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+
+*** START OF THE PROJECT GUTENBERG EBOOK 75399 ***
+
+
+
+
+
+THE SCIENCE OF BEAUTY.
+
+                                EDINBURGH:
+                    PRINTED BY BALLANTYNE AND COMPANY,
+                               PAUL’S WORK.
+
+
+
+
+                                   THE
+                            SCIENCE OF BEAUTY,
+                        AS DEVELOPED IN NATURE AND
+                             APPLIED IN ART.
+
+                                    BY
+                           D. R. HAY, F.R.S.E.
+
+    “The irregular combinations of fanciful invention may delight
+    awhile, by that novelty of which the common satiety of life
+    sends us all in quest; the pleasures of sudden wonder are soon
+    exhausted, and the mind can only repose on the stability of
+    truth.”
+
+                                                         DR JOHNSON.
+
+                       WILLIAM BLACKWOOD AND SONS,
+                          EDINBURGH AND LONDON.
+                                MDCCCLVI.
+
+
+
+
+                                    TO
+                  JOHN GOODSIR, ESQ., F.R.S S. L. & E.,
+           PROFESSOR OF ANATOMY IN THE UNIVERSITY OF EDINBURGH,
+          AS AN EXPRESSION OF GRATITUDE FOR VALUABLE ASSISTANCE,
+                AS ALSO OF HIGH ESTEEM AND SINCERE REGARD,
+                        THIS VOLUME IS DEDICATED,
+                                    BY
+
+                                                               THE AUTHOR.
+
+
+
+
+PREFACE.
+
+
+My theory of beauty in form and colour being now admitted by the best
+authorities to be based on truth, I have of late been often asked, by
+those who wished to become acquainted with its nature, and the manner of
+its being applied in art, which of my publications I would recommend for
+their perusal. This question I have always found difficulty in answering;
+for although the law upon which my theory is based is characterised by
+unity, yet the subjects in which it is applied, and the modes of its
+application, are equally characterised by variety, and consequently
+occupy several volumes.
+
+Under these circumstances, I consulted a highly respected friend, whose
+mathematical talents and good taste are well known, and to whom I have
+been greatly indebted for much valuable assistance during the course
+of my investigations. The advice I received on this occasion, was to
+publish a _résumé_ of my former works, of such a character as not only
+to explain the nature of my theory, but to exhibit to the general reader,
+by the most simple modes of illustration and description, how it is
+developed in nature, and how it may be extensively and with ease applied
+in those arts in which beauty forms an essential element.
+
+The following pages, with their illustrations, are the results of an
+attempt to accomplish this object.
+
+To those who are already acquainted, through my former works, with
+the nature, scope, and tendency of my theory, I have the satisfaction
+to intimate that I have been enabled to include in this _résumé_ much
+original matter, with reference both to form and colour.
+
+                                                                D. R. HAY.
+
+
+
+
+CONTENTS.
+
+
+                                                                       PAGE
+
+  INTRODUCTION                                                            1
+
+  THE SCIENCE OF BEAUTY, EVOLVED FROM THE HARMONIC LAW OF NATURE,
+    AGREEABLY TO THE PYTHAGOREAN SYSTEM OF NUMERICAL RATIO               15
+
+  THE SCIENCE OF BEAUTY, AS APPLIED TO SOUNDS                            28
+
+  THE SCIENCE OF BEAUTY, AS APPLIED TO FORMS                             34
+
+  THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE FORM OF THE HUMAN HEAD
+    AND COUNTENANCE                                                      54
+
+  THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE FORM OF THE HUMAN FIGURE    61
+
+  THE SCIENCE OF BEAUTY, AS DEVELOPED IN COLOURS                         67
+
+  THE SCIENCE OF BEAUTY APPLIED TO THE FORMS AND PROPORTIONS OF
+    ANCIENT GRECIAN VASES AND ORNAMENTS                                  82
+
+  APPENDIX, NO. I.                                                       91
+
+  APPENDIX, NO. II.                                                      99
+
+  APPENDIX, NO. III.                                                    100
+
+  APPENDIX, NO. IV.                                                     100
+
+  APPENDIX, NO. V.                                                      104
+
+  APPENDIX, NO. VI.                                                     105
+
+
+
+
+ILLUSTRATIONS.
