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.

ILLUSTRATIONS.

PLATES

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

XVII. Diagram shewing the Geometric Construction of an Ornament belonging to the Parthenon at Athens.

D. R. Hay delᵗ. G. Aikman sc.

XVIII. Diagram of the Geometrical Construction of the ancient Grecian Ornament called the Honeysuckle.

D. R. Hay delᵗ. G. Aikman sc.

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.

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.

XXI. Diagram of a variation on the Form of the Portland Vase.

D. R. Hay delᵗ. G. Aikman sc.

XXII. Diagram of a second variation on the Form of the Portland Vase.

D. R. Hay delᵗ. G. Aikman sc.

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 articisms 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:—

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.

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:—

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:—

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:—

Scale I. being now complete, we have only to divide these latter quantities by (2) downwards in order to complete the other three. Thus:—

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:—

The following are the principal numerical relations which the parts in each scale bear to one another:—

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.

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:—

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:—

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.

[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—

[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:—

[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:—

[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.

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.

Fig. 2.

The foci being in either case thus simply ascertained, the method of describing the curve on a small scale is equally simple.

[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.

[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.

[Plate VII.]

[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—

These angles, and the figures which belong to them, are thus arranged:—

[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:—

These angles are employed in the construction of a diagram, which determines the proportions of the parts throughout the whole figure. Thus:—

[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.

[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.