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Title: The science of beauty, as developed in nature and applied in art
Author: David Ramsay Hay
Release date: February 17, 2025 [eBook #75399]
Language: English
Original publication: Edinburgh: William Blackwood & Sons, 1856
Credits: Tim Lindell and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive/American Libraries.)
*** START OF THE PROJECT GUTENBERG EBOOK THE SCIENCE OF BEAUTY, AS DEVELOPED IN NATURE AND APPLIED IN ART ***
THE SCIENCE OF BEAUTY.
EDINBURGH:
PRINTED BY BALLANTYNE AND COMPANY,
PAUL’S WORK.
THE
SCIENCE OF BEAUTY,
AS DEVELOPED IN NATURE AND
APPLIED IN ART.
BY
D. R. HAY, F.R.S.E.
“The irregular combinations of fanciful invention may delight
awhile, by that novelty of which the common satiety of life
sends us all in quest; the pleasures of sudden wonder are soon
exhausted, and the mind can only repose on the stability of
truth.”
DR JOHNSON.
WILLIAM BLACKWOOD AND SONS,
EDINBURGH AND LONDON.
MDCCCLVI.
TO
JOHN GOODSIR, ESQ., F.R.S S. L. & E.,
PROFESSOR OF ANATOMY IN THE UNIVERSITY OF EDINBURGH,
AS AN EXPRESSION OF GRATITUDE FOR VALUABLE ASSISTANCE,
AS ALSO OF HIGH ESTEEM AND SINCERE REGARD,
THIS VOLUME IS DEDICATED,
BY
THE AUTHOR.
PREFACE.
My theory of beauty in form and colour being now admitted by the best
authorities to be based on truth, I have of late been often asked, by
those who wished to become acquainted with its nature, and the manner of
its being applied in art, which of my publications I would recommend for
their perusal. This question I have always found difficulty in answering;
for although the law upon which my theory is based is characterised by
unity, yet the subjects in which it is applied, and the modes of its
application, are equally characterised by variety, and consequently
occupy several volumes.
Under these circumstances, I consulted a highly respected friend, whose
mathematical talents and good taste are well known, and to whom I have
been greatly indebted for much valuable assistance during the course
of my investigations. The advice I received on this occasion, was to
publish a _résumé_ of my former works, of such a character as not only
to explain the nature of my theory, but to exhibit to the general reader,
by the most simple modes of illustration and description, how it is
developed in nature, and how it may be extensively and with ease applied
in those arts in which beauty forms an essential element.
The following pages, with their illustrations, are the results of an
attempt to accomplish this object.
To those who are already acquainted, through my former works, with
the nature, scope, and tendency of my theory, I have the satisfaction
to intimate that I have been enabled to include in this _résumé_ much
original matter, with reference both to form and colour.
D. R. HAY.
CONTENTS.
PAGE
INTRODUCTION 1
THE SCIENCE OF BEAUTY, EVOLVED FROM THE HARMONIC LAW OF NATURE,
AGREEABLY TO THE PYTHAGOREAN SYSTEM OF NUMERICAL RATIO 15
THE SCIENCE OF BEAUTY, AS APPLIED TO SOUNDS 28
THE SCIENCE OF BEAUTY, AS APPLIED TO FORMS 34
THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE FORM OF THE HUMAN HEAD
AND COUNTENANCE 54
THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE FORM OF THE HUMAN FIGURE 61
THE SCIENCE OF BEAUTY, AS DEVELOPED IN COLOURS 67
THE SCIENCE OF BEAUTY APPLIED TO THE FORMS AND PROPORTIONS OF
ANCIENT GRECIAN VASES AND ORNAMENTS 82
APPENDIX, NO. I. 91
APPENDIX, NO. II. 99
APPENDIX, NO. III. 100
APPENDIX, NO. IV. 100
APPENDIX, NO. V. 104
APPENDIX, NO. VI. 105
ILLUSTRATIONS.
PLATES
[Illustration: I. Three Scales of the Elementary Rectilinear Figures,
viz., the Scalene Triangle, the Isosceles Triangle, and the Rectangle,
comprising twenty-seven varieties of each, according to the harmonic
parts of the Right Angle from ¹⁄₂ to ¹⁄₁₆.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: II. Diagram of the Rectilinear Orthography of the
Principal Front of the Parthenon of Athens, in which its Proportions are
determined by harmonic parts of the Right Angle.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: III. Diagram of the Rectilinear Orthography of the
Portico of the Temple of Theseus at Athens, in which its Proportions are
determined by harmonic parts of the Right Angle.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: IV. Diagram of the Rectilinear Orthography of the East End
of Lincoln Cathedral, in which its Proportions are determined by harmonic
parts of the Right Angle.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: V. Four Ellipses described from Foci, determined by
harmonic parts of the Right Angle, shewing in each the Scalene Triangle,
the Isosceles Triangle, and the Rectangle to which it belongs.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: VI. The Composite Ellipse of ¹⁄₆ and ¹⁄₈ of the Right
Angle, shewing its greater and lesser Axis, its various Foci, and the
Isosceles Triangle in which they are placed.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: VII. The Composite Ellipse of ¹⁄₄₈ and ¹⁄₆₄ of the Right
Angle, shewing how it forms the Entasis of the Columns of the Parthenon
of Athens.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: VIII. Sectional Outlines of two Mouldings of the Parthenon
of Athens, full size, shewing the harmonic nature of their Curves, and
the simple manner of their Construction.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: IX. Three Diagrams, giving a Vertical, a Front, and a Side
Aspect of the Geometrical Construction of the Human Head and Countenance,
in which the Proportions are determined by harmonic parts of the Right
Angle.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: X. Diagram in which the Symmetrical Proportions of the
Human Figure are determined by harmonic parts of the Right Angle.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: XI. The Contour of the Human Figure as viewed in Front
and in Profile, its Curves being determined by Ellipses, whose Foci are
determined by harmonic parts of the Right Angle.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: XII. Rectilinear Diagram, shewing the Proportions of the
Portland Vase, as determined by harmonic parts of the Right Angle, and
the outline of its form by an Elliptic Curve harmonically described.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: XIII. Rectilinear Diagram of the Proportions and
Curvilinear Outline of the form of an ancient Grecian Vase, the
proportions determined by harmonic parts of the Right Angle, and the
melody of the form by Curves of two Ellipses.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: XIV. Rectilinear Diagram of the Proportions and
Curvilinear Outline of the form an ancient Grecian Vase, the proportions
determined by harmonic parts of the Right Angle, and the melody of the
form by an Elliptic Curve.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: XV. Two Diagrams of Etruscan Vases, the harmony of
Proportions and melody of the Contour determined, respectively, by parts
of the Right Angle and an Elliptic Curve.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: XVI. Two Diagrams of Etruscan Vases, whose harmony of
Proportion and melody of Contour are determined as above.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: XVII. Diagram shewing the Geometric Construction of an
Ornament belonging to the Parthenon at Athens.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: XVIII. Diagram of the Geometrical Construction of the
ancient Grecian Ornament called the Honeysuckle.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: XIX. An additional Illustration of the Contour of the
Human Figure, as viewed in Front and in Profile.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: XX. Diagram shewing the manner in which the Elliptic
Curves are arranged in order to produce an Outline of the Form of the
Human Figure as viewed in Front.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: XXI. Diagram of a variation on the Form of the Portland
Vase.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: XXII. Diagram of a second variation on the Form of the
Portland Vase.
_D. R. Hay delᵗ._ _G. Aikman sc._]
[Illustration: XXIII. Diagram of a third variation on the Form of the
Portland Vase.
_D. R. Hay delᵗ._ _G. Aikman sc._]
INTRODUCTION.
Twelve years ago, one of our most eminent philosophers,[1] through the
medium of the _Edinburgh Review_,[2] gave the following account of what
was then the state of the fine arts as connected with science:—“The
disposition to introduce into the intellectual community the principles
of free intercourse, is by no means general; but we are confident that
Art will not sufficiently develop her powers, nor Science attain her
most commanding position, till the practical knowledge of the one is
taken in return for the sound deductions of the other.... It is in the
fine arts, principally, and in the speculations with which they are
associated, that the controlling power of scientific truth has not
exercised its legitimate influence. In discussing the principles of
painting, sculpture, architecture, and landscape gardening, philosophers
have renounced science as a guide, and even as an auxiliary; and a
school has arisen whose speculations will brook no restraint, and whose
decisions stand in opposition to the strongest convictions of our
senses. That the external world, in its gay colours and lovely forms, is
exhibited to the mind only as a tinted mass, neither within nor without
the eye, neither touching it nor distant from it—an ubiquitous chaos,
which experience only can analyse and transform into the realities
which compose it; that the beautiful and sublime in nature and in art
derive their power over the mind from association alone, are among the
philosophical doctrines of the present day, which, if it be safe, it is
scarcely prudent to question. Nor are these opinions the emanations of
poetical or ill-trained minds, which ingenuity has elaborated, and which
fashion sustains. They are conclusions at which most of our distinguished
philosophers have arrived. They have been given to the world with all the
authority of demonstrated truth; and in proportion to the hold which they
have taken of the public mind, have they operated as a check upon the
progress of knowledge.”
Such, then, was the state of art as connected with science twelve years
ago. But although the causes which then placed science and the fine arts
at variance have since been gradually diminishing, yet they are still
far from being removed. In proof of this I may refer to what took place
at the annual distribution of the prizes to the students attending our
Scottish Metropolitan School of Design, in 1854, the pupils in which
amount to upwards of two hundred. The meeting on that occasion included,
besides the pupils, a numerous and highly respectable assemblage of
artists and men of science. The chairman, a Professor in our University,
and editor of one of the most voluminous works on art, science, and
literature ever produced in this country, after extolling the general
progress of the pupils, so far as evinced by the drawings exhibited on
the occasion, drew the attention of the meeting to a discovery made by
the head master of the architectural and ornamental department of the
school, viz.—That the ground-plan of the Parthenon at Athens had been
constructed by the application of the _mysterious_ ovoid or _Vesica
Piscis_ of the middle ages, subdivided by the _mythic_ numbers 3 and 7,
and their intermediate odd number 5. Now, it may be remarked, that the
figure thus referred to is not an ovoid, neither is it in any way of a
mysterious nature, being produced simply by two equal circles cutting
each other in their centres. Neither can it be shewn that the numbers 3
and 7 are in any way more mythic than other numbers. In fact, the terms
_mysterious_ and _mythic_ so applied, can only be regarded as a remnant
of an ancient terminology, calculated to obscure the simplicity of
scientific truth, and when used by those employed to teach—for doubtless
the chairman only gave the description he received—must tend to retard
the connexion of that truth with the arts of design. I shall now give
a specimen of the manner in which a knowledge of the philosophy of the
fine arts is at present inculcated upon the public mind generally. In
the same metropolis there has likewise existed for upwards of ten years
a Philosophical Institution of great importance and utility, whose
members amount to nearly three thousand, embracing a large proportion
of the higher classes of society, both in respect to talent and wealth.
At the close of the session of this Institution, in 1854, a learned and
eloquent philologus, who occasionally lectures upon beauty, was appointed
to deliver the closing address, and touching upon the subject of the
beautiful, he thus concluded—
“In the worship of the beautiful, and in that alone, we are inferior to
the Greeks. Let us therefore be glad to borrow from them; not slavishly,
but with a wise adaptation—not exclusively, but with a cunning selection;
in art, as in religion, let us learn to prove all things, and hold fast
that which is good—not merely one thing which is good, but all good
things—Classicalism, Mediævalism, Modernism—let us have and hold them
all in one wide and lusty embrace. Why should the world of art be
more narrow, more monotonous, than the world of nature? Did God make
all the flowers of one pattern, to please the devotees of the rose or
the lily; and did He make all the hills, with the green folds of their
queenly mantles, all at one slope, to suit the angleometer of the most
mathematical of decorators? I trow not. Let us go and do likewise.”
I here take for granted, that what the lecturer meant by “the worship of
the beautiful,” is the production and appreciation of works of art in
which beauty should be a primary element; and judging from the remains
which we possess of such works as were produced by the ancient Grecians,
our inferiority to them in these respects cannot certainly be denied.
But I must reiterate what I have often before asserted, that it is
not by borrowing from them, however cunning our selection, or however
wise our adaptations, that this inferiority is to be removed, but by
a re-discovery of the science which these ancient artists must have
employed in the production of that symmetrical beauty and chaste elegance
which pervaded all their works for a period of nearly three hundred
years. And I hold, that as in religion, so in art, there is only one
truth, a grain of which is worth any amount of philological eloquence.
I also take for granted, that what is meant by Classicalism in the
above quotation, is the ancient Grecian style of art; by Mediævalism,
the semi-barbaric style of the middle ages; and by Modernism, that
chaotic jumble of all previous styles and fashions of art, which is the
peculiar characteristic of our present school, and which is, doubtless,
the result of a system of education based upon plagiarism and mere
imitation. Therefore a recommendation to embrace with equal fervour “as
good things,” these very opposite artic_isms_ must be a doctrine as
mischievous in art as it would be in religion to recommend as equally
good things the various _isms_ into which it has also been split in
modern times.
Now, “the world of nature” and “the world of art” have not that equality
of scope which this lecturer on beauty ascribes to them, but differ very
decidedly in that particular. Neither will it be difficult to shew why
“the world of art _should_ be more narrow than the world of nature”—that
it should be thereby rendered more monotonous does not follow.
It is well known, that the “world of nature” consists of productions,
including objects of every degree of beauty from the very lowest to the
highest, and calculated to suit not only the tastes arising from various
degrees of intellect, but those arising from the natural instincts of the
lower animals. On the other hand, “the world of art,” being devoted to
the gratification and improvement of intelligent minds only, is therefore
narrowed in its scope by the exclusion from its productions of the lower
degrees of beauty—even mediocrity is inadmissible; and we know that the
science of the ancient Greek artists enabled them to excel the highest
individual productions of nature in the perfection of symmetrical beauty.
Consequently, all objects in nature are not equally well adapted for
artistic study, and it therefore requires, on the part of the artist,
besides true genius, much experience and care to enable him to choose
proper subjects from nature; and it is in the choice of such subjects,
and not in plagiarism from the ancients, that he should select with
knowledge and adapt with wisdom. Hence, all such latitudinarian doctrines
as those I have quoted must act as a check upon the progress of knowledge
in the scientific truth of art. I have observed in some of my works, that
in this country a course had been followed in our search for the true
science of beauty not differing from that by which the alchymists of the
middle ages conducted their investigations; for our ideas of visible
beauty are still undefined, and our attempts to produce it in the various
branches of art are left dependant, in a great measure, upon chance.
Our schools are conducted without reference to any first principles or
definite laws of beauty, and from the drawing of a simple architectural
moulding to the intricate combinations of form in the human figure, the
pupils trust to their hands and eyes alone, servilely and mechanically
copying the works of the ancients, instead of being instructed in the
unerring principles of science, upon which the beauty of those works
normally depends. The instruction they receive is imparted without
reference to the judgment or understanding, and they are thereby led to
imitate effects without investigating causes. Doubtless, men of great
genius sometimes arrive at excellence in the arts of design without a
knowledge of the principles upon which beauty of form is based; but it
should be kept in mind, that true genius includes an intuitive perception
of those principles along with its creative power. It is, therefore, to
the generality of mankind that instruction in the definite laws of beauty
will be of most service, not only in improving the practice of those who
follow the arts professionally, but in enabling all of us to distinguish
the true from the false, and to exercise a sound and discriminating
taste in forming our judgment upon artistic productions. Æsthetic
culture should consequently supersede servile copying, as the basis of
instruction in our schools of art. Many teachers of drawing, however,
still assert, that, by copying the great works of the ancients, the mind
of the pupil will become imbued with ideas similar to theirs—that he
will imbibe their feeling for the beautiful, and thereby become inspired
with their genius, and think as they thought. To study carefully and
to investigate the principles which constitute the excellence of the
works of the ancients, is no doubt of much benefit to the student; but
it would be as unreasonable to suppose that he should become inspired
with artistic genius by merely copying them, as it would be to imagine,
that, in literature, poetic inspiration could be created by making boys
transcribe or repeat the works of the ancient poets. Sir Joshua Reynolds
considered copying as a delusive kind of industry, and has observed, that
“Nature herself is not to be too closely copied,” asserting that “there
are excellences in the art of painting beyond what is commonly called the
imitation of nature,” and that “a mere copier of nature can never produce
any thing great.” Proclus, an eminent philosopher and mathematician of
the later Platonist school (A.D. 485), says, that “he who takes for his
model such forms as nature produces, and confines himself to an exact
imitation of these, will never attain to what is perfectly beautiful. For
the works of nature are full of disproportion, and fall very short of the
true standard of beauty.”
It is remarked by Mr. J. C. Daniel, in the introduction to his
translation of M. Victor Cousin’s “Philosophy of the Beautiful,” that
“the English writers have advocated no theory which allows the beautiful
to be universal and absolute; nor have they professedly founded their
views on original and ultimate principles. Thus the doctrine of the
English school has for the most part been, that beauty is mutable and
special, and the inference that has been drawn from this teaching is,
that all tastes are equally just, provided that each man speaks of what
he feels.” He then observes, that the German, and some of the French
writers, have thought far differently; for with them the beautiful is
“simple, immutable, absolute, though its _forms_ are manifold.”
So far back as the year 1725, the same truths advanced by the modern
German and French writers, and so eloquently illustrated by M. Cousin,
were given to the world by Hutchison in his “Inquiry into the Original
of our Ideas of Beauty and Virtue.” This author says—“We, by absolute
beauty, understand only that beauty which we perceive in objects, without
comparison to any thing external, of which the object is supposed an
imitation or picture, such as the beauty perceived from the works of
nature, artificial forms, figures, theorems. Comparative or relative
beauty is that which we perceive in objects commonly considered as
imitations or resemblances of something else.”
Dr. Reid also, in his “Intellectual Powers of Man,” says—“That taste,
which we may call rational, is that part of our constitution by which we
are made to receive pleasure from the contemplation of what we conceive
to be excellent in its kind, the pleasure being annexed to this judgment,
and regulated by it. This taste may be true or false, according as it is
founded on a true or false judgment. And if it may be true or false, it
must have first principles.”
M. Victor Cousin’s opinion upon this subject is, however, still more
conclusive. He observes—“If the idea of the beautiful is not absolute,
like the idea of the true—if it is nothing more than the expression of
individual sentiment, the rebound of a changing sensation, or the result
of each person’s fancy—then the discussions on the fine arts waver
without support, and will never end. For a theory of the fine arts to
be possible, there must be something absolute in beauty, just as there
must be something absolute in the idea of goodness, to render morals a
possible science.”
The basis of the science of beauty must thus be founded upon fixed
principles, and when these principles are evolved with the same care
which has characterised the labours of investigators in natural science,
and are applied in the fine arts as the natural sciences have been in
the useful arts, a solid foundation will be laid, not only for correct
practice, but also for a just appreciation of productions in every branch
of the arts of design.
We know that the mind receives pleasure through the sense of hearing,
not only from the music of nature, but from the euphony of prosaic
composition, the rhythm of poetic measure, the artistic composition
of successive harmony in simple melody, and the combined harmony of
counterpoint in the more complex works of that art. We know, also,
that the mind is similarly gratified through the sense of seeing, not
only by the visible beauties of nature, but by those of art, whether
in symmetrical or picturesque compositions of forms, or in harmonious
arrangements of gay or sombre colouring.
Now, in respect to the first of these modes of sensation, we know,
that from the time of Pythagoras, the fact has been established, that
in whatever manner nature or art may address the ear, the degree of
obedience paid to the fundamental law of harmony will determine the
presence and degree of that beauty with which a perfect organ can impress
a well-constituted mind; and it is my object in this, as it has been
in former attempts, to prove it consistent with scientific truth, that
that beauty which is addressed to the mind by objects of nature and art,
through the eye, is similarly governed. In short, to shew that, as in
compositions of sounds, there can be no true beauty in the absence of a
strict obedience to this great law of nature, neither can there exist,
in compositions of forms or colours, that principle of unity in variety
which constitutes beauty, unless such compositions are governed by the
same law.
Although in the songs of birds, the gurgling of brooks, the sighing of
the gentle summer winds, and all the other beautiful music of nature,
no analysis might be able to detect the operation of any precise system
of harmony, yet the pleasure thus afforded to the human mind we know to
arise from its responding to every development of an obedience to this
law. When, in like manner, we find even in those compositions of forms
and colours which constitute the wildest and most rugged of Nature’s
scenery, a species of picturesque grandeur and beauty to which the mind
as readily responds as to her more mild and pleasing aspects, or to her
sweetest music, we may rest assured that this beauty is simply another
development of, and response to, the same harmonic law, although the
precise nature of its operation may be too subtle to be easily detected.
The _résumé_ of the various works I have already published upon the
subject, along with the additional illustrations I am about to lay before
my readers, will, I trust, point out a system of harmony, which, in
formative art, as well as in that of colouring, will rise superior to the
idiosyncracies of different artists, and bring back to one common type
the sensations of the eye and the ear, thereby improving that knowledge
of the laws of the universe which it is as much the business of science
to combine with the ornamental as with the useful arts.
In attempting this, however, I beg it may be understood, that I do not
believe any system, based even upon the laws of nature, capable of
forming a royal road to the perfection of art, or of “mapping the mighty
maze of a creative mind.” At the same time, however, I must continue to
reiterate the fact, that the diffusion of a general knowledge of the
science of visible beauty will afford latent artistic genius just such a
vantage ground as that which the general knowledge of philology diffused
throughout this country affords its latent literary genius. Although
_mere learning_ and _true genius_ differ as much in the practice of art
as they do in the practice of literature, yet a precise and systematic
education in the true science of beauty must certainly be as useful in
promoting the practice and appreciation of the one, as a precise and
systematic education in the science of philology is in promoting the
practice and appreciation of the other.
As all beauty is the result of harmony, it will be requisite here to
remark, that harmony is not a simple quality, but, as Aristotle defines
it, “the union of contrary principles having a ratio to each other.”
Harmony thus operates in the production of all that is beautiful in
nature, whether in the combinations, in the motions, or in the affinities
of the elements of matter.
The contrary principles to which Aristotle alludes, are those of
uniformity and variety; for, according to the predominance of the one or
the other of these principles, every kind of beauty is characterised.
Hence the difference between symmetrical and picturesque beauty:—the
first allied to the principle of uniformity, in being based upon precise
laws that may be taught so as to enable men of ordinary capacity to
produce it in their works—the second allied to the principle of variety
often to so great a degree that they yield an obedience to the precise
principles of harmony so subtilely, that they cannot be detected in its
constitution, but are only felt in the response by which true genius
acknowledges their presence. The generality of mankind may be capable
of perceiving this latter kind of beauty, and of feeling its effects
upon the mind, but men of genius, only, can impart it to works of art,
whether addressed to the eye or the ear. Throughout the sounds, forms,
and colours of nature, these two kinds of beauty are found not only in
distinct developments, but in every degree of amalgamation. We find in
the songs of some birds, such as those of the chaffinch, thrush, &c.,
a rhythmical division, resembling in some measure the symmetrically
precise arrangements of parts which characterises all artistic musical
composition; while in the songs of other birds, and in the other numerous
melodies with which nature charms and soothes the mind, there is no
distinct regularity in the division of their parts. In the forms of
nature, too, we find amongst the innumerable flowers with which the
surface of the earth is so profusely decorated, an almost endless variety
of systematic arrangements of beautiful figures, often so perfectly
symmetrical in their combination, that the most careful application
of the angleometer could scarcely detect the slightest deviation from
geometrical precision; while, amongst the masses of foliage by which
the forms of many trees are divided and subdivided into parts, as also
amongst the hills and valleys, the mountains and ravines, which divide
the earth’s surface, we find in every possible variety of aspect the
beauty produced by that irregular species of symmetry which characterises
the picturesque.
