orld-war may for the moment, and to the near and narrow view appear to contradict, but the swee

rose, and whither it flows, we know not; but that it is governed by mathematical law all of our knowledge in every field confirms. Were i

n of the sky. It is a necessary concomitant alike of the sharp bargain, the chemical experiment, and the fine frenzy of the poet. Music is

an entire misconception of the nature and function of art. "Art is a fountain of spontaneous emotion"-what, therefore, can it have in common with the proverbially driest, least spontaneous preoccupation of the human mind? But the above definition concludes with the assertion that th

E XI. IMAGINARY COM

art for art's sake, that is, art for form's sake-art for the sake of mathematics? To this objection there is an answer, and as this ans

? The perception of the world-order by the artist arouses an emotion to which he can give vent only in terms of number; but number is itself the most abstract expression of the world order. The form and content of art are therefore not different, but the

on this alone, just as does Oriental art, even today. The secret essence, the archetypal idea of the subject is the preoccupation of the Oriental artist, as it was of the Egyptian, and of the Greek. We of the West today seek as eagerly to fix the accidental and ephemeral aspec

sense of the eternal secret heart of things as it presents itself to our young eyes. Confronted with this difficulty, we have shirked it, and our ambition has shrunk to the portrayal of those aspects which shuffle our poverty out of sight

(written) drama-employing vehicles more flexible, have been more fortunate, though they

ed in number, and one might almost say nowhere else in such perfection. Mathematics is not the dry and deadly thing that our teaching of it and the uses we put it to have made it see

hetic is beyond the province of this essay, being beyond the compass of its author, but h

ause they are highly conventionalized representations of natural forms which are themselves beautiful, but because they express cosmic truths. The honeysuckle and the acanthus leaf, for example, express the idea of successive impulses, mounting, attaining a maximum, and descending-expanding from some focus of force in the manner universal throughout nature. Science recognizes in the spiral

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nt of their manifold associations with a past which the war makes suddenly more remote than slow centuries have done; suppose that we determine to supplant thes

without knowing the reason why. All ornamentalists are subjective mathematicians-an amazing statement

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amiliar checkerboard of the days are numerical arrangements which are mysterious, "magical"; each separate number is as a spider at the center of an amazing mathematical web. That is to say, every number is discovered to be half of the sum of the pairs of numbers which surround it, vertically, horizontally, and diagonally: all of the pairs add to the same sum, and the central number divides this sum b

lumns of figures of which it is composed: vertical, horizontal and diagonal add to the same sum: 34. An analysis of this square reveals the fact that it is made up of the figures of two different orders of counting: the ordinary order, beginning at the left hand upper corner and reading across and down in the usual way, and the reverse-ordinary, beginning at the lower righ

thing, but they at least furnish a clue which the artist would be foolish not to follow up. Let him attack his problem this ti

merical. It is clear that the arrangement of numbers in a magic square is necessitous-they must be placed in a certain way in order that the summation of every column shall be the same. The problem then becomes to make that necessity reveal

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umbers is 10, which is twice the center number. The magic path is the endless line developed by following, free hand, the numbers in their natural order, from 1 to 9 and back to 1 again. The drawing at the right of Figure 4 is t

of the beauty of this one, some contain even richer decorative possibilities. But there are also other

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and up). If, in such a magic square, a simple graphic symbol be substituted for the numbers belonging to each order, pattern spontaneously springs to life. Figures 5 and 6 exemplify the method, and Figures 7 and 8 the translation of some of these squares into richer patterns by elaborating the symbols while respecting their arrangement. By only a slight stret

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nating some of the lines which give the summation of 2056 by different symbols, as shown in

. The squaring of a number-the raising of it to its second power-finds graphic expression in the plane figure of the square; and the cubing of a number-the raising of it to its third power-in the solid figure of the cube. Now squares and cubes have been recognized from time immemorial as useful ornamenta

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XII. IMAGINARY COMP

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on of space, because of their inherent symmetry. These projections would correspond to the network of lines seen in looking through a glass paperweight of the given shape, the lines being formed by the joining of the several faces. Figure 13

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exhaustible well of mathematics fresh beauty may be drawn. But what of its significance? Ornament must mean so

essure of those forces which shall create the new. What is nature's first visible creative act? The formation of a geometrical crystal. The artist should take this hint, and organize geometry i

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him, but the most cursory acquaintance with primitive art shows that he is much more apt to crudely geometrize. Now it is not necessary to assume that we are to revert to the conditions of savagery in order to believe that in this matter of a sound ?sthetic we must begin where

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es of slain animals long after the necessity for it is past, and some even murder innocent wild creatures, giving to their ferocity the name of sport. Our women bedeck themselves with furs and feathers, the fruit of mercenar

ropriate symbols, but for that other more developed state of consciousness-less apparent but more important-these will not do. Something more sophisticated and recondite must be sought for if we are to have an ornamental mode capable of expressing not only the simplicity but the complexity of present-day psychology. This need not be sought for outside the field of geometry, but within it, and by an extension o

s something about this form of words "the fourth dimension" which seems to produce a sort of mental-phobia in certain minds, rendering them incapable of perception or reason. Such people, because they cannot stick their cane into it contend

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