VI
THE ARITHMETIC OF BEAUTY
Although architecture is based primarily upon geometry, it is possible to express all spatial relations numerically: for arithmetic, not geometry, is the universal science of quantity. The relation of masses one to another—of voids to solids, and of heights and lengths to widths—forms ratios; and when such ratios are simple and harmonious, architecture may be said, in Walter Pater's famous phrase, to "aspire towards the condition of music." The trained eye, and not an arithmetical formula, determines what is, and what is not, beautiful proportion. Nevertheless the fact that the eye instinctively rejects certain proportions as unpleasing, and accepts others as satisfactory, is an indication of the existence of laws of space, based upon number, not unlike those which govern musical harmony. The secret of the deep reasonableness of such selection by the senses lies hidden in the very nature of number itself, for number is the invisible thread on which the worlds are strung—the universe abstractly symbolized.
Number is the within of all things—the "first form of Brahman." It is the measure of time and space; it lurks in the heart-beat and is blazoned upon the starred canopy of night. Substance, in a state of vibration, in other words conditioned by number, ceaselessly undergoes the myriad transmutations which produce phenomenal life. Elements separate and combine chemically according to numerical ratios: "Moon, plant, gas, crystal, are concrete geometry and number." By the Pythagoreans and by the ancient Egyptians sex was attributed to numbers, odd numbers being conceived of as masculine or generating, and even numbers as feminine or parturitive, on account of their infinite divisibility. Harmonious combinations were those involving the marriage of a masculine and a feminine—an odd and an even—number.
[Illustration 72: A GRAPHIC SYSTEM OF NOTATION]
Numbers progress from unity to infinity, and return again to unity as the soul, defined by Pythagoras as a self-moving number, goes forth from, and returns to God. These two acts, one of projection and the other of recall; these two forces, centrifugal and centripetal, are symbolized in the operations of addition and subtraction. Within them is embraced the whole of computation; but because every number, every aggregation of units, is also a new unit capable of being added or subtracted, there are also the operations of multiplication and division, which consists in one case of the addition of several equal numbers together, and in the other, of the subtraction of several equal numbers from a greater until that is exhausted. In order to think correctly it is necessary to consider the whole of numeration, computation, and all mathematical processes whatsoever as the division of the unit into its component parts and the establishment of relations between these parts.
[Illustration 73]
[Illustration 74]
The progression and retrogression of numbers in groups expressed by the multiplication table gives rise to what may be termed "numerical conjunctions." These are analogous to astronomical conjunctions: the planets, revolving around the sun at different rates of speed, and in widely separated orbits, at certain times come into line with one another and with the sun. They are then said to be in conjunction. Similarly, numbers, advancing toward infinity singly and in groups (expressed by the multiplication table), at certain stages of their progression come into relation with one another. For example, an important conjunction occurs in 12, for of a series of twos it is the sixth, of threes the fourth, of fours the third, and of sixes the second. It stands to 8 in the ratio of 3:2, and to 9, of 4:3. It is related to 7 through being the product of 3 and 4, of which numbers 7 is the sum. The numbers 11 and 13 are not conjunctive; 14 is so in the series of twos, and sevens; 15 is so in the series of fives and threes. The next conjunction after 12, of 3 and 4 and their first multiples, is in 24, and the next following is in 36, which numbers are respectively the two and three of a series of twelves, each end being but a new beginning.
[Illustration 75]
It will be seen that this discovery of numerical conjunctions consists merely of resolving numbers into their prime factors, and that a conjunctive number is a common multiple; but by naming it so, to dismiss the entire subject as known and exhausted, is to miss a sense of the wonder, beauty and rhythm of it all: a mental impression analogous to that made upon the eye by the swift-glancing balls of a juggler, the evolutions of drilling troops, or the intricate figures of a dance; for these things are number concrete and animate in time and space.
[Illustration 76]
The truths of number are of all truths the most interior, abstract and difficult of apprehension, and since knowledge becomes clear and definite to the extent that it can be made to enter the mind through the channels of physical sense, it is well to accustom oneself to conceiving of number graphically, by means of geometrical symbols (Illustration 72), rather than in terms of the familiar arabic notation which though admirable for purposes of computation, is of too condensed and arbitrary a character to reveal the properties of individual numbers. To state, for example, that 4 is the first square, and 8 the first cube, conveys but a vague idea to most persons, but if 4 be represented as a square enclosing four smaller squares, and 8 as a cube containing eight smaller cubes, the idea is apprehended immediately and without effort. The number 3 is of course the triangle; the irregular and vital beauty of the number 5 appears clearly in the heptalpha, or five-pointed star; the faultless symmetry of 6, its relation to 3 and 2, and its regular division of the circle, are portrayed in the familiar hexagram known as the Shield of David. Seven, when represented as a compact group of circles reveals itself as a number of singular beauty and perfection, worthy of the important place accorded to it in all mystical philosophy (Illustration 73). It is a curious fact that when asked to think of any number less than 10, most persons will choose 7.
