The Laws of Beauty



VOL. V.— APRIL, 1860.—NO. XXX.

THE fatal mistake of many inquirers concerning the line of beauty has been, that they have sought in that which is outward for that which is within. Beauty, perceived only by the mind, and, so far as we have any direct proof, perceived by man alone of all the animals, must be an expression of intelligence, the work of mind. It cannot spring from anything purely accidental ; it does not arise from material, but from spiritual forces. That the outline of a figure, and its surface, are capable of expressing the emotions of the mind is manifest from the art of the sculptor, which represents in cold, colorless marble the varied expressions of living faces,— or from the art of the engraver, who, by simple outlines, can soothe you with a swelling lowland landscape, or brace you with the cool air of the mountains.

Now the highest beauty is doubtless that which expresses the noblest emotion. A face that shines, like that of Moses, from communion with the Highest, is more truly beautiful than the most faultless features without moral expression. But there is a beauty which does not reveal emotion, but only thought,— a beauty which consists simply in the form, and which is admired for its form alone. Let us, for the present, confine our attention to this most limited species of beauty,— the beauty of configuration only.

This beauty of mere outline has, by some celebrated writers, been resolved into some certain curved line, or line of beauty ; by others into numerical proportion of dimensions ; and again by others into early pleasing associations with curvilinear forms. But, if we look at the subject in an intellectual light, we shall find a better explanation. Forms are the embodiment of thought or law. For the common eye they must be embodied in material shape ; while to the geometer and the artist, they may be so distinctly shadowed forth in conception as to need no material figure to render their beauty appreciable. Now this embodiment, or this conception, in all cases, demands some law in the mind, by which it is conceived or made ; and we must look at the nature of this law, in order to approach more nearly to understanding the nature of beauty.

We are thus led, through our search for beauty, into the temple of Geometry, the most ancient and venerable of sciences. From her oracles alone can we learn the generation of beauty, so far as it consists in form alone.

Maupertuis’ law of the least action is not simply a mechanical, but it is a universal axiom. The Divine Being does all things with the least possible expenditure of force ; and all hearts and all minds honor men in proportion as they approach to this divine economy. As gracefulness in motion consists in moving with the least waste of muscular power, so elegance in intellectual and literary exertions arises from the ease with which their achievements are accomplished. We seek in all things simplicity and unity. In Nature we have faith that there is such unity, even in the midst of the wildest diversity. We honor intellectual conceptions in proportion to the greatness of their consequences and to the simplicity of their assumptions. Laws of form are beautiful in proportion to their simplicity and to the variety which they can comprise in unity. The beauty of forms themselves is in proportion to the simplicity of their law and to the variety of their outline.

This last sentence we regard as the fundamental canon concerning beauty,— governing, with a slight change of terms, beauty in all its departments.

Beginning with the fundamental division of figures into curvilinear and rectilinear, this dictum decides, that, in general, a curved outline is more beautiful than a right-lined figure. For a straight-lined figure necessarily requires at least half as many laws as it has sides, while a curvilinear outline requires, in general, but a single law. In a true curve, every point in the whole line (or surface) is subject to one and the same law of position. Thus, in the circle, every point of the circumference is subject to one ahd the same law,— that it must be at a certain distance from the centre. Half a dozen other laws, equally simple, might be named, which in like manner govern every point in the circumference of a circle : for instance, the curve bends at every point by a certain fixed but infinitesimal amount, just enough to make the adjacent points to be equally near the centre. Or, to take another example, every point of the elastic curve, that is, of the curve in which a spring of uniform stiffness can be bent by a force applied at the ends of the spring, is subject to this very simple law, that the curve bends in exact proportion to its distance from a certain straight line. Now a straight line, or a plane, is by this definition a curve, since every point in it is subject to one and the same law of position. A plane may, indeed, be considered a part of any curved surface you please, if you only take that surface on a sufficiently large scale. Thus, the surface of water conforms to the surface of a sphere eight thousand miles in diameter ; but, as the are of such a circle would arch up from a chord ten feet long by only the ten-millionth part of an inch, the surface of water in a cistern may be considered a plane. But no figure or outline can be composed of a single plane or a single straight line ; nor can the position of more than two straight lines, not parallel, be defined by a single simple law of position of the points in them. We may, therefore, regard it as the first deduction from our fundamental canon, that figures with curving outline are in general more beautiful than those composed of straight lines. The laws of their formation are simpler, and the eye, sweeping round the outline, feels the ease and gracefulness of the motion, recognizes the simplicity of the law by which it is guided, and is pleased with the result.

