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THE SCIENCE OF BEAUTY. 
 
edinburgh : 
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 Paul's work. 
 
THE 
 
 SCIENCE OF BEAUTY, 
 
 AS DEVELOPED IN NATURE AND 
 
 APPLIED IN ART. 
 
 D. E. HAY, F.E.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.E.SS.L. &E„ 
 
 PROFESSOR OF ANATOMY IN THE UNIVERSITY OF EDINBURGH, 
 
 AS AN EXPRESSION OF GRATITUDE FOR VALUABLE ASSISTANCE, 
 AS ALSO OE HIGH ESTEEM AND SINCERE REGARD, 
 THIS VOLUME IS DEDICATED, 
 BY 
 
 THE AUTHOR 
 
Digitized by the Internet Archive 
 in 2014 
 
 https://archive.org/details/scienceofbeautyaOOhayd / 
 
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 resume 
 of my former works, of such a character as not only 
 
viii 
 
 to explain the nature of my theory, but to exhibit to 
 the general reader, by the most simple modes of illus- 
 tration 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 ten- 
 dency of my theory, I have the satisfaction to intimate 
 that I have been enabled to include in this resume 
 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 Nume- 
 rical 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. Ill . 100 
 
 Appendix, No. IV 100 
 
 Appendix, No. V 104 
 
 Appendix, No. VI 105 
 
ILLUSTRATIONS. 
 
 PLATKS 
 
 I. Three Scales of the Elementary Rectilinear Figures, viz., the Scalene 
 Triangle, the Isosecles Triangle, and the Rectangle, comprising 
 twenty-seven varieties of each, according to the harmonic parts of 
 the Right Angle from -J to 
 
 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. 
 
 III. Diagram of the Rectilinear Orthography of the Portico of the 
 
 Temple of Theseus at Athens, in which its Proportions are deter- 
 mined by harmonic parts of the Right Angle. 
 
 IV. Diagram of the Rectilinear Orthography of the East End of Lincoln 
 
 Cathedral, in which its Proportions are determined by har- 
 monic parts of the Right Angle. 
 
 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. 
 
 VI. The Composite Ellipse of i and | of the Right Angle, shewing its 
 greater and lesser Axis, its various Foci, and the Isosceles Triangle 
 in which they are placed. 
 
 VII. The Composite Ellipse of ^ and <k of the Right Angle, shewing 
 how it forms the Entasis of the Columns of the Parthenon of 
 Athens. 
 
 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. 
 
 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. 
 
XI 
 
 X. Diagram in which the Symmetrical Proportions of the Human 
 Figure are determined by harmonic parts of the Right Angle. 
 
 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. 
 
 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 de- 
 scribed. 
 
 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. 
 
 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. 
 
 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. 
 
 XVI. Two Diagrams of Etruscan Vases, whose harmony of Proportion and 
 melody of Contour are determined as above. 
 
 XVII. Diagram shewing the Geometric Construction of an Ornament 
 belonging to the Parthenon at Athens. 
 
 XVIII. Diagram of the Geometrical Construction of the ancient Grecian 
 Ornament called the Honeysuckle. 
 
 XIX. An additional Illustration of the Contour of the Human Figure, as 
 viewed in Front and in Profile. 
 
 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. 
 
 XXI. Diagram of a variation on the Form of the Portland Vase. 
 
 XX IT. Diagram of a second variation on the Form of the Portland Vase. 
 
 XXIII. Diagram of a third variation on the Form of the Portland Vase. 
 
INTRODUCTION. 
 
 Twelve years ago, one of our most eminent philosophers,* 
 through the medium of the Edinburgh Review^ 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 inter- 
 course, 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 control- 
 ling power of scientific truth has not exercised its legitimate 
 influence. In discussing the principles of painting, sculpture, 
 architecture, and landscape gardening, philosophers have re- 
 nounced 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 con- 
 victions 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 touch- 
 ing it nor distant from it — an ubiquitous chaos, which experi- 
 
 * Sir David Brewster. f No. CLVIII., October 1843. 
 
 A 
 
2 
 
 once 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 sus- 
 tains. 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 draw- 
 ings exhibited on the occasion, drew the attention of the 
 meeting to a discovery made by the head master of the archi- 
 tectural and ornamental department of the school, viz. — That 
 the ground-plan of the Parthenon at Athens had been con- 
 structed by the application of the mysterious ovoid or Vesica 
 
3 
 
 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 re- 
 ceived — 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, Medievalism, Modernism— let us have 
 and hold them all in one wide and lusty embrace. Whv 
 
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 appre- 
 ciation 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 in- 
 feriority 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 
 Medievalism, 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 pre- 
 sent school, and which is, doubtless, the result of a system of 
 education based upon plagiarism and mere imitation. There- 
 fore a recommendation to embrace with equal fervour " as 
 good things," these very opposite artic^ms must be a doctrine 
 as mischievous in art as it would be in religion to recommend 
 
5 
 
 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 pro- 
 ductions 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 
 
6 
 
 investigations ; for our ideas of visible beauty are still unde- 
 fined, 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 prin- 
 ciples 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. ^Esthetic culture 
 should consequently supersede servile copying, as the basis 
 of instruction in our schools of art. Many teachers of draw- 
 ing, 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 litera- 
 ture, poetic inspiration could be created by making boys 
 transcribe or repeat the works of the ancient poets. Sir 
 Joshua Eeynolds 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 philo- 
 sopher 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 
 
8 
 
 the modern German and French writers, and so eloquently 
 illustrated by M. Cousin, were given to the world by Hutchi- 
 son 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 per- 
 ceived 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 Eeid 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 re- 
 gulated 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 beau- 
 tiful 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 in- 
 
9 
 
 vestigators 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 com- 
 plex 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 sym- 
 metrical 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-con- 
 stituted 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 
 
10 
 
 music of nature, no analysis might be able to detect the opera- 
 tion of any precise system of harmony, yet the pleasure thus 
 afforded to the human mind we know to arise from its respond- 
 ing 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 resume 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 
 
11 
 
 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 sys- 
 tematic 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 pro- 
 duction of all that is beautiful in nature, whether in the com- 
 binations, 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, 
 
12 
 
 resembling in some measure the symmetrically precise arrange- 
 ments of parts which characterises all artistic musical com- 
 position ; 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 sym- 
 metrical 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 sub- 
 divided 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 distribu- 
 tions, 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 
 
13 
 
 its variety an apparently equal ratio. The harmony of com- 
 bination in the normal proportions of its parts, and the beau- 
 tifully 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 pic- 
 turesque 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 sym- 
 metrical ; but the type of its beauty will continue to be the 
 
14 
 
 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 sym- 
 metrical, 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. 
 4 
 
THE SCIENCE OF BEAUTY EVOLVED FROM THE HABMONIC 
 LAW OF NATUEE, AOEEEABLY TO THE PYTHAGOEEAN 
 SYSTEM OF NUMEEICAL EATIO. 
 
 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. 
 
 ^Esthetic 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 abso- 
 lutely physical nor absolutely metaphysical, but of an inter- 
 mediate 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 im- 
 pressions made upon the mind through the senses of hear- 
 ing and seeing. Thus, the aesthetic 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 
 
1G 
 
 developments of the great harmonic law upon which the 
 science is hased. 
 
 Although the eye and the ear are two different senses, 
 and, consequently, various in their modes of receiving im- 
 pressions; 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 aesthetic 
 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 aesthetic 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 har- 
 mony 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 
 
17 
 
 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 instanta- 
 neously 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 metaphysi- 
 cally aesthetic 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 de- 
 pendent 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. 
 
 B 
 
18 
 
 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 confound- 
 ing 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 irre- 
 fragable 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 im- 
 pression 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 syste- 
 matic, though it may be unrecognised, laws; but when 
 Oberon, in " Midsummer Night's Dream," is made to ex- 
 claim — 
 
 " 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 legiti- 
 mate, yet altogether distinct from, although accompanying 
 that which constitutes the more precise science of aesthetics 
 
19 
 
 as here defined. The composition of the rhythm is an ope- 
 ration of the understanding, but the beauty of the poetic 
 fancy is an operation of the imagination. 
 
