Chapter III: Introduction (1)

Twelve years ago, one of our most eminent philosophers,[1] through the medium of the _Edinburgh Review_,[2] gave the following account of what was then the state of the fine arts as connected with science:—“The disposition to introduce into the intellectual community the principles of free intercourse, is by no means general; but we are confident that Art will not sufficiently develop her powers, nor Science attain her most commanding position, till the practical knowledge of the one is taken in return for the sound deductions of the other.... It is in the fine arts, principally, and in the speculations with which they are associated, that the controlling power of scientific truth has not exercised its legitimate influence. In discussing the principles of painting, sculpture, architecture, and landscape gardening, philosophers have renounced science as a guide, and even as an auxiliary; and a school has arisen whose speculations will brook no restraint, and whose decisions stand in opposition to the strongest convictions of our senses. That the external world, in its gay colours and lovely forms, is exhibited to the mind only as a tinted mass, neither within nor without the eye, neither touching it nor distant from it—an ubiquitous chaos, which experience only can analyse and transform into the realities which compose it; that the beautiful and sublime in nature and in art derive their power over the mind from association alone, are among the philosophical doctrines of the present day, which, if it be safe, it is scarcely prudent to question. Nor are these opinions the emanations of poetical or ill-trained minds, which ingenuity has elaborated, and which fashion sustains. They are conclusions at which most of our distinguished philosophers have arrived. They have been given to the world with all the authority of demonstrated truth; and in proportion to the hold which they have taken of the public mind, have they operated as a check upon the progress of knowledge.”

Such, then, was the state of art as connected with science twelve years ago. But although the causes which then placed science and the fine arts at variance have since been gradually diminishing, yet they are still far from being removed. In proof of this I may refer to what took place at the annual distribution of the prizes to the students attending our Scottish Metropolitan School of Design, in 1854, the pupils in which amount to upwards of two hundred. The meeting on that occasion included, besides the pupils, a numerous and highly respectable assemblage of artists and men of science. The chairman, a Professor in our University, and editor of one of the most voluminous works on art, science, and literature ever produced in this country, after extolling the general progress of the pupils, so far as evinced by the drawings exhibited on the occasion, drew the attention of the meeting to a discovery made by the head master of the architectural and ornamental department of the school, viz.—That the ground-plan of the Parthenon at Athens had been constructed by the application of the _mysterious_ ovoid or _Vesica Piscis_ of the middle ages, subdivided by the _mythic_ numbers 3 and 7, and their intermediate odd number 5. Now, it may be remarked, that the figure thus referred to is not an ovoid, neither is it in any way of a mysterious nature, being produced simply by two equal circles cutting each other in their centres. Neither can it be shewn that the numbers 3 and 7 are in any way more mythic than other numbers. In fact, the terms _mysterious_ and _mythic_ so applied, can only be regarded as a remnant of an ancient terminology, calculated to obscure the simplicity of scientific truth, and when used by those employed to teach—for doubtless the chairman only gave the description he received—must tend to retard the connexion of that truth with the arts of design. I shall now give a specimen of the manner in which a knowledge of the philosophy of the fine arts is at present inculcated upon the public mind generally. In the same metropolis there has likewise existed for upwards of ten years a Philosophical Institution of great importance and utility, whose members amount to nearly three thousand, embracing a large proportion of the higher classes of society, both in respect to talent and wealth. At the close of the session of this Institution, in 1854, a learned and eloquent philologus, who occasionally lectures upon beauty, was appointed to deliver the closing address, and touching upon the subject of the beautiful, he thus concluded—

“In the worship of the beautiful, and in that alone, we are inferior to the Greeks. Let us therefore be glad to borrow from them; not slavishly, but with a wise adaptation—not exclusively, but with a cunning selection; in art, as in religion, let us learn to prove all things, and hold fast that which is good—not merely one thing which is good, but all good things—Classicalism, Mediævalism, Modernism—let us have and hold them all in one wide and lusty embrace. Why should the world of art be more narrow, more monotonous, than the world of nature? Did God make all the flowers of one pattern, to please the devotees of the rose or the lily; and did He make all the hills, with the green folds of their queenly mantles, all at one slope, to suit the angleometer of the most mathematical of decorators? I trow not. Let us go and do likewise.”

I here take for granted, that what the lecturer meant by “the worship of the beautiful,” is the production and appreciation of works of art in which beauty should be a primary element; and judging from the remains which we possess of such works as were produced by the ancient Grecians, our inferiority to them in these respects cannot certainly be denied. But I must reiterate what I have often before asserted, that it is not by borrowing from them, however cunning our selection, or however wise our adaptations, that this inferiority is to be removed, but by a re-discovery of the science which these ancient artists must have employed in the production of that symmetrical beauty and chaste elegance which pervaded all their works for a period of nearly three hundred years. And I hold, that as in religion, so in art, there is only one truth, a grain of which is worth any amount of philological eloquence.

I also take for granted, that what is meant by Classicalism in the above quotation, is the ancient Grecian style of art; by Mediævalism, the semi-barbaric style of the middle ages; and by Modernism, that chaotic jumble of all previous styles and fashions of art, which is the peculiar characteristic of our present school, and which is, doubtless, the result of a system of education based upon plagiarism and mere imitation. Therefore a recommendation to embrace with equal fervour “as good things,” these very opposite artic_isms_ must be a doctrine as mischievous in art as it would be in religion to recommend as equally good things the various _isms_ into which it has also been split in modern times.

