{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_75399/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_75399","id":"gutenberg_75399","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/75399","summary":null,"tags":["Aesthetics","Category: Art","Category: Philosophy & Ethics","Nature (Aesthetics)","Proportion (Art)"],"text":"THE SCIENCE OF BEAUTY.\n\n EDINBURGH:\n PRINTED BY BALLANTYNE AND COMPANY,\n PAUL\u2019S WORK.\n\n\n\n\n THE\n SCIENCE OF BEAUTY,\n AS DEVELOPED IN NATURE AND\n APPLIED IN ART.\n\n BY\n D. R. HAY, F.R.S.E.\n\n \u201cThe irregular combinations of fanciful invention may delight\n awhile, by that novelty of which the common satiety of life\n sends us all in quest; the pleasures of sudden wonder are soon\n exhausted, and the mind can only repose on the stability of\n truth.\u201d\n\n DR JOHNSON.\n\n WILLIAM BLACKWOOD AND SONS,\n EDINBURGH AND LONDON.\n MDCCCLVI.\n\n\n\n\n TO\n JOHN GOODSIR, ESQ., F.R.S S. L. & E.,\n PROFESSOR OF ANATOMY IN THE UNIVERSITY OF EDINBURGH,\n AS AN EXPRESSION OF GRATITUDE FOR VALUABLE ASSISTANCE,\n AS ALSO OF HIGH ESTEEM AND SINCERE REGARD,\n THIS VOLUME IS DEDICATED,\n BY\n\n THE AUTHOR.\n\n\n\n\nPREFACE.\n\n\nMy theory of beauty in form and colour being now admitted by the best\nauthorities to be based on truth, I have of late been often asked, by\nthose who wished to become acquainted with its nature, and the manner of\nits being applied in art, which of my publications I would recommend for\ntheir perusal. This question I have always found difficulty in answering;\nfor although the law upon which my theory is based is characterised by\nunity, yet the subjects in which it is applied, and the modes of its\napplication, are equally characterised by variety, and consequently\noccupy several volumes.\n\nUnder these circumstances, I consulted a highly respected friend, whose\nmathematical talents and good taste are well known, and to whom I have\nbeen greatly indebted for much valuable assistance during the course\nof my investigations. The advice I received on this occasion, was to\npublish a _r\u00e9sum\u00e9_ of my former works, of such a character as not only\nto explain the nature of my theory, but to exhibit to the general reader,\nby the most simple modes of illustration and description, how it is\ndeveloped in nature, and how it may be extensively and with ease applied\nin those arts in which beauty forms an essential element.\n\nThe following pages, with their illustrations, are the results of an\nattempt to accomplish this object.\n\nTo those who are already acquainted, through my former works, with\nthe nature, scope, and tendency of my theory, I have the satisfaction\nto intimate that I have been enabled to include in this _r\u00e9sum\u00e9_ much\noriginal matter, with reference both to form and colour.\n\n D. R. HAY.\n\n\n\n\nCONTENTS.\n\n\n PAGE\n\n INTRODUCTION 1\n\n THE SCIENCE OF BEAUTY, EVOLVED FROM THE HARMONIC LAW OF NATURE,\n AGREEABLY TO THE PYTHAGOREAN SYSTEM OF NUMERICAL RATIO 15\n\n THE SCIENCE OF BEAUTY, AS APPLIED TO SOUNDS 28\n\n THE SCIENCE OF BEAUTY, AS APPLIED TO FORMS 34\n\n THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE FORM OF THE HUMAN HEAD\n AND COUNTENANCE 54\n\n THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE FORM OF THE HUMAN FIGURE 61\n\n THE SCIENCE OF BEAUTY, AS DEVELOPED IN COLOURS 67\n\n THE SCIENCE OF BEAUTY APPLIED TO THE FORMS AND PROPORTIONS OF\n ANCIENT GRECIAN VASES AND ORNAMENTS 82\n\n APPENDIX, NO. I. 91\n\n APPENDIX, NO. II. 99\n\n APPENDIX, NO. III. 100\n\n APPENDIX, NO. IV. 100\n\n APPENDIX, NO. V. 104\n\n APPENDIX, NO. VI. 105\n\n\n\n\nILLUSTRATIONS.\n\n\nPLATES\n\n[Illustration: I. Three Scales of the Elementary Rectilinear Figures,\nviz., the Scalene Triangle, the Isosceles Triangle, and the Rectangle,\ncomprising twenty-seven varieties of each, according to the harmonic\nparts of the Right Angle from \u00b9\u2044\u2082 to \u00b9\u2044\u2081\u2086.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: II. Diagram of the Rectilinear Orthography of the\nPrincipal Front of the Parthenon of Athens, in which its Proportions are\ndetermined by harmonic parts of the Right Angle.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: III. Diagram of the Rectilinear Orthography of the\nPortico of the Temple of Theseus at Athens, in which its Proportions are\ndetermined by harmonic parts of the Right Angle.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: IV. Diagram of the Rectilinear Orthography of the East End\nof Lincoln Cathedral, in which its Proportions are determined by harmonic\nparts of the Right Angle.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: V. Four Ellipses described from Foci, determined by\nharmonic parts of the Right Angle, shewing in each the Scalene Triangle,\nthe Isosceles Triangle, and the Rectangle to which it belongs.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: VI. The Composite Ellipse of \u00b9\u2044\u2086 and \u00b9\u2044\u2088 of the Right\nAngle, shewing its greater and lesser Axis, its various Foci, and the\nIsosceles Triangle in which they are placed.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: VII. The Composite Ellipse of \u00b9\u2044\u2084\u2088 and \u00b9\u2044\u2086\u2084 of the Right\nAngle, shewing how it forms the Entasis of the Columns of the Parthenon\nof Athens.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: VIII. Sectional Outlines of two Mouldings of the Parthenon\nof Athens, full size, shewing the harmonic nature of their Curves, and\nthe simple manner of their Construction.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: IX. Three Diagrams, giving a Vertical, a Front, and a Side\nAspect of the Geometrical Construction of the Human Head and Countenance,\nin which the Proportions are determined by harmonic parts of the Right\nAngle.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: X. Diagram in which the Symmetrical Proportions of the\nHuman Figure are determined by harmonic parts of the Right Angle.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: XI. The Contour of the Human Figure as viewed in Front\nand in Profile, its Curves being determined by Ellipses, whose Foci are\ndetermined by harmonic parts of the Right Angle.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: XII. Rectilinear Diagram, shewing the Proportions of the\nPortland Vase, as determined by harmonic parts of the Right Angle, and\nthe outline of its form by an Elliptic Curve harmonically described.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: XIII. Rectilinear Diagram of the Proportions and\nCurvilinear Outline of the form of an ancient Grecian Vase, the\nproportions determined by harmonic parts of the Right Angle, and the\nmelody of the form by Curves of two Ellipses.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: XIV. Rectilinear Diagram of the Proportions and\nCurvilinear Outline of the form an ancient Grecian Vase, the proportions\ndetermined by harmonic parts of the Right Angle, and the melody of the\nform by an Elliptic Curve.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: XV. Two Diagrams of Etruscan Vases, the harmony of\nProportions and melody of the Contour determined, respectively, by parts\nof the Right Angle and an Elliptic Curve.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: XVI. Two Diagrams of Etruscan Vases, whose harmony of\nProportion and melody of Contour are determined as above.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: XVII. Diagram shewing the Geometric Construction of an\nOrnament belonging to the Parthenon at Athens.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: XVIII. Diagram of the Geometrical Construction of the\nancient Grecian Ornament called the Honeysuckle.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: XIX. An additional Illustration of the Contour of the\nHuman Figure, as viewed in Front and in Profile.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: XX. Diagram shewing the manner in which the Elliptic\nCurves are arranged in order to produce an Outline of the Form of the\nHuman Figure as viewed in Front.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: XXI. Diagram of a variation on the Form of the Portland\nVase.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: XXII. Diagram of a second variation on the Form of the\nPortland Vase.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n[Illustration: XXIII. Diagram of a third variation on the Form of the\nPortland Vase.\n\n_D. R. Hay del\u1d57._ _G. Aikman sc._]\n\n\n\n\nINTRODUCTION.\n\n\nTwelve years ago, one of our most eminent philosophers,[1] through the\nmedium of the _Edinburgh Review_,[2] gave the following account of what\nwas then the state of the fine arts as connected with science:\u2014\u201cThe\ndisposition to introduce into the intellectual community the principles\nof free intercourse, is by no means general; but we are confident that\nArt will not sufficiently develop her powers, nor Science attain her\nmost commanding position, till the practical knowledge of the one is\ntaken in return for the sound deductions of the other.... It is in the\nfine arts, principally, and in the speculations with which they are\nassociated, that the controlling power of scientific truth has not\nexercised its legitimate influence. In discussing the principles of\npainting, sculpture, architecture, and landscape gardening, philosophers\nhave renounced science as a guide, and even as an auxiliary; and a\nschool has arisen whose speculations will brook no restraint, and whose\ndecisions stand in opposition to the strongest convictions of our\nsenses. That the external world, in its gay colours and lovely forms, is\nexhibited to the mind only as a tinted mass, neither within nor without\nthe eye, neither touching it nor distant from it\u2014an ubiquitous chaos,\nwhich experience only can analyse and transform into the realities\nwhich compose it; that the beautiful and sublime in nature and in art\nderive their power over the mind from association alone, are among the\nphilosophical doctrines of the present day, which, if it be safe, it is\nscarcely prudent to question. Nor are these opinions the emanations of\npoetical or ill-trained minds, which ingenuity has elaborated, and which\nfashion sustains. They are conclusions at which most of our distinguished\nphilosophers have arrived. They have been given to the world with all the\nauthority of demonstrated truth; and in proportion to the hold which they\nhave taken of the public mind, have they operated as a check upon the\nprogress of knowledge.\u201d\n\nSuch, then, was the state of art as connected with science twelve years\nago. But although the causes which then placed science and the fine arts\nat variance have since been gradually diminishing, yet they are still\nfar from being removed. In proof of this I may refer to what took place\nat the annual distribution of the prizes to the students attending our\nScottish Metropolitan School of Design, in 1854, the pupils in which\namount to upwards of two hundred. The meeting on that occasion included,\nbesides the pupils, a numerous and highly respectable assemblage of\nartists and men of science. The chairman, a Professor in our University,\nand editor of one of the most voluminous works on art, science, and\nliterature ever produced in this country, after extolling the general\nprogress of the pupils, so far as evinced by the drawings exhibited on\nthe occasion, drew the attention of the meeting to a discovery made by\nthe head master of the architectural and ornamental department of the\nschool, viz.\u2014That the ground-plan of the Parthenon at Athens had been\nconstructed by the application of the _mysterious_ ovoid or _Vesica\nPiscis_ of the middle ages, subdivided by the _mythic_ numbers 3 and 7,\nand their intermediate odd number 5. Now, it may be remarked, that the\nfigure thus referred to is not an ovoid, neither is it in any way of a\nmysterious nature, being produced simply by two equal circles cutting\neach other in their centres. Neither can it be shewn that the numbers 3\nand 7 are in any way more mythic than other numbers. In fact, the terms\n_mysterious_ and _mythic_ so applied, can only be regarded as a remnant\nof an ancient terminology, calculated to obscure the simplicity of\nscientific truth, and when used by those employed to teach\u2014for doubtless\nthe chairman only gave the description he received\u2014must tend to retard\nthe connexion of that truth with the arts of design. I shall now give\na specimen of the manner in which a knowledge of the philosophy of the\nfine arts is at present inculcated upon the public mind generally. In\nthe same metropolis there has likewise existed for upwards of ten years\na Philosophical Institution of great importance and utility, whose\nmembers amount to nearly three thousand, embracing a large proportion\nof the higher classes of society, both in respect to talent and wealth.\nAt the close of the session of this Institution, in 1854, a learned and\neloquent philologus, who occasionally lectures upon beauty, was appointed\nto deliver the closing address, and touching upon the subject of the\nbeautiful, he thus concluded\u2014\n\n\u201cIn the worship of the beautiful, and in that alone, we are inferior to\nthe Greeks. Let us therefore be glad to borrow from them; not slavishly,\nbut with a wise adaptation\u2014not exclusively, but with a cunning selection;\nin art, as in religion, let us learn to prove all things, and hold fast\nthat which is good\u2014not merely one thing which is good, but all good\nthings\u2014Classicalism, Medi\u00e6valism, Modernism\u2014let us have and hold them\nall in one wide and lusty embrace. Why should the world of art be\nmore narrow, more monotonous, than the world of nature? Did God make\nall the flowers of one pattern, to please the devotees of the rose or\nthe lily; and did He make all the hills, with the green folds of their\nqueenly mantles, all at one slope, to suit the angleometer of the most\nmathematical of decorators? I trow not. Let us go and do likewise.\u201d\n\nI here take for granted, that what the lecturer meant by \u201cthe worship of\nthe beautiful,\u201d is the production and appreciation of works of art in\nwhich beauty should be a primary element; and judging from the remains\nwhich we possess of such works as were produced by the ancient Grecians,\nour inferiority to them in these respects cannot certainly be denied.\nBut I must reiterate what I have often before asserted, that it is\nnot by borrowing from them, however cunning our selection, or however\nwise our adaptations, that this inferiority is to be removed, but by\na re-discovery of the science which these ancient artists must have\nemployed in the production of that symmetrical beauty and chaste elegance\nwhich pervaded all their works for a period of nearly three hundred\nyears. And I hold, that as in religion, so in art, there is only one\ntruth, a grain of which is worth any amount of philological eloquence.\n\nI also take for granted, that what is meant by Classicalism in the\nabove quotation, is the ancient Grecian style of art; by Medi\u00e6valism,\nthe semi-barbaric style of the middle ages; and by Modernism, that\nchaotic jumble of all previous styles and fashions of art, which is the\npeculiar characteristic of our present school, and which is, doubtless,\nthe result of a system of education based upon plagiarism and mere\nimitation. Therefore a recommendation to embrace with equal fervour \u201cas\ngood things,\u201d these very opposite artic_isms_ must be a doctrine as\nmischievous in art as it would be in religion to recommend as equally\ngood things the various _isms_ into which it has also been split in\nmodern times.\n\nNow, \u201cthe world of nature\u201d and \u201cthe world of art\u201d have not that equality\nof scope which this lecturer on beauty ascribes to them, but differ very\ndecidedly in that particular. Neither will it be difficult to shew why\n\u201cthe world of art _should_ be more narrow than the world of nature\u201d\u2014that\nit should be thereby rendered more monotonous does not follow.\n\nIt is well known, that the \u201cworld of nature\u201d consists of productions,\nincluding objects of every degree of beauty from the very lowest to the\nhighest, and calculated to suit not only the tastes arising from various\ndegrees of intellect, but those arising from the natural instincts of the\nlower animals. On the other hand, \u201cthe world of art,\u201d being devoted to\nthe gratification and improvement of intelligent minds only, is therefore\nnarrowed in its scope by the exclusion from its productions of the lower\ndegrees of beauty\u2014even mediocrity is inadmissible; and we know that the\nscience of the ancient Greek artists enabled them to excel the highest\nindividual productions of nature in the perfection of symmetrical beauty.\nConsequently, all objects in nature are not equally well adapted for\nartistic study, and it therefore requires, on the part of the artist,\nbesides true genius, much experience and care to enable him to choose\nproper subjects from nature; and it is in the choice of such subjects,\nand not in plagiarism from the ancients, that he should select with\nknowledge and adapt with wisdom. Hence, all such latitudinarian doctrines\nas those I have quoted must act as a check upon the progress of knowledge\nin the scientific truth of art. I have observed in some of my works, that\nin this country a course had been followed in our search for the true\nscience of beauty not differing from that by which the alchymists of the\nmiddle ages conducted their investigations; for our ideas of visible\nbeauty are still undefined, and our attempts to produce it in the various\nbranches of art are left dependant, in a great measure, upon chance.\nOur schools are conducted without reference to any first principles or\ndefinite laws of beauty, and from the drawing of a simple architectural\nmoulding to the intricate combinations of form in the human figure, the\npupils trust to their hands and eyes alone, servilely and mechanically\ncopying the works of the ancients, instead of being instructed in the\nunerring principles of science, upon which the beauty of those works\nnormally depends. The instruction they receive is imparted without\nreference to the judgment or understanding, and they are thereby led to\nimitate effects without investigating causes. Doubtless, men of great\ngenius sometimes arrive at excellence in the arts of design without a\nknowledge of the principles upon which beauty of form is based; but it\nshould be kept in mind, that true genius includes an intuitive perception\nof those principles along with its creative power. It is, therefore, to\nthe generality of mankind that instruction in the definite laws of beauty\nwill be of most service, not only in improving the practice of those who\nfollow the arts professionally, but in enabling all of us to distinguish\nthe true from the false, and to exercise a sound and discriminating\ntaste in forming our judgment upon artistic productions. \u00c6sthetic\nculture should consequently supersede servile copying, as the basis of\ninstruction in our schools of art. Many teachers of drawing, however,\nstill assert, that, by copying the great works of the ancients, the mind\nof the pupil will become imbued with ideas similar to theirs\u2014that he\nwill imbibe their feeling for the beautiful, and thereby become inspired\nwith their genius, and think as they thought. To study carefully and\nto investigate the principles which constitute the excellence of the\nworks of the ancients, is no doubt of much benefit to the student; but\nit would be as unreasonable to suppose that he should become inspired\nwith artistic genius by merely copying them, as it would be to imagine,\nthat, in literature, poetic inspiration could be created by making boys\ntranscribe or repeat the works of the ancient poets. Sir Joshua Reynolds\nconsidered copying as a delusive kind of industry, and has observed, that\n\u201cNature herself is not to be too closely copied,\u201d asserting that \u201cthere\nare excellences in the art of painting beyond what is commonly called the\nimitation of nature,\u201d and that \u201ca mere copier of nature can never produce\nany thing great.\u201d Proclus, an eminent philosopher and mathematician of\nthe later Platonist school (A.D. 485), says, that \u201che who takes for his\nmodel such forms as nature produces, and confines himself to an exact\nimitation of these, will never attain to what is perfectly beautiful. For\nthe works of nature are full of disproportion, and fall very short of the\ntrue standard of beauty.\u201d\n\nIt is remarked by Mr. J. C. Daniel, in the introduction to his\ntranslation of M. Victor Cousin\u2019s \u201cPhilosophy of the Beautiful,\u201d that\n\u201cthe English writers have advocated no theory which allows the beautiful\nto be universal and absolute; nor have they professedly founded their\nviews on original and ultimate principles. Thus the doctrine of the\nEnglish school has for the most part been, that beauty is mutable and\nspecial, and the inference that has been drawn from this teaching is,\nthat all tastes are equally just, provided that each man speaks of what\nhe feels.\u201d He then observes, that the German, and some of the French\nwriters, have thought far differently; for with them the beautiful is\n\u201csimple, immutable, absolute, though its _forms_ are manifold.\u201d\n\nSo far back as the year 1725, the same truths advanced by the modern\nGerman and French writers, and so eloquently illustrated by M. Cousin,\nwere given to the world by Hutchison in his \u201cInquiry into the Original\nof our Ideas of Beauty and Virtue.\u201d This author says\u2014\u201cWe, by absolute\nbeauty, understand only that beauty which we perceive in objects, without\ncomparison to any thing external, of which the object is supposed an\nimitation or picture, such as the beauty perceived from the works of\nnature, artificial forms, figures, theorems. Comparative or relative\nbeauty is that which we perceive in objects commonly considered as\nimitations or resemblances of something else.\u201d\n\nDr. Reid also, in his \u201cIntellectual Powers of Man,\u201d says\u2014\u201cThat taste,\nwhich we may call rational, is that part of our constitution by which we\nare made to receive pleasure from the contemplation of what we conceive\nto be excellent in its kind, the pleasure being annexed to this judgment,\nand regulated by it. This taste may be true or false, according as it is\nfounded on a true or false judgment. And if it may be true or false, it\nmust have first principles.\u201d\n\nM. Victor Cousin\u2019s opinion upon this subject is, however, still more\nconclusive. He observes\u2014\u201cIf the idea of the beautiful is not absolute,\nlike the idea of the true\u2014if it is nothing more than the expression of\nindividual sentiment, the rebound of a changing sensation, or the result\nof each person\u2019s fancy\u2014then the discussions on the fine arts waver\nwithout support, and will never end. For a theory of the fine arts to\nbe possible, there must be something absolute in beauty, just as there\nmust be something absolute in the idea of goodness, to render morals a\npossible science.\u201d\n\nThe basis of the science of beauty must thus be founded upon fixed\nprinciples, and when these principles are evolved with the same care\nwhich has characterised the labours of investigators in natural science,\nand are applied in the fine arts as the natural sciences have been in\nthe useful arts, a solid foundation will be laid, not only for correct\npractice, but also for a just appreciation of productions in every branch\nof the arts of design.\n\nWe know that the mind receives pleasure through the sense of hearing,\nnot only from the music of nature, but from the euphony of prosaic\ncomposition, the rhythm of poetic measure, the artistic composition\nof successive harmony in simple melody, and the combined harmony of\ncounterpoint in the more complex works of that art. We know, also,\nthat the mind is similarly gratified through the sense of seeing, not\nonly by the visible beauties of nature, but by those of art, whether\nin symmetrical or picturesque compositions of forms, or in harmonious\narrangements of gay or sombre colouring.\n\nNow, in respect to the first of these modes of sensation, we know,\nthat from the time of Pythagoras, the fact has been established, that\nin whatever manner nature or art may address the ear, the degree of\nobedience paid to the fundamental law of harmony will determine the\npresence and degree of that beauty with which a perfect organ can impress\na well-constituted mind; and it is my object in this, as it has been\nin former attempts, to prove it consistent with scientific truth, that\nthat beauty which is addressed to the mind by objects of nature and art,\nthrough the eye, is similarly governed. In short, to shew that, as in\ncompositions of sounds, there can be no true beauty in the absence of a\nstrict obedience to this great law of nature, neither can there exist,\nin compositions of forms or colours, that principle of unity in variety\nwhich constitutes beauty, unless such compositions are governed by the\nsame law.\n\nAlthough in the songs of birds, the gurgling of brooks, the sighing of\nthe gentle summer winds, and all the other beautiful music of nature,\nno analysis might be able to detect the operation of any precise system\nof harmony, yet the pleasure thus afforded to the human mind we know to\narise from its responding to every development of an obedience to this\nlaw. When, in like manner, we find even in those compositions of forms\nand colours which constitute the wildest and most rugged of Nature\u2019s\nscenery, a species of picturesque grandeur and beauty to which the mind\nas readily responds as to her more mild and pleasing aspects, or to her\nsweetest music, we may rest assured that this beauty is simply another\ndevelopment of, and response to, the same harmonic law, although the\nprecise nature of its operation may be too subtle to be easily detected.\n\nThe _r\u00e9sum\u00e9_ of the various works I have already published upon the\nsubject, along with the additional illustrations I am about to lay before\nmy readers, will, I trust, point out a system of harmony, which, in\nformative art, as well as in that of colouring, will rise superior to the\nidiosyncracies of different artists, and bring back to one common type\nthe sensations of the eye and the ear, thereby improving that knowledge\nof the laws of the universe which it is as much the business of science\nto combine with the ornamental as with the useful arts.\n\nIn attempting this, however, I beg it may be understood, that I do not\nbelieve any system, based even upon the laws of nature, capable of\nforming a royal road to the perfection of art, or of \u201cmapping the mighty\nmaze of a creative mind.\u201d At the same time, however, I must continue to\nreiterate the fact, that the diffusion of a general knowledge of the\nscience of visible beauty will afford latent artistic genius just such a\nvantage ground as that which the general knowledge of philology diffused\nthroughout this country affords its latent literary genius. Although\n_mere learning_ and _true genius_ differ as much in the practice of art\nas they do in the practice of literature, yet a precise and systematic\neducation in the true science of beauty must certainly be as useful in\npromoting the practice and appreciation of the one, as a precise and\nsystematic education in the science of philology is in promoting the\npractice and appreciation of the other.\n\nAs all beauty is the result of harmony, it will be requisite here to\nremark, that harmony is not a simple quality, but, as Aristotle defines\nit, \u201cthe union of contrary principles having a ratio to each other.\u201d\nHarmony thus operates in the production of all that is beautiful in\nnature, whether in the combinations, in the motions, or in the affinities\nof the elements of matter.\n\nThe contrary principles to which Aristotle alludes, are those of\nuniformity and variety; for, according to the predominance of the one or\nthe other of these principles, every kind of beauty is characterised.\nHence the difference between symmetrical and picturesque beauty:\u2014the\nfirst allied to the principle of uniformity, in being based upon precise\nlaws that may be taught so as to enable men of ordinary capacity to\nproduce it in their works\u2014the second allied to the principle of variety\noften to so great a degree that they yield an obedience to the precise\nprinciples of harmony so subtilely, that they cannot be detected in its\nconstitution, but are only felt in the response by which true genius\nacknowledges their presence. The generality of mankind may be capable\nof perceiving this latter kind of beauty, and of feeling its effects\nupon the mind, but men of genius, only, can impart it to works of art,\nwhether addressed to the eye or the ear. Throughout the sounds, forms,\nand colours of nature, these two kinds of beauty are found not only in\ndistinct developments, but in every degree of amalgamation. We find in\nthe songs of some birds, such as those of the chaffinch, thrush, &c.,\na rhythmical division, resembling in some measure the symmetrically\nprecise arrangements of parts which characterises all artistic musical\ncomposition; while in the songs of other birds, and in the other numerous\nmelodies with which nature charms and soothes the mind, there is no\ndistinct regularity in the division of their parts. In the forms of\nnature, too, we find amongst the innumerable flowers with which the\nsurface of the earth is so profusely decorated, an almost endless variety\nof systematic arrangements of beautiful figures, often so perfectly\nsymmetrical in their combination, that the most careful application\nof the angleometer could scarcely detect the slightest deviation from\ngeometrical precision; while, amongst the masses of foliage by which\nthe forms of many trees are divided and subdivided into parts, as also\namongst the hills and valleys, the mountains and ravines, which divide\nthe earth\u2019s surface, we find in every possible variety of aspect the\nbeauty produced by that irregular species of symmetry which characterises\nthe picturesque.