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 and 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 arc 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 series 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 its 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
scaffolding 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 series 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 tree-like forms,
however diverse,--the sprig of mignonette, the rose-bush, the fir, the
cedar, the fan-shaped elm, the oval rock-maple, 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 alt 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
adjustment 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 encyclopedia 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.