Our second deduction relates principally to rectilinear figures ; it is, that symmetry is in general, and particularly in rectilinear figures, more beautiful than irregularity. It requires, in general, simpler laws to produce symmetry than to produce what is unsymmetrical ; since the corresponding parts in a symmetrical figure are instinctively recognized as flowing from one and the same law. This preference for symmetry is, however, frequently subordinated to higher demands of the fundamental canon. If the outline be rectilineal, simplicity of law produces symmetry, and variety of result can be attained only at the expense of simplicity in the law. But in curved outlines it frequently happens, that, with equally simple laws, we can obtain much greater variety by dispensing with symmetry ; and then, by the canon, we thus obtain the higher beauty.

The question may be asked, In what way does this canon decide the question of proportions ? Which of the two rectangles is, according to this dictum, more beautiful, that in which the sides are in simple ratio, or that in which the angles made with the sides by a diagonal are in such ratio ? — that, for instance, in which the shorter side is three-fifths of the longer, or that in which the shorter side is five hundred and seventy-seven thousandths of the longer ? Our own view was formerly in favor of a simple ratio between the sides ; but experiments have convinced us that persons of good taste, and who have never been prejudiced by reading Hay’s ingenious speculations, do nevertheless agree in preferring rectangles and ellipses which fulfil his law of simple ratio between the angles made by the diagonal. We acknowledge that we have not brought this result under the canon, but look upon it as indicating the necessity of another canon to somewhat this effect,—that in the laws of form direction is a more important element than distance.

We have said that a curved line is one in which every point is subject to one and the same law of position. Now it may be easily proved, that, in a series of points in a plane, each of which fulfils one and the same condition of position, any three, if taken sufficiently near each other, lie in one straight line. A fourth point near the third lies, then, in a straight line with the second and third,—a fifth with the third and fourth, and so on. The whole scries of points must, in short, form a line. But it may also be easily proved that any four of these points, taken sufficiently near each other, lie in the arc of a circle. How strange the paradox to which we are thus led ! Every law of a curve, however simple, leads to the same conclusion ; a curve must bend at every point, and yet not bend at any point ; it must be nowhere a straight line, and yet be a straight line at every part. The blacksmith, passing an iron bar between three rollers to make a tire for a wheel, bends every part of it infinitely little, so that the bending shall not be perceptible at any one spot, and shall yet in the whole length arch the tire to a full circle. It may be that in this paradox lies an additional charm of the curved outline. The eye is pleased to find itself deceived, lured insensibly round into a line running in a different direction from that on which it started.

The simplest law of position for a point would be, either to have it in a given direction from a given point,— a law which would manifestly generate a straight line, — or else to have it at a given distance from the given point, which would generate the surface of a sphere, the outline of which is the circumference of a circle. The straight line fulfils part of the conditions of beauty demanded by the first canon, but not the whole,—it has no variety, and must be combined in order to produce a large effect. The simplest combination of straight lines is in parallels, and this is its usual combination in works of Art. The circle also fulfils but imperfectly the demands of the fundamental canon. It is the simplest of all curves, and the standard or measure of curvature,— vastly more simple in its laws than any rectilineal figure, and therefore more beautiful than any simple figure of that kind. There is, however, a sort of monotony in its beauty,— it has no variety of parts.

The outline of a sphere, projected by the beholder against any plane surface behind it, is a circle, only when a perpendicular, let fall on the plane from the eye, passes through the centre of the sphere. In other positions the projection of the sphere becomes an ellipse, or one of its varieties, the parabola and hyperbola. The parabola is the boundary of the projection of a sphere upon a plane, when the eye is just as far from the plane as the outer edge of the sphere is, and the hyperbola is a similar curve formed by bringing the eye still nearer to the plane.