Now, “the world of nature” and “the world of art” have not that equality of scope which this lecturer on beauty ascribes to them, but differ very decidedly in that particular. Neither will it be difficult to shew why “the world of art _should_ be more narrow than the world of nature”—that it should be thereby rendered more monotonous does not follow.

It is well known, that the “world of nature” consists of productions, including objects of every degree of beauty from the very lowest to the highest, and calculated to suit not only the tastes arising from various degrees of intellect, but those arising from the natural instincts of the lower animals. On the other hand, “the world of art,” being devoted to the gratification and improvement of intelligent minds only, is therefore narrowed in its scope by the exclusion from its productions of the lower degrees of beauty—even mediocrity is inadmissible; and we know that the science of the ancient Greek artists enabled them to excel the highest individual productions of nature in the perfection of symmetrical beauty. Consequently, all objects in nature are not equally well adapted for artistic study, and it therefore requires, on the part of the artist, besides true genius, much experience and care to enable him to choose proper subjects from nature; and it is in the choice of such subjects, and not in plagiarism from the ancients, that he should select with knowledge and adapt with wisdom. Hence, all such latitudinarian doctrines as those I have quoted must act as a check upon the progress of knowledge in the scientific truth of art. I have observed in some of my works, that in this country a course had been followed in our search for the true science of beauty not differing from that by which the alchymists of the middle ages conducted their investigations; for our ideas of visible beauty are still undefined, and our attempts to produce it in the various branches of art are left dependant, in a great measure, upon chance. Our schools are conducted without reference to any first principles or definite laws of beauty, and from the drawing of a simple architectural moulding to the intricate combinations of form in the human figure, the pupils trust to their hands and eyes alone, servilely and mechanically copying the works of the ancients, instead of being instructed in the unerring principles of science, upon which the beauty of those works normally depends. The instruction they receive is imparted without reference to the judgment or understanding, and they are thereby led to imitate effects without investigating causes. Doubtless, men of great genius sometimes arrive at excellence in the arts of design without a knowledge of the principles upon which beauty of form is based; but it should be kept in mind, that true genius includes an intuitive perception of those principles along with its creative power. It is, therefore, to the generality of mankind that instruction in the definite laws of beauty will be of most service, not only in improving the practice of those who follow the arts professionally, but in enabling all of us to distinguish the true from the false, and to exercise a sound and discriminating taste in forming our judgment upon artistic productions. Æsthetic culture should consequently supersede servile copying, as the basis of instruction in our schools of art. Many teachers of drawing, however, still assert, that, by copying the great works of the ancients, the mind of the pupil will become imbued with ideas similar to theirs—that he will imbibe their feeling for the beautiful, and thereby become inspired with their genius, and think as they thought. To study carefully and to investigate the principles which constitute the excellence of the works of the ancients, is no doubt of much benefit to the student; but it would be as unreasonable to suppose that he should become inspired with artistic genius by merely copying them, as it would be to imagine, that, in literature, poetic inspiration could be created by making boys transcribe or repeat the works of the ancient poets. Sir Joshua Reynolds considered copying as a delusive kind of industry, and has observed, that “Nature herself is not to be too closely copied,” asserting that “there are excellences in the art of painting beyond what is commonly called the imitation of nature,” and that “a mere copier of nature can never produce any thing great.” Proclus, an eminent philosopher and mathematician of the later Platonist school (A.D. 485), says, that “he who takes for his model such forms as nature produces, and confines himself to an exact imitation of these, will never attain to what is perfectly beautiful. For the works of nature are full of disproportion, and fall very short of the true standard of beauty.”

It is remarked by Mr. J. C. Daniel, in the introduction to his translation of M. Victor Cousin’s “Philosophy of the Beautiful,” that “the English writers have advocated no theory which allows the beautiful to be universal and absolute; nor have they professedly founded their views on original and ultimate principles. Thus the doctrine of the English school has for the most part been, that beauty is mutable and special, and the inference that has been drawn from this teaching is, that all tastes are equally just, provided that each man speaks of what he feels.” He then observes, that the German, and some of the French writers, have thought far differently; for with them the beautiful is “simple, immutable, absolute, though its _forms_ are manifold.”

So far back as the year 1725, the same truths advanced by the modern German and French writers, and so eloquently illustrated by M. Cousin, were given to the world by Hutchison in his “Inquiry into the Original of our Ideas of Beauty and Virtue.” This author says—“We, by absolute beauty, understand only that beauty which we perceive in objects, without comparison to any thing external, of which the object is supposed an imitation or picture, such as the beauty perceived from the works of nature, artificial forms, figures, theorems. Comparative or relative beauty is that which we perceive in objects commonly considered as imitations or resemblances of something else.”

Dr. Reid also, in his “Intellectual Powers of Man,” says—“That taste, which we may call rational, is that part of our constitution by which we are made to receive pleasure from the contemplation of what we conceive to be excellent in its kind, the pleasure being annexed to this judgment, and regulated by it. This taste may be true or false, according as it is founded on a true or false judgment. And if it may be true or false, it must have first principles.”

