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diff --git a/75399-0.txt b/75399-0.txt new file mode 100644 index 0000000..e96d7c4 --- /dev/null +++ b/75399-0.txt @@ -0,0 +1,4191 @@ + +*** START OF THE PROJECT GUTENBERG EBOOK 75399 *** + + + + + +THE SCIENCE OF BEAUTY. + + EDINBURGH: + PRINTED BY BALLANTYNE AND COMPANY, + PAUL’S WORK. + + + + + THE + SCIENCE OF BEAUTY, + AS DEVELOPED IN NATURE AND + APPLIED IN ART. + + BY + D. R. HAY, F.R.S.E. + + “The irregular combinations of fanciful invention may delight + awhile, by that novelty of which the common satiety of life + sends us all in quest; the pleasures of sudden wonder are soon + exhausted, and the mind can only repose on the stability of + truth.” + + DR JOHNSON. + + WILLIAM BLACKWOOD AND SONS, + EDINBURGH AND LONDON. + MDCCCLVI. + + + + + TO + JOHN GOODSIR, ESQ., F.R.S S. L. & E., + PROFESSOR OF ANATOMY IN THE UNIVERSITY OF EDINBURGH, + AS AN EXPRESSION OF GRATITUDE FOR VALUABLE ASSISTANCE, + AS ALSO OF HIGH ESTEEM AND SINCERE REGARD, + THIS VOLUME IS DEDICATED, + BY + + THE AUTHOR. + + + + +PREFACE. + + +My theory of beauty in form and colour being now admitted by the best +authorities to be based on truth, I have of late been often asked, by +those who wished to become acquainted with its nature, and the manner of +its being applied in art, which of my publications I would recommend for +their perusal. This question I have always found difficulty in answering; +for although the law upon which my theory is based is characterised by +unity, yet the subjects in which it is applied, and the modes of its +application, are equally characterised by variety, and consequently +occupy several volumes. + +Under these circumstances, I consulted a highly respected friend, whose +mathematical talents and good taste are well known, and to whom I have +been greatly indebted for much valuable assistance during the course +of my investigations. The advice I received on this occasion, was to +publish a _résumé_ of my former works, of such a character as not only +to explain the nature of my theory, but to exhibit to the general reader, +by the most simple modes of illustration and description, how it is +developed in nature, and how it may be extensively and with ease applied +in those arts in which beauty forms an essential element. + +The following pages, with their illustrations, are the results of an +attempt to accomplish this object. + +To those who are already acquainted, through my former works, with +the nature, scope, and tendency of my theory, I have the satisfaction +to intimate that I have been enabled to include in this _résumé_ much +original matter, with reference both to form and colour. + + D. R. HAY. + + + + +CONTENTS. + + + PAGE + + INTRODUCTION 1 + + THE SCIENCE OF BEAUTY, EVOLVED FROM THE HARMONIC LAW OF NATURE, + AGREEABLY TO THE PYTHAGOREAN SYSTEM OF NUMERICAL RATIO 15 + + THE SCIENCE OF BEAUTY, AS APPLIED TO SOUNDS 28 + + THE SCIENCE OF BEAUTY, AS APPLIED TO FORMS 34 + + THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE FORM OF THE HUMAN HEAD + AND COUNTENANCE 54 + + THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE FORM OF THE HUMAN FIGURE 61 + + THE SCIENCE OF BEAUTY, AS DEVELOPED IN COLOURS 67 + + THE SCIENCE OF BEAUTY APPLIED TO THE FORMS AND PROPORTIONS OF + ANCIENT GRECIAN VASES AND ORNAMENTS 82 + + APPENDIX, NO. I. 91 + + APPENDIX, NO. II. 99 + + APPENDIX, NO. III. 100 + + APPENDIX, NO. IV. 100 + + APPENDIX, NO. V. 104 + + APPENDIX, NO. VI. 105 + + + + +ILLUSTRATIONS. + + +PLATES + +[Illustration: I. Three Scales of the Elementary Rectilinear Figures, +viz., the Scalene Triangle, the Isosceles Triangle, and the Rectangle, +comprising twenty-seven varieties of each, according to the harmonic +parts of the Right Angle from ¹⁄₂ to ¹⁄₁₆. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: II. Diagram of the Rectilinear Orthography of the +Principal Front of the Parthenon of Athens, in which its Proportions are +determined by harmonic parts of the Right Angle. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: III. Diagram of the Rectilinear Orthography of the +Portico of the Temple of Theseus at Athens, in which its Proportions are +determined by harmonic parts of the Right Angle. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: IV. Diagram of the Rectilinear Orthography of the East End +of Lincoln Cathedral, in which its Proportions are determined by harmonic +parts of the Right Angle. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: V. Four Ellipses described from Foci, determined by +harmonic parts of the Right Angle, shewing in each the Scalene Triangle, +the Isosceles Triangle, and the Rectangle to which it belongs. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: VI. The Composite Ellipse of ¹⁄₆ and ¹⁄₈ of the Right +Angle, shewing its greater and lesser Axis, its various Foci, and the +Isosceles Triangle in which they are placed. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: VII. The Composite Ellipse of ¹⁄₄₈ and ¹⁄₆₄ of the Right +Angle, shewing how it forms the Entasis of the Columns of the Parthenon +of Athens. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: VIII. Sectional Outlines of two Mouldings of the Parthenon +of Athens, full size, shewing the harmonic nature of their Curves, and +the simple manner of their Construction. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: IX. Three Diagrams, giving a Vertical, a Front, and a Side +Aspect of the Geometrical Construction of the Human Head and Countenance, +in which the Proportions are determined by harmonic parts of the Right +Angle. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: X. Diagram in which the Symmetrical Proportions of the +Human Figure are determined by harmonic parts of the Right Angle. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: XI. The Contour of the Human Figure as viewed in Front +and in Profile, its Curves being determined by Ellipses, whose Foci are +determined by harmonic parts of the Right Angle. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: XII. Rectilinear Diagram, shewing the Proportions of the +Portland Vase, as determined by harmonic parts of the Right Angle, and +the outline of its form by an Elliptic Curve harmonically described. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: XIII. Rectilinear Diagram of the Proportions and +Curvilinear Outline of the form of an ancient Grecian Vase, the +proportions determined by harmonic parts of the Right Angle, and the +melody of the form by Curves of two Ellipses. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: XIV. Rectilinear Diagram of the Proportions and +Curvilinear Outline of the form an ancient Grecian Vase, the proportions +determined by harmonic parts of the Right Angle, and the melody of the +form by an Elliptic Curve. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: XV. Two Diagrams of Etruscan Vases, the harmony of +Proportions and melody of the Contour determined, respectively, by parts +of the Right Angle and an Elliptic Curve. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: XVI. Two Diagrams of Etruscan Vases, whose harmony of +Proportion and melody of Contour are determined as above. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: XVII. Diagram shewing the Geometric Construction of an +Ornament belonging to the Parthenon at Athens. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: XVIII. Diagram of the Geometrical Construction of the +ancient Grecian Ornament called the Honeysuckle. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: XIX. An additional Illustration of the Contour of the +Human Figure, as viewed in Front and in Profile. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: XX. Diagram shewing the manner in which the Elliptic +Curves are arranged in order to produce an Outline of the Form of the +Human Figure as viewed in Front. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: XXI. Diagram of a variation on the Form of the Portland +Vase. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: XXII. Diagram of a second variation on the Form of the +Portland Vase. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + +[Illustration: XXIII. Diagram of a third variation on the Form of the +Portland Vase. + +_D. R. Hay delᵗ._ _G. Aikman sc._] + + + + +INTRODUCTION. + + +Twelve years ago, one of our most eminent philosophers,[1] through the +medium of the _Edinburgh Review_,[2] gave the following account of what +was then the state of the fine arts as connected with science:—“The +disposition to introduce into the intellectual community the principles +of free intercourse, is by no means general; but we are confident that +Art will not sufficiently develop her powers, nor Science attain her +most commanding position, till the practical knowledge of the one is +taken in return for the sound deductions of the other.... It is in the +fine arts, principally, and in the speculations with which they are +associated, that the controlling power of scientific truth has not +exercised its legitimate influence. In discussing the principles of +painting, sculpture, architecture, and landscape gardening, philosophers +have renounced science as a guide, and even as an auxiliary; and a +school has arisen whose speculations will brook no restraint, and whose +decisions stand in opposition to the strongest convictions of our +senses. That the external world, in its gay colours and lovely forms, is +exhibited to the mind only as a tinted mass, neither within nor without +the eye, neither touching it nor distant from it—an ubiquitous chaos, +which experience only can analyse and transform into the realities +which compose it; that the beautiful and sublime in nature and in art +derive their power over the mind from association alone, are among the +philosophical doctrines of the present day, which, if it be safe, it is +scarcely prudent to question. Nor are these opinions the emanations of +poetical or ill-trained minds, which ingenuity has elaborated, and which +fashion sustains. They are conclusions at which most of our distinguished +philosophers have arrived. They have been given to the world with all the +authority of demonstrated truth; and in proportion to the hold which they +have taken of the public mind, have they operated as a check upon the +progress of knowledge.” + +Such, then, was the state of art as connected with science twelve years +ago. But although the causes which then placed science and the fine arts +at variance have since been gradually diminishing, yet they are still +far from being removed. In proof of this I may refer to what took place +at the annual distribution of the prizes to the students attending our +Scottish Metropolitan School of Design, in 1854, the pupils in which +amount to upwards of two hundred. The meeting on that occasion included, +besides the pupils, a numerous and highly respectable assemblage of +artists and men of science. The chairman, a Professor in our University, +and editor of one of the most voluminous works on art, science, and +literature ever produced in this country, after extolling the general +progress of the pupils, so far as evinced by the drawings exhibited on +the occasion, drew the attention of the meeting to a discovery made by +the head master of the architectural and ornamental department of the +school, viz.—That the ground-plan of the Parthenon at Athens had been +constructed by the application of the _mysterious_ ovoid or _Vesica +Piscis_ of the middle ages, subdivided by the _mythic_ numbers 3 and 7, +and their intermediate odd number 5. Now, it may be remarked, that the +figure thus referred to is not an ovoid, neither is it in any way of a +mysterious nature, being produced simply by two equal circles cutting +each other in their centres. Neither can it be shewn that the numbers 3 +and 7 are in any way more mythic than other numbers. In fact, the terms +_mysterious_ and _mythic_ so applied, can only be regarded as a remnant +of an ancient terminology, calculated to obscure the simplicity of +scientific truth, and when used by those employed to teach—for doubtless +the chairman only gave the description he received—must tend to retard +the connexion of that truth with the arts of design. I shall now give +a specimen of the manner in which a knowledge of the philosophy of the +fine arts is at present inculcated upon the public mind generally. In +the same metropolis there has likewise existed for upwards of ten years +a Philosophical Institution of great importance and utility, whose +members amount to nearly three thousand, embracing a large proportion +of the higher classes of society, both in respect to talent and wealth. +At the close of the session of this Institution, in 1854, a learned and +eloquent philologus, who occasionally lectures upon beauty, was appointed +to deliver the closing address, and touching upon the subject of the +beautiful, he thus concluded— + +“In the worship of the beautiful, and in that alone, we are inferior to +the Greeks. Let us therefore be glad to borrow from them; not slavishly, +but with a wise adaptation—not exclusively, but with a cunning selection; +in art, as in religion, let us learn to prove all things, and hold fast +that which is good—not merely one thing which is good, but all good +things—Classicalism, Mediævalism, Modernism—let us have and hold them +all in one wide and lusty embrace. Why should the world of art be +more narrow, more monotonous, than the world of nature? Did God make +all the flowers of one pattern, to please the devotees of the rose or +the lily; and did He make all the hills, with the green folds of their +queenly mantles, all at one slope, to suit the angleometer of the most +mathematical of decorators? I trow not. Let us go and do likewise.” + +I here take for granted, that what the lecturer meant by “the worship of +the beautiful,” is the production and appreciation of works of art in +which beauty should be a primary element; and judging from the remains +which we possess of such works as were produced by the ancient Grecians, +our inferiority to them in these respects cannot certainly be denied. +But I must reiterate what I have often before asserted, that it is +not by borrowing from them, however cunning our selection, or however +wise our adaptations, that this inferiority is to be removed, but by +a re-discovery of the science which these ancient artists must have +employed in the production of that symmetrical beauty and chaste elegance +which pervaded all their works for a period of nearly three hundred +years. And I hold, that as in religion, so in art, there is only one +truth, a grain of which is worth any amount of philological eloquence. + +I also take for granted, that what is meant by Classicalism in the +above quotation, is the ancient Grecian style of art; by Mediævalism, +the semi-barbaric style of the middle ages; and by Modernism, that +chaotic jumble of all previous styles and fashions of art, which is the +peculiar characteristic of our present school, and which is, doubtless, +the result of a system of education based upon plagiarism and mere +imitation. Therefore a recommendation to embrace with equal fervour “as +good things,” these very opposite artic_isms_ must be a doctrine as +mischievous in art as it would be in religion to recommend as equally +good things the various _isms_ into which it has also been split in +modern times. + +Now, “the world of nature” and “the world of art” have not that equality +of scope which this lecturer on beauty ascribes to them, but differ very +decidedly in that particular. Neither will it be difficult to shew why +“the world of art _should_ be more narrow than the world of nature”—that +it should be thereby rendered more monotonous does not follow. + +It is well known, that the “world of nature” consists of productions, +including objects of every degree of beauty from the very lowest to the +highest, and calculated to suit not only the tastes arising from various +degrees of intellect, but those arising from the natural instincts of the +lower animals. On the other hand, “the world of art,” being devoted to +the gratification and improvement of intelligent minds only, is therefore +narrowed in its scope by the exclusion from its productions of the lower +degrees of beauty—even mediocrity is inadmissible; and we know that the +science of the ancient Greek artists enabled them to excel the highest +individual productions of nature in the perfection of symmetrical beauty. +Consequently, all objects in nature are not equally well adapted for +artistic study, and it therefore requires, on the part of the artist, +besides true genius, much experience and care to enable him to choose +proper subjects from nature; and it is in the choice of such subjects, +and not in plagiarism from the ancients, that he should select with +knowledge and adapt with wisdom. Hence, all such latitudinarian doctrines +as those I have quoted must act as a check upon the progress of knowledge +in the scientific truth of art. I have observed in some of my works, that +in this country a course had been followed in our search for the true +science of beauty not differing from that by which the alchymists of the +middle ages conducted their investigations; for our ideas of visible +beauty are still undefined, and our attempts to produce it in the various +branches of art are left dependant, in a great measure, upon chance. +Our schools are conducted without reference to any first principles or +definite laws of beauty, and from the drawing of a simple architectural +moulding to the intricate combinations of form in the human figure, the +pupils trust to their hands and eyes alone, servilely and mechanically +copying the works of the ancients, instead of being instructed in the +unerring principles of science, upon which the beauty of those works +normally depends. The instruction they receive is imparted without +reference to the judgment or understanding, and they are thereby led to +imitate effects without investigating causes. Doubtless, men of great +genius sometimes arrive at excellence in the arts of design without a +knowledge of the principles upon which beauty of form is based; but it +should be kept in mind, that true genius includes an intuitive perception +of those principles along with its creative power. It is, therefore, to +the generality of mankind that instruction in the definite laws of beauty +will be of most service, not only in improving the practice of those who +follow the arts professionally, but in enabling all of us to distinguish +the true from the false, and to exercise a sound and discriminating +taste in forming our judgment upon artistic productions. Æsthetic +culture should consequently supersede servile copying, as the basis of +instruction in our schools of art. Many teachers of drawing, however, +still assert, that, by copying the great works of the ancients, the mind +of the pupil will become imbued with ideas similar to theirs—that he +will imbibe their feeling for the beautiful, and thereby become inspired +with their genius, and think as they thought. To study carefully and +to investigate the principles which constitute the excellence of the +works of the ancients, is no doubt of much benefit to the student; but +it would be as unreasonable to suppose that he should become inspired +with artistic genius by merely copying them, as it would be to imagine, +that, in literature, poetic inspiration could be created by making boys +transcribe or repeat the works of the ancient poets. Sir Joshua Reynolds +considered copying as a delusive kind of industry, and has observed, that +“Nature herself is not to be too closely copied,” asserting that “there +are excellences in the art of painting beyond what is commonly called the +imitation of nature,” and that “a mere copier of nature can never produce +any thing great.” Proclus, an eminent philosopher and mathematician of +the later Platonist school (A.D. 485), says, that “he who takes for his +model such forms as nature produces, and confines himself to an exact +imitation of these, will never attain to what is perfectly beautiful. For +the works of nature are full of disproportion, and fall very short of the +true standard of beauty.” + +It is remarked by Mr. J. C. Daniel, in the introduction to his +translation of M. Victor Cousin’s “Philosophy of the Beautiful,” that +“the English writers have advocated no theory which allows the beautiful +to be universal and absolute; nor have they professedly founded their +views on original and ultimate principles. Thus the doctrine of the +English school has for the most part been, that beauty is mutable and +special, and the inference that has been drawn from this teaching is, +that all tastes are equally just, provided that each man speaks of what +he feels.” He then observes, that the German, and some of the French +writers, have thought far differently; for with them the beautiful is +“simple, immutable, absolute, though its _forms_ are manifold.” + +So far back as the year 1725, the same truths advanced by the modern +German and French writers, and so eloquently illustrated by M. Cousin, +were given to the world by Hutchison in his “Inquiry into the Original +of our Ideas of Beauty and Virtue.” This author says—“We, by absolute +beauty, understand only that beauty which we perceive in objects, without +comparison to any thing external, of which the object is supposed an +imitation or picture, such as the beauty perceived from the works of +nature, artificial forms, figures, theorems. Comparative or relative +beauty is that which we perceive in objects commonly considered as +imitations or resemblances of something else.” + +Dr. Reid also, in his “Intellectual Powers of Man,” says—“That taste, +which we may call rational, is that part of our constitution by which we +are made to receive pleasure from the contemplation of what we conceive +to be excellent in its kind, the pleasure being annexed to this judgment, +and regulated by it. This taste may be true or false, according as it is +founded on a true or false judgment. And if it may be true or false, it +must have first principles.” + +M. Victor Cousin’s opinion upon this subject is, however, still more +conclusive. He observes—“If the idea of the beautiful is not absolute, +like the idea of the true—if it is nothing more than the expression of +individual sentiment, the rebound of a changing sensation, or the result +of each person’s fancy—then the discussions on the fine arts waver +without support, and will never end. For a theory of the fine arts to +be possible, there must be something absolute in beauty, just as there +must be something absolute in the idea of goodness, to render morals a +possible science.” + +The basis of the science of beauty must thus be founded upon fixed +principles, and when these principles are evolved with the same care +which has characterised the labours of investigators in natural science, +and are applied in the fine arts as the natural sciences have been in +the useful arts, a solid foundation will be laid, not only for correct +practice, but also for a just appreciation of productions in every branch +of the arts of design. + +We know that the mind receives pleasure through the sense of hearing, +not only from the music of nature, but from the euphony of prosaic +composition, the rhythm of poetic measure, the artistic composition +of successive harmony in simple melody, and the combined harmony of +counterpoint in the more complex works of that art. We know, also, +that the mind is similarly gratified through the sense of seeing, not +only by the visible beauties of nature, but by those of art, whether +in symmetrical or picturesque compositions of forms, or in harmonious +arrangements of gay or sombre colouring. + +Now, in respect to the first of these modes of sensation, we know, +that from the time of Pythagoras, the fact has been established, that +in whatever manner nature or art may address the ear, the degree of +obedience paid to the fundamental law of harmony will determine the +presence and degree of that beauty with which a perfect organ can impress +a well-constituted mind; and it is my object in this, as it has been +in former attempts, to prove it consistent with scientific truth, that +that beauty which is addressed to the mind by objects of nature and art, +through the eye, is similarly governed. In short, to shew that, as in +compositions of sounds, there can be no true beauty in the absence of a +strict obedience to this great law of nature, neither can there exist, +in compositions of forms or colours, that principle of unity in variety +which constitutes beauty, unless such compositions are governed by the +same law. + +Although in the songs of birds, the gurgling of brooks, the sighing of +the gentle summer winds, and all the other beautiful music of nature, +no analysis might be able to detect the operation of any precise system +of harmony, yet the pleasure thus afforded to the human mind we know to +arise from its responding to every development of an obedience to this +law. When, in like manner, we find even in those compositions of forms +and colours which constitute the wildest and most rugged of Nature’s +scenery, a species of picturesque grandeur and beauty to which the mind +as readily responds as to her more mild and pleasing aspects, or to her +sweetest music, we may rest assured that this beauty is simply another +development of, and response to, the same harmonic law, although the +precise nature of its operation may be too subtle to be easily detected. + +The _résumé_ of the various works I have already published upon the +subject, along with the additional illustrations I am about to lay before +my readers, will, I trust, point out a system of harmony, which, in +formative art, as well as in that of colouring, will rise superior to the +idiosyncracies of different artists, and bring back to one common type +the sensations of the eye and the ear, thereby improving that knowledge +of the laws of the universe which it is as much the business of science +to combine with the ornamental as with the useful arts. + +In attempting this, however, I beg it may be understood, that I do not +believe any system, based even upon the laws of nature, capable of +forming a royal road to the perfection of art, or of “mapping the mighty +maze of a creative mind.” At the same time, however, I must continue to +reiterate the fact, that the diffusion of a general knowledge of the +science of visible beauty will afford latent artistic genius just such a +vantage ground as that which the general knowledge of philology diffused +throughout this country affords its latent literary genius. Although +_mere learning_ and _true genius_ differ as much in the practice of art +as they do in the practice of literature, yet a precise and systematic +education in the true science of beauty must certainly be as useful in +promoting the practice and appreciation of the one, as a precise and +systematic education in the science of philology is in promoting the +practice and appreciation of the other. + +As all beauty is the result of harmony, it will be requisite here to +remark, that harmony is not a simple quality, but, as Aristotle defines +it, “the union of contrary principles having a ratio to each other.” +Harmony thus operates in the production of all that is beautiful in +nature, whether in the combinations, in the motions, or in the affinities +of the elements of matter. + +The contrary principles to which Aristotle alludes, are those of +uniformity and variety; for, according to the predominance of the one or +the other of these principles, every kind of beauty is characterised. +Hence the difference between symmetrical and picturesque beauty:—the +first allied to the principle of uniformity, in being based upon precise +laws that may be taught so as to enable men of ordinary capacity to +produce it in their works—the second allied to the principle of variety +often to so great a degree that they yield an obedience to the precise +principles of harmony so subtilely, that they cannot be detected