHO
THE COLUMNS BELOW

24
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.
In the drama the repetition of a speech, or of an entire scene, but under
circumstances which give it a different meaning, is often very 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 varia
tions are common in architecture also. The channeled triglyphs of a Greek

38

THE BEAUTIFUL NECESSITY

in

THE LAW OF
CONSONANCE

ONE BAY CFTHE
"ANGEL CHOIRS OF
LINCOLN CATFEDRAL

Doric frieze echo the fluted columns
below (Illustration 24). The balus
trade which crowns a colonnade is
a repetition, in some sort, of the
colonnade itself. The modillions of
a Corinthian cornice are but elabor
ate 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 pointed
arch and gable, the column and entablature everywhere recur under different

THE SAME MOTIF
REPEATED WITH
VARIATIONS'——

25

THE- LAW OF CONSONANCE! REPETITION ^VARIATION
l

CHATEAU MAINTENON.~THk CENTRAL PAVILION WITH ITS TWO
TURRETS ECHOES THE ENTIRE. FACADE. WITH ITS TWO TOWE,R,$

26

Ill

CHANGELESS CHANGE

39

THE LAW OF
CONSONANCE

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 Chateau
Maintenon, with its two turrets, echoes the entire facade with its two towers.
Like the overture to an opera, it introduces themes
which find a more extended development elsewhere
(Illustration 26).
This law of Consonance is operative in archi
tecture 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 sub
stantially 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 manifesta
tions; hence, inevitably, echoes, resemblances — Consonance.

MV
PATTERN FROM &
GOIOMAL BEDSPREAD

27

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 similar 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

40

THE BEAUTIFUL NECESSITY

in

THE LAW OF DIVERSITY IN MONOTONY

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the presentment of the truth that identity does not exclude individuality.
The law is binding, yet the will is free: all men are brothers, bound by the
ties of brotherhood, yet each is unique, a free agent, and never so free as
Ill

CHANGELESS CHANGE

4i

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29
when most bound by the Good Law. This truth nature beautifully pro
claims, 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 similar 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 Mitelus, — each one a separate masterpiece of
various ornamentation adorning an established architectural form (Illus
tration 28) .
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 build
ers 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

42 THE BEAUTIFUL NECESSITY in
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 one
another. 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 higher
planes, 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 length and
elements, but involving a different succession 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
exhibition of the law occurs when the opposed elements do not exactly
match, but differ from one another, 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 towards refinement and com
pactness of form in space, or contrariwise, towards 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
in 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 has ever affirmed, 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 shape assumed by the ultimate physical atom, and the ulti
mate physical atom is the physical cosmos in miniature.

in CHANGELESS CHANGE 43
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 to appear to grow lighter, more intricate, finer,
from the ground upwards; 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 Colos
seum, or in the Palazzo Uguccioni, in Florence, to mention only two
examples out of a great number. In the Riccardi Palace an effect of increas
ing 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 elab
orately, 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 princi
ple, 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,

44

THE BEAUTIFUL NECESSITY

in

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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 towards

Ill

CHANGELESS CHANGE

45

which the lines of the walls and ceiling con
verge centering in the head of Christ. Puvis
de Chavannes, in his Boston Library decora
tion, leads the eye, by a system of triangula-
tion, to the small figure of the Genius of
Enlightenment above the central door (Illus
tration 32) ; and Ruskin, in his Elements of
Drawing, has shown how artfully • Turner
arranged some of his composition to attract
attention to a focal point.
This law of Radiation enters largely into
architecture. The Colosseum, based upon the

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46

THE BEAUTIFUL NECESSITY

in

THE UWCF ILADIATION
ILLUSTRATE!) IN NATURE

ellipse, a figure generated from two
points, or foci, and the Pantheon, based
upon the circle, a figure generated from
a single center, are familiar examples.
The distinctive characteristic of Gothic
construction, the concentration or focal-
ization of the weight of the vaults and
arches at certain points, is another illus
tration of the same principle applied to
architecture, beautifully exemplified in
the semi-circular 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.
The tracery of the great roses high
up in the facades of the cathedrals of 33
Paris and of Amiens illustrate Radiation, — in the one case masculine:
straight, angular, direct; in the other feminine: curved, flowing, sinuous.
The same Beautiful
Necessity determined the
characteristics of much
of the ornament of wide
ly separated styles and
periods : the Egyptian
lotus, the Greek honey
suckle, the Roman acan
thus, 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

SHOW CK/JTAL

WTERi ERNEST
THOMPSON SETON

STON . . *WWW£W>^ . 61

THE ART ANAT
OMY OF ANIMALS

'WJ

THE LAWS OF RADIATION AND OF R.HYTHM1G
DIMINUTION ILLUSTRATED BY A PEACOCK'S TRAIN

34

Ill

CHANGELESS CHANGE

47

THE UW OF RADIATION

FRENCH RENAISSANCE

BYZANTINE.