\n\nIn like manner, we find in wild as well as cultivated flowers the\nmost symmetrical distributions of colours accompanying an equally\nprecise species of harmony in their various kinds of contrasts, often\nas mathematically regular as the geometric diagrams by which writers\nupon colour sometimes illustrate their works; while in the general\ncolouring of the picturesque beauties of nature, there is an endless\nvariety in its distributions, its blendings, and its modifications. In\nthe forms and colouring of animals, too, the same endless variety of\nregular and irregular symmetry is to be found. But the highest degree of\nbeauty in nature is the result of an equal balance of uniformity with\nvariety. Of this the human figure is an example; because, when it is\nof those proportions universally acknowledged to be the most perfect,\nits uniformity bears to its variety an apparently equal ratio. The\nharmony of combination in the normal proportions of its parts, and the\nbeautifully simple harmony of succession in the normal melody of its\nsoftly undulating outline, are the perfection of symmetrical beauty,\nwhile the innumerable changes upon the contour which arise from the\nactions and attitudes occasioned by the various emotions of the mind,\nare calculated to produce every species of picturesque beauty, from the\nsoftest and most pleasing to the grandest and most sublime.\n\nAmongst the purely picturesque objects of inanimate nature, I may, as in\na former work, instance an ancient oak tree, for its beauty is enhanced\nby want of apparent symmetry. Thus, the more fantastically crooked its\nbranches, and the greater the dissimilarity and variety it exhibits in\nits masses of foliage, the more beautiful it appears to the artist and\nthe amateur; and, as in the human figure, any attempt to produce variety\nin the proportions of its lateral halves would be destructive of its\nsymmetrical beauty, so in the oak tree any attempt to produce palpable\nsimilarity between any of its opposite sides would equally deteriorate\nits picturesque beauty. But picturesque beauty is not the result of the\ntotal absence of symmetry; for, as none of the irregularly constructed\nmusic of nature could be pleasing to the ear unless there existed in\nthe arrangement of its notes an obedience, however subtle, to the great\nharmonic law of Nature, so neither could any object be picturesquely\nbeautiful, unless the arrangement of its parts yields, although it may be\nobscurely, an obedience to the same law.\n\nHowever symmetrically beautiful any architectural structure may be, when\nin a complete and perfect state, it must, as it proceeds towards ruin,\nblend the picturesque with the symmetrical; but the type of its beauty\nwill continue to be the latter, so long as a sufficient portion of it\nremains to convey an idea of its original perfection. It is the same with\nthe human form and countenance; for age does not destroy their original\nbeauty, but in both only lessens that which is symmetrical, while it\nincreases that which is picturesque.\n\nIn short, as a variety of simultaneously produced sounds, which do not\nrelate to each other agreeably to this law, can only convey to the mind\na feeling of mere noise; so a variety of forms or colours simultaneously\nexposed to the eye under similar circumstances, can only convey to the\nmind a feeling of chaotic confusion, or what may be termed _visible_\ndiscord. As, therefore, the two principles of uniformity and variety,\nor similarity and dissimilarity, are in operation in every harmonious\ncombination of the elements of sound, of form, and of colour, we must\nfirst have recourse to numbers in the abstract before we can form a\nproper basis for a universal science of beauty.\n\n\n\n\nTHE SCIENCE OF BEAUTY EVOLVED FROM THE HARMONIC LAW OF NATURE, AGREEABLY\nTO THE PYTHAGOREAN SYSTEM OF NUMERICAL RATIO.\n\n\nThe scientific principles of beauty appear to have been well known to the\nancient Greeks; and it must have been by the practical application of\nthat knowledge to the arts of Design, that that people continued for a\nperiod of upwards of three hundred years to execute, in every department\nof these arts, works surpassing in chaste beauty any that had ever before\nappeared, and which have not been equalled during the two thousand years\nwhich have since elapsed.\n\n\u00c6sthetic science, as the science of beauty is now termed, is based\nupon that great harmonic law of nature which pervades and governs the\nuniverse. It is in its nature neither absolutely physical nor absolutely\nmetaphysical, but of an intermediate nature, assimilating in various\ndegrees, more or less, to one or other of those opposite kinds of\nscience. It specially embodies the inherent principles which govern\nimpressions made upon the mind through the senses of hearing and seeing.\nThus, the \u00e6sthetic pleasure derived from listening to the beautiful in\nmusical composition, and from contemplating the beautiful in works of\nformative art, is in both cases simply a response in the human mind to\nartistic developments of the great harmonic law upon which the science\nis based.\n\nAlthough the eye and the ear are two different senses, and, consequently,\nvarious in their modes of receiving impressions; yet the sensorium is\nbut one, and the mind by which these impressions are perceived and\nappreciated is also characterised by unity. There appears, likewise,\na striking analogy between the natural constitution of the two kinds\nof beauty, which is this, that the more physically \u00e6sthetic elements\nof the highest works of musical composition are melody, harmony, and\ntone, whilst those of the highest works of formative art are contour,\nproportion, and colour. The melody or theme of a musical composition\nand its harmony are respectively analogous,\u20141st, To the outline of\nan artistic work of formative art; and 2d, To the proportion which\nexists amongst its parts. To the careful investigator these analogies\nbecome identities in their effect upon the mind, like those of the more\nmetaphysically \u00e6sthetic emotions produced by expression in either of\nthese arts.\n\nAgreeably to the first analogy, the outline and contour of an object,\nsuppose that of a building in shade when viewed against a light\nbackground, has a similar effect upon the mind with that of the simple\nmelody of a musical composition when addressed to the ear unaccompanied\nby the combined harmony of counterpoint. Agreeably to the second analogy,\nthe various parts into which the surface of the supposed elevation is\ndivided being simultaneously presented to the eye, will, if arranged\nagreeably to the same great law, affect the mind like that of an equally\nharmonious arrangement of musical notes accompanying the supposed melody.\n\nThere is, however, a difference between the construction of these two\norgans of sense, viz., that the ear must in a great degree receive its\nimpressions involuntarily; while the eye, on the other hand, is provided\nby nature with the power of either dwelling upon, or instantly shutting\nout or withdrawing itself from an object. The impression of a sound,\nwhether simple or complex, when made upon the ear, is instantaneously\nconveyed to the mind; but when the sound ceases, the power of observation\nalso ceases. But the eye can dwell upon objects presented to it so long\nas they are allowed to remain pictured on the retina; and the mind has\nthereby the power of leisurely examining and comparing them. Hence the\near guides more as a mere sense, at once and without reflection; whilst\nthe eye, receiving its impressions gradually, and part by part, is more\ndirectly under the influence of mental analysis, consequently producing\na more metaphysically \u00e6sthetic emotion. Hence, also, the acquired power\nof the mind in appreciating impressions made upon it through the organ of\nsight under circumstances, such as perspective, &c., which to those who\ntake a hasty view of the subject appear impossible.\n\nDealing as this science therefore does, alike with the sources and the\nresulting principles of beauty, it is scarcely less dependent on the\naccuracy of the senses than on the power of the understanding, inasmuch\nas the effect which it produces is as essential a property of objects,\nas are its laws inherent in the human mind. It necessarily comprehends a\nknowledge of those first principles in art, by which certain combinations\nof sounds, forms, and colours produce an effect upon the mind, connected,\nin the first instance, with sensation, and in the second with the\nreasoning faculty. It is, therefore, not only the basis of all true\npractice in art, but of all sound judgment on questions of artistic\ncriticism, and necessarily includes those laws whereon a correct taste\nmust be based. Doubtless many eloquent and ingenious treatises have been\nwritten upon beauty and taste; but in nearly every case, with no other\neffect than that of involving the subject in still greater uncertainty.\nEven when restricted to the arts of design, they have failed to exhibit\nany definite principles whereby the true may be distinguished from\nthe false, and some natural and recognised laws of beauty reduced to\ndemonstration. This may be attributed, in a great degree, to the neglect\nof a just discrimination between what is merely agreeable, or capable\nof exciting pleasurable sensations, and what is essentially beautiful;\nbut still more to the confounding of the operations of the understanding\nwith those of the imagination. Very slight reflection, however, will\nsuffice to shew how essentially distinct these two faculties of the mind\nare; the former being regulated, in matters of taste, by irrefragable\nprinciples existing in nature, and responded to by an inherent principle\nexisting in the human mind; while the latter operates in the production\nof ideal combinations of its own creation, altogether independent of any\nimmediate impression made upon the senses. The beauty of a flower, for\nexample, or of a dew-drop, depends on certain combinations of form and\ncolour, manifestly referable to definite and systematic, though it may\nbe unrecognised, laws; but when Oberon, in \u201cMidsummer Night\u2019s Dream,\u201d is\nmade to exclaim\u2014\n\n \u201cAnd that same dew, which sometimes on the buds\n Was wont to swell, like round and orient pearls,\n Stood now within the pretty floweret\u2019s eyes,\n Like tears that did their own disgrace bewail,\u201d\u2014\n\nthe poet introduces a new element of beauty equally legitimate, yet\naltogether distinct from, although accompanying that which constitutes\nthe more precise science of \u00e6sthetics as here defined. The composition\nof the rhythm is an operation of the understanding, but the beauty of the\npoetic fancy is an operation of the imagination.\n\nOur physical and mental powers, \u00e6sthetically considered, may therefore be\nclassed under three heads, in their relation to the fine arts, viz., the\nreceptive, the perceptive, and the conceptive.\n\nThe senses of hearing and seeing are respectively, in the degree of\ntheir physical power, receptive of impressions made upon them, and of\nthese impressions the sensorium, in the degree of its mental power, is\nperceptive. This perception enables the mind to form a judgment whereby\nit appreciates the nature and quality of the impression originally made\non the receptive organ. The mode of this operation is intuitive, and\nthe quickness and accuracy with which the nature and quality of the\nimpression is apprehended, will be in the degree of the intellectual\nvigour of the mind by which it is perceived. Thus we are, by the\ncultivation of these intuitive faculties, enabled to decide with accuracy\nas to harmony or discord, proportion or deformity, and assign sound\nreasons for our judgment in matters of taste. But mental conception is\nthe intuitive power of constructing original ideas from these materials;\nfor after the receptive power has acted, the perception operates in\nestablishing facts, and then the judgment is formed upon these operations\nby the reasoning powers, which lead, in their turn, to the creations of\nthe imagination.\n\nThe power of forming these creations is the true characteristic\nof genius, and determines the point at which art is placed beyond\nall determinable canons,\u2014at which, indeed, \u00e6sthetics give place to\nmetaphysics.\n\nIn the science of beauty, therefore, the human mind is the subject, and\nthe effect of external nature, as well as of works of art, the object.\nThe external world, and the individual mind, with all that lies within\nthe scope of its powers, may be considered as two separate existences,\nhaving a distinct relation to each other. The subject is affected by the\nobject, through that inherent faculty by which it is enabled to respond\nto every development of the all-governing harmonic law of nature; and the\nmedia of communication are the sensorium and its inlets\u2014the organs of\nsense.\n\nThis harmonic law of nature was either originally discovered by that\nillustrious philosopher Pythagoras, upwards of five hundred years before\nChrist, or a knowledge of it obtained by him about that period, from\nthe Egyptian or Chaldean priests. For after having been initiated into\nall the Grecian and barbarian sacred mysteries, he went to Egypt, where\nhe remained upwards of twenty years, studying in the colleges of its\npriests; and from Egypt he went into the East, and visited the Persian\nand Chaldean magi.[3]\n\nBy the generality of the biographers of Pythagoras, it is said to be\ndifficult to give a clear idea of his philosophy, as it is almost certain\nhe never committed it to writing, and that it has been disfigured by the\nfantastic dreams and chimeras of later Pythagoreans. Diogenes La\u00ebrtius,\nhowever, whose \u201cLives of the Philosophers\u201d was supposed to be written\nabout the end of the second century of our era, says \u201cthere are three\nvolumes extant written by Pythagoras. One on education, one on politics,\nand one on natural philosophy.\u201d And adds, that there were several other\nbooks extant, attributed to Pythagoras, but which were not written by\nhim. Also, in his \u201cLife of Philolaus,\u201d that Plato wrote to Dion to take\ncare and purchase the books of Pythagoras.[4] But whether this great\nphilosopher committed his discoveries to writing or not, his doctrines\nregarding the philosophy of beauty are well-known to be, that he\nconsidered numbers as the essence and the principle of all things, and\nattributed to them a real and distinct existence; so that, in his view,\nthey were the elements out of which the universe was constructed, and to\nwhich it owed its beauty. Diogenes La\u00ebrtius gives the following account\nof this law:\u2014\u201cThat the monad was the beginning of everything. From the\nmonad proceeds an indefinite duad, which is subordinate to the monad as\nto its cause. That from the monad and indefinite duad proceeds numbers.\nThat the part of science to which Pythagoras applied himself above all\nothers, was arithmetic; and that he taught \u2018that from numbers proceed\nsigns, and from these latter, lines, of which plane figures consist;\nthat from plane figures are derived solid bodies; that of all plane\nfigures the most beautiful was the circle, and of all solid bodies the\nmost beautiful was the sphere.\u2019 He discovered the numerical relations of\nsounds on a single string; and taught that everything owes its existence\nand consistency to harmony. In so far as I know, the most condensed\naccount of all that is known of the Pythagorian system of numbers is the\nfollowing:\u2014\u2018The monad or unity is that quantity, which, being deprived of\nall number, remains fixed. It is the fountain of all number. The duad is\nimperfect and passive, and the cause of increase and division. The triad,\ncomposed of the monad and duad, partakes of the nature of both. The\ntetrad, tetractys, or quaternion number is most perfect. The decad, which\nis the sum of the four former, comprehends all arithmetical and musical\nproportions.\u2019\u201d[5]\n\nThese short quotations, I believe, comprise all that is known, for\ncertain, of the manner in which Pythagoras systematised the law of\nnumbers. Yet, from the teachings of this great philosopher and his\ndisciples, the harmonic law of nature, in which the fundamental\nprinciples of beauty are embodied, became so generally understood and\nuniversally applied in practice throughout all Greece, that the fragments\nof their works, which have reached us through a period of two thousand\nyears, are still held to be examples of the highest artistic excellence\never attained by mankind. In the present state of art, therefore, a\nknowledge of this law, and of the manner in which it may again be applied\nin the production of beauty in all works of form and colour, must be\nof singular advantage; and the object of this work is to assist in the\nattainment of such a knowledge.\n\nIt has been remarked, with equal comprehensiveness and truth, by a\nwriter[6] in the _British and Foreign Medical Review_, that \u201cthere\nis harmony of numbers in all nature\u2014in the force of gravity\u2014in the\nplanetary movements\u2014in the laws of heat, light, electricity, and chemical\naffinity\u2014in the forms of animals and plants\u2014in the perceptions of the\nmind. The direction, indeed, of modern natural and physical science is\ntowards a generalization which shall express the fundamental laws of all\nby one simple numerical ratio. And we think modern science will soon shew\nthat the mysticism of Pythagoras was mystical only to the unlettered,\nand that it was a system of philosophy founded on the then existing\nmathematics, which latter seem to have comprised more of the philosophy\nof numbers than our present.\u201d Many years of careful investigation have\nconvinced me of the truth of this remark, and of the great advantage\nderivable from an application of the Pythagorean system in the arts\nof design. For so simple is its nature, that any one of an ordinary\ncapacity of mind, and having a knowledge of the most simple rules of\narithmetic, may, in a very short period, easily comprehend its nature,\nand be able to apply it in practice.\n\nThe elements of the Pythagorean system of harmonic number, so far as can\nbe gathered from the quotations I have given above, seem to be simply\nthe indivisible monad (1); the duad (2), arising from the union of one\nmonad with another; the triad (3), arising from the union of the monad\nwith the duad; and the tetrad (4), arising from the union of one duad\nwith another, which tetrad is considered a perfect number. From the\nunion of these four elements arises the decad (10), the number, which,\nagreeably to the Pythagorean system, comprehends all arithmetical and\nharmonic proportions. If, therefore, we take these elements and unite\nthem progressively in the following order, we shall find the series of\nharmonic numbers (2), (3), (5), and (7), which, with their multiples, are\nthe complete numerical elements of all harmony, thus:\u2014\n\n 1 + 1 = 2\n 1 + 2 = 3\n 2 + 3 = 5\n 3 + 4 = 7\n\nIn order to render an extended series of harmonic numbers useful, it\nmust be divided into scales; and it is a rule in the formation of these\nscales, that the first must begin with the monad (1) and end with the\nduad (2), the second begin with the duad (2) and end with the tetrad (4),\nand that the beginning and end of all other scales must be continued in\nthe same arithmetical progression. These primary elements will then form\nthe foundation of a series of such scales.\n\n I. (1) (2)\n II. (2) (3) (4)\n III. (4) (5) (6) (7) (8)\n IV. (8) (9) (10) ( ) (12) ( ) (14) (15) (16)\n\nThe first of these scales has in (1) and (2) a beginning and an end; but\nthe second has in (2), (3), and (4) the essential requisites demanded\nby Aristotle in every composition, viz., \u201ca beginning, a middle, and\nan end;\u201d while the third has not only these essential requisites, but\ntwo intermediate parts (5) and (7), by which the beginning, the middle,\nand the end are united. In the fourth scale, however, the arithmetical\nprogression is interrupted by the omission of numbers 11 and 13, which,\nnot being multiples of either (2), (3), (5), or (7), are inadmissible.\n\nSuch is the nature of the harmonic law which governs the progressive\nscales of numbers by the simple multiplication of the monad.\n\nI shall now use these numbers as divisors in the formation of a series\nof four such scales of parts, which has for its primary element, instead\nof the indivisible monad, a quantity which may be indefinitely divided,\nbut which cannot be added to or multiplied. Like the monad, however, this\nquantity is represented by (1). The following is this series of four\nscales of harmonic parts:\u2014\n\n I. (1) (\u00b9\u2044\u2082)\n II. (\u00b9\u2044\u2082) (\u00b9\u2044\u2083) (\u00b9\u2044\u2084)\n III. (\u00b9\u2044\u2084) (\u00b9\u2044\u2085) (\u00b9\u2044\u2086) (\u00b9\u2044\u2087) (\u00b9\u2044\u2088)\n IV. (\u00b9\u2044\u2088) (\u00b9\u2044\u2089) (\u00b9\u2044\u2081\u2080) ( ) (\u00b9\u2044\u2081\u2082) ( ) (\u00b9\u2044\u2081\u2084) (\u00b9\u2044\u2081\u2085) (\u00b9\u2044\u2081\u2086)\n\nThe scales I., II., and III. may now be rendered as complete as scale\nIV., simply by multiplying upwards by 2 from (\u00b9\u2044\u2089), (\u00b9\u2044\u2085), (\u00b9\u2044\u2083), (\u00b9\u2044\u2087),\nand (\u00b9\u2044\u2081\u2085), thus:\u2014\n\n I. (1) (\u2078\u2044\u2089) (\u2074\u2044\u2085) (\u00b2\u2044\u2083) (\u2074\u2044\u2087) (\u2078\u2044\u2081\u2085) (\u00b9\u2044\u2082)\n II. (\u00b9\u2044\u2082) (\u2074\u2044\u2089) (\u00b2\u2044\u2085) (\u00b9\u2044\u2083) (\u00b2\u2044\u2087) (\u2074\u2044\u2081\u2085) (\u00b9\u2044\u2084)\n III. (\u00b9\u2044\u2084) (\u00b2\u2044\u2089) (\u00b9\u2044\u2085) (\u00b9\u2044\u2086) (\u00b9\u2044\u2087) (\u00b2\u2044\u2081\u2085) (\u00b9\u2044\u2088)\n IV. (\u00b9\u2044\u2088) (\u00b9\u2044\u2089) (\u00b9\u2044\u2081\u2080) ( ) (\u00b9\u2044\u2081\u2082) ( ) (\u00b9\u2044\u2081\u2084) (\u00b9\u2044\u2081\u2085) (\u00b9\u2044\u2081\u2086)\n\nWe now find between the beginning and the end of scale I. the quantities\n(\u2078\u2044\u2089), (\u2074\u2044\u2085), (\u00b2\u2044\u2083), (\u2074\u2044\u2087), and (\u2078\u2044\u2081\u2085).\n\nThe three first of these quantities we find to be the remainders of the\nwhole indefinite quantity contained in (1), after subtracting from it\nthe primary harmonic quantities (\u00b9\u2044\u2089), (\u00b9\u2044\u2085), and (\u00b9\u2044\u2083); we, however,\nfind also amongst these harmonic quantities that of (\u00b9\u2044\u2084), which being\nsubtracted from (1) leaves (\u00b3\u2044\u2084), a quantity the most suitable whereby\nto fill up the hiatus between (\u2074\u2044\u2085) and (\u00b2\u2044\u2083) in scale I., which arises\nfrom the omission of (\u00b9\u2044\u2081\u2081) in scale IV. In like manner we find the two\nlast of these quantities, (\u2074\u2044\u2087) and (\u2078\u2044\u2081\u2085), are respectively the largest\nof the two parts into which 7 and 15 are susceptible of being divided.\nFinding the number 5 to be divisible into parts more unequal than (2)\nto (3) and less unequal than (4) to (7), (\u00b3\u2044\u2085) naturally fills up the\nhiatus between these quantities in scale I., which hiatus arises from the\nomission of (\u00b9\u2044\u2081\u2083) in scale IV. Thus:\u2014\n\n I. (1) (\u2078\u2044\u2089) (\u2074\u2044\u2085) (\u00b3\u2044\u2084) (\u00b2\u2044\u2083) (\u00b3\u2044\u2085) (\u2074\u2044\u2087) (\u2078\u2044\u2081\u2085) (\u00b9\u2044\u2082)\n II. (\u00b9\u2044\u2082) (\u2074\u2044\u2089) (\u00b2\u2044\u2085) ( ) (\u00b9\u2044\u2083) ( ) (\u00b2\u2044\u2087) (\u2074\u2044\u2081\u2085) (\u00b9\u2044\u2084)\n III. (\u00b9\u2044\u2084) (\u00b2\u2044\u2089) (\u00b9\u2044\u2085) ( ) (\u00b9\u2044\u2086) ( ) (\u00b9\u2044\u2087) (\u00b2\u2044\u2081\u2085) (\u00b9\u2044\u2088)\n IV. (\u00b9\u2044\u2088) (\u00b9\u2044\u2089) (\u00b9\u2044\u2081\u2080) ( ) (\u00b9\u2044\u2081\u2082) ( ) (\u00b9\u2044\u2081\u2084) (\u00b9\u2044\u2081\u2085) (\u00b9\u2044\u2081\u2086)\n\nScale I. being now complete, we have only to divide these latter\nquantities by (2) downwards in order to complete the other three. Thus:\u2014\n\n I. (1) (\u2078\u2044\u2089) (\u2074\u2044\u2085) (\u00b3\u2044\u2084) (\u00b2\u2044\u2083) (\u00b3\u2044\u2085) (\u2074\u2044\u2087) (\u2078\u2044\u2081\u2085) (\u00b9\u2044\u2082)\n II. (\u00b9\u2044\u2082) (\u2074\u2044\u2089) (\u00b2\u2044\u2085) (\u00b3\u2044\u2088) (\u00b9\u2044\u2083) (\u00b3\u2044\u2081\u2080) (\u00b2\u2044\u2087) (\u2074\u2044\u2081\u2085) (\u00b9\u2044\u2084)\n III. (\u00b9\u2044\u2084) (\u00b2\u2044\u2089) (\u00b9\u2044\u2085) (\u00b3\u2044\u2081\u2086) (\u00b9\u2044\u2086) (\u00b3\u2044\u2082\u2080) (\u00b9\u2044\u2087) (\u00b2\u2044\u2081\u2085) (\u00b9\u2044\u2088)\n IV. (\u00b9\u2044\u2088) (\u00b9\u2044\u2089) (\u00b9\u2044\u2081\u2080) (\u00b3\u2044\u2083\u2082) (\u00b9\u2044\u2081\u2082) (\u00b3\u2044\u2084\u2080) (\u00b9\u2044\u2081\u2084) (\u00b9\u2044\u2081\u2085) (\u00b9\u2044\u2081\u2086)\n\nThe harmony existing amongst these numbers or quantities consists of the\nnumerical relations which the parts bear to the whole and to each other;\nand the more simple these relations are, the more perfect is the harmony.\nThe following are the numerical harmonic ratios which the parts bear to\nthe whole:\u2014\n\n I. (1:1) (8:9) (4: 5) (3: 4) (2: 3) (3: 5) (4: 7) (8:15) (1: 2)\n II. (1:2) (4:9) (2: 5) (3: 8) (1: 3) (3:10) (2: 7) (4:15) (1: 4)\n III. (1:4) (2:9) (1: 5) (3:16) (1: 6) (3:20) (1: 7) (2:15) (1: 8)\n IV. (1:8) (1:9) (1:10) (3:32) (1:12) (3:40) (1:14) (1:15) (1:16)\n\nThe following are the principal numerical relations which the parts in\neach scale bear to one another:\u2014\n\n (\u00b9\u2044\u2082):(\u2074\u2044\u2087) = (7:8)\n (\u2074\u2044\u2085):(\u2078\u2044\u2089) = (9:10)\n (\u00b2\u2044\u2083):(\u2074\u2044\u2085) = (5:6)\n (\u2074\u2044\u2087):(\u00b2\u2044\u2083) = (6:7)\n (\u2078\u2044\u2081\u2085):(\u2074\u2044\u2087) = (14:15)\n (\u00b9\u2044\u2082):(\u2078\u2044\u2081\u2085) = (15:16)\n\nAlthough these relations are exemplified by parts of scale I., the same\nratios exist between the relative parts of scales II., III., and IV.,\nand would exist between the parts of any other scales that might be added\nto that series.\n\nThese are the simple elements of the science of that harmony which\npervades the universe, and by which the various kinds of beauty\n\u00e6sthetically impressed upon the senses of hearing and seeing are\ngoverned.\n\n\n\n\nTHE SCIENCE OF BEAUTY AS APPLIED TO SOUNDS.\n\n\nIt is well-known that all sounds arise from a peculiar action of the\nair, and that this action may be excited by the concussion resulting\nfrom the sudden displacement of a portion of the atmosphere itself, or\nby the rapid motions of bodies, or of confined columns of air; in all\nwhich cases, when the motions are irregular, and the force great, the\nsound conveyed to the sensorium is called a noise. But that musical\nsounds are the result of equal and regular vibratory motions, either\nof an elastic body, or of a column of air in a tube, exciting in the\nsurrounding atmosphere a regular and equal pulsation. The ear is the\nmedium of communication between those varieties of atmospheric action and\nthe seat of consciousness. To describe fully the beautiful arrangement\nof the various parts of this organ, and their adaptation to the purpose\nof collecting and conveying these undulatory motions of the atmosphere,\nis as much beyond the scope of my present attempt as it is beyond my\nanatomical knowledge; but I may simply remark, that within the ear, and\nmost carefully protected in the construction of that organ, there is a\nsmall cavity containing a pellucid fluid, in which the minute extremities\nof the auditory nerve float; and that this fluid is the last of the media\nthrough which the action producing the sensation of sound is conveyed\nto the nerve, and thence to the sensorium, where its nature becomes\nperceptible to the mind.\n\nThe impulses which produce musical notes must arrive at a certain\nfrequency before the ear loses the intervals of silence between them,\nand is impressed by only one continued sound; and as they increase\nin frequency the sound becomes more acute upon the ear. The pitch of\na musical note is, therefore, determined by the frequency of these\nimpulses; but, on the other hand, its intensity or loudness will depend\nupon the violence and the quality of its tone on the material employed in\nproducing them. All such sounds, therefore, whatever be their loudness\nor the quality of their tone in which the impulses occur with the same\nfrequency are in perfect unison, having the same pitch. Upon this the\nwhole doctrine of harmonies is founded, and by this the laws of numerical\nratio are found to operate in the production of harmony, and the theory\nof music rendered susceptible of exact reasoning.