in its +constitution, but are only felt in the response by which true genius +acknowledges their presence. The generality of mankind may be capable +of perceiving this latter kind of beauty, and of feeling its effects +upon the mind, but men of genius, only, can impart it to works of art, +whether addressed to the eye or the ear. Throughout the sounds, forms, +and colours of nature, these two kinds of beauty are found not only in +distinct developments, but in every degree of amalgamation. We find in +the songs of some birds, such as those of the chaffinch, thrush, &c., +a rhythmical division, resembling in some measure the symmetrically +precise arrangements of parts which characterises all artistic musical +composition; while in the songs of other birds, and in the other numerous +melodies with which nature charms and soothes the mind, there is no +distinct regularity in the division of their parts. In the forms of +nature, too, we find amongst the innumerable flowers with which the +surface of the earth is so profusely decorated, an almost endless variety +of systematic arrangements of beautiful figures, often so perfectly +symmetrical in their combination, that the most careful application +of the angleometer could scarcely detect the slightest deviation from +geometrical precision; while, amongst the masses of foliage by which +the forms of many trees are divided and subdivided into parts, as also +amongst the hills and valleys, the mountains and ravines, which divide +the earth’s surface, we find in every possible variety of aspect the +beauty produced by that irregular species of symmetry which characterises +the picturesque. + +In like manner, we find in wild as well as cultivated flowers the +most symmetrical distributions of colours accompanying an equally +precise species of harmony in their various kinds of contrasts, often +as mathematically regular as the geometric diagrams by which writers +upon colour sometimes illustrate their works; while in the general +colouring of the picturesque beauties of nature, there is an endless +variety in its distributions, its blendings, and its modifications. In +the forms and colouring of animals, too, the same endless variety of +regular and irregular symmetry is to be found. But the highest degree of +beauty in nature is the result of an equal balance of uniformity with +variety. Of this the human figure is an example; because, when it is +of those proportions universally acknowledged to be the most perfect, +its uniformity bears to its variety an apparently equal ratio. The +harmony of combination in the normal proportions of its parts, and the +beautifully simple harmony of succession in the normal melody of its +softly undulating outline, are the perfection of symmetrical beauty, +while the innumerable changes upon the contour which arise from the +actions and attitudes occasioned by the various emotions of the mind, +are calculated to produce every species of picturesque beauty, from the +softest and most pleasing to the grandest and most sublime. + +Amongst the purely picturesque objects of inanimate nature, I may, as in +a former work, instance an ancient oak tree, for its beauty is enhanced +by want of apparent symmetry. Thus, the more fantastically crooked its +branches, and the greater the dissimilarity and variety it exhibits in +its masses of foliage, the more beautiful it appears to the artist and +the amateur; and, as in the human figure, any attempt to produce variety +in the proportions of its lateral halves would be destructive of its +symmetrical beauty, so in the oak tree any attempt to produce palpable +similarity between any of its opposite sides would equally deteriorate +its picturesque beauty. But picturesque beauty is not the result of the +total absence of symmetry; for, as none of the irregularly constructed +music of nature could be pleasing to the ear unless there existed in +the arrangement of its notes an obedience, however subtle, to the great +harmonic law of Nature, so neither could any object be picturesquely +beautiful, unless the arrangement of its parts yields, although it may be +obscurely, an obedience to the same law. + +However symmetrically beautiful any architectural structure may be, when +in a complete and perfect state, it must, as it proceeds towards ruin, +blend the picturesque with the symmetrical; but the type of its beauty +will continue to be the latter, so long as a sufficient portion of it +remains to convey an idea of its original perfection. It is the same with +the human form and countenance; for age does not destroy their original +beauty, but in both only lessens that which is symmetrical, while it +increases that which is picturesque. + +In short, as a variety of simultaneously produced sounds, which do not +relate to each other agreeably to this law, can only convey to the mind +a feeling of mere noise; so a variety of forms or colours simultaneously +exposed to the eye under similar circumstances, can only convey to the +mind a feeling of chaotic confusion, or what may be termed _visible_ +discord. As, therefore, the two principles of uniformity and variety, +or similarity and dissimilarity, are in operation in every harmonious +combination of the elements of sound, of form, and of colour, we must +first have recourse to numbers in the abstract before we can form a +proper basis for a universal science of beauty. + + + + +THE SCIENCE OF BEAUTY EVOLVED FROM THE HARMONIC LAW OF NATURE, AGREEABLY +TO THE PYTHAGOREAN SYSTEM OF NUMERICAL RATIO. + + +The scientific principles of beauty appear to have been well known to the +ancient Greeks; and it must have been by the practical application of +that knowledge to the arts of Design, that that people continued for a +period of upwards of three hundred years to execute, in every department +of these arts, works surpassing in chaste beauty any that had ever before +appeared, and which have not been equalled during the two thousand years +which have since elapsed. + +Æsthetic science, as the science of beauty is now termed, is based +upon that great harmonic law of nature which pervades and governs the +universe. It is in its nature neither absolutely physical nor absolutely +metaphysical, but of an intermediate nature, assimilating in various +degrees, more or less, to one or other of those opposite kinds of +science. It specially embodies the inherent principles which govern +impressions made upon the mind through the senses of hearing and seeing. +Thus, the æsthetic pleasure derived from listening to the beautiful in +musical composition, and from contemplating the beautiful in works of +formative art, is in both cases simply a response in the human mind to +artistic developments of the great harmonic law upon which the science +is based. + +Although the eye and the ear are two different senses, and, consequently, +various in their modes of receiving impressions; yet the sensorium is +but one, and the mind by which these impressions are perceived and +appreciated is also characterised by unity. There appears, likewise, +a striking analogy between the natural constitution of the two kinds +of beauty, which is this, that the more physically æsthetic elements +of the highest works of musical composition are melody, harmony, and +tone, whilst those of the highest works of formative art are contour, +proportion, and colour. The melody or theme of a musical composition +and its harmony are respectively analogous,—1st, To the outline of +an artistic work of formative art; and 2d, To the proportion which +exists amongst its parts. To the careful investigator these analogies +become identities in their effect upon the mind, like those of the more +metaphysically æsthetic emotions produced by expression in either of +these arts. + +Agreeably to the first analogy, the outline and contour of an object, +suppose that of a building in shade when viewed against a light +background, has a similar effect upon the mind with that of the simple +melody of a musical composition when addressed to the ear unaccompanied +by the combined harmony of counterpoint. Agreeably to the second analogy, +the various parts into which the surface of the supposed elevation is +divided being simultaneously presented to the eye, will, if arranged +agreeably to the same great law, affect the mind like that of an equally +harmonious arrangement of musical notes accompanying the supposed melody. + +There is, however, a difference between the construction of these two +organs of sense, viz., that the ear must in a great degree receive its +impressions involuntarily; while the eye, on the other hand, is provided +by nature with the power of either dwelling upon, or instantly shutting +out or withdrawing itself from an object. The impression of a sound, +whether simple or complex, when made upon the ear, is instantaneously +conveyed to the mind; but when the sound ceases, the power of observation +also ceases. But the eye can dwell upon objects presented to it so long +as they are allowed to remain pictured on the retina; and the mind has +thereby the power of leisurely examining and comparing them. Hence the +ear guides more as a mere sense, at once and without reflection; whilst +the eye, receiving its impressions gradually, and part by part, is more +directly under the influence of mental analysis, consequently producing +a more metaphysically æsthetic emotion. Hence, also, the acquired power +of the mind in appreciating impressions made upon it through the organ of +sight under circumstances, such as perspective, &c., which to those who +take a hasty view of the subject appear impossible. + +Dealing as this science therefore does, alike with the sources and the +resulting principles of beauty, it is scarcely less dependent on the +accuracy of the senses than on the power of the understanding, inasmuch +as the effect which it produces is as essential a property of objects, +as are its laws inherent in the human mind. It necessarily comprehends a +knowledge of those first principles in art, by which certain combinations +of sounds, forms, and colours produce an effect upon the mind, connected, +in the first instance, with sensation, and in the second with the +reasoning faculty. It is, therefore, not only the basis of all true +practice in art, but of all sound judgment on questions of artistic +criticism, and necessarily includes those laws whereon a correct taste +must be based. Doubtless many eloquent and ingenious treatises have been +written upon beauty and taste; but in nearly every case, with no other +effect than that of involving the subject in still greater uncertainty. +Even when restricted to the arts of design, they have failed to exhibit +any definite principles whereby the true may be distinguished from +the false, and some natural and recognised laws of beauty reduced to +demonstration. This may be attributed, in a great degree, to the neglect +of a just discrimination between what is merely agreeable, or capable +of exciting pleasurable sensations, and what is essentially beautiful; +but still more to the confounding of the operations of the understanding +with those of the imagination. Very slight reflection, however, will +suffice to shew how essentially distinct these two faculties of the mind +are; the former being regulated, in matters of taste, by irrefragable +principles existing in nature, and responded to by an inherent principle +existing in the human mind; while the latter operates in the production +of ideal combinations of its own creation, altogether independent of any +immediate impression made upon the senses. The beauty of a flower, for +example, or of a dew-drop, depends on certain combinations of form and +colour, manifestly referable to definite and systematic, though it may +be unrecognised, laws; but when Oberon, in “Midsummer Night’s Dream,” is +made to exclaim— + + “And that same dew, which sometimes on the buds + Was wont to swell, like round and orient pearls, + Stood now within the pretty floweret’s eyes, + Like tears that did their own disgrace bewail,”— + +the poet introduces a new element of beauty equally legitimate, yet +altogether distinct from, although accompanying that which constitutes +the more precise science of æsthetics as here defined. The composition +of the rhythm is an operation of the understanding, but the beauty of the +poetic fancy is an operation of the imagination. + +Our physical and mental powers, æsthetically considered, may therefore be +classed under three heads, in their relation to the fine arts, viz., the +receptive, the perceptive, and the conceptive. + +The senses of hearing and seeing are respectively, in the degree of +their physical power, receptive of impressions made upon them, and of +these impressions the sensorium, in the degree of its mental power, is +perceptive. This perception enables the mind to form a judgment whereby +it appreciates the nature and quality of the impression originally made +on the receptive organ. The mode of this operation is intuitive, and +the quickness and accuracy with which the nature and quality of the +impression is apprehended, will be in the degree of the intellectual +vigour of the mind by which it is perceived. Thus we are, by the +cultivation of these intuitive faculties, enabled to decide with accuracy +as to harmony or discord, proportion or deformity, and assign sound +reasons for our judgment in matters of taste. But mental conception is +the intuitive power of constructing original ideas from these materials; +for after the receptive power has acted, the perception operates in +establishing facts, and then the judgment is formed upon these operations +by the reasoning powers, which lead, in their turn, to the creations of +the imagination. + +The power of forming these creations is the true characteristic +of genius, and determines the point at which art is placed beyond +all determinable canons,—at which, indeed, æsthetics give place to +metaphysics. + +In the science of beauty, therefore, the human mind is the subject, and +the effect of external nature, as well as of works of art, the object. +The external world, and the individual mind, with all that lies within +the scope of its powers, may be considered as two separate existences, +having a distinct relation to each other. The subject is affected by the +object, through that inherent faculty by which it is enabled to respond +to every development of the all-governing harmonic law of nature; and the +media of communication are the sensorium and its inlets—the organs of +sense. + +This harmonic law of nature was either originally discovered by that +illustrious philosopher Pythagoras, upwards of five hundred years before +Christ, or a knowledge of it obtained by him about that period, from +the Egyptian or Chaldean priests. For after having been initiated into +all the Grecian and barbarian sacred mysteries, he went to Egypt, where +he remained upwards of twenty years, studying in the colleges of its +priests; and from Egypt he went into the East, and visited the Persian +and Chaldean magi.[3] + +By the generality of the biographers of Pythagoras, it is said to be +difficult to give a clear idea of his philosophy, as it is almost certain +he never committed it to writing, and that it has been disfigured by the +fantastic dreams and chimeras of later Pythagoreans. Diogenes Laërtius, +however, whose “Lives of the Philosophers” was supposed to be written +about the end of the second century of our era, says “there are three +volumes extant written by Pythagoras. One on education, one on politics, +and one on natural philosophy.” And adds, that there were several other +books extant, attributed to Pythagoras, but which were not written by +him. Also, in his “Life of Philolaus,” that Plato wrote to Dion to take +care and purchase the books of Pythagoras.[4] But whether this great +philosopher committed his discoveries to writing or not, his doctrines +regarding the philosophy of beauty are well-known to be, that he +considered numbers as the essence and the principle of all things, and +attributed to them a real and distinct existence; so that, in his view, +they were the elements out of which the universe was constructed, and to +which it owed its beauty. Diogenes Laërtius gives the following account +of this law:—“That the monad was the beginning of everything. From the +monad proceeds an indefinite duad, which is subordinate to the monad as +to its cause. That from the monad and indefinite duad proceeds numbers. +That the part of science to which Pythagoras applied himself above all +others, was arithmetic; and that he taught ‘that from numbers proceed +signs, and from these latter, lines, of which plane figures consist; +that from plane figures are derived solid bodies; that of all plane +figures the most beautiful was the circle, and of all solid bodies the +most beautiful was the sphere.’ He discovered the numerical relations of +sounds on a single string; and taught that everything owes its existence +and consistency to harmony. In so far as I know, the most condensed +account of all that is known of the Pythagorian system of numbers is the +following:—‘The monad or unity is that quantity, which, being deprived of +all number, remains fixed. It is the fountain of all number. The duad is +imperfect and passive, and the cause of increase and division. The triad, +composed of the monad and duad, partakes of the nature of both. The +tetrad, tetractys, or quaternion number is most perfect. The decad, which +is the sum of the four former, comprehends all arithmetical and musical +proportions.’”[5] + +These short quotations, I believe, comprise all that is known, for +certain, of the manner in which Pythagoras systematised the law of +numbers. Yet, from the teachings of this great philosopher and his +disciples, the harmonic law of nature, in which the fundamental +principles of beauty are embodied, became so generally understood and +universally applied in practice throughout all Greece, that the fragments +of their works, which have reached us through a period of two thousand +years, are still held to be examples of the highest artistic excellence +ever attained by mankind. In the present state of art, therefore, a +knowledge of this law, and of the manner in which it may again be applied +in the production of beauty in all works of form and colour, must be +of singular advantage; and the object of this work is to assist in the +attainment of such a knowledge. + +It has been remarked, with equal comprehensiveness and truth, by a +writer[6] in the _British and Foreign Medical Review_, that “there +is harmony of numbers in all nature—in the force of gravity—in the +planetary movements—in the laws of heat, light, electricity, and chemical +affinity—in the forms of animals and plants—in the perceptions of the +mind. The direction, indeed, of modern natural and physical science is +towards a generalization which shall express the fundamental laws of all +by one simple numerical ratio. And we think modern science will soon shew +that the mysticism of Pythagoras was mystical only to the unlettered, +and that it was a system of philosophy founded on the then existing +mathematics, which latter seem to have comprised more of the philosophy +of numbers than our present.” Many years of careful investigation have +convinced me of the truth of this remark, and of the great advantage +derivable from an application of the Pythagorean system in the arts +of design. For so simple is its nature, that any one of an ordinary +capacity of mind, and having a knowledge of the most simple rules of +arithmetic, may, in a very short period, easily comprehend its nature, +and be able to apply it in practice. + +The elements of the Pythagorean system of harmonic number, so far as can +be gathered from the quotations I have given above, seem to be simply +the indivisible monad (1); the duad (2), arising from the union of one +monad with another; the triad (3), arising from the union of the monad +with the duad; and the tetrad (4), arising from the union of one duad +with another, which tetrad is considered a perfect number. From the +union of these four elements arises the decad (10), the number, which, +agreeably to the Pythagorean system, comprehends all arithmetical and +harmonic proportions. If, therefore, we take these elements and unite +them progressively in the following order, we shall find the series of +harmonic numbers (2), (3), (5), and (7), which, with their multiples, are +the complete numerical elements of all harmony, thus:— + + 1 + 1 = 2 + 1 + 2 = 3 + 2 + 3 = 5 + 3 + 4 = 7 + +In order to render an extended series of harmonic numbers useful, it +must be divided into scales; and it is a rule in the formation of these +scales, that the first must begin with the monad (1) and end with the +duad (2), the second begin with the duad (2) and end with the tetrad (4), +and that the beginning and end of all other scales must be continued in +the same arithmetical progression. These primary elements will then form +the foundation of a series of such scales. + + I. (1) (2) + II. (2) (3) (4) + III. (4) (5) (6) (7) (8) + IV. (8) (9) (10) ( ) (12) ( ) (14) (15) (16) + +The first of these scales has in (1) and (2) a beginning and an end; but +the second has in (2), (3), and (4) the essential requisites demanded +by Aristotle in every composition, viz., “a beginning, a middle, and +an end;” while the third has not only these essential requisites, but +two intermediate parts (5) and (7), by which the beginning, the middle, +and the end are united. In the fourth scale, however, the arithmetical +progression is interrupted by the omission of numbers 11 and 13, which, +not being multiples of either (2), (3), (5), or (7), are inadmissible. + +Such is the nature of the harmonic law which governs the progressive +scales of numbers by the simple multiplication of the monad. + +I shall now use these numbers as divisors in the formation of a series +of four such scales of parts, which has for its primary element, instead +of the indivisible monad, a quantity which may be indefinitely divided, +but which cannot be added to or multiplied. Like the monad, however, this +quantity is represented by (1). The following is this series of four +scales of harmonic parts:— + + I. (1) (¹⁄₂) + II. (¹⁄₂) (¹⁄₃) (¹⁄₄) + III. (¹⁄₄) (¹⁄₅) (¹⁄₆) (¹⁄₇) (¹⁄₈) + IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) ( ) (¹⁄₁₂) ( ) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆) + +The scales I., II., and III. may now be rendered as complete as scale +IV., simply by multiplying upwards by 2 from (¹⁄₉), (¹⁄₅), (¹⁄₃), (¹⁄₇), +and (¹⁄₁₅), thus:— + + I. (1) (⁸⁄₉) (⁴⁄₅) (²⁄₃) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂) + II. (¹⁄₂) (⁴⁄₉) (²⁄₅) (¹⁄₃) (²⁄₇) (⁴⁄₁₅) (¹⁄₄) + III. (¹⁄₄) (²⁄₉) (¹⁄₅) (¹⁄₆) (¹⁄₇) (²⁄₁₅) (¹⁄₈) + IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) ( ) (¹⁄₁₂) ( ) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆) + +We now find between the beginning and the end of scale I. the quantities +(⁸⁄₉), (⁴⁄₅), (²⁄₃), (⁴⁄₇), and (⁸⁄₁₅). + +The three first of these quantities we find to be the remainders of the +whole indefinite quantity contained in (1), after subtracting from it +the primary harmonic quantities (¹⁄₉), (¹⁄₅), and (¹⁄₃); we, however, +find also amongst these harmonic quantities that of (¹⁄₄), which being +subtracted from (1) leaves (³⁄₄), a quantity the most suitable whereby +to fill up the hiatus between (⁴⁄₅) and (²⁄₃) in scale I., which arises +from the omission of (¹⁄₁₁) in scale IV. In like manner we find the two +last of these quantities, (⁴⁄₇) and (⁸⁄₁₅), are respectively the largest +of the two parts into which 7 and 15 are susceptible of being divided. +Finding the number 5 to be divisible into parts more unequal than (2) +to (3) and less unequal than (4) to (7), (³⁄₅) naturally fills up the +hiatus between these quantities in scale I., which hiatus arises from the +omission of (¹⁄₁₃) in scale IV. Thus:— + + I. (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂) + II. (¹⁄₂) (⁴⁄₉) (²⁄₅) ( ) (¹⁄₃) ( ) (²⁄₇) (⁴⁄₁₅) (¹⁄₄) + III. (¹⁄₄) (²⁄₉) (¹⁄₅) ( ) (¹⁄₆) ( ) (¹⁄₇) (²⁄₁₅) (¹⁄₈) + IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) ( ) (¹⁄₁₂) ( ) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆) + +Scale I. being now complete, we have only to divide these latter +quantities by (2) downwards in order to complete the other three. Thus:— + + I. (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂) + II. (¹⁄₂) (⁴⁄₉) (²⁄₅) (³⁄₈) (¹⁄₃) (³⁄₁₀) (²⁄₇) (⁴⁄₁₅) (¹⁄₄) + III. (¹⁄₄) (²⁄₉) (¹⁄₅) (³⁄₁₆) (¹⁄₆) (³⁄₂₀) (¹⁄₇) (²⁄₁₅) (¹⁄₈) + IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) (³⁄₃₂) (¹⁄₁₂) (³⁄₄₀) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆) + +The harmony existing amongst these numbers or quantities consists of the +numerical relations which the parts bear to the whole and to each other; +and the more simple these relations are, the more perfect is the harmony. +The following are the numerical harmonic ratios which the parts bear to +the whole:— + + I. (1:1) (8:9) (4: 5) (3: 4) (2: 3) (3: 5) (4: 7) (8:15) (1: 2) + II. (1:2) (4:9) (2: 5) (3: 8) (1: 3) (3:10) (2: 7) (4:15) (1: 4) + III. (1:4) (2:9) (1: 5) (3:16) (1: 6) (3:20) (1: 7) (2:15) (1: 8) + IV. (1:8) (1:9) (1:10) (3:32) (1:12) (3:40) (1:14) (1:15) (1:16) + +The following are the principal numerical relations which the parts in +each scale bear to one another:— + + (¹⁄₂):(⁴⁄₇) = (7:8) + (⁴⁄₅):(⁸⁄₉) = (9:10) + (²⁄₃):(⁴⁄₅) = (5:6) + (⁴⁄₇):(²⁄₃) = (6:7) + (⁸⁄₁₅):(⁴⁄₇) = (14:15) + (¹⁄₂):(⁸⁄₁₅) = (15:16) + +Although these relations are exemplified by parts of scale I., the same +ratios exist between the relative parts of scales II., III., and IV., +and would exist between the parts of any other scales that might be added +to that series. + +These are the simple elements of the science of that harmony which +pervades the universe, and by which the various kinds of beauty +æsthetically impressed upon the senses of hearing and seeing are +governed. + + + + +THE SCIENCE OF BEAUTY AS APPLIED TO SOUNDS. + + +It is well-known that all sounds arise from a peculiar action of the +air, and that this action may be excited by the concussion resulting +from the sudden displacement of a portion of the atmosphere itself, or +by the rapid motions of bodies, or of confined columns of air; in all +which cases, when the motions are irregular, and the force great, the +sound conveyed to the sensorium is called a noise. But that musical +sounds are the result of equal and regular vibratory motions, either +of an elastic body, or of a column of air in a tube, exciting in the +surrounding atmosphere a regular and equal pulsation. The ear is the +medium of communication between those varieties of atmospheric action and +the seat of consciousness. To describe fully the beautiful arrangement +of the various parts of this organ, and their adaptation to the purpose +of collecting and conveying these undulatory motions of the atmosphere, +is as much beyond the scope of my present attempt as it is beyond my +anatomical knowledge; but I may simply remark, that within the ear, and +most carefully protected in the construction of that organ, there is a +small cavity containing a pellucid fluid, in which the minute extremities +of the auditory nerve float; and that this fluid is the last of the media +through which the action producing the sensation of sound is conveyed +to the nerve, and thence to the sensorium, where its nature becomes +perceptible to the mind. + +The impulses which produce musical notes must arrive at a certain +frequency before the ear loses the intervals of silence between them, +and is impressed by only one continued sound; and as they increase +in frequency the sound becomes more acute upon the ear. The pitch of +a musical note is, therefore, determined by the frequency of these +impulses; but, on the other hand, its intensity or loudness will depend +upon the violence and the quality of its tone on the material employed in +producing them. All such sounds, therefore, whatever be their loudness +or the quality of their tone in which the impulses occur with