 Our physical and mental powers, aesthetically 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 per- 
 ception operates in establishing facts, and then the judg- 
 ment 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 charac- 
 teristic of genius, and determines the point at which art is 
 placed beyond all determinable canons, — at which, indeed, 
 aesthetics give place to metaphysics. 
 
 In the science of beauty, therefore, the human mind is the 
 
20 
 
 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 senorium 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 bar- 
 barian 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.* 
 
 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 Laertius, 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, f But whether this great 
 
 * Diogenes Laertius's " Lives of the Philosophers," literally translated. Bohn : 
 London. t Ibid. 
 
21 
 
 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 dis- 
 tinct 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 Laertius gives the following ac- 
 count 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 num- 
 bers 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 im- 
 perfect 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.' " * 
 
 These short quotations, I believe, comprise all that is known, 
 
 * Rose's " Biographical Dictionary." 
 
22 
 
 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 thou- 
 sand 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* in the British and Foreign Medical Re- 
 view, 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 funda- 
 mental 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 applica- 
 tion of the Pythagorean system in the arts of design. For so 
 
 * Professor Laycock, now of the University of Edinburgh. 
 
23 
 
 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 propor- 
 tions. 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 begin- 
 ning 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. 
 
24 
 
 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 pro- 
 gression 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- (i) (h) 
 
 n. (i) (i) (i) 
 
 I"- (i) (i) (i) (+) (*) 
 
 IV. (i) (i) (A) ( ) ( T V) ( ) trV) (A) (A) 
 
 The scales I., IT., and III. may now be rendered as com- 
 
25 
 
 plete as scale IV., simply by multiplying upwards by 2 from 
 (*). (*)>(*). (+)» and ( T V), thus:- ' 
 
 I- (i) (t) (*) (I) (*) (A) (i) 
 
 ii. (*) (i) (i) (« m (a) (i) 
 
 in. (i) (i) (i) (*) (*) (A) (i) 
 IV. (i) (4) (A) ( ) (A) ( ) (A) (A) (A) 
 
 We now find between the beginning and the end of scale I. 
 the quantities (f ), (|), (f), ft), and (A). 
 
 The three first of these quantities we find to be the re- 
 mainders of the whole indefinite quantity contained in (1), after 
 subtracting from it the primary harmonic quantities (^), (i), 
 and (^) ; we, however, find also amongst these harmonic 
 quantities that of which being subtracted from (1) leaves 
 (f ), a quantity the most suitable whereby to fill up the hiatus 
 between (|) and (|- ) in scale I., which arises from the omission 
 of (y X ) in scale IV. In like manner we find the two last of 
 these quantities, (f ) and ( T %), 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), (f ) naturally 
 fills up the hiatus between these quantities in scale I., which 
 hiatus arises from the omission of (yHj) in scale IV. Thus : — 
 
 I- (i) (t) (I) (!) (I) (t) (i) (A) (i) 
 
 ii- (£) (I) (!) ( ) (*) ( ) (*) (A) (i) 
 
 in. (i) (|) (I) ( ) (i) ( ) W (A) (i) 
 
 IV. (i) (1) (A) ( ) (A) ( ) (A) (A) (A) 
 
 Scale I. being now complete, we have only to divide these 
 
26 
 
 latter quantities by (2) downwards in order to complete the 
 other three. Thus : — 
 
 I- (i) (t) (!) (I) (I) (t) (*) (A) (h) 
 
 n. (i) (I) (I) (f) tt) (A) W (t 4 5) (i) 
 
 in. (i) (|) ft) (A) (*) (A) (+) (A) (I) 
 
 IV. (i) (£) (A) (A) (A) (A) (A) (A) (A) 
 
 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 : — 
 
 9) (4 : 5) (3 : 4) (2 : 3) (3 : 5) (4 
 9) (2: 5) (3: 8) (1 : 3) (3 : 10) (2 
 9) (1: 5) (3:16) (1 : 6)(3:20)(1 
 9) (1:10) (3:32) (1:12) (3:40) (1 
 
 The following are the principal numerical relations which 
 the parts in each scale bear to one another : — 
 
 I. (1 : 1) (8 
 II. (1 : 2) (4 
 
 III. (1:4) (2 
 
 IV. (1 : 8) (1 
 
 (i) : (f) = 
 
 (*) : (f) = 
 
 (f) 2 (*) = 
 
 (+) (I) = 
 
 (A) : (♦) = 
 
 (i) : (A) = 
 
 (7 
 
 8) 
 
 (9 : 
 
 10) 
 
 (5 : 
 
 6) 
 
 (6 
 
 7) 
 
 (14 
 
 : 15) 
 
 (15 
 
 : 16) 
 
 7)(8:15)(1: 2) 
 7) (4:15)(1: 4) 
 7)(2:15)(1: 8) 
 14) (1 :15) (1 :16) 
 
 Although these relations are exemplified hy parts of scale I., 
 the same ratios exist between the relative parts of scales II., 
 
27 
 
 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 aesthetically 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 concus- 
 sion 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 de- 
 scribe 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 be- 
 yond my anatomical knowledge ; but I may simply remark, 
 that within the ear, and most carefully protected in the con- 
 struction 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 
 
20 
 
 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 im- 
 pulses 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 pro- 
 duced 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 sur- 
 rounding atmosphere a series of pulsations of the same fre- 
 quency. 
 
 These pulsations will communicate through the ear a musical 
 note which would, therefore, be the fundamental note of such 
 
30 
 
 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 pro- 
 gression, 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 har- 
 monics. 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 prin- 
 ciple 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 
 
31 
 
 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 length- 
 ened 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 Ger- 
 mans do, the note termed by us B flat as B natural, and the 
 note we term B natural as H. Now, although this arrange- 
 ment 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. 
 
 j f a) (i) (t) (i) (i) (!) (*) (a) (*>* 
 
 ' 1 C D E F G ABH c 
 
 II. 
 
 (i)* (*) (i) (t) ar (to) (f) (a) (4) 
 
 d 
 
 f 9 
 
 III. 
 
 ar (i) ar (a) a)* (*) ar (a) (*)• 
 
 c d e f g a b h c 
 
 IV. 
 
 (*)* (*)* (A)* (A) (A)* (A) (A)* (A)* (A) ; 
 
 G d e f g a b h c 
 
32 
 
 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 com- 
 binations of musical sounds, the aesthetic 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 agree- 
 ment* 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 pulsa- 
 tions 2 : 1, 3 : 2, 5 : 4, may thus be exemplified : — 
 
 Pulsation of 2 : 1 ■] | 
 
 G t . 
 
 Pulsation of 3 : 2 ] I 
 c ( - 
 
 Pulsation of 5 : 4 i I 
 c ( • 
 
 ! ! 1 ! 
 , . . . . 
 
 • • • 
 
33 
 
 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 per- 
 formance 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 tie 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. 
 
 o 
 
THE SCIENCE OF BEAUTY AS APPLIED TO FOEMS. 
 
 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 acquire- 
 ment 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 geome- 
 trical formula, entered into the education, and thereby facili- 
 tated the practice of ancient Greek art, I have in a former 
 work expressed my firm belief, which is founded on the re- 
 markable 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 sym- 
 
85 
 
 metrical 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 ad- 
 miration 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 appre- 
 ciation of art were disseminated amongst the people generally. 
 Indeed, Miiller, 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 
 
3G 
 
 brightest period to close, as this science fell into disuse, and 
 the Greek artists, instead of working for an enlightened com- 
 munity, 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 Yenus, — its opposite, the massive and 
 powerful mould of the Hercules, — or the characteristic repre- 
 sentation 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 Pytha- 
 gorean law of number to the harmony of form, it will be re- 
 quisite 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 some- 
 times physically within the eye. Like the atmospheric pulsa- 
 tions which produce sound, the action which produces light is 
 
37 
 
 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 aesthetically 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 with- 
 
38 
 
 out effort, and the mind with- equal facility receives and ap- 
 preciates 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. (i) (I) (*) (t) (*) (f) (*) (A) (!) 
 
 n. (!) (!) CD (f) (!) (A) (f) A) (!) 
 