\n\nThe mechanical means by which such sounds can be produced are extremely\nvarious; but, as it is my purpose simply to shew the nature of harmony\nof sound as related to, or as evolving numerical harmonic ratio, I shall\nconfine myself to the most simple mode of illustration\u2014namely, that of\nthe monochord. This is an instrument consisting of a string of a given\nlength stretched between two bridges standing upon a graduated scale.\nSuppose this string to be stretched until its tension is such that,\nwhen drawn a little to a side and suddenly let go, it would vibrate at\nthe rate of 64 vibrations in a second of time, producing to a certain\ndistance in the surrounding atmosphere a series of pulsations of the same\nfrequency.\n\nThese pulsations will communicate through the ear a musical note which\nwould, therefore, be the fundamental note of such a string. Now, the\nphenomenon said to be discovered by Pythagoras is well known to those\nacquainted with the science of acoustics, namely, that immediately after\nthe string is thus put into vibratory motion, it spontaneously divides\nitself, by a node, into two equal parts, the vibrations of each of which\noccur with a double frequency\u2014namely, 128 in a second of time, and,\nconsequently, produce a note doubly acute in pitch, although much weaker\nas to intensity or loudness; that it then, while performing these two\nseries of vibrations, divides itself, by two nodes, into three parts,\neach of which vibrates with a frequency triple that of the whole string;\nthat is, performs 192 vibrations in a second of time, and produces a\nnote corresponding in increase of acuteness, but still less intense than\nthe former, and that this continues to take place in the arithmetical\nprogression of 2, 3, 4, &c. Simultaneous vibrations, agreeably to the\nsame law of progression, which, however, seem to admit of no other primes\nthan the numbers 2, 3, 5, and 7, are easily excited upon any stringed\ninstrument, even by the lightest possible touch of any of its strings\nwhile in a state of vibratory motion, and the notes thus produced are\ndistinguished by the name of harmonics. It follows, then, that one-half\nof a musical string, when divided from the whole by the pressure of the\nfinger, or any other means, and put into vibratory motion, produces a\nnote doubly acute to that produced by the vibratory motion of the whole\nstring; the third part, similarly separated, a note trebly acute; and\nthe same with every part into which any musical string may be divided.\nThis is the fundamental principle by which all stringed instruments are\nmade to produce harmony. It is the same with wind instruments, the sounds\nof which are produced by the frequency of the pulsations occasioned in\nthe surrounding atmosphere by agitating a column of air confined within\na tube as in an organ, in which the frequency of pulsation becomes\ngreater in an inverse ratio to the length of the pipes. But the following\nseries of four successive scales of musical notes will give the reader\na more comprehensive view of the manner in which they follow the law of\nnumerical ratio just explained than any more lengthened exposition.\n\nIt is here requisite to mention, that in the construction of these\nscales, I have not only adopted the old German or literal mode of\nindicating the notes, but have included, as the Germans do, the note\ntermed by us B flat as B natural, and the note we term B natural as\nH. Now, although this arrangement differs from that followed in the\nconstruction of our modern Diatonic scale, yet as the ratio of 4:7\nis more closely related to that of 1:2 than that of 8:15, and as it\nis offered by nature in the spontaneous division of the monochord,\nI considered it quite admissible. The figures give the parts of the\nmonochord which would produce the notes.\n\n I. { (1) (\u2078\u2044\u2089) (\u2074\u2044\u2085) (\u00b3\u2044\u2084) (\u00b2\u2044\u2083) (\u00b3\u2044\u2085) (\u2074\u2044\u2087) (\u2078\u2044\u2081\u2085) (\u00b9\u2044\u2082)*\n { C D E F G A B H _c_\n\n II. { (\u00b9\u2044\u2082)* (\u2074\u2044\u2089) (\u00b2\u2044\u2085) (\u00b3\u2044\u2088) (\u00b9\u2044\u2083)* (\u00b3\u2044\u2081\u2080) (\u00b2\u2044\u2087) (\u00b2\u2044\u2081\u2085) (\u00b9\u2044\u2084)*\n { _c_ _d_ _e_ _f_ _g_ _a_ _b_ _h_ _c\u2032_\n\n III. { (\u00b9\u2044\u2084)* (\u00b2\u2044\u2089) (\u00b9\u2044\u2085)* (\u00b3\u2044\u2081\u2086) (\u00b9\u2044\u2086)* (\u00b3\u2044\u2082\u2080) (\u00b9\u2044\u2087)* (\u00b2\u2044\u2081\u2085) (\u00b9\u2044\u2088)*\n { _c\u2032_ _d\u2032_ _e\u2032_ _f\u2032_ _g\u2032_ _a\u2032_ _b\u2032_ _h\u2032_ _c\u2032\u2032_\n\n IV. { (\u00b9\u2044\u2088)* (\u00b9\u2044\u2089)* (\u00b9\u2044\u2081\u2080)* (\u00b3\u2044\u2083\u2082) (\u00b9\u2044\u2081\u2082)* (\u00b3\u2044\u2084\u2080) (\u00b9\u2044\u2081\u2084)* (\u00b9\u2044\u2081\u2085)* (\u00b9\u2044\u2081\u2086)*\n { _c\u2032\u2032_ _d\u2032\u2032_ _e\u2032\u2032_ _f\u2032\u2032_ _g\u2032\u2032_ _a\u2032\u2032_ _b\u2032\u2032_ _h\u2032\u2032_ _c\u2032\u2032\u2032_\n\nThe notes marked (*) are the harmonics which naturally arise from the\ndivision of the string by 2, 3, 5, and 7, and the multiples of these\nprimes.\n\nThus every musical sound is composed of a certain number of parts called\npulsations, and these parts must in every scale relate harmonically\nto some fundamental number. When these parts are multiples of the\nfundamental number by 2, 4, 8, &c., like the pulsations of the sounds\nindicated by _c_, _c\u2032_, _c\u2032\u2032_, _c\u2032\u2032\u2032_, they are called tonic notes, being\nthe most consonant; when the pulsations are similar multiples by 3, 6,\n12, &c., like those of the sounds indicated by _g_, _g\u2032_, _g\u2032\u2032_, they are\ncalled dominant notes, being the next most consonant; and multiples by\n5, 10, &c., like those of the sounds indicated by _e_, _e\u2032_, _e\u2032\u2032_, they\nare called mediant notes, from a similar cause. In harmonic combinations\nof musical sounds, the \u00e6sthetic feeling produced by their agreement\ndepends upon the relations they bear to each other with reference to the\nnumber of pulsations produced in a given time by the fundamental note of\nthe scale to which they belong; and it will be observed, that the more\nsimple the numerical ratios are amongst the pulsations of any number of\nnotes simultaneously produced, the more perfect their agreement. Hence\nthe origin of the common chord or fundamental concord in the united\nsounds of the tonic, the dominant, and the mediant notes, the ratios and\ncoincidences of whose pulsations 2:1, 3:2, 5:4, may thus be exemplified:\u2014\n\n[Illustration]\n\nIn musical composition, the law of number also governs its division\ninto parts, in order to produce upon the ear, along with the beauty of\nharmony, that of rhythm. Thus a piece of music is divided into parts\neach of which contains a certain number of other parts called bars,\nwhich may be divided and subdivided into any number of notes, and the\nperformance of each bar is understood to occupy the same portion of time,\nhowever numerous the notes it contains may be; so that the music of art\nis regularly symmetrical in its structure; while that of nature is in\ngeneral as irregular and indefinite in its rhythm as it is in its harmony.\n\nThus I have endeavoured briefly to explain the manner in which the law of\nnumerical ratio operates in that species of beauty perceived through the\near.\n\nThe definite principles of the art of music founded upon this law have\nbeen for ages so systematised that those who are instructed in them\nadvance steadily in proportion to their natural endowments, while those\nwho refuse this instruction rarely attain to any excellence. In the\nsister arts of form and colour, however, a system of tuition, founded\nupon this law, is still a desideratum, and a knowledge of the scientific\nprinciples by which these arts are governed is confined to a very few,\nand scarcely acknowledged amongst those whose professions most require\ntheir practical application.\n\n\n\n\nTHE SCIENCE OF BEAUTY AS APPLIED TO FORMS.\n\n\nIt is justly remarked, in the \u201cIllustrated Record of the New York\nExhibition of 1853,\u201d that \u201cit is a question worthy of consideration how\nfar the mediocrity of the present day is attributable to an overweening\nreliance on natural powers and a neglect of the lights of science;\u201d\nand there is expressed a thorough conviction of the fact that, besides\nthe evils of the copying system, \u201cmuch genius is now wasted in the\nacquirement of rudimentary knowledge in the slow school of practical\nexperiment, and that the excellence of the ancient Greek school of\ndesign arose from a thoroughly digested canon of form, and the use\nof geometrical formulas, which make the works even of the second and\nthird-rate genius of that period the wonder and admiration of the present\nday.\u201d\n\nThat such a canon of form, and that the use of such geometrical formula,\nentered into the education, and thereby facilitated the practice of\nancient Greek art, I have in a former work expressed my firm belief,\nwhich is founded on the remarkable fact, that for a period of nearly\nthree centuries, and throughout a whole country politically divided into\nstates often at war with each other, works of sculpture, architecture,\nand ornamental design were executed, which surpass in symmetrical\nbeauty any works of the kind produced during the two thousand years that\nhave since elapsed. So decided is this superiority, that the artistic\nremains of the extraordinary period I alluded to are, in all civilised\nnations, still held to be the most perfect specimens of formative art in\nthe world; and even when so fragmentary as to be denuded of everything\nthat can convey an idea of expression, they still excite admiration and\nwonder by the purity of their geometric beauty. And so universal was\nthis excellence, that it seems to have characterised every production of\nformative art, however humble the use to which it was applied.\n\nThe common supposition, that this excellence was the result of an\nextraordinary amount of genius existing among the Greek people during\nthat particular period, is not consistent with what we know of the\nprogress of mankind in any other direction, and is, in the present state\nof art, calculated to retard its progress, inasmuch as such an idea\nwould suggest that, instead of making any exertion to arrive at a like\ngeneral excellence, the world must wait for it until a similar supposed\npsychological phenomenon shall occur.\n\nBut history tends to prove that the long period of universal artistic\nexcellence throughout Greece could only be the result of an early\ninculcation of some well-digested system of correct elementary\nprinciples, by which the ordinary amount of genius allotted to mankind\nin every age was properly nurtured and cultivated; and by which, also, a\ncorrect knowledge and appreciation of art were disseminated amongst the\npeople generally. Indeed, M\u00fcller, in his \u201cAncient Art and its Remains,\u201d\nshews clearly that some certain fixed principles, constituting a science\nof proportions, were known in Greece, and that they formed the basis of\nall artists\u2019 education and practice during the period referred to; also,\nthat art began to decline, and its brightest period to close, as this\nscience fell into disuse, and the Greek artists, instead of working for\nan enlightened community, who understood the nature of the principles\nwhich guided them, were called upon to gratify the impatient whims of\npampered and tyrannical rulers.\n\nBy being instructed in this science of proportion, the Greek artists\nwere enabled to impart to their representations of the human figure\na mathematically correct species of symmetrical beauty; whether\naccompanying the slender and delicately undulated form of the Venus,\u2014its\nopposite, the massive and powerful mould of the Hercules,\u2014or the\ncharacteristic representation of any other deity in the heathen\nmythology. And this seems to have been done with equal ease in the minute\nfigure cut on a precious gem, and in the most colossal statue. The same\ninstruction likewise enabled the architects of Greece to institute those\nvarieties of proportions in structure called the Classical Orders of\nArchitecture; which are so perfect that, since the science which gave\nthem birth has been buried in oblivion, classical architecture has been\nlittle more than an imitative art; for all who have since written upon\nthe subject, from Vitruvius downwards, have arrived at nothing, in\nso far as the great elementary principles in question are concerned,\nbeyond the most vague and unsatisfactory conjectures. For a more clear\nunderstanding of the nature of this application of the Pythagorean law of\nnumber to the harmony of form, it will be requisite to repeat the fact,\nthat modern science has shewn that the cause of the impression, produced\nby external nature upon the sensorium, called light, may be traced to a\nmolecular or ethereal action. This action is excited naturally by the\nsun, artificially by the combustion of various substances, and sometimes\nphysically within the eye. Like the atmospheric pulsations which produce\nsound, the action which produces light is capable, within a limited\nsphere, of being reflected from some bodies and transmitted through\nothers; and by this reflection and transmission the visible nature of\nforms and figures is communicated to the sensorium. The eye is the\nmedium of this communication; and its structural beauty, and perfect\nadaptation to the purpose of conveying this action, must, like those of\nthe ear, be left to the anatomist fully to describe. It is here only\nnecessary to remark, that the optic nerve, like the auditory nerve,\nends in a carefully protected fluid, which is the last of the media\ninterposed between this peculiarly subtle action and the nerve upon\nwhich it impresses the presence of the object from which it is reflected\nor through which it is transmitted, and the nature of such object made\nperceptible to the mind. The eye and the ear are thus, in one essential\npoint, similar in their physiology, relatively to the means provided\nfor receiving impressions from external nature; it is, therefore, but\nreasonable to believe that the eye is capable of appreciating the exact\nsubdivision of spaces, just as the ear is capable of appreciating the\nexact subdivision of intervals of time; so that the division of space\ninto exact numbers of equal parts will \u00e6sthetically affect the mind\nthrough the medium of the eye.\n\nWe assume, therefore, that the standard of symmetry, so estimated, is\ndeduced from the simplest law that could have been conceived\u2014the law\nthat the angles of direction must all bear to some fixed angle the same\nsimple relations which the different notes in a chord of music bear to\nthe fundamental note; that is, relations expressed arithmetically by the\nsmallest natural numbers. Thus the eye, being guided in its estimate by\ndirection rather than by distance, just as the ear is guided by number\nof vibrations rather than by magnitude, both it and the ear convey\nsimplicity and harmony to the mind without effort, and the mind with\nequal facility receives and appreciates them.\n\n\n_On the Rectilinear Forms and Proportions of Architecture._\n\nAs we are accustomed in all cases to refer direction to the horizontal\nand vertical lines, and as the meeting of these lines makes the right\nangle, it naturally constitutes the fundamental angle, by the harmonic\ndivision of which a system of proportion may be established, and the\ntheory of symmetrical beauty, like that of music, rendered susceptible of\nexact reasoning.\n\nLet therefore the right angle be the fundamental angle, and let it be\ndivided upon the quadrant of a circle into the harmonic parts already\nexplained, thus:\u2014\n\n Super- Sub- Sub- Sub- Semi-sub-\n Right tonic Mediant dominant Dominant mediant tonic tonic Tonic\n Angle. Angles. Angles. Angles. Angles. Angles. Angles. Angles. Angles.\n I. (1) (\u2078\u2044\u2089) (\u2074\u2044\u2085) (\u00b3\u2044\u2084) (\u00b2\u2044\u2083) (\u00b3\u2044\u2085) (\u2074\u2044\u2087) (\u2078\u2044\u2081\u2085) (\u00b9\u2044\u2082)\n II. (\u00b9\u2044\u2082) (\u2074\u2044\u2089) (\u00b2\u2044\u2085) (\u00b3\u2044\u2088) (\u00b9\u2044\u2083) (\u00b3\u2044\u2081\u2080) (\u00b2\u2044\u2087) (\u2074\u2044\u2081\u2085) (\u00b9\u2044\u2084)\n III. (\u00b9\u2044\u2084) (\u00b2\u2044\u2089) (\u00b9\u2044\u2085) (\u00b3\u2044\u2081\u2086) (\u00b9\u2044\u2086) (\u00b3\u2044\u2082\u2080) (\u00b9\u2044\u2087) (\u00b2\u2044\u2081\u2085) (\u00b9\u2044\u2088)\n IV. (\u00b9\u2044\u2088) (\u00b9\u2044\u2089) (\u00b9\u2044\u2081\u2080) (\u00b3\u2044\u2083\u2082) (\u00b9\u2044\u2081\u2082) (\u00b3\u2044\u2084\u2080) (\u00b9\u2044\u2081\u2084) (\u00b9\u2044\u2081\u2085) (\u00b9\u2044\u2081\u2086)\n\nIn order that the analogy may be kept in view, I have given to the parts\nof each of these four scales the appropriate nomenclature of the notes\nwhich form the diatonic scale in music.\n\nWhen a right angled triangle is constructed so that its two smallest\nangles are equal, I term it simply the triangle of (\u00b9\u2044\u2082), because the\nsmaller angles are each one-half of the right angle. But when the two\nangles are unequal, the triangle may be named after the smallest. For\ninstance, when the smaller angle, which we shall here suppose to be\none-third of the right angle, is made with the vertical line, the\ntriangle may be called the vertical scalene triangle of (\u00b9\u2044\u2083); and when\nmade with the horizontal line, the horizontal scalene triangle of (\u00b9\u2044\u2083).\nAs every rectangle is made up of two of these right angled triangles,\nthe same terminology may also be applied to these figures. Thus, the\nequilateral rectangle or perfect square is simply the rectangle of (\u00b9\u2044\u2082),\nbeing composed of two similar right angled triangles of (\u00b9\u2044\u2082); and when\ntwo vertical scalene triangles of (\u00b9\u2044\u2083), and of similar dimensions, are\nunited by their hypothenuses, they form the vertical rectangle of (\u00b9\u2044\u2083),\nand in like manner the horizontal triangles of (\u00b9\u2044\u2083) similarly united\nwould form the horizontal rectangle of (\u00b9\u2044\u2083). As the isosceles triangle\nis in like manner composed of two right angled scalene triangles joined\nby one of their sides, the same terminology may be applied to every\nvariety of that figure. All the angles of the first of the above scales,\nexcept that of (\u00b9\u2044\u2082), give rectangles whose longest sides are in the\nhorizontal line, while the other three give rectangles whose longest\nsides are in the vertical line. I have illustrated in Plate I. the manner\nin which this harmonic law acts upon these elementary rectilinear figures\nby constructing a series agreeably to the angles of scales II., III.,\nIV. Throughout this series _a b c_ is the primary scalene triangle, of\nwhich the rectangle _a b c e_ is composed; _d c e_ the vertical isosceles\ntriangle; and when the plate is turned, _d e a_ the horizontal isosceles\ntriangle, both of which are composed of the same primary scalene triangle.\n\n[Sidenote: Plate I.]\n\nThus the most simple elements of symmetry in rectilinear forms are the\nthree following figures:\u2014\n\n The equilateral rectangle or perfect square,\n The oblong rectangle, and\n The isosceles triangle.\n\nIt has been shewn that in harmonic combinations of musical sounds, the\n\u00e6sthetic feeling produced by their agreement depends upon the relation\nthey bear to each other with reference to the number of pulsations\nproduced in a given time by the fundamental note of the scale to which\nthey belong; and that the more simply they relate to each other in this\nway the more perfect the harmony, as in the common chord of the first\nscale, the relations of whose parts are in the simple ratios of 2:1, 3:2,\nand 5:4. It is equally consistent with this law, that when applied to\nform in the composition of an assortment of figures of any kind, their\nrespective proportions should bear a very simple ratio to each other in\norder that a definite and pleasing harmony may be produced amongst the\nvarious parts. Now, this is as effectually done by forming them upon the\nharmonic divisions of the right angle as musical harmony is produced by\nsounds resulting from harmonic divisions of a vibratory body.\n\nHaving in previous works[7] given the requisite illustrations of this\nfact in full detail, I shall here confine myself to the most simple kind,\ntaking for my first example one of the finest specimens of classical\narchitecture in the world\u2014the front portico of the Parthenon of Athens.\n\nThe angles which govern the proportions of this beautiful elevation are\nthe following harmonic parts of the right angle\u2014\n\n Tonic Dominant Mediant Subtonic Supertonic\n Angles. Angles. Angles. Angle. Angles.\n (\u00b9\u2044\u2082) (\u00b9\u2044\u2083) (\u00b9\u2044\u2085) (\u00b9\u2044\u2087) (\u00b9\u2044\u2089)\n (\u00b9\u2044\u2084) (\u00b9\u2044\u2086) (\u00b9\u2044\u2081\u2080) (\u00b9\u2044\u2081\u2088)\n (\u00b9\u2044\u2088)\n (\u00b9\u2044\u2081\u2086)\n\n[Sidenote: Plate II.]\n\nIn Plate II. I give a diagram of its rectilinear orthography, which is\nsimply constructed by lines drawn, either horizontally, vertically, or\nobliquely, which latter make with either of the former lines one or other\nof the harmonic angles in the above series. For example, the horizontal\nline AB represents the length of the base or surface of the upper step of\nthe substructure of the building. The line AE, which makes an angle of\n(\u00b9\u2044\u2085) with the horizontal, determines the height of the colonnade. The\nline AD, which makes an angle of (\u00b9\u2044\u2084) with the horizontal, determines\nthe height of the portico, exclusive of the pediment. The line AC, which\nmakes an angle of (\u00b9\u2044\u2083) with the horizontal, determines the height of the\nportico, including the pediment. The line GD, which makes an angle of\n(\u00b9\u2044\u2087) with the horizontal, determines the form of the pediment. The lines\nEZ and LY, which respectively make angles of (\u00b9\u2044\u2081\u2086) and (\u00b9\u2044\u2081\u2088) with the\nhorizontal, determine the breadth of the architrave, frieze, and cornice.\nThe line _v n u_, which makes an angle of (\u00b9\u2044\u2083) with the vertical,\ndetermines the breadth of the triglyphs. The line _t d_, which makes an\nangle of (\u00b9\u2044\u2082), determines the breadth of the metops. The lines _c b\nr f_, and _a i_, which make each an angle of (\u00b9\u2044\u2086) with the vertical,\ndetermine the width of the five centre intercolumniations. The line _z\nk_, which makes an angle of (\u00b9\u2044\u2088) with the vertical, determines the width\nof the two remaining intercolumniations. The lines _c s_, _q x_, and _y\nh_, each of which makes an angle of (\u00b9\u2044\u2081\u2080) with the vertical, determine\nthe diameters of the three columns on each side of the centre. The line\n_w l_, which makes an angle of (\u00b9\u2044\u2089) with the vertical, determines the\ndiameter of the two remaining or corner columns.\n\nIn all this, the length and breadth of the parts are determined by\nhorizontal and vertical lines, which are necessarily at right angles with\neach other, and the position of which are determined by one or other of\nthe lines making the harmonic angles above enumerated.\n\nNow, the lengths and breadths thus so simply determined by these few\nangles, have been proved to be correct by their agreement with the most\ncareful measurements which could possibly be made of this exquisite\nspecimen of formative art. These measurements were obtained by the\n\u201cSociety of Dilettanti,\u201d London, who, expressly for that purpose, sent Mr\nF. C. Penrose, a highly educated architect, to Athens, where he remained\nfor about five months, engaged in the execution of this interesting\ncommission, the results of which are now published in a magnificent\nvolume by the Society.[8] The agreement was so striking, that Mr Penrose\nhas been publicly thanked by an eminent man of science for bearing\ntestimony to the truth of my theory, who in doing so observes, \u201cThe\ndimensions which he (Mr Penrose) gives are to me the surest verification\nof the theory I could have desired. The minute discrepancies form that\nvery element of practical incertitude, both as to execution and direct\nmeasurement, which always prevails in materialising a mathematical\ncalculation made under such conditions.\u201d[9]\n\nAlthough the measurements taken by Mr Penrose are undeniably correct, as\nall who examine the great work just referred to must acknowledge, and\nalthough they have afforded me the best possible means of testing the\naccuracy of my theory as applied to the Parthenon, yet the ideas of Mr\nPenrose as to the principles they evolve are founded upon the fallacious\ndoctrine which has so long prevailed, and still prevails, in the\n\u00e6sthetics of architecture, viz., that harmony may be imparted by ratios\nbetween the lengths and breadths of parts.\n\nI have taken for my second example an elevation which, although of\nsmaller dimensions, is no less celebrated for the beauty of its\nproportions than the Parthenon itself, viz., the front portico of the\ntemple of Theseus, which has also been measured by Mr Penrose.\n\nThe angles which govern the proportions of this elevation are the\nfollowing harmonic parts of the right angle:\u2014\n\n Tonic Dominant Mediant\n Angles. Angles. Angles.\n (\u00b9\u2044\u2082) (\u00b9\u2044\u2083) (\u00b2\u2044\u2085)\n (\u00b9\u2044\u2084) (\u00b9\u2044\u2086) (\u00b9\u2044\u2085)\n (\u00b9\u2044\u2081\u2082)\n\n[Sidenote: Plate III.]\n\nA diagram of the rectilinear orthography of this portico is given in\nPlate III. Its construction is similar to that of the Parthenon in\nrespect to the harmonic parts of the right angle, and I have therefore\nonly to observe, that the line A E makes an angle of (\u00b9\u2044\u2084); the line A D\nan angle of (\u00b9\u2044\u2083); the line A C an angle of (\u00b2\u2044\u2085); the line G D an angle\nof (\u00b9\u2044\u2086); and the lines E Z and L Y angles of (\u00b9\u2044\u2081\u2082) with the horizontal.\n\nAs to the colonnade or vertical part, the line _a b_, which determines\nthe three middle intercolumniations, makes an angle of (\u00b9\u2044\u2085); the line\n_c d_, which determines the two outer intercolumniations, makes an angle\nof (\u00b9\u2044\u2086); and the line _e f_, which determines the lesser diameter of\nthe columns, makes an angle of (\u00b9\u2044\u2081\u2082) with the vertical. I need give no\nfurther details here, as my intention is to shew the simplicity of the\nmethod by which this theory may be reduced to practice, and because I\nhave given in my other works ample details, in full illustration of the\northography of these two structures, especially the first.[10]\n\nThe foregoing examples being both horizontal rectangular compositions,\nthe proportions of their principal parts have necessarily been determined\nby lines drawn from the extremities of the base, making angles with\nthe horizontal line, and forming thereby the diagonals of the various\nrectangles into which, in their leading features, they are necessarily\nresolved. But the example I am now about to give is of another character,\nbeing a vertical pyramidal composition, and consequently the proportions\nof its principal parts are determined by the angles which the oblique\nlines make with the vertical line representing the height of the\nelevation, and forming a series of isosceles triangles; for the isosceles\ntriangle is the type of all pyramidal composition.\n\nThis third example is the east end of Lincoln Cathedral, a Gothic\nstructure, which is acknowledged to be one of the finest specimens of\nthat style of architecture existing in this country.\n\nThe angles which govern the proportions of this elevation are the\nfollowing harmonic parts of the right angle:\u2014\n\n Tonic. Dominant. Mediant. Subtonic. Supertonic.\n (\u00b9\u2044\u2082) (\u00b9\u2044\u2083) (\u00b9\u2044\u2085) (\u00b9\u2044\u2087) (\u00b2\u2044\u2089)\n (\u00b9\u2044\u2084) (\u00b9\u2044\u2086) (\u00b9\u2044\u2081\u2080) (\u00b9\u2044\u2089)\n (\u00b9\u2044\u2081\u2082)\n\n[Sidenote: Plate IV.]\n\nIn Plate IV. I give a diagram of the vertical, horizontal, and oblique\nlines, which compose the orthography of this beautiful elevation.\n\nThe line A B represents the full height of this structure. The line A C,\nwhich makes an angle of (\u00b2\u2044\u2089) with the vertical, determines the width of\nthe design, the tops of the aisle windows, and the bases of the pediments\non the inner buttresses; A G, (\u00b9\u2044\u2085) with the vertical, that of the outer\nbuttress; A F, (\u00b9\u2044\u2089) with the vertical, that of the space between the\nouter and inner buttresses and the width of the great centre window; and\nA E, (\u00b9\u2044\u2081\u2082) with vertical, that of both the inner buttresses and the\nspace between these. A H, which makes (\u00b9\u2044\u2084) with the vertical, determines\nthe form of the pediment of the centre, and the full height of the base\nand surbase. A I, which makes (\u00b9\u2044\u2083) with the vertical, determines the\nform of the pediment of the smaller gables, the base of the pediment on\nthe outer buttress, the base of the ornamental recess between the outer\nand inner buttresses, the spring of the arch of the centre window, the\ntops of the pediments on the inner buttresses, and the spring of the\narch of the upper window. A K, which makes (\u00b9\u2044\u2082), determines the height\nof the outer buttress; and A Z, which makes (\u00b9\u2044\u2086) with the horizontal,\ndetermines that of the inner buttresses. For the reasons already given,\nI need not here go into further detail.