the same +frequency are in perfect unison, having the same pitch. Upon this the +whole doctrine of harmonies is founded, and by this the laws of numerical +ratio are found to operate in the production of harmony, and the theory +of music rendered susceptible of exact reasoning. + +The mechanical means by which such sounds can be produced are extremely +various; but, as it is my purpose simply to shew the nature of harmony +of sound as related to, or as evolving numerical harmonic ratio, I shall +confine myself to the most simple mode of illustration—namely, that of +the monochord. This is an instrument consisting of a string of a given +length stretched between two bridges standing upon a graduated scale. +Suppose this string to be stretched until its tension is such that, +when drawn a little to a side and suddenly let go, it would vibrate at +the rate of 64 vibrations in a second of time, producing to a certain +distance in the surrounding atmosphere a series of pulsations of the same +frequency. + +These pulsations will communicate through the ear a musical note which +would, therefore, be the fundamental note of such a string. Now, the +phenomenon said to be discovered by Pythagoras is well known to those +acquainted with the science of acoustics, namely, that immediately after +the string is thus put into vibratory motion, it spontaneously divides +itself, by a node, into two equal parts, the vibrations of each of which +occur with a double frequency—namely, 128 in a second of time, and, +consequently, produce a note doubly acute in pitch, although much weaker +as to intensity or loudness; that it then, while performing these two +series of vibrations, divides itself, by two nodes, into three parts, +each of which vibrates with a frequency triple that of the whole string; +that is, performs 192 vibrations in a second of time, and produces a +note corresponding in increase of acuteness, but still less intense than +the former, and that this continues to take place in the arithmetical +progression of 2, 3, 4, &c. Simultaneous vibrations, agreeably to the +same law of progression, which, however, seem to admit of no other primes +than the numbers 2, 3, 5, and 7, are easily excited upon any stringed +instrument, even by the lightest possible touch of any of its strings +while in a state of vibratory motion, and the notes thus produced are +distinguished by the name of harmonics. It follows, then, that one-half +of a musical string, when divided from the whole by the pressure of the +finger, or any other means, and put into vibratory motion, produces a +note doubly acute to that produced by the vibratory motion of the whole +string; the third part, similarly separated, a note trebly acute; and +the same with every part into which any musical string may be divided. +This is the fundamental principle by which all stringed instruments are +made to produce harmony. It is the same with wind instruments, the sounds +of which are produced by the frequency of the pulsations occasioned in +the surrounding atmosphere by agitating a column of air confined within +a tube as in an organ, in which the frequency of pulsation becomes +greater in an inverse ratio to the length of the pipes. But the following +series of four successive scales of musical notes will give the reader +a more comprehensive view of the manner in which they follow the law of +numerical ratio just explained than any more lengthened exposition. + +It is here requisite to mention, that in the construction of these +scales, I have not only adopted the old German or literal mode of +indicating the notes, but have included, as the Germans do, the note +termed by us B flat as B natural, and the note we term B natural as +H. Now, although this arrangement differs from that followed in the +construction of our modern Diatonic scale, yet as the ratio of 4:7 +is more closely related to that of 1:2 than that of 8:15, and as it +is offered by nature in the spontaneous division of the monochord, +I considered it quite admissible. The figures give the parts of the +monochord which would produce the notes. + + I. { (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)* + { C D E F G A B H _c_ + + II. { (¹⁄₂)* (⁴⁄₉) (²⁄₅) (³⁄₈) (¹⁄₃)* (³⁄₁₀) (²⁄₇) (²⁄₁₅) (¹⁄₄)* + { _c_ _d_ _e_ _f_ _g_ _a_ _b_ _h_ _c′_ + + III. { (¹⁄₄)* (²⁄₉) (¹⁄₅)* (³⁄₁₆) (¹⁄₆)* (³⁄₂₀) (¹⁄₇)* (²⁄₁₅) (¹⁄₈)* + { _c′_ _d′_ _e′_ _f′_ _g′_ _a′_ _b′_ _h′_ _c′′_ + + IV. { (¹⁄₈)* (¹⁄₉)* (¹⁄₁₀)* (³⁄₃₂) (¹⁄₁₂)* (³⁄₄₀) (¹⁄₁₄)* (¹⁄₁₅)* (¹⁄₁₆)* + { _c′′_ _d′′_ _e′′_ _f′′_ _g′′_ _a′′_ _b′′_ _h′′_ _c′′′_ + +The notes marked (*) are the harmonics which naturally arise from the +division of the string by 2, 3, 5, and 7, and the multiples of these +primes. + +Thus every musical sound is composed of a certain number of parts called +pulsations, and these parts must in every scale relate harmonically +to some fundamental number. When these parts are multiples of the +fundamental number by 2, 4, 8, &c., like the pulsations of the sounds +indicated by _c_, _c′_, _c′′_, _c′′′_, they are called tonic notes, being +the most consonant; when the pulsations are similar multiples by 3, 6, +12, &c., like those of the sounds indicated by _g_, _g′_, _g′′_, they are +called dominant notes, being the next most consonant; and multiples by +5, 10, &c., like those of the sounds indicated by _e_, _e′_, _e′′_, they +are called mediant notes, from a similar cause. In harmonic combinations +of musical sounds, the æsthetic feeling produced by their agreement +depends upon the relations they bear to each other with reference to the +number of pulsations produced in a given time by the fundamental note of +the scale to which they belong; and it will be observed, that the more +simple the numerical ratios are amongst the pulsations of any number of +notes simultaneously produced, the more perfect their agreement. Hence +the origin of the common chord or fundamental concord in the united +sounds of the tonic, the dominant, and the mediant notes, the ratios and +coincidences of whose pulsations 2:1, 3:2, 5:4, may thus be exemplified:— + +[Illustration] + +In musical composition, the law of number also governs its division +into parts, in order to produce upon the ear, along with the beauty of +harmony, that of rhythm. Thus a piece of music is divided into parts +each of which contains a certain number of other parts called bars, +which may be divided and subdivided into any number of notes, and the +performance of each bar is understood to occupy the same portion of time, +however numerous the notes it contains may be; so that the music of art +is regularly symmetrical in its structure; while that of nature is in +general as irregular and indefinite in its rhythm as it is in its harmony. + +Thus I have endeavoured briefly to explain the manner in which the law of +numerical ratio operates in that species of beauty perceived through the +ear. + +The definite principles of the art of music founded upon this law have +been for ages so systematised that those who are instructed in them +advance steadily in proportion to their natural endowments, while those +who refuse this instruction rarely attain to any excellence. In the +sister arts of form and colour, however, a system of tuition, founded +upon this law, is still a desideratum, and a knowledge of the scientific +principles by which these arts are governed is confined to a very few, +and scarcely acknowledged amongst those whose professions most require +their practical application. + + + + +THE SCIENCE OF BEAUTY AS APPLIED TO FORMS. + + +It is justly remarked, in the “Illustrated Record of the New York +Exhibition of 1853,” that “it is a question worthy of consideration how +far the mediocrity of the present day is attributable to an overweening +reliance on natural powers and a neglect of the lights of science;” +and there is expressed a thorough conviction of the fact that, besides +the evils of the copying system, “much genius is now wasted in the +acquirement of rudimentary knowledge in the slow school of practical +experiment, and that the excellence of the ancient Greek school of +design arose from a thoroughly digested canon of form, and the use +of geometrical formulas, which make the works even of the second and +third-rate genius of that period the wonder and admiration of the present +day.” + +That such a canon of form, and that the use of such geometrical formula, +entered into the education, and thereby facilitated the practice of +ancient Greek art, I have in a former work expressed my firm belief, +which is founded on the remarkable fact, that for a period of nearly +three centuries, and throughout a whole country politically divided into +states often at war with each other, works of sculpture, architecture, +and ornamental design were executed, which surpass in symmetrical +beauty any works of the kind produced during the two thousand years that +have since elapsed. So decided is this superiority, that the artistic +remains of the extraordinary period I alluded to are, in all civilised +nations, still held to be the most perfect specimens of formative art in +the world; and even when so fragmentary as to be denuded of everything +that can convey an idea of expression, they still excite admiration and +wonder by the purity of their geometric beauty. And so universal was +this excellence, that it seems to have characterised every production of +formative art, however humble the use to which it was applied. + +The common supposition, that this excellence was the result of an +extraordinary amount of genius existing among the Greek people during +that particular period, is not consistent with what we know of the +progress of mankind in any other direction, and is, in the present state +of art, calculated to retard its progress, inasmuch as such an idea +would suggest that, instead of making any exertion to arrive at a like +general excellence, the world must wait for it until a similar supposed +psychological phenomenon shall occur. + +But history tends to prove that the long period of universal artistic +excellence throughout Greece could only be the result of an early +inculcation of some well-digested system of correct elementary +principles, by which the ordinary amount of genius allotted to mankind +in every age was properly nurtured and cultivated; and by which, also, a +correct knowledge and appreciation of art were disseminated amongst the +people generally. Indeed, Müller, in his “Ancient Art and its Remains,” +shews clearly that some certain fixed principles, constituting a science +of proportions, were known in Greece, and that they formed the basis of +all artists’ education and practice during the period referred to; also, +that art began to decline, and its brightest period to close, as this +science fell into disuse, and the Greek artists, instead of working for +an enlightened community, who understood the nature of the principles +which guided them, were called upon to gratify the impatient whims of +pampered and tyrannical rulers. + +By being instructed in this science of proportion, the Greek artists +were enabled to impart to their representations of the human figure +a mathematically correct species of symmetrical beauty; whether +accompanying the slender and delicately undulated form of the Venus,—its +opposite, the massive and powerful mould of the Hercules,—or the +characteristic representation of any other deity in the heathen +mythology. And this seems to have been done with equal ease in the minute +figure cut on a precious gem, and in the most colossal statue. The same +instruction likewise enabled the architects of Greece to institute those +varieties of proportions in structure called the Classical Orders of +Architecture; which are so perfect that, since the science which gave +them birth has been buried in oblivion, classical architecture has been +little more than an imitative art; for all who have since written upon +the subject, from Vitruvius downwards, have arrived at nothing, in +so far as the great elementary principles in question are concerned, +beyond the most vague and unsatisfactory conjectures. For a more clear +understanding of the nature of this application of the Pythagorean law of +number to the harmony of form, it will be requisite to repeat the fact, +that modern science has shewn that the cause of the impression, produced +by external nature upon the sensorium, called light, may be traced to a +molecular or ethereal action. This action is excited naturally by the +sun, artificially by the combustion of various substances, and sometimes +physically within the eye. Like the atmospheric pulsations which produce +sound, the action which produces light is capable, within a limited +sphere, of being reflected from some bodies and transmitted through +others; and by this reflection and transmission the visible nature of +forms and figures is communicated to the sensorium. The eye is the +medium of this communication; and its structural beauty, and perfect +adaptation to the purpose of conveying this action, must, like those of +the ear, be left to the anatomist fully to describe. It is here only +necessary to remark, that the optic nerve, like the auditory nerve, +ends in a carefully protected fluid, which is the last of the media +interposed between this peculiarly subtle action and the nerve upon +which it impresses the presence of the object from which it is reflected +or through which it is transmitted, and the nature of such object made +perceptible to the mind. The eye and the ear are thus, in one essential +point, similar in their physiology, relatively to the means provided +for receiving impressions from external nature; it is, therefore, but +reasonable to believe that the eye is capable of appreciating the exact +subdivision of spaces, just as the ear is capable of appreciating the +exact subdivision of intervals of time; so that the division of space +into exact numbers of equal parts will æsthetically affect the mind +through the medium of the eye. + +We assume, therefore, that the standard of symmetry, so estimated, is +deduced from the simplest law that could have been conceived—the law +that the angles of direction must all bear to some fixed angle the same +simple relations which the different notes in a chord of music bear to +the fundamental note; that is, relations expressed arithmetically by the +smallest natural numbers. Thus the eye, being guided in its estimate by +direction rather than by distance, just as the ear is guided by number +of vibrations rather than by magnitude, both it and the ear convey +simplicity and harmony to the mind without effort, and the mind with +equal facility receives and appreciates them. + + +_On the Rectilinear Forms and Proportions of Architecture._ + +As we are accustomed in all cases to refer direction to the horizontal +and vertical lines, and as the meeting of these lines makes the right +angle, it naturally constitutes the fundamental angle, by the harmonic +division of which a system of proportion may be established, and the +theory of symmetrical beauty, like that of music, rendered susceptible of +exact reasoning. + +Let therefore the right angle be the fundamental angle, and let it be +divided upon the quadrant of a circle into the harmonic parts already +explained, thus:— + + Super- Sub- Sub- Sub- Semi-sub- + Right tonic Mediant dominant Dominant mediant tonic tonic Tonic + Angle. Angles. Angles. Angles. Angles. Angles. Angles. Angles. Angles. + I. (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂) + II. (¹⁄₂) (⁴⁄₉) (²⁄₅) (³⁄₈) (¹⁄₃) (³⁄₁₀) (²⁄₇) (⁴⁄₁₅) (¹⁄₄) + III. (¹⁄₄) (²⁄₉) (¹⁄₅) (³⁄₁₆) (¹⁄₆) (³⁄₂₀) (¹⁄₇) (²⁄₁₅) (¹⁄₈) + IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) (³⁄₃₂) (¹⁄₁₂) (³⁄₄₀) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆) + +In order that the analogy may be kept in view, I have given to the parts +of each of these four scales the appropriate nomenclature of the notes +which form the diatonic scale in music. + +When a right angled triangle is constructed so that its two smallest +angles are equal, I term it simply the triangle of (¹⁄₂), because the +smaller angles are each one-half of the right angle. But when the two +angles are unequal, the triangle may be named after the smallest. For +instance, when the smaller angle, which we shall here suppose to be +one-third of the right angle, is made with the vertical line, the +triangle may be called the vertical scalene triangle of (¹⁄₃); and when +made with the horizontal line, the horizontal scalene triangle of (¹⁄₃). +As every rectangle is made up of two of these right angled triangles, +the same terminology may also be applied to these figures. Thus, the +equilateral rectangle or perfect square is simply the rectangle of (¹⁄₂), +being composed of two similar right angled triangles of (¹⁄₂); and when +two vertical scalene triangles of (¹⁄₃), and of similar dimensions, are +united by their hypothenuses, they form the vertical rectangle of (¹⁄₃), +and in like manner the horizontal triangles of (¹⁄₃) similarly united +would form the horizontal rectangle of (¹⁄₃). As the isosceles triangle +is in like manner composed of two right angled scalene triangles joined +by one of their sides, the same terminology may be applied to every +variety of that figure. All the angles of the first of the above scales, +except that of (¹⁄₂), give rectangles whose longest sides are in the +horizontal line, while the other three give rectangles whose longest +sides are in the vertical line. I have illustrated in Plate I. the manner +in which this harmonic law acts upon these elementary rectilinear figures +by constructing a series agreeably to the angles of scales II., III., +IV. Throughout this series _a b c_ is the primary scalene triangle, of +which the rectangle _a b c e_ is composed; _d c e_ the vertical isosceles +triangle; and when the plate is turned, _d e a_ the horizontal isosceles +triangle, both of which are composed of the same primary scalene triangle. + +[Sidenote: Plate I.] + +Thus the most simple elements of symmetry in rectilinear forms are the +three following figures:— + + The equilateral rectangle or perfect square, + The oblong rectangle, and + The isosceles triangle. + +It has been shewn that in harmonic combinations of musical sounds, the +æsthetic feeling produced by their agreement depends upon the relation +they bear to each other with reference to the number of pulsations +produced in a given time by the fundamental note of the scale to which +they belong; and that the more simply they relate to each other in this +way the more perfect the harmony, as in the common chord of the first +scale, the relations of whose parts are in the simple ratios of 2:1, 3:2, +and 5:4. It is equally consistent with this law, that when applied to +form in the composition of an assortment of figures of any kind, their +respective proportions should bear a very simple ratio to each other in +order that a definite and pleasing harmony may be produced amongst the +various parts. Now, this is as effectually done by forming them upon the +harmonic divisions of the right angle as musical harmony is produced by +sounds resulting from harmonic divisions of a vibratory body. + +Having in previous works[7] given the requisite illustrations of this +fact in full detail, I shall here confine myself to the most simple kind, +taking for my first example one of the finest specimens of classical +architecture in the world—the front portico of the Parthenon of Athens. + +The angles which govern the proportions of this beautiful elevation are +the following harmonic parts of the right angle— + + Tonic Dominant Mediant Subtonic Supertonic + Angles. Angles. Angles. Angle. Angles. + (¹⁄₂) (¹⁄₃) (¹⁄₅) (¹⁄₇) (¹⁄₉) + (¹⁄₄) (¹⁄₆) (¹⁄₁₀) (¹⁄₁₈) + (¹⁄₈) + (¹⁄₁₆) + +[Sidenote: Plate II.] + +In Plate II. I give a diagram of its rectilinear orthography, which is +simply constructed by lines drawn, either horizontally, vertically, or +obliquely, which latter make with either of the former lines one or other +of the harmonic angles in the above series. For example, the horizontal +line AB represents the length of the base or surface of the upper step of +the substructure of the building. The line AE, which makes an angle of +(¹⁄₅) with the horizontal, determines the height of the colonnade. The +line AD, which makes an angle of (¹⁄₄) with the horizontal, determines +the height of the portico, exclusive of the pediment. The line AC, which +makes an angle of (¹⁄₃) with the horizontal, determines the height of the +portico, including the pediment. The line GD, which makes an angle of +(¹⁄₇) with the horizontal, determines the form of the pediment. The lines +EZ and LY, which respectively make angles of (¹⁄₁₆) and (¹⁄₁₈) with the +horizontal, determine the breadth of the architrave, frieze, and cornice. +The line _v n u_, which makes an angle of (¹⁄₃) with the vertical, +determines the breadth of the triglyphs. The line _t d_, which makes an +angle of (¹⁄₂), determines the breadth of the metops. The lines _c b +r f_, and _a i_, which make each an angle of (¹⁄₆) with the vertical, +determine the width of the five centre intercolumniations. The line _z +k_, which makes an angle of (¹⁄₈) with the vertical, determines the width +of the two remaining intercolumniations. The lines _c s_, _q x_, and _y +h_, each of which makes an angle of (¹⁄₁₀) with the vertical, determine +the diameters of the three columns on each side of the centre. The line +_w l_, which makes an angle of (¹⁄₉) with the vertical, determines the +diameter of the two remaining or corner columns. + +In all this, the length and breadth of the parts are determined by +horizontal and vertical lines, which are necessarily at right angles with +each other, and the position of which are determined by one or other of +the lines making the harmonic angles above enumerated. + +Now, the lengths and breadths thus so simply determined by these few +angles, have been proved to be correct by their agreement with the most +careful measurements which could possibly be made of this exquisite +specimen of formative art. These measurements were obtained by the +“Society of Dilettanti,” London, who, expressly for that purpose, sent Mr +F. C. Penrose, a highly educated architect, to Athens, where he remained +for about five months, engaged in the execution of this interesting +commission, the results of which are now published in a magnificent +volume by the Society.[8] The agreement was so striking, that Mr Penrose +has been publicly thanked by an eminent man of science for bearing +testimony to the truth of my theory, who in doing so observes, “The +dimensions which he (Mr Penrose) gives are to me the surest verification +of the theory I could have desired. The minute discrepancies form that +very element of practical incertitude, both as to execution and direct +measurement, which always prevails in materialising a mathematical +calculation made under such conditions.”[9] + +Although the measurements taken by Mr Penrose are undeniably correct, as +all who examine the great work just referred to must acknowledge, and +although they have afforded me the best possible means of testing the +accuracy of my theory as applied to the Parthenon, yet the ideas of Mr +Penrose as to the principles they evolve are founded upon the fallacious +doctrine which has so long prevailed, and still prevails, in the +æsthetics of architecture, viz., that harmony may be imparted by ratios +between the lengths and breadths of parts. + +I have taken for my second example an elevation which, although of +smaller dimensions, is no less celebrated for the beauty of its +proportions than the Parthenon itself, viz., the front portico of the +temple of Theseus, which has also been measured by Mr Penrose. + +The angles which govern the proportions of this elevation are the +following harmonic parts of the right angle:— + + Tonic Dominant Mediant + Angles. Angles. Angles. + (¹⁄₂) (¹⁄₃) (²⁄₅) + (¹⁄₄) (¹⁄₆) (¹⁄₅) + (¹⁄₁₂) + +[Sidenote: Plate III.] + +A diagram of the rectilinear orthography of this portico is given in +Plate III. Its construction is similar to that of the Parthenon in +respect to the harmonic parts of the right angle, and I have therefore +only to observe, that the line A E makes an angle of (¹⁄₄); the line A D +an angle of (¹⁄₃); the line A C an angle of (²⁄₅); the line G D an angle +of (¹⁄₆); and the lines E Z and L Y angles of (¹⁄₁₂) with the horizontal. + +As to the colonnade or vertical part, the line _a b_, which determines +the three middle intercolumniations, makes an angle of (¹⁄₅); the line +_c d_, which determines the two outer intercolumniations, makes an angle +of (¹⁄₆); and the line _e f_, which determines the lesser diameter of +the columns, makes an angle of (¹⁄₁₂) with the vertical. I need give no +further details here, as my intention is to shew the simplicity of the +method by which this theory may be reduced to practice, and because I +have given in my other works ample details, in full illustration of the +orthography of these two structures, especially the first.[10] + +The foregoing examples being both horizontal rectangular compositions, +the proportions of their principal parts have necessarily been determined +by lines drawn from the extremities of the base, making angles with +the horizontal line, and forming thereby the diagonals of the various +rectangles into which, in their leading features, they are necessarily +resolved. But the example I am now about to give is of another character, +being a vertical pyramidal composition, and consequently the proportions +of its principal parts are determined by the angles which the oblique +lines make with the vertical line representing the height of the +elevation, and forming a series of isosceles triangles; for the isosceles +triangle is the type of all pyramidal composition. + +This third example is the east end of Lincoln Cathedral, a Gothic +structure, which is acknowledged to be one of the finest specimens of +that style of architecture existing in this country. + +The angles which govern the proportions of this elevation are the +following harmonic parts of the right angle:— + + Tonic. Dominant. Mediant. Subtonic. Supertonic. + (¹⁄₂) (¹⁄₃) (¹⁄₅) (¹⁄₇) (²⁄₉) + (¹⁄₄) (¹⁄₆) (¹⁄₁₀) (¹⁄₉) + (¹⁄₁₂) + +[Sidenote: Plate IV.] + +In Plate IV. I give a diagram of the vertical, horizontal, and oblique +lines, which compose the orthography of this beautiful elevation. + +The line A B represents the full height of this structure. The line A C, +which makes an angle of (²⁄₉) with the vertical, determines the width of +the design, the tops of the aisle windows, and the bases of the pediments +on the inner buttresses; A G, (¹⁄₅) with the vertical, that of the outer +buttress; A F, (¹⁄₉) with the vertical, that of the space between the +outer and inner buttresses and the width of the great centre window; and +A E, (¹⁄₁₂) with vertical, that of both the inner buttresses and the +space between these. A H, which makes (¹⁄₄) with the vertical, determines +the form of the pediment of the centre, and the full height of the base +and surbase. A I, which makes (¹⁄₃) with the vertical, determines the +form of the pediment of the smaller gables, the base of the pediment on +the outer buttress, the base of the ornamental recess between the outer +and inner buttresses, the spring of the arch of the centre window, the +tops of the pediments on the inner buttresses, and the spring of the +arch of the upper window. A K, which makes (¹⁄₂), determines the height +of the outer buttress; and A Z, which makes (¹⁄₆) with the horizontal, +determines that of the inner buttresses. For the reasons already given, +I need not here go into further detail.