M. Victor Cousin’s opinion upon this subject is, however, still more conclusive. He observes—“If the idea of the beautiful is not absolute, like the idea of the true—if it is nothing more than the expression of individual sentiment, the rebound of a changing sensation, or the result of each person’s fancy—then the discussions on the fine arts waver without support, and will never end. For a theory of the fine arts to be possible, there must be something absolute in beauty, just as there must be something absolute in the idea of goodness, to render morals a possible science.”

The basis of the science of beauty must thus be founded upon fixed principles, and when these principles are evolved with the same care which has characterised the labours of investigators in natural science, and are applied in the fine arts as the natural sciences have been in the useful arts, a solid foundation will be laid, not only for correct practice, but also for a just appreciation of productions in every branch of the arts of design.

We know that the mind receives pleasure through the sense of hearing, not only from the music of nature, but from the euphony of prosaic composition, the rhythm of poetic measure, the artistic composition of successive harmony in simple melody, and the combined harmony of counterpoint in the more complex works of that art. We know, also, that the mind is similarly gratified through the sense of seeing, not only by the visible beauties of nature, but by those of art, whether in symmetrical or picturesque compositions of forms, or in harmonious arrangements of gay or sombre colouring.

Now, in respect to the first of these modes of sensation, we know, that from the time of Pythagoras, the fact has been established, that in whatever manner nature or art may address the ear, the degree of obedience paid to the fundamental law of harmony will determine the presence and degree of that beauty with which a perfect organ can impress a well-constituted mind; and it is my object in this, as it has been in former attempts, to prove it consistent with scientific truth, that that beauty which is addressed to the mind by objects of nature and art, through the eye, is similarly governed. In short, to shew that, as in compositions of sounds, there can be no true beauty in the absence of a strict obedience to this great law of nature, neither can there exist, in compositions of forms or colours, that principle of unity in variety which constitutes beauty, unless such compositions are governed by the same law.

Although in the songs of birds, the gurgling of brooks, the sighing of the gentle summer winds, and all the other beautiful music of nature, no analysis might be able to detect the operation of any precise system of harmony, yet the pleasure thus afforded to the human mind we know to arise from its responding to every development of an obedience to this law. When, in like manner, we find even in those compositions of forms and colours which constitute the wildest and most rugged of Nature’s scenery, a species of picturesque grandeur and beauty to which the mind as readily responds as to her more mild and pleasing aspects, or to her sweetest music, we may rest assured that this beauty is simply another development of, and response to, the same harmonic law, although the precise nature of its operation may be too subtle to be easily detected.

The _résumé_ of the various works I have already published upon the subject, along with the additional illustrations I am about to lay before my readers, will, I trust, point out a system of harmony, which, in formative art, as well as in that of colouring, will rise superior to the idiosyncracies of different artists, and bring back to one common type the sensations of the eye and the ear, thereby improving that knowledge of the laws of the universe which it is as much the business of science to combine with the ornamental as with the useful arts.

In attempting this, however, I beg it may be understood, that I do not believe any system, based even upon the laws of nature, capable of forming a royal road to the perfection of art, or of “mapping the mighty maze of a creative mind.” At the same time, however, I must continue to reiterate the fact, that the diffusion of a general knowledge of the science of visible beauty will afford latent artistic genius just such a vantage ground as that which the general knowledge of philology diffused throughout this country affords its latent literary genius. Although _mere learning_ and _true genius_ differ as much in the practice of art as they do in the practice of literature, yet a precise and systematic education in the true science of beauty must certainly be as useful in promoting the practice and appreciation of the one, as a precise and systematic education in the science of philology is in promoting the practice and appreciation of the other.

As all beauty is the result of harmony, it will be requisite here to remark, that harmony is not a simple quality, but, as Aristotle defines it, “the union of contrary principles having a ratio to each other.” Harmony thus operates in the production of all that is beautiful in nature, whether in the combinations, in the motions, or in the affinities of the elements of matter.

The contrary principles to which Aristotle alludes, are those of uniformity and variety; for, according to the predominance of the one or the other of these principles, every kind of beauty is characterised. Hence the difference between symmetrical and picturesque beauty:—the first allied to the principle of uniformity, in being based upon precise laws that may be taught so as to enable men of ordinary capacity to produce it in their works—the second allied to the principle of variety often to so great a degree that they yield an obedience to the precise principles of harmony so subtilely, that they cannot be detected in its constitution, but are only felt in the response by which true genius acknowledges their presence. The generality of mankind may be capable of perceiving this latter kind of beauty, and of feeling its effects upon the mind, but men of genius, only, can impart it to works of art, whether addressed to the eye or the ear. Throughout the sounds, forms, and colours of nature, these two kinds of beauty are found not only in distinct developments, but in every degree of amalgamation. We find in the songs of some birds, such as those of the chaffinch, thrush, &c., a rhythmical division, resembling in some measure the symmetrically precise arrangements of parts which characterises all artistic musical composition; while in the songs of other birds, and in the other numerous melodies with which nature charms and soothes the mind, there is no distinct regularity in the division of their parts. In the forms of nature, too, we find amongst the innumerable flowers with which the surface of the earth is so profusely decorated, an almost endless variety of systematic arrangements of beautiful figures, often so perfectly symmetrical in their combination, that the most careful application of the angleometer could scarcely detect the slightest deviation from geometrical precision; while, amongst the masses of foliage by which the forms of many trees are divided and subdivided into parts, as also amongst the hills and valleys, the mountains and ravines, which divide the earth’s surface, we find in every possible variety of aspect the beauty produced by that irregular species of symmetry which characterises the picturesque.