35

gives that ornament a unity,
a coherence, and a kind of
beauty all its own (Illustra
tion 35).
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 ex
hausted, but it is not the
multiplicity and diversity of
laws which it is important to
keep in mind, so much as
their essential unity and co-ordination, for they are but different aspects of
the One Law, that whereby the Logos manifests himself 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.
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 react
ing upon one another to
produce a third, hence the
law of Trinity. Given this
third term, and the innumer
able combinations made pos
sible by its relations to and
reaction 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 combination 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,

48 THE BEAUTIFUL NECESSITY ni
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, as it were, and the aptness and complete
ness 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 neces
sary 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 every one 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 fulfillment 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 hori
zontal 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 sup
ports 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

iii CHANGELESS CHANGE 49
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 sub
ordinate 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 and the manner of their fulfilment. The architect
who is also an artist, however, will do this and something beyond. Working
for the most part unconsciously, harmoniously, joyfully, 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 missions 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 recon
cilement 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. Pur
sued 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 regard 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
[5°1

rv

THE BODILY TEMPLE

5i

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THE. LAW OP EJtVTHMlC
raNLLNUTION 1WU$TRKZD
USTTIfc TAPERING BOD*
J DIME*. FING&R>$ & TOES.

37

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 articula
tion; and in this way is effected that transition from unity
to multiplicity, from simplicity to complexity, which ap
pears to be so universal throughout nature, and of which a
tree is the perfect symbol.
The body is rich in veiled repetitions, echoes, conson
ances. 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

52

THE BEAUTIFUL NECESSITY

iv

ANTEFIX
V-THE HAND ODOJED

ACOMEARISON BETWEEN THE. HAND.CLOJBD
AND OPEN. AND THE. GEE.EK ANTEFDC..AND SHELLS

(Illustrations 37, 38). Finally, the limbs
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 (Illus
tration 39).
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 micro
cosm — 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 alto
gether fanciful between the animate body
of flesh and the inanimate body of stone.

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 diminu
tion is illustrated in the taper
ing of the entire body and of
the limbs, in the graduated
sizes and lengths of the
fingers and the toes, and in
the successively decreasing
lengths of the palm and of
the joints of the fingers, so
that in closing the hand the
fingers describe natural spirals
radiate as it were from the trunk,

A COMEAMSON BBTWEfW GIOTTO'S
C&MEAN1LR A COLUMN JfiOMTHE EAfc
THENON.AND THE. HOMAN JHGttR&~

40

IV

THE BODILY TEMPLE

53

THE BODY THE ARCHETYPE
OF SACRED EDIFICES ;°™an
" feqoTHiC

THE KANDAE1VA CHURCH <5P -ST.
TEMPLE, KHAJUEAHO OUEN AT ROUEN

Do we not all of us, consciously or
unconsciously, recognize the fact of
character and physiognomy in build
ings? Are they not, to our imagina
tion, masculine or feminine, winning
or forbidding — human, in point of
fact — to a greater degree that any
thing else of man's creating? They
are this certainly to the 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 across the city or 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 as

54

THE BEAUTIFUL NECESSITY

iv

unmistakably masculine: the ratio of their breadth to their height is the
ratio of the breadth to the height of a man (Illustration 40).
At certain periods of the world's history, periods of mystical enlighten
ment, 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

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43

great bronze Buddha of Japan, with its dreaming eyes; the little known
colossal figures of India — 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 propor
tions, 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

IV

THE BODILY TEMPLE

55

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established, the idea is sus
ceptible of almost infinite de
velopment. That the ancients
proportioned their temples
from the human figure is no
new idea, nor is it at all sur
prising. The sculpture of the
Egyptians and the Greeks re
veals the fact that they studied
the body abstractly, in its ex
terior presentment. It is clear
that the rules of its proportions
must have been established for
sculpture, and it is not un
reasonable to suppose that they
became canonical in architec
ture also. Vitruvius and Al-
berti both lay stress on the fact
that all sacred buildings should
be founded on the proportions
of the human body.
In France, during the Mid
dle 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 the 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 Saviour should be made a symbol, not of the cross only, but
of the body crucified ?
The vesica piscis (a figure formed by the developing arcs of two

SOUTH CELESTIAL POLE  >
SUN SETS— WEST
OMEGA
THE SYMBOLISM OF A GOTHIC CATHEDRAL
HCM'THB BOJICEUCIANSf HAKi&ft/E JENNINGS

56

THE BEAUTIFUL NECESSITY

iv

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-THIGHS
C» PATELLA

LEG Vi

THE ItELiATIOH OP THE. MOURE) TO
THE.^OtJAKE./TH& OII£CLE.,ANDTHE
BOTTn^ATEEAL. TEIAKGL1,

THE mOPPSStJOHS OFTHE/
FIGUJSE AS ESTABLISHED fV
DOGTOI6 lolMMEJi).

45

equilateral triangles having common side) which in
so many cases seems to have determined the main
proportion of a cathedral plan — the interior length
and the width across the transepts — appears as an
aureole around the figure of Christ in early repre
sentations, a fact which certainly points to a relation
between the two (Illustrations 42, 43). A curious
little book, The Rosicrucians, by Hargrave Jen
nings, 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 with the head of Christ, the north
transept to his right hand, the south transept to the left hand, the nave to the
body, and the north and south towers to the right and left feet respectively
(Illustration 44).
The cathedral builders excelled all others in the artfulness with which
they established and maintained a relation between their architecture and
the stature of a man. This is perhaps one reason why the French and

IV

THE BODILY TEMPLE

57

47

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 archi
tecture is not necessarily the most
awe-inspiring which gives the im
pression of having been built by
giants for the abode of pigmies; like the other arts, architecture is highest
when it is most human. The mediaeval builders, true to this dictum, em
ployed 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 circum
ference of each shaft of such a cluster approximated
the girth of a man; by this device the mouldings 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 column below, as in classic architecture, but
with relation to a man's stature.
It may be stated as a general rule that every work
of architecture, of whatever style, should have some
where 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 same 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