 HI. (i) (I) (!) (A) (!) (A) (!) (A) (I) 
 
 iv. (!) (!) (A) (A) (A) (A) (A) (A) (A) 
 
 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 Q), 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 
 
39 
 
 of the right angle, is made with the vertical line, the triangle 
 may be called the vertical scalene triangle of (J) ; and when 
 made with the horizontal line, the horizontal scalene triangle 
 of (^-). 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 (Jj), being composed of two 
 similar right angled triangles of (J>) ; and when two vertical 
 scalene triangles of (J), and of similar dimensions, are united 
 by their hypothenuses, they form the vertical rectangle of (^), 
 and in like manner the horizontal triangles of (J) similarly 
 united would form the horizontal rectangle of (^). 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 (i), 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 Plate I. 
 manner in which this harmonic law acts upon these elementary 
 rectilinear figures by constructing a series agreeably to the 
 angles of scales II., III., IY. Throughout this series ab 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. 
 
 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 
 
40 
 
 musical sounds, the aesthetic feeling produced by their agree- 
 ment 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 propor- 
 tions 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 * 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 
 Angles 
 
 ft) 
 
 Dominant 
 Angles. 
 
 ft) 
 
 Mediant 
 Angles. 
 
 ft) 
 
 Subtonic 
 Angle. 
 
 ft) 
 
 Supertonic 
 Angles. 
 
 ft) 
 
 ft) 
 
 ft) 
 
 (to) 
 
 
 (tV) 
 
 ft) 
 
 
 
 
 
 (rs) 
 
 
 
 
 
 Plate ii. In Plate II. I give a diagram of its rectilinear orthography, 
 which is simply constructed by lines drawn, either horizontally, 
 
 * " The Geometric Beauty of the Human Figure Defined," &c. 
 
41 
 
 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 sub- 
 structure of the building. The line AE, which makes an 
 angle of (^) with the horizontal, determines the height of the 
 colonnade. The line AD, which makes an angle of (^) with 
 the horizontal, determines the height of the portico, exclusive 
 of the pediment. The line A C, which makes an angle of (J) 
 with the horizontal, determines the height of the portico, in- 
 cluding the pediment. The line GD, which makes an angle 
 of (^) with the horizontal, determines the form of the pedi- 
 ment. The lines E Z and L Y, which respectively make angles 
 °f (A) an d (rs) w ^ n * ne horizontal, determine the breadth of 
 the architrave, frieze, and cornice. The line v nu, which makes 
 an angle of (J) with the vertical, determines the breadth of 
 the triglyphs. The line t d, which makes an angle of (^), 
 determines the breadth of the metops. The lines cb rf 9 and 
 a i, which make each an angle of (^) with the vertical, deter- 
 mine the width of the five centre intercolumniations. The 
 line z k, which makes an angle of (-g) with the vertical, deter- 
 mines the width of the two remaining intercolumniations. The 
 lines c s, q x, and y h, each of which makes an angle of {^) 
 with the vertical, determine the diameters of the three columns 
 on each side of the centre. The line wl, which makes an 
 angle of (^) with the vertical, determines the diameter of 
 the two remaining or corner columns. 
 
 In all this, the length and breadth of the parts are deter- 
 mined 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. 
 
42 
 
 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 Dilet- 
 tanti," 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.* 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." f 
 
 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 pre- 
 vails, in the aesthetics 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 
 * Longman and Co., London. f See Appendix. 
 
43 
 
 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. 
 
 (i) 
 
 Angles. 
 
 (D 
 (*) 
 
 (i) 
 
 (1) 
 ft) 
 
 
 (A) 
 
 
 A diagram of the rectilinear orthography of this portico is 
 given in Plate III. Its construction is similar to that of Plate in. 
 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 (-J-) ; the line A D an angle of ; the 
 line A C an angle of (f) ; the line Gr D an angle of ; and 
 the lines E Z and L Y angles of (^) with the horizontal. 
 
 As to the colonnade or vertical part, the line a b, which 
 determines the three middle intercolumniations, makes an 
 angle of (\) ; the line c d, which determines the two outer 
 intercolumniations, makes an angle of ; and the line e f, 
 which determines the lesser diameter of the columns, makes an 
 angle of (^) 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/"" 
 
 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 
 
 * " The Orthographic Beauty of the Parthenon," &c., and " The Harmonic Law 
 of Nature applied to Architectural Design." 
 
44 
 
 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. 
 
 (i) (*) (*) (+) (t) 
 
 (i) (i) (tt>) (i) 
 
 (A) 
 
 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 (f) with the vertical, 
 determines the width of the design, the tops of the aisle win- 
 dows, and the bases of the pediments on the inner buttresses ; 
 A Gr, (i) with the vertical, that of the outer buttress ; A F, 
 (i) with the vertical, that of the space between the outer and 
 inner buttresses and the width of the great centre window ; 
 and A E, (^) with vertical, that of both the inner buttresses 
 and the space between these. A H, which makes (J) with the 
 vertical, determines the form of the pediment of the centre, 
 
/ 
 
hREay del* 
 
 &.AUcman, sc. 
 
45 
 
 and the full height of the base and surbase. A I, which makes 
 (J) 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 (^), determines the height of the outer buttress ; and 
 A Z, which makes (^) with the horizontal, determines that of 
 the inner buttresses. For the reasons already given, I need not 
 here go into further detail.* 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 cor- 
 responding terminology. For instance, the circle belongs to 
 the equilateral rectangle ; that is, the rectangle of (|), 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 
 
 * For further details, see " Harmonic Law of Nature," &c. 
 
40 
 
 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. 
 
 Take AB 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. 
 
 Fig. 1. 
 
 
 G- 
 
 
 B 
 
 But, when the proportions of an ellipse are to be imparted 
 
47 
 
 by means of one of the harmonic angles, suppose the angle 
 of (J), then the following is the process : — 
 
 Let ABC (figure 2) represent the length of the intended 
 ellipse. Through B draw B e indefinitely, at right angles with 
 ABC; through C draw the line Gf indefinitely, making, with 
 B C, an angle of (J). 
 
 Take B C upon the compasses, and place the point of one 
 leg upon D where Gf intersects B e, and the point of the 
 other upon the line A B, it will meet it at F, which is one 
 focus. Keeping the point of one leg still upon D, remove 
 the point of the other to the line B C, and it will meet it at 
 G, which is the other focus. 
 
 Fig. 2. 
 
 A 
 
 The foci being in either case thus simply ascertained, the 
 method of describing the curve on a small scale is equally 
 simple. 
 
 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 ; 
 
48 
 
 thus simply the ellipse is described. When, however, the 
 governing angle is acute, say less than (^), it is requisite to 
 adopt a more accurate method of description,* as the archi- 
 tectural 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 
 (Hr) to (^~) 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 care- 
 fully 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 
 Plate v. it. To exemplify this, I give Plate V., which exhibits the 
 ellipses of (J), (1), (i), and 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 
 
 * By a very simple machine, which I have lately invented, an ellipse of any 
 given proportions, even to those of (&), 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. 
 
49 
 
 were either parabolic or hyperbolic ; but I believe such curves 
 can have no connexion with the elementary forms of archi- 
 tecture, for they are curves which represent motion, and do 
 not, by continued production, form closed figures. 
 
 But I have shewn, in a former work,* that the contours of 
 these mouldings are composed of curves of the composite 
 ellipse, — a figure which I so name because it is composed sim- 
 ply of arcs of various ellipses harmonically flowing into each 
 other. The composite ellipse, when drawn systematically upon 
 the isosceles triangle, resembles closely parabolic and hyper- 
 bolic 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 har- 
 monically circumscribe or be resolved into any regular figure. 
 
 The composite ellipse may be thus described. 
 