In like manner, we find in wild as well as cultivated flowers the most symmetrical distributions of colours accompanying an equally precise species of harmony in their various kinds of contrasts, often as mathematically regular as the geometric diagrams by which writers upon colour sometimes illustrate their works; while in the general colouring of the picturesque beauties of nature, there is an endless variety in its distributions, its blendings, and its modifications. In the forms and colouring of animals, too, the same endless variety of regular and irregular symmetry is to be found. But the highest degree of beauty in nature is the result of an equal balance of uniformity with variety. Of this the human figure is an example; because, when it is of those proportions universally acknowledged to be the most perfect, its uniformity bears to its variety an apparently equal ratio. The harmony of combination in the normal proportions of its parts, and the beautifully simple harmony of succession in the normal melody of its softly undulating outline, are the perfection of symmetrical beauty, while the innumerable changes upon the contour which arise from the actions and attitudes occasioned by the various emotions of the mind, are calculated to produce every species of picturesque beauty, from the softest and most pleasing to the grandest and most sublime.

Amongst the purely picturesque objects of inanimate nature, I may, as in a former work, instance an ancient oak tree, for its beauty is enhanced by want of apparent symmetry. Thus, the more fantastically crooked its branches, and the greater the dissimilarity and variety it exhibits in its masses of foliage, the more beautiful it appears to the artist and the amateur; and, as in the human figure, any attempt to produce variety in the proportions of its lateral halves would be destructive of its symmetrical beauty, so in the oak tree any attempt to produce palpable similarity between any of its opposite sides would equally deteriorate its picturesque beauty. But picturesque beauty is not the result of the total absence of symmetry; for, as none of the irregularly constructed music of nature could be pleasing to the ear unless there existed in the arrangement of its notes an obedience, however subtle, to the great harmonic law of Nature, so neither could any object be picturesquely beautiful, unless the arrangement of its parts yields, although it may be obscurely, an obedience to the same law.

However symmetrically beautiful any architectural structure may be, when in a complete and perfect state, it must, as it proceeds towards ruin, blend the picturesque with the symmetrical; but the type of its beauty will continue to be the latter, so long as a sufficient portion of it remains to convey an idea of its original perfection. It is the same with the human form and countenance; for age does not destroy their original beauty, but in both only lessens that which is symmetrical, while it increases that which is picturesque.

In short, as a variety of simultaneously produced sounds, which do not relate to each other agreeably to this law, can only convey to the mind a feeling of mere noise; so a variety of forms or colours simultaneously exposed to the eye under similar circumstances, can only convey to the mind a feeling of chaotic confusion, or what may be termed _visible_ discord. As, therefore, the two principles of uniformity and variety, or similarity and dissimilarity, are in operation in every harmonious combination of the elements of sound, of form, and of colour, we must first have recourse to numbers in the abstract before we can form a proper basis for a universal science of beauty.

THE SCIENCE OF BEAUTY EVOLVED FROM THE HARMONIC LAW OF NATURE, AGREEABLY TO THE PYTHAGOREAN SYSTEM OF NUMERICAL RATIO.

The scientific principles of beauty appear to have been well known to the ancient Greeks; and it must have been by the practical application of that knowledge to the arts of Design, that that people continued for a period of upwards of three hundred years to execute, in every department of these arts, works surpassing in chaste beauty any that had ever before appeared, and which have not been equalled during the two thousand years which have since elapsed.

Æsthetic science, as the science of beauty is now termed, is based upon that great harmonic law of nature which pervades and governs the universe. It is in its nature neither absolutely physical nor absolutely metaphysical, but of an intermediate nature, assimilating in various degrees, more or less, to one or other of those opposite kinds of science. It specially embodies the inherent principles which govern impressions made upon the mind through the senses of hearing and seeing. Thus, the æsthetic pleasure derived from listening to the beautiful in musical composition, and from contemplating the beautiful in works of formative art, is in both cases simply a response in the human mind to artistic developments of the great harmonic law upon which the science is based.

Although the eye and the ear are two different senses, and, consequently, various in their modes of receiving impressions; yet the sensorium is but one, and the mind by which these impressions are perceived and appreciated is also characterised by unity. There appears, likewise, a striking analogy between the natural constitution of the two kinds of beauty, which is this, that the more physically æsthetic elements of the highest works of musical composition are melody, harmony, and tone, whilst those of the highest works of formative art are contour, proportion, and colour. The melody or theme of a musical composition and its harmony are respectively analogous,—1st, To the outline of an artistic work of formative art; and 2d, To the proportion which exists amongst its parts. To the careful investigator these analogies become identities in their effect upon the mind, like those of the more metaphysically æsthetic emotions produced by expression in either of these arts.

Agreeably to the first analogy, the outline and contour of an object, suppose that of a building in shade when viewed against a light background, has a similar effect upon the mind with that of the simple melody of a musical composition when addressed to the ear unaccompanied by the combined harmony of counterpoint. Agreeably to the second analogy, the various parts into which the surface of the supposed elevation is divided being simultaneously presented to the eye, will, if arranged agreeably to the same great law, affect the mind like that of an equally harmonious arrangement of musical notes accompanying the supposed melody.