[11] It is, however, worthy of\nremark in this place, that notwithstanding the great difference which\nexists between the style of composition in this Gothic design, and in\nthat of the east end of the Parthenon, the harmonic elements upon which\nthe orthographic beauty of the one depends, are almost identical with\nthose of the other.\n\n\n_On the Curvilinear Forms and Proportions of Architecture._\n\nEach regular rectilinear figure has a curvilinear figure that exclusively\nbelongs to it, and to which may be applied a corresponding terminology.\nFor instance, the circle belongs to the equilateral rectangle; that\nis, the rectangle of (\u00b9\u2044\u2082), an ellipse to every other rectangle, and\na composite ellipse to every isosceles triangle. Thus the most simple\nelements of beauty in the curvilinear forms of architectural design are\nthe following three figures:\u2014\n\n The circle,\n The ellipse, and\n The composite ellipse.\n\nI find it necessary in this place to go into some details regarding\nthe specific character of the two latter figures, because the proper\nmode of describing these beautiful curves, and their high value in the\npractice of the architectural draughtsman and ornamental designer, seem\nas yet unknown. In proof of this assertion, I must again refer to Mr\nPenrose\u2019s great work published by the \u201cSociety of Dilettanti.\u201d At page\n52 of that work it is observed, that \u201cby whatever means an ellipse is to\nbe constructed mechanically, it is a work of time (if not of absolute\ndifficulty) so to arrange the foci, &c., as to produce an ellipse of any\nexact length and breadth which may be desired.\u201d Now, this is far from\nbeing the case, for the method of arranging the foci of an ellipse of any\ngiven length and breadth is extremely simple, being as follows:\u2014\n\nLet A B C (figure 1) be the length, and D B E the breadth of the desired\nellipse.\n\n[Illustration: Fig. 1.]\n\nTake A B upon the compasses, and place the point of one leg upon E and\nthe point of the other upon the line A B, it will meet it at F, which is\none focus: keeping the point of the one leg upon E, remove the point of\nthe other to the line B C, and it will meet it at G, which is the other\nfocus. But, when the proportions of an ellipse are to be imparted by\nmeans of one of the harmonic angles, suppose the angle of (\u00b9\u2044\u2083), then the\nfollowing is the process:\u2014\n\nLet A B C (figure 2) represent the length of the intended ellipse.\nThrough B draw B _e_ indefinitely, at right angles with A B C; through C\ndraw the line C _f_ indefinitely, making, with B C, an angle of (\u00b9\u2044\u2083).\n\nTake B C upon the compasses, and place the point of one leg upon D where\nC f intersects B _e_, and the point of the other upon the line A B, it\nwill meet it at F, which is one focus. Keeping the point of one leg still\nupon D, remove the point of the other to the line B C, and it will meet\nit at G, which is the other focus.\n\n[Illustration: Fig. 2.]\n\nThe foci being in either case thus simply ascertained, the method of\ndescribing the curve on a small scale is equally simple.\n\n[Sidenote: Plate V.]\n\nA pin is fixed into each of the two foci, and another into the point D.\nAround these three pins a waxed thread, flexible but not elastic, is\ntied, care being taken that the knot be of a kind that will not slip.\nThe pin at D is now removed, and a hard black lead pencil introduced\nwithin the thread band. The pencil is then moved around the pins fixed\nin the foci, keeping the thread band at a full and equal tension; thus\nsimply the ellipse is described. When, however, the governing angle is\nacute, say less than (\u00b9\u2044\u2086), it is requisite to adopt a more accurate\nmethod of description,[12] as the architectural examples which follow\nwill shew. But architectural draughtsmen and ornamental designers would\ndo well to supply themselves, for ordinary practice, with half a dozen\nseries of ellipses, varying in the proportions of their axes from (\u2074\u2044\u2089)\nto (\u00b9\u2044\u2086) of the scale, and the length of their major axes from 1 to 6\ninches. These should be described by the above simple process, upon\nvery strong drawing paper, and carefully cut out, the edge of the paper\nbeing kept smooth, and each ellipse having its greater and lesser axes,\nits foci, and the hypothenuse of its scalene triangle drawn upon it. To\nexemplify this, I give Plate V., which exhibits the ellipses of (\u00b9\u2044\u2083),\n(\u00b9\u2044\u2084), (\u00b9\u2044\u2085), and (\u00b9\u2044\u2086), inscribed in their rectangles, on which _a b_\nand _c d_ are respectively the greater and lesser axes, _o o_ the foci,\nand _d b_ the angle of each. Such a series of these beautiful figures\nwould be found particularly useful in drawing the mouldings of Grecian\narchitecture; for, to describe the curvilinear contour of such mouldings\nfrom single points, as has been done with those which embellish even our\nmost pretending attempts at the restoration of that classical style of\narchitecture, is to give the resemblance of an external form without the\nharmony which constitutes its real beauty.\n\nMr Penrose, owing to the supposed difficulty regarding the description of\nellipses just alluded to, endeavours to shew that the curves of all the\nmouldings throughout the Parthenon were either parabolic or hyperbolic;\nbut I believe such curves can have no connexion with the elementary forms\nof architecture, for they are curves which represent motion, and do not,\nby continued production, form closed figures.\n\nBut I have shewn, in a former work,[13] that the contours of these\nmouldings are composed of curves of the composite ellipse,\u2014a figure\nwhich I so name because it is composed simply of arcs of various\nellipses harmonically flowing into each other. The composite ellipse,\nwhen drawn systematically upon the isosceles triangle, resembles closely\nparabolic and hyperbolic curves\u2014only differing from these inasmuch as it\npossesses the essential quality of circumscribing harmonically one of the\nelementary rectilinear figures employed in architecture, while those of\nthe parabola and hyperbola, as I have just observed, are merely curves\nof motion, and, consequently, never can harmonically circumscribe or be\nresolved into any regular figure.\n\nThe composite ellipse may be thus described.\n\n[Sidenote: Plate VI.]\n\nLet A B C (Plate VI.) be a vertical isosceles triangle of (\u00b9\u2044\u2086), bisect A\nB in D, and through D draw indefinitely D _f_ perpendicular to A B, and\nthrough B draw indefinitely B _g_, making the angle D B _g_ (\u00b9\u2044\u2088), D _f_\nand B _g_ intersecting each other in M. Take B D and D M as semi-axes of\nan ellipse, the foci of which will be at _p_ and _q_, in each of these,\nand in each of the foci _h t_ and _k r_ in the lines A C and B C, fix\na pin, and one also in the point M, tie a thread around these pins,\nwithdraw the pin from M, and trace the composite ellipse in the manner\nalready described with respect to the simple ellipse.\n\nIn some of my earlier works I described this figure by taking the angles\nof the isosceles triangle as foci; but the above method is much more\ncorrect. As the elementary angle of the triangle is (\u00b9\u2044\u2086), and that of\nthe elliptic curve described around it (\u00b9\u2044\u2088), I call it the composite\nellipse of (\u00b9\u2044\u2086) and (\u00b9\u2044\u2088), their harmonic ratio being 4:3; and so on of\nall others, according to the difference that may thus exist between the\nelementary angles.\n\nThe visible curves which soften and beautify the melody of the outline\nof the front of the Parthenon, as given in Mr Penrose\u2019s great work, I\nhave carefully analysed, and have found them in as perfect agreement\nwith this system, as its rectilinear harmony has been shewn to be. This\nI demonstrated in the work just referred to[14] by a series of twelve\nplates, shewing that the entasis of the columns (a subject upon which\nthere has been much speculation) is simply an arc of an ellipse of\n(\u00b9\u2044\u2084\u2088), whose greater axis makes with the vertical an angle of (\u00b9\u2044\u2086\u2084);\nor simply, the form of one of these columns is the frustrum of an\nelliptic-sided or prolate-spheroidal cone, whose section is a composite\nellipse of (\u00b9\u2044\u2084\u2088) and (\u00b9\u2044\u2086\u2084), the harmonic ratio of these two angles\nbeing 4:3, the same as that of the angles of the composite ellipse just\nexemplified.\n\n[Sidenote: Plate VII.]\n\n[Sidenote: Plate VIII.]\n\nIn Plate VII. is represented the section of such a cone, of which A B C\nis the isosceles triangle of (\u00b9\u2044\u2084\u2088), and B D and D M the semi-axes of an\nellipse of (\u00b9\u2044\u2086\u2084). M N and O P are the entases of the column, and _d e f_\nthe normal construction of the capital. All these are fully illustrated\nin the work above referred to,[15] in which I have also shewn that the\ncurve of the neck of the column is that of an ellipse of (\u00b9\u2044\u2086); the curve\nof the capital or echinus, that of an ellipse of (\u00b9\u2044\u2081\u2084); the curve of\nthe moulding under the cymatium of the pediment, that of an ellipse of\n(\u00b9\u2044\u2083); and the curve of the bed-moulding of the cornice of the pediment,\nthat of an ellipse of (\u00b9\u2044\u2083). The curve of the cavetto of the soffit of\nthe corona is composed of ellipses of (\u00b9\u2044\u2086) and (\u00b9\u2044\u2081\u2084); the curve of the\ncymatium which surmounts the corona, is that of an ellipse of (\u00b9\u2044\u2083); the\ncurve of the moulding of the capital of the ant\u00e6 of the posticum, that of\nan ellipse of (\u00b9\u2044\u2083); the curves of the lower moulding of the same capital\nare composed of those of an ellipse of (\u00b9\u2044\u2083) and of the circle (\u00b9\u2044\u2082); the\ncurve of the moulding which is placed between the two latter is that of\nan ellipse of (\u00b9\u2044\u2083); the curve of the upper moulding of the band under\nthe beams of the ceiling of the peristyle, that of an ellipse of (\u00b9\u2044\u2083);\nthe curve of the lower moulding of the same band, that of an ellipse of\n(\u00b9\u2044\u2084); and the curves of the moulding at the bottom of the small step\nor podium between the columns, are those of the circle (\u00b9\u2044\u2082) and of an\nellipse of (\u00b9\u2044\u2083). I have also shewn the curve of the fluting of the\ncolumns to be that of (\u00b9\u2044\u2081\u2084). The greater axis of each of these ellipses,\nwhen not in the vertical or horizontal lines, makes an harmonic angle\nwith one or other of them. In Plate VIII., sections of the two last-named\nmouldings are represented full size, which will give the reader an idea\nof the simple manner in which the ellipses are employed in the production\nof those harmonic curves.\n\nThus we find that the system here adopted for applying this law of\nnature to the production of beauty in the abstract forms employed\nin architectural composition, so far from involving us in anything\ncomplicated, is characterised by extreme simplicity.\n\nIn concluding this part of my treatise, I may here repeat what I have\nadvanced in a late work,[16] viz., my conviction of the probability that\na system of applying this law of nature in architectural construction\nwas the only great practical secret of the Freemasons, all their other\nsecrets being connected, not with their art, but with the social\nconstitution of their society. This valuable secret, however, seems\nto have been lost, as its practical application fell into disuse; but,\nas that ancient society consisted of speculative as well as practical\nmasons, the secrets connected with their social union have still been\npreserved, along with the excellent laws by which the brotherhood is\ngoverned. It can scarcely be doubted that there was some such practically\nuseful secret amongst the Freemasons or early Gothic architects; for\nwe find in all the venerable remains of their art which exist in this\ncountry, symmetrical elegance of form pervading the general design,\nharmonious proportion amongst all the parts, beautiful geometrical\narrangements throughout all the tracery, as well as in the elegantly\nsymmetrised foliated decorations which belong to that style of\narchitecture. But it is at the same time worthy of remark, that whenever\nthey diverged from architecture to sculpture and painting, and attempted\nto represent the human figure, or even any of the lower animals, their\nproductions are such as to convince us that in this country these arts\nwere in a very degraded state of barbarism\u2014the figures are often much\ndisproportioned in their parts and distorted in their attitudes, while\ntheir representations of animals and chimeras are whimsically absurd.\nIt would, therefore, appear that architecture, as a fine art, must have\nbeen preserved by some peculiar influence from partaking of the barbarism\nso apparent in the sister arts of that period. Although its practical\nsecrets have been long lost, the Freemasons of the present day trace the\noriginal possession of them to Moses, who, they say, \u201cmodelled masonry\ninto a perfect system, and circumscribed its mysteries by _land-marks_\nsignificant and unalterable.\u201d Now, as Moses received his education in\nEgypt, where Pythagoras is said to have acquired his first knowledge of\nthe harmonic law of numbers, it is highly probable that this perfect\nsystem of the great Jewish legislator was based upon the same law of\nnature which constituted the foundation of the Pythagorean philosophy,\nand ultimately led to that excellence in art which is still the\nadmiration of the world.\n\nPythagoras, it would appear, formed a system much more perfect and\ncomprehensive than that practised by the Freemasons in the middle ages of\nChristianity; for it was as applicable to sculpture, painting, and music,\nas it was to architecture. This perfection in architecture is strikingly\nexemplified in the Parthenon, as compared with the Gothic structures\nof the middle ages; for it will be found that the whole six elementary\nfigures I have enumerated as belonging to architecture, are required in\ncompleting the orthographic beauty of that noble structure. And amongst\nthese, none conduce more to that beauty than the simple and composite\nellipses. Now, in the architecture of the best periods of Gothic, or,\nindeed, in that of any after period (Roman architecture included), these\nbeautiful curves seem to have been ignored, and that of the circle alone\nemployed.\n\nBe those matters as they may, however, the great law of numerical\nharmonic ratio remains unalterable, and a proper application of it in\nthe science of art will never fail to be as productive of effect, as its\noperation in nature is universal, certain, and continual.\n\n\n\n\nTHE SCIENCE OF BEAUTY, AS DEVELOPED IN THE HUMAN HEAD AND COUNTENANCE.\n\n\nThe most remarkable characteristics of the human head and countenance\nare the globular form of the cranium, united as it is with the prolate\nspheroidal form produced by the parts which constitute the face, and the\napproximation of the profile to the vertical; for in none of the lower\nanimals does the skull present so near a resemblance to a combination\nof these geometric forms, nor the plane of the face to this direction.\nWe also find that although these peculiar characteristics are variously\nmodified among the numerous races of mankind, yet one law appears to\ngovern the beauty of the whole. The highest and most cultivated of these\nraces, however, present only an approximation to the perfect development\nof those distinguishing marks of humanity; and therefore the beauty of\nform and proportion which in nature characterises the human head and\ncountenance, exhibits only a partial development of the harmonic law of\nvisible beauty. On the other hand, we find that, in their sculpture, the\nancient Greeks surpassed ordinary nature, and produced in their beau\nideal a species of beauty free from the imperfections and peculiarities\nthat constitute the individuality by which the countenances of men are\ndistinguished from each other. It may be requisite here to remark, that\nthis species of beauty is independent of the more intellectual quality\nof expression. For as Sir Charles Bell has said, \u201cBeauty of countenance\nmay be defined in words, as well as demonstrated in art. A face may\nbe beautiful in sleep, and a statue without expression may be highly\nbeautiful. But it will be said there is expression in the sleeping figure\nor in the statue. Is it not rather that we see in these the capacity\nfor expression?\u2014that our minds are active in imagining what may be the\nmotions of these features when awake or animated? Thus, we speak of an\nexpressive face before we have seen a movement grave or cheerful, or any\nindication in the features of what prevails in the heart.\u201d\n\nThis capacity for expression certainly enhances our admiration of the\nhuman countenance; but it is more a concomitant of the primary cause of\nits beauty than the cause itself. This cause rests on that simple and\nsecure basis\u2014the harmonic law of nature; for the nearer the countenance\napproximates to an harmonious combination of the most perfect figures in\ngeometry, or rather the more its general form and the relation of its\nindividual parts are arranged in obedience to that law, the higher its\ndegree of beauty, and the greater its capacity for the expression of the\npassions.\n\nVarious attempts have been made to define geometrically the difference\nbetween the ordinary and the ideal beauty of the human head and\ncountenance, the most prominent of which is that of Camper. He traced,\nupon a profile of the skull, a line in a horizontal direction, passing\nthrough the foramen of the ear and the exterior margin of the sockets of\nthe front teeth of the upper jaw, upon which he raised an oblique line,\ntangential to the margin of these sockets, and to the most prominent part\nof the forehead. Agreeably to the obliquity of this line, he determined\nthe relative proportion of the areas occupied by the brain and by the\nface, and hence inferred the degree of intellect. When he applied this\nmeasurement to the heads of the antique statues, he found the angle much\ngreater than in ordinary nature; but that this simple fact afforded no\nrule for the reproduction of the ideal beauty of ancient Greek art, is\nvery evident from the heads and countenances by which his treatise is\nillustrated. Sir Charles Bell justly remarks, that although, by Camper\u2019s\nmethod, the forehead may be thrown forward, yet, while the features\nof common nature are preserved, we refuse to acknowledge a similarity\nto the beautiful forms of the antique marbles. \u201cIt is true,\u201d he says,\n\u201cthat, by advancing the forehead, it is raised, the face is shortened,\nand the eye brought to the centre of the head. But with all this, there\nis much wanting\u2014that which measurement, or a mere line, will not shew\nus.\u201d\u2014\u201cThe truth is, that we are more moved by the features than by the\nform of the whole head. Unless there be a conformity in every feature to\nthe general shape of the head, throwing the forehead forward on the face\nproduces deformity; and the question returns with full force\u2014How is it\nthat we are led to concede that the antique head of the Apollo, or of the\nJupiter, is beautiful when the facial line makes a hundred degrees with\nthe horizontal line? In other words\u2014How do we admit that to be beautiful\nwhich is not natural? Simply for the same reason that, if we discover a\nbroken portion of an antique, a nose, or a chin of marble, we can say,\nwithout deliberation\u2014This must have belonged to a work of antiquity;\nwhich proves that the character is distinguishable in every part\u2014in each\nfeature, as well as in the whole head.\u201d\n\nDr Oken says upon this subject:[17]\u2014\u201cThe face is beautiful whose nose is\nparallel to the spine. No human face has grown into this estate; but\nevery nose makes an acute angle with the spine. The facial angle is, as\nis well known, 80\u00b0. What, as yet, no man has remarked, and what is not to\nbe remarked, either, without our view of the cranial signification, the\nold masters have felt through inspiration. They have not only made the\nfacial angle a right angle, but have even stepped beyond this\u2014the Romans\ngoing up to 96\u00b0, the Greeks even to 100\u00b0. Whence comes it that this\nunnatural face of the Grecian works of art is still more beautiful than\nthat of the Roman, when the latter comes nearer to nature? The reason\nthereof resides in the fact of the Grecian artistic face representing\nnature\u2019s design more than that of the Roman; for, in the former, the nose\nis placed quite perpendicular, or parallel to the spinal cord, and thus\nreturns whither it has been derived.\u201d\n\nOther various and conflicting opinions upon this subject have been\ngiven to the world; but we find that the principle from which arose\nthe ideal beauty of the head and countenance, as represented in works\nof ancient Greek art, is still a matter of dispute. When, however, we\nexamine carefully a fine specimen, we find its beauty and grandeur to\ndepend more upon the degree of harmony amongst its parts, as to their\nrelative proportions and mode of arrangement, than upon their excellence\ntaken individually. It is, therefore, clear that those (and they are\nmany) who attribute the beauty of ancient Greek sculpture merely to a\nselection of parts from various models, must be in error. No assemblage\nof parts from ordinary nature could have produced its principal\ncharacteristic, the excess in the angle of the facial line, much less\ncould it have led to that exquisite harmony of parts by which it is so\neminently distinguished; neither can we reasonably agree with Dr Oken and\nothers, who assert that it was produced by an exclusive degree of the\ninspiration of genius amongst the Greek people during a certain period.\n\nThat the inspiration of genius, combined with a careful study of nature,\nwere essential elements in the production of the great works which have\nbeen handed down to us, no one will deny; but these elements have existed\nin all ages, whilst the ideal head belongs exclusively to the Greeks\nduring the period in which the schools of Pythagoras and Plato were open.\nIs it not, therefore, reasonable to suppose, that, besides genius and the\nstudy of nature, another element was employed in the production of this\nexcellence, and that this element arose from the precise mathematical\ndoctrines taught in the schools of these philosophers?\n\nAn application of the great harmonic law seems to prove that there is no\nobject in nature in which the science of beauty is more clearly developed\nthan in the human head and countenance, nor to the representations of\nwhich the same science is more easily applied; and it is to the mode\nin which this is done that the varieties of sex and character may be\nimparted to works of art. Having gone into full detail, and given ample\nillustrations in a former work,[18] it is unnecessary for me to enter\nupon that part of the subject in this _r\u00e9sum\u00e9_; but only to shew the\ntypical structure of beauty by which this noble work of creation is\ndistinguished.\n\nThe angles which govern the form and proportions of the human head and\ncountenance are, with the right angle, a series of seven, which, from the\nsimplicity of their ratios to each other, are calculated to produce the\nmost perfect concord. It consists of the right angle and its following\nparts\u2014\n\n Tonic. Dominant. Mediant. Subtonic.\n (\u00b9\u2044\u2082) (\u00b9\u2044\u2083) (\u00b9\u2044\u2085) (\u00b9\u2044\u2087)\n (\u00b9\u2044\u2084) (\u00b9\u2044\u2086)\n\nThese angles, and the figures which belong to them, are thus arranged:\u2014\n\n[Sidenote: Plate IX.]\n\nThe vertical line A B (Plate IX. fig. 2) represents the full length of\nthe head and face. Taking this line as the greater axis of an ellipse of\n(\u00b9\u2044\u2083), such an ellipse is described around it. Through A the lines A G,\nA K, A L, A M, and A N, are drawn on each side of the line A B, making,\nwith the vertical, respectively the angles of (\u00b9\u2044\u2083), (\u00b9\u2044\u2084), (\u00b9\u2044\u2085), (\u00b9\u2044\u2086),\nand (\u00b9\u2044\u2087). Through the points G, K, L, M, and N, where these straight\nlines meet the curved line of the ellipse, horizontal lines are drawn\nby which the following isosceles triangles are formed, A G G, A K K, A\nL L, A M M, and A N N. From the centre X of the equilateral triangle\nA G G the curvilinear figure of (\u00b9\u2044\u2082), viz., the circle, is described\ncircumscribing that triangle.\n\nThe curvilinear plane figures of (\u00b9\u2044\u2082) and (\u00b9\u2044\u2083), respectively, represent\nthe solid bodies of which they are sections, viz., a sphere and a prolate\nspheroid. These bodies, from the manner in which they are here placed,\nare partially amalgamated, as shewn in figures 1 and 3 of the same plate,\nthus representing the form of the human head and countenance, both in\ntheir external appearance and osseous structure, more correctly than they\ncould be represented by any other geometrical figures. Thus, the angles\nof (\u00b9\u2044\u2082) and (\u00b9\u2044\u2083) determine the typical form.\n\nFrom each of the points _u_ and _n_, where A M cuts G G on both sides of\nA B, a circle is described through the points _p_ and _q_, where A K cuts\nG G on both sides of A B, and with the same radius a circle is described\nfrom the point _a_, where K K cuts A B.\n\nThe circles _u_ and _n_ determine the position and size of the eyeballs,\nand the circle _a_ the width of the nose, as also the horizontal width of\nthe mouth.\n\nThe lines G G and K K also determine the length of the joinings of the\near to the head. The lines L L and M M determine the vertical width of\nthe mouth and lips when at perfect repose, and the line N N the superior\nedge of the chin. Thus simply are the features arranged and proportioned\non the facial surface.\n\nIt must, however, be borne in mind, that in treating simply of the\n\u00e6sthetic beauty of the human head and countenance, we have only to do\nwith the external appearance. In this research, therefore, the system\nof Dr Camper, Dr Owen, and others, whose investigations were more of\na physiological than an \u00e6sthetic character, can be of little service;\nbecause, according to that system, the facial angle is determined by\ndrawing a line tangential to the exterior margin of the sockets of\nthe front teeth of the upper jaw, and the most prominent part of the\nforehead. Now, as these sockets are, when the skull is naturally clothed,\nand the features in repose, entirely concealed by the upper lip, we must\ntake the prominent part of it, instead of the sockets under it, in order\nto determine properly this distinguishing mark of humanity. And I believe\nit will be found, that when the head is properly poised, the nearer the\nangle which this line makes with the horizontal approaches 90\u00b0, the more\nsymmetrically beautiful will be the general arrangement of the parts (see\nline _y z_, figure 3, Plate IX.).\n\n\n\n\nTHE SCIENCE OF BEAUTY, AS DEVELOPED IN THE FORM OF THE HUMAN FIGURE.\n\n\nThe manner in which this science is developed in the symmetrical\nproportions of the entire human figure, is as remarkable for its\nsimplicity as it has been shewn to be in those of the head and\ncountenance. Having gone into very full details, and given ample\nillustration in two former works[19] upon this subject, I may here\nconfine myself to the illustration of one description of figure, and to a\nreiteration of some facts stated in these works. These facts are, _1st_,\nThat on a given line the human figure is developed, as to its principal\npoints, entirely by lines drawn either from the extremities of this line,\nor from some obvious or determined localities. _2d_, That the angles\nwhich these lines make with the given line, are all simple sub-multiples\nof some given fundamental angle, or bear to it a proportion expressible\nunder the most simple relations, such as those which constitute the scale\nof music. _3d_, That the contour is resolved into a series of ellipses\nof the same simple angles. And, _4th_, That these ellipses, like the\nlines, are inclined to the first given line by angles which are simple\nsub-multiples of the given fundamental angle. From which four facts,\nand agreeably to the hypothesis I have adopted, it results as a natural\nconsequence that the only effort which the mind exercises through the\neye, in order to put itself in possession of the data for forming its\njudgment, is this, that it compares the angles about a point, and thereby\nappreciates the simplicity of their relations. In selecting the prominent\nfeatures of a figure, the eye is not seeking to compare their relative\ndistances\u2014it is occupied solely with their relative positions. In tracing\nthe contour, in like manner, it is not left in vague uncertainty as\nto what is the curve which is presented to it; unconsciously it feels\nthe complete ellipse developed before it; and if that ellipse and its\nposition are both formed by angles of the same simple relative value as\nthose which aided its determination of the positions of the prominent\nfeatures, it is satisfied, and finds the symmetry perfect.\n\nM\u00fcller, and other investigators into the arch\u00e6ology of art, refer to the\ngreat difficulty which exists in discovering the principles which the\nancients followed in regard to the proportions of the human figure, from\nthe different sexes and characters to which they require to be applied.