In like manner, we find in wild as well as cultivated flowers the
most symmetrical distributions of colours accompanying an equally
precise species of harmony in their various kinds of contrasts, often
as mathematically regular as the geometric diagrams by which writers
upon colour sometimes illustrate their works; while in the general
colouring of the picturesque beauties of nature, there is an endless
variety in its distributions, its blendings, and its modifications. In
the forms and colouring of animals, too, the same endless variety of
regular and irregular symmetry is to be found. But the highest degree of
beauty in nature is the result of an equal balance of uniformity with
variety. Of this the human figure is an example; because, when it is
of those proportions universally acknowledged to be the most perfect,
its uniformity bears to its variety an apparently equal ratio. The
harmony of combination in the normal proportions of its parts, and the
beautifully simple harmony of succession in the normal melody of its
softly undulating outline, are the perfection of symmetrical beauty,
while the innumerable changes upon the contour which arise from the
actions and attitudes occasioned by the various emotions of the mind,
are calculated to produce every species of picturesque beauty, from the
softest and most pleasing to the grandest and most sublime.
Amongst the purely picturesque objects of inanimate nature, I may, as in
a former work, instance an ancient oak tree, for its beauty is enhanced
by want of apparent symmetry. Thus, the more fantastically crooked its
branches, and the greater the dissimilarity and variety it exhibits in
its masses of foliage, the more beautiful it appears to the artist and
the amateur; and, as in the human figure, any attempt to produce variety
in the proportions of its lateral halves would be destructive of its
symmetrical beauty, so in the oak tree any attempt to produce palpable
similarity between any of its opposite sides would equally deteriorate
its picturesque beauty. But picturesque beauty is not the result of the
total absence of symmetry; for, as none of the irregularly constructed
music of nature could be pleasing to the ear unless there existed in
the arrangement of its notes an obedience, however subtle, to the great
harmonic law of Nature, so neither could any object be picturesquely
beautiful, unless the arrangement of its parts yields, although it may be
obscurely, an obedience to the same law.
However symmetrically beautiful any architectural structure may be, when
in a complete and perfect state, it must, as it proceeds towards ruin,
blend the picturesque with the symmetrical; but the type of its beauty
will continue to be the latter, so long as a sufficient portion of it
remains to convey an idea of its original perfection. It is the same with
the human form and countenance; for age does not destroy their original
beauty, but in both only lessens that which is symmetrical, while it
increases that which is picturesque.
In short, as a variety of simultaneously produced sounds, which do not
relate to each other agreeably to this law, can only convey to the mind
a feeling of mere noise; so a variety of forms or colours simultaneously
exposed to the eye under similar circumstances, can only convey to the
mind a feeling of chaotic confusion, or what may be termed _visible_
discord. As, therefore, the two principles of uniformity and variety,
or similarity and dissimilarity, are in operation in every harmonious
combination of the elements of sound, of form, and of colour, we must
first have recourse to numbers in the abstract before we can form a
proper basis for a universal science of beauty.
THE SCIENCE OF BEAUTY EVOLVED FROM THE HARMONIC LAW OF NATURE, AGREEABLY
TO THE PYTHAGOREAN SYSTEM OF NUMERICAL RATIO.
The scientific principles of beauty appear to have been well known to the
ancient Greeks; and it must have been by the practical application of
that knowledge to the arts of Design, that that people continued for a
period of upwards of three hundred years to execute, in every department
of these arts, works surpassing in chaste beauty any that had ever before
appeared, and which have not been equalled during the two thousand years
which have since elapsed.
Æsthetic science, as the science of beauty is now termed, is based
upon that great harmonic law of nature which pervades and governs the
universe. It is in its nature neither absolutely physical nor absolutely
metaphysical, but of an intermediate nature, assimilating in various
degrees, more or less, to one or other of those opposite kinds of
science. It specially embodies the inherent principles which govern
impressions made upon the mind through the senses of hearing and seeing.
Thus, the æsthetic pleasure derived from listening to the beautiful in
musical composition, and from contemplating the beautiful in works of
formative art, is in both cases simply a response in the human mind to
artistic developments of the great harmonic law upon which the science
is based.
Although the eye and the ear are two different senses, and, consequently,
various in their modes of receiving impressions; yet the sensorium is
but one, and the mind by which these impressions are perceived and
appreciated is also characterised by unity. There appears, likewise,
a striking analogy between the natural constitution of the two kinds
of beauty, which is this, that the more physically æsthetic elements
of the highest works of musical composition are melody, harmony, and
tone, whilst those of the highest works of formative art are contour,
proportion, and colour. The melody or theme of a musical composition
and its harmony are respectively analogous,—1st, To the outline of
an artistic work of formative art; and 2d, To the proportion which
exists amongst its parts. To the careful investigator these analogies
become identities in their effect upon the mind, like those of the more
metaphysically æsthetic emotions produced by expression in either of
these arts.
Agreeably to the first analogy, the outline and contour of an object,
suppose that of a building in shade when viewed against a light
background, has a similar effect upon the mind with that of the simple
melody of a musical composition when addressed to the ear unaccompanied
by the combined harmony of counterpoint. Agreeably to the second analogy,
the various parts into which the surface of the supposed elevation is
divided being simultaneously presented to the eye, will, if arranged
agreeably to the same great law, affect the mind like that of an equally
harmonious arrangement of musical notes accompanying the supposed melody.
There is, however, a difference between the construction of these two
organs of sense, viz., that the ear must in a great degree receive its
impressions involuntarily; while the eye, on the other hand, is provided
by nature with the power of either dwelling upon, or instantly shutting
out or withdrawing itself from an object. The impression of a sound,
whether simple or complex, when made upon the ear, is instantaneously
conveyed to the mind; but when the sound ceases, the power of observation
also ceases. But the eye can dwell upon objects presented to it so long
as they are allowed to remain pictured on the retina; and the mind has
thereby the power of leisurely examining and comparing them. Hence the
ear guides more as a mere sense, at once and without reflection; whilst
the eye, receiving its impressions gradually, and part by part, is more
directly under the influence of mental analysis, consequently producing
a more metaphysically æsthetic emotion. Hence, also, the acquired power
of the mind in appreciating impressions made upon it through the organ of
sight under circumstances, such as perspective, &c., which to those who
take a hasty view of the subject appear impossible.
Dealing as this science therefore does, alike with the sources and the
resulting principles of beauty, it is scarcely less dependent on the
accuracy of the senses than on the power of the understanding, inasmuch
as the effect which it produces is as essential a property of objects,
as are its laws inherent in the human mind. It necessarily comprehends a
knowledge of those first principles in art, by which certain combinations
of sounds, forms, and colours produce an effect upon the mind, connected,
in the first instance, with sensation, and in the second with the
reasoning faculty. It is, therefore, not only the basis of all true
practice in art, but of all sound judgment on questions of artistic
criticism, and necessarily includes those laws whereon a correct taste
must be based. Doubtless many eloquent and ingenious treatises have been
written upon beauty and taste; but in nearly every case, with no other
effect than that of involving the subject in still greater uncertainty.
Even when restricted to the arts of design, they have failed to exhibit
any definite principles whereby the true may be distinguished from
the false, and some natural and recognised laws of beauty reduced to
demonstration. This may be attributed, in a great degree, to the neglect
of a just discrimination between what is merely agreeable, or capable
of exciting pleasurable sensations, and what is essentially beautiful;
but still more to the confounding of the operations of the understanding
with those of the imagination. Very slight reflection, however, will
suffice to shew how essentially distinct these two faculties of the mind
are; the former being regulated, in matters of taste, by irrefragable
principles existing in nature, and responded to by an inherent principle
existing in the human mind; while the latter operates in the production
of ideal combinations of its own creation, altogether independent of any
immediate impression made upon the senses. The beauty of a flower, for
example, or of a dew-drop, depends on certain combinations of form and
colour, manifestly referable to definite and systematic, though it may
be unrecognised, laws; but when Oberon, in “Midsummer Night’s Dream,” is
made to exclaim—
“And that same dew, which sometimes on the buds
Was wont to swell, like round and orient pearls,
Stood now within the pretty floweret’s eyes,
Like tears that did their own disgrace bewail,”—
the poet introduces a new element of beauty equally legitimate, yet
altogether distinct from, although accompanying that which constitutes
the more precise science of æsthetics as here defined. The composition
of the rhythm is an operation of the understanding, but the beauty of the
poetic fancy is an operation of the imagination.
Our physical and mental powers, æsthetically considered, may therefore be
classed under three heads, in their relation to the fine arts, viz., the
receptive, the perceptive, and the conceptive.
The senses of hearing and seeing are respectively, in the degree of
their physical power, receptive of impressions made upon them, and of
these impressions the sensorium, in the degree of its mental power, is
perceptive. This perception enables the mind to form a judgment whereby
it appreciates the nature and quality of the impression originally made
on the receptive organ. The mode of this operation is intuitive, and
the quickness and accuracy with which the nature and quality of the
impression is apprehended, will be in the degree of the intellectual
vigour of the mind by which it is perceived. Thus we are, by the
cultivation of these intuitive faculties, enabled to decide with accuracy
as to harmony or discord, proportion or deformity, and assign sound
reasons for our judgment in matters of taste. But mental conception is
the intuitive power of constructing original ideas from these materials;
for after the receptive power has acted, the perception operates in
establishing facts, and then the judgment is formed upon these operations
by the reasoning powers, which lead, in their turn, to the creations of
the imagination.
The power of forming these creations is the true characteristic
of genius, and determines the point at which art is placed beyond
all determinable canons,—at which, indeed, æsthetics give place to
metaphysics.
In the science of beauty, therefore, the human mind is the subject, and
the effect of external nature, as well as of works of art, the object.
The external world, and the individual mind, with all that lies within
the scope of its powers, may be considered as two separate existences,
having a distinct relation to each other. The subject is affected by the
object, through that inherent faculty by which it is enabled to respond
to every development of the all-governing harmonic law of nature; and the
media of communication are the sensorium and its inlets—the organs of
sense.
This harmonic law of nature was either originally discovered by that
illustrious philosopher Pythagoras, upwards of five hundred years before
Christ, or a knowledge of it obtained by him about that period, from
the Egyptian or Chaldean priests. For after having been initiated into
all the Grecian and barbarian sacred mysteries, he went to Egypt, where
he remained upwards of twenty years, studying in the colleges of its
priests; and from Egypt he went into the East, and visited the Persian
and Chaldean magi.[3]
By the generality of the biographers of Pythagoras, it is said to be
difficult to give a clear idea of his philosophy, as it is almost certain
he never committed it to writing, and that it has been disfigured by the
fantastic dreams and chimeras of later Pythagoreans. Diogenes Laërtius,
however, whose “Lives of the Philosophers” was supposed to be written
about the end of the second century of our era, says “there are three
volumes extant written by Pythagoras. One on education, one on politics,
and one on natural philosophy.” And adds, that there were several other
books extant, attributed to Pythagoras, but which were not written by
him. Also, in his “Life of Philolaus,” that Plato wrote to Dion to take
care and purchase the books of Pythagoras.[4] But whether this great
philosopher committed his discoveries to writing or not, his doctrines
regarding the philosophy of beauty are well-known to be, that he
considered numbers as the essence and the principle of all things, and
attributed to them a real and distinct existence; so that, in his view,
they were the elements out of which the universe was constructed, and to
which it owed its beauty. Diogenes Laërtius gives the following account
of this law:—“That the monad was the beginning of everything. From the
monad proceeds an indefinite duad, which is subordinate to the monad as
to its cause. That from the monad and indefinite duad proceeds numbers.
That the part of science to which Pythagoras applied himself above all
others, was arithmetic; and that he taught ‘that from numbers proceed
signs, and from these latter, lines, of which plane figures consist;
that from plane figures are derived solid bodies; that of all plane
figures the most beautiful was the circle, and of all solid bodies the
most beautiful was the sphere.’ He discovered the numerical relations of
sounds on a single string; and taught that everything owes its existence
and consistency to harmony. In so far as I know, the most condensed
account of all that is known of the Pythagorian system of numbers is the
following:—‘The monad or unity is that quantity, which, being deprived of
all number, remains fixed. It is the fountain of all number. The duad is
imperfect and passive, and the cause of increase and division. The triad,
composed of the monad and duad, partakes of the nature of both. The
tetrad, tetractys, or quaternion number is most perfect. The decad, which
is the sum of the four former, comprehends all arithmetical and musical
proportions.’”[5]
These short quotations, I believe, comprise all that is known, for
certain, of the manner in which Pythagoras systematised the law of
numbers. Yet, from the teachings of this great philosopher and his
disciples, the harmonic law of nature, in which the fundamental
principles of beauty are embodied, became so generally understood and
universally applied in practice throughout all Greece, that the fragments
of their works, which have reached us through a period of two thousand
years, are still held to be examples of the highest artistic excellence
ever attained by mankind. In the present state of art, therefore, a
knowledge of this law, and of the manner in which it may again be applied
in the production of beauty in all works of form and colour, must be
of singular advantage; and the object of this work is to assist in the
attainment of such a knowledge.
It has been remarked, with equal comprehensiveness and truth, by a
writer[6] in the _British and Foreign Medical Review_, that “there
is harmony of numbers in all nature—in the force of gravity—in the
planetary movements—in the laws of heat, light, electricity, and chemical
affinity—in the forms of animals and plants—in the perceptions of the
mind. The direction, indeed, of modern natural and physical science is
towards a generalization which shall express the fundamental laws of all
by one simple numerical ratio. And we think modern science will soon shew
that the mysticism of Pythagoras was mystical only to the unlettered,
and that it was a system of philosophy founded on the then existing
mathematics, which latter seem to have comprised more of the philosophy
of numbers than our present.” Many years of careful investigation have
convinced me of the truth of this remark, and of the great advantage
derivable from an application of the Pythagorean system in the arts
of design. For so simple is its nature, that any one of an ordinary
capacity of mind, and having a knowledge of the most simple rules of
arithmetic, may, in a very short period, easily comprehend its nature,
and be able to apply it in practice.
The elements of the Pythagorean system of harmonic number, so far as can
be gathered from the quotations I have given above, seem to be simply
the indivisible monad (1); the duad (2), arising from the union of one
monad with another; the triad (3), arising from the union of the monad
with the duad; and the tetrad (4), arising from the union of one duad
with another, which tetrad is considered a perfect number. From the
union of these four elements arises the decad (10), the number, which,
agreeably to the Pythagorean system, comprehends all arithmetical and
harmonic proportions. If, therefore, we take these elements and unite
them progressively in the following order, we shall find the series of
harmonic numbers (2), (3), (5), and (7), which, with their multiples, are
the complete numerical elements of all harmony, thus:—
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 4 = 7
In order to render an extended series of harmonic numbers useful, it
must be divided into scales; and it is a rule in the formation of these
scales, that the first must begin with the monad (1) and end with the
duad (2), the second begin with the duad (2) and end with the tetrad (4),
and that the beginning and end of all other scales must be continued in
the same arithmetical progression. These primary elements will then form
the foundation of a series of such scales.
I. (1) (2)
II. (2) (3) (4)
III. (4) (5) (6) (7) (8)
IV. (8) (9) (10) ( ) (12) ( ) (14) (15) (16)
The first of these scales has in (1) and (2) a beginning and an end; but
the second has in (2), (3), and (4) the essential requisites demanded
by Aristotle in every composition, viz., “a beginning, a middle, and
an end;” while the third has not only these essential requisites, but
two intermediate parts (5) and (7), by which the beginning, the middle,
and the end are united. In the fourth scale, however, the arithmetical
progression is interrupted by the omission of numbers 11 and 13, which,
not being multiples of either (2), (3), (5), or (7), are inadmissible.
Such is the nature of the harmonic law which governs the progressive
scales of numbers by the simple multiplication of the monad.
I shall now use these numbers as divisors in the formation of a series
of four such scales of parts, which has for its primary element, instead
of the indivisible monad, a quantity which may be indefinitely divided,
but which cannot be added to or multiplied. Like the monad, however, this
quantity is represented by (1). The following is this series of four
scales of harmonic parts:—
I. (1) (¹⁄₂)
II. (¹⁄₂) (¹⁄₃) (¹⁄₄)
III. (¹⁄₄) (¹⁄₅) (¹⁄₆) (¹⁄₇) (¹⁄₈)
IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) ( ) (¹⁄₁₂) ( ) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)
The scales I., II., and III. may now be rendered as complete as scale
IV., simply by multiplying upwards by 2 from (¹⁄₉), (¹⁄₅), (¹⁄₃), (¹⁄₇),
and (¹⁄₁₅), thus:—
I. (1) (⁸⁄₉) (⁴⁄₅) (²⁄₃) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)
II. (¹⁄₂) (⁴⁄₉) (²⁄₅) (¹⁄₃) (²⁄₇) (⁴⁄₁₅) (¹⁄₄)
III. (¹⁄₄) (²⁄₉) (¹⁄₅) (¹⁄₆) (¹⁄₇) (²⁄₁₅) (¹⁄₈)
IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) ( ) (¹⁄₁₂) ( ) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)
We now find between the beginning and the end of scale I. the quantities
(⁸⁄₉), (⁴⁄₅), (²⁄₃), (⁴⁄₇), and (⁸⁄₁₅).
The three first of these quantities we find to be the remainders of the
whole indefinite quantity contained in (1), after subtracting from it
the primary harmonic quantities (¹⁄₉), (¹⁄₅), and (¹⁄₃); we, however,
find also amongst these harmonic quantities that of (¹⁄₄), which being
subtracted from (1) leaves (³⁄₄), a quantity the most suitable whereby
to fill up the hiatus between (⁴⁄₅) and (²⁄₃) in scale I., which arises
from the omission of (¹⁄₁₁) in scale IV. In like manner we find the two
last of these quantities, (⁴⁄₇) and (⁸⁄₁₅), are respectively the largest
of the two parts into which 7 and 15 are susceptible of being divided.
Finding the number 5 to be divisible into parts more unequal than (2)
to (3) and less unequal than (4) to (7), (³⁄₅) naturally fills up the
hiatus between these quantities in scale I., which hiatus arises from the
omission of (¹⁄₁₃) in scale IV. Thus:—
I. (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)
II. (¹⁄₂) (⁴⁄₉) (²⁄₅) ( ) (¹⁄₃) ( ) (²⁄₇) (⁴⁄₁₅) (¹⁄₄)
III. (¹⁄₄) (²⁄₉) (¹⁄₅) ( ) (¹⁄₆) ( ) (¹⁄₇) (²⁄₁₅) (¹⁄₈)
IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) ( ) (¹⁄₁₂) ( ) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)
Scale I. being now complete, we have only to divide these latter
quantities by (2) downwards in order to complete the other three. Thus:—
I. (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)
II. (¹⁄₂) (⁴⁄₉) (²⁄₅) (³⁄₈) (¹⁄₃) (³⁄₁₀) (²⁄₇) (⁴⁄₁₅) (¹⁄₄)
III. (¹⁄₄) (²⁄₉) (¹⁄₅) (³⁄₁₆) (¹⁄₆) (³⁄₂₀) (¹⁄₇) (²⁄₁₅) (¹⁄₈)
IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) (³⁄₃₂) (¹⁄₁₂) (³⁄₄₀) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)
The harmony existing amongst these numbers or quantities consists of the
numerical relations which the parts bear to the whole and to each other;
and the more simple these relations are, the more perfect is the harmony.
The following are the numerical harmonic ratios which the parts bear to
the whole:—
I. (1:1) (8:9) (4: 5) (3: 4) (2: 3) (3: 5) (4: 7) (8:15) (1: 2)
II. (1:2) (4:9) (2: 5) (3: 8) (1: 3) (3:10) (2: 7) (4:15) (1: 4)
III. (1:4) (2:9) (1: 5) (3:16) (1: 6) (3:20) (1: 7) (2:15) (1: 8)
IV. (1:8) (1:9) (1:10) (3:32) (1:12) (3:40) (1:14) (1:15) (1:16)
The following are the principal numerical relations which the parts in
each scale bear to one another:—
(¹⁄₂):(⁴⁄₇) = (7:8)
(⁴⁄₅):(⁸⁄₉) = (9:10)
(²⁄₃):(⁴⁄₅) = (5:6)
(⁴⁄₇):(²⁄₃) = (6:7)
(⁸⁄₁₅):(⁴⁄₇) = (14:15)
(¹⁄₂):(⁸⁄₁₅) = (15:16)
Although these relations are exemplified by parts of scale I., the same
ratios exist between the relative parts of scales II., III., and IV.,
and would exist between the parts of any other scales that might be added
to that series.
These are the simple elements of the science of that harmony which
pervades the universe, and by which the various kinds of beauty
æsthetically impressed upon the senses of hearing and seeing are
governed.
THE SCIENCE OF BEAUTY AS APPLIED TO SOUNDS.
It is well-known that all sounds arise from a peculiar action of the
air, and that this action may be excited by the concussion resulting
from the sudden displacement of a portion of the atmosphere itself, or
by the rapid motions of bodies, or of confined columns of air; in all
which cases, when the motions are irregular, and the force great, the
sound conveyed to the sensorium is called a noise. But that musical
sounds are the result of equal and regular vibratory motions, either
of an elastic body, or of a column of air in a tube, exciting in the
surrounding atmosphere a regular and equal pulsation. The ear is the
medium of communication between those varieties of atmospheric action and
the seat of consciousness. To describe fully the beautiful arrangement
of the various parts of this organ, and their adaptation to the purpose
of collecting and conveying these undulatory motions of the atmosphere,
is as much beyond the scope of my present attempt as it is beyond my
anatomical knowledge; but I may simply remark, that within the ear, and
most carefully protected in the construction of that organ, there is a
small cavity containing a pellucid fluid, in which the minute extremities
of the auditory nerve float; and that this fluid is the last of the media
through which the action producing the sensation of sound is conveyed
to the nerve, and thence to the sensorium, where its nature becomes
perceptible to the mind.
The impulses which produce musical notes must arrive at a certain
frequency before the ear loses the intervals of silence between them,
and is impressed by only one continued sound; and as they increase
in frequency the sound becomes more acute upon the ear. The pitch of
a musical note is, therefore, determined by the frequency of these
impulses; but, on the other hand, its intensity or loudness will depend
upon the violence and the quality of its tone on the material employed in
producing them. All such sounds, therefore, whatever be their loudness
or the quality of their tone in which the impulses occur with the same
frequency are in perfect unison, having the same pitch. Upon this the
whole doctrine of harmonies is founded, and by this the laws of numerical
ratio are found to operate in the production of harmony, and the theory
of music rendered susceptible of exact reasoning.
The mechanical means by which such sounds can be produced are extremely
various; but, as it is my purpose simply to shew the nature of harmony
of sound as related to, or as evolving numerical harmonic ratio, I shall
confine myself to the most simple mode of illustration—namely, that of
the monochord. This is an instrument consisting of a string of a given
length stretched between two bridges standing upon a graduated scale.
Suppose this string to be stretched until its tension is such that,
when drawn a little to a side and suddenly let go, it would vibrate at
the rate of 64 vibrations in a second of time, producing to a certain
distance in the surrounding atmosphere a series of pulsations of the same
frequency.
These pulsations will communicate through the ear a musical note which
would, therefore, be the fundamental note of such a string. Now, the
phenomenon said to be discovered by Pythagoras is well known to those
acquainted with the science of acoustics, namely, that immediately after
the string is thus put into vibratory motion, it spontaneously divides
itself, by a node, into two equal parts, the vibrations of each of which
occur with a double frequency—namely, 128 in a second of time, and,
consequently, produce a note doubly acute in pitch, although much weaker
as to intensity or loudness; that it then, while performing these two
series of vibrations, divides itself, by two nodes, into three parts,
each of which vibrates with a frequency triple that of the whole string;
that is, performs 192 vibrations in a second of time, and produces a
note corresponding in increase of acuteness, but still less intense than
the former, and that this continues to take place in the arithmetical
progression of 2, 3, 4, &c. Simultaneous vibrations, agreeably to the
same law of progression, which, however, seem to admit of no other primes
than the numbers 2, 3, 5, and 7, are easily excited upon any stringed
instrument, even by the lightest possible touch of any of its strings
while in a state of vibratory motion, and the notes thus produced are
distinguished by the name of harmonics. It follows, then, that one-half
of a musical string, when divided from the whole by the pressure of the
finger, or any other means, and put into vibratory motion, produces a
note doubly acute to that produced by the vibratory motion of the whole
string; the third part, similarly separated, a note trebly acute; and
the same with every part into which any musical string may be divided.