 Let ABC (Plate VI.) be a vertical isosceles triangle of Plate vi. 
 (^), bisect AB in D, and through D draw indefinitely T>f 
 perpendicular to A B, and through B draw indefinitely B g, 
 making the angle DB^ (-|), 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 Orthographic Beauty of the Parthenon," &c. 
 D 
 
50 
 
 the triangle is (^), and that of the elliptic curve described 
 around it (^), I call it the composite ellipse of (^) and (-|), 
 their harmonic ratio being 4:3; and so on of all others, accord- 
 ing to the difference that may thus exist between the elemen- 
 tary 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 * 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 (^g), whose greater axis makes with the vertical an angle of 
 ) ; or simply, the form of one of these columns is the frus- 
 trum of an elliptic-sided or prolate-spheroidal cone, whose 
 section is a composite ellipse of (^g) and (-$t), the harmonic 
 ratio of these two angles being 4 : 3, the same as that of the 
 angles of the composite ellipse just exemplified. 
 Plate vii. In Plate VII. is represented the section of such a cone, of 
 which A B C is the isosceles triangle of and B D and 
 
 D M the semi-axes of an ellipse of (^). M IS and O P are 
 the entases of the column, and def the normal construction of 
 the capital. All these are fully illustrated in the work above 
 referred to,f in which I have also shewn that the curve of the 
 neck of the column is that of an ellipse of (^) ; the curve of 
 the capital or echinus, that of an ellipse of {-j^) ; the curve of 
 the moulding under the cymatium of the pediment, that of an 
 ellipse of (J); and the curve of the bed-moulding of the cornice 
 of the pediment, that of an ellipse of (J). The curve of the 
 cavetto of the soffit of the corona is composed of ellipses of (^) 
 
 * " The Orthographic Beauty of the Parthenon/' &c. t Ibid. 
 
51 
 
 and (xV)? the curve of the cymatium which surmounts the 
 corona, is that of an ellipse of the curve of the mould- 
 ing of the capital of the antse of the posticum, that of an 
 ellipse of (^); the curves of the lower moulding of the same 
 capital are composed of those of an ellipse of (^) and of the 
 circle (^) ; the curve of the moulding which is placed between 
 the two latter is that of an ellipse of (^) ; the curve of the upper 
 moulding of the band under the beams of the ceiling of the 
 peristyle, that of an ellipse of (J); the curve of the lower 
 moulding of the same band, that of an ejlipse of (J) ; and the 
 curves of the moulding at the bottom of the small step or 
 podium between the columns, are those of the circle (^) and of 
 an ellipse of (^). I have also shewn the curve of the fluting 
 of the columns to be that of (xV)- 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 Plate vm. 
 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,* 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. 
 
 * " The Harmonic Law of Nature applied to Architectural Design." 
 
52 
 
 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 brother- 
 hood is governed. It can scarcely be doubted that there was 
 some such practically useful secret amongst the Freemasons 
 or early Gothic architects ; for w T e find in all the venerable 
 remains of their art which exist in this country, symmetrical 
 elegance of form pervading the general design, harmonious pro- 
 portion amongst all the parts, beautiful geometrical arrange- 
 ments 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 whim- 
 sically absurd. It would, therefore, appear that achitecture, 
 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 
 
DJLEay eUt? 
 
 £.Aibha7t sc. 
 
D.R.lLcy del* 
 
 &.AJanan. sc 
 
53 
 
 great Jewish legislator was based upon the same law of nature 
 which constituted the foundation of the Pythagorean philo- 
 sophy, 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 Free- 
 masons 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 archi- 
 tecture 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, cer- 
 tain, 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 con- 
 stitute the individuality by which the countenances of men are 
 distinguished from each other. It may be requisite here to 
 
55 
 
 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 denned 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 admira- 
 tion 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 
 
56 
 
 brain and by tbe face, and hence inferred the degree of intel- 
 lect. 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 
 J upiter, is beautiful when the facial line makes a hundred de- 
 grees 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 deli- 
 beration — 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 : * — " The face is beautiful 
 whose nose is parallel to the spine. No human face has grown 
 
 * " Physio-philosophy." By Dr Oken. Translated by Talk ; and published by 
 the Ray Society. London, 1848. 
 
57 
 
 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 de- 
 rived." 
 
 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 pro- 
 portions 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. IsTo 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 distin- 
 guished ; neither can we reasonably agree with Dr Oken 
 and others, who assert that it was produced by an exclusive 
 
58 
 
 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 em- 
 ployed 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,* it is 
 unnecessary for me to enter upon that part of the subject in 
 this resume ; 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 — 
 
 * " The Science of those Proportions by which the Human Head and Counte- 
 nance, as represented in Works of ancient Greek Art, are distinguished from those 
 of ordinary Nature." 
 
59 
 
 Tonic. Dominant. Mediant. Subtonic. 
 
 (i) (i) G) 0-) 
 
 ft) (*) 
 
 These angles, and the figures which belong to them, are 
 thus arranged : — 
 
 The vertical line A B (Plate IX. fig. 2) represents the Plate ix. 
 full length of the head and face. Taking this line as the 
 greater axis of an ellipse of (-J-), such an ellipse is described 
 around it. Through A the lines AG, A K, A L, AM, and 
 A N", are drawn on each side of the line A B, making, with 
 the vertical, respectively the angles of (^), (J), and 
 (i). Through the points G, K, L, M, and 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 UST K From 
 the centre X of the equilateral triangle A G G the curvilinear 
 figure of viz., the circle, is described circumscribing that 
 triangle. 
 
 The curvilinear plane figures of (^) and (^), respectively, 
 represent the solid bodies of which they are sections, viz., a 
 sphere and a prolate spheroid. These bodies, from the man- 
 ner in which they are here placed, are partially amalgamated, 
 as shewn in figures 1 and 3 of the same plate, thus represent- 
 ing the form of the human head and countenance, both in 
 their external appearance and osseous structure, more cor- 
 rectly than they could be represented by any other geome- 
 trical figures. Thus, the angles of (£) and (£) 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 
 
GO 
 
 the same radius a circle is described from the point a, where 
 K K cuts AB. 
 
 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 Gr G and K K also determine the length of the 
 joinings of the ear to the head. The lines LL and M M de- 
 termine the vertical width of the mouth and lips when at 
 perfect repose, and the line N IT 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 sim- 
 ply of the aesthetic beauty of the human head and counte- 
 nance, 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 aesthetic 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 be- 
 lieve 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 
 FOEM OF THE HUMAN FIGUBE. 
 
 The manner in -which this science is developed in the symme- 
 trical 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 * 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 
 bv 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 rela- 
 tions, 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, \th, 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. 
 
 * "The Geometric Beauty of the Human Figure Defined," &c., and "The 
 Natural Principles of Beauty Developed in the Human Figure." 
 
62 
 
 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. 
 
 Miiller, and other investigators into the archaeology of art, 
 refer to the great difficulty which exists in discovering the 
 principles which the ancients followed in regard to the pro- 
 portions 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 Yenus. 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 Yenus, 
 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 
 
63 
 
 work* 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 pro- 
 portional width of the pelvis of the female, the centres 
 of that motion which the heads of the thigh bones per- 
 form 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 vertribal 
 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 gran- 
 deur 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 Geometric Beauty of the Human Figure Defined/' &c. 
 
64 
 
 The angles wbieli govern the form and proportions of such 
 a figure are, with the right angle, a series of twelve, as 
 follows : — 
 
 Tonic. 
 
 (*) 
 (i) 
 (i) 
 
 (*) 
 (£) 
 (A) 
 
 Mediant. 
 
 (i) 
 
 (tV) 
 
 Sub tonic. 
 
 (+) 
 (A) 
 
 Supertonic. 
 
 (*) 
 
 These angles are employed in the construction of a diagram, 
 which determines the proportions of the parts throughout the 
 whole figure. Thus : — 
 
 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 CAD (fr), FAG- (J), HAI (J), K A L (J), and 
 MAN" (|). At the point B, make the angles KBL (%), 
 UBA (i), and OB A (^). 
 