\nBut in the system thus founded upon the harmonic law of nature, no such\ndifficulty is felt, for it is as applicable to the massive proportions\nwhich characterise the ancient representations of the Hercules, as to the\ndelicate and perfectly symmetrical beauty of the Venus. This change is\neffected simply by an increase in the fundamental angle. For instance,\nin the construction of a figure of the exact proportions of the Venus,\nthe right angle is adopted. But in the construction of a figure of the\nmassive proportions of the Hercules, it is requisite to adopt an angle\nwhich bears to the right angle the ratio of 6:5. The adoption of this\nangle I have shewn in another work[20] to produce in the Hercules those\nproportions which are so characteristic of physical power. The ellipses\nwhich govern the outline, being also formed upon the same larger class\nof angles, give the contour of the muscles a more massive character. In\ncomparing the male and female forms thus geometrically constructed, it\nwill be found that that of the female is more harmoniously symmetrical,\nbecause the right angle is the fundamental angle for the trunk and the\nlimbs as well as for the head and countenance; while in that of the male,\nthe right angle is the fundamental angle for the head only. It may also\nbe observed, that, from the greater proportional width of the pelvis\nof the female, the centres of that motion which the heads of the thigh\nbones perform in the cotyloid cavities, and the centres of that still\nmore extensive range of motion which the arm is capable of performing at\nthe shoulder joints, are nearly in the same line which determines the\ncentral motion of the vertebral column, while those of the male are not;\nconsequently all the motions of the female are more graceful than those\nof the male.\n\nThis difference between the fundamental angles, which impart to the\nhuman figure, on the one hand, the beauty of feminine proportion and\ncontour, and on the other, the grandeur of masculine strength, being in\nthe ratio of 5:6, allows ample latitude for those intermediate classes of\nproportions which the ancients imparted to their various other deities\nin which these two qualities were blended. I therefore confine myself to\nan illustration of the external contour of the form, and the relative\nproportions of all the parts of a female figure, such as those of the\nstatues of the Venus of Melos and Venus of Medici.\n\nThe angles which govern the form and proportions of such a figure are,\nwith the right angle, a series of twelve, as follows:\u2014\n\n Tonic. Dominant. Mediant. Subtonic. Supertonic.\n (\u00b9\u2044\u2082) (\u00b9\u2044\u2083) (\u00b9\u2044\u2085) (\u00b9\u2044\u2087) (\u00b9\u2044\u2089)\n (\u00b9\u2044\u2084) (\u00b9\u2044\u2086) (\u00b9\u2044\u2081\u2080) (\u00b9\u2044\u2081\u2084)\n (\u00b9\u2044\u2088) (\u00b9\u2044\u2081\u2082)\n\nThese angles are employed in the construction of a diagram, which\ndetermines the proportions of the parts throughout the whole figure.\nThus:\u2014\n\n[Sidenote: Plate X.]\n\nLet the line A B (fig. 1, plate X.) represent the height of the figure\nto be constructed. At the point A, make the angles of C A D (\u00b9\u2044\u2083), F A G\n(\u00b9\u2044\u2084), H A I (\u00b9\u2044\u2085), K A L (\u00b9\u2044\u2086), and M A N (\u00b9\u2044\u2087). At the point B, make\nthe angles K B L (\u00b9\u2044\u2088), U B A (\u00b9\u2044\u2081\u2082), and O B A (\u00b9\u2044\u2081\u2084).\n\nThrough the point K, in which the lines A K and B K intersect one\nanother, draw P K O parallel to A B, and through C F H and M, where\nthis line meets A C, A F, A H, and A M, draw C D, F G, H I, and M N,\nperpendicular to A B; draw also K L perpendicular to A B; join B F and\nB H, and through C draw C E, making with A B the angle (\u00b9\u2044\u2082), which\ncompletes the arrangement of the eleven angles upon A B; for F B G is\nvery nearly (\u00b9\u2044\u2081\u2080), and H B I very nearly (\u00b9\u2044\u2089).\n\nAt the point _f_, where A C intersects O B, draw _f a_ perpendicular to\nA B; and through the point _i_, where B O intersects M N, draw S _i_ T\nparallel to A C.\n\nThrough _m_, where S _i_ T intersects F B, draw _m n_; through _\u03b2_, where\nS _i_ T intersects K B, draw _\u03b2 w_; through T draw T _g_, making an angle\nof (\u00b9\u2044\u2083) with O P. Join N P, M B, and _g_ P, and where N P intersects K\nB, draw Q R perpendicular to A B.\n\nOn A E as a diameter, describe a circle cutting A C in _r_, and draw _r\no_ perpendicular to A B.\n\nWith A _o_ and _o r_ as semi-axes, describe the ellipse A _r e_, cutting\nA H in _t_; and draw _t u_ perpendicular to A B. With A _u_ and _t u_, as\nsemi-axes describe the ellipse A _t d_. On _a_ L, as major axis, describe\nthe ellipse of (\u00b9\u2044\u2083).\n\nFor the side aspect or profile of the figure the diagram is thus\nconstructed\u2014\n\nOn one side of a line A B (fig. 2, Plate X.) construct the rectilinear\nportion of a diagram the same as fig. 1. Through _i_ draw W Y parallel to\nA B, and draw A _z_ perpendicular to A B. Make W _a_ equal to A _a_ (fig.\n1), and on _a l_, as major axis, describe the ellipse of (\u00b9\u2044\u2084). Through\n_a_ draw _a p_ parallel to A F, and through _p_ draw _p t_ perpendicular\nto W Y. Through _a_ draw _f a u_ perpendicular to W Y.\n\nUpon a diameter equal to A E describe a circle whose circumference shall\ntouch A B and A _z_. With semi-axes equal to A _o_ and _o r_ (fig.\n1), describe an ellipse with its major axis parallel to A B, and its\ncircumference touching O P and _z_ A.\n\n[Sidenote: Plate XI.]\n\nThus simply are the diagrams of the general proportions of the human\nfigure, as viewed in front and in profile, constructed; and Plate XI.\ngives the contour in both points of view, as composed entirely of the\ncurvilinear figures of (\u00b9\u2044\u2082), (\u00b9\u2044\u2083), (\u00b9\u2044\u2084), (\u00b9\u2044\u2085), and (\u00b9\u2044\u2086).\n\nFurther detail here would be out of place, and I shall therefore refer\nthose who require it to the Appendix, or the more elaborate works to\nwhich I have already referred.\n\nThe beauty derived from proportion, imparted by the system here pointed\nout, and from a contour of curves derived from the same harmonic angles,\nis not confined to the human figure, but is found in various minor\ndegrees of perfection in all the organic forms of nature, whether animate\nor inanimate, of which I have in other works given many examples.[21]\n\n\n\n\nTHE SCIENCE OF BEAUTY, AS DEVELOPED IN COLOURS.\n\n\nThere is not amongst the various phenomena of nature one that more\nreadily excites our admiration, or makes on the mind a more vivid\nimpression of the order, variety, and harmonious beauty of the creation,\nthan that of colour. On the general landscape this phenomenon is\ndisplayed in the production of that species of harmony in which colours\nare so variously blended, and in which they are by light, shade, and\ndistance modified in such an infinity of gradation and hue. Although\ngenius is continually struggling, with but partial success, to imitate\nthose effects, yet, through the Divine beneficence, all whose organs of\nsight are in an ordinary degree of perfection can appreciate and enjoy\nthem. In winter this pleasure is often to a certain extent withdrawn,\nwhen the colourless snow alone clothes the surface of the earth. But\nthis is only a pause in the general harmony, which, as the spring\nreturns, addresses itself the more pleasingly to our perception in its\nvernal melody, which, gradually resolving itself into the full rich hues\nof luxuriant beauty exhibited in the foliage and flowers of summer,\nsubsequently rises into the more vivid and powerful harmonies of autumn\u2019s\ncolouring. Thus the eye is prepared again to enjoy that rest which such\nexciting causes may be said to have rendered necessary.\n\nWhen we pass from the general colouring of nature to that of particular\nobjects, we are again wrapt in wonder and admiration by the beauty and\nharmony which so constantly, and in such infinite variety, present\nthemselves to our view, and which are so often found combined in the\nmost minute objects. And the systematic order and uniformity perceptible\namidst this endless variety in the colouring of animate and inanimate\nnature is thus another characteristic of beauty equally prevalent\nthroughout creation.\n\nBy this uniformity in colour, various species of animals are often\ndistinguished; and in each individual of most of these species, how much\nis this beauty enhanced when the uniformity prevails in the resemblance\nof their lateral halves! The human countenance exemplifies this in a\nstriking manner; the slightest variety of colour between one and another\nof the double parts is at once destructive of its symmetrical beauty.\nMany of the lower animals, whether inhabitants of the earth, the air, or\nthe water, owe much of their beauty to this kind of uniformity in the\ncolour of the furs, feathers, scales, or shells, with which they are\nclothed.\n\nIn the vegetable kingdom, we find a great degree of uniformity of colour\nin the leaves, flowers, and fruit of the same plant, combined with all\nthe harmonious beauty of variety which a little careful examination\ndevelops.\n\nIn the colours of minerals, too, the same may be observed. In short, in\nthe beauty of colouring, as in every other species of beauty, uniformity\nand variety are found to combine.\n\nAn appreciation of colour depends, in the first place, as much upon the\nphysical powers of the eye in conveying a proper impression to the mind,\nas that of music on those of the ear. But an ear for music, or an eye for\ncolour, are, in so far as beauty is concerned, erroneous expressions;\nbecause they are merely applicable to the impression made upon the\nsenses, and do not refer to the \u00e6sthetical principles of harmony, by\nwhich beauty can alone be understood.\n\nA good eye, combined with experience, may enable us to form a correct\nidea as to the purity of an individual colour, or of the relative\ndifference existing between two separate hues; but this sort of\ndiscrimination does not constitute that kind of appreciation of the\nharmony of colour by which we admire and enjoy its development in nature\nand art. The power of perceiving and appreciating beauty of any kind,\nis a principle inherent in the human mind, which may be improved by\ncultivation in the degree of the perfection of the art senses. Great\npains have been bestowed on the education of the ear, in assisting it to\nappreciate the melody and harmony of sound; but still much remains to be\ndone in regard to the cultivation of the eye, in appreciating colour as\nwell as form.\n\nIt is true, that there are individuals whose powers of vision are\nperfect, in so far as regards the appreciation of light, shade, and\nconfiguration, but who are totally incapable of perceiving effects\nproduced by the intermediate phenomenon of colour, every object appearing\nto them either white, black, or neutral gray; others, who are equally\nblind as to the effect of one of the three primary colours, but see the\nother two perfectly, either singly or combined; while there are many\nwho, having the full physical power of perceiving all the varieties of\nthe phenomenon, and who are even capable of making nice distinctions\namongst a variety of various colours, are yet incapable of appreciating\nthe \u00e6sthetic quality of harmony which exists in their proper combination.\nIt is the same with respect to the effects of sounds upon the ear\u2014some\nhave organs so constituted, that notes above or below a certain pitch\nare to them inaudible; while others, with physical powers otherwise\nperfect, are incapable of appreciating either melody or harmony in\nmusical composition. But perceptions so imperfectly constituted are,\nby the goodness of the Creator, of very rare occurrence; therefore all\nattempts at improvement in the science of \u00e6sthetics must be suited to the\ncapacities of the generality of mankind, amongst whom the perception of\ncolour exists in a variety as great as that by which their countenances\nare distinguished. Artists now and then appear who have this intuitive\nperception in such perfection, that they are capable of transferring to\ntheir works the most beautiful harmonies and most delicate gradations of\ncolours, in a manner that no acquired knowledge could have enabled them\nto impart. To those who possess such a gift, as well as to those to whom\nthe ordinary powers of perception are denied, it would be equally useless\nto offer an explanation of the various modes in which the harmony of\ncolour develops itself, or to attempt a definition of the many various\ncolours, hues, tints, and shades, arising out of the simple elements of\nthis phenomenon. But to those whose powers lie between these extremes,\nbeing neither above nor below cultivation, such an explanation and\ndefinition must form a step towards the improvement of that inherent\nprinciple which constitutes the basis of \u00e6sthetical science.\n\nAlthough the variety and harmony of colour which nature is continually\npresenting to our view, are apparent to all whose visual organs are in a\nnatural state, and thus to the generality of mankind; yet a knowledge of\nthe simplicity by which this variety and beauty are produced, is, after\nages of philosophic research and experimental inquiry, only beginning to\nbe properly understood.\n\nLight may be considered as an active, and darkness a passive principle in\nthe economy of Nature, and colour an intermediate phenomenon arising from\ntheir joint influence; and it is in the ratios in which these primary\nprinciples act upon each other, by which I here intend to explain the\nscience of beauty as evolved in colour. It has been usual to consider\ncolour as an inherent quality in light, and to suppose that coloured\nbodies absorb certain classes of its rays, and reflect or transmit the\nremainder; but it appears to me that colour is more probably the result\nof certain modes in which the opposite principles of motion and rest,\nor force and resistance, operate in the production, refraction, and\nreflection of light, and that each colour is mutually related, although\nin different degrees, to these active and passive principles.\n\nWhite and black are the representatives of light and darkness, or\nactivity and rest, and are therefore calculated as pigments to reduce\ncolours and hues to tints and shades.\n\nHaving, however, fully illustrated the nature of tints and shades in\na former work,[22] I shall here confine myself to colours in their\nfull intensity\u2014shewing the various modifications which their union\nwith each other produce, along with the harmonic relations which these\nmodifications bear to the primaries, and to each other in respect to\nwarmth and coolness of tone, as well as to light and shade.\n\nThe primary colours are red, yellow, and blue. Of these, yellow is\nmost allied to light, and blue to shade, while red is neutral in these\nrespects, being equally allied to both. In respect to tone, that of red\nis warm, and that of blue cool, while the tone of yellow is neutral. The\nratios of their relations to each other in these respects will appear in\nthe harmonic scales to which, for the first time, I am about to subject\ncolours, and to systematise their various simple and compound relations,\nwhich are as follow:\u2014\n\nFrom the binary union of the primary colours, the secondary colours arise\u2014\n\nOrange colour, from the union of yellow and red.\n\nGreen, from the union of yellow and blue.\n\nPurple, from the union of red and blue.\n\nFrom the binary union of the secondary colours, the primary hues arise\u2014\n\nYellow-hue, from the union of orange and green.\n\nRed-hue, from the union of orange and purple.\n\nBlue-hue, from the union of purple and green.\n\nFrom the binary union of the primary hues, the secondary hues arise\u2014\n\nOrange-hue, from the union of yellow-hue and red-hue.\n\nGreen-hue, from the union of yellow-hue and blue-hue.\n\nPurple-hue, from the union of red-hue and blue-hue.\n\nEach hue owes its characteristic distinction to the proportionate\npredominance or subordination of one or other of the three primary\ncolours in its composition.\n\nIt follows, that in every hue of _red_, yellow and blue are subordinate;\nin every hue of _yellow_, red and blue are subordinate; and in every\nhue of _blue_, red and yellow are subordinate. In like manner, in every\nhue of _green_, red is subordinate; in every hue of _orange_, blue is\nsubordinate; and in every hue of _purple_, yellow is subordinate.\n\nBy the union of two primary colours, in the production of a secondary\ncolour, the nature of both primaries is altered; and as there are only\nthree primary or simple colours in the scale, the two that are united\nharmonically in a compound colour, form the natural contrast to the\nremaining simple colour.\n\nNotwithstanding all the variety that extends beyond the six positive\ncolours, it may be said that there are only three proper contrasts of\ncolour in nature, and that all others are simply modifications of these.\n\nPure red is the most perfect contrast to pure green; because it is\ncharacterised amongst the primary colours by warmth of tone, while\namongst the secondary colours green is distinguished by coolness of tone,\nboth being equally related to the primary elements of light and shade.\n\nPure yellow is the most perfect contrast to pure purple; because it is\ncharacterised amongst the primary colours as most allied to light, whilst\npure purple is characterised amongst the secondaries as most allied to\nshade, both being equally neutral as to tone.\n\nPure blue is the most perfect contrast to pure orange; because it is\ncharacterised amongst the primary colours as not only the most allied\nto shade, but as being the coolest in tone, whilst pure orange is\ncharacterised amongst the secondaries as being the most allied to light\nand the warmest in tone. The same principle operates throughout all the\nmodifications of these primary and secondary colours.\n\nSuch is the simple nature of contrast upon which the beauty of colouring\nmainly depends.\n\nIt being now established as a scientific fact, that the effect of\nlight upon the eye is the result of an ethereal action, similar to the\natmospheric action by which the effect of sound is produced upon the\near; also, that the various colours which light assumes are the effect\nof certain modifications in this ethereal action;\u2014just as the various\nsounds, which constitute the scale of musical notes, are known to be the\neffect of certain modifications in the atmospheric action by which sounds\nin general are produced:\n\nTherefore, as harmony may thus be impressed upon the mind through\neither of these two art senses\u2014hearing and seeing\u2014the principles which\ngovern the modifications in the ethereal action of light, so as to\nproduce through the eye the effect of harmony, cannot be supposed to\ndiffer from those principles which we know govern the modifications of\nthe atmospheric action of sound, in producing through the ear a like\neffect. I shall therefore endeavour to illustrate the science of beauty\nas evolved in colours, by forming scales of their various modifications\nagreeably to the same Pythagorean system of numerical ratio from which\nthe harmonic elements of beauty in sounds were originally evolved, and by\nwhich I have endeavoured in this, as in previous works, to systematise\nthe harmonic beauty of forms.\n\n[Illustration]\n\nIt will be observed, that with a view to avoid complexity as much as\npossible, I have, in the arrangement of the above series of scales, not\nonly confined myself to the merely elementary parts of the Pythagorean\nsystem, but have left out the harmonic modifications upon (\u00b9\u2044\u2081\u2081)\nand (\u00b9\u2044\u2081\u2083), in order that the arithmetical progression might not be\ninterrupted.[23]\n\nThe above elementary process will, I trust, be found sufficient to\nexplain the progress, by harmonic union, of a primary colour to a\ntoned gray, and how the simple and compound colours naturally arrange\nthemselves into the elements of five scales, the parts of which continue\nfrom primary to secondary colour; from secondary colour to primary hue;\nfrom primary hue to secondary hue; from secondary hue to primary-toned\ngray; and from primary-toned gray to secondary-toned gray in the simple\nratio of 2:1; thereby producing a series of the most beautiful and\nperfect contrasts.\n\nThe natural arrangement of the primary colours upon the solar spectrum is\nred, yellow, blue, and I have therefore adopted the same arrangement on\nthe present occasion. Red being, consequently, the first tonic, and blue\nthe second, the divisions express the numerical ratios which the colours\nbear to one another, in respect to that colourific power for which red\nis pre-eminent. Thus, yellow is to red, as 2:3; blue to yellow, as 3:4;\npurple to orange, as 5:6; and green to purple, as 6:7.\n\nThe following series of completed scales are arranged upon the\nforegoing principle, with the natural connecting links of red-orange,\nyellow-orange, yellow-green, and blue-green, introduced in their proper\nplaces.\n\nThe appropriate terminology of musical notes has been adopted, and the\nscales are composed as follows:\u2014\n\n Scale I. consists of primary and secondary colours;\n Scale II. of secondary colours and primary hues;\n Scale III. of primary and secondary hues;\n Scale IV. of secondary hues and primary-toned grays; and\n Scale V. of primary and secondary-toned grays;\n\nAll the parts in each of these scales, from the first tonic to the\nsecond, relate to the same parts of the scale below them in the simple\nratio of 2:1; and serially to the first tonic in the following ratios:\u2014\n\n 8:9, 4:5, 3:4, 2:3, 3:5, 4:7, 8:15, 1:2.\n\n\n_First Series of Scales._\n\n ----+------+-------+-------+-------+-------+-------+------+-------+-------\n |Tonic.| | | | | | | |\n | |Supertonic. | | | | | |\n | | |Mediant. | | | | |\n | | | |Subdominant. | | | |\n | | | | |Dominant. | | |\n | | | | | |Submediant. | |\n | | | | | | |Subtonic. |\n | | | | | | | |Semi-subtonic.\n | | | | | | | | |Tonic.\n ----+------+-------+-------+-------+-------+-------+------+-------+-------\n I. |(\u00b9\u2044\u2082) |(\u2074\u2044\u2089) |(\u00b2\u2044\u2085) |(\u00b3\u2044\u2088) |(\u00b9\u2044\u2083) |(\u00b3\u2044\u2081\u2080) |(\u00b2\u2044\u2087) |(\u2074\u2044\u2081\u2085) |(\u00b9\u2044\u2084)\n |Red. |Red- |Orange.|Yellow-|Yellow.|Yellow-|Green.|Blue- |Blue.\n | |orange.| |orange.| |green. | |green. |\n ----+------+-------+-------+-------+-------+-------+------+-------+-------\n II. |(\u00b9\u2044\u2084) |(\u00b2\u2044\u2089) |(\u00b9\u2044\u2085) |(\u00b3\u2044\u2081\u2086) |(\u00b9\u2044\u2086) |(\u00b3\u2044\u2082\u2080) |(\u00b9\u2044\u2087) |(\u00b2\u2044\u2081\u2085) |(\u00b9\u2044\u2088)\n |Green.|Blue- |Blue |Blue- |Purple |Red- |Red |Red- |Orange.\n | |green |hue. |purple |hue. |purple |hue. |orange |\n | |hue. | |hue. | |hue. | |hue. |\n ----+------+-------+-------+-------+-------+-------+------+-------+-------\n III.|(\u00b9\u2044\u2088) |(\u00b9\u2044\u2089) |(\u00b9\u2044\u2081\u2080) |(\u00b3\u2044\u2083\u2082) |(\u00b9\u2044\u2081\u2082) |(\u00b3\u2044\u2084\u2080) |(\u00b9\u2044\u2081\u2084)|(\u00b9\u2044\u2081\u2085) |(\u00b9\u2044\u2081\u2086)\n |Red |Red- |Orange |Yellow-|Yellow |Yellow-|Green |Blue- |Blue\n |hue. |orange |hue. |orange |hue. |green |hue. |green |hue.\n | |hue. | |hue. | |hue. | |hue. |\n ----+------+-------+-------+-------+-------+-------+------+-------+-------\n IV. |(\u00b9\u2044\u2081\u2086)|(\u00b9\u2044\u2081\u2088) |(\u00b9\u2044\u2082\u2080) |(\u00b3\u2044\u2086\u2084) |(\u00b9\u2044\u2082\u2084) |(\u00b3\u2044\u2088\u2080) |(\u00b9\u2044\u2082\u2088)|(\u00b9\u2044\u2083\u2080) |(\u00b9\u2044\u2083\u2082)\n |Green |Blue- |Blue- |Blue- |Purple |Red- |Red- |Red- |Orange\n |hue. |green- |toned |purple-|hue. |purple-|toned |orange-|hue.\n | |toned |gray. |toned | |toned |gray. |toned |\n | |gray. | |gray. | |gray. | |gray. |\n ----+------+-------+-------+-------+-------+-------+------+-------+-------\n V. |(\u00b9\u2044\u2083\u2082)|(\u00b9\u2044\u2083\u2086) |(\u00b9\u2044\u2084\u2080) |(\u00b3\u2044\u2081\u2082\u2088)|(\u00b9\u2044\u2084\u2088) |(\u00b3\u2044\u2081\u2086\u2080)|(\u00b9\u2044\u2085\u2086)|(\u00b9\u2044\u2086\u2080) |(\u00b9\u2044\u2086\u2084)\n |Red- |Red- |Orange-|Yellow-|Yellow-|Yellow-|Green-| Blue- |Blue-\n |toned |orange-|toned |orange-|toned |green- |toned | green-|toned\n |gray. |toned |gray. |toned |gray. |toned |gray. | toned |gray.\n | |gray. | |gray. | | gray. | | gray. |\n ----+------+-------+-------+-------+-------+-------+------+-------+-------\n\nTo the scales of chromatic power I add another series of scales, in\nwhich yellow, being the first tonic, and blue the second, the numerical\ndivisions express the ratios which the colours in each scale bear to one\nanother in respect to light and shade. Thus red is to yellow, in respect\nto light, as 2:3; blue to red, as 3:4; green to orange, as 5:6, and\npurple to green, as 6:7.\n\nThese scales may therefore be termed scales for the colour-blind,\nbecause, in comparing colours, those whose sight is thus defective,\nnaturally compare the ratios of the light and shade of which different\ncolours are primarily constituted.\n\n[Illustration]\n\nThe following is a series of five complete scales of the harmonic parts\ninto which the light and shade in colours may be divided in each scale\naccording to the above arrangement:\u2014\n\n\n_Second Series of Scales._\n\n ----+-------+-------+-------+-------+------+-------+-------+-------+------\n |Tonic. | | | | | | | |\n | |Supertonic. | | | | | |\n | | |Mediant. | | | | |\n | | | |Subdominant. | | | |\n | | | | |Dominant. | | |\n | | | | | |Submediant. | |\n | | | | | | |Subtonic. |\n | | | | | | | |Semi-subtonic.\n | | | | | | | | |Tonic.\n ----+-------+-------+-------+-------+------+-------+-------+-------+------\n I. |(\u00b9\u2044\u2082) |(\u2074\u2044\u2089) |(\u00b2\u2044\u2085) |(\u00b3\u2044\u2088) |(\u00b9\u2044\u2083) |(\u00b3\u2044\u2081\u2080) |(\u00b2\u2044\u2087) |(\u2074\u2044\u2081\u2085) |(\u00b9\u2044\u2084)\n |Yellow.|Yellow-|Orange.|Red- |Red. |Red- |Purple.|Blue- |Blue.\n | |orange.| |orange.| |purple.| |purple.|\n ----+-------+-------+-------+-------+------+-------+-------+-------+------\n II. |(\u00b9\u2044\u2084) |(\u00b2\u2044\u2089) |(\u00b9\u2044\u2085) |(\u00b3\u2044\u2081\u2086) |(\u00b9\u2044\u2086) |(\u00b3\u2044\u2082\u2080) |(\u00b9\u2044\u2087) |(\u00b2\u2044\u2081\u2085) |(\u00b9\u2044\u2088)\n |Purple.|Blue- |Blue |Blue- |Green.|Yellow-|Yellow |Yellow-|Orange\n | |purple |hue. |green | |green |hue. |orange |\n | |hue. | |hue. | |hue. | |hue. |\n ----+-------+-------+-------+-------+------+-------+-------+-------+------\n III.|(\u00b9\u2044\u2088) |(\u00b9\u2044\u2089) |(\u00b9\u2044\u2081\u2080) |(\u00b3\u2044\u2083\u2082) |(\u00b9\u2044\u2081\u2082)|(\u00b3\u2044\u2084\u2080) |(\u00b9\u2044\u2081\u2084) |(\u00b9\u2044\u2081\u2085) |(\u00b9\u2044\u2081\u2086)\n |Yellow |Yellow-|Orange |Red- |Red |Red- |Purple |Blue- |Blue\n |hue. |orange |hue. |orange |hue. |purple |hue. |purple |hue.\n | |hue. | |hue. | |hue. | |hue. |\n ----+-------+-------+-------+-------+------+-------+-------+-------+------\n IV. |(\u00b9\u2044\u2081\u2086) |(\u00b9\u2044\u2081\u2088) |(\u00b9\u2044\u2082\u2080) |(\u00b3\u2044\u2086\u2084) |(\u00b9\u2044\u2082\u2084)|(\u00b3\u2044\u2088\u2080) |(\u00b9\u2044\u2082\u2088) |(\u00b9\u2044\u2083\u2080) |(\u00b9\u2044\u2083\u2082)\n |Purple |Blue- |Blue- |Blue- |Green |Yellow-|Yellow-|Yellow-|Orange\n |hue. |purple-|toned |green- |hue. |green- |toned |orange-|hue.\n | |toned |gray. |toned | |toned |gray. |toned |\n | |gray. | |gray. | |gray. | |gray. |\n ----+-------+-------+-------+-------+------+-------+-------+-------+------\n V. |(\u00b9\u2044\u2083\u2082) |(\u00b9\u2044\u2083\u2086) |(\u00b9\u2044\u2084\u2080) |(\u00b3\u2044\u2081\u2082\u2088)|(\u00b9\u2044\u2084\u2088)|(\u00b3\u2044\u2081\u2086\u2080)|(\u00b9\u2044\u2085\u2086) |(\u00b9\u2044\u2086\u2080) |(\u00b9\u2044\u2086\u2084)\n |Yellow-|Yellow-|Orange-|Red- |Red- |Red- |Purple-|Blue- |Blue-\n |toned |orange-|toned |orange-|toned |purple-|toned |green- |toned\n |gray. |toned |gray. |toned |gray. |toned |gray. |toned |gray.\n | |gray. | |gray. | |gray. | |gray. |\n ----+-------+-------+-------+-------+------+-------+-------+-------+------\n\nShould I be correct in arranging colours upon scales identical with those\nupon which musical notes have been arranged, and in assuming that colours\nhave the same ratios to each other, in respect to their harmonic power\nupon the eye, which musical notes have in respect to their harmonic power\nupon the ear, the colourist may yet be enabled to impart harmonic beauty\nto his works with as much certainty and ease, as the musician imparts the\nsame quality to his compositions: for the colourist has no more right to\ntrust exclusively to his eye in the arrangement of colours, than the\nmusician has to trust exclusively to his ear in the arrangement of sounds.\n\nWe find, in comparing the dominant parts in the first and second scales\nof the second series, that they are equal as to light and shade, so that\ntheir relative powers of contrast depend entirely upon colour. Hence it\nis that red and green are the two colours, the difference between which\nthe colour-blind are least able to appreciate. Professor George Wilson,\nin his excellent work, \u201cResearches on Colour-Blindness,\u201d mentions the\ncase of an engraver, which proves the power of the eye in being able to\nappreciate these original constituents of colour, irrespective of the\nintermediate phenomenon of tone. This engraver, instead of expressing\nregret on account of his being colour-blind, observed to the professor,\n\u201cMy defective vision is, to a certain extent, a useful and valuable\nquality. Thus, an engraver has two negatives to deal with, _i.e._, white\nand black. Now, when I look at a picture, I see it only in white and\nblack, or light and shade, or, as artists term it, the effect. I find\nat times many of my brother engravers in doubt how to translate certain\ncolours of pictures, which to me are matters of decided certainty and\nease. Thus to me it is valuable.