^IGfURiB nrvro&D
ACCOEDTNGTOTffi, EGYPTIAN- CANON

48

58

THE BEAUTIFUL NECESSITY

iv

TIE T!a0T%TffiHEK3HT

SCISSORS SHAPE

THE MEDIAEVAL METHOD
O? DRAWING THE HGUEE

49

arches below, but is of a height convenient for
lounging elbows. The door to Giotto's cam
panile reveals the true size of the tower as
nothing else could, because it is so evidently
related to the human figure and not to the
great windows higher up in the shaft.
The geometrical plane figures which play
the most important part in architectural
proportion are the square, the circle and the
triangle; and the human figure is intimately
related to these elementary forms. If a man
stand with heels together, and arms out
stretched horizontally in opposite directions,
he will be inscribed, as it were, within a
square, and his arms will mark, with fair
accuracy, the base of an inverted equilateral
triangle, the apex of which will touch the
ground at his feet. If the arms be extended upward at an angle, and the
legs correspondingly separated, the extremities will touch the circumferences
of a circle having its center in the navel (Illustrations 45, 46).
The figure has been variously analyzed
with a view to establishing numerical
ratios between its parts (Illustrations 47,
48, 49). Some of these are so simple
and easily remembered that they have
obtained a certain popular currency ; such
as that the length of the hand equals the
length of the face; that the span of the
horizontally extended arms equals the
height; and the well known rule that
twice around the wrist is once around
the neck, and twice around the neck is
once around the waist. The Roman
architect, Vitruvius, writing in the age
of Augustus Caesar, formulated the im
portant proportions of the statues of
.„ classical antiquity, and except that he

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iv THE BODILY TEMPLE 59
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 :io.
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 (Illus
tration 45).
Such analyses of the figure, however conducted, reveal 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 formulae, but present in the consciousness as a restraining
influence, acting and reacting upon the mind with a conscious intention
towards 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
THE analysis of the chemical elements by means of clairvoyant vision,
undertaken by Mrs. Besant and Mr. Leadbeater, and lately published
to the world in Occult Chemistry, makes plain the fact that 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 (Illus
trations si, 52, 53). .
By the true artist, in the crystal mirror of whose mind the universal
harmony is focused and reflected, this secret of the cause and source of
rhythm — that it dwells in a correlation of parts based on an ultimate sim
plicity — 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.
[60]

LATENT GEOMETRY

61

THE, HEXAGRAM AND EQU [LATERAL TRIANGLE. IN NATURE
TFE HUMAN© FIGURE,

SNOW CENTALS

HONEY COMB THE. FACE,

FLLSH FLY

The first painter who deliberately
rather than instinctively based his com
position on geometrical principles seems
to have been Fra Bartolomeo, in his Last
Judgment, in the church of St. Maria
Nuova, in Florence. Symonds says of
this picture, "Simple figures — the pyra
mid and triangle, upright, inverted, and
interwoven like the rhymes of a sonnet
— form the basis of the composition.
This system was

53

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 Bartolomeo
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 paint
ing. His Madonna of the Rocks, and Virgin on the

62

THE BEAUTIFUL NECESSITY

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 sym
metry 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 com
position known to their immediate predeces
sors, and the decadence of the art in the
ensuing period may be traced not alone to the false

TJUANqUIV^.rY'NOlVI.S' CFTHE
LAJT JUDqM^NT,F; BAclMOMMEO

THE. EMPLOYMENT OF THE. EQUILATERAL^
TRIANGLE- IN RENAISSANCE PAINTING-

THE.CANGIAW HOLY—
-^-FAMILY BY RAPHAEO,

^snz

Itw

THE LAST SUPPER., CENTER. THE MADONNA DEL. SACCO, BY
ONLY (RESTORED) BY D°c VINCI ANDREA Dbb S*ARiTO

55

54
sentiment and affectation
of the times, but also
in the abandonment by
the artists of those
obscurely geometrical
arrangements and group
ings which, in the works
of the greatest masters,
so satisfy the eye and
haunt the memory of
the beholder (Illustra
tions 55, 56).
Sculpture, even more
than painting, is based
on geometry. The colos
si of Egypt, the bas-
reliefs of Assyria, the
figured pediments and
metopes of the temples
of Greece, the carved
tombs of Ravenna, the
Delia Robbia lunettes,
the sculptured tympani
of Gothic church portals,

LATENT GEOMETRY

63

GEOMETRICAL bASIS OF THE S15TINE CEILING PAINTINGS

56

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, pyr
amids, and cones, — the cylinder appearing in
the column, and the hemisphere in the dome.
The plans, likewise, of the world's famous build
ings, reduced to their simplest expression, are discovered to resolve them
selves into a few simple geometrical figures (Illustration 58).
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

64

THE BEAUTIFUL NECESSITY

cathedrals of the Ile-de-France — are harmoniously proportioned, their prin
cipal 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 co-ordinating the entire fabric into one
easily remembered whole. Upon some such principle is surely founded what
Symonds calls "that severe and lofty art of composition which seeks the
highest beauty of design in architectural harmony supreme, above the
melodies of gracefulness of detail."
There is abundant evidence in support of the theory that the builders
of antiquity, the masonic guilds of the Middle Ages, and the architects of the
Italian Renaissance, knew and followed certain rules ; but though this theory
be denied, or even disproven, if after all these men obtained their results

THE GEOMETRICAL. BASIS CF THE PLAN IN MCHITECTUEAL DESIGN

KEAMANTES PLAN
FOE. JT. PETER'S

TEMPLE CT> ZEUS
AT ASEIQBNTUM

ST PAULS
.LONDON.

