all-pervasive harmony of parts, between which definite numerical
relations are traceable, and an apprehension of these should assist
the architectural designer to arrive at beauty of proportion by
methods of his own, not perhaps in the shape of rigid formulæ, but
present in the consciousness as a restraining influence, acting and
reacting upon the mind with a conscious intention toward rhythm and
harmony. By means of such exercises, he will approach nearer to an
understanding of that great mystery, the beauty and significance of
numbers, of which mystery music, architecture, and the human figure
are equally presentments--considered, that is, from the standpoint of
the occultist.
V
LATENT GEOMETRY
[Illustration 51: THE HEXAGRAM AND EQUILATERAL TRIANGLE IN NATURE]
It is a well known fact that in the microscopically minute of nature,
units everywhere tend to arrange themselves with relation to certain
simple geometrical solids, among which are the tetrahedron, the cube,
and the sphere. This process gives rise to harmony, which may be
defined as the relation between parts and unity, the simplicity latent
in the infinitely complex, the potential complexity of that which is
simple. Proceeding to things visible and tangible, this indwelling
harmony, rhythm, proportion, which has its basis in geometry and
number, is seen to exist in crystals, flower forms, leaf groups, and
the like, where it is obvious; and in the more highly organized world
of the animal kingdom also; though here the geometry is latent rather
than patent, eluding though not quite defying analysis, and thus
augmenting beauty, which like a woman is alluring in proportion as she
eludes (Illustrations 51, 52, 53).
[Illustration 52: PROPORTIONS OF THE HORSE]
[Illustration 53]
By the true artist, in the crystal mirror of whose mind the universal
harmony is focused and reflected, this secret of the cause and source
of rhythm--that it dwells in a correlation of parts based on an
ultimate simplicity--is instinctively apprehended. A knowledge of it
formed part of the equipment of the painters who made glorious the
golden noon of pictorial art in Italy during the Renaissance. The
problem which preoccupied them was, as Symonds says of Leonardo,
"to submit the freest play of form to simple figures of geometry in
grouping." Alberti held that the painter should above all things have
mastered geometry, and it is known that the study of perspective and
kindred subjects was widespread and popular.
[Illustration 54]
The first painter who deliberately rather than instinctively based
his compositions on geometrical principles seems to have been Fra
Bartolommeo, in his Last Judgment, in the church of St. Maria Nuova,
in Florence. Symonds says of this picture, "Simple figures--the
pyramid and triangle, upright, inverted, and interwoven like the
rhymes of a sonnet--form the basis of the composition. This system was
adhered to by the Fratre in all his subsequent works" (Illustration
54). Raphael, with that power of assimilation which distinguishes
him among men of genius, learned from Fra Bartolommeo this method
of disposing figures and combining them in masses with almost
mathematical precision. It would have been indeed surprising if
Leonardo da Vinci, in whom the artist and the man of science were
so wonderfully united, had not been greatly preoccupied with the
mathematics of the art of painting. His Madonna of the Rocks, and
Virgin on the Lap of Saint Anne, in the Louvre, exhibit the very
perfection of pyramidal composition. It is however in his masterpiece,
The Last Supper, that he combines geometrical symmetry and precision
with perfect naturalness and freedom in the grouping of individually
interesting and dramatic figures. Michael Angelo, Andrea del Sarto,
and the great Venetians, in whose work the art of painting may be said
to have culminated, recognized and obeyed those mathematical laws of
composition known to their immediate predecessors, and the decadence
of the art in the ensuing period may be traced not alone to the false
sentiment and affectation of the times, but also in the abandonment by
the artists of those obscurely geometrical arrangements and groupings
which in the works of the greatest masters so satisfy the eye and
haunt the memory of the beholder (Illustrations 55, 56).