\u201d\n\nThe colour-blind are therefore as incapable of receiving pleasure from\nthe harmonious union of various colours, as those who, to use a common\nterm, have no ear for music, are of being gratified by the \u201cmelody of\nsweet sounds.\u201d\n\nThe generality of mankind are, however, capable of appreciating the\nharmony of colour which, like that of both sound and form, arises from\nthe simultaneous exhibition of opposite principles having a ratio to each\nother. These principles are in continual operation throughout nature,\nand from them we often derive pleasure without being conscious of the\ncause. All who are not colour-blind must have felt themselves struck\nwith the harmonic beauty of a cloudless sky, although in it there is no\nconfiguration, and at first sight apparently but one colour. Now, as\nwe know that there can be no more impression of harmony made upon the\nmind by looking upon a single colour, than there could be by listening\nto a single continued musical note, however sweet its tone, we are apt\nat first to imagine that the organ of vision has, in some measure,\nconveyed a false impression to the mind. But it has not done so; for\nlight, when reflected from the atmosphere, produces those cool tones of\nblue, gray, and purple, which seem to clothe the distant mountains; but,\nwhen transmitted through the same atmosphere, it produces those numerous\nwarm tints, the most intense of which give the gorgeous effects which\nso often accompany the setting sun. We have, therefore, in the upper\npart of a clear sky, where the atmosphere may be said to be illuminated\nprincipally by reflection from the surface of the earth, a comparatively\ncool tone of blue, the result of reflection, which gradually blends into\nthe warm tints, the result of transmission through the same atmosphere.\nSuch a composition of harmonious colouring is to the eye what the voice\nof the soft breath of summer amongst the trees, the hum of insects on\na sultry day, or the simple harmony of the \u00c6olian harp, is to the ear.\nTo such a composition of chromatic harmony must also be referred the\nuniversal concurrence of mankind in appreciating the peculiar beauty of\nwhite marble statuary. That the principal constituent of beauty in such\nworks ought to be harmony of form, no one will deny; but this is not the\nonly element, as appears from the fact, that a cast in plaster of Paris,\nof a fine white marble statue, although identical in form, is far less\nbeautiful than the original. Now this undoubtedly must be the consequence\nof its having been changed from a semi-translucent substance, which,\nlike the atmosphere, can transmit as well as reflect light, to an opaque\nsubstance, which can only reflect it. Thus the opposite principles of\nchromatic warmth and coolness are equally balanced in white marble\u2014the\none being the natural result of the partial transmission of light, and\nthe other that of its reflection.\n\nAs a series of coloured illustrations would be beyond the scope of this\n_r\u00e9sum\u00e9_, I may refer those who wish to prosecute the inquiry, with the\nassistance of such a series, to my published works upon the subject.[24]\n\n\n\n\nTHE SCIENCE OF BEAUTY, APPLIED TO THE FORMS AND PROPORTIONS OF ANCIENT\nGRECIAN VASES AND ORNAMENTS.\n\n\nIn examining the remains of the ornamental works of the ancient Greek\nartists, it appears highly probable that the harmony of their proportions\nand melody of their contour are equally the result of a systematised\napplication of the same harmonic law. This probability not being fully\nelucidated in any of my former works, I will require to go into some\ndetail on the present occasion. I take for my first illustration an\nunexceptionable example, viz.:\u2014\n\n\n_The Portland Vase._\n\nAlthough this beautiful specimen of ancient art was found about the\nmiddle of the sixteenth century, inclosed in a marble sarcophagus within\na sepulchral chamber under the Monte del Grano, near Rome, and although\nthe date of its production is unknown, yet its being a work of ancient\nGrecian art is undoubted; and the exquisite beauty of its form has been\nuniversally acknowledged, both during the time it remained in the palace\nof the Barberini family at Rome, and since it was added to the treasures\nof the British Museum. The forms and proportions of this gem of art\nappear to me to yield an obedience to the great harmonic law of nature,\nsimilar to that which I have instanced in the proportions and contour of\nthe best specimens of ancient Grecian architecture.\n\n[Sidenote: Plate XII.]\n\nLet the line A B (Plate XII.) represent the full height of the vase.\nThrough A draw A _a_, and through B draw B _b_ indefinitely, A _a_ making\nan angle of (\u00b9\u2044\u2082), and B _b_ an angle of (\u00b9\u2044\u2083), with the vertical.\nThrough the point C, where A _a_ and B _b_ intersect one another, draw\nD C E vertical. Through A C and B respectively, draw A D, C F, and B\nE horizontal. Draw similar lines on the other side of A B, and the\nrectilinear portion of the diagram is complete.\n\nThe curvilinear contour may be thus added:\u2014\n\nTake a cut-out ellipse of (\u00b9\u2044\u2084), whose greater axis is equal to the line\nA B, and\n\n_1st._ Place it upon the diagram, so that its circumference may be\ntangential to the lines C E and C F, and its greater axis _m n_ may make\nan angle of (\u00b9\u2044\u2085) with the vertical, and trace its circumference.\n\n_2d._ Place it with its circumference tangential to that of the first at\nthe point m, while its greater axis (of which _o p_ is a part) is in the\nhorizontal, and trace the portion of its circumference _q o r_.\n\n_3d._ Place it with its circumference tangential to that of the above at\n_v_, while its greater axis (of which _u v_ is a part) makes an angle of\n(\u00b3\u2044\u2081\u2080) with the vertical, and trace the portion of its circumference _s v\nt_.\n\nThus the curvilinear contour of the body and neck are harmonically\ndetermined.\n\nThe curve of the handle may be determined by the same ellipse placed so\nthat its greater axis (of which _i k_ is a part) makes an angle of (\u00b9\u2044\u2086)\nwith the vertical.\n\nMake similar tracings on the other side of A B, and the diagram is\ncomplete. The inscribing rectangle D G E K is that of (\u00b2\u2044\u2085).\n\nThe outline resulting from this diagram, not only is in perfect agreement\nwith my recollection of the form, but with the measurements of the\noriginal given in the \u201cPenny Cyclop\u00e6dia;\u201d of the accuracy of which there\ncan be no doubt. They are stated thus:\u2014\u201cIt is about ten inches in height,\nand beautifully curved from the top downwards; the diameter at the top\nbeing about three inches and a-half; at the neck or smallest part, two\ninches; at the largest (mid-height), seven inches; and at the bottom,\nfive inches.\u201d\n\nThe harmonic elements of this beautiful form, therefore, appear to be the\nfollowing parts of the right angle:\u2014\n\n Tonic. Dominant. Mediant. Submediant.\n (\u00b9\u2044\u2082) (\u00b9\u2044\u2083) (\u00b9\u2044\u2085) (\u00b3\u2044\u2081\u2080)\n (\u00b9\u2044\u2084) (\u00b9\u2044\u2086)\n\nWhen we reflect upon the variety of harmonic ellipses that may\nbe described, and the innumerable positions in which they may be\nharmonically placed with respect to the horizontal and vertical lines,\nas well as upon the various modes in which their circumferences may be\ncombined, the variety which may be introduced amongst such forms as the\nforegoing appears almost endless. My second example is that of\u2014\n\n\n_An Ancient Grecian Marble Vase of a Vertical Composition._\n\nI shall now proceed to another class of the ancient Greek vase, the form\nof which is of a more complex character. The specimen I have chosen for\nthe first example of this class is one of those so correctly measured and\nbeautifully delineated by Tatham in his unequalled work.[25] This vase is\na work of ancient Grecian art in Parian marble, which he met with in the\ncollection at the Villa Albani, near Rome. Its height is 4 ft. 4\u00b9\u2044\u2082 in.\n\n[Sidenote: Plate XIII.]\n\nThe following is the formula by which I endeavour to develop its harmonic\nelements:\u2014\n\nLet A B (Plate XIII.) represent the full height of this vase. Through B\ndraw B D, making an angle of (\u00b9\u2044\u2085) with the vertical. Through D draw D O\nvertical, through A draw A C, making an angle of (\u00b2\u2044\u2085); through B draw\nB L, making an angle of (\u00b9\u2044\u2082), and B S, making an angle of (\u00b3\u2044\u2081\u2080), each\nwith the vertical. Through A draw A D, through B draw B O, through L draw\nL N, through C draw C F, and through S draw S P, all horizontal. Through\nA draw A H, making an angle of (\u00b9\u2044\u2081\u2080) with the vertical, and through\nH draw H M vertical. Draw similar lines on the other side of A B, and\nthe rectilinear portion of the diagram is complete, and its inscribing\nrectangle that of (\u00b3\u2044\u2088).\n\nThe curvilinear portion may thus be added\u2014\n\nTake a cut-out ellipse of (\u00b9\u2044\u2083), whose greater axis is about the length\nof the body of the intended vase, place it with its lesser axis upon the\nline S P, and its greater axis upon the line D O, and trace the part _a\nb_ of its circumference upon the diagram. Place the same ellipse with\none of its foci upon C, and its greater axis upon C F, and trace its\ncircumference upon the diagram. Take a cut-out ellipse of (\u00b9\u2044\u2085), whose\ngreater axis is nearly equal to that of the ellipse already used; place\nit with its greater axis upon M H, and its lesser axis upon L N, and\ntrace its circumference upon the diagram. Make similar tracings upon the\nother side of A B, and the diagram is complete. In this, as in the other\ndiagrams, the strong portions of the lines give the contour of the vase.\nThe harmonic elements of this classical form, therefore, appear to be the\nright angle and its following parts:\u2014\n\n Tonic. Dominant. Mediant. Submediant.\n (\u00b9\u2044\u2082) (\u00b9\u2044\u2083) (\u00b2\u2044\u2085) (\u00b3\u2044\u2081\u2080)\n (\u00b9\u2044\u2085)\n (\u00b9\u2044\u2081\u2080)\n\nMy third example is that of\u2014\n\n\n_An Ancient Grecian Vase of a Horizontal Composition._\n\nThis example belongs to the same class as the last, but it is of a\nhorizontal composition. It was carefully drawn from the original in the\nmuseum of the Vatican by Tatham, in whose etchings it will be found with\nits ornamental decorations. The diagram of its harmonic elements may be\nconstructed as follows:\u2014\n\n[Sidenote: Plate XIV.]\n\nLet A B (Plate XIV.) represent the full height of the vase. Through B\ndraw B D, making an angle of (\u00b2\u2044\u2085) with the vertical. Through A draw A H,\nA L, and A C, making respectively the following angles, (\u00b9\u2044\u2085) with the\nvertical, (\u2074\u2044\u2089) with the vertical, and (\u00b3\u2044\u2081\u2080) with the horizontal. These\nangles determine the horizontal lines H B, L N, and C F, which divide\nthe vase into its parts, and the inscribing rectangle D G K O is (\u00b3\u2044\u2088).\nThis completes the rectilinear portion of the diagram. The ellipse by\nwhich the curvilinear portion is added is one of (\u00b9\u2044\u2085), the greater axis\nof which, at _a b_, as also at _c d_, makes an angle of (\u00b9\u2044\u2081\u2082) with\nthe vertical, and the same axis at _e f_ an angle of (\u00b9\u2044\u2081\u2082) with the\nhorizontal.\n\nThe harmonic elements of this vase, therefore, appear to be:\u2014\n\n Tonic. Dominant. Mediant. Submediant. Supertonic.\n The Right (\u00b9\u2044\u2081\u2082) (\u00b2\u2044\u2085) (\u00b3\u2044\u2081\u2080) (\u2074\u2044\u2089)\n Angle. (\u00b9\u2044\u2085)\n\nMy remaining examples are those of\u2014\n\n\n_Etruscan Vases._\n\nOf these vases I give four examples, by which the simplicity of the\nmethod employed in applying the harmonic law will be apparent.\n\n[Sidenote: Plate XV.]\n\nThe inscribing rectangle D G E K of fig. 1, Plate XV., is one of (\u00b3\u2044\u2088),\nwithin which are arranged tracings from an ellipse of (\u00b3\u2044\u2081\u2080), whose\ngreater axis at _a b_ makes an angle of (\u00b9\u2044\u2081\u2082), at _c d_ an angle of\n(\u00b3\u2044\u2081\u2080), and at _e f_ an angle of (\u00b3\u2044\u2084), with the vertical. The harmonic\nelements of the contour of this vase, therefore, appear to be:\u2014\n\n Tonic. Dominant. Subdominants. Submediant.\n The Right (\u00b9\u2044\u2081\u2082) (\u00b3\u2044\u2084) (\u00b3\u2044\u2081\u2080)\n Angle. (\u00b3\u2044\u2088)\n\nThe inscribing rectangle L M N O of fig. 2 is that of (\u00b9\u2044\u2082), within which\nare arranged tracings from an ellipse of (\u00b9\u2044\u2083), whose greater axis, at\n_a b_ and _c d_ respectively, makes angles of (\u00b9\u2044\u2082) and (\u2074\u2044\u2089) with the\nhorizontal, while that at _e f_ is in the horizontal line. The harmonic\nelements of the contour of this vase, therefore, appear to be:\u2014\n\n Tonic. Dominant. Subtonic.\n (\u00b9\u2044\u2082) (\u00b9\u2044\u2083) (\u2074\u2044\u2089)\n\n[Sidenote: Plate XVI.]\n\nThe inscribing rectangle P Q R S of fig. 1, Plate XVI., is one of (\u2074\u2044\u2089),\nwithin which are arranged tracings from an ellipse of (\u00b3\u2044\u2088), whose\ngreater axis, at _a b_, _c d_, and _e f_, makes respectively angles\nof (\u00b9\u2044\u2086) with the horizontal, (\u00b3\u2044\u2085) and (\u2074\u2044\u2085) with the vertical. Its\nharmonic elements, therefore, appear to be:\u2014\n\n Tonic. Dominant. Mediant. Supertonic. Subdominant. Submediant.\n The Right (\u00b9\u2044\u2086) (\u2074\u2044\u2085) (\u2074\u2044\u2089) (\u00b3\u2044\u2088) (\u00b3\u2044\u2085)\n Angle.\n\nThe inscribing rectangle T U V X of fig. 2 is one of (\u2074\u2044\u2089), within which\nare arranged tracings from an ellipse of (\u00b3\u2044\u2088) whose greater axis at _a\nb_ is in the vertical line, and at _c d_ makes an angle of (\u00b9\u2044\u2082). The\nharmonic elements of the contour of this vase, therefore, appear to be:\u2014\n\n Tonic. Submediant. Supertonic.\n (\u00b9\u2044\u2082) (\u00b3\u2044\u2088) (\u2074\u2044\u2089)\n\nThese four Etruscan vases, the contours of which are thus reduced to the\nharmonic law of nature, are in the British Museum, and engravings of\nthem are to be found in the well-known work of Mr Henry Moses, Plates\n4, 6, 14, and 7, respectively, where they are represented with their\nappropriate decorations and colours.\n\nTo these, I add two examples of\u2014\n\n\n_Ancient Grecian Ornament._\n\nI have elsewhere shewn[26] that the elliptic curve pervades the Parthenon\nfrom the entases of the column to the smallest moulding, and we need not,\ntherefore, be surprised to find it employed in the construction of the\nonly two ornaments belonging to that great work.\n\n[Sidenote: Plate XVII.]\n\nIn the diagram (Plate XVII.), I endeavour to exhibit the geometric\nconstruction of the upper part of one of the ornamental apices, termed\nantefix\u00e6, which surmounted the cornice of the Parthenon.\n\nThe first ellipse employed is that of (\u00b9\u2044\u2083), whose greater axis _a b_ is\nin the vertical line; the second is also that of (\u00b9\u2044\u2083), whose greater\naxis _c d_ makes, with the vertical, an angle of (\u00b9\u2044\u2081\u2082); the third\nellipse is the same with its major axis _e f_ in the vertical line.\nThrough one of the foci of this ellipse at A the line A C is drawn, and\nupon the part of the circumference C _e_, the number of parts, 1, 2, 3,\n4, 5, 6, 7, of which the surmounting part of this ornament is to consist,\nare set off. That part of the circumference of the ellipse whose larger\naxis is _c d_ is divided from _g_ to _c_ into a like number of parts. The\nthird ellipse employed is one of (\u00b9\u2044\u2084).\n\nTake a cut-out ellipse of this kind, whose larger axis is equal in length\nto the inscribing rectangle. Place it with its vertex upon the same\nellipse at _g_, so that its circumference will pass through C, and trace\nit; remove its apix first to _p_, then to _q_, and proceed in the same\nway to _q_, _r_, _s_, _t_, _u_, and _v_, so that its circumference will\npass through the seven divisions on _c g_ and _e_ C: _v o_, _u n_, _t m_,\n_s i_, _r k_, _q j_, _p l_, and _g x_, are parts of the larger axes of\nthe ellipses from which the curves are traced. The small ellipse of which\nthe ends of the parts are formed is that of (\u00b9\u2044\u2083).\n\n[Sidenote: Plate XVIII.]\n\nIn the diagram (Plate XVIII.), I endeavour to exhibit the geometric\nconstruction of the ancient Grecian ornament, commonly called the\n_Honeysuckle_, from its resemblance to the flower of that name. The first\npart of the process is similar to that just explained with reference to\nthe antefix\u00e6 of the Parthenon, although the angles in some parts differ.\nThe contour is determined by the circumference of an ellipse of (\u00b9\u2044\u2083),\nwhose major axis A B makes an angle of (\u00b9\u2044\u2089) with the vertical, and\nthe leaves or petals are arranged upon a portion of the perimeter of a\nsimilar ellipse whose larger axis E F is in the vertical line, and these\nparts are again arranged upon a similar ellipse whose larger axis C D\nmakes an angle of (\u00b9\u2044\u2081\u2082) with the vertical. The first series of curved\nlines proceeding from 1, 2, 3, 4, 5, 6, 7, and 8, are between K E and H\nC, part of the circumference of an ellipse of (\u00b9\u2044\u2083); and those between C\nH and A G are parts of the circumference of four ellipses, each of (\u00b9\u2044\u2083),\nbut varying as to the lengths of their larger axes from 5 to 3 inches.\nThe change from the convex to the concave, which produces the ogie forms\nof which this ornament is composed, takes place upon the line C H, and\nthe lines _a b_, _c d_, _e f_, _g h_, _i k_, _l m_, _n o_, and _p q_, are\nparts of the larger axis of the four ellipses the circumference of which\ngive the upper parts of the petals or leaves.\n\nThis peculiar Grecian ornament is often, like the antefix\u00e6 of the\nParthenon, combined with the curve of the spiral scroll. But the volute\nis so well understood that I have not rendered my diagrams more complex\nby adding that figure. Many varieties of this union are to be found in\nTatham\u2019s etchings, already referred to. The antefix\u00e6 of the Parthenon,\nand its only other ornament the honeysuckle, as represented on the soffit\nof the cornice, are to be found in Stewart\u2019s \u201cAthens.\u201d\n\n\n\n\nAPPENDIX.\n\n\nNo. I.\n\nIn pages 34, 35, and 58, I have reiterated an opinion advanced in several\nof my former works, viz., that, besides genius, and the study of nature,\nan additional cause must be assigned for the general excellence which\ncharacterises such works of Grecian art as were executed during a period\ncommencing about 500 B.C., and ending about 200 B.C. And that this cause\nmost probably was, that the artists of that period were instructed in\na system of fixed principles, based upon the doctrines of Pythagoras\nand Plato. This opinion has not been objected to by the generality of\nthose critics who have reviewed my works; but has, however, met with an\nopponent, whose recondite researches and learned observations are worthy\nof particular attention. These are given in an essay by Mr C. Knight\nWatson, \u201cOn the Classical Authorities for Ancient Art,\u201d which appeared in\nthe _Cambridge Journal of Classical and Sacred Philology_ in June 1854.\nAs this essay is not otherwise likely to meet the eyes of the generality\nof my readers, and as the objections he raises to my opinion only occupy\ntwo out of the sixteen ample paragraphs which constitute the first part\nof the essay, I shall quote them fully:\u2014\n\n \u201cThe next name on our list is that of the famous Euphranor\n (B.C. 362). For the fact that to the practice of sculpture\n and of painting he added an exposition of the theory, we are\n indebted to Pliny, who says (xxxv. 11, 40), \u2018Volumina quoque\n composuit de symmetria et coloribus.\u2019 When we reflect on the\n _critical_ position occupied by Euphranor in the history\n of Greek art, as a connecting link between the idealism of\n Pheidias and the naturalism of Lysippus, we can scarcely\n overestimate the value of a treatise on art proceeding from\n such a quarter. This is especially the case with the first\n of the two works here assigned to Euphranor. The inquiries\n which of late years have been instituted by Mr D. R. Hay of\n Edinburgh, on the proportions of the human figure, and on the\n natural principles of beauty as illustrated by works of Greek\n art, constitute an epoch in the study of \u00e6sthetics and the\n philosophy of form. Now, in the presence of these inquiries,\n or of such less solid results as Mr Hay\u2019s predecessors in\n the same field have elicited, it naturally becomes an object\n of considerable interest to ascertain how far these laws of\n form and principles of beauty were consciously developed in\n the mind, and by the chisel, of the sculptor: how far any\n such system of curves and proportions as Mr Hay\u2019s was used\n by the Greek as a practical manual of his craft. Without in\n the least wishing to impugn the accuracy of that gentleman\u2019s\n results\u2014a piece of presumption I should do well to avoid\u2014I must\n be permitted to doubt whether the \u2018Symmetria\u2019 of Euphranor\n contained anything analogous to them in kind, or indeed equal\n in value. It must not be forgotten that the truth of Mr Hay\u2019s\n theory is perfectly compatible with the fact, that of such\n theory the Greek may have been utterly ignorant. It is on this\n fact I insist: it is here that I join issue with Mr Hay, and\n with his reviewer in a recent number of _Blackwood\u2019s Magazine_.\n Or, to speak more accurately,\u2014while I am quite prepared to find\n that the Elgin marbles will best of all stand the test which Mr\n Hay has hitherto applied, I believe, to works of a later age, I\n am none the less convinced that it is precisely that golden age\n of Hellenic art to which they belong, precisely that first and\n chief of Hellenic artists by whom they were executed, to which\n and to whom any such line of research on the laws of form would\n have been pre-eminently alien. Pheidias, remember, by the right\n of primogeniture, is the ruling spirit of idealism in art. Of\n spontaneity was that idealism begotten and nurtured: by any\n such system as Mr Hay\u2019s, that spontaneity would be smothered\n and paralysed. Pheidias copied an idea in his own mind\u2014\u2018Ipsius\n in mente insidebat species pulchritudinis eximia qu\u00e6dam\u2019\n (_Cic._);\u2014later ages copied _him_. He created: they criticised.\n He was the author of Iliads: they the authors of Poetics.\n Doubtless, if you unsphere the _spirit_ of Mr Hay\u2019s theories,\n you will find nothing discordant with what I have here said.\n That is a sound view of Beauty which makes it consist in that\n due subordination of the parts to the whole, that due relation\n of the parts to each other, which Mendelssohn had in his\n mind when he said that the essence of beauty was \u2018unity in\n variety\u2019\u2014variety beguiling the imagination, the perception of\n unity exercising the thewes and sinews of the intellect. On\n such a view of beauty, Mr Hay\u2019s theory may, _in spirit_, be\n said to rest. But here, as in higher things, it is the letter\n that killeth, while the spirit giveth life. And accordingly I\n must enter a protest against any endeavour to foist upon the\n palmy days of Hellenic art systems of geometrical proportions\n incompatible, as I believe, with those higher and broader\n principles by which the progress of ancient sculpture was\n ordered and governed\u2014systems which will bear nothing of that\n \u2018felicity and chance by which\u2019\u2014and not by rule\u2014\u2018Lord Bacon\n believed that a painter may make a better face than ever was:\u2019\n systems which take no account of that fundamental distinction\n between the schools of Athens and of Argos, and their\n respective disciples and descendants, without which you will\n make nonsense of the pages of Pliny, and\u2014what is worse\u2014sense\n of the pages of his commentators;\u2014systems, in short, which may\n have their value as instruments for the education of the eye,\n and for instructions in the arts of design, but must be cast\n aside as matters of learned trifling and curious disputation,\n where they profess to be royal roads to art, and to map the\n mighty maze of a creative mind. And even as regards the\n application of such a system of proportions to those works of\n sculpture which are posterior to the Pheidian age, only partial\n can have been the prevalence which it or any other _one_ system\n can have obtained. The discrepancies of different artists\n in the treatment of what was called, technically called,\n _Symmetria_ (as in the title of Euphranor\u2019s work) were, by the\n concurrent testimony of all ancient writers, far too salient\n and important to warrant the supposition of any uniform scale\n of proportions, as advocated by Mr Hay. Even in Egypt, where\n one might surely have expected that such uniformity would have\n been observed with far greater rigour than in Greece, the\n discoveries of Dr Lepsius (_Vorl\u00e4ufige Nachricht_, Berlin,\n 1849) have elicited three totally different \u03ba\u03b1\u03bd\u03cc\u03bd\u03b5\u03c2, one of\n which is identical with the system of proportions of the human\n figure detailed in Diodorus. While we thus venture to differ\n from Mr Hay on the historical data he has mixed up with his\n inquiries, we feel bound to pay him a large and glad tribute of\n praise for having devised a system of proportions which rises\n superior to the idiosyncracies of different artists, which\n brings back to one common type the sensations of eye and ear,\n and so makes a giant stride towards that _codification_, if\n I may so speak, of the laws of the universe which it is the\n business of the science to effect. I have no hesitation in\n saying, that, for scientific precision of method and importance\n of results, Albert Durer, Da Vinci, and Hogarth, not to mention\n less noteworthy writers, must all yield the palm to Mr Hay.\n\n \u201cI am quite aware that in the digression I have here allowed\n myself, on systems of proportions prevalent among ancient\n artists, and on the probable contents of such treatises as that\n of Euphranor, _De Symmetria_, I have laid myself open to the\n charge of treating an intricate question in a very perfunctory\n way. At present the exigencies of the subject more immediately\n in hand allow me only to urge in reply, that, as regards the\n point at issue\u2014I mean the \u2018solidarit\u00e9\u2019 between theories such as\n Mr Hay\u2019s and the practice of Pheidias\u2014the _onus probandi_ rests\n with my adversaries.\u201d\n\nI am bound, in the first place, gratefully to acknowledge the kind and\ncomplimentary notice which, notwithstanding our difference of opinion,\nthis author has been pleased to take of my works; and, in the second, to\nassure him that if any of them profess to be \u201croyal roads to art,\u201d or\nto \u201cmap the mighty maze of a creative mind,\u201d they certainly profess to\ndo more than I ever meant they should; for I never entertained the idea\nthat a system of \u00e6sthetic culture, even when based upon a law of nature,\nwas capable of effecting any such object. But I doubt not that this too\ncommon misapprehension of the scope and tendency of my works must arise\nfrom a want of perspicuity in my style.\n\nI have certainly, on one occasion,[27] gone the length of stating\nthat as poetic genius must yield obedience to the rules of rhythmical\nmeasure, even in the highest flights of her inspirations; and musical\ngenius must, in like manner, be subject to the strictly defined laws of\nharmony in the most delicate, as well as in the most powerfully grand\nof her compositions; so must genius, in the formative arts, either\nconsciously or unconsciously have clothed her creations of ideal beauty\nwith proportions strictly in accordance with the laws which nature has\nset up as inflexible standards. If, therefore, the laws of proportion, in\ntheir relation to the arts of design, constitute the harmony of geometry,\nas definitely as those that are applicable to poetry and music produce\nthe harmony of acoustics; the former ought, certainly, to hold the same\nrelative position in those arts which are addressed to the eye, that is\naccorded to the latter in those which are addressed to the ear. Until\nso much science be brought to bear upon the arts of design, the student\nmust continue to copy from individual and imperfect objects in nature, or\nfrom the few existing remains of ancient Greek art, in total ignorance of\nthe laws by which their proportions are produced, and, what is equally\ndetrimental to art, the accuracy of all criticism must continue to rest\nupon the indefinite and variable basis of mere opinion.