<

v

THE DUOMO
/arnatENCE
THE MNTHOTT TIE CHURCH a*
AT KONE S SIMEON STY1TES
r~\
MICHELANQEXO'S
PLANKX,}. PETERS
THECEKim AT PAVIA
NOTRfin^E AT PARIS
© H
•WM!N"$ FI&JT
riflNPCS-S DSU1.S
HOLLAND HOUSE
58
CATHEDRALATKZKAH
SALISBURYCATHEDRAL
o
INIQO JONES' PLAN
rcSLWHlTaHAII,
NF.NHEDM.
LATENT GEOMETRY

65

unconsciously, their creations so
lend themselves to a geometrical
analysis that the claim for the ex
istence of certain canons of propor
tion, based on geometry, remains
unimpeached. The plane figures, principally em
ployed in determining architectural
proportion are the circle, the equi
lateral 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 tetra
hedron, 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 re
lated 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 does not answer it. 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
i sufficient toranind the reader that the circle is the symbol of the universe,
the equilateral triangle of the higher trinity (atma, buddhi, manas), and the
square qfjthe lawer~a/ua?Erinary7^fTnaii's~sevenfold nalure.
The square is principally used in preliminary plotting : it is the determin
ing figure in many of the palaces of the Italian Renaissance ; the Arc d'EtoiUe
in Paris is a modern example of its use (Illustrations 59, 60). The circle is
most often employed in conjunction with the square and the triangle. In
Thomas Jefferson's Rotunda for the University of Virginia, a single great

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66

THE BEAUTIFUL NECESSITY

.EGYPTIAN

QR£EK„

ROMAN

MEDIAEVAL

60
point of geometry, is the equilateral triangle.
has an especial fondness for this figure, jtist
related sounds. Indeed, it might not be too
fjxijifjal^Q_asserJiJliat__tb^
any key (the tonic with its third and jftfthjjs.
the musical equivalent of the equilateral tri
angle. 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 three-fold thing; doubled, its

circle was the determin
ing figure, as his original
pen sketch of the build
ing shows. Some of the
best Roman triumphal
arches submit themselves
to a circular synopsis,
and a system of double
intersecting circles has
been applied, with inter
esting results, to facades
as widely different as
those of the Parthenon
and the Farnese Palace
in Rome, though it would
be fatuous to claim that
these figures determined
the proportions of these
facades. By far the most im
portant figure in archi
tectural proportion, con
sidered from the stand-
It would seem that the eye
as the ear has, for certain

jEPPF;]t(raN1r pen sketch
foilthe rotunda of the
UNIVgltiTTY OV VIRGINIA

LATENT GEOMETRY

67

68

THE BEAUTIFUL NECESSITY

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.

63
It may be stated as a general rule
that whenever three important points
in any architectural composition coin
cide (approximately or exactly) with
the three extremities of an equilateral
triangle, it makes for beauty of propor
tion. An ancient and notable example
occurs in the pyramids of Egypt, the
sides of which, in their original con
dition, are believed to have been equi
lateral triangles. It is a demonstrable
fact that certain geometrical intersec
tions 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 inter
sects the architrave, determines both
the projection of the cornice and the

THE. EOUILATBRALTRJANGLE, IN
ROMAN ARCHITECTUR£

A SECTION OF THE. PANTHEON, ROME.

64

LATENT GEOMETRY

69

THE EQUILATERAL TRIANGLE IN ITALIAN AleJOHITEGTURZ,
(RiNAI-S-SANCE,)

WINDOW IN AKQMAN BUACE. SECTION OF BASILICA OF SAN LORENZO, FLORENCE.

65
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 bv the Romans in determining the_
proportions of triumphal arches, basilicas and baths. That the same figure
was a factor inTEe designing of Gothic cathedrals is sufficiently indicated
in the accompanying facsimile reproduction 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
Itajian_Renaissance doubtless inherited certain of theJRoman canons of
architectural proportionTfor they seem very generally to have recognized
them as an essential principle of design.
Nevertheless, when all is said, it is easy to exaggerate the importance of
this matter of geometrical proportion. The designer who seeks the ultimate
secret of architectural harmony in mathematics rather than, in the trained

7°

THE BEAUTIFUL NECESSITY

eye, is following the wrong road to success. A happy inspiration is worth all
the formulas in the world — if it is really happy, the artist will probably find
that he has "followed the rules without knowing them." Even while formu-
lating^corjcepts-qf art the-authpr must again reiteratg_ that the concept is
unf ruitfulin art. The "mechanism" oLspatiaLheaufy is anJuiteresting study.
and within certain limits, a useful one ; but it can never take the place of
the creative faculty, it can only restrain and" TlrFecQtT" The study^oLp_rapor=
tiqnjs_to the architect what the study of harmony is to a musician, — it helps
his genius adequately to express itself.