This is the fundamental principle by which all stringed instruments are
made to produce harmony. It is the same with wind instruments, the sounds
of which are produced by the frequency of the pulsations occasioned in
the surrounding atmosphere by agitating a column of air confined within
a tube as in an organ, in which the frequency of pulsation becomes
greater in an inverse ratio to the length of the pipes. But the following
series of four successive scales of musical notes will give the reader
a more comprehensive view of the manner in which they follow the law of
numerical ratio just explained than any more lengthened exposition.
It is here requisite to mention, that in the construction of these
scales, I have not only adopted the old German or literal mode of
indicating the notes, but have included, as the Germans do, the note
termed by us B flat as B natural, and the note we term B natural as
H. Now, although this arrangement differs from that followed in the
construction of our modern Diatonic scale, yet as the ratio of 4:7
is more closely related to that of 1:2 than that of 8:15, and as it
is offered by nature in the spontaneous division of the monochord,
I considered it quite admissible. The figures give the parts of the
monochord which would produce the notes.
I. { (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)*
{ C D E F G A B H _c_
II. { (¹⁄₂)* (⁴⁄₉) (²⁄₅) (³⁄₈) (¹⁄₃)* (³⁄₁₀) (²⁄₇) (²⁄₁₅) (¹⁄₄)*
{ _c_ _d_ _e_ _f_ _g_ _a_ _b_ _h_ _c′_
III. { (¹⁄₄)* (²⁄₉) (¹⁄₅)* (³⁄₁₆) (¹⁄₆)* (³⁄₂₀) (¹⁄₇)* (²⁄₁₅) (¹⁄₈)*
{ _c′_ _d′_ _e′_ _f′_ _g′_ _a′_ _b′_ _h′_ _c′′_
IV. { (¹⁄₈)* (¹⁄₉)* (¹⁄₁₀)* (³⁄₃₂) (¹⁄₁₂)* (³⁄₄₀) (¹⁄₁₄)* (¹⁄₁₅)* (¹⁄₁₆)*
{ _c′′_ _d′′_ _e′′_ _f′′_ _g′′_ _a′′_ _b′′_ _h′′_ _c′′′_
The notes marked (*) are the harmonics which naturally arise from the
division of the string by 2, 3, 5, and 7, and the multiples of these
primes.
Thus every musical sound is composed of a certain number of parts called
pulsations, and these parts must in every scale relate harmonically
to some fundamental number. When these parts are multiples of the
fundamental number by 2, 4, 8, &c., like the pulsations of the sounds
indicated by _c_, _c′_, _c′′_, _c′′′_, they are called tonic notes, being
the most consonant; when the pulsations are similar multiples by 3, 6,
12, &c., like those of the sounds indicated by _g_, _g′_, _g′′_, they are
called dominant notes, being the next most consonant; and multiples by
5, 10, &c., like those of the sounds indicated by _e_, _e′_, _e′′_, they
are called mediant notes, from a similar cause. In harmonic combinations
of musical sounds, the æsthetic feeling produced by their agreement
depends upon the relations they bear to each other with reference to the
number of pulsations produced in a given time by the fundamental note of
the scale to which they belong; and it will be observed, that the more
simple the numerical ratios are amongst the pulsations of any number of
notes simultaneously produced, the more perfect their agreement. Hence
the origin of the common chord or fundamental concord in the united
sounds of the tonic, the dominant, and the mediant notes, the ratios and
coincidences of whose pulsations 2:1, 3:2, 5:4, may thus be exemplified:—
[Illustration]
In musical composition, the law of number also governs its division
into parts, in order to produce upon the ear, along with the beauty of
harmony, that of rhythm. Thus a piece of music is divided into parts
each of which contains a certain number of other parts called bars,
which may be divided and subdivided into any number of notes, and the
performance of each bar is understood to occupy the same portion of time,
however numerous the notes it contains may be; so that the music of art
is regularly symmetrical in its structure; while that of nature is in
general as irregular and indefinite in its rhythm as it is in its harmony.
Thus I have endeavoured briefly to explain the manner in which the law of
numerical ratio operates in that species of beauty perceived through the
ear.
The definite principles of the art of music founded upon this law have
been for ages so systematised that those who are instructed in them
advance steadily in proportion to their natural endowments, while those
who refuse this instruction rarely attain to any excellence. In the
sister arts of form and colour, however, a system of tuition, founded
upon this law, is still a desideratum, and a knowledge of the scientific
principles by which these arts are governed is confined to a very few,
and scarcely acknowledged amongst those whose professions most require
their practical application.
THE SCIENCE OF BEAUTY AS APPLIED TO FORMS.
It is justly remarked, in the “Illustrated Record of the New York
Exhibition of 1853,” that “it is a question worthy of consideration how
far the mediocrity of the present day is attributable to an overweening
reliance on natural powers and a neglect of the lights of science;”
and there is expressed a thorough conviction of the fact that, besides
the evils of the copying system, “much genius is now wasted in the
acquirement of rudimentary knowledge in the slow school of practical
experiment, and that the excellence of the ancient Greek school of
design arose from a thoroughly digested canon of form, and the use
of geometrical formulas, which make the works even of the second and
third-rate genius of that period the wonder and admiration of the present
day.”
That such a canon of form, and that the use of such geometrical formula,
entered into the education, and thereby facilitated the practice of
ancient Greek art, I have in a former work expressed my firm belief,
which is founded on the remarkable fact, that for a period of nearly
three centuries, and throughout a whole country politically divided into
states often at war with each other, works of sculpture, architecture,
and ornamental design were executed, which surpass in symmetrical
beauty any works of the kind produced during the two thousand years that
have since elapsed. So decided is this superiority, that the artistic
remains of the extraordinary period I alluded to are, in all civilised
nations, still held to be the most perfect specimens of formative art in
the world; and even when so fragmentary as to be denuded of everything
that can convey an idea of expression, they still excite admiration and
wonder by the purity of their geometric beauty. And so universal was
this excellence, that it seems to have characterised every production of
formative art, however humble the use to which it was applied.
The common supposition, that this excellence was the result of an
extraordinary amount of genius existing among the Greek people during
that particular period, is not consistent with what we know of the
progress of mankind in any other direction, and is, in the present state
of art, calculated to retard its progress, inasmuch as such an idea
would suggest that, instead of making any exertion to arrive at a like
general excellence, the world must wait for it until a similar supposed
psychological phenomenon shall occur.
But history tends to prove that the long period of universal artistic
excellence throughout Greece could only be the result of an early
inculcation of some well-digested system of correct elementary
principles, by which the ordinary amount of genius allotted to mankind
in every age was properly nurtured and cultivated; and by which, also, a
correct knowledge and appreciation of art were disseminated amongst the
people generally. Indeed, Müller, in his “Ancient Art and its Remains,”
shews clearly that some certain fixed principles, constituting a science
of proportions, were known in Greece, and that they formed the basis of
all artists’ education and practice during the period referred to; also,
that art began to decline, and its brightest period to close, as this
science fell into disuse, and the Greek artists, instead of working for
an enlightened community, who understood the nature of the principles
which guided them, were called upon to gratify the impatient whims of
pampered and tyrannical rulers.
By being instructed in this science of proportion, the Greek artists
were enabled to impart to their representations of the human figure
a mathematically correct species of symmetrical beauty; whether
accompanying the slender and delicately undulated form of the Venus,—its
opposite, the massive and powerful mould of the Hercules,—or the
characteristic representation of any other deity in the heathen
mythology. And this seems to have been done with equal ease in the minute
figure cut on a precious gem, and in the most colossal statue. The same
instruction likewise enabled the architects of Greece to institute those
varieties of proportions in structure called the Classical Orders of
Architecture; which are so perfect that, since the science which gave
them birth has been buried in oblivion, classical architecture has been
little more than an imitative art; for all who have since written upon
the subject, from Vitruvius downwards, have arrived at nothing, in
so far as the great elementary principles in question are concerned,
beyond the most vague and unsatisfactory conjectures. For a more clear
understanding of the nature of this application of the Pythagorean law of
number to the harmony of form, it will be requisite to repeat the fact,
that modern science has shewn that the cause of the impression, produced
by external nature upon the sensorium, called light, may be traced to a
molecular or ethereal action. This action is excited naturally by the
sun, artificially by the combustion of various substances, and sometimes
physically within the eye. Like the atmospheric pulsations which produce
sound, the action which produces light is capable, within a limited
sphere, of being reflected from some bodies and transmitted through
others; and by this reflection and transmission the visible nature of
forms and figures is communicated to the sensorium. The eye is the
medium of this communication; and its structural beauty, and perfect
adaptation to the purpose of conveying this action, must, like those of
the ear, be left to the anatomist fully to describe. It is here only
necessary to remark, that the optic nerve, like the auditory nerve,
ends in a carefully protected fluid, which is the last of the media
interposed between this peculiarly subtle action and the nerve upon
which it impresses the presence of the object from which it is reflected
or through which it is transmitted, and the nature of such object made
perceptible to the mind. The eye and the ear are thus, in one essential
point, similar in their physiology, relatively to the means provided
for receiving impressions from external nature; it is, therefore, but
reasonable to believe that the eye is capable of appreciating the exact
subdivision of spaces, just as the ear is capable of appreciating the
exact subdivision of intervals of time; so that the division of space
into exact numbers of equal parts will æsthetically affect the mind
through the medium of the eye.
We assume, therefore, that the standard of symmetry, so estimated, is
deduced from the simplest law that could have been conceived—the law
that the angles of direction must all bear to some fixed angle the same
simple relations which the different notes in a chord of music bear to
the fundamental note; that is, relations expressed arithmetically by the
smallest natural numbers. Thus the eye, being guided in its estimate by
direction rather than by distance, just as the ear is guided by number
of vibrations rather than by magnitude, both it and the ear convey
simplicity and harmony to the mind without effort, and the mind with
equal facility receives and appreciates them.
_On the Rectilinear Forms and Proportions of Architecture._
As we are accustomed in all cases to refer direction to the horizontal
and vertical lines, and as the meeting of these lines makes the right
angle, it naturally constitutes the fundamental angle, by the harmonic
division of which a system of proportion may be established, and the
theory of symmetrical beauty, like that of music, rendered susceptible of
exact reasoning.
Let therefore the right angle be the fundamental angle, and let it be
divided upon the quadrant of a circle into the harmonic parts already
explained, thus:—
Super- Sub- Sub- Sub- Semi-sub-
Right tonic Mediant dominant Dominant mediant tonic tonic Tonic
Angle. Angles. Angles. Angles. Angles. Angles. Angles. Angles. Angles.
I. (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)
II. (¹⁄₂) (⁴⁄₉) (²⁄₅) (³⁄₈) (¹⁄₃) (³⁄₁₀) (²⁄₇) (⁴⁄₁₅) (¹⁄₄)
III. (¹⁄₄) (²⁄₉) (¹⁄₅) (³⁄₁₆) (¹⁄₆) (³⁄₂₀) (¹⁄₇) (²⁄₁₅) (¹⁄₈)
IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) (³⁄₃₂) (¹⁄₁₂) (³⁄₄₀) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)
In order that the analogy may be kept in view, I have given to the parts
of each of these four scales the appropriate nomenclature of the notes
which form the diatonic scale in music.
When a right angled triangle is constructed so that its two smallest
angles are equal, I term it simply the triangle of (¹⁄₂), because the
smaller angles are each one-half of the right angle. But when the two
angles are unequal, the triangle may be named after the smallest. For
instance, when the smaller angle, which we shall here suppose to be
one-third of the right angle, is made with the vertical line, the
triangle may be called the vertical scalene triangle of (¹⁄₃); and when
made with the horizontal line, the horizontal scalene triangle of (¹⁄₃).
As every rectangle is made up of two of these right angled triangles,
the same terminology may also be applied to these figures. Thus, the
equilateral rectangle or perfect square is simply the rectangle of (¹⁄₂),
being composed of two similar right angled triangles of (¹⁄₂); and when
two vertical scalene triangles of (¹⁄₃), and of similar dimensions, are
united by their hypothenuses, they form the vertical rectangle of (¹⁄₃),
and in like manner the horizontal triangles of (¹⁄₃) similarly united
would form the horizontal rectangle of (¹⁄₃). As the isosceles triangle
is in like manner composed of two right angled scalene triangles joined
by one of their sides, the same terminology may be applied to every
variety of that figure. All the angles of the first of the above scales,
except that of (¹⁄₂), give rectangles whose longest sides are in the
horizontal line, while the other three give rectangles whose longest
sides are in the vertical line. I have illustrated in Plate I. the manner
in which this harmonic law acts upon these elementary rectilinear figures
by constructing a series agreeably to the angles of scales II., III.,
IV. Throughout this series _a b c_ is the primary scalene triangle, of
which the rectangle _a b c e_ is composed; _d c e_ the vertical isosceles
triangle; and when the plate is turned, _d e a_ the horizontal isosceles
triangle, both of which are composed of the same primary scalene triangle.
[Sidenote: Plate I.]
Thus the most simple elements of symmetry in rectilinear forms are the
three following figures:—
The equilateral rectangle or perfect square,
The oblong rectangle, and
The isosceles triangle.
It has been shewn that in harmonic combinations of musical sounds, the
æsthetic feeling produced by their agreement depends upon the relation
they bear to each other with reference to the number of pulsations
produced in a given time by the fundamental note of the scale to which
they belong; and that the more simply they relate to each other in this
way the more perfect the harmony, as in the common chord of the first
scale, the relations of whose parts are in the simple ratios of 2:1, 3:2,
and 5:4. It is equally consistent with this law, that when applied to
form in the composition of an assortment of figures of any kind, their
respective proportions should bear a very simple ratio to each other in
order that a definite and pleasing harmony may be produced amongst the
various parts. Now, this is as effectually done by forming them upon the
harmonic divisions of the right angle as musical harmony is produced by
sounds resulting from harmonic divisions of a vibratory body.
Having in previous works[7] given the requisite illustrations of this
fact in full detail, I shall here confine myself to the most simple kind,
taking for my first example one of the finest specimens of classical
architecture in the world—the front portico of the Parthenon of Athens.
The angles which govern the proportions of this beautiful elevation are
the following harmonic parts of the right angle—
Tonic Dominant Mediant Subtonic Supertonic
Angles. Angles. Angles. Angle. Angles.
(¹⁄₂) (¹⁄₃) (¹⁄₅) (¹⁄₇) (¹⁄₉)
(¹⁄₄) (¹⁄₆) (¹⁄₁₀) (¹⁄₁₈)
(¹⁄₈)
(¹⁄₁₆)
[Sidenote: Plate II.]
In Plate II. I give a diagram of its rectilinear orthography, which is
simply constructed by lines drawn, either horizontally, vertically, or
obliquely, which latter make with either of the former lines one or other
of the harmonic angles in the above series. For example, the horizontal
line AB represents the length of the base or surface of the upper step of
the substructure of the building. The line AE, which makes an angle of
(¹⁄₅) with the horizontal, determines the height of the colonnade. The
line AD, which makes an angle of (¹⁄₄) with the horizontal, determines
the height of the portico, exclusive of the pediment. The line AC, which
makes an angle of (¹⁄₃) with the horizontal, determines the height of the
portico, including the pediment. The line GD, which makes an angle of
(¹⁄₇) with the horizontal, determines the form of the pediment. The lines
EZ and LY, which respectively make angles of (¹⁄₁₆) and (¹⁄₁₈) with the
horizontal, determine the breadth of the architrave, frieze, and cornice.
The line _v n u_, which makes an angle of (¹⁄₃) with the vertical,
determines the breadth of the triglyphs. The line _t d_, which makes an
angle of (¹⁄₂), determines the breadth of the metops. The lines _c b
r f_, and _a i_, which make each an angle of (¹⁄₆) with the vertical,
determine the width of the five centre intercolumniations. The line _z
k_, which makes an angle of (¹⁄₈) with the vertical, determines the width
of the two remaining intercolumniations. The lines _c s_, _q x_, and _y
h_, each of which makes an angle of (¹⁄₁₀) with the vertical, determine
the diameters of the three columns on each side of the centre. The line
_w l_, which makes an angle of (¹⁄₉) with the vertical, determines the
diameter of the two remaining or corner columns.
In all this, the length and breadth of the parts are determined by
horizontal and vertical lines, which are necessarily at right angles with
each other, and the position of which are determined by one or other of
the lines making the harmonic angles above enumerated.
Now, the lengths and breadths thus so simply determined by these few
angles, have been proved to be correct by their agreement with the most
careful measurements which could possibly be made of this exquisite
specimen of formative art. These measurements were obtained by the
“Society of Dilettanti,” London, who, expressly for that purpose, sent Mr
F. C. Penrose, a highly educated architect, to Athens, where he remained
for about five months, engaged in the execution of this interesting
commission, the results of which are now published in a magnificent
volume by the Society.[8] The agreement was so striking, that Mr Penrose
has been publicly thanked by an eminent man of science for bearing
testimony to the truth of my theory, who in doing so observes, “The
dimensions which he (Mr Penrose) gives are to me the surest verification
of the theory I could have desired. The minute discrepancies form that
very element of practical incertitude, both as to execution and direct
measurement, which always prevails in materialising a mathematical
calculation made under such conditions.”[9]
Although the measurements taken by Mr Penrose are undeniably correct, as
all who examine the great work just referred to must acknowledge, and
although they have afforded me the best possible means of testing the
accuracy of my theory as applied to the Parthenon, yet the ideas of Mr
Penrose as to the principles they evolve are founded upon the fallacious
doctrine which has so long prevailed, and still prevails, in the
æsthetics of architecture, viz., that harmony may be imparted by ratios
between the lengths and breadths of parts.
I have taken for my second example an elevation which, although of
smaller dimensions, is no less celebrated for the beauty of its
proportions than the Parthenon itself, viz., the front portico of the
temple of Theseus, which has also been measured by Mr Penrose.
The angles which govern the proportions of this elevation are the
following harmonic parts of the right angle:—
Tonic Dominant Mediant
Angles. Angles. Angles.
(¹⁄₂) (¹⁄₃) (²⁄₅)
(¹⁄₄) (¹⁄₆) (¹⁄₅)
(¹⁄₁₂)
[Sidenote: Plate III.]
A diagram of the rectilinear orthography of this portico is given in
Plate III. Its construction is similar to that of the Parthenon in
respect to the harmonic parts of the right angle, and I have therefore
only to observe, that the line A E makes an angle of (¹⁄₄); the line A D
an angle of (¹⁄₃); the line A C an angle of (²⁄₅); the line G D an angle
of (¹⁄₆); and the lines E Z and L Y angles of (¹⁄₁₂) with the horizontal.
As to the colonnade or vertical part, the line _a b_, which determines
the three middle intercolumniations, makes an angle of (¹⁄₅); the line
_c d_, which determines the two outer intercolumniations, makes an angle
of (¹⁄₆); and the line _e f_, which determines the lesser diameter of
the columns, makes an angle of (¹⁄₁₂) with the vertical. I need give no
further details here, as my intention is to shew the simplicity of the
method by which this theory may be reduced to practice, and because I
have given in my other works ample details, in full illustration of the
orthography of these two structures, especially the first.[10]
The foregoing examples being both horizontal rectangular compositions,
the proportions of their principal parts have necessarily been determined
by lines drawn from the extremities of the base, making angles with
the horizontal line, and forming thereby the diagonals of the various
rectangles into which, in their leading features, they are necessarily
resolved. But the example I am now about to give is of another character,
being a vertical pyramidal composition, and consequently the proportions
of its principal parts are determined by the angles which the oblique
lines make with the vertical line representing the height of the
elevation, and forming a series of isosceles triangles; for the isosceles
triangle is the type of all pyramidal composition.
This third example is the east end of Lincoln Cathedral, a Gothic
structure, which is acknowledged to be one of the finest specimens of
that style of architecture existing in this country.
The angles which govern the proportions of this elevation are the
following harmonic parts of the right angle:—
Tonic. Dominant. Mediant. Subtonic. Supertonic.
(¹⁄₂) (¹⁄₃) (¹⁄₅) (¹⁄₇) (²⁄₉)
(¹⁄₄) (¹⁄₆) (¹⁄₁₀) (¹⁄₉)
(¹⁄₁₂)
[Sidenote: Plate IV.]
In Plate IV. I give a diagram of the vertical, horizontal, and oblique
lines, which compose the orthography of this beautiful elevation.
The line A B represents the full height of this structure. The line A C,
which makes an angle of (²⁄₉) with the vertical, determines the width of
the design, the tops of the aisle windows, and the bases of the pediments
on the inner buttresses; A G, (¹⁄₅) with the vertical, that of the outer
buttress; A F, (¹⁄₉) with the vertical, that of the space between the
outer and inner buttresses and the width of the great centre window; and
A E, (¹⁄₁₂) with vertical, that of both the inner buttresses and the
space between these. A H, which makes (¹⁄₄) with the vertical, determines
the form of the pediment of the centre, and the full height of the base
and surbase. A I, which makes (¹⁄₃) with the vertical, determines the
form of the pediment of the smaller gables, the base of the pediment on
the outer buttress, the base of the ornamental recess between the outer
and inner buttresses, the spring of the arch of the centre window, the
tops of the pediments on the inner buttresses, and the spring of the
arch of the upper window. A K, which makes (¹⁄₂), determines the height
of the outer buttress; and A Z, which makes (¹⁄₆) with the horizontal,
determines that of the inner buttresses. For the reasons already given,
I need not here go into further detail.[11] It is, however, worthy of
remark in this place, that notwithstanding the great difference which
exists between the style of composition in this Gothic design, and in
that of the east end of the Parthenon, the harmonic elements upon which
the orthographic beauty of the one depends, are almost identical with
those of the other.
_On the Curvilinear Forms and Proportions of Architecture._
Each regular rectilinear figure has a curvilinear figure that exclusively
belongs to it, and to which may be applied a corresponding terminology.
For instance, the circle belongs to the equilateral rectangle; that
is, the rectangle of (¹⁄₂), an ellipse to every other rectangle, and
a composite ellipse to every isosceles triangle. Thus the most simple
elements of beauty in the curvilinear forms of architectural design are
the following three figures:—
The circle,
The ellipse, and
The composite ellipse.
I find it necessary in this place to go into some details regarding
the specific character of the two latter figures, because the proper
mode of describing these beautiful curves, and their high value in the
practice of the architectural draughtsman and ornamental designer, seem
as yet unknown. In proof of this assertion, I must again refer to Mr
Penrose’s great work published by the “Society of Dilettanti.” At page
52 of that work it is observed, that “by whatever means an ellipse is to
be constructed mechanically, it is a work of time (if not of absolute
difficulty) so to arrange the foci, &c., as to produce an ellipse of any
exact length and breadth which may be desired.” Now, this is far from
being the case, for the method of arranging the foci of an ellipse of any
given length and breadth is extremely simple, being as follows:—
Let A B C (figure 1) be the length, and D B E the breadth of the desired
ellipse.
[Illustration: Fig. 1.]
Take A B upon the compasses, and place the point of one leg upon E and
the point of the other upon the line A B, it will meet it at F, which is
one focus: keeping the point of the one leg upon E, remove the point of
the other to the line B C, and it will meet it at G, which is the other
focus. But, when the proportions of an ellipse are to be imparted by
means of one of the harmonic angles, suppose the angle of (¹⁄₃), then the
following is the process:—
Let A B C (figure 2) represent the length of the intended ellipse.
Through B draw B _e_ indefinitely, at right angles with A B C; through C
draw the line C _f_ indefinitely, making, with B C, an angle of (¹⁄₃).