\n\nIt cannot be denied that men of great artistic genius are possessed of\nan intuitive feeling of appreciation for what is beautiful, by means\nof which they impart to their works the most perfect proportions,\nindependently of any knowledge of the definite laws which govern that\nspecies of beauty. But they often do so at the expense of much labour,\nmaking many trials before they can satisfy themselves in imparting to\nthem the true proportions which their minds can conceive, and which,\nalong with those other qualities of expression, action, or attitude,\nwhich belong more exclusively to the province of genius. In such\ncases, an acquaintance with the rules which constitute the science of\nproportion, instead of proving fetters to genius, would doubtless afford\nher such a vantage ground as would promote the more free exercise of\nher powers, and give confidence and precision in the embodiment of her\ninspirations; qualities which, although quite compatible with genius, are\nnot always intuitively developed along with that gift.\n\nIt is also true that the operations of the conceptive faculty of the mind\nare uncontrolled by definite laws, and that, therefore, there cannot\nexist any rules by the inculcation of which an ordinary mind can be\nimbued with genius sufficient to produce works of high art. Nevertheless,\nsuch a mind may be improved in its perceptive faculty by instruction in\nthe science of proportion, so as to be enabled to exercise as correct and\njust an appreciation of the conceptions of others, in works of plastic\nart, as that manifested by the educated portion of mankind in respect\nto poetry and music. In short, it appears that, in those arts which are\naddressed to the ear, men of genius communicate the original conceptions\nof their minds under the control of certain scientific laws, by means\nof which the educated easily distinguish the true from the false, and\nby which the works of the poet and musical composer may be placed above\nmere imitations of nature, or of the works of others; while, in those\narts that are addressed to the eye in their own peculiar language, such\nas sculpture, architecture, painting, and ornamental design, no such laws\nare as yet acknowledged.\n\nAlthough I am, and ever have been, far from endeavouring \u201cto foist upon\nthe palmy days of Hellenic art\u201d any system incompatible with those higher\nand more intellectual qualities which genius alone can impart; yet, from\nwhat has been handed down to us by writers on the subject, meagre as it\nis, I cannot help continuing to believe that, along with the physical and\nmetaphysical sciences, \u00e6sthetic science was taught in the early schools\nof Greece.\n\nI shall here take the liberty to repeat the proofs I advanced in a former\nwork as the ground of this belief, and to which the author, from whose\nessay I have quoted, undoubtedly refers. It is well known that, in the\ntime of Pythagoras, the treasures of science were veiled in mystery to\nall but the properly initiated, and the results of its various branches\nonly given to the world in the works of those who had acquired this\nknowledge. So strictly was this secresy maintained amongst the disciples\nand pupils of Pythagoras, that any one divulging the sacred doctrines\nto the profane, was expelled the community, and none of his former\nassociates allowed to hold further intercourse with him; it is even\nsaid, that one of his pupils incurred the displeasure of the philosopher\nfor having published the solution of a problem in geometry.[28] The\ndifficulty, therefore, which is expressed by writers, shortly after the\nperiod in which Pythagoras lived, regarding a precise knowledge of his\ntheories, is not to be wondered at, more especially when it is considered\nthat he never committed them to writing. It would appear, however, that\nhe proceeded upon the principle, that the order and beauty so apparent\nthroughout the whole universe, must compel men to believe in, and refer\nthem to, an intelligible cause. Pythagoras and his disciples sought for\nproperties in the science of numbers, by the knowledge of which they\nmight attain to that of nature; and they conceived those properties to\nbe indicated in the phenomena of sonorous bodies. Observing that Nature\nherself had thus irrevocably fixed the numerical value of the intervals\nof musical tones, they justly concluded that, as she is always uniform\nin her works, the same laws must regulate the general system of the\nuniverse.[29] Pythagoras, therefore, considered numerical proportion as\nthe great principle inherent in all things, and traced the various forms\nand phenomena of the world to numbers as their basis and essence.\n\nHow the principles of numbers were applied in the arts is not recorded,\nfarther than what transpires in the works of Plato, whose doctrines were\nfrom the school of Pythagoras. In explaining the principle of beauty,\nas developed in the elements of the material world, he commences in the\nfollowing words:\u2014\u201cBut when the Artificer began to adorn the universe,\nhe first of all figured with forms and numbers, fire and earth, water\nand air\u2014which possessed, indeed, certain traces of the true elements,\nbut were in every respect so constituted as it becomes anything to be\nfrom which Deity is absent. But we should always persevere in asserting\nthat Divinity rendered them, as much as possible, the most beautiful\nand the best, when they were in a state of existence opposite to such\na condition.\u201d Plato goes on further to say, that these elementary\nbodies must have forms; and as it is necessary that every depth should\ncomprehend the nature of a plane, and as of plane figures the triangle\nis the most elementary, he adopts two triangles as the originals or\nrepresentatives of the isosceles and the scalene kinds. The first\ntriangle of Plato is that which forms the half of the square, and is\nregulated by the number, 2; and the second, that which forms the half\nof the equilateral triangle, which is regulated by the number, 3;\nfrom various combinations of these, he formed the bodies of which he\nconsidered the elements to be composed. To these elementary figures I\nhave already referred.\n\nVitruvius, who studied architecture ages after the arts of Greece had\nbeen buried in the oblivion which succeeded her conquest, gives the\nmeasurements of various details of monuments of Greek art then existing.\nBut he seems to have had but a vague traditionary knowledge of the\nprinciple of harmony and proportion from which these measurements\nresulted. He says\u2014\u201cThe several parts which constitute a temple ought\nto be subject to the laws of symmetry; the principles of which should\nbe familiar to all who profess the science of architecture. Symmetry\nresults from proportion, which, in the Greek language, is termed\nanalogy. Proportion is the commensuration of the various constituent\nparts with the whole; in the existence of which symmetry is found to\nconsist. For no building can possess the attributes of composition\nin which symmetry and proportion are disregarded; nor unless there\nexist that perfect conformation of parts which may be observed in a\nwell-formed human being.\u201d After going at some length into details, he\nadds\u2014\u201cSince, therefore, the human figure appears to have been formed\nwith such propriety, that the several members are commensurate with\nthe whole, the artists of antiquity (meaning those of Greece at the\nperiod of her highest refinement) must be allowed to have followed the\ndictates of a judgment the most rational, when, transferring to works of\nart principles derived from nature, every part was so regulated as to\nbear a just proportion to the whole. Now, although the principles were\nuniversally acted upon, yet they were more particularly attended to in\nthe construction of temples and sacred edifices, the beauties or defects\nof which were destined to remain as a perpetual testimony of their skill\nor of their inability.\u201d\n\nVitruvius, however, gives no explanation of this ancient principle\nof proportion, as derived from the human form; but plainly shews his\nuncertainty upon the subject, by concluding this part of his essay in the\nfollowing words: \u201cIf it be true, therefore, that the decenary notation\nwas suggested by the members of man, and that the laws of proportion\narose from the relative measures existing between certain parts of each\nmember and the whole body, it will follow, that those are entitled to our\ncommendation who, in building temples to their deities, proportioned the\nedifices, so that the several parts of them might be commensurate with\nthe whole.\u201d It thus appears certain that the Grecians, at the period of\ntheir highest excellence, had arrived at a knowledge of some definite\nmathematical law of proportion, which formed a standard of perfectly\nsymmetrical beauty, not only in the representation of the human figure\nin sculpture and painting, but in architectural design, and indeed in\nall works where beauty of form and harmony of proportion constituted\nexcellence. That this law was not deduced from the proportions of\nthe human figure, as supposed by Vitruvius, but had its origin in\nmathematical science, seems equally certain; for in no other way can we\nsatisfactorily account for the proportions of the beau ideal forms of the\nancient Greek deities, or of those of their architectural structures,\nsuch as the Parthenon, the temple of Theseus, &c., or for the beauty that\npervades all the other formative art of the period.\n\nThis system of geometrical harmony, founded, as I have shewn it to be,\nupon numerical relations, must consequently have formed part of the Greek\nphilosophy of the period, by means of which the arts began to progress\ntowards that great excellence which they soon after attained. A little\nfurther investigation will shew, that immediately after this period a\ntheory connected with art was acknowledged and taught, and also that\nthere existed a Science of Proportion.\n\nPamphilus, the celebrated painter, who flourished about four hundred\nyears before the Christian era, from whom Apelles received the\nrudiments of his art, and whose school was distinguished for scientific\ncultivation, artistic knowledge, and the greatest accuracy in drawing,\nwould admit no pupil unacquainted with geometry.[30] The terms upon which\nhe engaged with his students were, that each should pay him one talent\n(\u00a3225 sterling) previous to receiving his instructions; for this he\nengaged \u201cto give them, _for ten years_, lessons founded on an excellent\ntheory.\u201d[31]\n\nIt was by the advice of Pamphilus that the magistrates of Sicyon ordained\nthat the study of drawing should constitute part of the education of the\ncitizens\u2014\u201ca law,\u201d says the Abb\u00e9 Barth\u00e9l\u00e9mie, \u201cwhich rescued the fine arts\nfrom servile hands.\u201d\n\nIt is stated of Parrhasius, the rival of Zeuxis, who flourished about\nthe same period as Pamphilus, that he accelerated the progress of art by\npurity and correctness of design; \u201cfor he was acquainted with the science\nof Proportions. Those he gave his gods and heroes were so happy, that\nartists did not hesitate to adopt them.\u201d Parrhasius, it is also stated,\nwas so admired by his contemporaries, that they decreed him the name of\nLegislator.[32] The whole history of the arts in Egypt and Greece concurs\nto prove that they were based on geometric precision, and were perfected\nby a continued application of the same science; while in all other\ncountries we find them originating in rude and misshapen imitations of\nnature.\n\nIn the earliest stages of Greek art, the gods\u2014then the only statues\u2014were\nrepresented in a tranquil and fixed posture, with the features exhibiting\na stiff inflexible earnestness, their only claim to excellence being\nsymmetrical proportion; and this attention to geometric precision\ncontinued as art advanced towards its culminating point, and was\nthereafter still exhibited in the neatly and regularly folded drapery,\nand in the curiously braided and symmetrically arranged hair.[33]\n\nThese researches, imperfect as they are, cannot fail to exhibit the\ngreat contrast that exists between the system of elementary education\nin art practised in ancient Greece, and that adopted in this country at\nthe present period. But it would be of very little service to point out\nthis contrast, were it not accompanied by some attempt to develop the\nprinciples which seem to have formed the basis of this excellence amongst\nthe Greeks.\n\nBut in making such an attempt, I cannot accuse myself of assuming\nanything incompatible with the free exercise of that spontaneity of\ngenius which the learned essayist says is the parent and nurse of\nidealism. For it is in no way more incompatible with the free exercise of\nartistic genius, that those who are so gifted should have the advantage\nof an elementary education in the science of \u00e6sthetics, than it is\nincompatible with the free exercise of literary or poetic genius, that\nthose who possess it should have the advantage of such an elementary\neducation in the science of philology as our literary schools and\ncolleges so amply afford.\n\n\nNo. II.\n\nThe letter from which I have made a quotation at page 42, arose out of\nthe following circumstance:\u2014In order that my theory, as applied to the\northographic beauty of the Parthenon, might be brought before the highest\ntribunal which this country afforded, I sent a paper upon the subject,\naccompanied by ample illustrations, to the Royal Institute of British\nArchitects, and it was read at a meeting of that learned body on the 7th\nof February 1853; at the conclusion of which, Mr Penrose kindly undertook\nto examine my theoretical views, in connexion with the measurements he\nhad taken of that beautiful structure by order of the Dilettanti Society,\nand report upon the subject to the Royal Institute. This report was\npublished by Mr Penrose, vol. xi., No. 539 of _The Builder_, and the\nletter from which I have quoted appeared in No. 542 of the same journal.\nIt was as follows:\u2014\n\n \u201cGEOMETRICAL RELATIONS IN ARCHITECTURE.\n\n \u201cWill you allow me, through the medium of your columns, to\n thank Mr Penrose for his testimony to the truth of Mr Hay\u2019s\n revival of Pythagoras? The dimensions which he gives are to\n me the surest verification of the theory that I could have\n desired. The minute discrepancies form that very element\n of practical incertitude, both as to execution and direct\n measurement, which always prevails in materialising a\n mathematical calculation under such conditions.\n\n \u201cIt is time that the scattered computations by which\n architecture has been analysed\u2014more than enough\u2014be synthetised\n into those formul\u00e6 which, as Mrs Somerville tells us, \u2018are\n emblematic of omniscience.\u2019 The young architects of our day\n feel trembling beneath their feet the ground whence men are\n about to evoke the great and slumbering corpse of art. Sir, it\n is food of this kind a reviving poetry demands.\n\n \u2014\u2014\u2018Give us truths,\n For we are weary of the surfaces,\n And die of inanition.\u2019\n\n \u201cI, for one, as I listen to such demonstrations, whose scope\n extends with every research into them, feel as if listening to\n those words of Pythagoras, which sowed in the mind of Greece\n the poetry whose manifestation in beauty has enchained the\n world in worship ever since its birth. And I am sure that in\n such a quarter, and in such thoughts, _we_ must look for the\n first shining of that lamp of art, which even now is prepared\n to burn.\n\n \u201cI know that this all sounds rhapsodical; but I know also that\n until the architect becomes a poet, and not a tradesman, we\n may look in vain for architecture: and I know that valuable\n as isolated and detailed investigations are in their proper\n bearings, yet that such purposes and bearings are to be found\n in the enunciation of principles sublime as the generalities of\n \u2018mathematical beauty.\u2019\n\n \u201cAUTOCTHON.\u201d\n\n\nNo. III.\n\nOf the work alluded to at page 58 I was favoured with two opinions\u2014the\none referring to the theory it propounds, and the other to its anatomical\naccuracy\u2014both of which I have been kindly permitted to publish.\n\nThe first is from Sir WILLIAM HAMILTON, Bart., professor of logic and\nmetaphysics in the University of Edinburgh, and is as follows:\u2014\n\n \u201cYour very elegant volume is to me extremely interesting, as\n affording an able contribution to what is the ancient, and,\n I conceive, the true theory of the Beautiful. But though\n your doctrine coincides with the one prevalent through all\n antiquity, it appears to me quite independent and original\n in you; and I esteem it the more, that it stands opposed to\n the hundred one-sided and exclusive views prevalent in modern\n times.\u2014_16 Great King Street, March 5, 1849._\u201d\n\nThe second is from JOHN GOODSIR, Esq., professor of anatomy in the\nUniversity of Edinburgh, and is as follows:\u2014\n\n \u201cI have examined the plates in your work on the proportions\n of the human head and countenance, and find the head you have\n given as typical of human beauty to be anatomically correct\n in its structure, only differing from ordinary nature in its\n proportions being more mathematically precise, and consequently\n more symmetrically beautiful.\u2014_College, Edinburgh, 17th April\n 1849._\u201d\n\n\nNo. IV.\n\nI shall here shew, as I have done in a former work, how the curvilinear\noutline of the figure is traced upon the rectilinear diagrams by portions\nof the ellipse of (\u00b9\u2044\u2083), (\u00b9\u2044\u2084), (\u00b9\u2044\u2085), and (\u00b9\u2044\u2086).\n\n[Sidenote: Plate XIX.]\n\nThe outline of the head and face, from points (1) to (3) (fig. 1, Plate\nXIX.), takes the direction of the two first curves of the diagram. From\npoint (3), the outline of the sterno-mastoid muscle continues to (4),\nwhere, joining the outline of the trapezius muscle, at first concave, it\nbecomes convex after passing through (5), reaches the point (6), where\nthe convex outline of the deltoid muscle commences, and, passing through\n(7), takes the outline of the arm as far as (8). The outline of the\nmuscles on the side, the latissimus dorsi and serratus magnus, commences\nunder the arm at the point (9), and joins the outline of the oblique\nmuscle of the abdomen by a concave curve at (10), which, rising into\nconvexity as it passes through the points (11) and (12), ends at (13),\nwhere it joins the outline of the gluteus medius muscle. The outline of\nthis latter muscle passes convexly through the point (14), and ends at\n(15), where the outline of the tensor vagin\u00e6 femoris and vastus externus\nmuscle of the thigh commences. This convex outline joins the concave\noutline of the biceps of the thigh at (16), which ends in that of the\nslight convexity of the condyles of the thigh-bone at (17). From this\npoint, the outline of the outer surface of the leg, which includes the\nbiceps, peroneus longus, and soleus muscles, after passing through the\npoint (18), continues convexly to (19), where the concave outline of the\ntendons of the peroneus longus continues to (20), whence the outline of\nthe outer ankle and foot commences.\n\nThe outline of the mamma and fold of the arm-pit commences at (21),\nand passes through the points (22) and (23). The circle at (24) is the\noutline of the areola, in the centre of which the nipple is placed.\n\nThe outline of the pubes commences at (25), and ends at the point (26),\nfrom which the outline of the inner surface of the thigh proceeds over\nthe gracilis, the sartorius, and vastus internus muscles, until it meets\nthe internal face of the knee-joint at (27), the outline of which ends\nat (28). The outline of the inside of the leg commences by proceeding\nover the gastrocnemius muscle as far as (29), where it meets that of the\nsoleus muscle, and continues over the tendons of the heel until it meets\nthe outline of the inner ankle and foot at (30).\n\nThe outline of the outer surface of the arm, as viewed in front, proceeds\nfrom (8) over the remainder of the deltoid, in which there is here a\nslight concavity, and, next, from (31) over the biceps muscle till (32),\nwhere it takes the line of the long supinator, and passing concavely, and\nalmost imperceptibly, into the long and short radial extensor muscles,\nreaches the wrist at (33). The outline of the inner surface of the arm\nfrom opposite (9) commences by passing over the triceps extensor, which\nends at (34), then over the gentle convexity of the condyles of the bone\nof the arm at (35), and, lastly, over the flexor sublimis which ends at\nthe wrist-joint (36).\n\nThe outline of the front of the figure commences at the point (1), (fig.\n2, Plate II.), and, passing almost vertically over the platzsma-myoidis\nmuscles, reaches the point (2), where the neck ends by a concave curve.\nFrom (2) the outline rises convexly over the ends of the clavicles,\nand continues so over the pectoral muscle till it reaches (3), where\nthe mamma swells out convexly to (4), and returns convexly towards\n(5), where the curve becomes concave. From (5) the outline follows the\nundulations of the rectus muscle of the abdomen, concave at the points\n(6) and (7), and having its greatest convexity at (8). This series\nof curves ends with a slight concavity at the point (9), where the\nhorizontal branch of the pubes is situated, over which the outline is\nconvex and ends at (10).\n\nThe outline of the thigh commences at the point (11) with a slight\nconcave curve, and then swells out convexly over the extensors of the\nleg, and, reaching (12), becomes gently concave, and, passing over the\npatella at (13), becomes again convex until it reaches the ligament of\nthat bone, where it becomes gently concave towards the point (14), whence\nit follows the slightly convex curve of the shin-bone, and then, becoming\nas slightly concave, ends with the muscles in front of the leg at (15).\n\nThe outline of the back commences at the point (16), and, following with\na concave curve the muscles of the neck as far as (17), swells into a\nconvex curve over the trapezius muscle towards the point (18); passing\nthrough which point, it continues to swell outward until it reaches\n(19), half way between (18) and (20); whence the convexity, becoming\nless and less, falls into the concave curve of the muscles of the loins\nat (21), and passing through the point (22), it rises into convexity. It\nthen passes through the point (23), follows the outline of the gluteus\nmaximus, the convex curve of which rises to the point (24), and then\nreturns inwards to that of (25), where it ends in the fold of the hip.\nFrom this point the outline follows the curve of the hamstring muscles\nby a slight concavity as far as (26), and then, becoming gently convex,\nit reaches (27); whence it becomes again gently concave, with a slight\nindication of the condyle of the thigh-bone at (28), and, reaching (29),\nfollows the convex curve of the gastrocnemius muscle through the point\n(30), and falling into the convex curve of the tendo Achilles at (31),\nends in the concavity over the heel at (32).\n\nThe outline of the front of the arm commences at the point (33), by a\ngentle concavity at the arm-pit, and then swells out in a convex curve\nover the biceps, reaching (34), where it becomes concave, and passing\nthrough (35), again becomes convex in passing over the long supinator,\nand, becoming gently concave as it passes the radial extensors, rises\nslightly at (36), and ends at (37), where the outline of the wrist\ncommences. The outline of the back of the arm commences with a concave\ncurve at (38), which becomes convex as it passes from the deltoid to the\nlong extensor and ends at the elbow (39), from below which the outline\nfollows the convex curve of the extensor ulnaris, reaching the wrist at\nthe point (40).\n\nIt will be seen that the various undulations of the outline are regulated\nby points which are determined generally by the intersections and\nsometimes by directions and extensions of the lines of the diagram, in\nthe same manner in which I shewed proportion to be imparted, in a late\nwork, to the osseous structure. The mode in which the curves of (\u00b9\u2044\u2082),\n(\u00b9\u2044\u2083), (\u00b9\u2044\u2084), (\u00b9\u2044\u2085), and (\u00b9\u2044\u2086) are thus so harmoniously blended in the\noutline of the female figure, only remains to be explained.\n\nThe curves which compose the outline of the female form are therefore\nsimply those of (\u00b9\u2044\u2082), (\u00b9\u2044\u2083), (\u00b9\u2044\u2084), (\u00b9\u2044\u2085), and (\u00b9\u2044\u2086).\n\nManner in which these curves are disposed in the lateral outline (figure\n1, Plate XIX.):\u2014\n\n Points. Curves.\n Head from 1 to 2 (\u00b9\u2044\u2082)\n Face \u201d 2 \u201d 3 (\u00b9\u2044\u2083)\n Neck \u201d 3 \u201d 4 (\u00b9\u2044\u2085)\n Shoulder \u201d 4 \u201d 6 (\u00b9\u2044\u2086)\n \u201d \u201d 6 \u201d 8 (\u00b9\u2044\u2084)\n Trunk \u201d 9 \u201d 15 (\u00b9\u2044\u2084)\n \u201d \u201d 21 \u201d 24 (\u00b9\u2044\u2082)\n Outer surface of thigh and leg \u201d 15 \u201d 20 (\u00b9\u2044\u2086)\n Inner surface of thigh and leg \u201d 25 \u201d 30 (\u00b9\u2044\u2086)\n Outer surface of the arm \u201d 8 \u201d 33 (\u00b9\u2044\u2086)\n Inner surface of the arm \u201d 9 \u201d 36 (\u00b9\u2044\u2086)\n\nManner in which they are disposed in the outline (figure 2, Plate XIX.):\u2014\n\n Points. Curves.\n Front of neck from 1 to 2 (\u00b9\u2044\u2086)\n \u201d trunk \u201d 2 \u201d 10 (\u00b9\u2044\u2084)\n Back of neck \u201d 16 \u201d 18 (\u00b9\u2044\u2086)\n \u201d trunk \u201d 18 \u201d 23 (\u00b9\u2044\u2084)\n \u201d \u201d \u201d 23 \u201d 25 (\u00b9\u2044\u2083)\n Front of thigh and leg \u201d 11 \u201d 13 (\u00b9\u2044\u2084)\n \u201d \u201d \u201d \u201d 13 \u201d 15 (\u00b9\u2044\u2086)\n Back of thigh and leg \u201d 25 \u201d 32 (\u00b9\u2044\u2086)\n Front of the arm \u201d 33 \u201d 37 (\u00b9\u2044\u2086)\n Back of the arm \u201d 38 \u201d 40 (\u00b9\u2044\u2086)\n Foot \u201d 0 \u201d 0 (\u00b9\u2044\u2086)\n\n[Sidenote: Plate XX.]\n\nIn order to exemplify more clearly the manner in which these various\ncurves appear in the outline of the figure, I give in Plate XX. the whole\ncurvilinear figures, complete, to which these portions belong that form\nthe outline of the sides of the head, neck, and trunk, and of the outer\nsurface of the thighs and legs.\n\nThe various angles which the axes of these ellipses form with the\nvertical, will be found amongst other details in the works I have just\nreferred to.\n\n\nNo. V.\n\nAt page 85 I have remarked upon the variety that may be introduced into\nany particular form of vase; and, in order to give the reader an idea of\nthe ease with which this may be done without violating the harmonic law,\nI shall here give three examples:\u2014\n\n[Sidenote: Plate XXI.]\n\nThe first of these (Plate XXI.) differs from the Portland vase, in the\nconcave curve of the neck flowing more gradually into the convex curve of\nthe body.\n\n[Sidenote: Plate XXII.]\n\nThe second (Plate XXII.) differs from the same vase in the same change\nof contour, as also in being of a smaller diameter at the top and at the\nbottom.\n\n[Sidenote: Plate XXIII.]\n\nThe third (Plate XXIII.) is the most simple arrangement of the elliptic\ncurve by which this kind of form may be produced; and it differs from the\nPortland vase in the relative proportions of height and diameter, and in\nhaving a fuller curve of contour.\n\nThe following comparison of the angles employed in these examples, with\nthe angles employed in the original, will shew that the changes of\ncontour in these forms, arise more from the mode in which the angles are\narranged than in a change of the angles themselves:\u2014\n\n Line Line Line Line\n Plate VIII. _A C_ (\u00b9\u2044\u2082) _B C_ (\u00b9\u2044\u2083) _o p_ (H) _v u_ (\u00b3\u2044\u2081\u2080)\n Plate XXI. (\u00b9\u2044\u2082) (\u00b9\u2044\u2083) (\u00b2\u2044\u2089) (\u00b9\u2044\u2084)\n Plate XXII. (\u00b9\u2044\u2082) (\u00b9\u2044\u2083) (\u00b9\u2044\u2088) (\u2074\u2044\u2089)\n Plate XXIII. (\u00b9\u2044\u2082) (\u00b9\u2044\u2084) (H) (-)\n\n Line Line\n Plate VIII. _m n_ (\u00b9\u2044\u2083) _i k_ (\u00b9\u2044\u2085) ellipse (\u00b9\u2044\u2084) rectangle (\u00b2\u2044\u2085)\n Plate XXI. (\u00b2\u2044\u2089) (\u00b9\u2044\u2085) (\u00b9\u2044\u2084) (\u00b2\u2044\u2085)\n Plate XXII. (\u00b9\u2044\u2083) (\u00b9\u2044\u2085) (\u00b9\u2044\u2084) (\u00b2\u2044\u2085)\n Plate XXIII. (\u00b9\u2044\u2085) (\u00b9\u2044\u2085) ellipses {(\u00b9\u2044\u2083)} (\u00b9\u2044\u2083)\n {(\u00b9\u2044\u2084)}\n\nThe harmonic elements of each are therefore simply the following parts of\nthe right angle:\u2014\n\n Tonic. Dominant. Mediant. Submediant.\n Plate VIII. (\u00b9\u2044\u2082) (\u00b9\u2044\u2083) (\u00b9\u2044\u2085) (\u00b3\u2044\u2081\u2080)\n (\u00b9\u2044\u2084)\n\n Tonic. Dominant. Mediant. Supertonic.\n Plate XXI. (\u00b9\u2044\u2082) (\u00b9\u2044\u2083) (\u00b9\u2044\u2085) (\u00b2\u2044\u2089)\n (\u00b9\u2044\u2084)\n\n Tonic. Dominant. Mediant. Supertonic.\n Plate XXII. (\u00b9\u2044\u2082) (\u00b9\u2044\u2083) (\u00b9\u2044\u2085) (\u2074\u2044\u2089)\n (\u00b9\u2044\u2084)\n (\u00b9\u2044\u2088)\n\n Tonic. Dominant. Mediant.\n Plate XXIII. (\u00b9\u2044\u2082) (\u00b9\u2044\u2083) (\u00b9\u2044\u2085)\n (\u00b9\u2044\u2084)\n\n\nNo. VI.\n\nSo far as I know, there has been only one attempt in modern times,\nbesides my own, to establish a universal system of proportion, based on a\nlaw of nature, and applicable to art. This attempt consists of a work of\n457 pages, with 166 engraved illustrations, by Dr Zeising, a professor in\nLeipzic, where it was published in 1854.