 Through the point K, in which the lines A K and B K 
 intersect one another, draw P K parallel to A B, and 
 through CFH and M, where this line meets AC, A F, AH, 
 and A M, draw CD, F G, HI, 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 (J), 
 which completes the arrangement of the eleven angles upon 
 A B ; for F B Gr is very nearly (f^), and H B I very 
 nearly (i). 
 
 At the point f 9 where A C intersects B, draw fa 
 perpendicular to AB; and through the point i, where B O 
 intersects M draw S i T parallel to A C. 
 
 Through m, where S i T intersects F B, draw m n ; through 
 ft where S i T intersects K B, draw /3 w; through T draw T g, 
 making an angle of (J) with O P. Join N P, M B, and g P, 
 
65 
 
 and where TS P intersects K B, draw Q E perpendicular to 
 A B. 
 
 On A E as a diameter, describe a circle cutting AC in r, 
 and draw r o perpendicular to A B. 
 
 With A o and or as semi-axes, describe the ellipse Are, 
 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 (-J-). 
 
 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 7, as major axis, 
 describe the ellipse of (J). Through a draw a p parallel to 
 A F, and through p drawj? t perpendicular to W Y. Through 
 a draw/ 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 P and 
 z A. 
 
 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 (J), (J), (J), 
 U>) and (I). 
 
 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 
 
 E 
 
CG 
 
 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.* 
 
 * " Essay on Ornamental Design," &c, and " The Geometric Beauty of the Hu- 
 man Figure," &c. 
 
D.R.Zay del? 
 
 t.Aikmasi, s-c 
 
XI. 
 
THE SIENCE OF BEAUTY, AS DEVELOPED IN COLOUES. 
 
 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. 
 
68 
 
 When we pass from the general colouring of nature to that 
 of particular objects, we are again wrapt in wonder and admi- 
 ration 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 preva- 
 lent 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 unifor- 
 mity 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 unifor- 
 mity 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 pro- 
 per 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 
 
69 
 
 are merely applicable to the impression made upon the senses, 
 and do not refer to the sesthetical 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 cul- 
 tivation 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 phenome- 
 non, and who are even capable of making nice distinctions 
 amongst a variety of various colours, are yet incapable of appre- 
 ciating the aesthetic 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 
 
70 
 
 of appreciating either melody or harmony in musical compo- 
 sition. 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 eesthetics 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 grada- 
 tions 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 prin- 
 ciple which constitutes the basis of sesthetical 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 pro- 
 perly 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 ; 
 
71 
 
 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 dark- 
 ness, 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,* I shall here confine myself to 
 colours in their full intensity — shewing the various modifica- 
 tions 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 rela- 
 tions 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 rela- 
 tions, which are as follow : — 
 
 * " A Nomenclature of Colours, applicable to the Arts and Natural Sciences," 
 &c, &c. 
 
72 
 
 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 pri- 
 mary 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 aud blue-hue. 
 
 Purple-hue, from the union of red- hue and blue-hue. 
 
 Each hue owes its characteristic distinction to the propor- 
 tionate 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 sub- 
 ordinate ; 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 
 
73 
 
 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 distin- 
 guished 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 second- 
 aries as being the most allied to light and the warmest in 
 tone. The same principle operates throughout all the modi- 
 fications 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 
 
74 
 
 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 system- 
 atise the harmonic beauty of forms. 
 
 WARM TONE LIGHT C00LT0NE 
 
 TONED 
 GREY 
 
 It will be observed, that with a view to avoid complexity 
 
75 
 
 as much as possible, I have, in the arrangement of the above 
 series of scales, not only confined myself to the merely ele- 
 mentary parts of the Pythagorean system, but have left out 
 the harmonic modifications upon (^ T ) and ( X V)> in order that 
 the arithmetical progression might not be interrupted.'* 
 
 The above elementary process will, I trust, be found suffi- 
 cient 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. Bed 
 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, intro- 
 duced in their proper places. 
 
 The appropriate terminology of musical notes has been 
 adopted, and the scales are composed as follows : — 
 
 * See pp. 24 and 25. 
 
76 
 
 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. 
 
 Super- 
 tonic. 
 
 Mediant. 
 
 Sub- 
 dominant. 
 
 Dominant. 
 
 Sub- 
 mediant. 
 
 Sub- 
 tonic. 
 
 Semi- 
 Subtouic. 
 
 Tonic. 
 
 
 (*) 
 
 (*) 
 
 (1) 
 
 (1) 
 
 (i) 
 
 (A) 
 
 (f) 
 
 (A) 
 
 (i) 
 
 
 Red. 
 
 Red- 
 orange. 
 
 Orange. 
 
 Yellow- 
 orange. 
 
 Yellow. 
 
 Yellow- 
 green. 
 
 Green. 
 
 Blue- 
 green. 
 
 Blue. 
 
 
 (i) 
 
 (1) 
 
 (i) 
 
 (A) 
 
 tt) 
 
 (A) 
 
 (!) 
 
 (A) 
 
 (i) 
 
 II. 
 
 Green. 
 
 Blue- 
 green 
 hue. 
 
 Blue 
 hue. 
 
 Blue- 
 purple 
 hue. 
 
 Purple. 
 
 Red- 
 purple 
 hue. 
 
 Red 
 hue. 
 
 Red- 
 orange 
 hue. 
 
 Orange. 
 
 Ill 
 
 tt) 
 
 (*) 
 
 (A) 
 
 (A) 
 
 (A) 
 
 (A) 
 
 (A) 
 
 (A) 
 
 (A) 
 
 Red 
 hue. 
 
 Red- 
 orange 
 hue. 
 
 Orange 
 hue. 
 
 Ytllow- 
 orange 
 hue. 
 
 Yellow 
 hue. 
 
 Yellow- 
 green 
 hue. 
 
 Green 
 hue. 
 
 Blue- 
 green 
 hue. 
 
 Blue 
 hue. 
 
 
 (A) 
 
 (A) 
 
 (A) 
 
 (A) 
 
 (A) 
 
 (A) 
 
 (A) 
 
 (A) 
 
 (A) 
 
 IV. 
 
 Green 
 hue. 
 
 Blue- 
 green - 
 toned 
 gray. 
 
 Blue- 
 toned 
 gray. 
 
 Blue- 
 purple- 
 toned 
 gray. 
 
 Purple 
 hue. 
 
 Red-' 
 purple- 
 toned 
 gray. 
 
 Red- 
 toned 
 gray. 
 
 Red. 
 orange- 
 toned 
 gray. 
 
 Orange 
 hue. 
 
 
 (A) 
 
 (A) 
 
 (A) 
 
 ,(l2 8) 
 
 (A) 
 
 (i~fo) 
 
 (A) 
 
 (A) 
 
 (A) 
 
 V. 
 
 Red- 
 toned 
 gray. 
 
 Red- 
 orange- 
 toned 
 gray. 
 
 Orange- 
 toned 
 gray. 
 
 Yellow. 
 
 orange- 
 toned 
 gray. 
 
 Yellow- 
 toned 
 gray. 
 
 Yellow- 
 green - 
 toned 
 gray. 
 
 Green- 
 toned 
 . gray. 
 
 Blue- 
 green - 
 toned 
 gray. 
 
 Blue- 
 toned 
 gray. 
 
77 
 
 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. 
 
 LIGHT WARM TONE SHADE 
 
 V(fc) (is) : (A) 
 
 RED 
 TONED 
 GREY 
 
 The following is a series of five complete scales of the har- 
 
78 
 
 monic 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. 
 
 Super- 
 tonic. 
 
 Mediant. 
 
 Sub- 
 dominant. 
 
 Dominant. 
 
 Sub- 
 mediant. 
 
 Sub- 
 tonic. 
 
 Semi- 
 subtonic. 
 
 Tonic. 
 
 
 (\) 
 
 \2/ 
 
 (4t) 
 
 \9/ 
 
 \5j 
 
 (%) 
 \8/ 
 
 (1) 
 
 Vio/ 
 
 XTJ 
 
 (-*-) 
 VI 5/ 
 
 \4/ 
 
 I. 
 