[11] It is, however, worthy of +remark in this place, that notwithstanding the great difference which +exists between the style of composition in this Gothic design, and in +that of the east end of the Parthenon, the harmonic elements upon which +the orthographic beauty of the one depends, are almost identical with +those of the other. + + +_On the Curvilinear Forms and Proportions of Architecture._ + +Each regular rectilinear figure has a curvilinear figure that exclusively +belongs to it, and to which may be applied a corresponding terminology. +For instance, the circle belongs to the equilateral rectangle; that +is, the rectangle of (¹⁄₂), an ellipse to every other rectangle, and +a composite ellipse to every isosceles triangle. Thus the most simple +elements of beauty in the curvilinear forms of architectural design are +the following three figures:— + + The circle, + The ellipse, and + The composite ellipse. + +I find it necessary in this place to go into some details regarding +the specific character of the two latter figures, because the proper +mode of describing these beautiful curves, and their high value in the +practice of the architectural draughtsman and ornamental designer, seem +as yet unknown. In proof of this assertion, I must again refer to Mr +Penrose’s great work published by the “Society of Dilettanti.” At page +52 of that work it is observed, that “by whatever means an ellipse is to +be constructed mechanically, it is a work of time (if not of absolute +difficulty) so to arrange the foci, &c., as to produce an ellipse of any +exact length and breadth which may be desired.” Now, this is far from +being the case, for the method of arranging the foci of an ellipse of any +given length and breadth is extremely simple, being as follows:— + +Let A B C (figure 1) be the length, and D B E the breadth of the desired +ellipse. + +[Illustration: Fig. 1.] + +Take A B upon the compasses, and place the point of one leg upon E and +the point of the other upon the line A B, it will meet it at F, which is +one focus: keeping the point of the one leg upon E, remove the point of +the other to the line B C, and it will meet it at G, which is the other +focus. But, when the proportions of an ellipse are to be imparted by +means of one of the harmonic angles, suppose the angle of (¹⁄₃), then the +following is the process:— + +Let A B C (figure 2) represent the length of the intended ellipse. +Through B draw B _e_ indefinitely, at right angles with A B C; through C +draw the line C _f_ indefinitely, making, with B C, an angle of (¹⁄₃). + +Take B C upon the compasses, and place the point of one leg upon D where +C f intersects B _e_, and the point of the other upon the line A B, it +will meet it at F, which is one focus. Keeping the point of one leg still +upon D, remove the point of the other to the line B C, and it will meet +it at G, which is the other focus. + +[Illustration: Fig. 2.] + +The foci being in either case thus simply ascertained, the method of +describing the curve on a small scale is equally simple. + +[Sidenote: Plate V.] + +A pin is fixed into each of the two foci, and another into the point D. +Around these three pins a waxed thread, flexible but not elastic, is +tied, care being taken that the knot be of a kind that will not slip. +The pin at D is now removed, and a hard black lead pencil introduced +within the thread band. The pencil is then moved around the pins fixed +in the foci, keeping the thread band at a full and equal tension; thus +simply the ellipse is described. When, however, the governing angle is +acute, say less than (¹⁄₆), it is requisite to adopt a more accurate +method of description,[12] as the architectural examples which follow +will shew. But architectural draughtsmen and ornamental designers would +do well to supply themselves, for ordinary practice, with half a dozen +series of ellipses, varying in the proportions of their axes from (⁴⁄₉) +to (¹⁄₆) of the scale, and the length of their major axes from 1 to 6 +inches. These should be described by the above simple process, upon +very strong drawing paper, and carefully cut out, the edge of the paper +being kept smooth, and each ellipse having its greater and lesser axes, +its foci, and the hypothenuse of its scalene triangle drawn upon it. To +exemplify this, I give Plate V., which exhibits the ellipses of (¹⁄₃), +(¹⁄₄), (¹⁄₅), and (¹⁄₆), inscribed in their rectangles, on which _a b_ +and _c d_ are respectively the greater and lesser axes, _o o_ the foci, +and _d b_ the angle of each. Such a series of these beautiful figures +would be found particularly useful in drawing the mouldings of Grecian +architecture; for, to describe the curvilinear contour of such mouldings +from single points, as has been done with those which embellish even our +most pretending attempts at the restoration of that classical style of +architecture, is to give the resemblance of an external form without the +harmony which constitutes its real beauty. + +Mr Penrose, owing to the supposed difficulty regarding the description of +ellipses just alluded to, endeavours to shew that the curves of all the +mouldings throughout the Parthenon were either parabolic or hyperbolic; +but I believe such curves can have no connexion with the elementary forms +of architecture, for they are curves which represent motion, and do not, +by continued production, form closed figures. + +But I have shewn, in a former work,[13] that the contours of these +mouldings are composed of curves of the composite ellipse,—a figure +which I so name because it is composed simply of arcs of various +ellipses harmonically flowing into each other. The composite ellipse, +when drawn systematically upon the isosceles triangle, resembles closely +parabolic and hyperbolic curves—only differing from these inasmuch as it +possesses the essential quality of circumscribing harmonically one of the +elementary rectilinear figures employed in architecture, while those of +the parabola and hyperbola, as I have just observed, are merely curves +of motion, and, consequently, never can harmonically circumscribe or be +resolved into any regular figure. + +The composite ellipse may be thus described. + +[Sidenote: Plate VI.] + +Let A B C (Plate VI.) be a vertical isosceles triangle of (¹⁄₆), bisect A +B in D, and through D draw indefinitely D _f_ perpendicular to A B, and +through B draw indefinitely B _g_, making the angle D B _g_ (¹⁄₈), D _f_ +and B _g_ intersecting each other in M. Take B D and D M as semi-axes of +an ellipse, the foci of which will be at _p_ and _q_, in each of these, +and in each of the foci _h t_ and _k r_ in the lines A C and B C, fix +a pin, and one also in the point M, tie a thread around these pins, +withdraw the pin from M, and trace the composite ellipse in the manner +already described with respect to the simple ellipse. + +In some of my earlier works I described this figure by taking the angles +of the isosceles triangle as foci; but the above method is much more +correct. As the elementary angle of the triangle is (¹⁄₆), and that of +the elliptic curve described around it (¹⁄₈), I call it the composite +ellipse of (¹⁄₆) and (¹⁄₈), their harmonic ratio being 4:3; and so on of +all others, according to the difference that may thus exist between the +elementary angles. + +The visible curves which soften and beautify the melody of the outline +of the front of the Parthenon, as given in Mr Penrose’s great work, I +have carefully analysed, and have found them in as perfect agreement +with this system, as its rectilinear harmony has been shewn to be. This +I demonstrated in the work just referred to[14] by a series of twelve +plates, shewing that the entasis of the columns (a subject upon which +there has been much speculation) is simply an arc of an ellipse of +(¹⁄₄₈), whose greater axis makes with the vertical an angle of (¹⁄₆₄); +or simply, the form of one of these columns is the frustrum of an +elliptic-sided or prolate-spheroidal cone, whose section is a composite +ellipse of (¹⁄₄₈) and (¹⁄₆₄), the harmonic ratio of these two angles +being 4:3, the same as that of the angles of the composite ellipse just +exemplified. + +[Sidenote: Plate VII.] + +[Sidenote: Plate VIII.] + +In Plate VII. is represented the section of such a cone, of which A B C +is the isosceles triangle of (¹⁄₄₈), and B D and D M the semi-axes of an +ellipse of (¹⁄₆₄). M N and O P are the entases of the column, and _d e f_ +the normal construction of the capital. All these are fully illustrated +in the work above referred to,[15] in which I have also shewn that the +curve of the neck of the column is that of an ellipse of (¹⁄₆); the curve +of the capital or echinus, that of an ellipse of (¹⁄₁₄); the curve of +the moulding under the cymatium of the pediment, that of an ellipse of +(¹⁄₃); and the curve of the bed-moulding of the cornice of the pediment, +that of an ellipse of (¹⁄₃). The curve of the cavetto of the soffit of +the corona is composed of ellipses of (¹⁄₆) and (¹⁄₁₄); the curve of the +cymatium which surmounts the corona, is that of an ellipse of (¹⁄₃); the +curve of the moulding of the capital of the antæ of the posticum, that of +an ellipse of (¹⁄₃); the curves of the lower moulding of the same capital +are composed of those of an ellipse of (¹⁄₃) and of the circle (¹⁄₂); the +curve of the moulding which is placed between the two latter is that of +an ellipse of (¹⁄₃); the curve of the upper moulding of the band under +the beams of the ceiling of the peristyle, that of an ellipse of (¹⁄₃); +the curve of the lower moulding of the same band, that of an ellipse of +(¹⁄₄); and the curves of the moulding at the bottom of the small step +or podium between the columns, are those of the circle (¹⁄₂) and of an +ellipse of (¹⁄₃). I have also shewn the curve of the fluting of the +columns to be that of (¹⁄₁₄). The greater axis of each of these ellipses, +when not in the vertical or horizontal lines, makes an harmonic angle +with one or other of them. In Plate VIII., sections of the two last-named +mouldings are represented full size, which will give the reader an idea +of the simple manner in which the ellipses are employed in the production +of those harmonic curves. + +Thus we find that the system here adopted for applying this law of +nature to the production of beauty in the abstract forms employed +in architectural composition, so far from involving us in anything +complicated, is characterised by extreme simplicity. + +In concluding this part of my treatise, I may here repeat what I have +advanced in a late work,[16] viz., my conviction of the probability that +a system of applying this law of nature in architectural construction +was the only great practical secret of the Freemasons, all their other +secrets being connected, not with their art, but with the social +constitution of their society. This valuable secret, however, seems +to have been lost, as its practical application fell into disuse; but, +as that ancient society consisted of speculative as well as practical +masons, the secrets connected with their social union have still been +preserved, along with the excellent laws by which the brotherhood is +governed. It can scarcely be doubted that there was some such practically +useful secret amongst the Freemasons or early Gothic architects; for +we find in all the venerable remains of their art which exist in this +country, symmetrical elegance of form pervading the general design, +harmonious proportion amongst all the parts, beautiful geometrical +arrangements throughout all the tracery, as well as in the elegantly +symmetrised foliated decorations which belong to that style of +architecture. But it is at the same time worthy of remark, that whenever +they diverged from architecture to sculpture and painting, and attempted +to represent the human figure, or even any of the lower animals, their +productions are such as to convince us that in this country these arts +were in a very degraded state of barbarism—the figures are often much +disproportioned in their parts and distorted in their attitudes, while +their representations of animals and chimeras are whimsically absurd. +It would, therefore, appear that architecture, as a fine art, must have +been preserved by some peculiar influence from partaking of the barbarism +so apparent in the sister arts of that period. Although its practical +secrets have been long lost, the Freemasons of the present day trace the +original possession of them to Moses, who, they say, “modelled masonry +into a perfect system, and circumscribed its mysteries by _land-marks_ +significant and unalterable.” Now, as Moses received his education in +Egypt, where Pythagoras is said to have acquired his first knowledge of +the harmonic law of numbers, it is highly probable that this perfect +system of the great Jewish legislator was based upon the same law of +nature which constituted the foundation of the Pythagorean philosophy, +and ultimately led to that excellence in art which is still the +admiration of the world. + +Pythagoras, it would appear, formed a system much more perfect and +comprehensive than that practised by the Freemasons in the middle ages of +Christianity; for it was as applicable to sculpture, painting, and music, +as it was to architecture. This perfection in architecture is strikingly +exemplified in the Parthenon, as compared with the Gothic structures +of the middle ages; for it will be found that the whole six elementary +figures I have enumerated as belonging to architecture, are required in +completing the orthographic beauty of that noble structure. And amongst +these, none conduce more to that beauty than the simple and composite +ellipses. Now, in the architecture of the best periods of Gothic, or, +indeed, in that of any after period (Roman architecture included), these +beautiful curves seem to have been ignored, and that of the circle alone +employed. + +Be those matters as they may, however, the great law of numerical +harmonic ratio remains unalterable, and a proper application of it in +the science of art will never fail to be as productive of effect, as its +operation in nature is universal, certain, and continual. + + + + +THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE HUMAN HEAD AND COUNTENANCE. + + +The most remarkable characteristics of the human head and countenance +are the globular form of the cranium, united as it is with the prolate +spheroidal form produced by the parts which constitute the face, and the +approximation of the profile to the vertical; for in none of the lower +animals does the skull present so near a resemblance to a combination +of these geometric forms, nor the plane of the face to this direction. +We also find that although these peculiar characteristics are variously +modified among the numerous races of mankind, yet one law appears to +govern the beauty of the whole. The highest and most cultivated of these +races, however, present only an approximation to the perfect development +of those distinguishing marks of humanity; and therefore the beauty of +form and proportion which in nature characterises the human head and +countenance, exhibits only a partial development of the harmonic law of +visible beauty. On the other hand, we find that, in their sculpture, the +ancient Greeks surpassed ordinary nature, and produced in their beau +ideal a species of beauty free from the imperfections and peculiarities +that constitute the individuality by which the countenances of men are +distinguished from each other. It may be requisite here to remark, that +this species of beauty is independent of the more intellectual quality +of expression. For as Sir Charles Bell has said, “Beauty of countenance +may be defined in words, as well as demonstrated in art. A face may +be beautiful in sleep, and a statue without expression may be highly +beautiful. But it will be said there is expression in the sleeping figure +or in the statue. Is it not rather that we see in these the capacity +for expression?—that our minds are active in imagining what may be the +motions of these features when awake or animated? Thus, we speak of an +expressive face before we have seen a movement grave or cheerful, or any +indication in the features of what prevails in the heart.” + +This capacity for expression certainly enhances our admiration of the +human countenance; but it is more a concomitant of the primary cause of +its beauty than the cause itself. This cause rests on that simple and +secure basis—the harmonic law of nature; for the nearer the countenance +approximates to an harmonious combination of the most perfect figures in +geometry, or rather the more its general form and the relation of its +individual parts are arranged in obedience to that law, the higher its +degree of beauty, and the greater its capacity for the expression of the +passions. + +Various attempts have been made to define geometrically the difference +between the ordinary and the ideal beauty of the human head and +countenance, the most prominent of which is that of Camper. He traced, +upon a profile of the skull, a line in a horizontal direction, passing +through the foramen of the ear and the exterior margin of the sockets of +the front teeth of the upper jaw, upon which he raised an oblique line, +tangential to the margin of these sockets, and to the most prominent part +of the forehead. Agreeably to the obliquity of this line, he determined +the relative proportion of the areas occupied by the brain and by the +face, and hence inferred the degree of intellect. When he applied this +measurement to the heads of the antique statues, he found the angle much +greater than in ordinary nature; but that this simple fact afforded no +rule for the reproduction of the ideal beauty of ancient Greek art, is +very evident from the heads and countenances by which his treatise is +illustrated. Sir Charles Bell justly remarks, that although, by Camper’s +method, the forehead may be thrown forward, yet, while the features +of common nature are preserved, we refuse to acknowledge a similarity +to the beautiful forms of the antique marbles. “It is true,” he says, +“that, by advancing the forehead, it is raised, the face is shortened, +and the eye brought to the centre of the head. But with all this, there +is much wanting—that which measurement, or a mere line, will not shew +us.”—“The truth is, that we are more moved by the features than by the +form of the whole head. Unless there be a conformity in every feature to +the general shape of the head, throwing the forehead forward on the face +produces deformity; and the question returns with full force—How is it +that we are led to concede that the antique head of the Apollo, or of the +Jupiter, is beautiful when the facial line makes a hundred degrees with +the horizontal line? In other words—How do we admit that to be beautiful +which is not natural? Simply for the same reason that, if we discover a +broken portion of an antique, a nose, or a chin of marble, we can say, +without deliberation—This must have belonged to a work of antiquity; +which proves that the character is distinguishable in every part—in each +feature, as well as in the whole head.” + +Dr Oken says upon this subject:[17]—“The face is beautiful whose nose is +parallel to the spine. No human face has grown into this estate; but +every nose makes an acute angle with the spine. The facial angle is, as +is well known, 80°. What, as yet, no man has remarked, and what is not to +be remarked, either, without our view of the cranial signification, the +old masters have felt through inspiration. They have not only made the +facial angle a right angle, but have even stepped beyond this—the Romans +going up to 96°, the Greeks even to 100°. Whence comes it that this +unnatural face of the Grecian works of art is still more beautiful than +that of the Roman, when the latter comes nearer to nature? The reason +thereof resides in the fact of the Grecian artistic face representing +nature’s design more than that of the Roman; for, in the former, the nose +is placed quite perpendicular, or parallel to the spinal cord, and thus +returns whither it has been derived.” + +Other various and conflicting opinions upon this subject have been +given to the world; but we find that the principle from which arose +the ideal beauty of the head and countenance, as represented in works +of ancient Greek art, is still a matter of dispute. When, however, we +examine carefully a fine specimen, we find its beauty and grandeur to +depend more upon the degree of harmony amongst its parts, as to their +relative proportions and mode of arrangement, than upon their excellence +taken individually. It is, therefore, clear that those (and they are +many) who attribute the beauty of ancient Greek sculpture merely to a +selection of parts from various models, must be in error. No assemblage +of parts from ordinary nature could have produced its principal +characteristic, the excess in the angle of the facial line, much less +could it have led to that exquisite harmony of parts by which it is so +eminently distinguished; neither can we reasonably agree with Dr Oken and +others, who assert that it was produced by an exclusive degree of the +inspiration of genius amongst the Greek people during a certain period. + +That the inspiration of genius, combined with a careful study of nature, +were essential elements in the production of the great works which have +been handed down to us, no one will deny; but these elements have existed +in all ages, whilst the ideal head belongs exclusively to the Greeks +during the period in which the schools of Pythagoras and Plato were open. +Is it not, therefore, reasonable to suppose, that, besides genius and the +study of nature, another element was employed in the production of this +excellence, and that this element arose from the precise mathematical +doctrines taught in the schools of these philosophers? + +An application of the great harmonic law seems to prove that there is no +object in nature in which the science of beauty is more clearly developed +than in the human head and countenance, nor to the representations of +which the same science is more easily applied; and it is to the mode +in which this is done that the varieties of sex and character may be +imparted to works of art. Having gone into full detail, and given ample +illustrations in a former work,[18] it is unnecessary for me to enter +upon that part of the subject in this _résumé_; but only to shew the +typical structure of beauty by which this noble work of creation is +distinguished. + +The angles which govern the form and proportions of the human head and +countenance are, with the right angle, a series of seven, which, from the +simplicity of their ratios to each other, are calculated to produce the +most perfect concord. It consists of the right angle and its following +parts— + + Tonic. Dominant. Mediant. Subtonic. + (¹⁄₂) (¹⁄₃) (¹⁄₅) (¹⁄₇) + (¹⁄₄) (¹⁄₆) + +These angles, and the figures which belong to them, are thus arranged:— + +[Sidenote: Plate IX.] + +The vertical line A B (Plate IX. fig. 2) represents the full length of +the head and face. Taking this line as the greater axis of an ellipse of +(¹⁄₃), such an ellipse is described around it. Through A the lines A G, +A K, A L, A M, and A N, are drawn on each side of the line A B, making, +with the vertical, respectively the angles of (¹⁄₃), (¹⁄₄), (¹⁄₅), (¹⁄₆), +and (¹⁄₇). Through the points G, K, L, M, and N, where these straight +lines meet the curved line of the ellipse, horizontal lines are drawn +by which the following isosceles triangles are formed, A G G, A K K, A +L L, A M M, and A N N. From the centre X of the equilateral triangle +A G G the curvilinear figure of (¹⁄₂), viz., the circle, is described +circumscribing that triangle. + +The curvilinear plane figures of (¹⁄₂) and (¹⁄₃), respectively, represent +the solid bodies of which they are sections, viz., a sphere and a prolate +spheroid. These bodies, from the manner in which they are here placed, +are partially amalgamated, as shewn in figures 1 and 3 of the same plate, +thus representing the form of the human head and countenance, both in +their external appearance and osseous structure, more correctly than they +could be represented by any other geometrical figures. Thus, the angles +of (¹⁄₂) and (¹⁄₃) determine the typical form. + +From each of the points _u_ and _n_, where A M cuts G G on both sides of +A B, a circle is described through the points _p_ and _q_, where A K cuts +G G on both sides of A B, and with the same radius a circle is described +from the point _a_, where K K cuts A B. + +The circles _u_ and _n_ determine the position and size of the eyeballs, +and the circle _a_ the width of the nose, as also the horizontal width of +the mouth. + +The lines G G and K K also determine the length of the joinings of the +ear to the head. The lines L L and M M determine the vertical width of +the mouth and lips when at perfect repose, and the line N N the superior +edge of the chin. Thus simply are the features arranged and proportioned +on the facial surface. + +It must, however, be borne in mind, that in treating simply of the +æsthetic beauty of the human head and countenance, we have only to do +with the external appearance. In this research, therefore, the system +of Dr Camper, Dr Owen, and others, whose investigations were more of +a physiological than an æsthetic character, can be of little service; +because, according to that system, the facial angle is determined by +drawing a line tangential to the exterior margin of the sockets of +the front teeth of the upper jaw, and the most prominent part of the +forehead. Now, as these sockets are, when the skull is naturally clothed, +and the features in repose, entirely concealed by the upper lip, we must +take the prominent part of it, instead of the sockets under it, in order +to determine properly this distinguishing mark of humanity. And I believe +it will be found, that when the head is properly poised, the nearer the +angle which this line makes with the horizontal approaches 90°, the more +symmetrically beautiful will be the general arrangement of the parts (see +line _y z_, figure 3, Plate IX.). + + + + +THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE FORM OF THE HUMAN FIGURE. + + +The manner in which this science is developed in the symmetrical +proportions of the entire human figure, is as remarkable for its +simplicity as it has been shewn to be in those of the head and +countenance. Having gone into very full details, and given ample +illustration in two former works[19] upon this subject, I may here +confine myself to the illustration of one description of figure, and to a +reiteration of some facts stated in these works. These facts are, _1st_, +That on a given line the human figure is developed, as to its principal +points, entirely by lines drawn either from the extremities of this line, +or from some obvious or determined localities. _2d_, That the angles +which these lines make with the given line, are all simple sub-multiples +of some given fundamental angle, or bear to it a proportion expressible +under the most simple relations, such as those which constitute the scale +of music. _3d_, That the contour is resolved into a series of ellipses +of the same simple angles. And, _4th_, That these ellipses, like the +lines, are inclined to the first given line by angles which are simple +sub-multiples of the given fundamental angle. From which four facts, +and agreeably to the hypothesis I have adopted, it results as a natural +consequence that the only effort which the mind exercises through the +eye, in order to put itself in possession of the data for forming its +judgment, is this, that it compares the angles about a point, and thereby +appreciates the simplicity of their relations. In selecting the prominent +features of a figure, the eye is not seeking to compare their relative +distances—it is occupied solely with their relative positions. In tracing +the contour, in like manner, it is not left in vague uncertainty as +to what is the curve which is presented to it; unconsciously it feels +the complete ellipse developed before it; and if that ellipse and its +position are both formed by angles of the same simple relative value as +those which aided its determination of the positions of the prominent +features, it is satisfied, and finds the symmetry perfect. + +Müller, and other investigators into the archæology of art, refer to the +great difficulty which exists in discovering the principles