Take B C upon the compasses, and place the point of one leg upon D where
C f intersects B _e_, and the point of the other upon the line A B, it
will meet it at F, which is one focus. Keeping the point of one leg still
upon D, remove the point of the other to the line B C, and it will meet
it at G, which is the other focus.
[Illustration: Fig. 2.]
The foci being in either case thus simply ascertained, the method of
describing the curve on a small scale is equally simple.
[Sidenote: Plate V.]
A pin is fixed into each of the two foci, and another into the point D.
Around these three pins a waxed thread, flexible but not elastic, is
tied, care being taken that the knot be of a kind that will not slip.
The pin at D is now removed, and a hard black lead pencil introduced
within the thread band. The pencil is then moved around the pins fixed
in the foci, keeping the thread band at a full and equal tension; thus
simply the ellipse is described. When, however, the governing angle is
acute, say less than (¹⁄₆), it is requisite to adopt a more accurate
method of description,[12] as the architectural examples which follow
will shew. But architectural draughtsmen and ornamental designers would
do well to supply themselves, for ordinary practice, with half a dozen
series of ellipses, varying in the proportions of their axes from (⁴⁄₉)
to (¹⁄₆) of the scale, and the length of their major axes from 1 to 6
inches. These should be described by the above simple process, upon
very strong drawing paper, and carefully cut out, the edge of the paper
being kept smooth, and each ellipse having its greater and lesser axes,
its foci, and the hypothenuse of its scalene triangle drawn upon it. To
exemplify this, I give Plate V., which exhibits the ellipses of (¹⁄₃),
(¹⁄₄), (¹⁄₅), and (¹⁄₆), inscribed in their rectangles, on which _a b_
and _c d_ are respectively the greater and lesser axes, _o o_ the foci,
and _d b_ the angle of each. Such a series of these beautiful figures
would be found particularly useful in drawing the mouldings of Grecian
architecture; for, to describe the curvilinear contour of such mouldings
from single points, as has been done with those which embellish even our
most pretending attempts at the restoration of that classical style of
architecture, is to give the resemblance of an external form without the
harmony which constitutes its real beauty.
Mr Penrose, owing to the supposed difficulty regarding the description of
ellipses just alluded to, endeavours to shew that the curves of all the
mouldings throughout the Parthenon were either parabolic or hyperbolic;
but I believe such curves can have no connexion with the elementary forms
of architecture, for they are curves which represent motion, and do not,
by continued production, form closed figures.
But I have shewn, in a former work,[13] that the contours of these
mouldings are composed of curves of the composite ellipse,—a figure
which I so name because it is composed simply of arcs of various
ellipses harmonically flowing into each other. The composite ellipse,
when drawn systematically upon the isosceles triangle, resembles closely
parabolic and hyperbolic curves—only differing from these inasmuch as it
possesses the essential quality of circumscribing harmonically one of the
elementary rectilinear figures employed in architecture, while those of
the parabola and hyperbola, as I have just observed, are merely curves
of motion, and, consequently, never can harmonically circumscribe or be
resolved into any regular figure.
The composite ellipse may be thus described.
[Sidenote: Plate VI.]
Let A B C (Plate VI.) be a vertical isosceles triangle of (¹⁄₆), bisect A
B in D, and through D draw indefinitely D _f_ perpendicular to A B, and
through B draw indefinitely B _g_, making the angle D B _g_ (¹⁄₈), D _f_
and B _g_ intersecting each other in M. Take B D and D M as semi-axes of
an ellipse, the foci of which will be at _p_ and _q_, in each of these,
and in each of the foci _h t_ and _k r_ in the lines A C and B C, fix
a pin, and one also in the point M, tie a thread around these pins,
withdraw the pin from M, and trace the composite ellipse in the manner
already described with respect to the simple ellipse.
In some of my earlier works I described this figure by taking the angles
of the isosceles triangle as foci; but the above method is much more
correct. As the elementary angle of the triangle is (¹⁄₆), and that of
the elliptic curve described around it (¹⁄₈), I call it the composite
ellipse of (¹⁄₆) and (¹⁄₈), their harmonic ratio being 4:3; and so on of
all others, according to the difference that may thus exist between the
elementary angles.
The visible curves which soften and beautify the melody of the outline
of the front of the Parthenon, as given in Mr Penrose’s great work, I
have carefully analysed, and have found them in as perfect agreement
with this system, as its rectilinear harmony has been shewn to be. This
I demonstrated in the work just referred to[14] by a series of twelve
plates, shewing that the entasis of the columns (a subject upon which
there has been much speculation) is simply an arc of an ellipse of
(¹⁄₄₈), whose greater axis makes with the vertical an angle of (¹⁄₆₄);
or simply, the form of one of these columns is the frustrum of an
elliptic-sided or prolate-spheroidal cone, whose section is a composite
ellipse of (¹⁄₄₈) and (¹⁄₆₄), the harmonic ratio of these two angles
being 4:3, the same as that of the angles of the composite ellipse just
exemplified.
[Sidenote: Plate VII.]
[Sidenote: Plate VIII.]
In Plate VII. is represented the section of such a cone, of which A B C
is the isosceles triangle of (¹⁄₄₈), and B D and D M the semi-axes of an
ellipse of (¹⁄₆₄). M N and O P are the entases of the column, and _d e f_
the normal construction of the capital. All these are fully illustrated
in the work above referred to,[15] in which I have also shewn that the
curve of the neck of the column is that of an ellipse of (¹⁄₆); the curve
of the capital or echinus, that of an ellipse of (¹⁄₁₄); the curve of
the moulding under the cymatium of the pediment, that of an ellipse of
(¹⁄₃); and the curve of the bed-moulding of the cornice of the pediment,
that of an ellipse of (¹⁄₃). The curve of the cavetto of the soffit of
the corona is composed of ellipses of (¹⁄₆) and (¹⁄₁₄); the curve of the
cymatium which surmounts the corona, is that of an ellipse of (¹⁄₃); the
curve of the moulding of the capital of the antæ of the posticum, that of
an ellipse of (¹⁄₃); the curves of the lower moulding of the same capital
are composed of those of an ellipse of (¹⁄₃) and of the circle (¹⁄₂); the
curve of the moulding which is placed between the two latter is that of
an ellipse of (¹⁄₃); the curve of the upper moulding of the band under
the beams of the ceiling of the peristyle, that of an ellipse of (¹⁄₃);
the curve of the lower moulding of the same band, that of an ellipse of
(¹⁄₄); and the curves of the moulding at the bottom of the small step
or podium between the columns, are those of the circle (¹⁄₂) and of an
ellipse of (¹⁄₃). I have also shewn the curve of the fluting of the
columns to be that of (¹⁄₁₄). The greater axis of each of these ellipses,
when not in the vertical or horizontal lines, makes an harmonic angle
with one or other of them. In Plate VIII., sections of the two last-named
mouldings are represented full size, which will give the reader an idea
of the simple manner in which the ellipses are employed in the production
of those harmonic curves.
Thus we find that the system here adopted for applying this law of
nature to the production of beauty in the abstract forms employed
in architectural composition, so far from involving us in anything
complicated, is characterised by extreme simplicity.
In concluding this part of my treatise, I may here repeat what I have
advanced in a late work,[16] viz., my conviction of the probability that
a system of applying this law of nature in architectural construction
was the only great practical secret of the Freemasons, all their other
secrets being connected, not with their art, but with the social
constitution of their society. This valuable secret, however, seems
to have been lost, as its practical application fell into disuse; but,
as that ancient society consisted of speculative as well as practical
masons, the secrets connected with their social union have still been
preserved, along with the excellent laws by which the brotherhood is
governed. It can scarcely be doubted that there was some such practically
useful secret amongst the Freemasons or early Gothic architects; for
we find in all the venerable remains of their art which exist in this
country, symmetrical elegance of form pervading the general design,
harmonious proportion amongst all the parts, beautiful geometrical
arrangements throughout all the tracery, as well as in the elegantly
symmetrised foliated decorations which belong to that style of
architecture. But it is at the same time worthy of remark, that whenever
they diverged from architecture to sculpture and painting, and attempted
to represent the human figure, or even any of the lower animals, their
productions are such as to convince us that in this country these arts
were in a very degraded state of barbarism—the figures are often much
disproportioned in their parts and distorted in their attitudes, while
their representations of animals and chimeras are whimsically absurd.
It would, therefore, appear that architecture, as a fine art, must have
been preserved by some peculiar influence from partaking of the barbarism
so apparent in the sister arts of that period. Although its practical
secrets have been long lost, the Freemasons of the present day trace the
original possession of them to Moses, who, they say, “modelled masonry
into a perfect system, and circumscribed its mysteries by _land-marks_
significant and unalterable.” Now, as Moses received his education in
Egypt, where Pythagoras is said to have acquired his first knowledge of
the harmonic law of numbers, it is highly probable that this perfect
system of the great Jewish legislator was based upon the same law of
nature which constituted the foundation of the Pythagorean philosophy,
and ultimately led to that excellence in art which is still the
admiration of the world.
Pythagoras, it would appear, formed a system much more perfect and
comprehensive than that practised by the Freemasons in the middle ages of
Christianity; for it was as applicable to sculpture, painting, and music,
as it was to architecture. This perfection in architecture is strikingly
exemplified in the Parthenon, as compared with the Gothic structures
of the middle ages; for it will be found that the whole six elementary
figures I have enumerated as belonging to architecture, are required in
completing the orthographic beauty of that noble structure. And amongst
these, none conduce more to that beauty than the simple and composite
ellipses. Now, in the architecture of the best periods of Gothic, or,
indeed, in that of any after period (Roman architecture included), these
beautiful curves seem to have been ignored, and that of the circle alone
employed.
Be those matters as they may, however, the great law of numerical
harmonic ratio remains unalterable, and a proper application of it in
the science of art will never fail to be as productive of effect, as its
operation in nature is universal, certain, and continual.
THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE HUMAN HEAD AND COUNTENANCE.
The most remarkable characteristics of the human head and countenance
are the globular form of the cranium, united as it is with the prolate
spheroidal form produced by the parts which constitute the face, and the
approximation of the profile to the vertical; for in none of the lower
animals does the skull present so near a resemblance to a combination
of these geometric forms, nor the plane of the face to this direction.
We also find that although these peculiar characteristics are variously
modified among the numerous races of mankind, yet one law appears to
govern the beauty of the whole. The highest and most cultivated of these
races, however, present only an approximation to the perfect development
of those distinguishing marks of humanity; and therefore the beauty of
form and proportion which in nature characterises the human head and
countenance, exhibits only a partial development of the harmonic law of
visible beauty. On the other hand, we find that, in their sculpture, the
ancient Greeks surpassed ordinary nature, and produced in their beau
ideal a species of beauty free from the imperfections and peculiarities
that constitute the individuality by which the countenances of men are
distinguished from each other. It may be requisite here to remark, that
this species of beauty is independent of the more intellectual quality
of expression. For as Sir Charles Bell has said, “Beauty of countenance
may be defined in words, as well as demonstrated in art. A face may
be beautiful in sleep, and a statue without expression may be highly
beautiful. But it will be said there is expression in the sleeping figure
or in the statue. Is it not rather that we see in these the capacity
for expression?—that our minds are active in imagining what may be the
motions of these features when awake or animated? Thus, we speak of an
expressive face before we have seen a movement grave or cheerful, or any
indication in the features of what prevails in the heart.”
This capacity for expression certainly enhances our admiration of the
human countenance; but it is more a concomitant of the primary cause of
its beauty than the cause itself. This cause rests on that simple and
secure basis—the harmonic law of nature; for the nearer the countenance
approximates to an harmonious combination of the most perfect figures in
geometry, or rather the more its general form and the relation of its
individual parts are arranged in obedience to that law, the higher its
degree of beauty, and the greater its capacity for the expression of the
passions.
Various attempts have been made to define geometrically the difference
between the ordinary and the ideal beauty of the human head and
countenance, the most prominent of which is that of Camper. He traced,
upon a profile of the skull, a line in a horizontal direction, passing
through the foramen of the ear and the exterior margin of the sockets of
the front teeth of the upper jaw, upon which he raised an oblique line,
tangential to the margin of these sockets, and to the most prominent part
of the forehead. Agreeably to the obliquity of this line, he determined
the relative proportion of the areas occupied by the brain and by the
face, and hence inferred the degree of intellect. When he applied this
measurement to the heads of the antique statues, he found the angle much
greater than in ordinary nature; but that this simple fact afforded no
rule for the reproduction of the ideal beauty of ancient Greek art, is
very evident from the heads and countenances by which his treatise is
illustrated. Sir Charles Bell justly remarks, that although, by Camper’s
method, the forehead may be thrown forward, yet, while the features
of common nature are preserved, we refuse to acknowledge a similarity
to the beautiful forms of the antique marbles. “It is true,” he says,
“that, by advancing the forehead, it is raised, the face is shortened,
and the eye brought to the centre of the head. But with all this, there
is much wanting—that which measurement, or a mere line, will not shew
us.”—“The truth is, that we are more moved by the features than by the
form of the whole head. Unless there be a conformity in every feature to
the general shape of the head, throwing the forehead forward on the face
produces deformity; and the question returns with full force—How is it
that we are led to concede that the antique head of the Apollo, or of the
Jupiter, is beautiful when the facial line makes a hundred degrees with
the horizontal line? In other words—How do we admit that to be beautiful
which is not natural? Simply for the same reason that, if we discover a
broken portion of an antique, a nose, or a chin of marble, we can say,
without deliberation—This must have belonged to a work of antiquity;
which proves that the character is distinguishable in every part—in each
feature, as well as in the whole head.”
Dr Oken says upon this subject:[17]—“The face is beautiful whose nose is
parallel to the spine. No human face has grown into this estate; but
every nose makes an acute angle with the spine. The facial angle is, as
is well known, 80°. What, as yet, no man has remarked, and what is not to
be remarked, either, without our view of the cranial signification, the
old masters have felt through inspiration. They have not only made the
facial angle a right angle, but have even stepped beyond this—the Romans
going up to 96°, the Greeks even to 100°. Whence comes it that this
unnatural face of the Grecian works of art is still more beautiful than
that of the Roman, when the latter comes nearer to nature? The reason
thereof resides in the fact of the Grecian artistic face representing
nature’s design more than that of the Roman; for, in the former, the nose
is placed quite perpendicular, or parallel to the spinal cord, and thus
returns whither it has been derived.”
Other various and conflicting opinions upon this subject have been
given to the world; but we find that the principle from which arose
the ideal beauty of the head and countenance, as represented in works
of ancient Greek art, is still a matter of dispute. When, however, we
examine carefully a fine specimen, we find its beauty and grandeur to
depend more upon the degree of harmony amongst its parts, as to their
relative proportions and mode of arrangement, than upon their excellence
taken individually. It is, therefore, clear that those (and they are
many) who attribute the beauty of ancient Greek sculpture merely to a
selection of parts from various models, must be in error. No assemblage
of parts from ordinary nature could have produced its principal
characteristic, the excess in the angle of the facial line, much less
could it have led to that exquisite harmony of parts by which it is so
eminently distinguished; neither can we reasonably agree with Dr Oken and
others, who assert that it was produced by an exclusive degree of the
inspiration of genius amongst the Greek people during a certain period.
That the inspiration of genius, combined with a careful study of nature,
were essential elements in the production of the great works which have
been handed down to us, no one will deny; but these elements have existed
in all ages, whilst the ideal head belongs exclusively to the Greeks
during the period in which the schools of Pythagoras and Plato were open.
Is it not, therefore, reasonable to suppose, that, besides genius and the
study of nature, another element was employed in the production of this
excellence, and that this element arose from the precise mathematical
doctrines taught in the schools of these philosophers?
An application of the great harmonic law seems to prove that there is no
object in nature in which the science of beauty is more clearly developed
than in the human head and countenance, nor to the representations of
which the same science is more easily applied; and it is to the mode
in which this is done that the varieties of sex and character may be
imparted to works of art. Having gone into full detail, and given ample
illustrations in a former work,[18] it is unnecessary for me to enter
upon that part of the subject in this _résumé_; but only to shew the
typical structure of beauty by which this noble work of creation is
distinguished.
The angles which govern the form and proportions of the human head and
countenance are, with the right angle, a series of seven, which, from the
simplicity of their ratios to each other, are calculated to produce the
most perfect concord. It consists of the right angle and its following
parts—
Tonic. Dominant. Mediant. Subtonic.
(¹⁄₂) (¹⁄₃) (¹⁄₅) (¹⁄₇)
(¹⁄₄) (¹⁄₆)
These angles, and the figures which belong to them, are thus arranged:—
[Sidenote: Plate IX.]
The vertical line A B (Plate IX. fig. 2) represents the full length of
the head and face. Taking this line as the greater axis of an ellipse of
(¹⁄₃), such an ellipse is described around it. Through A the lines A G,
A K, A L, A M, and A N, are drawn on each side of the line A B, making,
with the vertical, respectively the angles of (¹⁄₃), (¹⁄₄), (¹⁄₅), (¹⁄₆),
and (¹⁄₇). Through the points G, K, L, M, and N, where these straight
lines meet the curved line of the ellipse, horizontal lines are drawn
by which the following isosceles triangles are formed, A G G, A K K, A
L L, A M M, and A N N. From the centre X of the equilateral triangle
A G G the curvilinear figure of (¹⁄₂), viz., the circle, is described
circumscribing that triangle.
The curvilinear plane figures of (¹⁄₂) and (¹⁄₃), respectively, represent
the solid bodies of which they are sections, viz., a sphere and a prolate
spheroid. These bodies, from the manner in which they are here placed,
are partially amalgamated, as shewn in figures 1 and 3 of the same plate,
thus representing the form of the human head and countenance, both in
their external appearance and osseous structure, more correctly than they
could be represented by any other geometrical figures. Thus, the angles
of (¹⁄₂) and (¹⁄₃) determine the typical form.
From each of the points _u_ and _n_, where A M cuts G G on both sides of
A B, a circle is described through the points _p_ and _q_, where A K cuts
G G on both sides of A B, and with the same radius a circle is described
from the point _a_, where K K cuts A B.
The circles _u_ and _n_ determine the position and size of the eyeballs,
and the circle _a_ the width of the nose, as also the horizontal width of
the mouth.
The lines G G and K K also determine the length of the joinings of the
ear to the head. The lines L L and M M determine the vertical width of
the mouth and lips when at perfect repose, and the line N N the superior
edge of the chin. Thus simply are the features arranged and proportioned
on the facial surface.
It must, however, be borne in mind, that in treating simply of the
æsthetic beauty of the human head and countenance, we have only to do
with the external appearance. In this research, therefore, the system
of Dr Camper, Dr Owen, and others, whose investigations were more of
a physiological than an æsthetic character, can be of little service;
because, according to that system, the facial angle is determined by
drawing a line tangential to the exterior margin of the sockets of
the front teeth of the upper jaw, and the most prominent part of the
forehead. Now, as these sockets are, when the skull is naturally clothed,
and the features in repose, entirely concealed by the upper lip, we must
take the prominent part of it, instead of the sockets under it, in order
to determine properly this distinguishing mark of humanity. And I believe
it will be found, that when the head is properly poised, the nearer the
angle which this line makes with the horizontal approaches 90°, the more
symmetrically beautiful will be the general arrangement of the parts (see
line _y z_, figure 3, Plate IX.).
THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE FORM OF THE HUMAN FIGURE.
The manner in which this science is developed in the symmetrical
proportions of the entire human figure, is as remarkable for its
simplicity as it has been shewn to be in those of the head and
countenance. Having gone into very full details, and given ample
illustration in two former works[19] upon this subject, I may here
confine myself to the illustration of one description of figure, and to a
reiteration of some facts stated in these works. These facts are, _1st_,
That on a given line the human figure is developed, as to its principal
points, entirely by lines drawn either from the extremities of this line,
or from some obvious or determined localities. _2d_, That the angles
which these lines make with the given line, are all simple sub-multiples
of some given fundamental angle, or bear to it a proportion expressible
under the most simple relations, such as those which constitute the scale
of music. _3d_, That the contour is resolved into a series of ellipses
of the same simple angles. And, _4th_, That these ellipses, like the
lines, are inclined to the first given line by angles which are simple
sub-multiples of the given fundamental angle. From which four facts,
and agreeably to the hypothesis I have adopted, it results as a natural
consequence that the only effort which the mind exercises through the
eye, in order to put itself in possession of the data for forming its
judgment, is this, that it compares the angles about a point, and thereby
appreciates the simplicity of their relations. In selecting the prominent
features of a figure, the eye is not seeking to compare their relative
distances—it is occupied solely with their relative positions. In tracing
the contour, in like manner, it is not left in vague uncertainty as
to what is the curve which is presented to it; unconsciously it feels
the complete ellipse developed before it; and if that ellipse and its
position are both formed by angles of the same simple relative value as
those which aided its determination of the positions of the prominent
features, it is satisfied, and finds the symmetry perfect.
Müller, and other investigators into the archæology of art, refer to the
great difficulty which exists in discovering the principles which the
ancients followed in regard to the proportions of the human figure, from
the different sexes and characters to which they require to be applied.
But in the system thus founded upon the harmonic law of nature, no such
difficulty is felt, for it is as applicable to the massive proportions
which characterise the ancient representations of the Hercules, as to the
delicate and perfectly symmetrical beauty of the Venus. This change is
effected simply by an increase in the fundamental angle. For instance,
in the construction of a figure of the exact proportions of the Venus,
the right angle is adopted. But in the construction of a figure of the
massive proportions of the Hercules, it is requisite to adopt an angle
which bears to the right angle the ratio of 6:5. The adoption of this
angle I have shewn in another work[20] to produce in the Hercules those
proportions which are so characteristic of physical power. The ellipses
which govern the outline, being also formed upon the same larger class
of angles, give the contour of the muscles a more massive character. In
comparing the male and female forms thus geometrically constructed, it
will be found that that of the female is more harmoniously symmetrical,
because the right angle is the fundamental angle for the trunk and the
limbs as well as for the head and countenance; while in that of the male,
the right angle is the fundamental angle for the head only. It may also
be observed, that, from the greater proportional width of the pelvis
of the female, the centres of that motion which the heads of the thigh
bones perform in the cotyloid cavities, and the centres of that still
more extensive range of motion which the arm is capable of performing at
the shoulder joints, are nearly in the same line which determines the
central motion of the vertebral column, while those of the male are not;
consequently all the motions of the female are more graceful than those
of the male.
This difference between the fundamental angles, which impart to the
human figure, on the one hand, the beauty of feminine proportion and
contour, and on the other, the grandeur of masculine strength, being in
the ratio of 5:6, allows ample latitude for those intermediate classes of
proportions which the ancients imparted to their various other deities
in which these two qualities were blended. I therefore confine myself to
an illustration of the external contour of the form, and the relative
proportions of all the parts of a female figure, such as those of the
statues of the Venus of Melos and Venus of Medici.
The angles which govern the form and proportions of such a figure are,
with the right angle, a series of twelve, as follows:—
Tonic. Dominant. Mediant. Subtonic. Supertonic.
(¹⁄₂) (¹⁄₃) (¹⁄₅) (¹⁄₇) (¹⁄₉)
(¹⁄₄) (¹⁄₆) (¹⁄₁₀) (¹⁄₁₄)
(¹⁄₈) (¹⁄₁₂)
These angles are employed in the construction of a diagram, which
determines the proportions of the parts throughout the whole figure.
Thus:—
[Sidenote: Plate X.]
Let the line A B (fig. 1, plate X.) represent the height of the figure
to be constructed. At the point A, make the angles of C A D (¹⁄₃), F A G
(¹⁄₄), H A I (¹⁄₅), K A L (¹⁄₆), and M A N (¹⁄₇). At the point B, make
the angles K B L (¹⁄₈), U B A (¹⁄₁₂), and O B A (¹⁄₁₄).
Through the point K, in which the lines A K and B K intersect one
another, draw P K O parallel to A B, and through C F H and M, where
this line meets A C, A F, A H, and A M, draw C D, F G, H I, and M N,
perpendicular to A B; draw also K L perpendicular to A B; join B F and
B H, and through C draw C E, making with A B the angle (¹⁄₂), which
completes the arrangement of the eleven angles upon A B; for F B G is
very nearly (¹⁄₁₀), and H B I very nearly (¹⁄₉).