[Illustration 55: THE EMPLOYMENT OF THE EQUILATERAL TRIANGLE IN
RENAISSANCE PAINTING]
[Illustration 56: GEOMETRICAL BASIS OF THE SISTINE CEILING PAINTINGS]
[Illustration 57: ASSYRIAN; GREEK]
[Illustration 58: THE GEOMETRICAL BASIS OF THE PLAN IN ARCHITECTURAL
DESIGN]
[Illustration 59]
Sculpture, even more than painting, is based on geometry. The colossi
of Egypt, the bas-reliefs of Assyria, the figured pediments and
metopes of the temples of Greece, the carved tombs of Revenna,
the Della Robbia lunettes, the sculptured tympani of Gothic church
portals, all alike lend themselves in greater or less degree to a
geometrical synopsis (Illustration 57). Whenever sculpture suffered
divorce from architecture the geometrical element became less
prominent, doubtless because of all the arts architecture is the most
clearly and closely related to geometry. Indeed, it may be said that
architecture is geometry made visible, in the same sense that music
is number made audible. A building is an aggregation of the commonest
geometrical forms: parallelograms, prisms, pyramids and cones--the
cylinder appearing in the column, and the hemisphere in the dome.
The plans likewise of the world's famous buildings reduced to their
simplest expression are discovered to resolve themselves into a few
simple geometrical figures. (Illustration 58). This is the "bed rock"
of all excellent design.
[Illustration 60: EGYPTIAN; GREEK; ROMAN; MEDIÆVAL]
[Illustration 61: JEFFERSON'S PEN SKETCH FOR THE ROTUNDA OF THE
UNIVERSITY OF VIRGINIA]
[Illustration 62: APPLICATION OF THE EQUILATERAL TRIANGLE TO THE
ERECHTHEUM AT ATHENS]
[Illustration 63]
[Illustration 64: THE EQUILATERAL TRIANGLE IN ROMAN ARCHITECTURE]
But architecture is geometrical in another and a higher sense than
this. Emerson says: "The pleasure a palace or a temple gives the eye
is that an order and a method has been communicated to stones, so that
they speak and geometrize, become tender or sublime with expression."
All truly great and beautiful works of architecture from the Egyptian
pyramids to the cathedrals of Ile-de-France--are harmoniously
proportioned, their principal and subsidiary masses being related,
sometimes obviously, more often obscurely, to certain symmetrical
figures of geometry, which though invisible to the sight and not
consciously present in the mind of the beholder, yet perform the
important function of coördinating the entire fabric into one easily
remembered whole. Upon some such principle is surely founded what
Symonds calls "that severe and lofty art of composition which seeks
the highest beauty of design in architectural harmony supreme, above
the melodies of gracefulness of detail."
[Illustration 65: THE EQUILATERAL TRIANGLE IN ITALIAN ARCHITECTURE]
[Illustration 66: THE HEXAGRAM IN GOTHIC ARCHITECTURE]
There is abundant evidence in support of the theory that the
builders of antiquity, the masonic guilds of the Middle Ages, and
the architects of the Italian Renaissance, knew and followed certain
rules, but though this theory be denied or even disproved--if after
all these men obtained their results unconsciously--their creations
so lend themselves to a geometrical analysis that the claim for the
existence of certain canons of proportion, based on geometry, remains
unimpeached.
[Illustration 67]
[Illustration 68]
The plane figures principally employed in determining architectural
proportion are the circle, the equilateral triangle, and the
square--which also yields the right angled isosceles triangle. It
will be noted that these are the two dimensional correlatives of the
sphere, the tetrahedron and the cube, mentioned as being among the
determining forms in molecular structure. The question naturally
arises, why the circle, the equilateral triangle and the square?
Because, aside from the fact that they are of all plane figures the
most elementary, they are intimately related to the body of man, as
has been shown (Illustration 45), and the body of man is as it were
the architectural archetype. But this simply removes the inquiry to
a different field, it is not an answer. Why is the body of man so
constructed and related? This leads us, as does every question, to
the threshold of a mystery upon which theosophy alone is able to throw
light. Any extended elucidation would be out of place here: it is
sufficient to remind the reader that the circle is the symbol of the
universe; the equilateral triangle, of the higher trinity (_atma,
buddhi, manas_); and the square, of the lower quaternary of man's
sevenfold nature.