There is, however, a difference between the construction of these two organs of sense, viz., that the ear must in a great degree receive its impressions involuntarily; while the eye, on the other hand, is provided by nature with the power of either dwelling upon, or instantly shutting out or withdrawing itself from an object. The impression of a sound, whether simple or complex, when made upon the ear, is instantaneously conveyed to the mind; but when the sound ceases, the power of observation also ceases. But the eye can dwell upon objects presented to it so long as they are allowed to remain pictured on the retina; and the mind has thereby the power of leisurely examining and comparing them. Hence the ear guides more as a mere sense, at once and without reflection; whilst the eye, receiving its impressions gradually, and part by part, is more directly under the influence of mental analysis, consequently producing a more metaphysically æsthetic emotion. Hence, also, the acquired power of the mind in appreciating impressions made upon it through the organ of sight under circumstances, such as perspective, &c., which to those who take a hasty view of the subject appear impossible.

Dealing as this science therefore does, alike with the sources and the resulting principles of beauty, it is scarcely less dependent on the accuracy of the senses than on the power of the understanding, inasmuch as the effect which it produces is as essential a property of objects, as are its laws inherent in the human mind. It necessarily comprehends a knowledge of those first principles in art, by which certain combinations of sounds, forms, and colours produce an effect upon the mind, connected, in the first instance, with sensation, and in the second with the reasoning faculty. It is, therefore, not only the basis of all true practice in art, but of all sound judgment on questions of artistic criticism, and necessarily includes those laws whereon a correct taste must be based. Doubtless many eloquent and ingenious treatises have been written upon beauty and taste; but in nearly every case, with no other effect than that of involving the subject in still greater uncertainty. Even when restricted to the arts of design, they have failed to exhibit any definite principles whereby the true may be distinguished from the false, and some natural and recognised laws of beauty reduced to demonstration. This may be attributed, in a great degree, to the neglect of a just discrimination between what is merely agreeable, or capable of exciting pleasurable sensations, and what is essentially beautiful; but still more to the confounding of the operations of the understanding with those of the imagination. Very slight reflection, however, will suffice to shew how essentially distinct these two faculties of the mind are; the former being regulated, in matters of taste, by irrefragable principles existing in nature, and responded to by an inherent principle existing in the human mind; while the latter operates in the production of ideal combinations of its own creation, altogether independent of any immediate impression made upon the senses. The beauty of a flower, for example, or of a dew-drop, depends on certain combinations of form and colour, manifestly referable to definite and systematic, though it may be unrecognised, laws; but when Oberon, in “Midsummer Night’s Dream,” is made to exclaim—

“And that same dew, which sometimes on the buds
Was wont to swell, like round and orient pearls,
Stood now within the pretty floweret’s eyes,
Like tears that did their own disgrace bewail,”—

the poet introduces a new element of beauty equally legitimate, yet altogether distinct from, although accompanying that which constitutes the more precise science of æsthetics as here defined. The composition of the rhythm is an operation of the understanding, but the beauty of the poetic fancy is an operation of the imagination.

Our physical and mental powers, æsthetically considered, may therefore be classed under three heads, in their relation to the fine arts, viz., the receptive, the perceptive, and the conceptive.

The senses of hearing and seeing are respectively, in the degree of their physical power, receptive of impressions made upon them, and of these impressions the sensorium, in the degree of its mental power, is perceptive. This perception enables the mind to form a judgment whereby it appreciates the nature and quality of the impression originally made on the receptive organ. The mode of this operation is intuitive, and the quickness and accuracy with which the nature and quality of the impression is apprehended, will be in the degree of the intellectual vigour of the mind by which it is perceived. Thus we are, by the cultivation of these intuitive faculties, enabled to decide with accuracy as to harmony or discord, proportion or deformity, and assign sound reasons for our judgment in matters of taste. But mental conception is the intuitive power of constructing original ideas from these materials; for after the receptive power has acted, the perception operates in establishing facts, and then the judgment is formed upon these operations by the reasoning powers, which lead, in their turn, to the creations of the imagination.

The power of forming these creations is the true characteristic of genius, and determines the point at which art is placed beyond all determinable canons,—at which, indeed, æsthetics give place to metaphysics.

In the science of beauty, therefore, the human mind is the subject, and the effect of external nature, as well as of works of art, the object. The external world, and the individual mind, with all that lies within the scope of its powers, may be considered as two separate existences, having a distinct relation to each other. The subject is affected by the object, through that inherent faculty by which it is enabled to respond to every development of the all-governing harmonic law of nature; and the media of communication are the sensorium and its inlets—the organs of sense.