+
+
+PLATES
+
+[Illustration: I. Three Scales of the Elementary Rectilinear Figures,
+viz., the Scalene Triangle, the Isosceles Triangle, and the Rectangle,
+comprising twenty-seven varieties of each, according to the harmonic
+parts of the Right Angle from ¹⁄₂ to ¹⁄₁₆.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: II. Diagram of the Rectilinear Orthography of the
+Principal Front of the Parthenon of Athens, in which its Proportions are
+determined by harmonic parts of the Right Angle.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: III. Diagram of the Rectilinear Orthography of the
+Portico of the Temple of Theseus at Athens, in which its Proportions are
+determined by harmonic parts of the Right Angle.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: IV. Diagram of the Rectilinear Orthography of the East End
+of Lincoln Cathedral, in which its Proportions are determined by harmonic
+parts of the Right Angle.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: V. Four Ellipses described from Foci, determined by
+harmonic parts of the Right Angle, shewing in each the Scalene Triangle,
+the Isosceles Triangle, and the Rectangle to which it belongs.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: VI. The Composite Ellipse of ¹⁄₆ and ¹⁄₈ of the Right
+Angle, shewing its greater and lesser Axis, its various Foci, and the
+Isosceles Triangle in which they are placed.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: VII. The Composite Ellipse of ¹⁄₄₈ and ¹⁄₆₄ of the Right
+Angle, shewing how it forms the Entasis of the Columns of the Parthenon
+of Athens.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: VIII. Sectional Outlines of two Mouldings of the Parthenon
+of Athens, full size, shewing the harmonic nature of their Curves, and
+the simple manner of their Construction.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: IX. Three Diagrams, giving a Vertical, a Front, and a Side
+Aspect of the Geometrical Construction of the Human Head and Countenance,
+in which the Proportions are determined by harmonic parts of the Right
+Angle.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: X. Diagram in which the Symmetrical Proportions of the
+Human Figure are determined by harmonic parts of the Right Angle.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: XI. The Contour of the Human Figure as viewed in Front
+and in Profile, its Curves being determined by Ellipses, whose Foci are
+determined by harmonic parts of the Right Angle.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: XII. Rectilinear Diagram, shewing the Proportions of the
+Portland Vase, as determined by harmonic parts of the Right Angle, and
+the outline of its form by an Elliptic Curve harmonically described.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: XIII. Rectilinear Diagram of the Proportions and
+Curvilinear Outline of the form of an ancient Grecian Vase, the
+proportions determined by harmonic parts of the Right Angle, and the
+melody of the form by Curves of two Ellipses.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: XIV. Rectilinear Diagram of the Proportions and
+Curvilinear Outline of the form an ancient Grecian Vase, the proportions
+determined by harmonic parts of the Right Angle, and the melody of the
+form by an Elliptic Curve.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: XV. Two Diagrams of Etruscan Vases, the harmony of
+Proportions and melody of the Contour determined, respectively, by parts
+of the Right Angle and an Elliptic Curve.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: XVI. Two Diagrams of Etruscan Vases, whose harmony of
+Proportion and melody of Contour are determined as above.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: XVII. Diagram shewing the Geometric Construction of an
+Ornament belonging to the Parthenon at Athens.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: XVIII. Diagram of the Geometrical Construction of the
+ancient Grecian Ornament called the Honeysuckle.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: XIX. An additional Illustration of the Contour of the
+Human Figure, as viewed in Front and in Profile.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: XX. Diagram shewing the manner in which the Elliptic
+Curves are arranged in order to produce an Outline of the Form of the
+Human Figure as viewed in Front.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: XXI. Diagram of a variation on the Form of the Portland
+Vase.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: XXII. Diagram of a second variation on the Form of the
+Portland Vase.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+[Illustration: XXIII. Diagram of a third variation on the Form of the
+Portland Vase.
+
+_D. R. Hay delᵗ._ _G. Aikman sc._]
+
+
+
+
+INTRODUCTION.