[Illustration 69]
[Illustration 70]
The square is principally used in preliminary plotting: it is the
determining figure in many of the palaces of the Italian Renaissance;
the Arc de Triomphe, in Paris is a modern example of its use
(Illustrations 59, 60). The circle is often employed in conjunction
with the square and the triangle. In Thomas Jefferson's Rotunda for
the University of Virginia, a single great circle was the determining
figure, as his original pen sketch of the building shows (Illustration
61). Some of the best Roman triumphal arches submit themselves to a
circular synopsis, and a system of double intersecting circles has
been applied, with interesting results, to façades as widely different
as those of the Parthenon and the Farnese Palace in Rome, though
it would be fatuous to claim that these figures determined the
proportions of the façades.
By far the most important figure in architectural proportion,
considered from the standpoint of geometry, is the equilateral
triangle. It would seem that the eye has an especial fondness for
this figure, just as the ear has for certain related sounds. Indeed it
might not be too fanciful to assert that the common chord of any key
(the tonic with its third and fifth) is the musical equivalent of
the equilateral triangle. It is scarcely necessary to dwell upon
the properties and unique perfection of this figure. Of all regular
polygons it is the simplest: its three equal sides subtend equal
angles, each of 60 degrees; it trisects the circumference of a circle;
it is the graphic symbol of the number three, and hence of every
threefold thing; doubled, its generating arcs form the _vesica
piscis_, of so frequent occurrence in early Christian art; two
symmetrically intersecting equilateral triangles yield the figure
known as "Solomon's Seal," or the "Shield of David," to which mystic
properties have always been ascribed.
It may be stated as a general rule that whenever three important
points in any architectural composition coincide (approximately or
exactly) with the three extremities of an equilateral triangle, it
makes for beauty of proportion. An ancient and notable example occurs
in the pyramids of Egypt, the sides of which, in their original
condition, are believed to have been equilateral triangles. It is a
demonstrable fact that certain geometrical intersections yield the
important proportions of Greek architecture. The perfect little
Erechtheum would seem to have been proportioned by means of the
equilateral triangle and the angle of 60 degrees, both in general and
in detail (Illustration 62). The same angle, erected from the central
axis of a column at the point where it intersects the architrave,
determines both the projection of the cornice and the height of
the architrave in many of the finest Greek and Roman temples
(Illustrations 67-70). The equilateral triangle used in conjunction
with the circle and the square was employed by the Romans in
determining the proportions of triumphal arches, basilicas and
baths. That the same figure was a factor in the designing of Gothic
cathedrals is sufficiently indicated in the accompanying facsimile
reproductions of an illustration from the Como Vitruvius, published in
Milan in 1521, which shows a vertical section of the Milan cathedral
and the system of equilateral triangles which determined its various
parts (Illustration 71). The _vesica piscis_ was often used to
establish the two main internal dimensions of the cathedral plan: the
greatest diameter of the figure corresponding with the width across
the transepts, the upper apex marking the limit of the apse, and the
lower, the termination of the nave. Such a proportion is seen to be
both subtle and simple, and possesses the advantage of being easily
laid out. The architects of the Italian Renaissance doubtless
inherited certain of the Roman canons of architectural proportion,
for they seem very generally to have recognized them as an essential
principle of design.
[Illustration 71]
Nevertheless, when all is said, it is easy to exaggerate the
importance of this matter of geometrical proportion. The designer
who seeks the ultimate secret of architectural harmony in mathematics
rather than in the trained eye, is following the wrong road to
success. A happy inspiration is worth all the formulæ in the world--if
it be really happy, the artist will probably find that he has
"followed the rules without knowing them." Even while formulating
concepts of art, the author must reiterate Schopenhauer's dictum
that the _concept_ is unfruitful in art. The mathematical analysis
of spatial beauty is an interesting study, and a useful one to the
artist; but it can never take the place of the creative faculty, it
can only supplement, restrain, direct it. The study of proportion is
to the architect what the study of harmony is to a musician--it helps
his genius adequately to express itself.