This harmonic law of nature was either originally discovered by that illustrious philosopher Pythagoras, upwards of five hundred years before Christ, or a knowledge of it obtained by him about that period, from the Egyptian or Chaldean priests. For after having been initiated into all the Grecian and barbarian sacred mysteries, he went to Egypt, where he remained upwards of twenty years, studying in the colleges of its priests; and from Egypt he went into the East, and visited the Persian and Chaldean magi.[3]

By the generality of the biographers of Pythagoras, it is said to be difficult to give a clear idea of his philosophy, as it is almost certain he never committed it to writing, and that it has been disfigured by the fantastic dreams and chimeras of later Pythagoreans. Diogenes Laërtius, however, whose “Lives of the Philosophers” was supposed to be written about the end of the second century of our era, says “there are three volumes extant written by Pythagoras. One on education, one on politics, and one on natural philosophy.” And adds, that there were several other books extant, attributed to Pythagoras, but which were not written by him. Also, in his “Life of Philolaus,” that Plato wrote to Dion to take care and purchase the books of Pythagoras.[4] But whether this great philosopher committed his discoveries to writing or not, his doctrines regarding the philosophy of beauty are well-known to be, that he considered numbers as the essence and the principle of all things, and attributed to them a real and distinct existence; so that, in his view, they were the elements out of which the universe was constructed, and to which it owed its beauty. Diogenes Laërtius gives the following account of this law:—“That the monad was the beginning of everything. From the monad proceeds an indefinite duad, which is subordinate to the monad as to its cause. That from the monad and indefinite duad proceeds numbers. That the part of science to which Pythagoras applied himself above all others, was arithmetic; and that he taught ‘that from numbers proceed signs, and from these latter, lines, of which plane figures consist; that from plane figures are derived solid bodies; that of all plane figures the most beautiful was the circle, and of all solid bodies the most beautiful was the sphere.’ He discovered the numerical relations of sounds on a single string; and taught that everything owes its existence and consistency to harmony. In so far as I know, the most condensed account of all that is known of the Pythagorian system of numbers is the following:—‘The monad or unity is that quantity, which, being deprived of all number, remains fixed. It is the fountain of all number. The duad is imperfect and passive, and the cause of increase and division. The triad, composed of the monad and duad, partakes of the nature of both. The tetrad, tetractys, or quaternion number is most perfect. The decad, which is the sum of the four former, comprehends all arithmetical and musical proportions.’”[5]

These short quotations, I believe, comprise all that is known, for certain, of the manner in which Pythagoras systematised the law of numbers. Yet, from the teachings of this great philosopher and his disciples, the harmonic law of nature, in which the fundamental principles of beauty are embodied, became so generally understood and universally applied in practice throughout all Greece, that the fragments of their works, which have reached us through a period of two thousand years, are still held to be examples of the highest artistic excellence ever attained by mankind. In the present state of art, therefore, a knowledge of this law, and of the manner in which it may again be applied in the production of beauty in all works of form and colour, must be of singular advantage; and the object of this work is to assist in the attainment of such a knowledge.

It has been remarked, with equal comprehensiveness and truth, by a writer[6] in the _British and Foreign Medical Review_, that “there is harmony of numbers in all nature—in the force of gravity—in the planetary movements—in the laws of heat, light, electricity, and chemical affinity—in the forms of animals and plants—in the perceptions of the mind. The direction, indeed, of modern natural and physical science is towards a generalization which shall express the fundamental laws of all by one simple numerical ratio. And we think modern science will soon shew that the mysticism of Pythagoras was mystical only to the unlettered, and that it was a system of philosophy founded on the then existing mathematics, which latter seem to have comprised more of the philosophy of numbers than our present.” Many years of careful investigation have convinced me of the truth of this remark, and of the great advantage derivable from an application of the Pythagorean system in the arts of design. For so simple is its nature, that any one of an ordinary capacity of mind, and having a knowledge of the most simple rules of arithmetic, may, in a very short period, easily comprehend its nature, and be able to apply it in practice.

The elements of the Pythagorean system of harmonic number, so far as can be gathered from the quotations I have given above, seem to be simply the indivisible monad (1); the duad (2), arising from the union of one monad with another; the triad (3), arising from the union of the monad with the duad; and the tetrad (4), arising from the union of one duad with another, which tetrad is considered a perfect number. From the union of these four elements arises the decad (10), the number, which, agreeably to the Pythagorean system, comprehends all arithmetical and harmonic proportions. If, therefore, we take these elements and unite them progressively in the following order, we shall find the series of harmonic numbers (2), (3), (5), and (7), which, with their multiples, are the complete numerical elements of all harmony, thus:—

1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 4 = 7

In order to render an extended series of harmonic numbers useful, it must be divided into scales; and it is a rule in the formation of these scales, that the first must begin with the monad (1) and end with the duad (2), the second begin with the duad (2) and end with the tetrad (4), and that the beginning and end of all other scales must be continued in the same arithmetical progression. These primary elements will then form the foundation of a series of such scales.

I. (1) (2)
II. (2) (3) (4)
III. (4) (5) (6) (7) (8)
IV. (8) (9) (10) ( ) (12) ( ) (14) (15) (16)

The first of these scales has in (1) and (2) a beginning and an end; but the second has in (2), (3), and (4) the essential requisites demanded by Aristotle in every composition, viz., “a beginning, a middle, and an end;” while the third has not only these essential requisites, but two intermediate parts (5) and (7), by which the beginning, the middle, and the end are united. In the fourth scale, however, the arithmetical progression is interrupted by the omission of numbers 11 and 13, which, not being multiples of either (2), (3), (5), or (7), are inadmissible.

Such is the nature of the harmonic law which governs the progressive scales of numbers by the simple multiplication of the monad.