which the +ancients followed in regard to the proportions of the human figure, from +the different sexes and characters to which they require to be applied. +But in the system thus founded upon the harmonic law of nature, no such +difficulty is felt, for it is as applicable to the massive proportions +which characterise the ancient representations of the Hercules, as to the +delicate and perfectly symmetrical beauty of the Venus. This change is +effected simply by an increase in the fundamental angle. For instance, +in the construction of a figure of the exact proportions of the Venus, +the right angle is adopted. But in the construction of a figure of the +massive proportions of the Hercules, it is requisite to adopt an angle +which bears to the right angle the ratio of 6:5. The adoption of this +angle I have shewn in another work[20] to produce in the Hercules those +proportions which are so characteristic of physical power. The ellipses +which govern the outline, being also formed upon the same larger class +of angles, give the contour of the muscles a more massive character. In +comparing the male and female forms thus geometrically constructed, it +will be found that that of the female is more harmoniously symmetrical, +because the right angle is the fundamental angle for the trunk and the +limbs as well as for the head and countenance; while in that of the male, +the right angle is the fundamental angle for the head only. It may also +be observed, that, from the greater proportional width of the pelvis +of the female, the centres of that motion which the heads of the thigh +bones perform in the cotyloid cavities, and the centres of that still +more extensive range of motion which the arm is capable of performing at +the shoulder joints, are nearly in the same line which determines the +central motion of the vertebral column, while those of the male are not; +consequently all the motions of the female are more graceful than those +of the male. + +This difference between the fundamental angles, which impart to the +human figure, on the one hand, the beauty of feminine proportion and +contour, and on the other, the grandeur of masculine strength, being in +the ratio of 5:6, allows ample latitude for those intermediate classes of +proportions which the ancients imparted to their various other deities +in which these two qualities were blended. I therefore confine myself to +an illustration of the external contour of the form, and the relative +proportions of all the parts of a female figure, such as those of the +statues of the Venus of Melos and Venus of Medici. + +The angles which govern the form and proportions of such a figure are, +with the right angle, a series of twelve, as follows:— + + Tonic. Dominant. Mediant. Subtonic. Supertonic. + (¹⁄₂) (¹⁄₃) (¹⁄₅) (¹⁄₇) (¹⁄₉) + (¹⁄₄) (¹⁄₆) (¹⁄₁₀) (¹⁄₁₄) + (¹⁄₈) (¹⁄₁₂) + +These angles are employed in the construction of a diagram, which +determines the proportions of the parts throughout the whole figure. +Thus:— + +[Sidenote: Plate X.] + +Let the line A B (fig. 1, plate X.) represent the height of the figure +to be constructed. At the point A, make the angles of C A D (¹⁄₃), F A G +(¹⁄₄), H A I (¹⁄₅), K A L (¹⁄₆), and M A N (¹⁄₇). At the point B, make +the angles K B L (¹⁄₈), U B A (¹⁄₁₂), and O B A (¹⁄₁₄). + +Through the point K, in which the lines A K and B K intersect one +another, draw P K O parallel to A B, and through C F H and M, where +this line meets A C, A F, A H, and A M, draw C D, F G, H I, and M N, +perpendicular to A B; draw also K L perpendicular to A B; join B F and +B H, and through C draw C E, making with A B the angle (¹⁄₂), which +completes the arrangement of the eleven angles upon A B; for F B G is +very nearly (¹⁄₁₀), and H B I very nearly (¹⁄₉). + +At the point _f_, where A C intersects O B, draw _f a_ perpendicular to +A B; and through the point _i_, where B O intersects M N, draw S _i_ T +parallel to A C. + +Through _m_, where S _i_ T intersects F B, draw _m n_; through _β_, where +S _i_ T intersects K B, draw _β w_; through T draw T _g_, making an angle +of (¹⁄₃) with O P. Join N P, M B, and _g_ P, and where N P intersects K +B, draw Q R perpendicular to A B. + +On A E as a diameter, describe a circle cutting A C in _r_, and draw _r +o_ perpendicular to A B. + +With A _o_ and _o r_ as semi-axes, describe the ellipse A _r e_, cutting +A H in _t_; and draw _t u_ perpendicular to A B. With A _u_ and _t u_, as +semi-axes describe the ellipse A _t d_. On _a_ L, as major axis, describe +the ellipse of (¹⁄₃). + +For the side aspect or profile of the figure the diagram is thus +constructed— + +On one side of a line A B (fig. 2, Plate X.) construct the rectilinear +portion of a diagram the same as fig. 1. Through _i_ draw W Y parallel to +A B, and draw A _z_ perpendicular to A B. Make W _a_ equal to A _a_ (fig. +1), and on _a l_, as major axis, describe the ellipse of (¹⁄₄). Through +_a_ draw _a p_ parallel to A F, and through _p_ draw _p t_ perpendicular +to W Y. Through _a_ draw _f a u_ perpendicular to W Y. + +Upon a diameter equal to A E describe a circle whose circumference shall +touch A B and A _z_. With semi-axes equal to A _o_ and _o r_ (fig. +1), describe an ellipse with its major axis parallel to A B, and its +circumference touching O P and _z_ A. + +[Sidenote: Plate XI.] + +Thus simply are the diagrams of the general proportions of the human +figure, as viewed in front and in profile, constructed; and Plate XI. +gives the contour in both points of view, as composed entirely of the +curvilinear figures of (¹⁄₂), (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆). + +Further detail here would be out of place, and I shall therefore refer +those who require it to the Appendix, or the more elaborate works to +which I have already referred. + +The beauty derived from proportion, imparted by the system here pointed +out, and from a contour of curves derived from the same harmonic angles, +is not confined to the human figure, but is found in various minor +degrees of perfection in all the organic forms of nature, whether animate +or inanimate, of which I have in other works given many examples.[21] + + + + +THE SCIENCE OF BEAUTY, AS DEVELOPED IN COLOURS. + + +There is not amongst the various phenomena of nature one that more +readily excites our admiration, or makes on the mind a more vivid +impression of the order, variety, and harmonious beauty of the creation, +than that of colour. On the general landscape this phenomenon is +displayed in the production of that species of harmony in which colours +are so variously blended, and in which they are by light, shade, and +distance modified in such an infinity of gradation and hue. Although +genius is continually struggling, with but partial success, to imitate +those effects, yet, through the Divine beneficence, all whose organs of +sight are in an ordinary degree of perfection can appreciate and enjoy +them. In winter this pleasure is often to a certain extent withdrawn, +when the colourless snow alone clothes the surface of the earth. But +this is only a pause in the general harmony, which, as the spring +returns, addresses itself the more pleasingly to our perception in its +vernal melody, which, gradually resolving itself into the full rich hues +of luxuriant beauty exhibited in the foliage and flowers of summer, +subsequently rises into the more vivid and powerful harmonies of autumn’s +colouring. Thus the eye is prepared again to enjoy that rest which such +exciting causes may be said to have rendered necessary. + +When we pass from the general colouring of nature to that of particular +objects, we are again wrapt in wonder and admiration by the beauty and +harmony which so constantly, and in such infinite variety, present +themselves to our view, and which are so often found combined in the +most minute objects. And the systematic order and uniformity perceptible +amidst this endless variety in the colouring of animate and inanimate +nature is thus another characteristic of beauty equally prevalent +throughout creation. + +By this uniformity in colour, various species of animals are often +distinguished; and in each individual of most of these species, how much +is this beauty enhanced when the uniformity prevails in the resemblance +of their lateral halves! The human countenance exemplifies this in a +striking manner; the slightest variety of colour between one and another +of the double parts is at once destructive of its symmetrical beauty. +Many of the lower animals, whether inhabitants of the earth, the air, or +the water, owe much of their beauty to this kind of uniformity in the +colour of the furs, feathers, scales, or shells, with which they are +clothed. + +In the vegetable kingdom, we find a great degree of uniformity of colour +in the leaves, flowers, and fruit of the same plant, combined with all +the harmonious beauty of variety which a little careful examination +develops. + +In the colours of minerals, too, the same may be observed. In short, in +the beauty of colouring, as in every other species of beauty, uniformity +and variety are found to combine. + +An appreciation of colour depends, in the first place, as much upon the +physical powers of the eye in conveying a proper impression to the mind, +as that of music on those of the ear. But an ear for music, or an eye for +colour, are, in so far as beauty is concerned, erroneous expressions; +because they are merely applicable to the impression made upon the +senses, and do not refer to the æsthetical principles of harmony, by +which beauty can alone be understood. + +A good eye, combined with experience, may enable us to form a correct +idea as to the purity of an individual colour, or of the relative +difference existing between two separate hues; but this sort of +discrimination does not constitute that kind of appreciation of the +harmony of colour by which we admire and enjoy its development in nature +and art. The power of perceiving and appreciating beauty of any kind, +is a principle inherent in the human mind, which may be improved by +cultivation in the degree of the perfection of the art senses. Great +pains have been bestowed on the education of the ear, in assisting it to +appreciate the melody and harmony of sound; but still much remains to be +done in regard to the cultivation of the eye, in appreciating colour as +well as form. + +It is true, that there are individuals whose powers of vision are +perfect, in so far as regards the appreciation of light, shade, and +configuration, but who are totally incapable of perceiving effects +produced by the intermediate phenomenon of colour, every object appearing +to them either white, black, or neutral gray; others, who are equally +blind as to the effect of one of the three primary colours, but see the +other two perfectly, either singly or combined; while there are many +who, having the full physical power of perceiving all the varieties of +the phenomenon, and who are even capable of making nice distinctions +amongst a variety of various colours, are yet incapable of appreciating +the æsthetic quality of harmony which exists in their proper combination. +It is the same with respect to the effects of sounds upon the ear—some +have organs so constituted, that notes above or below a certain pitch +are to them inaudible; while others, with physical powers otherwise +perfect, are incapable of appreciating either melody or harmony in +musical composition. But perceptions so imperfectly constituted are, +by the goodness of the Creator, of very rare occurrence; therefore all +attempts at improvement in the science of æsthetics must be suited to the +capacities of the generality of mankind, amongst whom the perception of +colour exists in a variety as great as that by which their countenances +are distinguished. Artists now and then appear who have this intuitive +perception in such perfection, that they are capable of transferring to +their works the most beautiful harmonies and most delicate gradations of +colours, in a manner that no acquired knowledge could have enabled them +to impart. To those who possess such a gift, as well as to those to whom +the ordinary powers of perception are denied, it would be equally useless +to offer an explanation of the various modes in which the harmony of +colour develops itself, or to attempt a definition of the many various +colours, hues, tints, and shades, arising out of the simple elements of +this phenomenon. But to those whose powers lie between these extremes, +being neither above nor below cultivation, such an explanation and +definition must form a step towards the improvement of that inherent +principle which constitutes the basis of æsthetical science. + +Although the variety and harmony of colour which nature is continually +presenting to our view, are apparent to all whose visual organs are in a +natural state, and thus to the generality of mankind; yet a knowledge of +the simplicity by which this variety and beauty are produced, is, after +ages of philosophic research and experimental inquiry, only beginning to +be properly understood. + +Light may be considered as an active, and darkness a passive principle in +the economy of Nature, and colour an intermediate phenomenon arising from +their joint influence; and it is in the ratios in which these primary +principles act upon each other, by which I here intend to explain the +science of beauty as evolved in colour. It has been usual to consider +colour as an inherent quality in light, and to suppose that coloured +bodies absorb certain classes of its rays, and reflect or transmit the +remainder; but it appears to me that colour is more probably the result +of certain modes in which the opposite principles of motion and rest, +or force and resistance, operate in the production, refraction, and +reflection of light, and that each colour is mutually related, although +in different degrees, to these active and passive principles. + +White and black are the representatives of light and darkness, or +activity and rest, and are therefore calculated as pigments to reduce +colours and hues to tints and shades. + +Having, however, fully illustrated the nature of tints and shades in +a former work,[22] I shall here confine myself to colours in their +full intensity—shewing the various modifications which their union +with each other produce, along with the harmonic relations which these +modifications bear to the primaries, and to each other in respect to +warmth and coolness of tone, as well as to light and shade. + +The primary colours are red, yellow, and blue. Of these, yellow is +most allied to light, and blue to shade, while red is neutral in these +respects, being equally allied to both. In respect to tone, that of red +is warm, and that of blue cool, while the tone of yellow is neutral. The +ratios of their relations to each other in these respects will appear in +the harmonic scales to which, for the first time, I am about to subject +colours, and to systematise their various simple and compound relations, +which are as follow:— + +From the binary union of the primary colours, the secondary colours arise— + +Orange colour, from the union of yellow and red. + +Green, from the union of yellow and blue. + +Purple, from the union of red and blue. + +From the binary union of the secondary colours, the primary hues arise— + +Yellow-hue, from the union of orange and green. + +Red-hue, from the union of orange and purple. + +Blue-hue, from the union of purple and green. + +From the binary union of the primary hues, the secondary hues arise— + +Orange-hue, from the union of yellow-hue and red-hue. + +Green-hue, from the union of yellow-hue and blue-hue. + +Purple-hue, from the union of red-hue and blue-hue. + +Each hue owes its characteristic distinction to the proportionate +predominance or subordination of one or other of the three primary +colours in its composition. + +It follows, that in every hue of _red_, yellow and blue are subordinate; +in every hue of _yellow_, red and blue are subordinate; and in every +hue of _blue_, red and yellow are subordinate. In like manner, in every +hue of _green_, red is subordinate; in every hue of _orange_, blue is +subordinate; and in every hue of _purple_, yellow is subordinate. + +By the union of two primary colours, in the production of a secondary +colour, the nature of both primaries is altered; and as there are only +three primary or simple colours in the scale, the two that are united +harmonically in a compound colour, form the natural contrast to the +remaining simple colour. + +Notwithstanding all the variety that extends beyond the six positive +colours, it may be said that there are only three proper contrasts of +colour in nature, and that all others are simply modifications of these. + +Pure red is the most perfect contrast to pure green; because it is +characterised amongst the primary colours by warmth of tone, while +amongst the secondary colours green is distinguished by coolness of tone, +both being equally related to the primary elements of light and shade. + +Pure yellow is the most perfect contrast to pure purple; because it is +characterised amongst the primary colours as most allied to light, whilst +pure purple is characterised amongst the secondaries as most allied to +shade, both being equally neutral as to tone. + +Pure blue is the most perfect contrast to pure orange; because it is +characterised amongst the primary colours as not only the most allied +to shade, but as being the coolest in tone, whilst pure orange is +characterised amongst the secondaries as being the most allied to light +and the warmest in tone. The same principle operates throughout all the +modifications of these primary and secondary colours. + +Such is the simple nature of contrast upon which the beauty of colouring +mainly depends. + +It being now established as a scientific fact, that the effect of +light upon the eye is the result of an ethereal action, similar to the +atmospheric action by which the effect of sound is produced upon the +ear; also, that the various colours which light assumes are the effect +of certain modifications in this ethereal action;—just as the various +sounds, which constitute the scale of musical notes, are known to be the +effect of certain modifications in the atmospheric action by which sounds +in general are produced: + +Therefore, as harmony may thus be impressed upon the mind through +either of these two art senses—hearing and seeing—the principles which +govern the modifications in the ethereal action of light, so as to +produce through the eye the effect of harmony, cannot be supposed to +differ from those principles which we know govern the modifications of +the atmospheric action of sound, in producing through the ear a like +effect. I shall therefore endeavour to illustrate the science of beauty +as evolved in colours, by forming scales of their various modifications +agreeably to the same Pythagorean system of numerical ratio from which +the harmonic elements of beauty in sounds were originally evolved, and by +which I have endeavoured in this, as in previous works, to systematise +the harmonic beauty of forms. + +[Illustration] + +It will be observed, that with a view to avoid complexity as much as +possible, I have, in the arrangement of the above series of scales, not +only confined myself to the merely elementary parts of the Pythagorean +system, but have left out the harmonic modifications upon (¹⁄₁₁) +and (¹⁄₁₃), in order that the arithmetical progression might not be +interrupted.[23] + +The above elementary process will, I trust, be found sufficient to +explain the progress, by harmonic union, of a primary colour to a +toned gray, and how the simple and compound colours naturally arrange +themselves into the elements of five scales, the parts of which continue +from primary to secondary colour; from secondary colour to primary hue; +from primary hue to secondary hue; from secondary hue to primary-toned +gray; and from primary-toned gray to secondary-toned gray in the simple +ratio of 2:1; thereby producing a series of the most beautiful and +perfect contrasts. + +The natural arrangement of the primary colours upon the solar spectrum is +red, yellow, blue, and I have therefore adopted the same arrangement on +the present occasion. Red being, consequently, the first tonic, and blue +the second, the divisions express the numerical ratios which the colours +bear to one another, in respect to that colourific power for which red +is pre-eminent. Thus, yellow is to red, as 2:3; blue to yellow, as 3:4; +purple to orange, as 5:6; and green to purple, as 6:7. + +The following series of completed scales are arranged upon the +foregoing principle, with the natural connecting links of red-orange, +yellow-orange, yellow-green, and blue-green, introduced in their proper +places. + +The appropriate terminology of musical notes has been adopted, and the +scales are composed as follows:— + + Scale I. consists of primary and secondary colours; + Scale II. of secondary colours and primary hues; + Scale III. of primary and secondary hues; + Scale IV. of secondary hues and primary-toned grays; and + Scale V. of primary and secondary-toned grays; + +All the parts in each of these scales, from the first tonic to the +second, relate to the same parts of the scale below them in the simple +ratio of 2:1; and serially to the first tonic in the following ratios:— + + 8:9, 4:5, 3:4, 2:3, 3:5, 4:7, 8:15, 1:2. + + +_First Series of Scales._ + + ----+------+-------+-------+-------+-------+-------+------+-------+------- + |Tonic.| | | | | | | | + | |Supertonic. | | | | | | + | | |Mediant. | | | | | + | | | |Subdominant. | | | | + | | | | |Dominant. | | | + | | | | | |Submediant. | | + | | | | | | |Subtonic. | + | | | | | | | |Semi-subtonic. + | | | | | | | | |Tonic. + ----+------+-------+-------+-------+-------+-------+------+-------+------- + I. |(¹⁄₂) |(⁴⁄₉) |(²⁄₅) |(³⁄₈) |(¹⁄₃) |(³⁄₁₀) |(²⁄₇) |(⁴⁄₁₅) |(¹⁄₄) + |Red. |Red- |Orange.|Yellow-|Yellow.|Yellow-|Green.|Blue- |Blue. + | |orange.| |orange.| |green. | |green. | + ----+------+-------+-------+-------+-------+-------+------+-------+------- + II. |(¹⁄₄) |(²⁄₉) |(¹⁄₅) |(³⁄₁₆) |(¹⁄₆) |(³⁄₂₀) |(¹⁄₇) |(²⁄₁₅) |(¹⁄₈) + |Green.|Blue- |Blue |Blue- |Purple |Red- |Red |Red- |Orange. + | |green |hue. |purple |hue. |purple |hue. |orange | + | |hue. | |hue. | |hue. | |hue. | + ----+------+-------+-------+-------+-------+-------+------+-------+------- + III.|(¹⁄₈) |(¹⁄₉) |(¹⁄₁₀) |(³⁄₃₂) |(¹⁄₁₂) |(³⁄₄₀) |(¹⁄₁₄)|(¹⁄₁₅) |(¹⁄₁₆) + |Red |Red- |Orange |Yellow-|Yellow |Yellow-|Green |Blue- |Blue + |hue. |orange |hue. |orange |hue. |green |hue. |green |hue. + | |hue. | |hue. | |hue. | |hue. | + ----+------+-------+-------+-------+-------+-------+------+-------+------- + IV. |(¹⁄₁₆)|(¹⁄₁₈) |(¹⁄₂₀) |(³⁄₆₄) |(¹⁄₂₄) |(³⁄₈₀) |(¹⁄₂₈)|(¹⁄₃₀) |(¹⁄₃₂) + |Green |Blue- |Blue- |Blue- |Purple |Red- |Red- |Red- |Orange + |hue. |green- |toned |purple-|hue. |purple-|toned |orange-|hue. + | |toned |gray. |toned | |toned |gray. |toned | + | |gray. | |gray. | |gray. | |gray. | + ----+------+-------+-------+-------+-------+-------+------+-------+------- + V. |(¹⁄₃₂)|(¹⁄₃₆) |(¹⁄₄₀) |(³⁄₁₂₈)|(¹⁄₄₈) |(³⁄₁₆₀)|(¹⁄₅₆)|(¹⁄₆₀) |(¹⁄₆₄) + |Red- |Red- |Orange-|Yellow-|Yellow-|Yellow-|Green-| Blue- |Blue- + |toned |orange-|toned |orange-|toned |green- |toned | green-|toned + |gray. |toned |gray. |toned |gray. |toned |gray. | toned |gray. + | |gray. | |gray. | | gray. | | gray. | + ----+------+-------+-------+-------+-------+-------+------+-------+------- + +To the scales of chromatic power I add another series of scales, in +which yellow, being the first tonic, and blue the second, the numerical +divisions express the ratios which the colours in each scale bear to one +another in respect to light and shade. Thus red is to yellow, in respect +to light, as 2:3; blue to red, as 3:4; green to orange, as 5:6, and +purple to green, as 6:7. + +These scales may therefore be termed scales for the colour-blind, +because, in comparing colours, those whose sight is thus defective, +naturally compare the ratios of the light and shade of which different +colours are primarily constituted. + +[Illustration] + +The following is a series of five complete scales of the harmonic parts +into which the light and shade in colours may be divided in each scale +according to the above arrangement:— + + +_Second Series of Scales._ + + ----+-------+-------+-------+-------+------+-------+-------+-------+------ + |Tonic. | | | | | | | | + | |Supertonic. | | | | | | + | | |Mediant. | | | | | + | | | |Subdominant. | | | | + | | | | |Dominant. | | | + | | | | | |Submediant. | | + | | | | | | |Subtonic. | + | | | | | | | |Semi-subtonic. + | | | | | | | | |Tonic. + ----+-------+-------+-------+-------+------+-------+-------+-------+------ + I. |(¹⁄₂) |(⁴⁄₉) |(²⁄₅) |(³⁄₈) |(¹⁄₃) |(³⁄₁₀) |(²⁄₇) |(⁴⁄₁₅) |(¹⁄₄) + |Yellow.|Yellow-|Orange.|Red- |Red. |Red- |Purple.|Blue- |Blue. + | |orange.| |orange.| |purple.| |purple.| + ----+-------+-------+-------+-------+------+-------+-------+-------+------ + II. |(¹⁄₄) |(²⁄₉) |(¹⁄₅) |(³⁄₁₆) |(¹⁄₆) |(³⁄₂₀) |(¹⁄₇) |(²⁄₁₅) |(¹⁄₈) + |Purple.|Blue- |Blue |Blue- |Green.|Yellow-|Yellow |Yellow-|Orange + | |purple |hue. |green | |green |hue. |orange | + | |hue. | |hue. | |hue. | |hue. | + ----+-------+-------+-------+-------+------+-------+-------+-------+------ + III.