THE, HE.XAGRAM IN GOTHIC ARCHITECTURE
SECTION OF WINDOW MULLlONS IN THE.
CDEE.BSTOR.Y, WINCHESTER. CATHEDRAL
(FROMOWILT),

ROSE. WINDOW IN SOUTH TRANSEPT
OFROUE.N CATHEDRAL (.FROM OWITj)
66

£9

ll

A^iaHOSO LN3LVT

72

THE BEAUTIFUL NECESSITY

LATENT GEOMETRY

73

69

74

THE BEAUTIFUL NECESSITY

70

LATENT GEOMETRY

75

LIBER.

PRIMVS

Idea geometricae architectonicae AB ICHNOSRAPHIA SVMPTA-VTPERAMVSSINEAS POSSnTT'
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VI
THE ARITHMETIC OF BEAUTY
ALTHOUGH architecture is based primarily upon geometry, it is pos
sible to express all spatial relations numerically, for arithmetic, and
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 — form 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. Never
theless the fact that the eye instinctively rejects certain proportions as un-
pleasing, and accepts others as satisfactory, is an indication of the existence
of spatial laws 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, that
is, 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
[76]

VI

THE ARITHMETIC OF BEAUTY

77

i = ©

Ii

2 =
3 =
4=

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+4 ^ £
£ £* £

MASG.lF&M.lNEUT
SE&lESlS&KJBSl SERIES

their infinite divisibility. Harmonious
combinations were those involving the
marriage of a masculine and a femin
ine — an odd and an even — number.
Number proceeds from unity towards
infinity, and returns 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 num
ber, every aggregation of units, is also
a new unit capable of being added
or subtracted, there are also the opera
tions of multiplication and division,
which consist, in the 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.
The progression and retrogression of numbers in groups expressed by
the multiplication table gives rise to what may be termed "numerical con
junctions." These are analogous to astronomical conjunctions : the planets,
revolving around the sun at dif
ferent rates of speed, and in wide
ly separated orbits, at certain times
come into line with each other
and with the sun. They are then
said to be in conjunction. Simi
larly, numbers, advancing towards
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

49. ORj.
7TAKEN7TIMES

"3. GRAPHIC SYSTEM OF NOTATION.

72

A ? it
A ffl eg?

D

O

73

78

THE BEAUTIFUL NECESSITY

vi

THE, NUMERICAL BASIS OFTr-E
ARCHITECTURAL ]

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. Eleven and thirteen
are not conjunctive numbers. Four
teen is so in the series of twos, fours,
and sevens; 15 is so in the series of
fives and threes. The next conjunc
tion after 12, of 3 and 4 and their
first multiples, is in 24, and the next
following is 36, which numbers are
respectively the two and three of a
series of twelves, each end being but
a new beginning.
It will be seen that this discovery
of numerical conjunctions consists
merely of resolving numbers into their
prime factors, and that a conjunctive
number is a common multiple ; but by
naming it so, to dismiss the entire
subject as known and exhausted, is to miss a sense of the wonder, beauty,
and rhythm of it all, a mental impression analogous to that made upon the

THE. TUXGAN, DCWO. AND IONIC OE.DE.R.S
ACCOEDINa TO VKjNOLE-~ PROPORTIONS
DETERMINED BY THE- NUMBER* 3. 4. ATD
THE.II& CONJUNCTIVE- NUMBER 12 •  ¦

74

THE. RE-IATION 6LTWE£.N THL
SUBMINOR. SEVLNTH (4 '.7) AND
THL EQU1LATE.RAL TR.[ANOLE.~
75

VI

THE ARITHMETIC OF BEAUTY

79

eye by the swift glancing balls of a juggler,
the evolutions of drilling troops, or the
intricate figures of a dance, for these things
are number concrete and animate in time
and space.
The truths of number are of all truths
the most interior, abstract, and difficult of
apprehension, and since knowledge becomes
clear and definite to the extent that it can
be made to enter the mind through the
channels of physical sense, it is well to
accustom oneself to conceiving of number
graphically, by means of geometrical sym
bols (Illustration 72), rather than in terms
of the familiar Arabic notation which,
though admirable for purposes of com
putation, is of too condensed and arbitrary

THE. BANQUETING HOUJE..WHnt«AI.L
3»- dHI 3X7

OLD .TOMEJWET KXirL>

77
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 en
closing four smaller squares,
and 8 as a cube containing
eight smaller cubes, the idea
is apprehended immediately
and without effort. Three 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 regu
lar division of the circle, are portrayed in the familiar hexagram known as the

PALACE- IK VICENZAT PALACE. IN ROME.,
7AS 2.2.AND3 1 8AS 3AMr>5~»

78

8o

THE BEAUTIFUL NECESSITY

vi

THE. BR0LLTTA AT MANTUA \ EALKZZO UOUCCIONI AT FLORENCE.

j #) »'

PALAZZO BASCTOLINl. FLORENCE. EUAZZO TACCONI, BOLOGNA
VARIOUS EMACE. FAQADES~ 3 USED AS A MULTIPLE.