I shall now use these numbers as divisors in the formation of a series of four such scales of parts, which has for its primary element, instead of the indivisible monad, a quantity which may be indefinitely divided, but which cannot be added to or multiplied. Like the monad, however, this quantity is represented by (1). The following is this series of four scales of harmonic parts:—

I. (1) (¹⁄₂)
II. (¹⁄₂) (¹⁄₃) (¹⁄₄)
III. (¹⁄₄) (¹⁄₅) (¹⁄₆) (¹⁄₇) (¹⁄₈)
IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) ( ) (¹⁄₁₂) ( ) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)

The scales I., II., and III. may now be rendered as complete as scale IV., simply by multiplying upwards by 2 from (¹⁄₉), (¹⁄₅), (¹⁄₃), (¹⁄₇), and (¹⁄₁₅), thus:—

I. (1) (⁸⁄₉) (⁴⁄₅) (²⁄₃) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)
II. (¹⁄₂) (⁴⁄₉) (²⁄₅) (¹⁄₃) (²⁄₇) (⁴⁄₁₅) (¹⁄₄)
III. (¹⁄₄) (²⁄₉) (¹⁄₅) (¹⁄₆) (¹⁄₇) (²⁄₁₅) (¹⁄₈)
IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) ( ) (¹⁄₁₂) ( ) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)

We now find between the beginning and the end of scale I. the quantities (⁸⁄₉), (⁴⁄₅), (²⁄₃), (⁴⁄₇), and (⁸⁄₁₅).

The three first of these quantities we find to be the remainders of the whole indefinite quantity contained in (1), after subtracting from it the primary harmonic quantities (¹⁄₉), (¹⁄₅), and (¹⁄₃); we, however, find also amongst these harmonic quantities that of (¹⁄₄), which being subtracted from (1) leaves (³⁄₄), a quantity the most suitable whereby to fill up the hiatus between (⁴⁄₅) and (²⁄₃) in scale I., which arises from the omission of (¹⁄₁₁) in scale IV. In like manner we find the two last of these quantities, (⁴⁄₇) and (⁸⁄₁₅), are respectively the largest of the two parts into which 7 and 15 are susceptible of being divided. Finding the number 5 to be divisible into parts more unequal than (2) to (3) and less unequal than (4) to (7), (³⁄₅) naturally fills up the hiatus between these quantities in scale I., which hiatus arises from the omission of (¹⁄₁₃) in scale IV. Thus:—

I. (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)
II. (¹⁄₂) (⁴⁄₉) (²⁄₅) ( ) (¹⁄₃) ( ) (²⁄₇) (⁴⁄₁₅) (¹⁄₄)
III. (¹⁄₄) (²⁄₉) (¹⁄₅) ( ) (¹⁄₆) ( ) (¹⁄₇) (²⁄₁₅) (¹⁄₈)
IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) ( ) (¹⁄₁₂) ( ) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)

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

I. (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)
II. (¹⁄₂) (⁴⁄₉) (²⁄₅) (³⁄₈) (¹⁄₃) (³⁄₁₀) (²⁄₇) (⁴⁄₁₅) (¹⁄₄)
III. (¹⁄₄) (²⁄₉) (¹⁄₅) (³⁄₁₆) (¹⁄₆) (³⁄₂₀) (¹⁄₇) (²⁄₁₅) (¹⁄₈)
IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) (³⁄₃₂) (¹⁄₁₂) (³⁄₄₀) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)

The harmony existing amongst these numbers or quantities consists of the numerical relations which the parts bear to the whole and to each other; and the more simple these relations are, the more perfect is the harmony. The following are the numerical harmonic ratios which the parts bear to the whole:—

I. (1:1) (8:9) (4: 5) (3: 4) (2: 3) (3: 5) (4: 7) (8:15) (1: 2)
II. (1:2) (4:9) (2: 5) (3: 8) (1: 3) (3:10) (2: 7) (4:15) (1: 4)
III. (1:4) (2:9) (1: 5) (3:16) (1: 6) (3:20) (1: 7) (2:15) (1: 8)
IV. (1:8) (1:9) (1:10) (3:32) (1:12) (3:40) (1:14) (1:15) (1:16)

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

(¹⁄₂):(⁴⁄₇) = (7:8)
(⁴⁄₅):(⁸⁄₉) = (9:10)
(²⁄₃):(⁴⁄₅) = (5:6)
(⁴⁄₇):(²⁄₃) = (6:7)
(⁸⁄₁₅):(⁴⁄₇) = (14:15)
(¹⁄₂):(⁸⁄₁₅) = (15:16)

Although these relations are exemplified by parts of scale I., the same ratios exist between the relative parts of scales II., III., and IV., and would exist between the parts of any other scales that might be added to that series.

These are the simple elements of the science of that harmony which pervades the universe, and by which the various kinds of beauty æsthetically impressed upon the senses of hearing and seeing are governed.

THE SCIENCE OF BEAUTY AS APPLIED TO SOUNDS.

It is well-known that all sounds arise from a peculiar action of the air, and that this action may be excited by the concussion resulting from the sudden displacement of a portion of the atmosphere itself, or by the rapid motions of bodies, or of confined columns of air; in all which cases, when the motions are irregular, and the force great, the sound conveyed to the sensorium is called a noise. But that musical sounds are the result of equal and regular vibratory motions, either of an elastic body, or of a column of air in a tube, exciting in the surrounding atmosphere a regular and equal pulsation. The ear is the medium of communication between those varieties of atmospheric action and the seat of consciousness. To describe fully the beautiful arrangement of the various parts of this organ, and their adaptation to the purpose of collecting and conveying these undulatory motions of the atmosphere, is as much beyond the scope of my present attempt as it is beyond my anatomical knowledge; but I may simply remark, that within the ear, and most carefully protected in the construction of that organ, there is a small cavity containing a pellucid fluid, in which the minute extremities of the auditory nerve float; and that this fluid is the last of the media through which the action producing the sensation of sound is conveyed to the nerve, and thence to the sensorium, where its nature becomes perceptible to the mind.