VI
THE ARITHMETIC OF BEAUTY
Although architecture is based primarily upon geometry, it is possible
to express all spatial relations numerically: for arithmetic, not
geometry, is the universal science of quantity. The relation of masses
one to another--of voids to solids, and of heights and lengths to
widths--forms ratios; and when such ratios are simple and harmonious,
architecture may be said, in Walter Pater's famous phrase, to
"aspire towards the condition of music." The trained eye, and not an
arithmetical formula, determines what is, and what is not, beautiful
proportion. Nevertheless the fact that the eye instinctively rejects
certain proportions as unpleasing, and accepts others as satisfactory,
is an indication of the existence of laws of space, based upon number,
not unlike those which govern musical harmony. The secret of the deep
reasonableness of such selection by the senses lies hidden in the very
nature of number itself, for number is the invisible thread on which
the worlds are strung--the universe abstractly symbolized.
Number is the within of all things--the "first form of Brahman." It
is the measure of time and space; it lurks in the heart-beat and is
blazoned upon the starred canopy of night. Substance, in a state of
vibration, in other words conditioned by number, ceaselessly undergoes
the myriad transmutations which produce phenomenal life. Elements
separate and combine chemically according to numerical ratios:
"Moon, plant, gas, crystal, are concrete geometry and number." By
the Pythagoreans and by the ancient Egyptians sex was attributed to
numbers, odd numbers being conceived of as masculine or generating,
and even numbers as feminine or parturitive, on account of their
infinite divisibility. Harmonious combinations were those
involving the marriage of a masculine and a feminine--an odd and an
even--number.
[Illustration 72: A GRAPHIC SYSTEM OF NOTATION]
Numbers progress from unity to infinity, and return again to unity as
the soul, defined by Pythagoras as a self-moving number, goes forth
from, and returns to God. These two acts, one of projection and the
other of recall; these two forces, centrifugal and centripetal, are
symbolized in the operations of addition and subtraction. Within them
is embraced the whole of computation; but because every number, every
aggregation of units, is also a new unit capable of being added
or subtracted, there are also the operations of multiplication and
division, which consists in one case of the addition of several equal
numbers together, and in the other, of the subtraction of several
equal numbers from a greater until that is exhausted. In order to
think correctly it is necessary to consider the whole of numeration,
computation, and all mathematical processes whatsoever as _the
division of the unit_ into its component parts and the establishment
of relations between these parts.
[Illustration 73]
[Illustration 74]
The progression and retrogression of numbers in groups expressed by
the multiplication table gives rise to what may be termed "numerical
conjunctions." These are analogous to astronomical conjunctions: the
planets, revolving around the sun at different rates of speed, and
in widely separated orbits, at certain times come into line with one
another and with the sun. They are then said to be in conjunction.
Similarly, numbers, advancing toward infinity singly and in groups
(expressed by the multiplication table), at certain stages of their
progression come into relation with one another. For example, an
important conjunction occurs in 12, for of a series of twos it is
the sixth, of threes the fourth, of fours the third, and of sixes the
second. It stands to 8 in the ratio of 3:2, and to 9, of 4:3. It is
related to 7 through being the product of 3 and 4, of which numbers 7
is the sum. The numbers 11 and 13 are not conjunctive; 14 is so in
the series of twos, and sevens; 15 is so in the series of fives and
threes. The next conjunction after 12, of 3 and 4 and their first
multiples, is in 24, and the next following is in 36, which numbers
are respectively the two and three of a series of twelves, each end
being but a new beginning.
[Illustration 75]
It will be seen that this discovery of numerical conjunctions consists
merely of resolving numbers into their prime factors, and that a
conjunctive number is a common multiple; but by naming it so, to
dismiss the entire subject as known and exhausted, is to miss a
sense of the wonder, beauty and rhythm of it all: a mental impression
analogous to that made upon the eye by the swift-glancing balls of a
juggler, the evolutions of drilling troops, or the intricate figures
of a dance; for these things are number concrete and animate in time
and space.