79
the nature of color; music, of sound in wood,
strings ; architecture shows forth the qualities of
beauty of materials. All of the arts,
and particularly music and architec
ture, portray in different manners and
degrees the truths of number. Archi
tecture does this in two ways : esoteri-
cally, as it were, in the form of har
monic proportions; and exoterically
in the form of symbols which repre
sent numbers and groups of numbers.
The fact that a series of threes and a
series of fours mutually conjoin in 12,

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 phi
losophy. It is a curious
fact that when asked to
think of any number less
than 10, most persons
will choose 7 (Illustra
tion 73).
Every form of art,
though primarily a vehi
cle for the expression
and transmission of par
ticular ideas and emo
tions, has subsidiary of
fices, just as a musical
tone has harmonics
which render it more
sweet. Painting reveals
in brass, and in stretched
light, and the strength and

VI

THE ARITHMETIC OF BEAUTY

81

finds an architectural ex
pression 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 the
ratio between the base and
the altitude of the right-
angled triangle which forms
half of an equilateral, and
the musical intervals of the
diminished seventh (Illustration 75) is architecturally suggested in the
Palazzo Giraud, which is four stories in height with seven openings in each
story (Illustration 76).
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 (Illustra
tions 77-83). Three, 5,
and 7 — the numbers
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

82

THE BEAUTIFUL NECESSITY

vi

B B M B H H H
l¥l¥l 1 I

COLLEGE, CFTHE.iAPr£.NZA,iSriSCIME/  AN
ILUJf TEiSTlON CB THE. VALUE, OF A VERTICAL
AND HORIZONTAL DIVISION OF A FAlJADB TO AS
SIST THE EYE- TO AN APPREHENSION OF THE.
NUMBE.B OBNUMBE&S OFWHICH IT IS THE.
EXPREWtON. IN THIS CASE. 21 .THE PRODUCT
OF0AND7.

83

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 them
selves as factors of some higher
quantity. There is a raison d'etre for string
courses other than to mark the posi
tion 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 some
times serve the useful purpose of so
subdividing a facade that the eye
estimates the number of its openings
without conscious effort and consequent fatigue (Illustration 83). The
tracery of Gothic rose-windows forms perhaps the highest and finest archi
tectural expression of number (Illustration 84). Just as thirst makes water
more sweet, so does
Gothic tracery confuse
the eye with its com
plexity only the more
greatly to gratify the
sight by revealing the
inherent simplicity in
which this complexity
has its root. Some
times, as in the case
of the Venetian Ducal
Palace, the numbers in
volved are too great
for counting, but other
and different arithmetical truths are portrayed ; for example, the multiplica
tion of the first arcade by 2 in the second, and this by 3 in the cusped arches-,

BEAUWUJ GOKDRAt,
BASED ON THE. HEX-
AGRAM.OE NUMBER.
6.WTH MUBl1PLE5a.4. ROSE. WINDOW IN WEJT
TRANJIvPT OF THv CHURCH
OF STOntN.ROUEN: BASED
ON THE. GRAPHIC SYMBOL
FOR) THE- NUMBER, SEVEN
ANUME.BICAL ANALYSIS OF GOTHIC TRAGEDY

84

VI

THE ARITHMETIC OF BEAUTY

83

NUMERATION IN CJROUPJ' EXPRESSED AECHITECTUKALIV

ANJND ARCADE- IN THE- SOUTH
TRANSEPT OF-THECHAPEI. OF-
LINCOLN CATHEDRAL/

and by 4 in the quatrefoils im
mediately above.
Seven is proverbially the perfect
number. It is of a quantity suf
ficiently complex to stimulate the
eye to resolve it, and yet so simple
that it can be so 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 78). Fifteen,
also, is a number of great perfec
tion. It is possible to arrange the first 9 numbers in the form of a "magic"
square so that the sum of each line, read across or up or down, will be 15.
Thus:

rALAZZO PRATEXE.S1-
OIMK5NANO

-SAN

RENAISSANCE- ORNAMENT

85

AEjCHITEjCTURAL. ornament
CONSIDERED AS TK.
OBJECTIF1CATION OF NUMELR__-

TWO

MtrrrrpucATioN
INCtROURTOFFIVL,

urn

THR.E6

ALTERNATION OP
THESE AND JE-VEN

4

9

2 = 15

3

5

7 = i5

8

1

6 = 15

15 15 is
Its beauty is portrayed geometrically in the
accompanying figure which expresses it,
being 15 triangles in three groups of 5
(Illustration 87). Few arrangements of
openings in a facade better satisfy the eye
than three superim
posed groups of five
(Illustrations 77,
81). 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

84 THE BEAUTIFUL NECESSITY vi
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 towards the
condition of music." All aspire to express the law, but music, being unin
cumbered by the leaden burden of gross physical matter, expresses it most
easily and adequately. This being so, there is nothing unreasonable in at
tempting to apply the known facts of musical harmony and rhythm to any
other art, and since these essays concern themselves primarily with archi
tecture, the final aspect in which that art will be presented here is as "frozen
music" — ponderable matter 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
[851

86

THE BEAUTIFUL NECESSITY

VII

THE. NORMAN PORCH CANT£.R>~
BURV~»AN ARCHITECTURAL. E.X~
PRiSSION OF A NOTE & HARMONICS

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 con
sonant 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 archi
tectural 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 that inner Self, in which all senses are
one. Containing within Itself the mystery
of number, It thrills responsive to every
audible or visible presentment of that
mystery. If a vibrating string yielding a certain
musical note be stopped in its center, that
is, divided by half, it will then sound the
octave of that note. The numerical ratio
which expresses the interval of the octave
is therefore i :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 15 the major third ; and 5 :6 the minor third. These comprise the
principal consonant intervals within the scope of one octave. The ratios of
inverted intervals, so called, are found by doubling the smaller number of
the original interval as given above. 2 :3, the fifth, gives 3 4, the fourth ;
4:5, the major third, gives 5:8, the minor sixth; 5:6, the minor third, gives
6:10, or 3:5, the major sixth.
Of these various consonant intervals the octave, fifth, and major third
are the most important, in the sense of being the most perfect, and they are
expressed by numbers of the smallest quantity, an odd number and an even.
It will be noted that all of 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

FROZEN MUSIC

87

ARCHITECTURE, AS HARMONY (m sw£)
VAMOUJ RENAIJIANOE.'WINDawJ INEQMEJ

•a (the, octave)

2:3 (THE. FIFTH)
47 (SUbMINOR) 7tl^

BY BRAMANTI.