At the point _f_, where A C intersects O B, draw _f a_ perpendicular to
A B; and through the point _i_, where B O intersects M N, draw S _i_ T
parallel to A C.
Through _m_, where S _i_ T intersects F B, draw _m n_; through _β_, where
S _i_ T intersects K B, draw _β w_; through T draw T _g_, making an angle
of (¹⁄₃) with O P. Join N P, M B, and _g_ P, and where N P intersects K
B, draw Q R perpendicular to A B.
On A E as a diameter, describe a circle cutting A C in _r_, and draw _r
o_ perpendicular to A B.
With A _o_ and _o r_ as semi-axes, describe the ellipse A _r e_, cutting
A H in _t_; and draw _t u_ perpendicular to A B. With A _u_ and _t u_, as
semi-axes describe the ellipse A _t d_. On _a_ L, as major axis, describe
the ellipse of (¹⁄₃).
For the side aspect or profile of the figure the diagram is thus
constructed—
On one side of a line A B (fig. 2, Plate X.) construct the rectilinear
portion of a diagram the same as fig. 1. Through _i_ draw W Y parallel to
A B, and draw A _z_ perpendicular to A B. Make W _a_ equal to A _a_ (fig.
1), and on _a l_, as major axis, describe the ellipse of (¹⁄₄). Through
_a_ draw _a p_ parallel to A F, and through _p_ draw _p t_ perpendicular
to W Y. Through _a_ draw _f a u_ perpendicular to W Y.
Upon a diameter equal to A E describe a circle whose circumference shall
touch A B and A _z_. With semi-axes equal to A _o_ and _o r_ (fig.
1), describe an ellipse with its major axis parallel to A B, and its
circumference touching O P and _z_ A.
[Sidenote: Plate XI.]
Thus simply are the diagrams of the general proportions of the human
figure, as viewed in front and in profile, constructed; and Plate XI.
gives the contour in both points of view, as composed entirely of the
curvilinear figures of (¹⁄₂), (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆).
Further detail here would be out of place, and I shall therefore refer
those who require it to the Appendix, or the more elaborate works to
which I have already referred.
The beauty derived from proportion, imparted by the system here pointed
out, and from a contour of curves derived from the same harmonic angles,
is not confined to the human figure, but is found in various minor
degrees of perfection in all the organic forms of nature, whether animate
or inanimate, of which I have in other works given many examples.[21]
THE SCIENCE OF BEAUTY, AS DEVELOPED IN COLOURS.
There is not amongst the various phenomena of nature one that more
readily excites our admiration, or makes on the mind a more vivid
impression of the order, variety, and harmonious beauty of the creation,
than that of colour. On the general landscape this phenomenon is
displayed in the production of that species of harmony in which colours
are so variously blended, and in which they are by light, shade, and
distance modified in such an infinity of gradation and hue. Although
genius is continually struggling, with but partial success, to imitate
those effects, yet, through the Divine beneficence, all whose organs of
sight are in an ordinary degree of perfection can appreciate and enjoy
them. In winter this pleasure is often to a certain extent withdrawn,
when the colourless snow alone clothes the surface of the earth. But
this is only a pause in the general harmony, which, as the spring
returns, addresses itself the more pleasingly to our perception in its
vernal melody, which, gradually resolving itself into the full rich hues
of luxuriant beauty exhibited in the foliage and flowers of summer,
subsequently rises into the more vivid and powerful harmonies of autumn’s
colouring. Thus the eye is prepared again to enjoy that rest which such
exciting causes may be said to have rendered necessary.
When we pass from the general colouring of nature to that of particular
objects, we are again wrapt in wonder and admiration by the beauty and
harmony which so constantly, and in such infinite variety, present
themselves to our view, and which are so often found combined in the
most minute objects. And the systematic order and uniformity perceptible
amidst this endless variety in the colouring of animate and inanimate
nature is thus another characteristic of beauty equally prevalent
throughout creation.
By this uniformity in colour, various species of animals are often
distinguished; and in each individual of most of these species, how much
is this beauty enhanced when the uniformity prevails in the resemblance
of their lateral halves! The human countenance exemplifies this in a
striking manner; the slightest variety of colour between one and another
of the double parts is at once destructive of its symmetrical beauty.
Many of the lower animals, whether inhabitants of the earth, the air, or
the water, owe much of their beauty to this kind of uniformity in the
colour of the furs, feathers, scales, or shells, with which they are
clothed.
In the vegetable kingdom, we find a great degree of uniformity of colour
in the leaves, flowers, and fruit of the same plant, combined with all
the harmonious beauty of variety which a little careful examination
develops.
In the colours of minerals, too, the same may be observed. In short, in
the beauty of colouring, as in every other species of beauty, uniformity
and variety are found to combine.
An appreciation of colour depends, in the first place, as much upon the
physical powers of the eye in conveying a proper impression to the mind,
as that of music on those of the ear. But an ear for music, or an eye for
colour, are, in so far as beauty is concerned, erroneous expressions;
because they are merely applicable to the impression made upon the
senses, and do not refer to the æsthetical principles of harmony, by
which beauty can alone be understood.
A good eye, combined with experience, may enable us to form a correct
idea as to the purity of an individual colour, or of the relative
difference existing between two separate hues; but this sort of
discrimination does not constitute that kind of appreciation of the
harmony of colour by which we admire and enjoy its development in nature
and art. The power of perceiving and appreciating beauty of any kind,
is a principle inherent in the human mind, which may be improved by
cultivation in the degree of the perfection of the art senses. Great
pains have been bestowed on the education of the ear, in assisting it to
appreciate the melody and harmony of sound; but still much remains to be
done in regard to the cultivation of the eye, in appreciating colour as
well as form.
It is true, that there are individuals whose powers of vision are
perfect, in so far as regards the appreciation of light, shade, and
configuration, but who are totally incapable of perceiving effects
produced by the intermediate phenomenon of colour, every object appearing
to them either white, black, or neutral gray; others, who are equally
blind as to the effect of one of the three primary colours, but see the
other two perfectly, either singly or combined; while there are many
who, having the full physical power of perceiving all the varieties of
the phenomenon, and who are even capable of making nice distinctions
amongst a variety of various colours, are yet incapable of appreciating
the æsthetic quality of harmony which exists in their proper combination.
It is the same with respect to the effects of sounds upon the ear—some
have organs so constituted, that notes above or below a certain pitch
are to them inaudible; while others, with physical powers otherwise
perfect, are incapable of appreciating either melody or harmony in
musical composition. But perceptions so imperfectly constituted are,
by the goodness of the Creator, of very rare occurrence; therefore all
attempts at improvement in the science of æsthetics must be suited to the
capacities of the generality of mankind, amongst whom the perception of
colour exists in a variety as great as that by which their countenances
are distinguished. Artists now and then appear who have this intuitive
perception in such perfection, that they are capable of transferring to
their works the most beautiful harmonies and most delicate gradations of
colours, in a manner that no acquired knowledge could have enabled them
to impart. To those who possess such a gift, as well as to those to whom
the ordinary powers of perception are denied, it would be equally useless
to offer an explanation of the various modes in which the harmony of
colour develops itself, or to attempt a definition of the many various
colours, hues, tints, and shades, arising out of the simple elements of
this phenomenon. But to those whose powers lie between these extremes,
being neither above nor below cultivation, such an explanation and
definition must form a step towards the improvement of that inherent
principle which constitutes the basis of æsthetical science.
Although the variety and harmony of colour which nature is continually
presenting to our view, are apparent to all whose visual organs are in a
natural state, and thus to the generality of mankind; yet a knowledge of
the simplicity by which this variety and beauty are produced, is, after
ages of philosophic research and experimental inquiry, only beginning to
be properly understood.
Light may be considered as an active, and darkness a passive principle in
the economy of Nature, and colour an intermediate phenomenon arising from
their joint influence; and it is in the ratios in which these primary
principles act upon each other, by which I here intend to explain the
science of beauty as evolved in colour. It has been usual to consider
colour as an inherent quality in light, and to suppose that coloured
bodies absorb certain classes of its rays, and reflect or transmit the
remainder; but it appears to me that colour is more probably the result
of certain modes in which the opposite principles of motion and rest,
or force and resistance, operate in the production, refraction, and
reflection of light, and that each colour is mutually related, although
in different degrees, to these active and passive principles.
White and black are the representatives of light and darkness, or
activity and rest, and are therefore calculated as pigments to reduce
colours and hues to tints and shades.
Having, however, fully illustrated the nature of tints and shades in
a former work,[22] I shall here confine myself to colours in their
full intensity—shewing the various modifications which their union
with each other produce, along with the harmonic relations which these
modifications bear to the primaries, and to each other in respect to
warmth and coolness of tone, as well as to light and shade.
The primary colours are red, yellow, and blue. Of these, yellow is
most allied to light, and blue to shade, while red is neutral in these
respects, being equally allied to both. In respect to tone, that of red
is warm, and that of blue cool, while the tone of yellow is neutral. The
ratios of their relations to each other in these respects will appear in
the harmonic scales to which, for the first time, I am about to subject
colours, and to systematise their various simple and compound relations,
which are as follow:—
From the binary union of the primary colours, the secondary colours arise—
Orange colour, from the union of yellow and red.
Green, from the union of yellow and blue.
Purple, from the union of red and blue.
From the binary union of the secondary colours, the primary hues arise—
Yellow-hue, from the union of orange and green.
Red-hue, from the union of orange and purple.
Blue-hue, from the union of purple and green.
From the binary union of the primary hues, the secondary hues arise—
Orange-hue, from the union of yellow-hue and red-hue.
Green-hue, from the union of yellow-hue and blue-hue.
Purple-hue, from the union of red-hue and blue-hue.
Each hue owes its characteristic distinction to the proportionate
predominance or subordination of one or other of the three primary
colours in its composition.
It follows, that in every hue of _red_, yellow and blue are subordinate;
in every hue of _yellow_, red and blue are subordinate; and in every
hue of _blue_, red and yellow are subordinate. In like manner, in every
hue of _green_, red is subordinate; in every hue of _orange_, blue is
subordinate; and in every hue of _purple_, yellow is subordinate.
By the union of two primary colours, in the production of a secondary
colour, the nature of both primaries is altered; and as there are only
three primary or simple colours in the scale, the two that are united
harmonically in a compound colour, form the natural contrast to the
remaining simple colour.
Notwithstanding all the variety that extends beyond the six positive
colours, it may be said that there are only three proper contrasts of
colour in nature, and that all others are simply modifications of these.
Pure red is the most perfect contrast to pure green; because it is
characterised amongst the primary colours by warmth of tone, while
amongst the secondary colours green is distinguished by coolness of tone,
both being equally related to the primary elements of light and shade.
Pure yellow is the most perfect contrast to pure purple; because it is
characterised amongst the primary colours as most allied to light, whilst
pure purple is characterised amongst the secondaries as most allied to
shade, both being equally neutral as to tone.
Pure blue is the most perfect contrast to pure orange; because it is
characterised amongst the primary colours as not only the most allied
to shade, but as being the coolest in tone, whilst pure orange is
characterised amongst the secondaries as being the most allied to light
and the warmest in tone. The same principle operates throughout all the
modifications of these primary and secondary colours.
Such is the simple nature of contrast upon which the beauty of colouring
mainly depends.
It being now established as a scientific fact, that the effect of
light upon the eye is the result of an ethereal action, similar to the
atmospheric action by which the effect of sound is produced upon the
ear; also, that the various colours which light assumes are the effect
of certain modifications in this ethereal action;—just as the various
sounds, which constitute the scale of musical notes, are known to be the
effect of certain modifications in the atmospheric action by which sounds
in general are produced:
Therefore, as harmony may thus be impressed upon the mind through
either of these two art senses—hearing and seeing—the principles which
govern the modifications in the ethereal action of light, so as to
produce through the eye the effect of harmony, cannot be supposed to
differ from those principles which we know govern the modifications of
the atmospheric action of sound, in producing through the ear a like
effect. I shall therefore endeavour to illustrate the science of beauty
as evolved in colours, by forming scales of their various modifications
agreeably to the same Pythagorean system of numerical ratio from which
the harmonic elements of beauty in sounds were originally evolved, and by
which I have endeavoured in this, as in previous works, to systematise
the harmonic beauty of forms.
[Illustration]
It will be observed, that with a view to avoid complexity as much as
possible, I have, in the arrangement of the above series of scales, not
only confined myself to the merely elementary parts of the Pythagorean
system, but have left out the harmonic modifications upon (¹⁄₁₁)
and (¹⁄₁₃), in order that the arithmetical progression might not be
interrupted.[23]
The above elementary process will, I trust, be found sufficient to
explain the progress, by harmonic union, of a primary colour to a
toned gray, and how the simple and compound colours naturally arrange
themselves into the elements of five scales, the parts of which continue
from primary to secondary colour; from secondary colour to primary hue;
from primary hue to secondary hue; from secondary hue to primary-toned
gray; and from primary-toned gray to secondary-toned gray in the simple
ratio of 2:1; thereby producing a series of the most beautiful and
perfect contrasts.
The natural arrangement of the primary colours upon the solar spectrum is
red, yellow, blue, and I have therefore adopted the same arrangement on
the present occasion. Red being, consequently, the first tonic, and blue
the second, the divisions express the numerical ratios which the colours
bear to one another, in respect to that colourific power for which red
is pre-eminent. Thus, yellow is to red, as 2:3; blue to yellow, as 3:4;
purple to orange, as 5:6; and green to purple, as 6:7.
The following series of completed scales are arranged upon the
foregoing principle, with the natural connecting links of red-orange,
yellow-orange, yellow-green, and blue-green, introduced in their proper
places.
The appropriate terminology of musical notes has been adopted, and the
scales are composed as follows:—
Scale I. consists of primary and secondary colours;
Scale II. of secondary colours and primary hues;
Scale III. of primary and secondary hues;
Scale IV. of secondary hues and primary-toned grays; and
Scale V. of primary and secondary-toned grays;
All the parts in each of these scales, from the first tonic to the
second, relate to the same parts of the scale below them in the simple
ratio of 2:1; and serially to the first tonic in the following ratios:—
8:9, 4:5, 3:4, 2:3, 3:5, 4:7, 8:15, 1:2.
_First Series of Scales._
----+------+-------+-------+-------+-------+-------+------+-------+-------
|Tonic.| | | | | | | |
| |Supertonic. | | | | | |
| | |Mediant. | | | | |
| | | |Subdominant. | | | |
| | | | |Dominant. | | |
| | | | | |Submediant. | |
| | | | | | |Subtonic. |
| | | | | | | |Semi-subtonic.
| | | | | | | | |Tonic.
----+------+-------+-------+-------+-------+-------+------+-------+-------
I. |(¹⁄₂) |(⁴⁄₉) |(²⁄₅) |(³⁄₈) |(¹⁄₃) |(³⁄₁₀) |(²⁄₇) |(⁴⁄₁₅) |(¹⁄₄)
|Red. |Red- |Orange.|Yellow-|Yellow.|Yellow-|Green.|Blue- |Blue.
| |orange.| |orange.| |green. | |green. |
----+------+-------+-------+-------+-------+-------+------+-------+-------
II. |(¹⁄₄) |(²⁄₉) |(¹⁄₅) |(³⁄₁₆) |(¹⁄₆) |(³⁄₂₀) |(¹⁄₇) |(²⁄₁₅) |(¹⁄₈)
|Green.|Blue- |Blue |Blue- |Purple |Red- |Red |Red- |Orange.
| |green |hue. |purple |hue. |purple |hue. |orange |
| |hue. | |hue. | |hue. | |hue. |
----+------+-------+-------+-------+-------+-------+------+-------+-------
III.|(¹⁄₈) |(¹⁄₉) |(¹⁄₁₀) |(³⁄₃₂) |(¹⁄₁₂) |(³⁄₄₀) |(¹⁄₁₄)|(¹⁄₁₅) |(¹⁄₁₆)
|Red |Red- |Orange |Yellow-|Yellow |Yellow-|Green |Blue- |Blue
|hue. |orange |hue. |orange |hue. |green |hue. |green |hue.
| |hue. | |hue. | |hue. | |hue. |
----+------+-------+-------+-------+-------+-------+------+-------+-------
IV. |(¹⁄₁₆)|(¹⁄₁₈) |(¹⁄₂₀) |(³⁄₆₄) |(¹⁄₂₄) |(³⁄₈₀) |(¹⁄₂₈)|(¹⁄₃₀) |(¹⁄₃₂)
|Green |Blue- |Blue- |Blue- |Purple |Red- |Red- |Red- |Orange
|hue. |green- |toned |purple-|hue. |purple-|toned |orange-|hue.
| |toned |gray. |toned | |toned |gray. |toned |
| |gray. | |gray. | |gray. | |gray. |
----+------+-------+-------+-------+-------+-------+------+-------+-------
V. |(¹⁄₃₂)|(¹⁄₃₆) |(¹⁄₄₀) |(³⁄₁₂₈)|(¹⁄₄₈) |(³⁄₁₆₀)|(¹⁄₅₆)|(¹⁄₆₀) |(¹⁄₆₄)
|Red- |Red- |Orange-|Yellow-|Yellow-|Yellow-|Green-| Blue- |Blue-
|toned |orange-|toned |orange-|toned |green- |toned | green-|toned
|gray. |toned |gray. |toned |gray. |toned |gray. | toned |gray.
| |gray. | |gray. | | gray. | | gray. |
----+------+-------+-------+-------+-------+-------+------+-------+-------
To the scales of chromatic power I add another series of scales, in
which yellow, being the first tonic, and blue the second, the numerical
divisions express the ratios which the colours in each scale bear to one
another in respect to light and shade. Thus red is to yellow, in respect
to light, as 2:3; blue to red, as 3:4; green to orange, as 5:6, and
purple to green, as 6:7.
These scales may therefore be termed scales for the colour-blind,
because, in comparing colours, those whose sight is thus defective,
naturally compare the ratios of the light and shade of which different
colours are primarily constituted.
[Illustration]
The following is a series of five complete scales of the harmonic parts
into which the light and shade in colours may be divided in each scale
according to the above arrangement:—
_Second Series of Scales._
----+-------+-------+-------+-------+------+-------+-------+-------+------
|Tonic. | | | | | | | |
| |Supertonic. | | | | | |
| | |Mediant. | | | | |
| | | |Subdominant. | | | |
| | | | |Dominant. | | |
| | | | | |Submediant. | |
| | | | | | |Subtonic. |
| | | | | | | |Semi-subtonic.
| | | | | | | | |Tonic.
----+-------+-------+-------+-------+------+-------+-------+-------+------
I. |(¹⁄₂) |(⁴⁄₉) |(²⁄₅) |(³⁄₈) |(¹⁄₃) |(³⁄₁₀) |(²⁄₇) |(⁴⁄₁₅) |(¹⁄₄)
|Yellow.|Yellow-|Orange.|Red- |Red. |Red- |Purple.|Blue- |Blue.
| |orange.| |orange.| |purple.| |purple.|
----+-------+-------+-------+-------+------+-------+-------+-------+------
II. |(¹⁄₄) |(²⁄₉) |(¹⁄₅) |(³⁄₁₆) |(¹⁄₆) |(³⁄₂₀) |(¹⁄₇) |(²⁄₁₅) |(¹⁄₈)
|Purple.|Blue- |Blue |Blue- |Green.|Yellow-|Yellow |Yellow-|Orange
| |purple |hue. |green | |green |hue. |orange |
| |hue. | |hue. | |hue. | |hue. |
----+-------+-------+-------+-------+------+-------+-------+-------+------
III.|(¹⁄₈) |(¹⁄₉) |(¹⁄₁₀) |(³⁄₃₂) |(¹⁄₁₂)|(³⁄₄₀) |(¹⁄₁₄) |(¹⁄₁₅) |(¹⁄₁₆)
|Yellow |Yellow-|Orange |Red- |Red |Red- |Purple |Blue- |Blue
|hue. |orange |hue. |orange |hue. |purple |hue. |purple |hue.
| |hue. | |hue. | |hue. | |hue. |
----+-------+-------+-------+-------+------+-------+-------+-------+------
IV. |(¹⁄₁₆) |(¹⁄₁₈) |(¹⁄₂₀) |(³⁄₆₄) |(¹⁄₂₄)|(³⁄₈₀) |(¹⁄₂₈) |(¹⁄₃₀) |(¹⁄₃₂)
|Purple |Blue- |Blue- |Blue- |Green |Yellow-|Yellow-|Yellow-|Orange
|hue. |purple-|toned |green- |hue. |green- |toned |orange-|hue.
| |toned |gray. |toned | |toned |gray. |toned |
| |gray. | |gray. | |gray. | |gray. |
----+-------+-------+-------+-------+------+-------+-------+-------+------
V. |(¹⁄₃₂) |(¹⁄₃₆) |(¹⁄₄₀) |(³⁄₁₂₈)|(¹⁄₄₈)|(³⁄₁₆₀)|(¹⁄₅₆) |(¹⁄₆₀) |(¹⁄₆₄)
|Yellow-|Yellow-|Orange-|Red- |Red- |Red- |Purple-|Blue- |Blue-
|toned |orange-|toned |orange-|toned |purple-|toned |green- |toned
|gray. |toned |gray. |toned |gray. |toned |gray. |toned |gray.
| |gray. | |gray. | |gray. | |gray. |
----+-------+-------+-------+-------+------+-------+-------+-------+------
Should I be correct in arranging colours upon scales identical with those
upon which musical notes have been arranged, and in assuming that colours
have the same ratios to each other, in respect to their harmonic power
upon the eye, which musical notes have in respect to their harmonic power
upon the ear, the colourist may yet be enabled to impart harmonic beauty
to his works with as much certainty and ease, as the musician imparts the
same quality to his compositions: for the colourist has no more right to
trust exclusively to his eye in the arrangement of colours, than the
musician has to trust exclusively to his ear in the arrangement of sounds.
We find, in comparing the dominant parts in the first and second scales
of the second series, that they are equal as to light and shade, so that
their relative powers of contrast depend entirely upon colour. Hence it
is that red and green are the two colours, the difference between which
the colour-blind are least able to appreciate. Professor George Wilson,
in his excellent work, “Researches on Colour-Blindness,” mentions the
case of an engraver, which proves the power of the eye in being able to
appreciate these original constituents of colour, irrespective of the
intermediate phenomenon of tone. This engraver, instead of expressing
regret on account of his being colour-blind, observed to the professor,
“My defective vision is, to a certain extent, a useful and valuable
quality. Thus, an engraver has two negatives to deal with, _i.e._, white
and black. Now, when I look at a picture, I see it only in white and
black, or light and shade, or, as artists term it, the effect. I find
at times many of my brother engravers in doubt how to translate certain
colours of pictures, which to me are matters of decided certainty and
ease. Thus to me it is valuable.”
The colour-blind are therefore as incapable of receiving pleasure from
the harmonious union of various colours, as those who, to use a common
term, have no ear for music, are of being gratified by the “melody of
sweet sounds.”
The generality of mankind are, however, capable of appreciating the
harmony of colour which, like that of both sound and form, arises from
the simultaneous exhibition of opposite principles having a ratio to each
other. These principles are in continual operation throughout nature,
and from them we often derive pleasure without being conscious of the
cause. All who are not colour-blind must have felt themselves struck
with the harmonic beauty of a cloudless sky, although in it there is no
configuration, and at first sight apparently but one colour. Now, as
we know that there can be no more impression of harmony made upon the
mind by looking upon a single colour, than there could be by listening
to a single continued musical note, however sweet its tone, we are apt
at first to imagine that the organ of vision has, in some measure,
conveyed a false impression to the mind. But it has not done so; for
light, when reflected from the atmosphere, produces those cool tones of
blue, gray, and purple, which seem to clothe the distant mountains; but,
when transmitted through the same atmosphere, it produces those numerous
warm tints, the most intense of which give the gorgeous effects which
so often accompany the setting sun. We have, therefore, in the upper
part of a clear sky, where the atmosphere may be said to be illuminated
principally by reflection from the surface of the earth, a comparatively
cool tone of blue, the result of reflection, which gradually blends into
the warm tints, the result of transmission through the same atmosphere.