[Illustration 76]
The truths of number are of all truths the most interior, abstract
and difficult of apprehension, and since knowledge becomes clear and
definite to the extent that it can be made to enter the mind through
the channels of physical sense, it is well to accustom oneself to
conceiving of number graphically, by means of geometrical symbols
(Illustration 72), rather than in terms of the familiar arabic
notation which though admirable for purposes of computation, is of
too condensed and arbitrary a character to reveal the properties of
individual numbers. To state, for example, that 4 is the first square,
and 8 the first cube, conveys but a vague idea to most persons, but if
4 be represented as a square enclosing four smaller squares, and 8
as a cube containing eight smaller cubes, the idea is apprehended
immediately and without effort. The number 3 is of course the
triangle; the irregular and vital beauty of the number 5 appears
clearly in the heptalpha, or five-pointed star; the faultless symmetry
of 6, its relation to 3 and 2, and its regular division of the circle,
are portrayed in the familiar hexagram known as the Shield of David.
Seven, when represented as a compact group of circles reveals itself
as a number of singular beauty and perfection, worthy of the important
place accorded to it in all mystical philosophy (Illustration 73). It
is a curious fact that when asked to think of any number less than 10,
most persons will choose 7.
[Illustration 77]
Every form of art, though primarily a vehicle for the expression and
transmission of particular ideas and emotions, has subsidiary offices,
just as a musical tone has harmonics which render it more sweet.
Painting reveals the nature of color; music, of sound--in wood,
in brass, and in stretched strings; architecture shows forth the
qualities of light, and the strength and beauty of materials. All
of the arts, and particularly music and architecture, portray in
different manners and degrees the truths of number. Architecture does
this in two ways: esoterically as it were in the form of harmonic
proportions; and exoterically in the form of symbols which represent
numbers and groups of numbers. The fact that a series of threes and
a series of fours mutually conjoin in 12, finds an architectural
expression in the Tuscan, the Doric, and the Ionic orders according to
Vignole, for in them all the stylobate is four parts, the entablature
3, and the intermediate column 12 (Illustration 74). The affinity
between 4 and 7, revealed in the fact that they express (very nearly)
the ratio between the base and the altitude of the right-angled
triangle which forms half of an equilateral, and the musical interval
of the diminished seventh, is architecturally suggested in the Palazzo
Giraud, which is four stories in height with seven openings in each
story (Illustration 75).
[Illustration 78]
[Illustration 79]
[Illustration 80]
[Illustration 81]
Every building is a symbol of some number or group of numbers, and
other things being equal the more perfect the numbers involved the
more beautiful will be the building (Illustrations 76-82). The numbers
5 and 7--those which occur oftenest--are the most satisfactory because
being of small quantity, they are easily grasped by the eye, and being
odd, they yield a center or axis, so necessary in every architectural
composition. Next in value are the lowest multiples of these numbers
and the least common multiples of any two of them, because the
eye, with a little assistance, is able to resolve them into their
constituent factors. It is part of the art of architecture to
render such assistance, for the eye counts always, consciously or
unconsciously, and when it is confronted with a number of units
greater than it can readily resolve, it is refreshed and rested if
these units are so grouped and arranged that they reveal themselves as
factors of some higher quantity.
[Illustration 82]
[Illustration 83: A NUMERICAL ANALYSIS OF GOTHIC TRACERY]
There is a raison d'être for string courses other than to mark the
position of a floor on the interior of a building, and for quoins and
pilasters other than to indicate the presence of a transverse wall.
These sometimes serve the useful purpose of so subdividing a façade
that the eye estimates the number of its openings without conscious
effort and consequent fatigue (Illustration 82). The tracery of Gothic
windows forms perhaps the highest and finest architectural expression
of number (Illustration 83). Just as thirst makes water more sweet, so
does Gothic tracery confuse the eye with its complexity only the more
greatly to gratify the sight by revealing the inherent simplicity in
which this complexity has its root. Sometimes, as in the case of
the Venetian Ducal Palace, the numbers involved are too great for
counting, but other and different arithmetical truths are portrayed;
for example, the multiplication of the first arcade by 2 in
the second, and this by 3 in the cusped arches, and by 4 in the
quatrefoils immediately above.
[Illustration 84: NUMERATION IN GROUPS EXPRESSED ARCHITECTURALLY]
[Illustration 85: ARCHITECTURAL ORNAMENT CONSIDERED AS THE
OBJECTIFICATION OF NUMBER. MULTIPLICATION IN GROUPS OF FIVE; TWO;
THREE; ALTERNATION OF THREE AND SEVEN]
[Illustration 86]