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,

stence 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 s

goes 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 n

2: A GRAPHIC SY

tion. 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, o

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rly, 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 thro

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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 e

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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

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ay 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 f

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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 th

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NUMERICAL ANALYSIS

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 whic

ERATION IN GROUPS EXP

RCHITECTURAL ORNAME

MBER. MULTIPLICATION

NATION OF TH

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, 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, al

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7

6

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of 5 (Illustration 86). Few arrangements of openings in a fa?ade better satisfy the eye than three superimposed groups

ure'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 i