[Illustration 77]
Every form of art, though primarily a vehicle for the expression and transmission of particular ideas and emotions, has subsidiary offices, just as a musical tone has harmonics which render it more sweet. Painting reveals the nature of color; music, of sound—in wood, in brass, and in stretched strings; architecture shows forth the qualities of light, and the strength and beauty of materials. All of the arts, and particularly music and architecture, portray in different manners and degrees the truths of number. Architecture does this in two ways: esoterically as it were in the form of harmonic proportions; and exoterically in the form of symbols which represent numbers and groups of numbers. The fact that a series of threes and a series of fours mutually conjoin in 12, finds an architectural expression in the Tuscan, the Doric, and the Ionic orders according to Vignole, for in them all the stylobate is four parts, the entablature 3, and the intermediate column 12 (Illustration 74). The affinity between 4 and 7, revealed in the fact that they express (very nearly) the ratio between the base and the altitude of the right-angled triangle which forms half of an equilateral, and the musical interval of the diminished seventh, is architecturally suggested in the Palazzo Giraud, which is four stories in height with seven openings in each story (Illustration 75).
[Illustration 78]
[Illustration 79]
[Illustration 80]
[Illustration 81]
Every building is a symbol of some number or group of numbers, and other things being equal the more perfect the numbers involved the more beautiful will be the building (Illustrations 76-82). The numbers 5 and 7—those which occur oftenest—are the most satisfactory because being of small quantity, they are easily grasped by the eye, and being odd, they yield a center or axis, so necessary in every architectural composition. Next in value are the lowest multiples of these numbers and the least common multiples of any two of them, because the eye, with a little assistance, is able to resolve them into their constituent factors. It is part of the art of architecture to render such assistance, for the eye counts always, consciously or unconsciously, and when it is confronted with a number of units greater than it can readily resolve, it is refreshed and rested if these units are so grouped and arranged that they reveal themselves as factors of some higher quantity.
[Illustration 82]
[Illustration 83: A NUMERICAL ANALYSIS OF GOTHIC TRACERY]
There is a raison d'être for string courses other than to mark the position of a floor on the interior of a building, and for quoins and pilasters other than to indicate the presence of a transverse wall. These sometimes serve the useful purpose of so subdividing a façade that the eye estimates the number of its openings without conscious effort and consequent fatigue (Illustration 82). The tracery of Gothic windows forms perhaps the highest and finest architectural expression of number (Illustration 83). Just as thirst makes water more sweet, so does Gothic tracery confuse the eye with its complexity only the more greatly to gratify the sight by revealing the inherent simplicity in which this complexity has its root. Sometimes, as in the case of the Venetian Ducal Palace, the numbers involved are too great for counting, but other and different arithmetical truths are portrayed; for example, the multiplication of the first arcade by 2 in the second, and this by 3 in the cusped arches, and by 4 in the quatrefoils immediately above.
[Illustration 84: NUMERATION IN GROUPS EXPRESSED ARCHITECTURALLY]
[Illustration 85: ARCHITECTURAL ORNAMENT CONSIDERED AS THE
OBJECTIFICATION OF NUMBER. MULTIPLICATION IN GROUPS OF FIVE; TWO;
THREE; ALTERNATION OF THREE AND SEVEN]
[Illustration 86]
Seven is proverbially the perfect number. It is of a quantity sufficiently complex to stimulate the eye to resolve it, and yet so simple that it can be analyzed at a glance; as a center with two equal sides, it is possessed of symmetry, and as the sum of an odd and even number (3 and 4) it has vitality and variety. All these properties a work of architecture can variously reveal (Illustration 77). Fifteen, also, is a number of great perfection. It is possible to arrange the first 9 numbers in the form of a "magic" square so that the sum of each line, read vertically, horizontally or diagonally, will be 15. Thus:
4 9 2 = 15
3 5 7 = 15
8 1 6 = 15
— — —
15 15 15
Its beauty is portrayed geometrically in the accompanying figure which expresses it, being 15 triangles in three groups of 5 (Illustration 86). Few arrangements of openings in a façade better satisfy the eye than three superimposed groups of five (Illustrations 76-80). May not one source of this satisfaction dwell in the intrinsic beauty of the number 15?
In conclusion, it is perhaps well that the reader be again reminded that these are the by-ways, and not the highways of architecture: that the highest beauty comes always, not from beautiful numbers, nor from likenesses to Nature's eternal patterns of the world, but from utility, fitness, economy, and the perfect adaptation of means to ends. But along with this truth there goes another: that in every excellent work of architecture, in addition to its obvious and individual beauty, there dwells an esoteric and universal beauty, following as it does the archetypal pattern laid down by the Great Architect for the building of that temple which is the world wherein we dwell.