Section 7

forms: parallelograms, prisms, pyr- amids and cones — the cylinder ap- pearing in the column, and the hemi- sphere in the dome. The plans like- wise of the world's famous buildings reduced to their simplest expression are discovered to resolve themselves into a few simple geometrical fig- ures. (Illustration's). This is the g^ "bed rock" of all excellent design.
ROMAN
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JEPF^'e^N'J" PENJ'IOJltH FDR-'THE ROTUNDA OF THE UNIVER-flTY OP VIRGINIA
82
THE BEAUTIFUL NECESSITY
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porx:h of the- caryatides
LATENT GEOMETRY 83
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,
63
become tender or sublime with expression." All truly great and beautiful works of architecture — from the Egyp- tian pyramids to the cathedrals of Ile-de-France — are harmo- niously proportioned, their prin- cipal and subsidiary masses being related, sometimes ob- viously, more often obscurely, to certain symmetrical figures of geometry, which though in- visible to the sight and not con- sciously 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
THE. BQUILATE-RALTRIANOLtil RQA\AN ARCHlTECrURB
A SECTION OF THi PANTHEON, RCME
64
84
THE BEAUTIFUL NECESSITY
THE E^JUILATBRAL TRIANGLE IN ITALIAN ARjGHITEX:TUR£.
(R£NAI55ANCE()
WINDOW IN A EQMAN PALACE, SECTION OF EiMILICA OF SAN LORENZO, FLOIttNCE,
THE. HEXAGRAM IN GOTHIC ARCHITECTURE SExrriON OF windcw/ mullions in the.
CteRKTOR/Y, WIN0HBSTE.R.CATHDDRAL1 (FROfAOWlUr)
calls "that severe and lofty art of com- position which seeks the highest beauty of design in archi- tectural harmony supreme, above the melodies of grace- fulness of detail."
There is abun- dant evidence in support of the theory that the builders of antiquity, the ma- sonic 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
POSESWJflDOW IN, SOUTH TRANiEBT" OfiBpUE*! CATHBORAL (FROM OWJU)
66
LATENT GEOMETRY
— if after all these men ob- tained their results uncon- sciously — their creations so lend themselves to a geomet- rical analysis that the claim for the existence of certain canons of proportion, based on geometry, remains unim- peached.
The plane figures princi- pally employed in deter- mining architectural pro- portion are the circle, the equilateral triangle, and the square — ^which also
85
yields the right angled isosceles triangle. It will be noted that these are the two dimensional correlatives of the sphere, the tetrahe- dron and the cube, mentioned as being among the deter- mining forms in molecular structure. The question naturally arises, why the circle, the equilateral tri- angle and the square? Be- cause, aside from the fact that they are of all plane fig- ures the most elementary^
86
THE BEAUTIFUL NECESSITY
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 dif- ferent field, it is not an an- swer. Why is the body of man so constructed and re- lated? This leads us, as does every question, to the threshold of a mystery upon which theosophy alone is able to throw light. Any
69
extended elucidation would be out of place here: it is sufficient to remind the reader that the circle is the symbol of the universe; the equilateral triangle, of the higher trinity {atma,buddhi, manas) ; and the square, of the lower quaternary of man's sevenfold nature.
The square is principally used in preliminary plotting: it is the determining figure in many of the palaces of the Italian Renaissance; the
LATENT GEOMETRY 87
Arc de Triomphe, in Paris is a modern example of its use (Illustrations 59, 60). The circle is often employed in con- junction 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 Ro- man triumphal arches submit themselves to a circular synopsis, and a system of double intersecting circles has been applied, with interesting results, to fagades 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 propor- tions of the fagades.
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. In- deed 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 circum- ference 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 Chris- tian 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 impor- tant points in any architectural composition coincide (approxi- mately or exactly) with the three extremities of an equilateral triangle, it makes for beauty of proportion. An ancient and
88
THE BEAUTIFUL NECESSITY
LIBER.
PRIMVS
Idea GiOMnMCAL jmcHrrecroNicAE ab ichnosiiawiia, svMnA.vTPEHAMvssiNuis rossnrr
lEK OKIHOOBAPHIAM AC SCAENOOIUIFHIAM PEKOVCERE OMNTS SOLVM AD CIRCINI CENTKyM. SED Q\A£ ATRIQONO FT PlHVEhnVNTPOSJINTSWM HABERE RESPONSVM
TioNATAM qywrrVM etiam* symmetriae qvantiiate m oiudctjariam ac per.