By these metamorphoses the circle loses much of its monotony, without losing much of its simplicity. The law of the projection of a sphere upon a plane is simple, in whatever position the plane may be. And if we seek a law for the ellipse, or either of the conic sections, which shall confine our attention to the plane, the laws remain simple. There are for these curves two centres, which come together for the circle, and recede to an infinite distance for the parabola ; and the simple law of their formation is, that the curve everywhere makes equal angles with the lines drawn to these two centres. According to the fundamental canon, a conic section should be a beautiful curve ; and the proof that it is so is to be found in the attention which these curves have always drawn upon themselves from artists and from mathematicians. Plato, equally great in mathematics and in metaphysics, is said to have been the first to investigate the properties of the ellipse. For about a century and a half, to the time of Apollonius, the beauty of this curve, and of irs variations, the parabola and hyperbola, so fascinated the minds of Plato’s followers, that Apollonius found theorems and problems relating to these figures sufficient to fill eight books with condensed truths concerning them. The study of the conic sections has been a part of polite learning from his day downward. All men confess their beauty, which so entrances those of mathematical genius as entirely to absorb them. For eighteen centuries the finest spirits of our race drew some of their best means of intellectual discipline from the study of the ellipse. Then came a new era in the history of this curve. Hitherto it had been an abstract form, a geometrical speculation. But Kepler, by some fortunate guess, was led to examine whether the orbits of the planets might not be elliptical, and, lo ! it was found that this curve, whose beauty had so fascinated so many men for so many ages, had been deemed by the great Architect of the Heavens beautiful enough to introduce into Nature on the grandest scale; the morning stars had been for countless ages tracing diagrams beforehand in illustration of Apollonius’s conic sections. It seemed that this must have been the design of Providence in leading Plato and his followers to investigate the ellipse, that Kepler might be prepared to guide men to a knowledge of the movements of the heavenly bodies. " And,” said Kepler, " if the Creator has waited so many years for an observer, I may wait a century for a reader.” But in less than a century a reader arose in the person of the English Newton. The ellipse again appeared in human history, playing a no less important part than before. For, as it was only by a profound knowledge of ellipses that Kepler could establish his three beautiful facts with regard to the motions of the planets, so also was it only through a still more perfect and intimate acquaintance with the minute peculiarities of that curve that Sir Isaac Newton could demonstrate that these three facts were perfectly accounted for only by his theory of universal gravitation,—the most beautiful theory ever devised., and the most firmly established of all scientific hypotheses. If the ellipse, as a simply geometrical speculation, has had so much power in the education of the race, what are the intellectual relations of its beauty through its connection with astronomy ? Who can estimate the influence which this oldest of physical sciences has had upon human destiny ? Who can tell how much intellectual life and self-reliance, how much also of humility and reverential awe, how much adoration of Divine Wisdom, have been gained by man through his study of these heavenly diagrams, marked out by the sun and the moon, by the planets and the comets, upon the tablets of the sky ? Yet, without the ellipse, without the conic sections of Plato and Apollonius, astronomy would have been to this day a sealed science, and the labors of Hipparchus, Ptolemy, Tycho, and Copernicus would have waited in vain for the genius of Kepler and of Newton to educe divine order from the seeming chaos of motions.

But the obligations of man to the ellipse do not end here. The eighteenth and nineteenth centuries also owe it a debt of gratitude. Even where the knowledge of conic sections does not enter as a direct component of that analytical power which was the glory of a Lagrange, a Laplace, and a Gauss, and which is the glory of a Leverrier, a Peirce, and their companions in science, it serves as a part of the necessary scatfolding by which that skill is attained,— of the necessary discipline by which their power was exercised and made available for the solution of the great problems of astronomy, optics, and thermotics, which have been solved in our century.

There is another curve, generated by a simple law from a circle, which has played an important part at various epochs in the intellectual history of our race. A spot on the tire of a wheel running on a straight, level road, will describe in the air a scries of peculiar arches, called the cycloid. The law of its formation is simple ; the law of its curvature is also simple. The path in which the spot moves curves exactly in proportion to its nearness to the lowest point of the wheel. By the simplicity of its law, it ought, according to the canon, to be a beautiful curve. Now, although artists have not shown any admiration for the cycloid, as they have for the ellipse, yet the mathematicians have gazed upon it with great eagerness, and found it rich in intellectual treasures. Chasles, in his History, says that the cycloid interweaves itself with all the great discoveries of the seventeenth century.

A curve which fulfils more perfectly the demands of our dictum is that of an elastic thread, to which we have already alluded. If the two ends of a straight steel hair be brought towards each other by simple pressure, the intervening spring may be put into a series of various forms, — simple undulations, and those more complicated, a figure 8, loops turning alternately opposite ways, loops turning all one way, and finally a circle. Now the whole of this variety is the result of subjecting each part of the curve to a law more simple than that of the cycloid. The elastic curve is a curve which bends or curves exactly in proportion to its distance from a given straight line. According to the canon, therefore, this curve should be beautiful ; and it is acknowledged to be so in the examples given by the bending osier and the waving grain, — also by the few who have seen full drawings of all the forms. And the mathematician finds in it a new beauty, from its marvellous correspondence with the motions of a pendulum,— the algebraic expression of the two being identical.

The forms of organic life afford, however, the best examples of the dominion of our fundamental canon. The infinite variety of vegetable forms, all beautiful, and each one different in its beauty, is all the result of simple laws. It is true that these simple laws are not as yet all discovered ; but the one great discovery of Phyllotaxis, which shows that all plants follow one law in the arrangement of their leaves upon the stem, thereby intimates in unmistakable language the simplicity and unity of all organic vegetable laws ; and a similar assurance is given by the morphological reduction of all parts to a metamorphosed leaf.