 Yellow. 
 
 Yellow- 
 orange. 
 
 Orange. 
 
 Red- 
 orange. 
 
 Red. 
 
 Red- 
 purple. 
 
 Purple. 
 
 Blue- 
 purple. 
 
 Blue. 
 
 — 
 
 ft) 
 
 (%) 
 
 \5/ 
 
 (A) 
 
 \1 6/ 
 
 \6/ 
 
 (■Ar) 
 \2 0/ 
 
 (1) 
 
 \1 5/ 
 
 
 II. 
 
 Purple. 
 
 Blue- 
 purple 
 hue. 
 
 Blue 
 hue. 
 
 Blue- 
 green 
 hue. 
 
 Green. 
 
 Yellow- 
 green 
 hue. 
 
 Yellow 
 hue. 
 
 Yellow- 
 orange 
 hue. 
 
 Orange. 
 
 — 
 
 (i) 
 
 \ 0/ 
 
 (£) 
 
 \ y / 
 
 (A). 
 
 (A) 
 
 (A") 
 
 (A) 
 
 (tt) 
 
 \ J- 4: / 
 
 (A) 
 
 (A) 
 
 III. 
 
 Yellow 
 hue. 
 
 Yellow- 
 orange 
 hue. 
 
 Orange 
 hue. 
 
 Red- 
 orange 
 hue. 
 
 Red 
 hue. 
 
 Red- 
 purple 
 hue. 
 
 Purple 
 hue. 
 
 Blue- 
 purple 
 hue. 
 
 Blue 
 hue. 
 
 
 (tts) 
 
 (A) 
 
 (A) 
 
 (A) 
 
 (21) 
 
 (w) 
 
 (is) 
 
 
 (A) 
 
 IV. 
 
 Purple 
 hue. 
 
 Blue- 
 purple- 
 toned 
 gray. 
 
 Blue- 
 toned 
 gray 
 
 Blue- 
 green - 
 toned 
 gray. 
 
 Green 
 hue. 
 
 Yellow- 
 green- 
 toned 
 gray. 
 
 Yellow, 
 toned 
 gray. 
 
 Yellow- 
 orange- 
 toned 
 gray. 
 
 Orange 
 hue. 
 
 
 (s\) 
 
 (A) 
 
 (A) 
 
 (xfs") 
 
 (A) 
 
 (l6o) 
 
 (A) 
 
 (A) 
 
 «*) 
 
 v. 
 
 Yellow- 
 toned 
 gray. 
 
 Yellow- 
 orange- 
 toned 
 gray. 
 
 Orange- 
 toned 
 gray. 
 
 Red- 
 orange- 
 toned 
 gray. 
 
 Red- 
 toned 
 gray. 
 
 Red- 
 purple- 
 toned 
 gray. 
 
 Purple- 
 toned 
 gray. 
 
 Blue- 
 purple- 
 toned 
 gray. 
 
 Blue- 
 toned 
 gray. 
 
 Should I be correct in arranging colours upon scales iden- 
 tical with those upon which musical notes have been ar- 
 ranged, 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 com- 
 positions : for the colourist has no more right to trust exclu- 
 
79 
 
 sively 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 con- 
 trast 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 inter- 
 mediate 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 vinion 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 appre- 
 ciating 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 
 
80 
 
 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 impres- 
 sion 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 atmos- 
 phere, 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 iEolian 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 
 
81 
 
 form, is far less beautiful than the original. ]STow this un- 
 doubtedly 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 trans- 
 mission of light, and the other that of its reflection. 
 
 As a series of coloured illustrations would be beyond the 
 scope of this resume, I may refer those who wish to prosecute 
 the inquiry, with the assistance of such a series, to my pub- 
 lished works upon the subject.* 
 
 * " 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. 
 
 F 
 
THE SCIENCE OF BEAUTY, APPLIED TO THE FOEMS AND 
 PEOPOETIONS OF ANCIENT GEECIAN YASES 
 AND OENAMENTS. 
 
 In examining the remains of the ornamental works of the 
 ancient Greek artists, it appears highly probable that the har- 
 mony 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 Eome, 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 Eome, and since it 
 was added to the treasures of the British Museum. The 
 
S3 
 
 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. 
 
 Let the line A B (Plate XII.) represent the full height Plate xn. 
 of the vase. Through A draw A a, and through B draw B b 
 indefinitely, A a making an angle of (l), and B b an angle of 
 (\), with the vertical. Through the point C, where A a and 
 B b intersect one another, draw D C E vertical. Through 
 AC and B respectively, draw AD, CF, and BE horizontal. 
 Draw similar lines on the other side of AB, and the recti- 
 linear portion of the diagram is complete. 
 
 The curvilinear contour may be thus added : — 
 
 Take a cut-out ellipse of (^), 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 (^) 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 op is a 
 part) is in the horizontal, and trace the portion of its circum- 
 ference qor. 
 
 2>d. 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 (^) with the vertical, and trace the portion of its 
 circumference svt. 
 
 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 % k is a part) 
 makes an angle of (^) with the vertical. 
 
84 
 
 Make similar tracings on the other side of A B, and the 
 diagram is complete. The inscribing rectangle D Gr E K is 
 that of (|). 
 
 The outline resulting from this diagram, not only is in per- 
 fect agreement with my recollection of the form, but with 
 the measurements of the original given in the " Penny Cyclo- 
 paedia ; " 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. 
 
 (h) (i) (*) (to) 
 
 (D (i) 
 
 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 
 
85 
 
 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/'" This vase is a work of 
 ancient Grecian art in Parian marble, which he met with in 
 the collection at the Villa Albani, near Koine. Its height is 
 4 ft. 41 in. 
 
 The following is the formula by which I endeavour to de- 
 velop its harmonic elements : — 
 
 Let A B (Plate XIII.) represent the full height of this vase. Plate xnr. 
 Through B draw B D, making an angle of (-J-) with the verti- 
 cal. Through D draw T> vertical, through A draw A C, 
 making an angle of (§) ; through B draw B L, making an 
 angle of (^), and B S, making an angle of (j^j), each with the 
 vertical. Through A draw A D, through B draw B 0, through 
 L draw L If, through C draw C F, and through S draw S P, 
 all horizontal. Through A draw AH, making an angle of 
 (to) w ^ n the vertical, and through H draw H M vertical. 
 Draw similar lines on the other side of AB, and the recti- 
 linear portion of the diagram is complete, and its inscribing 
 rectangle that of (f). 
 
 The curvilinear portion may thus be added — 
 
 Take a cut-out ellipse of (J), 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 0, 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 (J), whose 
 greater axis is nearly equal to that of the ellipse already used ; 
 
 * " Etchings Representing the Best Examples of Grecian and Roman Architec- 
 tural Ornament, drawn from the Originals," &c. By Charles Heathcote Tatham, 
 Architect. London : Priestly and Weale. 1826. 
 
86 
 
 place it with its greater axis upon MH, and its lesser axis 
 upon LN, and trace its circumference upon the diagram. 
 Make similar tracings upon the other side of AB, 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, ap- 
 pear to be the right angle and its following parts : — 
 
 Tonic. Dominant. Mediant. 
 
 (i) (J) (!) 
 
 a) 
 (A) 
 
 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 : — 
 
 xiv. Let AB (Plate XIV.) represent the full height of the vase. 
 Through B draw B D, making an angle of (f ) with the ver- 
 tical. Through A draw AH, A L, and A C, making respec- 
 tively the following angles, (i) with the vertical, ) with the 
 vertical, and (^) with the horizontal. These angles determine 
 the horizontal lines H B, L BT, and C F, which divide the vase 
 into its parts, and the inscribing rectangle D6K0 is (f ). 
 This completes the rectilinear portion of the diagram. The 
 ellipse by which the curvilinear portion is added is one of (^), 
 the greater axis of which, at a b, as also at c d, makes an angle 
 
 Submediant. 
 
 (to) 
 
87 
 
 of (^) with the vertical, and the same axis at ef an angle of 
 (r^) w i*h ^ ne horizontal. 
 