OPERlS'.CECOKMnoNEM OSTENDERE KIVNT DISTRIBVINTVR PENH Q.VEMADMODVM SACRA CATHEORAIIS AESES MJEDIOLANI
71
LATENT GEOMETRY 89
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 o:f 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 (Illus- tration 62). The same angle, erected from the central axis of a column at the point where it intersects the architrave, deter- mines 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 conjunc- tion 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 in- ternal 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 pro- portion, for they seem very generally to have recognized them as an essential principle of design.
Nevertheless, when all is said, it is easy to exaggerate the im- portance of this matter of geometrical proportion. The de-
90 THE BEAUTIFUL NECESSITY
signer who seeks the ultimate secret of architectural harmony in mathematics rather than in the trained eye, is following the wrong road to success. A happy inspiration is worth all the formulae in the world — if it be really happy, the artist will probably find that he has "followed the rules without knowing them." Even while formulating concepts of art, the author must reiterate Schopenhauer's dictum that the concept is un- fruitful 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 supple- ment, restrain, direct it. The study of proportion is to the archi- tect 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 univer- sal 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, archi- tecture 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 reason- ableness 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 symbol- ized.
Number is the within of all things — the "first form of Brah- man." 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 condi- tioned by number, ceaselessly undergoes the myriad transmuta- tions which produce phenomenal life. Elements separate and combine chemically according to numerical ratios : "Moon, plant,
91
92
THE BEAUTIFUL NECESSITY
3=
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maso.Ifbm.]neut:
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gas, crystal, are concrete geome- try and number." By the Pytha- goreans and by the ancient Egyp- tians sex was attributed to num- bers, odd numbers being con- cieved of as masculine or gen- erating, and even numbers as feminine or parturitive, on ac- count of their infinite divisibility. Harmonious combinations were those involving the marriage of a masculine and a feminine — an odd and an even — number.
Numbers progress from unity to infinity, and return again to unity as the soul, defined by Pythagoras as a self-moving num- ber, goes forth from, and re- turns to God. These two acts, one of projection and the other of recall ; these two forces, centrifugal and centripetal, are sym- bolized 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 mul- tiplication and division, which consists in one case of the addition of several equal numbers together, and in the other, of the sub- traction of several equal numbers from a greater until that is ex- hausted. 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 compo- nent parts and the establishment of relations between these parts.
49, ORj. 7TAKEM 7TIMES
•AQEiAJRHIC SVrrEM Cff NOTATION.
72
THE ARITHMETIC OF BEAUTY
93
The progression and re- trogression of numbers in groups expressed by the mul- tiplication 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 v^^idely separated orbits, at certain times come into line with one another and with the sun. They are then said to be in con- junction. Similarly, numbers, advancing toward infinity singly and in groups (expressed by the multiplication table), at cer- tain stages of their progression come into relation with one an- other. For example, an impor- tant 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 tj. 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 con- junctive; 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 mul- tiples, is in 24, and the next fol-
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94 THE BEAUTIFUL NECESSITY
lowing is in 36, which numbers are respec- tively 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 76 consists merely of re-
solving 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.
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 fa- miliar arabic notation which though admirable for purposes of computa-
THB OANQUETINa HOUJE.WHm«AR,
76
THE ARITHMETIC OF BEAUTY
95
rALACE-lNVICENZ*.] PALACE. tN EOME^ 7 AS 2.2>AND3 I 8 AS 3AHC>Sr>
77
tion, is of too condensed and arbitrary a char- acter to reveal the prop- erties 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 rep- resented as a square en- closing four smaller squares, and 8 as a cube containing eight smaller cubes, the idea is appre- hended 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.
Every form of art, though primarily a vehicle for the ex- pression 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 ; ar- chitecture shows forth the qualities of light, and the strength and
96
THE.Bta£nAArMAmUA.lEALKr2D VOJCClOm fH TUXiUO,
THE BEAUTIFUL NECESSITY
beauty of materials. All of the arts, and particularly music and architecture, portray in dififerent manners and degrees the truths of number. Architecture does this in two ways: esoterically as it were in the form of harmonic propor- tions; and exoteric- ally in the form of symbols which rep- resent numbers and groups of numbers. The fact that a series of threes and a series of fours mutually co'njoin 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 en- tablature 3, and the intermed- iate 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 "
Buxeio tMBcaLan. noRmcx. sujozo taccom. ooloona. Vff^RiDW EMACE. FACAr*5~ 3 U5E0 AS A MUmPLE/
78
97
1
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eo
THE ARITHMETIC OF BEAUTY
triangle which forms half of an equilateral, and the musical in- terval of the dimin- ished seventh, is ar- chitecturally suggest- ed in the Palazzo Giraud , which is four stories in height with seven openings in each story (Illustration 75). Every building is a symbol of some num- ber or group of num- bers, 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