|(¹⁄₈) |(¹⁄₉) |(¹⁄₁₀) |(³⁄₃₂) |(¹⁄₁₂)|(³⁄₄₀) |(¹⁄₁₄) |(¹⁄₁₅) |(¹⁄₁₆) + |Yellow |Yellow-|Orange |Red- |Red |Red- |Purple |Blue- |Blue + |hue. |orange |hue. |orange |hue. |purple |hue. |purple |hue. + | |hue. | |hue. | |hue. | |hue. | + ----+-------+-------+-------+-------+------+-------+-------+-------+------ + IV. |(¹⁄₁₆) |(¹⁄₁₈) |(¹⁄₂₀) |(³⁄₆₄) |(¹⁄₂₄)|(³⁄₈₀) |(¹⁄₂₈) |(¹⁄₃₀) |(¹⁄₃₂) + |Purple |Blue- |Blue- |Blue- |Green |Yellow-|Yellow-|Yellow-|Orange + |hue. |purple-|toned |green- |hue. |green- |toned |orange-|hue. + | |toned |gray. |toned | |toned |gray. |toned | + | |gray. | |gray. | |gray. | |gray. | + ----+-------+-------+-------+-------+------+-------+-------+-------+------ + V. |(¹⁄₃₂) |(¹⁄₃₆) |(¹⁄₄₀) |(³⁄₁₂₈)|(¹⁄₄₈)|(³⁄₁₆₀)|(¹⁄₅₆) |(¹⁄₆₀) |(¹⁄₆₄) + |Yellow-|Yellow-|Orange-|Red- |Red- |Red- |Purple-|Blue- |Blue- + |toned |orange-|toned |orange-|toned |purple-|toned |green- |toned + |gray. |toned |gray. |toned |gray. |toned |gray. |toned |gray. + | |gray. | |gray. | |gray. | |gray. | + ----+-------+-------+-------+-------+------+-------+-------+-------+------ + +Should I be correct in arranging colours upon scales identical with those +upon which musical notes have been arranged, and in assuming that colours +have the same ratios to each other, in respect to their harmonic power +upon the eye, which musical notes have in respect to their harmonic power +upon the ear, the colourist may yet be enabled to impart harmonic beauty +to his works with as much certainty and ease, as the musician imparts the +same quality to his compositions: for the colourist has no more right to +trust exclusively to his eye in the arrangement of colours, than the +musician has to trust exclusively to his ear in the arrangement of sounds. + +We find, in comparing the dominant parts in the first and second scales +of the second series, that they are equal as to light and shade, so that +their relative powers of contrast depend entirely upon colour. Hence it +is that red and green are the two colours, the difference between which +the colour-blind are least able to appreciate. Professor George Wilson, +in his excellent work, “Researches on Colour-Blindness,” mentions the +case of an engraver, which proves the power of the eye in being able to +appreciate these original constituents of colour, irrespective of the +intermediate phenomenon of tone. This engraver, instead of expressing +regret on account of his being colour-blind, observed to the professor, +“My defective vision is, to a certain extent, a useful and valuable +quality. Thus, an engraver has two negatives to deal with, _i.e._, white +and black. Now, when I look at a picture, I see it only in white and +black, or light and shade, or, as artists term it, the effect. I find +at times many of my brother engravers in doubt how to translate certain +colours of pictures, which to me are matters of decided certainty and +ease. Thus to me it is valuable.” + +The colour-blind are therefore as incapable of receiving pleasure from +the harmonious union of various colours, as those who, to use a common +term, have no ear for music, are of being gratified by the “melody of +sweet sounds.” + +The generality of mankind are, however, capable of appreciating the +harmony of colour which, like that of both sound and form, arises from +the simultaneous exhibition of opposite principles having a ratio to each +other. These principles are in continual operation throughout nature, +and from them we often derive pleasure without being conscious of the +cause. All who are not colour-blind must have felt themselves struck +with the harmonic beauty of a cloudless sky, although in it there is no +configuration, and at first sight apparently but one colour. Now, as +we know that there can be no more impression of harmony made upon the +mind by looking upon a single colour, than there could be by listening +to a single continued musical note, however sweet its tone, we are apt +at first to imagine that the organ of vision has, in some measure, +conveyed a false impression to the mind. But it has not done so; for +light, when reflected from the atmosphere, produces those cool tones of +blue, gray, and purple, which seem to clothe the distant mountains; but, +when transmitted through the same atmosphere, it produces those numerous +warm tints, the most intense of which give the gorgeous effects which +so often accompany the setting sun. We have, therefore, in the upper +part of a clear sky, where the atmosphere may be said to be illuminated +principally by reflection from the surface of the earth, a comparatively +cool tone of blue, the result of reflection, which gradually blends into +the warm tints, the result of transmission through the same atmosphere. +Such a composition of harmonious colouring is to the eye what the voice +of the soft breath of summer amongst the trees, the hum of insects on +a sultry day, or the simple harmony of the Æolian harp, is to the ear. +To such a composition of chromatic harmony must also be referred the +universal concurrence of mankind in appreciating the peculiar beauty of +white marble statuary. That the principal constituent of beauty in such +works ought to be harmony of form, no one will deny; but this is not the +only element, as appears from the fact, that a cast in plaster of Paris, +of a fine white marble statue, although identical in form, is far less +beautiful than the original. Now this undoubtedly must be the consequence +of its having been changed from a semi-translucent substance, which, +like the atmosphere, can transmit as well as reflect light, to an opaque +substance, which can only reflect it. Thus the opposite principles of +chromatic warmth and coolness are equally balanced in white marble—the +one being the natural result of the partial transmission of light, and +the other that of its reflection. + +As a series of coloured illustrations would be beyond the scope of this +_résumé_, I may refer those who wish to prosecute the inquiry, with the +assistance of such a series, to my published works upon the subject.[24] + + + + +THE SCIENCE OF BEAUTY, APPLIED TO THE FORMS AND PROPORTIONS OF ANCIENT +GRECIAN VASES AND ORNAMENTS. + + +In examining the remains of the ornamental works of the ancient Greek +artists, it appears highly probable that the harmony of their proportions +and melody of their contour are equally the result of a systematised +application of the same harmonic law. This probability not being fully +elucidated in any of my former works, I will require to go into some +detail on the present occasion. I take for my first illustration an +unexceptionable example, viz.:— + + +_The Portland Vase._ + +Although this beautiful specimen of ancient art was found about the +middle of the sixteenth century, inclosed in a marble sarcophagus within +a sepulchral chamber under the Monte del Grano, near Rome, and although +the date of its production is unknown, yet its being a work of ancient +Grecian art is undoubted; and the exquisite beauty of its form has been +universally acknowledged, both during the time it remained in the palace +of the Barberini family at Rome, and since it was added to the treasures +of the British Museum. The forms and proportions of this gem of art +appear to me to yield an obedience to the great harmonic law of nature, +similar to that which I have instanced in the proportions and contour of +the best specimens of ancient Grecian architecture. + +[Sidenote: Plate XII.] + +Let the line A B (Plate XII.) represent the full height of the vase. +Through A draw A _a_, and through B draw B _b_ indefinitely, A _a_ making +an angle of (¹⁄₂), and B _b_ an angle of (¹⁄₃), with the vertical. +Through the point C, where A _a_ and B _b_ intersect one another, draw +D C E vertical. Through A C and B respectively, draw A D, C F, and B +E horizontal. Draw similar lines on the other side of A B, and the +rectilinear portion of the diagram is complete. + +The curvilinear contour may be thus added:— + +Take a cut-out ellipse of (¹⁄₄), whose greater axis is equal to the line +A B, and + +_1st._ Place it upon the diagram, so that its circumference may be +tangential to the lines C E and C F, and its greater axis _m n_ may make +an angle of (¹⁄₅) with the vertical, and trace its circumference. + +_2d._ Place it with its circumference tangential to that of the first at +the point m, while its greater axis (of which _o p_ is a part) is in the +horizontal, and trace the portion of its circumference _q o r_. + +_3d._ Place it with its circumference tangential to that of the above at +_v_, while its greater axis (of which _u v_ is a part) makes an angle of +(³⁄₁₀) with the vertical, and trace the portion of its circumference _s v +t_. + +Thus the curvilinear contour of the body and neck are harmonically +determined. + +The curve of the handle may be determined by the same ellipse placed so +that its greater axis (of which _i k_ is a part) makes an angle of (¹⁄₆) +with the vertical. + +Make similar tracings on the other side of A B, and the diagram is +complete. The inscribing rectangle D G E K is that of (²⁄₅). + +The outline resulting from this diagram, not only is in perfect agreement +with my recollection of the form, but with the measurements of the +original given in the “Penny Cyclopædia;” of the accuracy of which there +can be no doubt. They are stated thus:—“It is about ten inches in height, +and beautifully curved from the top downwards; the diameter at the top +being about three inches and a-half; at the neck or smallest part, two +inches; at the largest (mid-height), seven inches; and at the bottom, +five inches.” + +The harmonic elements of this beautiful form, therefore, appear to be the +following parts of the right angle:— + + Tonic. Dominant. Mediant. Submediant. + (¹⁄₂) (¹⁄₃) (¹⁄₅) (³⁄₁₀) + (¹⁄₄) (¹⁄₆) + +When we reflect upon the variety of harmonic ellipses that may +be described, and the innumerable positions in which they may be +harmonically placed with respect to the horizontal and vertical lines, +as well as upon the various modes in which their circumferences may be +combined, the variety which may be introduced amongst such forms as the +foregoing appears almost endless. My second example is that of— + + +_An Ancient Grecian Marble Vase of a Vertical Composition._ + +I shall now proceed to another class of the ancient Greek vase, the form +of which is of a more complex character. The specimen I have chosen for +the first example of this class is one of those so correctly measured and +beautifully delineated by Tatham in his unequalled work.[25] This vase is +a work of ancient Grecian art in Parian marble, which he met with in the +collection at the Villa Albani, near Rome. Its height is 4 ft. 4¹⁄₂ in. + +[Sidenote: Plate XIII.] + +The following is the formula by which I endeavour to develop its harmonic +elements:— + +Let A B (Plate XIII.) represent the full height of this vase. Through B +draw B D, making an angle of (¹⁄₅) with the vertical. Through D draw D O +vertical, through A draw A C, making an angle of (²⁄₅); through B draw +B L, making an angle of (¹⁄₂), and B S, making an angle of (³⁄₁₀), each +with the vertical. Through A draw A D, through B draw B O, through L draw +L N, through C draw C F, and through S draw S P, all horizontal. Through +A draw A H, making an angle of (¹⁄₁₀) with the vertical, and through +H draw H M vertical. Draw similar lines on the other side of A B, and +the rectilinear portion of the diagram is complete, and its inscribing +rectangle that of (³⁄₈). + +The curvilinear portion may thus be added— + +Take a cut-out ellipse of (¹⁄₃), whose greater axis is about the length +of the body of the intended vase, place it with its lesser axis upon the +line S P, and its greater axis upon the line D O, and trace the part _a +b_ of its circumference upon the diagram. Place the same ellipse with +one of its foci upon C, and its greater axis upon C F, and trace its +circumference upon the diagram. Take a cut-out ellipse of (¹⁄₅), whose +greater axis is nearly equal to that of the ellipse already used; place +it with its greater axis upon M H, and its lesser axis upon L N, and +trace its circumference upon the diagram. Make similar tracings upon the +other side of A B, and the diagram is complete. In this, as in the other +diagrams, the strong portions of the lines give the contour of the vase. +The harmonic elements of this classical form, therefore, appear to be the +right angle and its following parts:— + + Tonic. Dominant. Mediant. Submediant. + (¹⁄₂) (¹⁄₃) (²⁄₅) (³⁄₁₀) + (¹⁄₅) + (¹⁄₁₀) + +My third example is that of— + + +_An Ancient Grecian Vase of a Horizontal Composition._ + +This example belongs to the same class as the last, but it is of a +horizontal composition. It was carefully drawn from the original in the +museum of the Vatican by Tatham, in whose etchings it will be found with +its ornamental decorations. The diagram of its harmonic elements may be +constructed as follows:— + +[Sidenote: Plate XIV.] + +Let A B (Plate XIV.) represent the full height of the vase. Through B +draw B D, making an angle of (²⁄₅) with the vertical. Through A draw A H, +A L, and A C, making respectively the following angles, (¹⁄₅) with the +vertical, (⁴⁄₉) with the vertical, and (³⁄₁₀) with the horizontal. These +angles determine the horizontal lines H B, L N, and C F, which divide +the vase into its parts, and the inscribing rectangle D G K O is (³⁄₈). +This completes the rectilinear portion of the diagram. The ellipse by +which the curvilinear portion is added is one of (¹⁄₅), the greater axis +of which, at _a b_, as also at _c d_, makes an angle of (¹⁄₁₂) with +the vertical, and the same axis at _e f_ an angle of (¹⁄₁₂) with the +horizontal. + +The harmonic elements of this vase, therefore, appear to be:— + + Tonic. Dominant. Mediant. Submediant. Supertonic. + The Right (¹⁄₁₂) (²⁄₅) (³⁄₁₀) (⁴⁄₉) + Angle. (¹⁄₅) + +My remaining examples are those of— + + +_Etruscan Vases._ + +Of these vases I give four examples, by which the simplicity of the +method employed in applying the harmonic law will be apparent. + +[Sidenote: Plate XV.] + +The inscribing rectangle D G E K of fig. 1, Plate XV., is one of (³⁄₈), +within which are arranged tracings from an ellipse of (³⁄₁₀), whose +greater axis at _a b_ makes an angle of (¹⁄₁₂), at _c d_ an angle of +(³⁄₁₀), and at _e f_ an angle of (³⁄₄), with the vertical. The harmonic +elements of the contour of this vase, therefore, appear to be:— + + Tonic. Dominant. Subdominants. Submediant. + The Right (¹⁄₁₂) (³⁄₄) (³⁄₁₀) + Angle. (³⁄₈) + +The inscribing rectangle L M N O of fig. 2 is that of (¹⁄₂), within which +are arranged tracings from an ellipse of (¹⁄₃), whose greater axis, at +_a b_ and _c d_ respectively, makes angles of (¹⁄₂) and (⁴⁄₉) with the +horizontal, while that at _e f_ is in the horizontal line. The harmonic +elements of the contour of this vase, therefore, appear to be:— + + Tonic. Dominant. Subtonic. + (¹⁄₂) (¹⁄₃) (⁴⁄₉) + +[Sidenote: Plate XVI.] + +The inscribing rectangle P Q R S of fig. 1, Plate XVI., is one of (⁴⁄₉), +within which are arranged tracings from an ellipse of (³⁄₈), whose +greater axis, at _a b_, _c d_, and _e f_, makes respectively angles +of (¹⁄₆) with the horizontal, (³⁄₅) and (⁴⁄₅) with the vertical. Its +harmonic elements, therefore, appear to be:— + + Tonic. Dominant. Mediant. Supertonic. Subdominant. Submediant. + The Right (¹⁄₆) (⁴⁄₅) (⁴⁄₉) (³⁄₈) (³⁄₅) + Angle. + +The inscribing rectangle T U V X of fig. 2 is one of (⁴⁄₉), within which +are arranged tracings from an ellipse of (³⁄₈) whose greater axis at _a +b_ is in the vertical line, and at _c d_ makes an angle of (¹⁄₂). The +harmonic elements of the contour of this vase, therefore, appear to be:— + + Tonic. Submediant. Supertonic. + (¹⁄₂) (³⁄₈) (⁴⁄₉) + +These four Etruscan vases, the contours of which are thus reduced to the +harmonic law of nature, are in the British Museum, and engravings of +them are to be found in the well-known work of Mr Henry Moses, Plates +4, 6, 14, and 7, respectively, where they are represented with their +appropriate decorations and colours. + +To these, I add two examples of— + + +_Ancient Grecian Ornament._ + +I have elsewhere shewn[26] that the elliptic curve pervades the Parthenon +from the entases of the column to the smallest moulding, and we need not, +therefore, be surprised to find it employed in the construction of the +only two ornaments belonging to that great work. + +[Sidenote: Plate XVII.] + +In the diagram (Plate XVII.), I endeavour to exhibit the geometric +construction of the upper part of one of the ornamental apices, termed +antefixæ, which surmounted the cornice of the Parthenon. + +The first ellipse employed is that of (¹⁄₃), whose greater axis _a b_ is +in the vertical line; the second is also that of (¹⁄₃), whose greater +axis _c d_ makes, with the vertical, an angle of (¹⁄₁₂); the third +ellipse is the same with its major axis _e f_ in the vertical line. +Through one of the foci of this ellipse at A the line A C is drawn, and +upon the part of the circumference C _e_, the number of parts, 1, 2, 3, +4, 5, 6, 7, of which the surmounting part of this ornament is to consist, +are set off. That part of the circumference of the ellipse whose larger +axis is _c d_ is divided from _g_ to _c_ into a like number of parts. The +third ellipse employed is one of (¹⁄₄). + +Take a cut-out ellipse of this kind, whose larger axis is equal in length +to the inscribing rectangle. Place it with its vertex upon the same +ellipse at _g_, so that its circumference will pass through C, and trace +it; remove its apix first to _p_, then to _q_, and proceed in the same +way to _q_, _r_, _s_, _t_, _u_, and _v_, so that its circumference will +pass through the seven divisions on _c g_ and _e_ C: _v o_, _u n_, _t m_, +_s i_, _r k_, _q j_, _p l_, and _g x_, are parts of the larger axes of +the ellipses from which the curves are traced. The small ellipse of which +the ends of the parts are formed is that of (¹⁄₃). + +[Sidenote: Plate XVIII.] + +In the diagram (Plate XVIII.), I endeavour to exhibit the geometric +construction of the ancient Grecian ornament, commonly called the +_Honeysuckle_, from its resemblance to the flower of that name. The first +part of the process is similar to that just explained with reference to +the antefixæ of the Parthenon, although the angles in some parts differ. +The contour is determined by the circumference of an ellipse of (¹⁄₃), +whose major axis A B makes an angle of (¹⁄₉) with the vertical, and +the leaves or petals are arranged upon a portion of the perimeter of a +similar ellipse whose larger axis E F is in the vertical line, and these +parts are again arranged upon a similar ellipse whose larger axis C D +makes an angle of (¹⁄₁₂) with the vertical. The first series of curved +lines proceeding from 1, 2, 3, 4, 5, 6, 7, and 8, are between K E and H +C, part of the circumference of an ellipse of (¹⁄₃); and those between C +H and A G are parts of the circumference of four ellipses, each of (¹⁄₃), +but varying as to the lengths of their larger axes from 5 to 3 inches. +The change from the convex to the concave, which produces the ogie forms +of which this ornament is composed, takes place upon the line C H, and +the lines _a b_, _c d_, _e f_, _g h_, _i k_, _l m_, _n o_, and _p q_, are +parts of the larger axis of the four ellipses the circumference of which +give the upper parts of the petals or leaves. + +This peculiar Grecian ornament is often, like the antefixæ of the +Parthenon, combined with the curve of the spiral scroll. But the volute +is so well understood that I have not rendered my diagrams more complex +by adding that figure. Many varieties of this union are to be found in +Tatham’s etchings, already referred to. The antefixæ of the Parthenon, +and its only other ornament the honeysuckle, as represented on the soffit +of the cornice, are to be found in Stewart’s “Athens.” + + + + +APPENDIX. + + +No. I. + +In pages 34, 35, and 58, I have reiterated an opinion advanced in several +of my former works, viz., that, besides genius, and the study of nature, +an additional cause must be assigned for the general excellence which +characterises such works of Grecian art as were executed during a period +commencing about 500 B.C., and ending about 200 B.C. And that this cause +most probably was, that the artists of that period were instructed in +a system of fixed principles, based upon the doctrines of Pythagoras +and Plato. This opinion has not been objected to by the generality of +those critics who have reviewed my works; but has, however, met with an +opponent, whose recondite researches and learned observations are worthy +of particular attention. These are given in an essay by Mr C. Knight +Watson, “On the Classical Authorities for Ancient Art,” which appeared in +the _Cambridge Journal of Classical and Sacred Philology_ in June 1854. +As this essay is not otherwise likely to meet the eyes of the generality +of my readers, and as the objections he raises to my opinion only occupy +two out of the sixteen ample paragraphs which constitute the first part +of the essay, I shall quote them fully:— + + “The next name on our list is that of the famous Euphranor + (B.C. 362). For the fact that to the practice of sculpture + and of painting he added an exposition of the theory, we are + indebted to Pliny, who says (xxxv. 11, 40), ‘Volumina quoque + composuit de symmetria et coloribus.’ When we reflect on the + _critical_ position occupied by Euphranor in the history + of Greek art, as a connecting link between the idealism of + Pheidias and the naturalism of Lysippus, we can scarcely + overestimate the value of a treatise on art proceeding from + such a quarter. This is especially the case with the first + of the two works here assigned to Euphranor. The inquiries + which of late years have been instituted by Mr D. R. Hay of + Edinburgh, on the proportions of the human figure, and on the + natural principles of beauty as illustrated by works of Greek + art, constitute an epoch in the study of æsthetics and the + philosophy of form. Now, in the presence of these inquiries, + or of such less solid results as Mr Hay’s predecessors in + the same field have elicited, it naturally becomes an object + of considerable interest to ascertain how far these laws of + form and principles of beauty were consciously developed in + the mind, and by the chisel, of the sculptor: how far any + such system of curves and proportions as Mr Hay’s was used + by the Greek as a practical manual of his craft. Without in + the least wishing to impugn the accuracy of that gentleman’s + results—a piece of presumption I should do well to avoid—I must + be permitted to doubt whether the ‘Symmetria’ of Euphranor + contained anything analogous to them in kind, or indeed equal + in value. It must not be forgotten that the truth of Mr Hay’s + theory is perfectly compatible with the fact, that of such + theory the Greek may have been utterly ignorant. It is on this + fact I insist: it is here that I join issue with Mr Hay, and + with his reviewer in a recent number of _Blackwood’s Magazine_. + Or, to speak more accurately,—while I am quite prepared to find + that the Elgin marbles will best of all stand the test which Mr + Hay has hitherto applied, I believe, to works of a later age, I + am none the less convinced that it is precisely that golden age + of Hellenic art to which they belong, precisely that first and + chief of Hellenic artists by whom they were executed, to which + and to whom any such line of research on the laws of form would + have been pre-eminently alien. Pheidias, remember, by the right + of primogeniture, is the ruling spirit of idealism in art. Of + spontaneity was that idealism begotten and nurtured: by any + such system as Mr Hay’s, that spontaneity would be smothered + and paralysed. Pheidias copied an idea in his own mind—‘Ipsius + in mente insidebat species pulchritudinis eximia quædam’ + (_Cic._);—later ages copied _him_. He created: they criticised. + He was the author of Iliads: they the authors of Poetics. + Doubtless, if you unsphere the _spirit_ of Mr Hay’s theories, + you will find nothing discordant with what I have here said. + That is a sound view of Beauty which makes it consist in that + due subordination of the parts to the whole, that due relation + of the parts to each other, which Mendelssohn had in his + mind when he said that the essence of beauty was ‘unity in + variety’—variety beguiling the imagination, the perception of + unity exercising the thewes and sinews of the intellect. On + such a view of beauty, Mr Hay’s theory may, _in spirit_, be + said to rest. But here, as in higher things, it is the letter + that killeth, while the spirit giveth life. And accordingly I + must enter a protest against any endeavour to foist upon the + palmy days of Hellenic art systems of geometrical proportions + incompatible, as I believe, with those higher and broader + principles by which the progress of ancient sculpture was + ordered and governed—systems which will bear nothing of that + ‘felicity and chance by which’—and not by rule—‘Lord Bacon + believed that a painter may make a better face than ever was:’ + systems which take no account of that fundamental distinction + between the schools of Athens and of Argos, and their + respective disciples and descendants, without which you will + make nonsense of the pages of Pliny, and—what is worse—sense + of the pages of his commentators;—systems, in short, which may + have their value as instruments for the education of the eye, + and for instructions in the arts of design, but must be cast + aside as matters of learned trifling and curious disputation, + where they profess to be royal roads to art, and to map the + mighty maze of a creative mind. And even as regards the + application of such a system of proportions to those works of + sculpture which are posterior to the Pheidian age, only partial + can have been the prevalence which it or any other _one_ system + can have obtained. The discrepancies of different artists + in the treatment of what was called, technically called, + _Symmetria_ (as in the title of Euphranor’s work) were, by the + concurrent testimony of all ancient writers, far too salient + and important to warrant the supposition of any uniform scale + of proportions, as advocated by Mr Hay. Even in Egypt, where + one might surely have expected that such uniformity would have + been observed with far greater rigour than in Greece, the + discoveries of Dr Lepsius (_Vorläufige Nachricht_, Berlin, + 1849) have elicited three totally different κανόνες, one of + which is identical with the system of proportions of the human + figure detailed in Diodorus. While we thus venture to differ + from Mr Hay on the historical data he has mixed up with his + inquiries, we feel bound to pay him a large and glad tribute of + praise for having devised a system of proportions which rises + superior to the idiosyncracies of different artists, which + brings back to one common type the sensations of eye and ear, + and so makes a giant stride towards that _codification_, if + I may so speak, of the laws of the universe which it is the + business of the science to effect. I have no hesitation in + saying, that, for scientific precision of method and importance + of results, Albert Durer, Da Vinci, and Hogarth, not to mention + less noteworthy writers, must all yield the palm to Mr Hay. + + “I am quite aware that in the digression I have here allowed + myself, on systems of proportions prevalent among ancient + artists, and on the probable contents of such treatises as that + of Euphranor, _De Symmetria_, I have laid myself open to the + charge of treating an intricate question in a very perfunctory + way. At present the exigencies of the subject more immediately + in hand allow me only to urge in reply, that, as regards the + point at issue—I mean the ‘solidarité’ between theories such as + Mr Hay’s and the practice of Pheidias—the _onus probandi_ rests + with my adversaries.” + +I am bound, in the first place, gratefully to acknowledge the kind and +complimentary notice which, notwithstanding our difference of opinion, +this author has been pleased to take of my works; and, in the second, to +assure him that if any of them profess to be “royal roads to art,” or +to “map the mighty maze of a creative mind,” they certainly profess to +do more than I ever meant they should; for I never entertained the idea +that a system of æsthetic culture, even when based upon a law of nature, +was capable of effecting any such object. But I doubt not that this too +common misapprehension of the scope and tendency of my works must arise +from a want of perspicuity in my style. + +I have certainly, on one occasion,[27] gone the length of stating +that as poetic genius must yield obedience to the rules of rhythmical +measure, even in the highest flights of her inspirations; and musical +genius must, in like manner, be subject to the strictly defined laws of +harmony in the most delicate, as well as in the most powerfully grand +of her compositions; so must genius, in the formative arts, either +consciously or unconsciously have clothed her creations of ideal beauty +with proportions strictly in accordance with the laws which nature has +set up as inflexible standards. If, therefore, the laws of proportion, in +their relation to the arts of design, constitute the harmony of geometry, +as definitely as those that are applicable to poetry and music produce +the harmony of acoustics; the former ought, certainly, to hold the same +relative position in those arts which are addressed to the eye, that is +accorded to the latter in those which are addressed to the ear. Until +so much science be brought to bear upon the arts of design, the student +must continue to copy from individual and imperfect objects in nature, or +from the few existing remains of ancient Greek art, in total ignorance of +the laws by which their proportions are produced, and, what is equally +detrimental to art, the accuracy of all criticism must continue to rest +upon the indefinite and variable basis of mere opinion. + +It cannot be denied that men of great artistic genius are possessed of +an intuitive feeling of appreciation for what is beautiful, by means +of which they impart to their works the most perfect proportions, +independently of any knowledge of the definite laws which govern that +species of beauty. But they often do so at the expense of much labour, +making many trials before they can satisfy themselves in imparting to +them the true proportions which their minds can conceive, and which, +along with those other qualities of expression, action, or attitude, +which belong more exclusively to the province of genius. In such +cases, an acquaintance with the rules which constitute the science of +proportion, instead of proving fetters to genius, would doubtless afford +her such a vantage ground as would promote the more free exercise of +her powers, and give confidence and precision in the embodiment of her +inspirations; qualities which, although quite compatible with genius, are +not always intuitively developed along with that gift. + +It is also true that the operations of the conceptive faculty of the mind +are uncontrolled by definite laws, and that, therefore, there cannot +exist any rules by the inculcation of which an ordinary mind can be +imbued with genius sufficient to produce works of high art. Nevertheless, +such a mind may be improved in its perceptive faculty by instruction in +the science of proportion, so as to be enabled to exercise as correct and +just an appreciation of the conceptions of others, in works of plastic +art, as that manifested by the educated portion of mankind in respect +to poetry and music. In short, it appears that, in those arts which are +addressed to the ear, men of genius communicate the original conceptions +of their minds under the control of certain scientific laws, by means +of which the educated easily distinguish the true from the false, and +by which the works of the poet and musical composer may be placed above +mere imitations of nature, or of the works of others; while, in those +arts that are addressed to the eye in their own peculiar language, such +as sculpture, architecture, painting, and ornamental design, no such laws +are as yet acknowledged. + +Although I am, and ever have been, far from endeavouring “to foist upon +the palmy days of Hellenic art” any system incompatible with those higher +and more intellectual qualities which genius alone can impart; yet, from +what has been handed down to us by writers on the subject, meagre as it +is, I cannot help continuing to believe that, along with the physical and +metaphysical sciences, æsthetic science was taught in the early schools +of Greece. + +I shall here take the liberty to repeat the proofs I advanced in a former +work as the ground of this belief, and to which the author, from whose +essay I have quoted, undoubtedly refers. It is well known that, in the +time of Pythagoras, the treasures of science were veiled in mystery to +all but the properly initiated, and the results of its various branches +only given to the world in the works of those who had acquired this +knowledge. So strictly was this secresy maintained amongst the disciples +and pupils of Pythagoras, that any one divulging the sacred doctrines +to the profane, was expelled the community, and none of his former +associates allowed to hold further intercourse with him; it is even +said, that one of his pupils incurred the displeasure of the philosopher +for having published the solution of a problem in geometry.