VII by the ratio 4:7, though
included among the dis
sonances, forms, accord
ing to Helmholtz, a more
perfect consonance with
the tonic than the .minor
sixth. A natural deduction
from these facts is that
relations of architectural
length and breadth,
heighth and width, to be
"musical" should be cap
able of being expressed
by ratios of quantitively
small numbers, prefer
ably an odd number and
an even. Although, gen
erally speaking, the sim
pler the ratio the more
perfect the consonance,
yet the intervals of the
fifth and major third
(2:3 and 4:5), are con
sidered 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 archi
tecture simple ratios should be employed in connection with those more
complex. Harmonics are those tones which sound with and re-enforce any musical
note when it is sounded. The distinguishable harmonics of the tonic yield
the ratios 1 :2, 2 .-3, 3 14, 4 :5 and 4 -.7. A note and its harmonics form a
natural chord. They may be compared to the widening circles which appear

1:2 (THE. OCTAVE)
4:7§UBMINOS,7IK)

1'ifrHE.OCTAVF,)Z:3CTH5> FIFTH)
*'-5<THE THIKD)

PALAZZO FIK^O.

DOOR; IN S. LORjLNZO
IN QAMASCO.— Emiv

THE BEAUTIFUL NECESSITY

VII

SIENA

CHAPEL OP THE
PALAZZO PUBLICO
THE SUBMINOR. SEVENTH

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 ex
press these overtones, if translated into terms
of space, would yield proportions agreeable
to the eye, and such is the fact, as the accom
panying examples sufficiently indicate (Illus
trations 88-91).
The interval of the sub-minor seventh
(4:7), used in this way, in connection with
the simpler intervals of the octave (1:2), and
the fifth (2:3), is particularly pleasing be
cause it is neither too obvious nor too subtle.
This ratio of 4:7 is important for the reason
that it expresses the angle of sixty degrees,
that is, the numbers 4 and 7 represent (very
nearly) the ratio between one-half the base
and the altitude of an equilateral triangle;
also because they form part of the numerical
series 1, 4, 7, 10, etc. Both are "mystic"
numbers, and in Gothic architecture, particu
larly, proportions were frequently determined
by numbers to which a mystic value was at
tached. According to Gwilt, the Gothic
chapels of Windsor and Oxford are divided
longitudinally by four, and transversely by
seven equal parts. The arcade above the
roses in the facade of the cathedral of Tours
shows seven principal units across the front of the nave, and four in each
of the towers.
A distinguishing characteristic of the series of ratios which represent
the consonant intervals within the compass of an octave is that it advances
by the addition of 1 to both terms : 1 :2, 2 13, 3 -.4, 4 15, and 5 :6. Such a

HOUSE IN ROME

90

VII

FROZEN MUSIC

89

series always approaches unity, just as,
represented graphically by means of paral
lelograms, it tends towards a square. Ac
cording to W. Watkins Lloyd — in an article
published in The American Architect of
March 31, 1888 — the scale of ratios which
determined all the important proportions of
. the Parthenon is of this order, advancing
by consecutive differences of 5. The author
has no means of verifying the truth of this
statement, but gives it here for what it is
worth (Illustration 92). Alberti in his book
presents a design for a tower showing his
idea for its general proportions. It consists
of six stories, in a sequence of orders. The
lowest story is a perfect cube and each of
the other stories is 11- 12 of the story below,
or diminishing practically in the proportion
of 8, 7, 6, 5, 4, 3, allowing in each case for
the amount hidden by the projection of the
cornice below ; each order being accurate as
regards column, entablature, etc. It is of
interest to compare this with Ruskin's idea
in his "Seven Lamps," where he takes the
case of a plant called Alisma Plantago, in
which the various branches diminish in the proportion of 7, 6, 5, 4, 3, respec
tively, and so carry out the same idea; on which Ruskin observes that
diminution in a building should be after the manner of Nature.
It would be a profitless
task to formulate exact rules
of architectural proportion
based upon the laws of
musical harmony. The two
arts are too different from
each other for that, and
moreover the last appeal
must always be to the eye,

ASC^DB oFthe CANCtfUJEKlA

j:e=

MINOR, THIKIC''

io-.it 8:ia

¦*;5-=

MAJOR.3H>'

3:4- =

FOUE.TJT'

7: IE

2:3= 1:2.0c

STTA

w' 'E,

5" '.IO
4:3 3:8 2,:7 1 :G

GRAPHICAL EXPRESSION
OFMUSICAL INTERVALS.