+
+
+Twelve years ago, one of our most eminent philosophers,[1] through the
+medium of the _Edinburgh Review_,[2] gave the following account of what
+was then the state of the fine arts as connected with science:—“The
+disposition to introduce into the intellectual community the principles
+of free intercourse, is by no means general; but we are confident that
+Art will not sufficiently develop her powers, nor Science attain her
+most commanding position, till the practical knowledge of the one is
+taken in return for the sound deductions of the other.... It is in the
+fine arts, principally, and in the speculations with which they are
+associated, that the controlling power of scientific truth has not
+exercised its legitimate influence. In discussing the principles of
+painting, sculpture, architecture, and landscape gardening, philosophers
+have renounced science as a guide, and even as an auxiliary; and a
+school has arisen whose speculations will brook no restraint, and whose
+decisions stand in opposition to the strongest convictions of our
+senses. That the external world, in its gay colours and lovely forms, is
+exhibited to the mind only as a tinted mass, neither within nor without
+the eye, neither touching it nor distant from it—an ubiquitous chaos,
+which experience only can analyse and transform into the realities
+which compose it; that the beautiful and sublime in nature and in art
+derive their power over the mind from association alone, are among the
+philosophical doctrines of the present day, which, if it be safe, it is
+scarcely prudent to question. Nor are these opinions the emanations of
+poetical or ill-trained minds, which ingenuity has elaborated, and which
+fashion sustains. They are conclusions at which most of our distinguished
+philosophers have arrived. They have been given to the world with all the
+authority of demonstrated truth; and in proportion to the hold which they
+have taken of the public mind, have they operated as a check upon the
+progress of knowledge.”
+
+Such, then, was the state of art as connected with science twelve years
+ago. But although the causes which then placed science and the fine arts
+at variance have since been gradually diminishing, yet they are still
+far from being removed. In proof of this I may refer to what took place
+at the annual distribution of the prizes to the students attending our
+Scottish Metropolitan School of Design, in 1854, the pupils in which
+amount to upwards of two hundred. The meeting on that occasion included,
+besides the pupils, a numerous and highly respectable assemblage of
+artists and men of science. The chairman, a Professor in our University,
+and editor of one of the most voluminous works on art, science, and
+literature ever produced in this country, after extolling the general
+progress of the pupils, so far as evinced by the drawings exhibited on
+the occasion, drew the attention of the meeting to a discovery made by
+the head master of the architectural and ornamental department of the
+school, viz.—That the ground-plan of the Parthenon at Athens had been
+constructed by the application of the _mysterious_ ovoid or _Vesica
+Piscis_ of the middle ages, subdivided by the _mythic_ numbers 3 and 7,
+and their intermediate odd number 5. Now, it may be remarked, that the
+figure thus referred to is not an ovoid, neither is it in any way of a
+mysterious nature, being produced simply by two equal circles cutting
+each other in their centres. Neither can it be shewn that the numbers 3
+and 7 are in any way more mythic than other numbers. In fact, the terms
+_mysterious_ and _mythic_ so applied, can only be regarded as a remnant
+of an ancient terminology, calculated to obscure the simplicity of
+scientific truth, and when used by those employed to teach—for doubtless
+the chairman only gave the description he received—must tend to retard
+the connexion of that truth with the arts of design. I shall now give
+a specimen of the manner in which a knowledge of the philosophy of the
+fine arts is at present inculcated upon the public mind generally. In
+the same metropolis there has likewise existed for upwards of ten years
+a Philosophical Institution of great importance and utility, whose
+members amount to nearly three thousand, embracing a large proportion
+of the higher classes of society, both in respect to talent and wealth.
+At the close of the session of this Institution, in 1854, a learned and
+eloquent philologus, who occasionally lectures upon beauty, was appointed
+to deliver the closing address, and touching upon the subject of the
+beautiful, he thus concluded—
+
+“In the worship of the beautiful, and in that alone, we are inferior to
+the Greeks. Let us therefore be glad to borrow from them; not slavishly,
+but with a wise adaptation—not exclusively, but with a cunning selection;
+in art, as in religion, let us learn to prove all things, and hold fast
+that which is good—not merely one thing which is good, but all good
+things—Classicalism, Mediævalism, Modernism—let us have and hold them
+all in one wide and lusty embrace. Why should the world of art be
+more narrow, more monotonous, than the world of nature? Did God make
+all the flowers of one pattern, to please the devotees of the rose or
+the lily; and did He make all the hills, with the green folds of their
+queenly mantles, all at one slope, to suit the angleometer of the most
+mathematical of decorators? I trow not. Let us go and do likewise.”