The law of phyllotaxis, like that of the elastic curve, is carried out in time as well as in space. As the formula for the elastic curve is the same as that for the pendulum, so the law by which the spaces of the leaves are divided in scattering them round the stem, to give each its opportunity for light and air, is the same as that by which the times of the planets are proportioned to keep them scattered about the sun, and prevent them from gathering on one side of their central orb.

The forms of plants and trees are dependent upon the arrangement of the branches, and the arrangement of the branches depends upon that of the buds or leaves. The leaves are arranged by this numerical law,—that the angular distance about the stem between two successive leaves shall be in such ratio to the whole circumference as may be expressed by a continued fraction composed wholly of the figure 1. It is, then, true, that all the beauty of the vegetable world which depends on the arrangement of parts — the graceful symmetry or more graceful apparent disregard of symmetry in the general form of plants, all the charm of the varying forms of forest trees, which adds such loveliness to the winter landscape, and such a refined source of pleasure to the exhilaration of the winter morning walk—is the result of the simplest variations in a simple numerical law ; and is thus clearly brought under our fundamental canon. It is the perception of this unity in diversity, of this similarity of plan, for instance, in all treelike forms, however diverse, — the sprig of mignonette, the rose-bush, the fir, the cedar, the fan-shaped elm, the oval rockmaple, the columnar hickory, the dense and slender shaft of the poplar,—which charms the eye of those who have never heard in what algebraic or arithmetical terms this unity may be defined, in what geometrical or architectural figures this diversity may be expressed.

When we look at the animal kingdom, we recognize there also the presence of simple, all-pervading laws. The four great types of animal structures are readily discerned by the dullest eye : no man fails to see the likeness among all vertebrates, or the likeness among all articulates, the likeness among all mollusks, or the likeness among all radiates. These four types show, moreover, a certain unity, even to the untaught eye : we call them all by one name, animals, and feel that there is a likeness between them deeper than the widest differences in their structure ; there are analogies where there are not homologies.

The difference between the four types of animals is marked at a very early period in the embryo,— the embryo taking one of four different forms, according to the department to which it belongs; and Peirce has shown that these four forms are all embodiments of one single law of position. If, then, one single algebraic law of form includes the four diverse forms of the four great branches of the animal kingdom, is it extravagant to suppose that the diversities in each branch are also capable of being included in simple generalizations of form ? Is it unreasonable to believe that the exceeding beauty of animated forms, and of the highest, the human form, arises from the fact that these forms are the result of some simple intellectual law, a simple conception of the Divine Geometer, assuming varied developments in the great series of animated beings ? It is the unity of the form, arising from the simplicity of its law, and the multiplicity of its manifestations or details, arising from the generality of its law, that, intuitively perceived by the eye, although the intellect may not apprehend them, give the charm to the figures of the animate creation.

The subject, even in the narrow limits which we have imposed upon ourselves, would admit of a much longer discussion. The various animals might, for instance, be compared with each other, and the beauty of the most beautiful could be clearly shown to be owing to the greater variety in the outline, or the greater variety of position, which they included in equal unity of general effect. And should we step outside the bounds which we have prescribed to ourselves, we should find that in other things than questions of mere form the general canon holds true, that laws produce beauty in proportion to their own simplicity and to the variety of their effects. As a single example, take the most beautiful of the fine arts, the art which is free from the laws of space, and subject only to those of time, and in which, therefore, we find a beauty removed as far as possible from that of curvilinear outlines. How exceedingly simple are the fundamental laws of music, of simple rhythm and simple harmony ! yet how infinitely varied, and how inexpressibly touching are its effects ! In studying music as a mere matter of intellectual science, all is simple ; it is only an easy chapter in acoustics. But in studying it on the side of the emotions, in studying the laws of counterpoint and of musical form, which are governed by the effect upon the ear and the heart, we find intricacy and difficulties, increased beyond our power of understanding.

So in the harmony of the spheres, in the varied beauty which clothes the earth and pervades the heavens, in the beauty which addresses itself to eye and ear, and in the beauty which addresses only the inward sense, — the harmonious arrangements of the social world, and the adj ustment of domestic, civil, and political relations,— there is an infinite diversity of result, infinitely varied in its effect upon the observer. But could we behold the Kosmos as it is beheld by its Creator, we should perchance find the whole encyclopædia of our science resting upon a few great, but simple laws ; we should see that the whole universe, in all its infinite complication, is the fulfilment of perhaps a single simple thought of the Divine Mind, and that it is this unity pervading the diversity which makes it the Kosmos, Beauty.