Such a composition of harmonious colouring is to the eye what the voice
of the soft breath of summer amongst the trees, the hum of insects on
a sultry day, or the simple harmony of the Æolian harp, is to the ear.
To such a composition of chromatic harmony must also be referred the
universal concurrence of mankind in appreciating the peculiar beauty of
white marble statuary. That the principal constituent of beauty in such
works ought to be harmony of form, no one will deny; but this is not the
only element, as appears from the fact, that a cast in plaster of Paris,
of a fine white marble statue, although identical in form, is far less
beautiful than the original. Now this undoubtedly must be the consequence
of its having been changed from a semi-translucent substance, which,
like the atmosphere, can transmit as well as reflect light, to an opaque
substance, which can only reflect it. Thus the opposite principles of
chromatic warmth and coolness are equally balanced in white marble—the
one being the natural result of the partial transmission of light, and
the other that of its reflection.
As a series of coloured illustrations would be beyond the scope of this
_résumé_, I may refer those who wish to prosecute the inquiry, with the
assistance of such a series, to my published works upon the subject.[24]
THE SCIENCE OF BEAUTY, APPLIED TO THE FORMS AND PROPORTIONS OF ANCIENT
GRECIAN VASES AND ORNAMENTS.
In examining the remains of the ornamental works of the ancient Greek
artists, it appears highly probable that the harmony of their proportions
and melody of their contour are equally the result of a systematised
application of the same harmonic law. This probability not being fully
elucidated in any of my former works, I will require to go into some
detail on the present occasion. I take for my first illustration an
unexceptionable example, viz.:—
_The Portland Vase._
Although this beautiful specimen of ancient art was found about the
middle of the sixteenth century, inclosed in a marble sarcophagus within
a sepulchral chamber under the Monte del Grano, near Rome, and although
the date of its production is unknown, yet its being a work of ancient
Grecian art is undoubted; and the exquisite beauty of its form has been
universally acknowledged, both during the time it remained in the palace
of the Barberini family at Rome, and since it was added to the treasures
of the British Museum. The forms and proportions of this gem of art
appear to me to yield an obedience to the great harmonic law of nature,
similar to that which I have instanced in the proportions and contour of
the best specimens of ancient Grecian architecture.
[Sidenote: Plate XII.]
Let the line A B (Plate XII.) represent the full height of the vase.
Through A draw A _a_, and through B draw B _b_ indefinitely, A _a_ making
an angle of (¹⁄₂), and B _b_ an angle of (¹⁄₃), with the vertical.
Through the point C, where A _a_ and B _b_ intersect one another, draw
D C E vertical. Through A C and B respectively, draw A D, C F, and B
E horizontal. Draw similar lines on the other side of A B, and the
rectilinear portion of the diagram is complete.
The curvilinear contour may be thus added:—
Take a cut-out ellipse of (¹⁄₄), whose greater axis is equal to the line
A B, and
_1st._ Place it upon the diagram, so that its circumference may be
tangential to the lines C E and C F, and its greater axis _m n_ may make
an angle of (¹⁄₅) with the vertical, and trace its circumference.
_2d._ Place it with its circumference tangential to that of the first at
the point m, while its greater axis (of which _o p_ is a part) is in the
horizontal, and trace the portion of its circumference _q o r_.
_3d._ Place it with its circumference tangential to that of the above at
_v_, while its greater axis (of which _u v_ is a part) makes an angle of
(³⁄₁₀) with the vertical, and trace the portion of its circumference _s v
t_.
Thus the curvilinear contour of the body and neck are harmonically
determined.
The curve of the handle may be determined by the same ellipse placed so
that its greater axis (of which _i k_ is a part) makes an angle of (¹⁄₆)
with the vertical.
Make similar tracings on the other side of A B, and the diagram is
complete. The inscribing rectangle D G E K is that of (²⁄₅).
The outline resulting from this diagram, not only is in perfect agreement
with my recollection of the form, but with the measurements of the
original given in the “Penny Cyclopædia;” of the accuracy of which there
can be no doubt. They are stated thus:—“It is about ten inches in height,
and beautifully curved from the top downwards; the diameter at the top
being about three inches and a-half; at the neck or smallest part, two
inches; at the largest (mid-height), seven inches; and at the bottom,
five inches.”
The harmonic elements of this beautiful form, therefore, appear to be the
following parts of the right angle:—
Tonic. Dominant. Mediant. Submediant.
(¹⁄₂) (¹⁄₃) (¹⁄₅) (³⁄₁₀)
(¹⁄₄) (¹⁄₆)
When we reflect upon the variety of harmonic ellipses that may
be described, and the innumerable positions in which they may be
harmonically placed with respect to the horizontal and vertical lines,
as well as upon the various modes in which their circumferences may be
combined, the variety which may be introduced amongst such forms as the
foregoing appears almost endless. My second example is that of—
_An Ancient Grecian Marble Vase of a Vertical Composition._
I shall now proceed to another class of the ancient Greek vase, the form
of which is of a more complex character. The specimen I have chosen for
the first example of this class is one of those so correctly measured and
beautifully delineated by Tatham in his unequalled work.[25] This vase is
a work of ancient Grecian art in Parian marble, which he met with in the
collection at the Villa Albani, near Rome. Its height is 4 ft. 4¹⁄₂ in.
[Sidenote: Plate XIII.]
The following is the formula by which I endeavour to develop its harmonic
elements:—
Let A B (Plate XIII.) represent the full height of this vase. Through B
draw B D, making an angle of (¹⁄₅) with the vertical. Through D draw D O
vertical, through A draw A C, making an angle of (²⁄₅); through B draw
B L, making an angle of (¹⁄₂), and B S, making an angle of (³⁄₁₀), each
with the vertical. Through A draw A D, through B draw B O, through L draw
L N, through C draw C F, and through S draw S P, all horizontal. Through
A draw A H, making an angle of (¹⁄₁₀) with the vertical, and through
H draw H M vertical. Draw similar lines on the other side of A B, and
the rectilinear portion of the diagram is complete, and its inscribing
rectangle that of (³⁄₈).
The curvilinear portion may thus be added—
Take a cut-out ellipse of (¹⁄₃), whose greater axis is about the length
of the body of the intended vase, place it with its lesser axis upon the
line S P, and its greater axis upon the line D O, and trace the part _a
b_ of its circumference upon the diagram. Place the same ellipse with
one of its foci upon C, and its greater axis upon C F, and trace its
circumference upon the diagram. Take a cut-out ellipse of (¹⁄₅), whose
greater axis is nearly equal to that of the ellipse already used; place
it with its greater axis upon M H, and its lesser axis upon L N, and
trace its circumference upon the diagram. Make similar tracings upon the
other side of A B, and the diagram is complete. In this, as in the other
diagrams, the strong portions of the lines give the contour of the vase.
The harmonic elements of this classical form, therefore, appear to be the
right angle and its following parts:—
Tonic. Dominant. Mediant. Submediant.
(¹⁄₂) (¹⁄₃) (²⁄₅) (³⁄₁₀)
(¹⁄₅)
(¹⁄₁₀)
My third example is that of—
_An Ancient Grecian Vase of a Horizontal Composition._
This example belongs to the same class as the last, but it is of a
horizontal composition. It was carefully drawn from the original in the
museum of the Vatican by Tatham, in whose etchings it will be found with
its ornamental decorations. The diagram of its harmonic elements may be
constructed as follows:—
[Sidenote: Plate XIV.]
Let A B (Plate XIV.) represent the full height of the vase. Through B
draw B D, making an angle of (²⁄₅) with the vertical. Through A draw A H,
A L, and A C, making respectively the following angles, (¹⁄₅) with the
vertical, (⁴⁄₉) with the vertical, and (³⁄₁₀) with the horizontal. These
angles determine the horizontal lines H B, L N, and C F, which divide
the vase into its parts, and the inscribing rectangle D G K O is (³⁄₈).
This completes the rectilinear portion of the diagram. The ellipse by
which the curvilinear portion is added is one of (¹⁄₅), the greater axis
of which, at _a b_, as also at _c d_, makes an angle of (¹⁄₁₂) with
the vertical, and the same axis at _e f_ an angle of (¹⁄₁₂) with the
horizontal.
The harmonic elements of this vase, therefore, appear to be:—
Tonic. Dominant. Mediant. Submediant. Supertonic.
The Right (¹⁄₁₂) (²⁄₅) (³⁄₁₀) (⁴⁄₉)
Angle. (¹⁄₅)
My remaining examples are those of—
_Etruscan Vases._
Of these vases I give four examples, by which the simplicity of the
method employed in applying the harmonic law will be apparent.
[Sidenote: Plate XV.]
The inscribing rectangle D G E K of fig. 1, Plate XV., is one of (³⁄₈),
within which are arranged tracings from an ellipse of (³⁄₁₀), whose
greater axis at _a b_ makes an angle of (¹⁄₁₂), at _c d_ an angle of
(³⁄₁₀), and at _e f_ an angle of (³⁄₄), with the vertical. The harmonic
elements of the contour of this vase, therefore, appear to be:—
Tonic. Dominant. Subdominants. Submediant.
The Right (¹⁄₁₂) (³⁄₄) (³⁄₁₀)
Angle. (³⁄₈)
The inscribing rectangle L M N O of fig. 2 is that of (¹⁄₂), within which
are arranged tracings from an ellipse of (¹⁄₃), whose greater axis, at
_a b_ and _c d_ respectively, makes angles of (¹⁄₂) and (⁴⁄₉) with the
horizontal, while that at _e f_ is in the horizontal line. The harmonic
elements of the contour of this vase, therefore, appear to be:—
Tonic. Dominant. Subtonic.
(¹⁄₂) (¹⁄₃) (⁴⁄₉)
[Sidenote: Plate XVI.]
The inscribing rectangle P Q R S of fig. 1, Plate XVI., is one of (⁴⁄₉),
within which are arranged tracings from an ellipse of (³⁄₈), whose
greater axis, at _a b_, _c d_, and _e f_, makes respectively angles
of (¹⁄₆) with the horizontal, (³⁄₅) and (⁴⁄₅) with the vertical. Its
harmonic elements, therefore, appear to be:—
Tonic. Dominant. Mediant. Supertonic. Subdominant. Submediant.
The Right (¹⁄₆) (⁴⁄₅) (⁴⁄₉) (³⁄₈) (³⁄₅)
Angle.
The inscribing rectangle T U V X of fig. 2 is one of (⁴⁄₉), within which
are arranged tracings from an ellipse of (³⁄₈) whose greater axis at _a
b_ is in the vertical line, and at _c d_ makes an angle of (¹⁄₂). The
harmonic elements of the contour of this vase, therefore, appear to be:—
Tonic. Submediant. Supertonic.
(¹⁄₂) (³⁄₈) (⁴⁄₉)
These four Etruscan vases, the contours of which are thus reduced to the
harmonic law of nature, are in the British Museum, and engravings of
them are to be found in the well-known work of Mr Henry Moses, Plates
4, 6, 14, and 7, respectively, where they are represented with their
appropriate decorations and colours.
To these, I add two examples of—
_Ancient Grecian Ornament._
I have elsewhere shewn[26] that the elliptic curve pervades the Parthenon
from the entases of the column to the smallest moulding, and we need not,
therefore, be surprised to find it employed in the construction of the
only two ornaments belonging to that great work.
[Sidenote: Plate XVII.]
In the diagram (Plate XVII.), I endeavour to exhibit the geometric
construction of the upper part of one of the ornamental apices, termed
antefixæ, which surmounted the cornice of the Parthenon.
The first ellipse employed is that of (¹⁄₃), whose greater axis _a b_ is
in the vertical line; the second is also that of (¹⁄₃), whose greater
axis _c d_ makes, with the vertical, an angle of (¹⁄₁₂); the third
ellipse is the same with its major axis _e f_ in the vertical line.
Through one of the foci of this ellipse at A the line A C is drawn, and
upon the part of the circumference C _e_, the number of parts, 1, 2, 3,
4, 5, 6, 7, of which the surmounting part of this ornament is to consist,
are set off. That part of the circumference of the ellipse whose larger
axis is _c d_ is divided from _g_ to _c_ into a like number of parts. The
third ellipse employed is one of (¹⁄₄).
Take a cut-out ellipse of this kind, whose larger axis is equal in length
to the inscribing rectangle. Place it with its vertex upon the same
ellipse at _g_, so that its circumference will pass through C, and trace
it; remove its apix first to _p_, then to _q_, and proceed in the same
way to _q_, _r_, _s_, _t_, _u_, and _v_, so that its circumference will
pass through the seven divisions on _c g_ and _e_ C: _v o_, _u n_, _t m_,
_s i_, _r k_, _q j_, _p l_, and _g x_, are parts of the larger axes of
the ellipses from which the curves are traced. The small ellipse of which
the ends of the parts are formed is that of (¹⁄₃).
[Sidenote: Plate XVIII.]
In the diagram (Plate XVIII.), I endeavour to exhibit the geometric
construction of the ancient Grecian ornament, commonly called the
_Honeysuckle_, from its resemblance to the flower of that name. The first
part of the process is similar to that just explained with reference to
the antefixæ of the Parthenon, although the angles in some parts differ.
The contour is determined by the circumference of an ellipse of (¹⁄₃),
whose major axis A B makes an angle of (¹⁄₉) with the vertical, and
the leaves or petals are arranged upon a portion of the perimeter of a
similar ellipse whose larger axis E F is in the vertical line, and these
parts are again arranged upon a similar ellipse whose larger axis C D
makes an angle of (¹⁄₁₂) with the vertical. The first series of curved
lines proceeding from 1, 2, 3, 4, 5, 6, 7, and 8, are between K E and H
C, part of the circumference of an ellipse of (¹⁄₃); and those between C
H and A G are parts of the circumference of four ellipses, each of (¹⁄₃),
but varying as to the lengths of their larger axes from 5 to 3 inches.
The change from the convex to the concave, which produces the ogie forms
of which this ornament is composed, takes place upon the line C H, and
the lines _a b_, _c d_, _e f_, _g h_, _i k_, _l m_, _n o_, and _p q_, are
parts of the larger axis of the four ellipses the circumference of which
give the upper parts of the petals or leaves.
This peculiar Grecian ornament is often, like the antefixæ of the
Parthenon, combined with the curve of the spiral scroll. But the volute
is so well understood that I have not rendered my diagrams more complex
by adding that figure. Many varieties of this union are to be found in
Tatham’s etchings, already referred to. The antefixæ of the Parthenon,
and its only other ornament the honeysuckle, as represented on the soffit
of the cornice, are to be found in Stewart’s “Athens.”
APPENDIX.
No. I.
In pages 34, 35, and 58, I have reiterated an opinion advanced in several
of my former works, viz., that, besides genius, and the study of nature,
an additional cause must be assigned for the general excellence which
characterises such works of Grecian art as were executed during a period
commencing about 500 B.C., and ending about 200 B.C. And that this cause
most probably was, that the artists of that period were instructed in
a system of fixed principles, based upon the doctrines of Pythagoras
and Plato. This opinion has not been objected to by the generality of
those critics who have reviewed my works; but has, however, met with an
opponent, whose recondite researches and learned observations are worthy
of particular attention. These are given in an essay by Mr C. Knight
Watson, “On the Classical Authorities for Ancient Art,” which appeared in
the _Cambridge Journal of Classical and Sacred Philology_ in June 1854.
As this essay is not otherwise likely to meet the eyes of the generality
of my readers, and as the objections he raises to my opinion only occupy
two out of the sixteen ample paragraphs which constitute the first part
of the essay, I shall quote them fully:—
“The next name on our list is that of the famous Euphranor
(B.C. 362). For the fact that to the practice of sculpture
and of painting he added an exposition of the theory, we are
indebted to Pliny, who says (xxxv. 11, 40), ‘Volumina quoque
composuit de symmetria et coloribus.’ When we reflect on the
_critical_ position occupied by Euphranor in the history
of Greek art, as a connecting link between the idealism of
Pheidias and the naturalism of Lysippus, we can scarcely
overestimate the value of a treatise on art proceeding from
such a quarter. This is especially the case with the first
of the two works here assigned to Euphranor. The inquiries
which of late years have been instituted by Mr D. R. Hay of
Edinburgh, on the proportions of the human figure, and on the
natural principles of beauty as illustrated by works of Greek
art, constitute an epoch in the study of æsthetics and the
philosophy of form. Now, in the presence of these inquiries,
or of such less solid results as Mr Hay’s predecessors in
the same field have elicited, it naturally becomes an object
of considerable interest to ascertain how far these laws of
form and principles of beauty were consciously developed in
the mind, and by the chisel, of the sculptor: how far any
such system of curves and proportions as Mr Hay’s was used
by the Greek as a practical manual of his craft. Without in
the least wishing to impugn the accuracy of that gentleman’s
results—a piece of presumption I should do well to avoid—I must
be permitted to doubt whether the ‘Symmetria’ of Euphranor
contained anything analogous to them in kind, or indeed equal
in value. It must not be forgotten that the truth of Mr Hay’s
theory is perfectly compatible with the fact, that of such
theory the Greek may have been utterly ignorant. It is on this
fact I insist: it is here that I join issue with Mr Hay, and
with his reviewer in a recent number of _Blackwood’s Magazine_.
Or, to speak more accurately,—while I am quite prepared to find
that the Elgin marbles will best of all stand the test which Mr
Hay has hitherto applied, I believe, to works of a later age, I
am none the less convinced that it is precisely that golden age
of Hellenic art to which they belong, precisely that first and
chief of Hellenic artists by whom they were executed, to which
and to whom any such line of research on the laws of form would
have been pre-eminently alien. Pheidias, remember, by the right
of primogeniture, is the ruling spirit of idealism in art. Of
spontaneity was that idealism begotten and nurtured: by any
such system as Mr Hay’s, that spontaneity would be smothered
and paralysed. Pheidias copied an idea in his own mind—‘Ipsius
in mente insidebat species pulchritudinis eximia quædam’
(_Cic._);—later ages copied _him_. He created: they criticised.
He was the author of Iliads: they the authors of Poetics.
Doubtless, if you unsphere the _spirit_ of Mr Hay’s theories,
you will find nothing discordant with what I have here said.
That is a sound view of Beauty which makes it consist in that
due subordination of the parts to the whole, that due relation
of the parts to each other, which Mendelssohn had in his
mind when he said that the essence of beauty was ‘unity in
variety’—variety beguiling the imagination, the perception of
unity exercising the thewes and sinews of the intellect. On
such a view of beauty, Mr Hay’s theory may, _in spirit_, be
said to rest. But here, as in higher things, it is the letter
that killeth, while the spirit giveth life. And accordingly I
must enter a protest against any endeavour to foist upon the
palmy days of Hellenic art systems of geometrical proportions
incompatible, as I believe, with those higher and broader
principles by which the progress of ancient sculpture was
ordered and governed—systems which will bear nothing of that
‘felicity and chance by which’—and not by rule—‘Lord Bacon
believed that a painter may make a better face than ever was:’
systems which take no account of that fundamental distinction
between the schools of Athens and of Argos, and their
respective disciples and descendants, without which you will
make nonsense of the pages of Pliny, and—what is worse—sense
of the pages of his commentators;—systems, in short, which may
have their value as instruments for the education of the eye,
and for instructions in the arts of design, but must be cast
aside as matters of learned trifling and curious disputation,
where they profess to be royal roads to art, and to map the
mighty maze of a creative mind. And even as regards the
application of such a system of proportions to those works of
sculpture which are posterior to the Pheidian age, only partial
can have been the prevalence which it or any other _one_ system
can have obtained. The discrepancies of different artists
in the treatment of what was called, technically called,
_Symmetria_ (as in the title of Euphranor’s work) were, by the
concurrent testimony of all ancient writers, far too salient
and important to warrant the supposition of any uniform scale
of proportions, as advocated by Mr Hay. Even in Egypt, where
one might surely have expected that such uniformity would have
been observed with far greater rigour than in Greece, the
discoveries of Dr Lepsius (_Vorläufige Nachricht_, Berlin,
1849) have elicited three totally different κανόνες, one of
which is identical with the system of proportions of the human
figure detailed in Diodorus. While we thus venture to differ
from Mr Hay on the historical data he has mixed up with his
inquiries, we feel bound to pay him a large and glad tribute of
praise for having devised a system of proportions which rises
superior to the idiosyncracies of different artists, which
brings back to one common type the sensations of eye and ear,
and so makes a giant stride towards that _codification_, if
I may so speak, of the laws of the universe which it is the
business of the science to effect. I have no hesitation in
saying, that, for scientific precision of method and importance
of results, Albert Durer, Da Vinci, and Hogarth, not to mention
less noteworthy writers, must all yield the palm to Mr Hay.
“I am quite aware that in the digression I have here allowed
myself, on systems of proportions prevalent among ancient
artists, and on the probable contents of such treatises as that
of Euphranor, _De Symmetria_, I have laid myself open to the
charge of treating an intricate question in a very perfunctory
way. At present the exigencies of the subject more immediately
in hand allow me only to urge in reply, that, as regards the
point at issue—I mean the ‘solidarité’ between theories such as
Mr Hay’s and the practice of Pheidias—the _onus probandi_ rests
with my adversaries.”
I am bound, in the first place, gratefully to acknowledge the kind and
complimentary notice which, notwithstanding our difference of opinion,
this author has been pleased to take of my works; and, in the second, to
assure him that if any of them profess to be “royal roads to art,” or
to “map the mighty maze of a creative mind,” they certainly profess to
do more than I ever meant they should; for I never entertained the idea
that a system of æsthetic culture, even when based upon a law of nature,
was capable of effecting any such object. But I doubt not that this too
common misapprehension of the scope and tendency of my works must arise
from a want of perspicuity in my style.
I have certainly, on one occasion,[27] gone the length of stating
that as poetic genius must yield obedience to the rules of rhythmical
measure, even in the highest flights of her inspirations; and musical
genius must, in like manner, be subject to the strictly defined laws of
harmony in the most delicate, as well as in the most powerfully grand
of her compositions; so must genius, in the formative arts, either
consciously or unconsciously have clothed her creations of ideal beauty
with proportions strictly in accordance with the laws which nature has
set up as inflexible standards. If, therefore, the laws of proportion, in
their relation to the arts of design, constitute the harmony of geometry,
as definitely as those that are applicable to poetry and music produce
the harmony of acoustics; the former ought, certainly, to hold the same
relative position in those arts which are addressed to the eye, that is
accorded to the latter in those which are addressed to the ear. Until
so much science be brought to bear upon the arts of design, the student
must continue to copy from individual and imperfect objects in nature, or
from the few existing remains of ancient Greek art, in total ignorance of
the laws by which their proportions are produced, and, what is equally
detrimental to art, the accuracy of all criticism must continue to rest
upon the indefinite and variable basis of mere opinion.
It cannot be denied that men of great artistic genius are possessed of
an intuitive feeling of appreciation for what is beautiful, by means
of which they impart to their works the most perfect proportions,
independently of any knowledge of the definite laws which govern that
species of beauty. But they often do so at the expense of much labour,
making many trials before they can satisfy themselves in imparting to
them the true proportions which their minds can conceive, and which,
along with those other qualities of expression, action, or attitude,
which belong more exclusively to the province of genius. In such
cases, an acquaintance with the rules which constitute the science of
proportion, instead of proving fetters to genius, would doubtless afford
her such a vantage ground as would promote the more free exercise of
her powers, and give confidence and precision in the embodiment of her
inspirations; qualities which, although quite compatible with genius, are
not always intuitively developed along with that gift.