\n\nOne of the most learned and talented professors in our Edinburgh\nUniversity has reviewed that work as follows:\u2014\n\n\u201cIt has been rather cleverly said that the intellectual distinction\nbetween an Englishman and a Scotchman is this\u2014\u2018Give an Englishman two\nfacts, and he looks out for a third; give a Scotchman two facts, and he\nlooks out for a theory.\u2019 Neither of these tests distinguishes the German;\nhe is as likely to seek for a third fact as for a theory, and as likely\nto build a theory on two facts as to look abroad for further information.\nBut once let him have a theory in his mind, and he will ransack heaven\nand earth until he almost buries it under the weight of accumulated\nfacts. This remark applies with more than common force to a treatise\npublished last year by Dr Zeising, a professor in Leipsic, \u2018On a law of\nproportion which rules all nature.\u2019 The ingenious author, after proving\nfrom the writings of ancient and modern philosophers that there always\nexisted the belief (whence derived it is difficult to say), that some\nlaw does bind into one formula all the visible works of God, proceeds to\ncriticise the opinions of individual writers respecting that connecting\nlaw. It is not our purpose to follow him through his lengthy examination.\nSuffice it to say that he believes he has found the lost treasure in the\n_Tim\u00e6us_ of Plato, c. 31. The passage is confessedly an obscure one, and\nwill not bear a literal translation. The interpretation which Dr Zeising\nputs on it is certainly a little strained, but we are disposed to admit\nthat he does it with considerable reason. Agreeably to him, the passage\nruns thus:\u2014\u2018That bond is the most beautiful which binds the things as\nmuch as possible into one; and proportion effects this most perfectly\nwhen three things are so united that the greater bears to the middle the\nsame ratio that the middle bears to the less.\u2019\n\n\u201cWe must do Dr Zeising the justice to say that he has not made more\nthan a legitimate use of the materials which were presented to him in\nthe writings of the ancients, in his endeavour to establish the fact\nof the existence of this law amongst them. The canon of Polycletes,\nthe tradition of Varro mentioned by Pliny relative to that canon, the\nwritings of Galen and others, are all brought to bear on the same point\nwith more or less force. The sum of this portion of the argument is\nfairly this,\u2014that the ancient sculptors had _some_ law of proportion\u2014some\nauthorised examplar to which they referred as their work proceeded. That\nit was the law here attributed to Plato is by no means made out; but,\nconsidering the incidental manner in which that law is referred to, and\nthe obscurity of the passages as they exist, it is, perhaps, too much to\nexpect more than this broad feature of coincidence\u2014the fact that some\nlaw was known and appealed to. Dr Zeising now proceeds to examine modern\ntheories, and it is fair to state that he appears generally to take a\nvery just view of them.\n\n\u201cLet us now turn to Dr Zeising\u2019s own theory. It is this\u2014that in every\nbeautiful form lines are divided in extreme and mean ratio; or, that\nany line considered as a whole, bears to its larger part the same\nproportion that the larger bears to the smaller\u2014thus, a line of 5\ninches will be divided into parts which are very nearly 2 and 3 inches\nrespectively (1\u00b79 and 3\u00b71 inches). This is a well-known division of a\nline, and has been called the GOLDEN rule, but when or why, it is not\neasy to ascertain. With this rule in his hand, Dr Zeising proceeds to\nexamine all nature and art; nay, he even ventures beyond the threshold\nof nature to scan Deity. We will not follow him so far. Let us turn over\nthe pages of his carefully illustrated work, and see how he applies his\nline. We meet first with the Apollo Belvidere\u2014the golden line divides\nhim happily. We cannot say the same of the division of a handsome face\nwhich occurs a little further on. Our preconceived notions have made the\nface terminate with the chin, and not with the centre of the throat.\nIt is evident that, with such a rule as this, a little latitude as to\nthe extreme point of the object to be measured, relieves its inventor\nfrom a world of perplexities. This remark is equally applicable to the\n_arm_ which follows, to which the rule appears to apply admirably, yet\nwe have tried it on sundry plates of arms, both fleshy and bony, without\na shadow of success. Whether the rule was made for the arm or the arm\nfor the rule, we do not pretend to decide. But let us pass hastily on\nto page 284, where the Venus de Medicis and Raphael\u2019s Eve are presented\nto us. They bear the application of the line right well. It might,\nperhaps, be objected that it is remarkable that the same rule applies\nso exactly to the existing position of the figures, such as the Apollo\nand the Venus, the one of which is upright, and the other crouching.\nBut let that pass. We find Dr Zeising next endeavouring to square his\ntheory with the distances of the planets, with wofully scanty success.\nDescending from his lofty position, he spans the earth from corner to\ncorner, at which occupation we will leave him for a moment, whilst we\noffer a suggestion which is equally applicable to poets, painters,\nnovelists, and theorisers. Never err in excess\u2014defect is the safe side\u2014it\nis seldom a fault, often a real merit. Leave something for the student\nof your works to do\u2014don\u2019t chew the cud for him. Be assured he will not\nomit to pay you for every little thing which you have enabled him to\ndiscover. Poor Professor Zeising! he is far too German to leave any\nfield of discovery open for his readers. But let us return to him; we\nleft him on his back, lost for a time in a hopeless attempt to double\nCape Horn. We will be kind to him, as the child is to his man in the\nNoah\u2019s ark, and set him on his legs amongst his toys again. He is now in\nthe vegetable kingdom, amidst oak leaves and sections of the stems of\ndivers plants. He is in his element once more, and it were ungenerous\nnot to admit the merit of his endeavours, and the success which now and\nthen attends it. We will pass over his horses and their riders, together\nwith that portly personage, the Durham ox, for we have caught a glimpse\nof a form familiar to our eyes, the ever-to-be-admired Parthenon. This\nis the true test of a theory. Unlike the Durham ox just passed before\nus, the Parthenon will stand still to be measured. It has so stood for\ntwenty centuries, and every one that has scanned its proportions has\npronounced them exquisite. Beauty is not an adaptation to the acquired\ntaste of a single nation, or the conventionality of a single generation.\nIt emanates from a deep-rooted principle in nature, and appeals to the\nverdict of our whole humanity. We don\u2019t find fault with the Durham ox\u2014his\nproportions are probably good, though they be the result of breeding\nand cross-breeding; still we are not sure whether, in the march of\nagriculture, our grandchildren may not think him a very wretched beast.\nBut there is no mistake about the Parthenon; as a type of proportion it\nstands, has stood, and shall stand. Well, then, let us see how Dr Zeising\nsucceeds with his rule here. Alas! not a single point comes right. The\nParthenon is condemned, or its condemnation condemns the theory. Choose\nyour part. We choose the latter alternative; and now, our choice being\nmade, we need proceed no further. But a question or two have presented\nthemselves as we went along, which demand an answer. It may be asked\u2014How\ndo you account for the esteem in which this law of the section in\nextreme and mean ratio was held? We reply\u2014That it was esteemed just in\nthe same way that a tree is esteemed for its fruit. To divide a right\nangle into two or three, four or six, equal parts was easy enough. But\nto divide it into five or ten such parts was a real difficulty. And how\nwas the difficulty got over? It was effected by means of this golden\nrule. This is its great, its ruling application; and if we adopt the\nnotion that the ancients were possessed with the idea of the existence\nof angular symmetry, we shall have no difficulty in accounting for their\nappreciation of this problem. Nay, we may even go further, and admit,\nwith Dr Zeising, the interpretation of the passage of Plato,\u2014only with\nthis limitation, that Plato, as a geometer, was carried away by the\ngeometry of \u00e6sthetics from the thing itself. It may be asked again\u2014Is it\nnot probable that _some_ proportionality does exist amongst the parts of\nnatural objects? We reply\u2014That, _\u00e0 priori_, we expect some such system\nto exist, but that it is inconsistent with the scheme of _least effort_,\nwhich pervades and characterises all natural succession in space or in\ntime, that that system should be a complicated one. Whatever it is, its\nessence must be simplicity. And no system of simple linear proportion\nis found in nature; quite the contrary. We are, therefore, driven to\nanother hypothesis, viz.\u2014that the simplicity is one of angles, not of\nlines; that the eye estimates by search round a point, not by ascending\nand descending, going to the right and to the left,\u2014a theory which we\nconceive all nature conspires to prove. Beauty was not created for the\neye of man, but the eye of man and his mental eye were created for the\nappreciation of beauty. Examine the forms of animals and plants so minute\nthat nothing short of the most recent improvements in the microscope can\nsucceed in detecting their symmetry; or examine the forms of those little\nsilicious creations which grew thousands of years before Man was placed\non the earth, and, with forms of marvellous and varied beauty, they all\npoint to its source in angular symmetry. This is the keystone of formal\nbeauty, alike in the minutest animalcule, and in the noblest of God\u2019s\nworks, his own image\u2014Man.\u201d\n\n\nTHE END.\n\n BALLANTYNE AND COMPANY, PRINTERS, EDINBURGH.\n\n\n\n\nFOOTNOTES\n\n\n[1] Sir David Brewster.\n\n[2] No. CLVIII., October 1843.\n\n[3] Diogenes La\u00ebrtius\u2019s \u201cLives of the Philosophers,\u201d literally\ntranslated. Bohn: London.\n\n[4] Ibid.\n\n[5] Rose\u2019s \u201cBiographical Dictionary.\u201d\n\n[6] Professor Laycock, now of the University of Edinburgh.\n\n[7] \u201cThe Geometric Beauty of the Human Figure Defined,\u201d &c.\n\n[8] Longman and Co., London.\n\n[9] See Appendix.\n\n[10] \u201cThe Orthographic Beauty of the Parthenon,\u201d &c., and \u201cThe Harmonic\nLaw of Nature applied to Architectural Design.\u201d\n\n[11] For further details, see \u201cHarmonic Law of Nature,\u201d &c.\n\n[12] By a very simple machine, which I have lately invented, an ellipse\nof any given proportions, even to those of (\u00b9\u2044\u2086\u2084), which is the curve of\nthe entases of the columns of the Parthenon (see Plate VII.), and of any\nlength, from half an inch to fifty feet or upwards, may be easily and\ncorrectly described; the length and angle of the required ellipse being\nall that need be given.\n\n[13] \u201cThe Orthographic Beauty of the Parthenon,\u201d &c.\n\n[14] \u201cThe Orthographic Beauty of the Parthenon,\u201d &c.\n\n[15] Ibid.\n\n[16] \u201cThe Harmonic Law of Nature applied to Architectural Design.\u201d\n\n[17] \u201cPhysio-philosophy.\u201d By Dr Oken. Translated by Talk; and published\nby the Ray Society. London, 1848.\n\n[18] \u201cThe Science of those Proportions by which the Human Head and\nCountenance, as represented in Works of ancient Greek Art, are\ndistinguished from those of ordinary Nature.\u201d\n\n[19] \u201cThe Geometric Beauty of the Human Figure Defined,\u201d &c., and \u201cThe\nNatural Principles of Beauty Developed in the Human Figure.\u201d\n\n[20] \u201cThe Geometric Beauty of the Human Figure Defined,\u201d &c.\n\n[21] \u201cEssay on Ornamental Design,\u201d &c., and \u201cThe Geometric Beauty of the\nHuman Figure,\u201d &c.\n\n[22] \u201cA Nomenclature of Colours, applicable to the Arts and Natural\nSciences,\u201d &c., &c.\n\n[23] See pp. 24 and 25.\n\n[24] \u201cThe Principles of Beauty in Colouring Systematised,\u201d Fourteen\nDiagrams, each containing Six Colours and Hues.\n\n\u201cA Nomenclature of Colours,\u201d &c., Forty Diagrams, each containing Twelve\nExamples of Colours, Hues, Tints, and Shades.\n\n\u201cThe Laws of Harmonious Colouring,\u201d &c., One Diagram, containing Eighteen\nColours and Hues.\n\n[25] \u201cEtchings Representing the Best Examples of Grecian and Roman\nArchitectural Ornament, drawn from the Originals,\u201d &c. By Charles\nHeathcote Tatham, Architect. London: Priestly and Weale. 1826.\n\n[26] \u201cThe Orthographic Beauty of the Parthenon,\u201d &c.\n\n[27] \u201cScience of those Proportions,\u201d &c.\n\n[28] Abb\u00e9 Barth\u00e9l\u00e9mie\u2019s \u201cTravels of Anacharsis in Greece,\u201d vol iv., pp.\n193, 195.\n\n[29] Abb\u00e9 Barth\u00e9l\u00e9mie (vol. ii., pp. 168, 169), who cites as his\nauthorities, Cicer. De Nat. Deor., lib. i., cap. ii., t. 2, p. 405;\nJustin Mart., Ovat. ad Gent., p. 10; Aristot. Metaph., lib. i., cap. v.,\nt. 2, p. 845.\n\n[30] M\u00fcller\u2019s \u201cAncient Art and its Remains.\u201d\n\n[31] \u201cAnacharsis\u2019 Travels in Greece.\u201d By the Abb\u00e9 Barth\u00e9l\u00e9mie, vol. ii.,\np. 325.\n\n[32] \u201cAnacharsis\u2019 Travels in Greece.\u201d By the Abb\u00e9 Barth\u00e9l\u00e9mie, vol. vi.,\np. 225. The authorities the Abb\u00e9 quotes are\u2014Quintil., lib. xii., cap. x.,\np. 744; Plin., lib. xxxv., cap. ix., p. 691.\n\n[33] M\u00fcller\u2019s \u201cArch\u00e6ology of Art,\u201d &c.\n\n\n\n\nWorks by the Same Author.\n\n\nI.\n\nIn royal 8vo, with Copperplate Illustrations, price 2s. 6d.,\n\nTHE HARMONIC LAW OF NATURE APPLIED TO ARCHITECTURAL DESIGN.\n\n_From the Athen\u00e6um._\n\nThe beauty of the theory is its universality, and its simplicity. In\nnature, the Creator accomplished his purposes by the simplest means\u2014the\nharmony of nature is indestructible and self-restoring. Mr Hay\u2019s book on\nthe \u201cParthenon,\u201d on the \u201cNatural Principles of Beauty as developed in the\nHuman Figure,\u201d his \u201cPrinciples of Symmetrical Beauty,\u201d his \u201cPrinciples of\nColouring, and Nomenclature of Colours,\u201d his \u201cScience of Proportion,\u201d and\n\u201cEssay on Ornamental Design,\u201d we have already noticed with praise as the\nresults of philosophical and original thought.\n\n_From the Daily News._\n\nThis essay is a new application to Lincoln cathedral in Gothic\narchitecture, and to the Temple of Theseus in Greek architecture, of the\nprinciples of symmetrical beauty already so profusely illustrated and\ndemonstrated by Mr Hay. The theory which Mr Hay has propounded in so many\nvolumes is not only a splendid contribution towards a science of \u00e6sthetic\nproportions, but, for the first time in the history of art, proves the\npossibility, and lays the foundations of such a science. To those who are\nnot acquainted with the facts, these expressions will sound hyperbolical,\nbut they are most true.\n\n\nII.\n\nIn royal 8vo, with Copperplate Illustrations, price 5s.,\n\nTHE NATURAL PRINCIPLES OF BEAUTY, AS DEVELOPED IN THE HUMAN FIGURE.\n\n_From the Spectator._\n\nWe cannot refuse to entertain Mr Hay\u2019s system as of singular intrinsic\nexcellence. The simplicity of his law and its generality impress\nthemselves more deeply on the conviction with each time of enforcement.\nHis theory proceeds from the idea, that in nature every thing is effected\nby means more simple than any other that could have been conceived,\u2014an\nidea certainly consistent with whatever we can trace out or imagine of\nthe all-wise framing of the universe.\n\n_From the Sun._\n\nBy founding (if we may so phrase it) this noble theory, Mr Hay has\ncovered his name with distinction, and has laid the basis, we conceive,\nof no ephemeral reputation. By illustrating it anew, through the\nmedium of this graceful treatise, he has conferred a real boon upon\nthe community, for he has afforded the public another opportunity of\nfollowing the golden rule of the poet\u2014by looking through the holy and\nawful mystery of creation to the holier and yet more awful mystery of\nOmnipotence.\n\n_From the Cambridge Journal of Classical and Sacred Philology._\n\nThe inquiries which of late years have been instituted by Mr D. R. Hay\nof Edinburgh, on the proportions of the human figure, and on the natural\nprinciples of beauty, as illustrated by works of Greek art, constitute an\nepoch in the study of \u00e6sthetics and the philosophy of form.\n\n\nIII.\n\nIn royal 8vo, with Copperplate Illustrations, price 5s.,\n\nTHE ORTHOGRAPHIC BEAUTY OF THE PARTHENON REFERRED TO A LAW OF NATURE.\n\nTo which are prefixed, a few Observations on the Importance of \u00c6sthetic\nScience as an Element in Architectural Education.\n\n_From the Scottish Literary Gazette._\n\nWe think this work will satisfy every impartial mind that Mr Hay has\ndeveloped the true theory of the Parthenon\u2014that he has, in fact, to use\na kindred phraseology, both _parsed_ and _scanned_ this exquisitely\nbeautiful piece of architectural composition, and that, in doing so, he\nhas provided the true key by which the treasures of Greek art may be\nfurther unlocked, and rendered the means of correcting, improving, and\nelevating modern practice.\n\n_From the Edinburgh Guardian._\n\nAgain and again the attempt has been made to detect harmonic ratios in\nthe measurement of Athenian architecture, but ever without reward. Mr Hay\nhas, however, made the discovery, and to an extent of which no one had\npreviously dreamt.\n\n\nIV.\n\nIn 8vo, 100 Plates, price 6s.,\n\nFIRST PRINCIPLES OF SYMMETRICAL BEAUTY.\n\n_From the Spectator._\n\nThis is a grammar of pure form, in which the elements of symmetrical, as\ndistinguished from picturesque beauty, are demonstrated, by reducing the\noutlines or planes of curvilinear and rectilinear forms to their origin\nin the principles of geometrical proportion. In thus analysing symmetry\nof outline in natural and artificial objects, Mr Hay determines the fixed\nprinciples of beauty in positive shape, and shews how beautiful forms may\nbe reproduced and infinitely varied with mathematical precision. Hitherto\nthe originating and copying of beautiful forms have been alike empirical;\nthe production of a new design for a vase or a jug has been a matter of\nchance between the eye and the hand; and the copying of a Greek moulding\nor ornament, a merely mechanical process. By a series of problems, Mr\nHay places both the invention and imitation of beautiful forms on a sure\nbasis of science, giving to the fancy of original minds a clue to the\nevolving of new and elegant shapes, in which the infinite resources of\nnature are made subservient to the uses of art.\n\nThe volume is illustrated by one hundred diagrams beautifully executed,\nthat serve to explain the text, and suggest new ideas of beauty of\ncontour in common objects. To designers of pottery, hardware, and\narchitectural ornaments, this work is particularly valuable; but artists\nof every kind, and workmen of intelligence, will find it of great utility.\n\n_From the Athen\u00e6um._\n\nThe volume before us is the seventh of Mr Hay\u2019s works. It is the most\npractical and systematic, and likely to be one of the most useful.\nIt is, in short, a grammar of form, or a spelling-book of beauty.\nThis is beginning at the right end of the matter; and the necessity\nfor this kind of knowledge will inevitably, though gradually, be\nfelt. The work will, therefore, be ultimately appreciated and\nadopted as an introduction to the study of beautiful forms.\n\nThe third part of the work treats of the Greek oval or composite ellipse,\nas Mr Hay calls it. It is an ellipse of three foci, and gives practical\nforms for vases and architectural mouldings, which are curious to the\narchitect, and will be very useful both to the potter, the moulder,\nand the pattern-drawer. A fourth part contains applications of this to\npractice. Of the details worked out with so much judgment and ingenuity\nby Mr Hay, we should in vain attempt to communicate just notions without\nthe engravings of which his book is full. We must, therefore, refer to\nthe work itself. The forms there given are full of beauty, and so far\ntend to prove the system.\n\n\nV.\n\nIn 8vo, 14 Coloured Diagrams, Second Edition, price 15s.,\n\nTHE PRINCIPLES OF BEAUTY IN COLOURING SYSTEMATISED.\n\n_From the Spectator._\n\nIn this new analysis of the harmonies of colour, Mr Hay has performed the\nuseful service of tracing to the operation of certain fixed principles\nthe sources of beauty in particular combinations of hues and tints; so\nthat artists may, by the aid of this book, produce, with mathematical\ncertainty, the richest effects, hitherto attainable by genius alone. Mr\nHay has reduced this branch of art to a perfect system, and proved that\nan offence against good taste in respect to combinations of colour is, in\neffect, a violation of natural laws.\n\n\nVI.\n\nIn 8vo, 228 Examples of Colours, Hues, Tints, and Shades, price 63s.,\n\nA NOMENCLATURE OF COLOURS, APPLICABLE TO THE ARTS AND NATURAL SCIENCES.\n\n_From the Daily News._\n\nIn this work Mr Hay has brought a larger amount of practical knowledge\nto bear on the subject of colour than any other writer with whom we are\nacquainted, and in proportion to this practical knowledge is, as might be\nexpected, the excellence of his treatise. There is much in this volume\nwhich we would most earnestly recommend to the notice of artists, house\ndecorators, and, indeed, to all whose business or profession requires a\nknowledge of the management of colour. The work is replete with hints\nwhich they might turn to profitable account, and which they will find\nnowhere else.\n\n_From the Athen\u00e6um._\n\nWe have formerly stated the high opinion we entertain of Mr Hay\u2019s\nprevious exertions for the improvement of decorative art in this country.\nWe have already awarded him the merit of invention and creation of the\nnew and the beautiful in form. In his former treatises he furnished a\ntheory of definite proportions for the creation of the beautiful in form.\nIn the present work he proposes to supply a scale of definite proportions\nfor chromatic beauty. For this purpose he sets out very properly with a\nprecise nomenclature of colour. In this he has constructed a vocabulary\nfor the artist\u2014an alphabet for the artizan. He has gone further\u2014he\nconstructs words for three syllables. From this time, it will be possible\nto write a letter in Edinburgh about a coloured composition, which shall\nbe read off in London, Paris, St Petersburg, or Pekin, and shall so\nexpress its nature that it can be reproduced in perfect identity. This Mr\nHay has done, or at least so nearly, as to deserve our thanks on behalf\nof art, and artists of all grades, even to the decorative artizan\u2014not one\nof whom, be he house-painter, china pattern-drawer, or calico printer,\nshould be without the simple manual of \u201cwords for colours.\u201d\n\n\nVII.\n\nIn post 8vo, with a Coloured Diagram, Sixth Edition, price 7s. 6d.,\n\nTHE LAWS OF HARMONIOUS COLOURING ADAPTED TO INTERIOR DECORATIONS.\n\n_From the Atlas._\n\nEvery line of this useful book shews that the author has contrived to\nintellectualise his subject in a very interesting manner. The principles\nof harmony in colour as applied to decorative purposes, are explained and\nenforced in a lucid and practical style, and the relations of the various\ntints and shades to each other, so as to produce a harmonious result, are\ndescanted upon most satisfactorily and originally.\n\n_From the Edinburgh Review._\n\nIn so far as we know, Mr Hay is the first and only modern artist who\nhas entered upon the study of these subjects without the trammels of\nprejudice and authority. Setting aside the ordinances of fashion, as\nwell as the dicta of speculation, he has sought the foundation of\nhis profession in the properties of light, and in the laws of visual\nsensation, by which these properties are recognised and modified. The\ntruths to which he has appealed are fundamental and irrefragable.\n\n_From the Athen\u00e6um._\n\nWe have regarded, and do still regard, the production of Mr Hay\u2019s works\nas a remarkable psychological phenomenon\u2014one which is instructive both\nfor the philosopher and the critic to study with care and interest, not\nunmingled with respect. We see how his mind has been gradually guided\nby Nature herself out of one track, and into another, and ever and anon\nleading him to some vein of the beautiful and true, hitherto unworked.\n\n\nVIII.\n\nIn 4to, 25 Plates, price 36s.,\n\nON THE SCIENCE OF THOSE PROPORTIONS BY WHICH THE HUMAN HEAD AND\nCOUNTENANCE, AS REPRESENTED IN ANCIENT GREEK ART, ARE DISTINGUISHED FROM\nTHOSE OF ORDINARY NATURE.\n\n(PRINTED BY PERMISSION.)\n\n_From a Letter to the Author by Sir William Hamilton, Bart., Professor of\nLogic and Metaphysics in the Edinburgh University._\n\nYour very elegant volume, \u201cScience of those Proportions,\u201d &c., is to me\nextremely interesting, as affording an able contribution to what is the\nancient, and, I conceive, the true theory of the beautiful. But though\nyour doctrine coincides with the one prevalent through all antiquity, it\nappears to me quite independent and original in you; and I esteem it the\nmore that it stands opposed to the hundred one-sided and exclusive views\nprevalent in modern times.\n\n_From Chambers\u2019s Edinburgh Journal._\n\nWe now come to another, and much more remarkable corroboration, which\ncalls upon us to introduce to our readers one of the most valuable and\noriginal contributions that have ever been made to the Philosophy of Art,\nviz., Mr Hay\u2019s work \u201cOn the Science of those Proportions,\u201d &c. Mr Hay\u2019s\nplan is simply to form a scale composed of the well-known vibrations of\nthe monochord, which are the alphabet of music, and then to draw upon the\nquadrant of a circle angles _answering to these vibrations_. With the\nseries of triangles thus obtained he combines a circle and an ellipse,\nthe proportions of which are derived from the triangles themselves; and\nthus he obtains an infallible rule for the composition of the head of\nideal beauty.\n\n\nIX.\n\nIn 4to, 16 Plates, price 30s.,\n\nTHE GEOMETRIC BEAUTY OF THE HUMAN FIGURE DEFINED.\n\nTo which is prefixed, a SYSTEM of \u00c6STHETIC PROPORTION applicable to\nARCHITECTURE and the other FORMATIVE ARTS.\n\n_From the Cambridge Journal of Classical and Sacred Philology._\n\nWe feel bound to pay Mr Hay a large and glad tribute of praise for\nhaving devised a system of proportions which rises superior to the\nidiosyncrasies of different artists, which brings back to one common type\nthe sensations of Eye and Ear, and so makes a giant stride towards that\n_codification_ of the laws of the universe which it is the business of\nscience to effect. We have no hesitation in saying that, for scientific\nprecision of method and importance of results, Albert Durer, Da Vinci,\nand Hogarth\u2014not to mention less noteworthy writers\u2014must all yield the\npalm to Mr Hay.\n\n\nX.\n\nIn oblong folio, 57 Plates and numerous Woodcuts, price 42s.,\n\nAN ESSAY ON ORNAMENTAL DESIGN, IN WHICH ITS TRUE PRINCIPLES ARE DEVELOPED\nAND ELUCIDATED, &c.\n\n_From the Athen\u00e6um._\n\nIn conclusion, Mr Hay\u2019s book goes forth with our best wishes. It must be\ngood. It must be prolific of thought\u2014stimulant of invention. It is to be\nacknowledged as a benefit of an unusual character conferred on the arts\nof ornamental design.\n\n_From the Spectator._\n\nMr Hay has studied the subject deeply and scientifically. In this\ntreatise on ornamental design, the student will find a clue to the\ndiscovery of the source of an endless variety of beautiful forms and\ncombinations of lines, in the application of certain fixed laws of\nharmonious proportion to the purposes of art. Mr Hay also exemplifies\nthe application of his theory of linear harmony to the production of\nbeautiful forms generally, testing its soundness by applying it to the\nhuman figure, and the purest creations of Greek art.\n\n_From Fraser\u2019s Magazine._\n\nEach part of this work is enriched by diagrams of great beauty, direct\nemanations of principle, and, consequently, presenting entirely new\ncombinations of form. Had our space permitted, we should have made some\nextracts from this \u201cEssay on Ornamental Design;\u201d and we would have done\nso, because of the discriminating taste by which it is pervaded, and the\nforcible observations which it contains; but we cannot venture on the\nindulgence.\n\n\nXI.\n\nIn 4to, 17 Plates and 38 Woodcuts, price 25s.,\n\nPROPORTION, OR THE GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED.\n\n\nXII.\n\nIn 4to, 18 Plates and numerous Woodcuts, price 15s.,\n\nTHE NATURAL PRINCIPLES AND ANALOGY OF THE HARMONY OF FORM.\n\n_From the Edinburgh Review._\n\nNotwithstanding some trivial points of difference between Mr Hay\u2019s views\nand our own, we have derived the greatest pleasure from the perusal of\nthese works. They are all composed with accuracy and even elegance. His\nopinions and views are distinctly brought before the reader, and stated\nwith that modesty which characterises genius, and that firmness which\nindicates truth.\n\n_From Blackwood\u2019s Magazine._\n\nWe have no doubt that when Mr Hay\u2019s Art-discovery is duly developed and\ntaught, as it should be, in our schools, it will do more to improve the\ngeneral taste than anything which has yet been devised.","title":"The science of beauty, as developed in nature and applied in art","tradition":"philosophy-ethics","translated_text":null,"word_count":37762}