+
+I here take for granted, that what the lecturer meant by “the worship of
+the beautiful,” is the production and appreciation of works of art in
+which beauty should be a primary element; and judging from the remains
+which we possess of such works as were produced by the ancient Grecians,
+our inferiority to them in these respects cannot certainly be denied.
+But I must reiterate what I have often before asserted, that it is
+not by borrowing from them, however cunning our selection, or however
+wise our adaptations, that this inferiority is to be removed, but by
+a re-discovery of the science which these ancient artists must have
+employed in the production of that symmetrical beauty and chaste elegance
+which pervaded all their works for a period of nearly three hundred
+years. And I hold, that as in religion, so in art, there is only one
+truth, a grain of which is worth any amount of philological eloquence.
+
+I also take for granted, that what is meant by Classicalism in the
+above quotation, is the ancient Grecian style of art; by Mediævalism,
+the semi-barbaric style of the middle ages; and by Modernism, that
+chaotic jumble of all previous styles and fashions of art, which is the
+peculiar characteristic of our present school, and which is, doubtless,
+the result of a system of education based upon plagiarism and mere
+imitation. Therefore a recommendation to embrace with equal fervour “as
+good things,” these very opposite artic_isms_ must be a doctrine as
+mischievous in art as it would be in religion to recommend as equally
+good things the various _isms_ into which it has also been split in
+modern times.
+
+Now, “the world of nature” and “the world of art” have not that equality
+of scope which this lecturer on beauty ascribes to them, but differ very
+decidedly in that particular. Neither will it be difficult to shew why
+“the world of art _should_ be more narrow than the world of nature”—that
+it should be thereby rendered more monotonous does not follow.
+
+It is well known, that the “world of nature” consists of productions,
+including objects of every degree of beauty from the very lowest to the
+highest, and calculated to suit not only the tastes arising from various
+degrees of intellect, but those arising from the natural instincts of the
+lower animals. On the other hand, “the world of art,” being devoted to
+the gratification and improvement of intelligent minds only, is therefore
+narrowed in its scope by the exclusion from its productions of the lower
+degrees of beauty—even mediocrity is inadmissible; and we know that the
+science of the ancient Greek artists enabled them to excel the highest
+individual productions of nature in the perfection of symmetrical beauty.
+Consequently, all objects in nature are not equally well adapted for
+artistic study, and it therefore requires, on the part of the artist,
+besides true genius, much experience and care to enable him to choose
+proper subjects from nature; and it is in the choice of such subjects,
+and not in plagiarism from the ancients, that he should select with
+knowledge and adapt with wisdom. Hence, all such latitudinarian doctrines
+as those I have quoted must act as a check upon the progress of knowledge
+in the scientific truth of art. I have observed in some of my works, that
+in this country a course had been followed in our search for the true
+science of beauty not differing from that by which the alchymists of the
+middle ages conducted their investigations; for our ideas of visible
+beauty are still undefined, and our attempts to produce it in the various
+branches of art are left dependant, in a great measure, upon chance.