SCALL SHOWING PRINCIPAL PEO-
PQK.TIONS OFTHE PARTHENON--

go

THE BEAUTIFUL NECESSITY

VII

THE. PALAZZO VERZI AT VERONA (LOWER. PORTION
ONLY}. A COMPOSITION FOUNDED ON THE EQUAL AND
REGULAR, DIVISION OF SPACE, A$ MUSIC IS FOUNDED
ON THE EQUAL AND REGULAR; DIVISION OF TIME"-—

'ARCHITECTURE/ AS RHYTHM a division
OF SPACE/ OOR.R£,SPONDINO TO % AND 4A TIME,'

and not to a mathemati
cal formula, just as in
music the last appeal is
to the ear. Laws there
are, but they discover
themselves to the artist
as he proceeds, and are
for the most part incom
municable. Rules and
formulas are useful and
valuable not as a substi
tute for inspiration, but
as a guide : not as wings,
but as a tail. In this
connection perhaps all
that is necessary for the
architectural designer to
bear in mind is that im
portant ratios of length
94 and breadth, height and
and width, to be "musical" should be expressed by quantitatively small
numbers, and that if possible they should obey some simple law of numerical

VII

FROZEN MUSIC

9i

ARCHITECTURE AS RHYTHM
A DIVISION OF SPACE. CDEIJEScPONDINaTD
•V4 TIME. THE PRINCIPAL AND SaBQEDDSIATE
ACCENTS PALLING- ON SS&ttlQWTWtiXJm

PALMZZO VENDKAMIN CALERQI, VENICE

PALAZZO OIEAUD, IrONlE^'IOPMOST 5TOEY

^ w-^c-^ca^, *-iii>-ry*MS, x\J~tL-J±j - lc_cL ivjj^o I cJ ic-fcj
wlfLr^[r[rl';[rLrlriririrLS

progression. From this basic
simplicity complexity will follow,
but it will be an ordered and
harmonious complexity, like that
of a tree, or of a symphony.
In the same way that a musical
composition implies the division
of time into equal and regular
beats, so a work of architecture
should have for its basis some
unit of space. This unit should
be nowhere too obvious and may
be varied within certain limits,
just as musical time is retarded
or accelerated. The underlying
rhythm and symmetry will thus
give value and distinction to such
variation. Vasari tells how
Brunelleschi, Bramante and
Leonardo da Vinci used to work
on paper ruled in squares,
describing it as a "truly ingenious thing, and of great utility in the work
of design." By this means they developed proportions according to a definite
scheme. They set to work with a division of space analogous to the musician's
division of time. The examples given herewith indicate how close a parallel
may exist between music and architecture in this matter of ryhthm (Illus
trations 93-95).
It is a demonstrable fact that musical sounds weave invisible patterns
in the air. Architecture, correspondingly, in one of its aspects, is geometric
pattern made fixed and enduring. What could be more essentially musical,
for example, than the sea arcade of the Venetian Ducal Palace? The sand
forms traced by sound-waves oh a musically vibrating steel plate might easily
suggest architectural ornament did not the differences of scale and of material
tend to confuse the mind. The architect should occupy himself with
identities, not differences. If he will but bear in mind that architecture is
pattern in space, just as music is pattern in time, he will come to perceive the
essential identity between, say a Greek rosette and a Gothic rose-window;

CQENICg CFTHE MLXAEAJMSINA^cCIMB

95

92

THE BEAUTIFUL NECESSITY

VII

an arcade and an egg and dart moulding (Illustration 96). All architectural
forms and arrangements which give enduring pleasure are in their essence
musical. Every well composed f acade makes harmony in three dimensions ;
every good roof line sings a melody against the sky.

ARCHITECTURE AS PATTERN..(no scale)

OR,E£K~ FROM.THE.
E&ECHTHEION AT
ATHENS

ITALIAN GOTHIC~FROM THE.
DUCAL- PALACE. AT VENICE.

OOTHIC~>FACADE rTALIAN BLENAISSANCE^THE
OF NOTRE, DAME PALAZZO STOPPIANI AT ROME.

96

CONCLUSION
IN taking leave of the reader at the end of this excursion together among
the by-ways of a beautiful art, the author must needs add a final word or
two touching upon the purpose and scope of these essays. Architecture
(like everything else) has two aspects : it may be viewed from the standpoint
of utility, that is, as construction : or from the standpoint of expressiveness,
that is, as decoration. No attempt has been made here to deal' with its first
aspect, and of the second (which is again two-fold) only the universal, not
the particular expressiveness has been sought. The literature of architecture
is rich in works dealing with the utilitarian and constructive side of the art :
indeed, it may be said that to this side that literature is almost exclusively
devoted. This being so, it has seemed worth while to attempt to show the
obverse of the medal, even though it be "tails" instead of "heads."
One possible criticism the author meets, not with apologies, but with
defiance. The inductive method has not, in these pages, been honored by
a due observance. It would have been easy to have treated the subject
inductively, amassing facts and drawing conclusions, but to have done so
the author would have been false to the very principle about which the work
came into being. With the acceptance of the Ancient Wisdom, the inductive
method becomes a thing of the past. Facts are no longer useful in order to
establish a hypothesis, they are used rather to elucidate a known and accepted
truth; and when theosophy shall have become the universal religion of
mankind, this work, if it survives at all, will be chiefly, perhaps solely,
remarkable by reason of the fact that it was among the first in which the
attempt was made to again unify science, art, and religion, as they were
unified in those ancient times and among those ancient peoples when the
Wisdom swayed the hearts and minds of men. [93l