It is also true that the operations of the conceptive faculty of the mind
are uncontrolled by definite laws, and that, therefore, there cannot
exist any rules by the inculcation of which an ordinary mind can be
imbued with genius sufficient to produce works of high art. Nevertheless,
such a mind may be improved in its perceptive faculty by instruction in
the science of proportion, so as to be enabled to exercise as correct and
just an appreciation of the conceptions of others, in works of plastic
art, as that manifested by the educated portion of mankind in respect
to poetry and music. In short, it appears that, in those arts which are
addressed to the ear, men of genius communicate the original conceptions
of their minds under the control of certain scientific laws, by means
of which the educated easily distinguish the true from the false, and
by which the works of the poet and musical composer may be placed above
mere imitations of nature, or of the works of others; while, in those
arts that are addressed to the eye in their own peculiar language, such
as sculpture, architecture, painting, and ornamental design, no such laws
are as yet acknowledged.
Although I am, and ever have been, far from endeavouring “to foist upon
the palmy days of Hellenic art” any system incompatible with those higher
and more intellectual qualities which genius alone can impart; yet, from
what has been handed down to us by writers on the subject, meagre as it
is, I cannot help continuing to believe that, along with the physical and
metaphysical sciences, æsthetic science was taught in the early schools
of Greece.
I shall here take the liberty to repeat the proofs I advanced in a former
work as the ground of this belief, and to which the author, from whose
essay I have quoted, undoubtedly refers. It is well known that, in the
time of Pythagoras, the treasures of science were veiled in mystery to
all but the properly initiated, and the results of its various branches
only given to the world in the works of those who had acquired this
knowledge. So strictly was this secresy maintained amongst the disciples
and pupils of Pythagoras, that any one divulging the sacred doctrines
to the profane, was expelled the community, and none of his former
associates allowed to hold further intercourse with him; it is even
said, that one of his pupils incurred the displeasure of the philosopher
for having published the solution of a problem in geometry.[28] The
difficulty, therefore, which is expressed by writers, shortly after the
period in which Pythagoras lived, regarding a precise knowledge of his
theories, is not to be wondered at, more especially when it is considered
that he never committed them to writing. It would appear, however, that
he proceeded upon the principle, that the order and beauty so apparent
throughout the whole universe, must compel men to believe in, and refer
them to, an intelligible cause. Pythagoras and his disciples sought for
properties in the science of numbers, by the knowledge of which they
might attain to that of nature; and they conceived those properties to
be indicated in the phenomena of sonorous bodies. Observing that Nature
herself had thus irrevocably fixed the numerical value of the intervals
of musical tones, they justly concluded that, as she is always uniform
in her works, the same laws must regulate the general system of the
universe.[29] Pythagoras, therefore, considered numerical proportion as
the great principle inherent in all things, and traced the various forms
and phenomena of the world to numbers as their basis and essence.
How the principles of numbers were applied in the arts is not recorded,
farther than what transpires in the works of Plato, whose doctrines were
from the school of Pythagoras. In explaining the principle of beauty,
as developed in the elements of the material world, he commences in the
following words:—“But when the Artificer began to adorn the universe,
he first of all figured with forms and numbers, fire and earth, water
and air—which possessed, indeed, certain traces of the true elements,
but were in every respect so constituted as it becomes anything to be
from which Deity is absent. But we should always persevere in asserting
that Divinity rendered them, as much as possible, the most beautiful
and the best, when they were in a state of existence opposite to such
a condition.” Plato goes on further to say, that these elementary
bodies must have forms; and as it is necessary that every depth should
comprehend the nature of a plane, and as of plane figures the triangle
is the most elementary, he adopts two triangles as the originals or
representatives of the isosceles and the scalene kinds. The first
triangle of Plato is that which forms the half of the square, and is
regulated by the number, 2; and the second, that which forms the half
of the equilateral triangle, which is regulated by the number, 3;
from various combinations of these, he formed the bodies of which he
considered the elements to be composed. To these elementary figures I
have already referred.
Vitruvius, who studied architecture ages after the arts of Greece had
been buried in the oblivion which succeeded her conquest, gives the
measurements of various details of monuments of Greek art then existing.
But he seems to have had but a vague traditionary knowledge of the
principle of harmony and proportion from which these measurements
resulted. He says—“The several parts which constitute a temple ought
to be subject to the laws of symmetry; the principles of which should
be familiar to all who profess the science of architecture. Symmetry
results from proportion, which, in the Greek language, is termed
analogy. Proportion is the commensuration of the various constituent
parts with the whole; in the existence of which symmetry is found to
consist. For no building can possess the attributes of composition
in which symmetry and proportion are disregarded; nor unless there
exist that perfect conformation of parts which may be observed in a
well-formed human being.” After going at some length into details, he
adds—“Since, therefore, the human figure appears to have been formed
with such propriety, that the several members are commensurate with
the whole, the artists of antiquity (meaning those of Greece at the
period of her highest refinement) must be allowed to have followed the
dictates of a judgment the most rational, when, transferring to works of
art principles derived from nature, every part was so regulated as to
bear a just proportion to the whole. Now, although the principles were
universally acted upon, yet they were more particularly attended to in
the construction of temples and sacred edifices, the beauties or defects
of which were destined to remain as a perpetual testimony of their skill
or of their inability.”
Vitruvius, however, gives no explanation of this ancient principle
of proportion, as derived from the human form; but plainly shews his
uncertainty upon the subject, by concluding this part of his essay in the
following words: “If it be true, therefore, that the decenary notation
was suggested by the members of man, and that the laws of proportion
arose from the relative measures existing between certain parts of each
member and the whole body, it will follow, that those are entitled to our
commendation who, in building temples to their deities, proportioned the
edifices, so that the several parts of them might be commensurate with
the whole.” It thus appears certain that the Grecians, at the period of
their highest excellence, had arrived at a knowledge of some definite
mathematical law of proportion, which formed a standard of perfectly
symmetrical beauty, not only in the representation of the human figure
in sculpture and painting, but in architectural design, and indeed in
all works where beauty of form and harmony of proportion constituted
excellence. That this law was not deduced from the proportions of
the human figure, as supposed by Vitruvius, but had its origin in
mathematical science, seems equally certain; for in no other way can we
satisfactorily account for the proportions of the beau ideal forms of the
ancient Greek deities, or of those of their architectural structures,
such as the Parthenon, the temple of Theseus, &c., or for the beauty that
pervades all the other formative art of the period.
This system of geometrical harmony, founded, as I have shewn it to be,
upon numerical relations, must consequently have formed part of the Greek
philosophy of the period, by means of which the arts began to progress
towards that great excellence which they soon after attained. A little
further investigation will shew, that immediately after this period a
theory connected with art was acknowledged and taught, and also that
there existed a Science of Proportion.
Pamphilus, the celebrated painter, who flourished about four hundred
years before the Christian era, from whom Apelles received the
rudiments of his art, and whose school was distinguished for scientific
cultivation, artistic knowledge, and the greatest accuracy in drawing,
would admit no pupil unacquainted with geometry.[30] The terms upon which
he engaged with his students were, that each should pay him one talent
(£225 sterling) previous to receiving his instructions; for this he
engaged “to give them, _for ten years_, lessons founded on an excellent
theory.”[31]
It was by the advice of Pamphilus that the magistrates of Sicyon ordained
that the study of drawing should constitute part of the education of the
citizens—“a law,” says the Abbé Barthélémie, “which rescued the fine arts
from servile hands.”
It is stated of Parrhasius, the rival of Zeuxis, who flourished about
the same period as Pamphilus, that he accelerated the progress of art by
purity and correctness of design; “for he was acquainted with the science
of Proportions. Those he gave his gods and heroes were so happy, that
artists did not hesitate to adopt them.” Parrhasius, it is also stated,
was so admired by his contemporaries, that they decreed him the name of
Legislator.[32] The whole history of the arts in Egypt and Greece concurs
to prove that they were based on geometric precision, and were perfected
by a continued application of the same science; while in all other
countries we find them originating in rude and misshapen imitations of
nature.
In the earliest stages of Greek art, the gods—then the only statues—were
represented in a tranquil and fixed posture, with the features exhibiting
a stiff inflexible earnestness, their only claim to excellence being
symmetrical proportion; and this attention to geometric precision
continued as art advanced towards its culminating point, and was
thereafter still exhibited in the neatly and regularly folded drapery,
and in the curiously braided and symmetrically arranged hair.[33]
These researches, imperfect as they are, cannot fail to exhibit the
great contrast that exists between the system of elementary education
in art practised in ancient Greece, and that adopted in this country at
the present period. But it would be of very little service to point out
this contrast, were it not accompanied by some attempt to develop the
principles which seem to have formed the basis of this excellence amongst
the Greeks.
But in making such an attempt, I cannot accuse myself of assuming
anything incompatible with the free exercise of that spontaneity of
genius which the learned essayist says is the parent and nurse of
idealism. For it is in no way more incompatible with the free exercise of
artistic genius, that those who are so gifted should have the advantage
of an elementary education in the science of æsthetics, than it is
incompatible with the free exercise of literary or poetic genius, that
those who possess it should have the advantage of such an elementary
education in the science of philology as our literary schools and
colleges so amply afford.
No. II.
The letter from which I have made a quotation at page 42, arose out of
the following circumstance:—In order that my theory, as applied to the
orthographic beauty of the Parthenon, might be brought before the highest
tribunal which this country afforded, I sent a paper upon the subject,
accompanied by ample illustrations, to the Royal Institute of British
Architects, and it was read at a meeting of that learned body on the 7th
of February 1853; at the conclusion of which, Mr Penrose kindly undertook
to examine my theoretical views, in connexion with the measurements he
had taken of that beautiful structure by order of the Dilettanti Society,
and report upon the subject to the Royal Institute. This report was
published by Mr Penrose, vol. xi., No. 539 of _The Builder_, and the
letter from which I have quoted appeared in No. 542 of the same journal.
It was as follows:—
“GEOMETRICAL RELATIONS IN ARCHITECTURE.
“Will you allow me, through the medium of your columns, to
thank Mr Penrose for his testimony to the truth of Mr Hay’s
revival of Pythagoras? The dimensions which he gives are to
me the surest verification of the theory that I could have
desired. The minute discrepancies form that very element
of practical incertitude, both as to execution and direct
measurement, which always prevails in materialising a
mathematical calculation under such conditions.
“It is time that the scattered computations by which
architecture has been analysed—more than enough—be synthetised
into those formulæ which, as Mrs Somerville tells us, ‘are
emblematic of omniscience.’ The young architects of our day
feel trembling beneath their feet the ground whence men are
about to evoke the great and slumbering corpse of art. Sir, it
is food of this kind a reviving poetry demands.
——‘Give us truths,
For we are weary of the surfaces,
And die of inanition.’
“I, for one, as I listen to such demonstrations, whose scope
extends with every research into them, feel as if listening to
those words of Pythagoras, which sowed in the mind of Greece
the poetry whose manifestation in beauty has enchained the
world in worship ever since its birth. And I am sure that in
such a quarter, and in such thoughts, _we_ must look for the
first shining of that lamp of art, which even now is prepared
to burn.
“I know that this all sounds rhapsodical; but I know also that
until the architect becomes a poet, and not a tradesman, we
may look in vain for architecture: and I know that valuable
as isolated and detailed investigations are in their proper
bearings, yet that such purposes and bearings are to be found
in the enunciation of principles sublime as the generalities of
‘mathematical beauty.’
“AUTOCTHON.”
No. III.
Of the work alluded to at page 58 I was favoured with two opinions—the
one referring to the theory it propounds, and the other to its anatomical
accuracy—both of which I have been kindly permitted to publish.
The first is from Sir WILLIAM HAMILTON, Bart., professor of logic and
metaphysics in the University of Edinburgh, and is as follows:—
“Your very elegant volume is to me extremely interesting, as
affording an able contribution to what is the ancient, and,
I conceive, the true theory of the Beautiful. But though
your doctrine coincides with the one prevalent through all
antiquity, it appears to me quite independent and original
in you; and I esteem it the more, that it stands opposed to
the hundred one-sided and exclusive views prevalent in modern
times.—_16 Great King Street, March 5, 1849._”
The second is from JOHN GOODSIR, Esq., professor of anatomy in the
University of Edinburgh, and is as follows:—
“I have examined the plates in your work on the proportions
of the human head and countenance, and find the head you have
given as typical of human beauty to be anatomically correct
in its structure, only differing from ordinary nature in its
proportions being more mathematically precise, and consequently
more symmetrically beautiful.—_College, Edinburgh, 17th April
1849._”
No. IV.
I shall here shew, as I have done in a former work, how the curvilinear
outline of the figure is traced upon the rectilinear diagrams by portions
of the ellipse of (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆).
[Sidenote: Plate XIX.]
The outline of the head and face, from points (1) to (3) (fig. 1, Plate
XIX.), takes the direction of the two first curves of the diagram. From
point (3), the outline of the sterno-mastoid muscle continues to (4),
where, joining the outline of the trapezius muscle, at first concave, it
becomes convex after passing through (5), reaches the point (6), where
the convex outline of the deltoid muscle commences, and, passing through
(7), takes the outline of the arm as far as (8). The outline of the
muscles on the side, the latissimus dorsi and serratus magnus, commences
under the arm at the point (9), and joins the outline of the oblique
muscle of the abdomen by a concave curve at (10), which, rising into
convexity as it passes through the points (11) and (12), ends at (13),
where it joins the outline of the gluteus medius muscle. The outline of
this latter muscle passes convexly through the point (14), and ends at
(15), where the outline of the tensor vaginæ femoris and vastus externus
muscle of the thigh commences. This convex outline joins the concave
outline of the biceps of the thigh at (16), which ends in that of the
slight convexity of the condyles of the thigh-bone at (17). From this
point, the outline of the outer surface of the leg, which includes the
biceps, peroneus longus, and soleus muscles, after passing through the
point (18), continues convexly to (19), where the concave outline of the
tendons of the peroneus longus continues to (20), whence the outline of
the outer ankle and foot commences.
The outline of the mamma and fold of the arm-pit commences at (21),
and passes through the points (22) and (23). The circle at (24) is the
outline of the areola, in the centre of which the nipple is placed.
The outline of the pubes commences at (25), and ends at the point (26),
from which the outline of the inner surface of the thigh proceeds over
the gracilis, the sartorius, and vastus internus muscles, until it meets
the internal face of the knee-joint at (27), the outline of which ends
at (28). The outline of the inside of the leg commences by proceeding
over the gastrocnemius muscle as far as (29), where it meets that of the
soleus muscle, and continues over the tendons of the heel until it meets
the outline of the inner ankle and foot at (30).
The outline of the outer surface of the arm, as viewed in front, proceeds
from (8) over the remainder of the deltoid, in which there is here a
slight concavity, and, next, from (31) over the biceps muscle till (32),
where it takes the line of the long supinator, and passing concavely, and
almost imperceptibly, into the long and short radial extensor muscles,
reaches the wrist at (33). The outline of the inner surface of the arm
from opposite (9) commences by passing over the triceps extensor, which
ends at (34), then over the gentle convexity of the condyles of the bone
of the arm at (35), and, lastly, over the flexor sublimis which ends at
the wrist-joint (36).
The outline of the front of the figure commences at the point (1), (fig.
2, Plate II.), and, passing almost vertically over the platzsma-myoidis
muscles, reaches the point (2), where the neck ends by a concave curve.
From (2) the outline rises convexly over the ends of the clavicles,
and continues so over the pectoral muscle till it reaches (3), where
the mamma swells out convexly to (4), and returns convexly towards
(5), where the curve becomes concave. From (5) the outline follows the
undulations of the rectus muscle of the abdomen, concave at the points
(6) and (7), and having its greatest convexity at (8). This series
of curves ends with a slight concavity at the point (9), where the
horizontal branch of the pubes is situated, over which the outline is
convex and ends at (10).
The outline of the thigh commences at the point (11) with a slight
concave curve, and then swells out convexly over the extensors of the
leg, and, reaching (12), becomes gently concave, and, passing over the
patella at (13), becomes again convex until it reaches the ligament of
that bone, where it becomes gently concave towards the point (14), whence
it follows the slightly convex curve of the shin-bone, and then, becoming
as slightly concave, ends with the muscles in front of the leg at (15).
The outline of the back commences at the point (16), and, following with
a concave curve the muscles of the neck as far as (17), swells into a
convex curve over the trapezius muscle towards the point (18); passing
through which point, it continues to swell outward until it reaches
(19), half way between (18) and (20); whence the convexity, becoming
less and less, falls into the concave curve of the muscles of the loins
at (21), and passing through the point (22), it rises into convexity. It
then passes through the point (23), follows the outline of the gluteus
maximus, the convex curve of which rises to the point (24), and then
returns inwards to that of (25), where it ends in the fold of the hip.
From this point the outline follows the curve of the hamstring muscles
by a slight concavity as far as (26), and then, becoming gently convex,
it reaches (27); whence it becomes again gently concave, with a slight
indication of the condyle of the thigh-bone at (28), and, reaching (29),
follows the convex curve of the gastrocnemius muscle through the point
(30), and falling into the convex curve of the tendo Achilles at (31),
ends in the concavity over the heel at (32).
The outline of the front of the arm commences at the point (33), by a
gentle concavity at the arm-pit, and then swells out in a convex curve
over the biceps, reaching (34), where it becomes concave, and passing
through (35), again becomes convex in passing over the long supinator,
and, becoming gently concave as it passes the radial extensors, rises
slightly at (36), and ends at (37), where the outline of the wrist
commences. The outline of the back of the arm commences with a concave
curve at (38), which becomes convex as it passes from the deltoid to the
long extensor and ends at the elbow (39), from below which the outline
follows the convex curve of the extensor ulnaris, reaching the wrist at
the point (40).
It will be seen that the various undulations of the outline are regulated
by points which are determined generally by the intersections and
sometimes by directions and extensions of the lines of the diagram, in
the same manner in which I shewed proportion to be imparted, in a late
work, to the osseous structure. The mode in which the curves of (¹⁄₂),
(¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆) are thus so harmoniously blended in the
outline of the female figure, only remains to be explained.
The curves which compose the outline of the female form are therefore
simply those of (¹⁄₂), (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆).
Manner in which these curves are disposed in the lateral outline (figure
1, Plate XIX.):—
Points. Curves.
Head from 1 to 2 (¹⁄₂)
Face ” 2 ” 3 (¹⁄₃)
Neck ” 3 ” 4 (¹⁄₅)
Shoulder ” 4 ” 6 (¹⁄₆)
” ” 6 ” 8 (¹⁄₄)
Trunk ” 9 ” 15 (¹⁄₄)
” ” 21 ” 24 (¹⁄₂)
Outer surface of thigh and leg ” 15 ” 20 (¹⁄₆)
Inner surface of thigh and leg ” 25 ” 30 (¹⁄₆)
Outer surface of the arm ” 8 ” 33 (¹⁄₆)
Inner surface of the arm ” 9 ” 36 (¹⁄₆)
Manner in which they are disposed in the outline (figure 2, Plate XIX.):—
Points. Curves.
Front of neck from 1 to 2 (¹⁄₆)
” trunk ” 2 ” 10 (¹⁄₄)
Back of neck ” 16 ” 18 (¹⁄₆)
” trunk ” 18 ” 23 (¹⁄₄)
” ” ” 23 ” 25 (¹⁄₃)
Front of thigh and leg ” 11 ” 13 (¹⁄₄)
” ” ” ” 13 ” 15 (¹⁄₆)
Back of thigh and leg ” 25 ” 32 (¹⁄₆)
Front of the arm ” 33 ” 37 (¹⁄₆)
Back of the arm ” 38 ” 40 (¹⁄₆)
Foot ” 0 ” 0 (¹⁄₆)
[Sidenote: Plate XX.]
In order to exemplify more clearly the manner in which these various
curves appear in the outline of the figure, I give in Plate XX. the whole
curvilinear figures, complete, to which these portions belong that form
the outline of the sides of the head, neck, and trunk, and of the outer
surface of the thighs and legs.
The various angles which the axes of these ellipses form with the
vertical, will be found amongst other details in the works I have just
referred to.
No. V.
At page 85 I have remarked upon the variety that may be introduced into
any particular form of vase; and, in order to give the reader an idea of
the ease with which this may be done without violating the harmonic law,
I shall here give three examples:—
[Sidenote: Plate XXI.]
The first of these (Plate XXI.) differs from the Portland vase, in the
concave curve of the neck flowing more gradually into the convex curve of
the body.
[Sidenote: Plate XXII.]
The second (Plate XXII.) differs from the same vase in the same change
of contour, as also in being of a smaller diameter at the top and at the
bottom.
[Sidenote: Plate XXIII.]
The third (Plate XXIII.) is the most simple arrangement of the elliptic
curve by which this kind of form may be produced; and it differs from the
Portland vase in the relative proportions of height and diameter, and in
having a fuller curve of contour.
The following comparison of the angles employed in these examples, with
the angles employed in the original, will shew that the changes of
contour in these forms, arise more from the mode in which the angles are
arranged than in a change of the angles themselves:—
Line Line Line Line
Plate VIII. _A C_ (¹⁄₂) _B C_ (¹⁄₃) _o p_ (H) _v u_ (³⁄₁₀)
Plate XXI. (¹⁄₂) (¹⁄₃) (²⁄₉) (¹⁄₄)
Plate XXII. (¹⁄₂) (¹⁄₃) (¹⁄₈) (⁴⁄₉)
Plate XXIII. (¹⁄₂) (¹⁄₄) (H) (-)
Line Line
Plate VIII. _m n_ (¹⁄₃) _i k_ (¹⁄₅) ellipse (¹⁄₄) rectangle (²⁄₅)
Plate XXI. (²⁄₉) (¹⁄₅) (¹⁄₄) (²⁄₅)
Plate XXII. (¹⁄₃) (¹⁄₅) (¹⁄₄) (²⁄₅)
Plate XXIII. (¹⁄₅) (¹⁄₅) ellipses {(¹⁄₃)} (¹⁄₃)
{(¹⁄₄)}
The harmonic elements of each are therefore simply the following parts of
the right angle:—
Tonic. Dominant. Mediant. Submediant.
Plate VIII. (¹⁄₂) (¹⁄₃) (¹⁄₅) (³⁄₁₀)
(¹⁄₄)
Tonic. Dominant. Mediant. Supertonic.
Plate XXI. (¹⁄₂) (¹⁄₃) (¹⁄₅) (²⁄₉)
(¹⁄₄)
Tonic. Dominant. Mediant. Supertonic.
Plate XXII. (¹⁄₂) (¹⁄₃) (¹⁄₅) (⁴⁄₉)
(¹⁄₄)
(¹⁄₈)
Tonic. Dominant. Mediant.
Plate XXIII. (¹⁄₂) (¹⁄₃) (¹⁄₅)
(¹⁄₄)
No. VI.
So far as I know, there has been only one attempt in modern times,
besides my own, to establish a universal system of proportion, based on a
law of nature, and applicable to art. This attempt consists of a work of
457 pages, with 166 engraved illustrations, by Dr Zeising, a professor in
Leipzic, where it was published in 1854.
One of the most learned and talented professors in our Edinburgh
University has reviewed that work as follows:—
“It has been rather cleverly said that the intellectual distinction
between an Englishman and a Scotchman is this—‘Give an Englishman two
facts, and he looks out for a third; give a Scotchman two facts, and he
looks out for a theory.’ Neither of these tests distinguishes the German;
he is as likely to seek for a third fact as for a theory, and as likely
to build a theory on two facts as to look abroad for further information.
But once let him have a theory in his mind, and he will ransack heaven
and earth until he almost buries it under the weight of accumulated
facts. This remark applies with more than common force to a treatise
published last year by Dr Zeising, a professor in Leipsic, ‘On a law of
proportion which rules all nature.’ The ingenious author, after proving
from the writings of ancient and modern philosophers that there always
existed the belief (whence derived it is difficult to say), that some
law does bind into one formula all the visible works of God, proceeds to
criticise the opinions of individual writers respecting that connecting
law. It is not our purpose to follow him through his lengthy examination.