 The harmonic elements of this vase, therefore, appear to 
 be:— 
 
 Tonic. Dominant. Mediant. Submediant. Supertonic. 
 
 (A) (I) (to) (I) 
 
 (*) 
 
 The Right 
 Angle. 
 
 My remaining examples are those of — 
 Etruscan Vases. 
 
 Of these vases I give four examples, by which the sim- 
 plicity of the method employed in applying the harmonic law 
 will be apparent. 
 
 The inscribing rectangle D G E K of fig. 1, Plate XV., p 
 is one of (f), within which are arranged tracings from an 
 ellipse of (3^), whose greater axis at a b makes an angle of 
 (tt)> a * an angle of (1%-), and at ef an angle of (f ), with 
 the vertical. The harmonic elements of the contour of this 
 vase, therefore, appear to be : — 
 
 Tonic. Dominant. Subdominants. Submediant. 
 
 a) (i) (a) 
 (i) 
 
 The inscribing rectangle L M S of fig. 2 is that of (^), 
 within which are arranged tracings from an ellipse of (^), 
 whose greater axis, at a b and c d respectively, makes angles of 
 (I) and (f) with the horizontal, while that at ef is in the 
 horizontal line. The harmonic elements of the contour of 
 this vase, therefore, appear to be : — 
 
 Tonic. Dominant. Subtonic. 
 
 (*) <*) (♦) 
 
88 
 
 The inscribing rectangle PQRS of fig. 1, Plate XVI., 
 is one of (f), within which are arranged tracings from an 
 ellipse of (§), whose greater axis, at ab, cd, and ef, makes re- 
 spectively angles of (^) with the horizontal, (|) and (-J) with 
 the vertical. Its harmonic elements, therefore, appear to be : — 
 
 Tonic. Dominant. Mediant. Supertonic. Subdominant. Submediant. 
 
 W d) (*) (i) (i) (t) 
 
 The inscribing rectangle TUYX of fig. 2 is one of (f ), 
 within which are arranged tracings from an ellipse of (f) 
 whose greater axis at ab is in the vertical line, and at cd 
 makes an angle of (■£). The harmonic elements of the con- 
 tour of this vase, therefore, appear to be : — 
 
 Tonic. Submediant. Supertonic. 
 
 (i) (I) (*) 
 
 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, re- 
 spectively, where they are represented with their appropriate 
 decorations and colours. 
 
 To these, I add two examples of — 
 
 Ancient Grecian Ornament. 
 
 I have elsewhere shewn* 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 
 
 * " The Orthographic Beauty of the Parthenon," &c. 
 
89 
 
 employed in the construction of the only two ornaments 
 belonging to that great work. 
 
 In the diagram (Plate XVII.), I endeavour to exhibit the Plate xvn, 
 geometric construction of the upper part of one of the orna- 
 mental apices, termed antefixc©, which surmounted the cornice 
 of the Parthenon. 
 
 The first ellipse employed is that of (J), whose greater axis 
 a b is in the vertical line ; the second is also that of whose 
 greater axis c d makes, with the vertical, an angle of (y 1 ^) ; 
 the third ellipse is the same with its major axis ef 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 sur- 
 mounting 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 (J). 
 
 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 
 onc^ and eC: v o,un,tm, si, rk, qj, p I, 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 (^). 
 
 In the diagram (Plate XVIII.), I endeavour to exhibit the Plate xtiil 
 geometric construction of the ancient Grecian ornament, com- 
 monly 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 antefixee of the Parthe- 
 
90 
 
 non, although the angles in some parts differ. The contour is 
 determined by the circumference of an ellipse of Q), whose 
 major axis A B makes an angle of (i) 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 ( T y 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 circum- 
 ference of an ellipse of (J) ; and those between C H and A Gl- 
 are parts of the circumference of four ellipses, each of (J), 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 ab, cd, ef, g h, i k, I 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 antefixse 
 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 antefixae 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." 
 
May d»l l 
 
 is.Axlarum sc 
 
0. Alkrrum. 
 
REay it' 1 
 
 Cr AiAman sc. 
 
APPENDIX. 
 
 2n t o. 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 charac- 
 terises such works of Grecian art as were executed during a period com- 
 mencing 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 doctrinces 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), 
 1 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 propor- 
 tions of the human figure, and on the natural principles of beauty as ill us- 
 
92 
 
 trated by works of Greek art, constitute an epoch in the study of aesthetics 
 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 con- 
 vinced 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 spon- 
 taneity 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 pulchri- 
 tudinis eximia qusedam ' (Cic.) ; — later ages copied him. He created : they 
 criticised. He was the author of Iliads : they the authors of Poetics. Doubt- 
 less, 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 
 
93 
 
 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 applica- 
 tion 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 dis- 
 coveries of Dr Lepsius (Vorldufige Nachricht, Berlin, 1849) have elicited 
 three totally different Kavoves, one of which is identical with the system of 
 proportions of the human figure detailed in Diodorus. While we thus ven- 
 ture 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 idiosyn- 
 cracies of different artists, which brings back to one common type the 
 sensations of eye and ear, and so makes a giant stride towards that codifica- 
 tion, if I may so speak, of the laws of the universe which it is the business 
 of the science to effect. 1 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 per- 
 functory 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 ' solidarite ' 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 
 
94 
 
 system of aesthetic 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,* gone the length of stating that 
 as poetic genius must yield obedience to the rules of rythmical measure, 
 even in the highest nights of her inspirations ; and musical genius must, in 
 like manner, be subject to the strictly denned 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 upas 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 pro- 
 portion, so as to be enabled to exercise as correct and just an appreciation of 
 
 * rt Science of those Proportions," &c. 
 
95 
 
 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 com- 
 poser 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 pecu- 
 liar 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 meta- 
 physical sciences, aesthetic 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 inter- 
 course with him ; it is even said, that one of his pupils incurred the dis- 
 pleasure of the philosopher for having published the solution of a problem 
 in geometry.* 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.f 
 
 * Abbe Barthelemie's " Travels of Anacharsis in Greece," vol iv., pp. 193, 195. 
 
 t Abbe Barthelemie (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. 
 
96 
 
 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, far- 
 ther than what transpires in the works of Plato, whose doctrines were from the 
 school of Pythagoras. In explaining the principle of beaut}^ 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 con- 
 stituted 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 measure- 
 ments 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 com- 
 mensuration 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 com- 
 mensurate 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 univer- 
 
97 
 
 sally acted upon, yet they were more particularly attended to in the con- 
 struction 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 pro- 
 portion, 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 paint- 
 ing, 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 architec- 
 tural 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.* 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." t 
 
 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 
 
 * Miiller's "Ancient Art and its Remains." 
 
 t " Anacharsis' Travels in Greece." By the Abbe Barthelemie, vol. ii., p. 325. 
 
 G 
 
98 
 
 citizens — " a law," says the Abbe Barthelemie, " 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 Legis- 
 lator.* 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 con- 
 tinued 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.t 
 
 These researches, imperfect as they are, cannot fail to exhibit the great 
 contrast that exists between the system of elementary education in art prac- 
 tised 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 any- 
 thing 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 educa- 
 tion in the science of aesthetics, 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. 
 
 * " Anacharsis' Travels in Greece." By the Abbe Barthelemie, vol. vi., p. 225. The 
 authorities the Abbe quotes are— Quintil., lib. xii., cap. x., p. 744; Plin., lib. xxxv., cap. ix., 
 p. 691. 
 
 f Miiller's " Archaeology of Art," &c. 
 
99 
 
 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 Archi- 
 tects, 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 Dilletanti 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 1 
 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 formulae 
 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 
 
100 
 
 architecture : and I know that valuable as isolated and detailed investiga- 
 tions 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 preva- 
 lent 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." 
 
 JS T o. 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 (J), (i), (*), and (£). 
 
 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), 
 
101 
 
 the outline of the stern o-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 vagina? 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 out- 
 line 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 con- 
 tinues 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 
 
102 
 
 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 con- 
 cave 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 ontward 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 some- 
 times 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 (^), (^), (I), (£), and 
 
103 
 
 (J) 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 (J), (i), (|), and (l). 
 