[28] The +difficulty, therefore, which is expressed by writers, shortly after the +period in which Pythagoras lived, regarding a precise knowledge of his +theories, is not to be wondered at, more especially when it is considered +that he never committed them to writing. It would appear, however, that +he proceeded upon the principle, that the order and beauty so apparent +throughout the whole universe, must compel men to believe in, and refer +them to, an intelligible cause. Pythagoras and his disciples sought for +properties in the science of numbers, by the knowledge of which they +might attain to that of nature; and they conceived those properties to +be indicated in the phenomena of sonorous bodies. Observing that Nature +herself had thus irrevocably fixed the numerical value of the intervals +of musical tones, they justly concluded that, as she is always uniform +in her works, the same laws must regulate the general system of the +universe.[29] Pythagoras, therefore, considered numerical proportion as +the great principle inherent in all things, and traced the various forms +and phenomena of the world to numbers as their basis and essence. + +How the principles of numbers were applied in the arts is not recorded, +farther than what transpires in the works of Plato, whose doctrines were +from the school of Pythagoras. In explaining the principle of beauty, +as developed in the elements of the material world, he commences in the +following words:—“But when the Artificer began to adorn the universe, +he first of all figured with forms and numbers, fire and earth, water +and air—which possessed, indeed, certain traces of the true elements, +but were in every respect so constituted as it becomes anything to be +from which Deity is absent. But we should always persevere in asserting +that Divinity rendered them, as much as possible, the most beautiful +and the best, when they were in a state of existence opposite to such +a condition.” Plato goes on further to say, that these elementary +bodies must have forms; and as it is necessary that every depth should +comprehend the nature of a plane, and as of plane figures the triangle +is the most elementary, he adopts two triangles as the originals or +representatives of the isosceles and the scalene kinds. The first +triangle of Plato is that which forms the half of the square, and is +regulated by the number, 2; and the second, that which forms the half +of the equilateral triangle, which is regulated by the number, 3; +from various combinations of these, he formed the bodies of which he +considered the elements to be composed. To these elementary figures I +have already referred. + +Vitruvius, who studied architecture ages after the arts of Greece had +been buried in the oblivion which succeeded her conquest, gives the +measurements of various details of monuments of Greek art then existing. +But he seems to have had but a vague traditionary knowledge of the +principle of harmony and proportion from which these measurements +resulted. He says—“The several parts which constitute a temple ought +to be subject to the laws of symmetry; the principles of which should +be familiar to all who profess the science of architecture. Symmetry +results from proportion, which, in the Greek language, is termed +analogy. Proportion is the commensuration of the various constituent +parts with the whole; in the existence of which symmetry is found to +consist. For no building can possess the attributes of composition +in which symmetry and proportion are disregarded; nor unless there +exist that perfect conformation of parts which may be observed in a +well-formed human being.” After going at some length into details, he +adds—“Since, therefore, the human figure appears to have been formed +with such propriety, that the several members are commensurate with +the whole, the artists of antiquity (meaning those of Greece at the +period of her highest refinement) must be allowed to have followed the +dictates of a judgment the most rational, when, transferring to works of +art principles derived from nature, every part was so regulated as to +bear a just proportion to the whole. Now, although the principles were +universally acted upon, yet they were more particularly attended to in +the construction of temples and sacred edifices, the beauties or defects +of which were destined to remain as a perpetual testimony of their skill +or of their inability.” + +Vitruvius, however, gives no explanation of this ancient principle +of proportion, as derived from the human form; but plainly shews his +uncertainty upon the subject, by concluding this part of his essay in the +following words: “If it be true, therefore, that the decenary notation +was suggested by the members of man, and that the laws of proportion +arose from the relative measures existing between certain parts of each +member and the whole body, it will follow, that those are entitled to our +commendation who, in building temples to their deities, proportioned the +edifices, so that the several parts of them might be commensurate with +the whole.” It thus appears certain that the Grecians, at the period of +their highest excellence, had arrived at a knowledge of some definite +mathematical law of proportion, which formed a standard of perfectly +symmetrical beauty, not only in the representation of the human figure +in sculpture and painting, but in architectural design, and indeed in +all works where beauty of form and harmony of proportion constituted +excellence. That this law was not deduced from the proportions of +the human figure, as supposed by Vitruvius, but had its origin in +mathematical science, seems equally certain; for in no other way can we +satisfactorily account for the proportions of the beau ideal forms of the +ancient Greek deities, or of those of their architectural structures, +such as the Parthenon, the temple of Theseus, &c., or for the beauty that +pervades all the other formative art of the period. + +This system of geometrical harmony, founded, as I have shewn it to be, +upon numerical relations, must consequently have formed part of the Greek +philosophy of the period, by means of which the arts began to progress +towards that great excellence which they soon after attained. A little +further investigation will shew, that immediately after this period a +theory connected with art was acknowledged and taught, and also that +there existed a Science of Proportion. + +Pamphilus, the celebrated painter, who flourished about four hundred +years before the Christian era, from whom Apelles received the +rudiments of his art, and whose school was distinguished for scientific +cultivation, artistic knowledge, and the greatest accuracy in drawing, +would admit no pupil unacquainted with geometry.[30] The terms upon which +he engaged with his students were, that each should pay him one talent +(£225 sterling) previous to receiving his instructions; for this he +engaged “to give them, _for ten years_, lessons founded on an excellent +theory.”[31] + +It was by the advice of Pamphilus that the magistrates of Sicyon ordained +that the study of drawing should constitute part of the education of the +citizens—“a law,” says the Abbé Barthélémie, “which rescued the fine arts +from servile hands.” + +It is stated of Parrhasius, the rival of Zeuxis, who flourished about +the same period as Pamphilus, that he accelerated the progress of art by +purity and correctness of design; “for he was acquainted with the science +of Proportions. Those he gave his gods and heroes were so happy, that +artists did not hesitate to adopt them.” Parrhasius, it is also stated, +was so admired by his contemporaries, that they decreed him the name of +Legislator.[32] The whole history of the arts in Egypt and Greece concurs +to prove that they were based on geometric precision, and were perfected +by a continued application of the same science; while in all other +countries we find them originating in rude and misshapen imitations of +nature. + +In the earliest stages of Greek art, the gods—then the only statues—were +represented in a tranquil and fixed posture, with the features exhibiting +a stiff inflexible earnestness, their only claim to excellence being +symmetrical proportion; and this attention to geometric precision +continued as art advanced towards its culminating point, and was +thereafter still exhibited in the neatly and regularly folded drapery, +and in the curiously braided and symmetrically arranged hair.[33] + +These researches, imperfect as they are, cannot fail to exhibit the +great contrast that exists between the system of elementary education +in art practised in ancient Greece, and that adopted in this country at +the present period. But it would be of very little service to point out +this contrast, were it not accompanied by some attempt to develop the +principles which seem to have formed the basis of this excellence amongst +the Greeks. + +But in making such an attempt, I cannot accuse myself of assuming +anything incompatible with the free exercise of that spontaneity of +genius which the learned essayist says is the parent and nurse of +idealism. For it is in no way more incompatible with the free exercise of +artistic genius, that those who are so gifted should have the advantage +of an elementary education in the science of æsthetics, than it is +incompatible with the free exercise of literary or poetic genius, that +those who possess it should have the advantage of such an elementary +education in the science of philology as our literary schools and +colleges so amply afford. + + +No. II. + +The letter from which I have made a quotation at page 42, arose out of +the following circumstance:—In order that my theory, as applied to the +orthographic beauty of the Parthenon, might be brought before the highest +tribunal which this country afforded, I sent a paper upon the subject, +accompanied by ample illustrations, to the Royal Institute of British +Architects, and it was read at a meeting of that learned body on the 7th +of February 1853; at the conclusion of which, Mr Penrose kindly undertook +to examine my theoretical views, in connexion with the measurements he +had taken of that beautiful structure by order of the Dilettanti Society, +and report upon the subject to the Royal Institute. This report was +published by Mr Penrose, vol. xi., No. 539 of _The Builder_, and the +letter from which I have quoted appeared in No. 542 of the same journal. +It was as follows:— + + “GEOMETRICAL RELATIONS IN ARCHITECTURE. + + “Will you allow me, through the medium of your columns, to + thank Mr Penrose for his testimony to the truth of Mr Hay’s + revival of Pythagoras? The dimensions which he gives are to + me the surest verification of the theory that I could have + desired. The minute discrepancies form that very element + of practical incertitude, both as to execution and direct + measurement, which always prevails in materialising a + mathematical calculation under such conditions. + + “It is time that the scattered computations by which + architecture has been analysed—more than enough—be synthetised + into those formulæ which, as Mrs Somerville tells us, ‘are + emblematic of omniscience.’ The young architects of our day + feel trembling beneath their feet the ground whence men are + about to evoke the great and slumbering corpse of art. Sir, it + is food of this kind a reviving poetry demands. + + ——‘Give us truths, + For we are weary of the surfaces, + And die of inanition.’ + + “I, for one, as I listen to such demonstrations, whose scope + extends with every research into them, feel as if listening to + those words of Pythagoras, which sowed in the mind of Greece + the poetry whose manifestation in beauty has enchained the + world in worship ever since its birth. And I am sure that in + such a quarter, and in such thoughts, _we_ must look for the + first shining of that lamp of art, which even now is prepared + to burn. + + “I know that this all sounds rhapsodical; but I know also that + until the architect becomes a poet, and not a tradesman, we + may look in vain for architecture: and I know that valuable + as isolated and detailed investigations are in their proper + bearings, yet that such purposes and bearings are to be found + in the enunciation of principles sublime as the generalities of + ‘mathematical beauty.’ + + “AUTOCTHON.” + + +No. III. + +Of the work alluded to at page 58 I was favoured with two opinions—the +one referring to the theory it propounds, and the other to its anatomical +accuracy—both of which I have been kindly permitted to publish. + +The first is from Sir WILLIAM HAMILTON, Bart., professor of logic and +metaphysics in the University of Edinburgh, and is as follows:— + + “Your very elegant volume is to me extremely interesting, as + affording an able contribution to what is the ancient, and, + I conceive, the true theory of the Beautiful. But though + your doctrine coincides with the one prevalent through all + antiquity, it appears to me quite independent and original + in you; and I esteem it the more, that it stands opposed to + the hundred one-sided and exclusive views prevalent in modern + times.—_16 Great King Street, March 5, 1849._” + +The second is from JOHN GOODSIR, Esq., professor of anatomy in the +University of Edinburgh, and is as follows:— + + “I have examined the plates in your work on the proportions + of the human head and countenance, and find the head you have + given as typical of human beauty to be anatomically correct + in its structure, only differing from ordinary nature in its + proportions being more mathematically precise, and consequently + more symmetrically beautiful.—_College, Edinburgh, 17th April + 1849._” + + +No. IV. + +I shall here shew, as I have done in a former work, how the curvilinear +outline of the figure is traced upon the rectilinear diagrams by portions +of the ellipse of (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆). + +[Sidenote: Plate XIX.] + +The outline of the head and face, from points (1) to (3) (fig. 1, Plate +XIX.), takes the direction of the two first curves of the diagram. From +point (3), the outline of the sterno-mastoid muscle continues to (4), +where, joining the outline of the trapezius muscle, at first concave, it +becomes convex after passing through (5), reaches the point (6), where +the convex outline of the deltoid muscle commences, and, passing through +(7), takes the outline of the arm as far as (8). The outline of the +muscles on the side, the latissimus dorsi and serratus magnus, commences +under the arm at the point (9), and joins the outline of the oblique +muscle of the abdomen by a concave curve at (10), which, rising into +convexity as it passes through the points (11) and (12), ends at (13), +where it joins the outline of the gluteus medius muscle. The outline of +this latter muscle passes convexly through the point (14), and ends at +(15), where the outline of the tensor vaginæ femoris and vastus externus +muscle of the thigh commences. This convex outline joins the concave +outline of the biceps of the thigh at (16), which ends in that of the +slight convexity of the condyles of the thigh-bone at (17). From this +point, the outline of the outer surface of the leg, which includes the +biceps, peroneus longus, and soleus muscles, after passing through the +point (18), continues convexly to (19), where the concave outline of the +tendons of the peroneus longus continues to (20), whence the outline of +the outer ankle and foot commences. + +The outline of the mamma and fold of the arm-pit commences at (21), +and passes through the points (22) and (23). The circle at (24) is the +outline of the areola, in the centre of which the nipple is placed. + +The outline of the pubes commences at (25), and ends at the point (26), +from which the outline of the inner surface of the thigh proceeds over +the gracilis, the sartorius, and vastus internus muscles, until it meets +the internal face of the knee-joint at (27), the outline of which ends +at (28). The outline of the inside of the leg commences by proceeding +over the gastrocnemius muscle as far as (29), where it meets that of the +soleus muscle, and continues over the tendons of the heel until it meets +the outline of the inner ankle and foot at (30). + +The outline of the outer surface of the arm, as viewed in front, proceeds +from (8) over the remainder of the deltoid, in which there is here a +slight concavity, and, next, from (31) over the biceps muscle till (32), +where it takes the line of the long supinator, and passing concavely, and +almost imperceptibly, into the long and short radial extensor muscles, +reaches the wrist at (33). The outline of the inner surface of the arm +from opposite (9) commences by passing over the triceps extensor, which +ends at (34), then over the gentle convexity of the condyles of the bone +of the arm at (35), and, lastly, over the flexor sublimis which ends at +the wrist-joint (36). + +The outline of the front of the figure commences at the point (1), (fig. +2, Plate II.), and, passing almost vertically over the platzsma-myoidis +muscles, reaches the point (2), where the neck ends by a concave curve. +From (2) the outline rises convexly over the ends of the clavicles, +and continues so over the pectoral muscle till it reaches (3), where +the mamma swells out convexly to (4), and returns convexly towards +(5), where the curve becomes concave. From (5) the outline follows the +undulations of the rectus muscle of the abdomen, concave at the points +(6) and (7), and having its greatest convexity at (8). This series +of curves ends with a slight concavity at the point (9), where the +horizontal branch of the pubes is situated, over which the outline is +convex and ends at (10). + +The outline of the thigh commences at the point (11) with a slight +concave curve, and then swells out convexly over the extensors of the +leg, and, reaching (12), becomes gently concave, and, passing over the +patella at (13), becomes again convex until it reaches the ligament of +that bone, where it becomes gently concave towards the point (14), whence +it follows the slightly convex curve of the shin-bone, and then, becoming +as slightly concave, ends with the muscles in front of the leg at (15). + +The outline of the back commences at the point (16), and, following with +a concave curve the muscles of the neck as far as (17), swells into a +convex curve over the trapezius muscle towards the point (18); passing +through which point, it continues to swell outward until it reaches +(19), half way between (18) and (20); whence the convexity, becoming +less and less, falls into the concave curve of the muscles of the loins +at (21), and passing through the point (22), it rises into convexity. It +then passes through the point (23), follows the outline of the gluteus +maximus, the convex curve of which rises to the point (24), and then +returns inwards to that of (25), where it ends in the fold of the hip. +From this point the outline follows the curve of the hamstring muscles +by a slight concavity as far as (26), and then, becoming gently convex, +it reaches (27); whence it becomes again gently concave, with a slight +indication of the condyle of the thigh-bone at (28), and, reaching (29), +follows the convex curve of the gastrocnemius muscle through the point +(30), and falling into the convex curve of the tendo Achilles at (31), +ends in the concavity over the heel at (32). + +The outline of the front of the arm commences at the point (33), by a +gentle concavity at the arm-pit, and then swells out in a convex curve +over the biceps, reaching (34), where it becomes concave, and passing +through (35), again becomes convex in passing over the long supinator, +and, becoming gently concave as it passes the radial extensors, rises +slightly at (36), and ends at (37), where the outline of the wrist +commences. The outline of the back of the arm commences with a concave +curve at (38), which becomes convex as it passes from the deltoid to the +long extensor and ends at the elbow (39), from below which the outline +follows the convex curve of the extensor ulnaris, reaching the wrist at +the point (40). + +It will be seen that the various undulations of the outline are regulated +by points which are determined generally by the intersections and +sometimes by directions and extensions of the lines of the diagram, in +the same manner in which I shewed proportion to be imparted, in a late +work, to the osseous structure. The mode in which the curves of (¹⁄₂), +(¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆) are thus so harmoniously blended in the +outline of the female figure, only remains to be explained. + +The curves which compose the outline of the female form are therefore +simply those of (¹⁄₂), (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆). + +Manner in which these curves are disposed in the lateral outline (figure +1, Plate XIX.):— + + Points. Curves. + Head from 1 to 2 (¹⁄₂) + Face ” 2 ” 3 (¹⁄₃) + Neck ” 3 ” 4 (¹⁄₅) + Shoulder ” 4 ” 6 (¹⁄₆) + ” ” 6 ” 8 (¹⁄₄) + Trunk ” 9 ” 15 (¹⁄₄) + ” ” 21 ” 24 (¹⁄₂) + Outer surface of thigh and leg ” 15 ” 20 (¹⁄₆) + Inner surface of thigh and leg ” 25 ” 30 (¹⁄₆) + Outer surface of the arm ” 8 ” 33 (¹⁄₆) + Inner surface of the arm ” 9 ” 36 (¹⁄₆) + +Manner in which they are disposed in the outline (figure 2, Plate XIX.):— + + Points. Curves. + Front of neck from 1 to 2 (¹⁄₆) + ” trunk ” 2 ” 10 (¹⁄₄) + Back of neck ” 16 ” 18 (¹⁄₆) + ” trunk ” 18 ” 23 (¹⁄₄) + ” ” ” 23 ” 25 (¹⁄₃) + Front of thigh and leg ” 11 ” 13 (¹⁄₄) + ” ” ” ” 13 ” 15 (¹⁄₆) + Back of thigh and leg ” 25 ” 32 (¹⁄₆) + Front of the arm ” 33 ” 37 (¹⁄₆) + Back of the arm ” 38 ” 40 (¹⁄₆) + Foot ” 0 ” 0 (¹⁄₆) + +[Sidenote: Plate XX.] + +In order to exemplify more clearly the manner in which these various +curves appear in the outline of the figure, I give in Plate XX. the whole +curvilinear figures, complete, to which these portions belong that form +the outline of the sides of the head, neck, and trunk, and of the outer +surface of the thighs and legs. + +The various angles which the axes of these ellipses form with the +vertical, will be found amongst other details in the works I have just +referred to. + + +No. V. + +At page 85 I have remarked upon the variety that may be introduced into +any particular form of vase; and, in order to give the reader an idea of +the ease with which this may be done without violating the harmonic law, +I shall here give three examples:— + +[Sidenote: Plate XXI.] + +The first of these (Plate XXI.) differs from the Portland vase, in the +concave curve of the neck flowing more gradually into the convex curve of +the body. + +[Sidenote: Plate XXII.] + +The second (Plate XXII.) differs from the same vase in the same change +of contour, as also in being of a smaller diameter at the top and at the +bottom. + +[Sidenote: Plate XXIII.] + +The third (Plate XXIII.) is the most simple arrangement of the elliptic +curve by which this kind of form may be produced; and it differs from the +Portland vase in the relative proportions of height and diameter, and in +having a fuller curve of contour. + +The following comparison of the angles employed in these examples, with +the angles employed in the original, will shew that the changes of +contour in these forms, arise more from the mode in which the angles are +arranged than in a change of the angles themselves:— + + Line Line Line Line + Plate VIII. _A C_ (¹⁄₂) _B C_ (¹⁄₃) _o p_ (H) _v u_ (³⁄₁₀) + Plate XXI. (¹⁄₂) (¹⁄₃) (²⁄₉) (¹⁄₄) + Plate XXII. (¹⁄₂) (¹⁄₃) (¹⁄₈) (⁴⁄₉) + Plate XXIII. (¹⁄₂) (¹⁄₄) (H) (-) + + Line Line + Plate VIII. _m n_ (¹⁄₃) _i k_ (¹⁄₅) ellipse (¹⁄₄) rectangle (²⁄₅) + Plate XXI. (²⁄₉) (¹⁄₅) (¹⁄₄) (²⁄₅) + Plate XXII. (¹⁄₃) (¹⁄₅) (¹⁄₄) (²⁄₅) + Plate XXIII. (¹⁄₅) (¹⁄₅) ellipses {(¹⁄₃)} (¹⁄₃) + {(¹⁄₄)} + +The harmonic elements of each are therefore simply the following parts of +the right angle:— + + Tonic. Dominant. Mediant. Submediant. + Plate VIII. (¹⁄₂) (¹⁄₃) (¹⁄₅) (³⁄₁₀) + (¹⁄₄) + + Tonic. Dominant. Mediant. Supertonic. + Plate XXI. (¹⁄₂) (¹⁄₃) (¹⁄₅) (²⁄₉) + (¹⁄₄) + + Tonic. Dominant. Mediant. Supertonic. + Plate XXII. (¹⁄₂) (¹⁄₃) (¹⁄₅) (⁴⁄₉) + (¹⁄₄) + (¹⁄₈) + + Tonic. Dominant. Mediant. + Plate XXIII. (¹⁄₂) (¹⁄₃) (¹⁄₅) + (¹⁄₄) + + +No. VI. + +So far as I know, there has been only one attempt in modern times, +besides my own, to establish a universal system of proportion, based on a +law of nature, and applicable to art. This attempt consists of a work of +457 pages, with 166 engraved illustrations, by Dr Zeising, a professor in +Leipzic, where it was published in 1854. + +One of the most learned and talented professors in our Edinburgh +University has reviewed that work as follows:— + +“It has been rather cleverly said that the intellectual distinction +between an Englishman and a Scotchman is this—‘Give an Englishman two +facts, and he looks out for a third; give a Scotchman two facts, and he +looks out for a theory.’ Neither of these tests distinguishes the German; +he is as likely to seek for a third fact as for a theory, and as likely +to build a theory on two facts as to look abroad for further information. +But once let him have a theory in his mind, and he will ransack heaven +and earth until he almost buries it under the weight of accumulated +facts. This remark applies with more than common force to a treatise +published last year by Dr Zeising, a professor in Leipsic, ‘On a law of +proportion which rules all nature.’ The ingenious author, after proving +from the writings of ancient and modern philosophers that there always +existed the belief (whence derived it is difficult to say), that some +law does bind into one formula all the visible works of God, proceeds to +criticise the opinions of individual writers respecting that connecting +law. It is not our purpose to follow him through his lengthy examination. +Suffice it to say that he believes he has found the lost treasure in the +_Timæus_ of Plato, c. 31. The passage is confessedly an obscure one, and +will not bear a literal translation. The interpretation which Dr Zeising +puts on it is certainly a little strained, but we are disposed to admit +that he does it with considerable reason. Agreeably to him, the passage +runs thus:—‘That bond is the most beautiful which binds the things as +much as possible into one; and proportion effects this most perfectly +when three things are so united that the greater bears to the middle the +same ratio that the middle bears to the less.’ + +“We must do Dr Zeising the justice to say that he has not made more +than a legitimate use of the materials which were presented to him in +the writings of the ancients, in his endeavour to establish the fact +of the existence of this law amongst them. The canon of Polycletes, +the tradition of Varro mentioned by Pliny relative to that canon, the +writings of Galen and others, are all brought to bear on the same point +with more or less force. The sum of this portion of the argument is +fairly this,—that the ancient sculptors had _some_ law of proportion—some +authorised examplar to which they referred as their work proceeded. That +it was the law here attributed to Plato is by no means made out; but, +considering the incidental manner in which that law is referred to, and +the obscurity of the passages as they exist, it is, perhaps, too much to +expect more than this broad feature of coincidence—the fact that some +law was known and appealed to. Dr Zeising now proceeds to examine modern +theories, and it is fair to state that he appears generally to take a +very just view of them. + +“Let us now turn to Dr Zeising’s own theory. It is this—that in every +beautiful form lines are divided in extreme and mean ratio; or, that +any line considered as a whole, bears to its larger part the same +proportion that the larger bears to the smaller—thus, a line of 5 +inches will be divided into parts which are very nearly 2 and 3 inches +respectively (1·9 and 3·1 inches). This is a well-known division of a +line, and has been called the GOLDEN rule, but when or why, it is not +easy to ascertain. With this rule in his hand, Dr Zeising proceeds to +examine all nature and art; nay, he even ventures beyond the threshold +of nature to scan Deity. We will not follow him so far. Let us turn over +the pages of his carefully illustrated work, and see how he applies his +line. We meet first with the Apollo Belvidere—the golden line divides +him happily. We cannot say the same of the division of a handsome face +which occurs a little further on. Our preconceived notions have made the +face terminate with the chin, and not with the centre of the throat. +It is evident that, with such a rule as this, a little latitude as to +the extreme point of the object to be measured, relieves its inventor +from a world of perplexities. This remark is equally applicable to the +_arm_ which follows, to which the rule appears to apply admirably, yet +we have tried it on sundry plates of arms, both fleshy and bony, without +a shadow of success. Whether the rule was made for the arm or the arm +for the rule, we do not pretend to decide. But let us pass hastily on +to page 284, where the Venus de Medicis and Raphael’s Eve are presented +to us. They bear the application of the line right well. It might, +perhaps, be objected that it is remarkable that the same rule applies +so exactly to the existing position of the figures, such as the Apollo +and the Venus, the one of which is upright, and the other crouching. +But let that pass. We find Dr Zeising next endeavouring to square his +theory with the distances of the planets, with wofully scanty success. +Descending from his lofty position, he spans the earth from corner to +corner, at which occupation we will leave him for a moment, whilst we +offer a suggestion which is equally applicable to poets, painters, +novelists, and theorisers. Never err in excess—defect is the safe side—it +is seldom a fault, often a real merit. Leave something for the student +of your works to do—don’t chew the cud for him. Be assured he will not +omit to pay you for every little thing which you have enabled him to +discover. Poor Professor Zeising! he is far too German to leave any +field of discovery open for his readers. But let us return to him; we +left him on his back, lost for a time in a hopeless attempt to double +Cape Horn. We will be kind to him, as the child is to his man in the +Noah’s ark, and set him on his legs amongst his toys again. He is now in +the vegetable kingdom, amidst oak leaves and sections of the stems of +divers plants. He is in his element once more, and it were ungenerous +not to admit the merit of his endeavours, and the success which now and +then attends it. We will pass over his horses and their riders, together +with that portly personage, the Durham ox, for we have caught a glimpse +of a form familiar to our eyes, the ever-to-be-admired Parthenon. This +is the true test of a theory. Unlike the Durham ox just passed before +us, the Parthenon will stand still to be measured. It has so stood for +twenty centuries, and every one that has scanned its proportions has +pronounced them exquisite. Beauty is not an adaptation to the acquired +taste of a single nation, or the conventionality of a single generation. +It emanates from a deep-rooted principle in nature, and appeals to the +verdict of our whole humanity. We don’t find fault with the Durham ox—his +proportions are probably good, though they be the result of breeding +and cross-breeding; still we are not sure whether, in the march of +agriculture, our grandchildren may not think him a very wretched beast. +But there is no mistake about the Parthenon; as a type of proportion it +stands, has stood, and shall stand. Well, then, let us see how Dr Zeising +succeeds with his rule here. Alas! not a single point comes right. The +Parthenon is condemned, or its condemnation condemns the theory. Choose +your part. We choose the latter alternative; and now, our choice being +made, we need proceed no further. But a question or two have presented +themselves as we went along, which demand an answer. It may be asked—How +do you account for the esteem in which this law of the section in +extreme and mean ratio was held? We reply—That it was esteemed just in +the same way that a tree is esteemed for its fruit. To divide a right +angle into two or three, four or six, equal parts was easy enough. But +to divide it into five or ten such parts was a real difficulty. And how +was the difficulty got over? It was effected by means of this golden +rule. This is its great, its ruling application; and if we adopt the +notion that the ancients were possessed with the idea of the existence +of angular symmetry, we shall have no difficulty in accounting for their +appreciation of this problem. Nay, we may even go further, and admit, +with Dr Zeising, the interpretation of the passage of Plato,—only with +this limitation, that Plato, as a geometer, was carried away by the +geometry of æsthetics from the thing itself. It may be asked again—Is it +not probable that _some_ proportionality does exist amongst the parts of +natural objects? We reply—That, _à priori_, we expect some such system +to exist, but that it is inconsistent with the scheme of _least effort_, +which pervades and characterises all natural succession in space or in +time, that that system should be a complicated one. Whatever it is, its +essence must be simplicity. And no system of simple linear proportion +is found in nature; quite the contrary. We are, therefore, driven to +another hypothesis, viz.—that the simplicity is one of angles, not of +lines; that the eye estimates by search round a point, not by ascending +and descending, going to the right and to the left,—a theory which we +conceive all nature conspires to prove. Beauty was not created for the +eye of man, but the eye of man and his mental eye were created for the +appreciation of beauty. Examine the forms of animals and plants so minute +that nothing short of the most recent improvements in the microscope can +succeed in detecting their symmetry; or examine the forms of those little +silicious creations which grew thousands of years before Man was placed +on the earth, and, with forms of marvellous and varied beauty, they all +point to its source in angular symmetry. This is the keystone of formal +beauty, alike in the minutest animalcule, and in the noblest of God’s +works, his own image—Man.” + + +THE END. + + BALLANTYNE AND COMPANY, PRINTERS, EDINBURGH. + + + + +FOOTNOTES + + +[1] Sir David Brewster. + +[2] No. CLVIII., October 1843. + +[3] Diogenes Laërtius’s “Lives of the Philosophers,” literally +translated. Bohn: London. + +[4] Ibid. + +[5] Rose’s “Biographical Dictionary.” + +[6] Professor Laycock, now of the University of Edinburgh. + +[7] “The Geometric Beauty of the Human Figure Defined,” &c. + +[8] Longman and Co., London. + +[9] See Appendix. + +[10] “The Orthographic Beauty of the Parthenon,” &c., and “The Harmonic +Law of Nature applied to Architectural Design.” + +[11] For further details, see “Harmonic Law of Nature,” &c. + +[12] By a very simple machine, which I have lately invented, an ellipse +of any given proportions, even to those of (¹⁄₆₄), which is the curve of +the entases of the columns of the Parthenon (see Plate VII.), and of any +length, from half an inch to fifty feet or upwards, may be easily and +correctly described; the length and angle of the required ellipse being +all that need be given. + +[13] “The Orthographic Beauty of the Parthenon,” &c. + +[14] “The Orthographic Beauty of the Parthenon,” &c. + +[15] Ibid. + +[16] “The Harmonic Law of Nature applied to Architectural Design.” + +[17] “Physio-philosophy.” By Dr Oken. Translated by Talk; and published +by the Ray Society. London, 1848. + +[18] “The Science of those Proportions by which the Human Head and +Countenance, as represented in Works of ancient Greek Art, are +distinguished from those of ordinary Nature.” + +[19] “The Geometric Beauty of the Human Figure Defined,” &c., and “The +Natural Principles of Beauty Developed in the Human Figure.” + +[20] “The Geometric Beauty of the Human Figure Defined,” &c. + +[21] “Essay on Ornamental Design,” &c., and “The Geometric Beauty of the +Human Figure,” &c. + +[22] “A Nomenclature of Colours, applicable to the Arts and Natural +Sciences,” &c., &c. + +[23] See pp. 24 and 25. + +[24] “The Principles of Beauty in Colouring Systematised,” Fourteen +Diagrams, each containing Six Colours and Hues. + +“A Nomenclature of Colours,” &c., Forty Diagrams, each containing Twelve +Examples of Colours, Hues, Tints, and Shades. + +“The Laws of Harmonious Colouring,” &c., One Diagram, containing Eighteen +Colours and Hues. + +[25] “Etchings Representing the Best Examples of Grecian and Roman +Architectural Ornament, drawn from the Originals,” &c. By Charles +Heathcote Tatham, Architect. London: Priestly and Weale. 1826. + +[26] “The Orthographic Beauty of the Parthenon,” &c. + +[27] “Science of those Proportions,” &c. + +[28] Abbé Barthélémie’s “Travels of Anacharsis in Greece,” vol iv., pp. +193, 195. + +[29] Abbé Barthélémie (vol. ii., pp. 168, 169), who cites as his +authorities, Cicer. De Nat. Deor., lib. i., cap. ii., t. 2, p. 405; +Justin Mart., Ovat. ad Gent., p. 10; Aristot. Metaph., lib. i., cap. v., +t. 2, p. 845. + +[30] Müller’s “Ancient Art and its Remains.” + +[31] “Anacharsis’ Travels in Greece.” By the Abbé Barthélémie, vol. ii., +p. 325. + +[32] “Anacharsis’ Travels in Greece.” By the Abbé Barthélémie, vol. vi., +p. 225. The authorities the Abbé quotes are—Quintil., lib. xii., cap. x., +p. 744; Plin., lib. xxxv., cap. ix., p. 691. + +[33] Müller’s “Archæology of Art,” &c. + + + + +Works by the Same Author. + + +I. + +In royal 8vo, with Copperplate Illustrations, price 2s. 6d., + +THE HARMONIC LAW OF NATURE APPLIED TO ARCHITECTURAL DESIGN. + +_From the Athenæum._ + +The beauty of the theory is its universality, and its simplicity. In +nature, the Creator accomplished his purposes by the simplest means—the +harmony of nature is indestructible and self-restoring. Mr Hay’s book on +the “Parthenon,” on the “Natural Principles of Beauty as developed in the +Human Figure,” his “Principles of Symmetrical Beauty,” his “Principles of +Colouring, and Nomenclature of Colours,” his “Science of Proportion,” and +“Essay on Ornamental Design,” we have already noticed with praise as the +results of philosophical and original thought. + +_From the Daily News._ + +This essay is a new application to Lincoln cathedral in Gothic +architecture, and to the Temple of Theseus in Greek architecture, of the +principles of symmetrical beauty already so profusely illustrated and +demonstrated by Mr Hay. The theory which Mr Hay has propounded in so many +volumes is not only a splendid contribution towards a science of æsthetic +proportions, but, for the first time in the history of art, proves the +possibility, and lays the foundations of such a science. To those who are +not acquainted with the facts, these expressions will sound hyperbolical, +but they are most true. + + +II. + +In royal 8vo, with Copperplate Illustrations, price 5s., + +THE NATURAL PRINCIPLES OF BEAUTY, AS DEVELOPED IN THE HUMAN FIGURE. + +_From the Spectator._ + +We cannot refuse to entertain Mr Hay’s system as of singular intrinsic +excellence. The simplicity of his law and its generality impress +themselves more deeply on the conviction with each time of enforcement. +His theory proceeds from the idea, that in nature every thing is effected +by means more simple than any other that could have been conceived,—an +idea certainly consistent with whatever we can trace out or imagine of +the all-wise framing of the universe. + +_From the Sun._ + +By founding (if we may so phrase it) this noble theory, Mr Hay has +covered his name with distinction, and has laid the basis, we conceive, +of no ephemeral reputation. By illustrating it anew, through the +medium of this graceful treatise, he has conferred a real boon upon +the community, for he has afforded the public another opportunity of +following the golden rule of the poet—by looking through the holy and +awful mystery of creation to the holier and yet more awful mystery of +Omnipotence. + +_From the Cambridge Journal of Classical and Sacred Philology._ + +The inquiries which of late years have been instituted by Mr D. R. Hay +of Edinburgh, on the proportions of the human figure, and on the natural +principles of beauty, as illustrated by works of Greek art, constitute an +epoch in the study of æsthetics and the philosophy of form. + + +III. + +In royal 8vo, with Copperplate Illustrations, price 5s., + +THE ORTHOGRAPHIC BEAUTY OF THE PARTHENON REFERRED TO A LAW OF NATURE. + +To which are prefixed, a few Observations on the Importance of Æsthetic +Science as an Element in Architectural Education. + +_From the Scottish Literary Gazette._ + +We think this work will satisfy every impartial mind that Mr Hay has +developed the true theory of the Parthenon—that he has, in fact, to use +a kindred phraseology, both _parsed_ and _scanned_ this exquisitely +beautiful piece of architectural composition, and that, in doing so, he +has provided the true key by which the treasures of Greek art may be +further unlocked, and rendered the means of correcting, improving, and +elevating modern practice. + +_From the Edinburgh Guardian._ + +Again and again the attempt has been made to detect harmonic ratios in +the measurement of Athenian architecture, but ever without reward. Mr Hay +has, however, made the discovery, and to an extent of which no one had +previously dreamt. + + +IV. + +In 8vo, 100 Plates, price 6s., + +FIRST PRINCIPLES OF SYMMETRICAL BEAUTY. + +_From the Spectator._ + +This is a grammar of pure form, in which the elements of symmetrical, as +distinguished from picturesque beauty, are demonstrated, by reducing the +outlines or planes of curvilinear and rectilinear forms to their origin +in the principles of geometrical proportion. In thus analysing symmetry +of outline in natural and artificial objects, Mr Hay determines the fixed +principles of beauty in positive shape, and shews how beautiful forms may +be reproduced and infinitely varied with mathematical precision. Hitherto +the originating and copying of beautiful forms have been alike empirical; +the production of a new design for a vase or a jug has been a matter of +chance between the eye and the hand; and the copying of a Greek moulding +or ornament, a merely mechanical process. By a series of problems, Mr +Hay places both the invention and imitation of beautiful forms on a sure +basis of science, giving to the fancy of original minds a clue to the +evolving of new and elegant shapes, in which the infinite resources of +nature are made subservient to the uses of art. + +The volume is illustrated by one hundred diagrams beautifully executed, +that serve to explain the text, and suggest new ideas of beauty of +contour in common objects. To designers of pottery, hardware, and +architectural ornaments, this work is particularly valuable; but artists +of every kind, and workmen of intelligence, will find it of great utility. + +_From the Athenæum._ + +The volume before us is the seventh of Mr Hay’s works. It is the most +practical and systematic, and likely to be one of the most useful. +It is, in short, a grammar of form, or a spelling-book of beauty. +This is beginning at the right end of the matter; and the necessity +for this kind of knowledge will inevitably, though gradually, be +felt. The work will, therefore, be ultimately appreciated and +adopted as an introduction to the study of beautiful forms. + +The third part of the work treats of the Greek oval or composite ellipse, +as Mr Hay calls it. It is an ellipse of three foci, and gives practical +forms for vases and architectural mouldings, which are curious to the +architect, and will be very useful both to the potter, the moulder, +and the pattern-drawer. A fourth part contains applications of this to +practice. Of the details worked out with so much judgment and ingenuity +by Mr Hay, we should in vain attempt to communicate just notions without +the engravings of which his book is full. We must, therefore, refer to +the work itself. The forms there given are full of beauty, and so far +tend to prove the system. + + +V. + +In 8vo, 14 Coloured Diagrams, Second Edition, price 15s., + +THE PRINCIPLES OF BEAUTY IN COLOURING SYSTEMATISED. + +_From the Spectator._ + +In this new analysis of the harmonies of colour, Mr Hay has performed the +useful service of tracing to the operation of certain fixed principles +the sources of beauty in particular combinations of hues and tints; so +that artists may, by the aid of this book, produce, with mathematical +certainty, the richest effects, hitherto attainable by genius alone. Mr +Hay has reduced this branch of art to a perfect system, and proved that +an offence against good taste in respect to combinations of colour is, in +effect, a violation of natural laws. + + +VI. + +In 8vo, 228 Examples of Colours, Hues, Tints, and Shades, price 63s., + +A NOMENCLATURE OF COLOURS, APPLICABLE TO THE ARTS AND NATURAL SCIENCES. + +_From the Daily News._ + +In this work Mr Hay has brought a larger amount of practical knowledge +to bear on the subject of colour than any other writer with whom we are +acquainted, and in proportion to this practical knowledge is, as might be +expected, the excellence of his treatise. There is much in this volume +which we would most earnestly recommend to the notice of artists, house +decorators, and, indeed, to all whose business or profession requires a +knowledge of the management of colour. The work is replete with hints +which they might turn to profitable account, and which they will find +nowhere else. + +_From the Athenæum._ + +We have formerly stated the high opinion we entertain of Mr Hay’s +previous exertions for the improvement of decorative art in this country. +We have already awarded him the merit of invention and creation of the +new and the beautiful in form. In his former treatises he furnished a +theory of definite proportions for the creation of the beautiful in form. +In the present work he proposes to supply a scale of definite proportions +for chromatic beauty. For this purpose he sets out very properly with a +precise nomenclature of colour. In this he has constructed a vocabulary +for the artist—an alphabet for the artizan. He has gone further—he +constructs words for three syllables. From this time, it will be possible +to write a letter in Edinburgh about a coloured composition, which shall +be read off in London, Paris, St Petersburg, or Pekin, and shall so +express its nature that it can be reproduced in perfect identity. This Mr +Hay has done, or at least so nearly, as to deserve our thanks on behalf +of art, and artists of all grades, even to the decorative artizan—not one +of whom, be he house-painter, china pattern-drawer, or calico printer, +should be without the simple manual of “words for colours.” + + +VII. + +In post 8vo, with a Coloured Diagram, Sixth Edition, price 7s. 6d., + +THE LAWS OF HARMONIOUS COLOURING ADAPTED TO INTERIOR DECORATIONS. + +_From the Atlas._ + +Every line of this useful book shews that the author has contrived to +intellectualise his subject in a very interesting manner. The principles +of harmony in colour as applied to decorative purposes, are explained and +enforced in a lucid and practical style, and the relations of the various +tints and shades to each other, so as to produce a harmonious result, are +descanted upon most satisfactorily and originally. + +_From the Edinburgh Review._ + +In so far as we know, Mr Hay is the first and only modern artist who +has entered upon the study of these subjects without the trammels of +prejudice and authority. Setting aside the ordinances of fashion, as +well as the dicta of speculation, he has sought the foundation of +his profession in the properties of light, and in the laws of visual +sensation, by which these properties are recognised and modified. The +truths to which he has appealed are fundamental and irrefragable. + +_From the Athenæum._ + +We have regarded, and do still regard, the production of Mr Hay’s works +as a remarkable psychological phenomenon—one which is instructive both +for the philosopher and the critic to study with care and interest, not +unmingled with respect. We see how his mind has been gradually guided +by Nature herself out of one track, and into another, and ever and anon +leading him to some vein of the beautiful and true, hitherto unworked. + + +VIII. + +In 4to, 25 Plates, price 36s., + +ON THE SCIENCE OF THOSE PROPORTIONS BY WHICH THE HUMAN HEAD AND +COUNTENANCE, AS REPRESENTED IN ANCIENT GREEK ART, ARE DISTINGUISHED FROM +THOSE OF ORDINARY NATURE. + +(PRINTED BY PERMISSION.) + +_From a Letter to the Author by Sir William Hamilton, Bart., Professor of +Logic and Metaphysics in the Edinburgh University._ + +Your very elegant volume, “Science of those Proportions,” &c., is to me +extremely interesting, as affording an able contribution to what is the +ancient, and, I conceive, the true theory of the beautiful. But though +your doctrine coincides with the one prevalent through all antiquity, it +appears to me quite independent and original in you; and I esteem it the +more that it stands opposed to the hundred one-sided and exclusive views +prevalent in modern times. + +_From Chambers’s Edinburgh Journal._ + +We now come to another, and much more remarkable corroboration, which +calls upon us to introduce to our readers one of the most valuable and +original contributions that have ever been made to the Philosophy of Art, +viz., Mr Hay’s work “On the Science of those Proportions,” &c. Mr Hay’s +plan is simply to form a scale composed of the well-known vibrations of +the monochord, which are the alphabet of music, and then to draw upon the +quadrant of a circle angles _answering to these vibrations_. With the +series of triangles thus obtained he combines a circle and an ellipse, +the proportions of which are derived from the triangles themselves; and +thus he obtains an infallible rule for the composition of the head of +ideal beauty. + + +IX. + +In 4to, 16 Plates, price 30s., + +THE GEOMETRIC BEAUTY OF THE HUMAN FIGURE DEFINED. + +To which is prefixed, a SYSTEM of ÆSTHETIC PROPORTION applicable to +ARCHITECTURE and the other FORMATIVE ARTS. + +_From the Cambridge Journal of Classical and Sacred Philology._ + +We feel bound to pay Mr Hay a large and glad tribute of praise for +having devised a system of proportions which rises superior to the +idiosyncrasies of different artists, which brings back to one common type +the sensations of Eye and Ear, and so makes a giant stride towards that +_codification_ of the laws of the universe which it is the business of +science to effect. We have no hesitation in saying that, for scientific +precision of method and importance of results, Albert Durer, Da Vinci, +and Hogarth—not to mention less noteworthy writers—must all yield the +palm to Mr Hay. + + +X. + +In oblong folio, 57 Plates and numerous Woodcuts, price 42s., + +AN ESSAY ON ORNAMENTAL DESIGN, IN WHICH ITS TRUE PRINCIPLES ARE DEVELOPED +AND ELUCIDATED, &c. + +_From the Athenæum._ + +In conclusion, Mr Hay’s book goes forth with our best wishes. It must be +good. It must be prolific of thought—stimulant of invention. It is to be +acknowledged as a benefit of an unusual character conferred on the arts +of ornamental design. + +_From the Spectator._ + +Mr Hay has studied the subject deeply and scientifically. In this +treatise on ornamental design, the student will find a clue to the +discovery of the source of an endless variety of beautiful forms and +combinations of lines, in the application of certain fixed laws of +harmonious proportion to the purposes of art. Mr Hay also exemplifies +the application of his theory of linear harmony to the production of +beautiful forms generally, testing its soundness by applying it to the +human figure, and the purest creations of Greek art. + +_From Fraser’s Magazine._ + +Each part of this work is enriched by diagrams of great beauty, direct +emanations of principle, and, consequently, presenting entirely new +combinations of form. Had our space permitted, we should have made some +extracts from this “Essay on Ornamental Design;” and we would have done +so, because of the discriminating taste by which it is pervaded, and the +forcible observations which it contains; but we cannot venture on the +indulgence. + + +XI. + +In 4to, 17 Plates and 38 Woodcuts, price 25s., + +PROPORTION, OR THE GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. + + +XII. + +In 4to, 18 Plates and numerous Woodcuts, price 15s., + +THE NATURAL PRINCIPLES AND ANALOGY OF THE HARMONY OF FORM. + +_From the Edinburgh Review._ + +Notwithstanding some trivial points of difference between Mr Hay’s views +and our own, we have derived the greatest pleasure from the perusal of +these works. They are all composed with accuracy and even elegance. His +opinions and views are distinctly brought before the reader, and stated +with that modesty which characterises genius, and that firmness which +indicates truth. + +_From Blackwood’s Magazine._ + +We have no doubt that when Mr Hay’s Art-discovery is duly developed and +taught, as it should be, in our schools, it will do more to improve the +general taste than anything which has yet been devised. + + + +*** END OF THE PROJECT GUTENBERG EBOOK 75399 *** |