The impulses which produce musical notes must arrive at a certain frequency before the ear loses the intervals of silence between them, and is impressed by only one continued sound; and as they increase in frequency the sound becomes more acute upon the ear. The pitch of a musical note is, therefore, determined by the frequency of these impulses; but, on the other hand, its intensity or loudness will depend upon the violence and the quality of its tone on the material employed in producing them. All such sounds, therefore, whatever be their loudness or the quality of their tone in which the impulses occur with the same frequency are in perfect unison, having the same pitch. Upon this the whole doctrine of harmonies is founded, and by this the laws of numerical ratio are found to operate in the production of harmony, and the theory of music rendered susceptible of exact reasoning.

The mechanical means by which such sounds can be produced are extremely various; but, as it is my purpose simply to shew the nature of harmony of sound as related to, or as evolving numerical harmonic ratio, I shall confine myself to the most simple mode of illustration—namely, that of the monochord. This is an instrument consisting of a string of a given length stretched between two bridges standing upon a graduated scale. Suppose this string to be stretched until its tension is such that, when drawn a little to a side and suddenly let go, it would vibrate at the rate of 64 vibrations in a second of time, producing to a certain distance in the surrounding atmosphere a series of pulsations of the same frequency.

These pulsations will communicate through the ear a musical note which would, therefore, be the fundamental note of such a string. Now, the phenomenon said to be discovered by Pythagoras is well known to those acquainted with the science of acoustics, namely, that immediately after the string is thus put into vibratory motion, it spontaneously divides itself, by a node, into two equal parts, the vibrations of each of which occur with a double frequency—namely, 128 in a second of time, and, consequently, produce a note doubly acute in pitch, although much weaker as to intensity or loudness; that it then, while performing these two series of vibrations, divides itself, by two nodes, into three parts, each of which vibrates with a frequency triple that of the whole string; that is, performs 192 vibrations in a second of time, and produces a note corresponding in increase of acuteness, but still less intense than the former, and that this continues to take place in the arithmetical progression of 2, 3, 4, &c. Simultaneous vibrations, agreeably to the same law of progression, which, however, seem to admit of no other primes than the numbers 2, 3, 5, and 7, are easily excited upon any stringed instrument, even by the lightest possible touch of any of its strings while in a state of vibratory motion, and the notes thus produced are distinguished by the name of harmonics. It follows, then, that one-half of a musical string, when divided from the whole by the pressure of the finger, or any other means, and put into vibratory motion, produces a note doubly acute to that produced by the vibratory motion of the whole string; the third part, similarly separated, a note trebly acute; and the same with every part into which any musical string may be divided. This is the fundamental principle by which all stringed instruments are made to produce harmony. It is the same with wind instruments, the sounds of which are produced by the frequency of the pulsations occasioned in the surrounding atmosphere by agitating a column of air confined within a tube as in an organ, in which the frequency of pulsation becomes greater in an inverse ratio to the length of the pipes. But the following series of four successive scales of musical notes will give the reader a more comprehensive view of the manner in which they follow the law of numerical ratio just explained than any more lengthened exposition.

It is here requisite to mention, that in the construction of these scales, I have not only adopted the old German or literal mode of indicating the notes, but have included, as the Germans do, the note termed by us B flat as B natural, and the note we term B natural as H. Now, although this arrangement differs from that followed in the construction of our modern Diatonic scale, yet as the ratio of 4:7 is more closely related to that of 1:2 than that of 8:15, and as it is offered by nature in the spontaneous division of the monochord, I considered it quite admissible. The figures give the parts of the monochord which would produce the notes.

I. { (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)*
{ C D E F G A B H _c_

II. { (¹⁄₂)* (⁴⁄₉) (²⁄₅) (³⁄₈) (¹⁄₃)* (³⁄₁₀) (²⁄₇) (²⁄₁₅) (¹⁄₄)*
{ _c_ _d_ _e_ _f_ _g_ _a_ _b_ _h_ _c′_

III. { (¹⁄₄)* (²⁄₉) (¹⁄₅)* (³⁄₁₆) (¹⁄₆)* (³⁄₂₀) (¹⁄₇)* (²⁄₁₅) (¹⁄₈)*
{ _c′_ _d′_ _e′_ _f′_ _g′_ _a′_ _b′_ _h′_ _c′′_

IV. { (¹⁄₈)* (¹⁄₉)* (¹⁄₁₀)* (³⁄₃₂) (¹⁄₁₂)* (³⁄₄₀) (¹⁄₁₄)* (¹⁄₁₅)* (¹⁄₁₆)*
{ _c′′_ _d′′_ _e′′_ _f′′_ _g′′_ _a′′_ _b′′_ _h′′_ _c′′′_

The notes marked (*) are the harmonics which naturally arise from the division of the string by 2, 3, 5, and 7, and the multiples of these primes.