 Manner in which these curves are disposed in the lateral outline (figure 1, 
 Plate XIX.) :— 
 
 
 
 Points. 
 
 Curves. 
 
 Head . 
 
 from 
 
 1 to 
 
 2 
 
 ft) 
 
 Face 
 
 
 2 „ 
 
 3 
 
 (i) 
 
 Neck 
 
 
 3 M 
 
 4 
 
 (i) 
 
 Shoulder .... 
 
 » 
 
 4 „ 
 
 6 
 
 <*> 
 
 
 ;> ■ 
 
 6 „ 
 
 8 
 
 (i) 
 
 Trunk 
 
 55 
 
 9 „ 
 
 15 
 
 (1) 
 
 ,, ..... 
 
 » 
 
 21 „ 
 
 24 
 
 (i) 
 
 Outer surface of thigh and leg . 
 
 )f 
 
 15 „ 
 
 20 
 
 (*) 
 
 Inner surface of thigh and leg . 
 
 35 
 
 25 „ 
 
 30 
 
 (» 
 
 Outer surface of the arm 
 
 » 
 
 8 „ 
 
 33 
 
 (i) 
 
 Inner surface of the arm 
 
 33 
 
 „ 
 
 36 
 
 (*) 
 
 Manner in which they are disposed in the outline (figure 2, Plate XIX.) : — 
 
 
 
 Points. 
 
 Curves. 
 
 Front of neck 
 
 from 
 
 1 to 2 
 
 (fi 
 
 „ „ trunk 
 
 • 35 
 
 2 „ 10 
 
 (i) 
 
 Back of neck 
 
 55 
 
 16 „ 18 
 
 (i) 
 
 „ „ trunk 
 
 • 55 
 
 18 „ 23 
 
 (i) 
 
 55 » •)■) 
 
 55 
 
 23 „ 25 
 
 0) 
 
 Front of thigh and leg 
 
 55 
 
 11 „ 13 
 
 (1) 
 
 55 33 33 
 
 35 
 
 13 „ 15 
 
 (*) 
 
 Back of thigh and leg 
 
 35 
 
 25 „ 32 
 
 tt) 
 
 Front of the arm 
 
 55 
 
 33 „ 37 
 
 ft) 
 
 Back of the arm 
 
 33 
 
 38 „ 40 
 
 (I) 
 
 Foot . 
 
 • )i 
 
 „ 
 
 ft) 
 
 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 Plate XX. 
 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. 
 
104 
 
 i 
 
 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 : — 
 
 Plate XXI. The first of these (Plate XXI.) differs from the Portland vase, in the con- 
 cave curve of the neck flowing more gradually into the convex curve of the 
 body. 
 
 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. 
 
 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 
 
 Line 
 
 Line 
 
 Plate VIII. A G BC (l) op (h) vu (y^-) mn (i) ik (1) ellipse (l) rectangle (|-) 
 
 "'»<—'• ••• Q) - (£) -(f)- (!) - (!)•■■(+) - (*) ••• (!) 
 
 Plate XXII. ... (1) ... (1) ... (-1) ... (-4-) ... (1) ... (1) ... (1) ... (f) 
 
 P.ate XXIII.... (1) ... (1) ... («) ... (-) ... (1) ... (lapses 
 
 ;(*>; 
 m 
 
 (*) 
 
 The harmonic elements of each are therefore simply the following parts 
 of the right angle : — 
 
 Plate VIII. (i) 
 (i) 
 
 Tonic. 
 
 Plate XXI.. (i) 
 (i) 
 
 Dominant. 
 
 (*) 
 
 (*) 
 
 Mediant. 
 
 (i) 
 
 Mediant. 
 
 (*) 
 
 Submediant. 
 
 (A) 
 
 Supertonic. 
 
 (I) 
 
XIX. 
 
U.S.. Say del c 
 
 G.Aikrnajh sc 
 
&. Ajkmai 
 
D.RHoij del* 
 
 0- Ail-man sc 
 
105 
 
 Tonic. 
 
 Mediant. 
 
 Supertonic. 
 
 tt) 
 
 Tonic. 
 
 Dominant. 
 
 Mediant. 
 
 Plate XXI II. (i) 
 
 m 
 
 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 pro- 
 fessor in Leipzic, where it was published in 1854. 
 
 One of the most learned and talented professors in our Edinburgh Uni- 
 versity has reviewed that work as follows : — 
 
 " It has been rather cleverly said that the intellectual distinction between an 
 Englishman and a Scotchman is this — 1 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 rales 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. 
 Tt 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 
 Timoeus 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 
 
106 
 
 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 men- 
 tioned 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 divi- 
 sion 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 applica- 
 tion of the line right well. It might, perhaps, be objected that it is re- 
 markable 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 
 
107 
 
 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 stu- 
 dent 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 }'ou 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 Par- 
 thenon will stand still to be measured. It has so stood for twenty centuries, 
 and every one that has scanned its proportions has pronounced them exqui- 
 site. 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 Par- 
 thenon ; 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 con- 
 demnation condemns the theory. Choose your part. We choose the 
 latter alternative ; and now, our choice being made, w r e 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 1 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 difficuty got over 1 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 
 
108 
 
 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 aesthetics 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, a priori, we expect some such system to 
 exist, but that it is inconsistent with the scheme of least effort, which per- 
 vades 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 
 
i. 
 
 In royal 8vo, with Copperplate Illustrations, price 2s. 6d., 
 
 THE HARMONIC LAW OF NATURE APPLIED TO 
 ARCHITECTURAL DESIGN. 
 
 From the Athenceum. 
 
 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 lias propound- 
 ed in so many volumes is not only a splendid contribution towards a science of aesthetic pro- 
 portions, but, for the first time in the history of art, proves the possibility, and lays the foun- 
 dations 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 cer- 
 tainly 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 illustrat- 
 ing 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 aesthetics 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 iEsthetic 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. 
 
11 
 
 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 dis- 
 covery, and to an extent of which no one had previously dreamt. 
 
 IY. 
 
 In 8vo, 100 Plates, price 6s., 
 
 ■piRST PRINCIPLES OF SYMMETRICAL BEAUTY. 
 
 From the Spectator. 
 
 This is a grammar of pure form, iu 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 Athenaeum. 
 
 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 mould- 
 ings, 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, there- 
 fore, 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 com- 
 binations of hues and tints ; so that artists may, by the aid of this book, produce, with mathe- 
 matical 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. 
 
 A 
 
 YI. 
 
 In 8vo, 2'28 Examples of Colours, Hues, Tints, and Shades, price 63s., 
 
 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. 
 
Ill 
 
 From the Athenaeum. 
 
 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. Frdni 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 deco- 
 rative 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 proper- 
 ties are recognised and modified. The truths to which he has appealed are fundamental and 
 irrefragable. 
 
 From the Athenatum. 
 
 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 umningled 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 lead- 
 ing 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 REPRE- 
 SENTED 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 inte- 
 resting, 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. 
 
IV 
 
 IX. 
 
 Ia 4to, 16 Plates, price 30s., 
 
 THE GEOMETRIC BEAUTY OF THE HUMAN FIGURE 
 DEFINED. 
 
 To which is prefixed, a SYSTEM of ESTHETIC 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 ELUCI- 
 DATED, &c. 
 
 From the Athenceum. 
 
 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 au 
 unusual character conferred on the arts of ornamental desigu. 
 
 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 beau- 
 tiful 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 prin- 
 ciple, and, consequently, presenting entirely new combinations of fcrm. Had our space per- 
 mitted, 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 4 to, 17 Plates and 38 Woodcuts, price 25s., 
 
 PROPORTION, OR THE GEOMETRIC PRINCIPLE OF BEAUTY 
 ANALYSED. 
 
 XII. 
 
 In 4 to, 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 in- 
 dicates 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. 
 
GETTY RESEARCH INSTITUTE 
 
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