Suffice it to say that he believes he has found the lost treasure in the
_Timæus_ of Plato, c. 31. The passage is confessedly an obscure one, and
will not bear a literal translation. The interpretation which Dr Zeising
puts on it is certainly a little strained, but we are disposed to admit
that he does it with considerable reason. Agreeably to him, the passage
runs thus:—‘That bond is the most beautiful which binds the things as
much as possible into one; and proportion effects this most perfectly
when three things are so united that the greater bears to the middle the
same ratio that the middle bears to the less.’
“We must do Dr Zeising the justice to say that he has not made more
than a legitimate use of the materials which were presented to him in
the writings of the ancients, in his endeavour to establish the fact
of the existence of this law amongst them. The canon of Polycletes,
the tradition of Varro mentioned by Pliny relative to that canon, the
writings of Galen and others, are all brought to bear on the same point
with more or less force. The sum of this portion of the argument is
fairly this,—that the ancient sculptors had _some_ law of proportion—some
authorised examplar to which they referred as their work proceeded. That
it was the law here attributed to Plato is by no means made out; but,
considering the incidental manner in which that law is referred to, and
the obscurity of the passages as they exist, it is, perhaps, too much to
expect more than this broad feature of coincidence—the fact that some
law was known and appealed to. Dr Zeising now proceeds to examine modern
theories, and it is fair to state that he appears generally to take a
very just view of them.
“Let us now turn to Dr Zeising’s own theory. It is this—that in every
beautiful form lines are divided in extreme and mean ratio; or, that
any line considered as a whole, bears to its larger part the same
proportion that the larger bears to the smaller—thus, a line of 5
inches will be divided into parts which are very nearly 2 and 3 inches
respectively (1·9 and 3·1 inches). This is a well-known division of a
line, and has been called the GOLDEN rule, but when or why, it is not
easy to ascertain. With this rule in his hand, Dr Zeising proceeds to
examine all nature and art; nay, he even ventures beyond the threshold
of nature to scan Deity. We will not follow him so far. Let us turn over
the pages of his carefully illustrated work, and see how he applies his
line. We meet first with the Apollo Belvidere—the golden line divides
him happily. We cannot say the same of the division of a handsome face
which occurs a little further on. Our preconceived notions have made the
face terminate with the chin, and not with the centre of the throat.
It is evident that, with such a rule as this, a little latitude as to
the extreme point of the object to be measured, relieves its inventor
from a world of perplexities. This remark is equally applicable to the
_arm_ which follows, to which the rule appears to apply admirably, yet
we have tried it on sundry plates of arms, both fleshy and bony, without
a shadow of success. Whether the rule was made for the arm or the arm
for the rule, we do not pretend to decide. But let us pass hastily on
to page 284, where the Venus de Medicis and Raphael’s Eve are presented
to us. They bear the application of the line right well. It might,
perhaps, be objected that it is remarkable that the same rule applies
so exactly to the existing position of the figures, such as the Apollo
and the Venus, the one of which is upright, and the other crouching.
But let that pass. We find Dr Zeising next endeavouring to square his
theory with the distances of the planets, with wofully scanty success.
Descending from his lofty position, he spans the earth from corner to
corner, at which occupation we will leave him for a moment, whilst we
offer a suggestion which is equally applicable to poets, painters,
novelists, and theorisers. Never err in excess—defect is the safe side—it
is seldom a fault, often a real merit. Leave something for the student
of your works to do—don’t chew the cud for him. Be assured he will not
omit to pay you for every little thing which you have enabled him to
discover. Poor Professor Zeising! he is far too German to leave any
field of discovery open for his readers. But let us return to him; we
left him on his back, lost for a time in a hopeless attempt to double
Cape Horn. We will be kind to him, as the child is to his man in the
Noah’s ark, and set him on his legs amongst his toys again. He is now in
the vegetable kingdom, amidst oak leaves and sections of the stems of
divers plants. He is in his element once more, and it were ungenerous
not to admit the merit of his endeavours, and the success which now and
then attends it. We will pass over his horses and their riders, together
with that portly personage, the Durham ox, for we have caught a glimpse
of a form familiar to our eyes, the ever-to-be-admired Parthenon. This
is the true test of a theory. Unlike the Durham ox just passed before
us, the Parthenon will stand still to be measured. It has so stood for
twenty centuries, and every one that has scanned its proportions has
pronounced them exquisite. Beauty is not an adaptation to the acquired
taste of a single nation, or the conventionality of a single generation.
It emanates from a deep-rooted principle in nature, and appeals to the
verdict of our whole humanity. We don’t find fault with the Durham ox—his
proportions are probably good, though they be the result of breeding
and cross-breeding; still we are not sure whether, in the march of
agriculture, our grandchildren may not think him a very wretched beast.
But there is no mistake about the Parthenon; as a type of proportion it
stands, has stood, and shall stand. Well, then, let us see how Dr Zeising
succeeds with his rule here. Alas! not a single point comes right. The
Parthenon is condemned, or its condemnation condemns the theory. Choose
your part. We choose the latter alternative; and now, our choice being
made, we need proceed no further. But a question or two have presented
themselves as we went along, which demand an answer. It may be asked—How
do you account for the esteem in which this law of the section in
extreme and mean ratio was held? We reply—That it was esteemed just in
the same way that a tree is esteemed for its fruit. To divide a right
angle into two or three, four or six, equal parts was easy enough. But
to divide it into five or ten such parts was a real difficulty. And how
was the difficulty got over? It was effected by means of this golden
rule. This is its great, its ruling application; and if we adopt the
notion that the ancients were possessed with the idea of the existence
of angular symmetry, we shall have no difficulty in accounting for their
appreciation of this problem. Nay, we may even go further, and admit,
with Dr Zeising, the interpretation of the passage of Plato,—only with
this limitation, that Plato, as a geometer, was carried away by the
geometry of æsthetics from the thing itself. It may be asked again—Is it
not probable that _some_ proportionality does exist amongst the parts of
natural objects? We reply—That, _à priori_, we expect some such system
to exist, but that it is inconsistent with the scheme of _least effort_,
which pervades and characterises all natural succession in space or in
time, that that system should be a complicated one. Whatever it is, its
essence must be simplicity. And no system of simple linear proportion
is found in nature; quite the contrary. We are, therefore, driven to
another hypothesis, viz.—that the simplicity is one of angles, not of
lines; that the eye estimates by search round a point, not by ascending
and descending, going to the right and to the left,—a theory which we
conceive all nature conspires to prove. Beauty was not created for the
eye of man, but the eye of man and his mental eye were created for the
appreciation of beauty. Examine the forms of animals and plants so minute
that nothing short of the most recent improvements in the microscope can
succeed in detecting their symmetry; or examine the forms of those little
silicious creations which grew thousands of years before Man was placed
on the earth, and, with forms of marvellous and varied beauty, they all
point to its source in angular symmetry. This is the keystone of formal
beauty, alike in the minutest animalcule, and in the noblest of God’s
works, his own image—Man.”
THE END.
BALLANTYNE AND COMPANY, PRINTERS, EDINBURGH.
FOOTNOTES
[1] Sir David Brewster.
[2] No. CLVIII., October 1843.
[3] Diogenes Laërtius’s “Lives of the Philosophers,” literally
translated. Bohn: London.
[4] Ibid.
[5] Rose’s “Biographical Dictionary.”
[6] Professor Laycock, now of the University of Edinburgh.
[7] “The Geometric Beauty of the Human Figure Defined,” &c.
[8] Longman and Co., London.
[9] See Appendix.
[10] “The Orthographic Beauty of the Parthenon,” &c., and “The Harmonic
Law of Nature applied to Architectural Design.”
[11] For further details, see “Harmonic Law of Nature,” &c.
[12] By a very simple machine, which I have lately invented, an ellipse
of any given proportions, even to those of (¹⁄₆₄), which is the curve of
the entases of the columns of the Parthenon (see Plate VII.), and of any
length, from half an inch to fifty feet or upwards, may be easily and
correctly described; the length and angle of the required ellipse being
all that need be given.
[13] “The Orthographic Beauty of the Parthenon,” &c.
[14] “The Orthographic Beauty of the Parthenon,” &c.
[15] Ibid.
[16] “The Harmonic Law of Nature applied to Architectural Design.”
[17] “Physio-philosophy.” By Dr Oken. Translated by Talk; and published
by the Ray Society. London, 1848.
[18] “The Science of those Proportions by which the Human Head and
Countenance, as represented in Works of ancient Greek Art, are
distinguished from those of ordinary Nature.”
[19] “The Geometric Beauty of the Human Figure Defined,” &c., and “The
Natural Principles of Beauty Developed in the Human Figure.”
[20] “The Geometric Beauty of the Human Figure Defined,” &c.
[21] “Essay on Ornamental Design,” &c., and “The Geometric Beauty of the
Human Figure,” &c.
[22] “A Nomenclature of Colours, applicable to the Arts and Natural
Sciences,” &c., &c.
[23] See pp. 24 and 25.
[24] “The Principles of Beauty in Colouring Systematised,” Fourteen
Diagrams, each containing Six Colours and Hues.
“A Nomenclature of Colours,” &c., Forty Diagrams, each containing Twelve
Examples of Colours, Hues, Tints, and Shades.
“The Laws of Harmonious Colouring,” &c., One Diagram, containing Eighteen
Colours and Hues.
[25] “Etchings Representing the Best Examples of Grecian and Roman
Architectural Ornament, drawn from the Originals,” &c. By Charles
Heathcote Tatham, Architect. London: Priestly and Weale. 1826.
[26] “The Orthographic Beauty of the Parthenon,” &c.
[27] “Science of those Proportions,” &c.
[28] Abbé Barthélémie’s “Travels of Anacharsis in Greece,” vol iv., pp.
193, 195.
[29] Abbé Barthélémie (vol. ii., pp. 168, 169), who cites as his
authorities, Cicer. De Nat. Deor., lib. i., cap. ii., t. 2, p. 405;
Justin Mart., Ovat. ad Gent., p. 10; Aristot. Metaph., lib. i., cap. v.,
t. 2, p. 845.
[30] Müller’s “Ancient Art and its Remains.”
[31] “Anacharsis’ Travels in Greece.” By the Abbé Barthélémie, vol. ii.,
p. 325.
[32] “Anacharsis’ Travels in Greece.” By the Abbé Barthélémie, vol. vi.,
p. 225. The authorities the Abbé quotes are—Quintil., lib. xii., cap. x.,
p. 744; Plin., lib. xxxv., cap. ix., p. 691.
[33] Müller’s “Archæology of Art,” &c.
Works by the Same Author.
I.
In royal 8vo, with Copperplate Illustrations, price 2s. 6d.,
THE HARMONIC LAW OF NATURE APPLIED TO ARCHITECTURAL DESIGN.
_From the Athenæum._
The beauty of the theory is its universality, and its simplicity. In
nature, the Creator accomplished his purposes by the simplest means—the
harmony of nature is indestructible and self-restoring. Mr Hay’s book on
the “Parthenon,” on the “Natural Principles of Beauty as developed in the
Human Figure,” his “Principles of Symmetrical Beauty,” his “Principles of
Colouring, and Nomenclature of Colours,” his “Science of Proportion,” and
“Essay on Ornamental Design,” we have already noticed with praise as the
results of philosophical and original thought.
_From the Daily News._
This essay is a new application to Lincoln cathedral in Gothic
architecture, and to the Temple of Theseus in Greek architecture, of the
principles of symmetrical beauty already so profusely illustrated and
demonstrated by Mr Hay. The theory which Mr Hay has propounded in so many
volumes is not only a splendid contribution towards a science of æsthetic
proportions, but, for the first time in the history of art, proves the
possibility, and lays the foundations of such a science. To those who are
not acquainted with the facts, these expressions will sound hyperbolical,
but they are most true.
II.
In royal 8vo, with Copperplate Illustrations, price 5s.,
THE NATURAL PRINCIPLES OF BEAUTY, AS DEVELOPED IN THE HUMAN FIGURE.
_From the Spectator._
We cannot refuse to entertain Mr Hay’s system as of singular intrinsic
excellence. The simplicity of his law and its generality impress
themselves more deeply on the conviction with each time of enforcement.
His theory proceeds from the idea, that in nature every thing is effected
by means more simple than any other that could have been conceived,—an
idea certainly consistent with whatever we can trace out or imagine of
the all-wise framing of the universe.
_From the Sun._
By founding (if we may so phrase it) this noble theory, Mr Hay has
covered his name with distinction, and has laid the basis, we conceive,
of no ephemeral reputation. By illustrating it anew, through the
medium of this graceful treatise, he has conferred a real boon upon
the community, for he has afforded the public another opportunity of
following the golden rule of the poet—by looking through the holy and
awful mystery of creation to the holier and yet more awful mystery of
Omnipotence.
_From the Cambridge Journal of Classical and Sacred Philology._
The inquiries which of late years have been instituted by Mr D. R. Hay
of Edinburgh, on the proportions of the human figure, and on the natural
principles of beauty, as illustrated by works of Greek art, constitute an
epoch in the study of æsthetics and the philosophy of form.
III.
In royal 8vo, with Copperplate Illustrations, price 5s.,
THE ORTHOGRAPHIC BEAUTY OF THE PARTHENON REFERRED TO A LAW OF NATURE.
To which are prefixed, a few Observations on the Importance of Æsthetic
Science as an Element in Architectural Education.
_From the Scottish Literary Gazette._
We think this work will satisfy every impartial mind that Mr Hay has
developed the true theory of the Parthenon—that he has, in fact, to use
a kindred phraseology, both _parsed_ and _scanned_ this exquisitely
beautiful piece of architectural composition, and that, in doing so, he
has provided the true key by which the treasures of Greek art may be
further unlocked, and rendered the means of correcting, improving, and
elevating modern practice.
_From the Edinburgh Guardian._
Again and again the attempt has been made to detect harmonic ratios in
the measurement of Athenian architecture, but ever without reward. Mr Hay
has, however, made the discovery, and to an extent of which no one had
previously dreamt.
IV.
In 8vo, 100 Plates, price 6s.,
FIRST PRINCIPLES OF SYMMETRICAL BEAUTY.
_From the Spectator._
This is a grammar of pure form, in which the elements of symmetrical, as
distinguished from picturesque beauty, are demonstrated, by reducing the
outlines or planes of curvilinear and rectilinear forms to their origin
in the principles of geometrical proportion. In thus analysing symmetry
of outline in natural and artificial objects, Mr Hay determines the fixed
principles of beauty in positive shape, and shews how beautiful forms may
be reproduced and infinitely varied with mathematical precision. Hitherto
the originating and copying of beautiful forms have been alike empirical;
the production of a new design for a vase or a jug has been a matter of
chance between the eye and the hand; and the copying of a Greek moulding
or ornament, a merely mechanical process. By a series of problems, Mr
Hay places both the invention and imitation of beautiful forms on a sure
basis of science, giving to the fancy of original minds a clue to the
evolving of new and elegant shapes, in which the infinite resources of
nature are made subservient to the uses of art.
The volume is illustrated by one hundred diagrams beautifully executed,
that serve to explain the text, and suggest new ideas of beauty of
contour in common objects. To designers of pottery, hardware, and
architectural ornaments, this work is particularly valuable; but artists
of every kind, and workmen of intelligence, will find it of great utility.
_From the Athenæum._
The volume before us is the seventh of Mr Hay’s works. It is the most
practical and systematic, and likely to be one of the most useful.
It is, in short, a grammar of form, or a spelling-book of beauty.
This is beginning at the right end of the matter; and the necessity
for this kind of knowledge will inevitably, though gradually, be
felt. The work will, therefore, be ultimately appreciated and
adopted as an introduction to the study of beautiful forms.
The third part of the work treats of the Greek oval or composite ellipse,
as Mr Hay calls it. It is an ellipse of three foci, and gives practical
forms for vases and architectural mouldings, which are curious to the
architect, and will be very useful both to the potter, the moulder,
and the pattern-drawer. A fourth part contains applications of this to
practice. Of the details worked out with so much judgment and ingenuity
by Mr Hay, we should in vain attempt to communicate just notions without
the engravings of which his book is full. We must, therefore, refer to
the work itself. The forms there given are full of beauty, and so far
tend to prove the system.
V.
In 8vo, 14 Coloured Diagrams, Second Edition, price 15s.,
THE PRINCIPLES OF BEAUTY IN COLOURING SYSTEMATISED.
_From the Spectator._
In this new analysis of the harmonies of colour, Mr Hay has performed the
useful service of tracing to the operation of certain fixed principles
the sources of beauty in particular combinations of hues and tints; so
that artists may, by the aid of this book, produce, with mathematical
certainty, the richest effects, hitherto attainable by genius alone. Mr
Hay has reduced this branch of art to a perfect system, and proved that
an offence against good taste in respect to combinations of colour is, in
effect, a violation of natural laws.
VI.
In 8vo, 228 Examples of Colours, Hues, Tints, and Shades, price 63s.,
A NOMENCLATURE OF COLOURS, APPLICABLE TO THE ARTS AND NATURAL SCIENCES.
_From the Daily News._
In this work Mr Hay has brought a larger amount of practical knowledge
to bear on the subject of colour than any other writer with whom we are
acquainted, and in proportion to this practical knowledge is, as might be
expected, the excellence of his treatise. There is much in this volume
which we would most earnestly recommend to the notice of artists, house
decorators, and, indeed, to all whose business or profession requires a
knowledge of the management of colour. The work is replete with hints
which they might turn to profitable account, and which they will find
nowhere else.
_From the Athenæum._
We have formerly stated the high opinion we entertain of Mr Hay’s
previous exertions for the improvement of decorative art in this country.
We have already awarded him the merit of invention and creation of the
new and the beautiful in form. In his former treatises he furnished a
theory of definite proportions for the creation of the beautiful in form.
In the present work he proposes to supply a scale of definite proportions
for chromatic beauty. For this purpose he sets out very properly with a
precise nomenclature of colour. In this he has constructed a vocabulary
for the artist—an alphabet for the artizan. He has gone further—he
constructs words for three syllables. From this time, it will be possible
to write a letter in Edinburgh about a coloured composition, which shall
be read off in London, Paris, St Petersburg, or Pekin, and shall so
express its nature that it can be reproduced in perfect identity. This Mr
Hay has done, or at least so nearly, as to deserve our thanks on behalf
of art, and artists of all grades, even to the decorative artizan—not one
of whom, be he house-painter, china pattern-drawer, or calico printer,
should be without the simple manual of “words for colours.”
VII.
In post 8vo, with a Coloured Diagram, Sixth Edition, price 7s. 6d.,
THE LAWS OF HARMONIOUS COLOURING ADAPTED TO INTERIOR DECORATIONS.
_From the Atlas._
Every line of this useful book shews that the author has contrived to
intellectualise his subject in a very interesting manner. The principles
of harmony in colour as applied to decorative purposes, are explained and
enforced in a lucid and practical style, and the relations of the various
tints and shades to each other, so as to produce a harmonious result, are
descanted upon most satisfactorily and originally.
_From the Edinburgh Review._
In so far as we know, Mr Hay is the first and only modern artist who
has entered upon the study of these subjects without the trammels of
prejudice and authority. Setting aside the ordinances of fashion, as
well as the dicta of speculation, he has sought the foundation of
his profession in the properties of light, and in the laws of visual
sensation, by which these properties are recognised and modified. The
truths to which he has appealed are fundamental and irrefragable.
_From the Athenæum._
We have regarded, and do still regard, the production of Mr Hay’s works
as a remarkable psychological phenomenon—one which is instructive both
for the philosopher and the critic to study with care and interest, not
unmingled with respect. We see how his mind has been gradually guided
by Nature herself out of one track, and into another, and ever and anon
leading him to some vein of the beautiful and true, hitherto unworked.
VIII.
In 4to, 25 Plates, price 36s.,
ON THE SCIENCE OF THOSE PROPORTIONS BY WHICH THE HUMAN HEAD AND
COUNTENANCE, AS REPRESENTED IN ANCIENT GREEK ART, ARE DISTINGUISHED FROM
THOSE OF ORDINARY NATURE.
(PRINTED BY PERMISSION.)
_From a Letter to the Author by Sir William Hamilton, Bart., Professor of
Logic and Metaphysics in the Edinburgh University._
Your very elegant volume, “Science of those Proportions,” &c., is to me
extremely interesting, as affording an able contribution to what is the
ancient, and, I conceive, the true theory of the beautiful. But though
your doctrine coincides with the one prevalent through all antiquity, it
appears to me quite independent and original in you; and I esteem it the
more that it stands opposed to the hundred one-sided and exclusive views
prevalent in modern times.
_From Chambers’s Edinburgh Journal._
We now come to another, and much more remarkable corroboration, which
calls upon us to introduce to our readers one of the most valuable and
original contributions that have ever been made to the Philosophy of Art,
viz., Mr Hay’s work “On the Science of those Proportions,” &c. Mr Hay’s
plan is simply to form a scale composed of the well-known vibrations of
the monochord, which are the alphabet of music, and then to draw upon the
quadrant of a circle angles _answering to these vibrations_. With the
series of triangles thus obtained he combines a circle and an ellipse,
the proportions of which are derived from the triangles themselves; and
thus he obtains an infallible rule for the composition of the head of
ideal beauty.
IX.
In 4to, 16 Plates, price 30s.,
THE GEOMETRIC BEAUTY OF THE HUMAN FIGURE DEFINED.
To which is prefixed, a SYSTEM of ÆSTHETIC PROPORTION applicable to
ARCHITECTURE and the other FORMATIVE ARTS.
_From the Cambridge Journal of Classical and Sacred Philology._
We feel bound to pay Mr Hay a large and glad tribute of praise for
having devised a system of proportions which rises superior to the
idiosyncrasies of different artists, which brings back to one common type
the sensations of Eye and Ear, and so makes a giant stride towards that
_codification_ of the laws of the universe which it is the business of
science to effect. We have no hesitation in saying that, for scientific
precision of method and importance of results, Albert Durer, Da Vinci,
and Hogarth—not to mention less noteworthy writers—must all yield the
palm to Mr Hay.
X.
In oblong folio, 57 Plates and numerous Woodcuts, price 42s.,
AN ESSAY ON ORNAMENTAL DESIGN, IN WHICH ITS TRUE PRINCIPLES ARE DEVELOPED
AND ELUCIDATED, &c.
_From the Athenæum._
In conclusion, Mr Hay’s book goes forth with our best wishes. It must be
good. It must be prolific of thought—stimulant of invention. It is to be
acknowledged as a benefit of an unusual character conferred on the arts
of ornamental design.
_From the Spectator._
Mr Hay has studied the subject deeply and scientifically. In this
treatise on ornamental design, the student will find a clue to the
discovery of the source of an endless variety of beautiful forms and
combinations of lines, in the application of certain fixed laws of
harmonious proportion to the purposes of art. Mr Hay also exemplifies
the application of his theory of linear harmony to the production of
beautiful forms generally, testing its soundness by applying it to the
human figure, and the purest creations of Greek art.
_From Fraser’s Magazine._
Each part of this work is enriched by diagrams of great beauty, direct
emanations of principle, and, consequently, presenting entirely new
combinations of form. Had our space permitted, we should have made some
extracts from this “Essay on Ornamental Design;” and we would have done
so, because of the discriminating taste by which it is pervaded, and the
forcible observations which it contains; but we cannot venture on the
indulgence.
XI.
In 4to, 17 Plates and 38 Woodcuts, price 25s.,
PROPORTION, OR THE GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED.
XII.
In 4to, 18 Plates and numerous Woodcuts, price 15s.,
THE NATURAL PRINCIPLES AND ANALOGY OF THE HARMONY OF FORM.
_From the Edinburgh Review._
Notwithstanding some trivial points of difference between Mr Hay’s views
and our own, we have derived the greatest pleasure from the perusal of
these works. They are all composed with accuracy and even elegance. His
opinions and views are distinctly brought before the reader, and stated
with that modesty which characterises genius, and that firmness which
indicates truth.
_From Blackwood’s Magazine._
We have no doubt that when Mr Hay’s Art-discovery is duly developed and
taught, as it should be, in our schools, it will do more to improve the
general taste than anything which has yet been devised.
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