+Our schools are conducted without reference to any first principles or
+definite laws of beauty, and from the drawing of a simple architectural
+moulding to the intricate combinations of form in the human figure, the
+pupils trust to their hands and eyes alone, servilely and mechanically
+copying the works of the ancients, instead of being instructed in the
+unerring principles of science, upon which the beauty of those works
+normally depends. The instruction they receive is imparted without
+reference to the judgment or understanding, and they are thereby led to
+imitate effects without investigating causes. Doubtless, men of great
+genius sometimes arrive at excellence in the arts of design without a
+knowledge of the principles upon which beauty of form is based; but it
+should be kept in mind, that true genius includes an intuitive perception
+of those principles along with its creative power. It is, therefore, to
+the generality of mankind that instruction in the definite laws of beauty
+will be of most service, not only in improving the practice of those who
+follow the arts professionally, but in enabling all of us to distinguish
+the true from the false, and to exercise a sound and discriminating
+taste in forming our judgment upon artistic productions. Æsthetic
+culture should consequently supersede servile copying, as the basis of
+instruction in our schools of art. Many teachers of drawing, however,
+still assert, that, by copying the great works of the ancients, the mind
+of the pupil will become imbued with ideas similar to theirs—that he
+will imbibe their feeling for the beautiful, and thereby become inspired
+with their genius, and think as they thought. To study carefully and
+to investigate the principles which constitute the excellence of the
+works of the ancients, is no doubt of much benefit to the student; but
+it would be as unreasonable to suppose that he should become inspired
+with artistic genius by merely copying them, as it would be to imagine,
+that, in literature, poetic inspiration could be created by making boys
+transcribe or repeat the works of the ancient poets. Sir Joshua Reynolds
+considered copying as a delusive kind of industry, and has observed, that
+“Nature herself is not to be too closely copied,” asserting that “there
+are excellences in the art of painting beyond what is commonly called the
+imitation of nature,” and that “a mere copier of nature can never produce
+any thing great.” Proclus, an eminent philosopher and mathematician of
+the later Platonist school (A.D. 485), says, that “he who takes for his
+model such forms as nature produces, and confines himself to an exact
+imitation of these, will never attain to what is perfectly beautiful. For
+the works of nature are full of disproportion, and fall very short of the
+true standard of beauty.”
+
+It is remarked by Mr. J. C. Daniel, in the introduction to his
+translation of M. Victor Cousin’s “Philosophy of the Beautiful,” that
+“the English writers have advocated no theory which allows the beautiful
+to be universal and absolute; nor have they professedly founded their
+views on original and ultimate principles. Thus the doctrine of the
+English school has for the most part been, that beauty is mutable and
+special, and the inference that has been drawn from this teaching is,
+that all tastes are equally just, provided that each man speaks of what
+he feels.” He then observes, that the German, and some of the French
+writers, have thought far differently; for with them the beautiful is
+“simple, immutable, absolute, though its _forms_ are manifold.”
+
+So far back as the year 1725, the same truths advanced by the modern
+German and French writers, and so eloquently illustrated by M. Cousin,
+were given to the world by Hutchison in his “Inquiry into the Original
+of our Ideas of Beauty and Virtue.” This author says—“We, by absolute
+beauty, understand only that beauty which we perceive in objects, without
+comparison to any thing external, of which the object is supposed an
+imitation or picture, such as the beauty perceived from the works of
+nature, artificial forms, figures, theorems. Comparative or relative
+beauty is that which we perceive in objects commonly considered as
+imitations or resemblances of something else.”
+
+Dr. Reid also, in his “Intellectual Powers of Man,” says—“That taste,
+which we may call rational, is that part of our constitution by which we
+are made to receive pleasure from the contemplation of what we conceive
+to be excellent in its kind, the pleasure being annexed to this judgment,
+and regulated by it. This taste may be true or false, according as it is
+founded on a true or false judgment. And if it may be true or false, it
+must have first principles.”
+
+M. Victor Cousin’s opinion upon this subject is, however, still more
+conclusive. He observes—“If the idea of the beautiful is not absolute,
+like the idea of the true—if it is nothing more than the expression of
+individual sentiment, the rebound of a changing sensation, or the result
+of each person’s fancy—then the discussions on the fine arts waver
+without support, and will never end. For a theory of the fine arts to
+be possible, there must be something absolute in beauty, just as there
+must be something absolute in the idea of goodness, to render morals a
+possible science.”
+
+The basis of the science of beauty must thus be founded upon fixed
+principles, and when these principles are evolved with the same care
+which has characterised the labours of investigators in natural science,
+and are applied in the fine arts as the natural sciences have been in
+the useful arts, a solid foundation will be laid, not only for correct
+practice, but also for a just appreciation of productions in every branch
+of the arts of design.
+
+We know that the mind receives pleasure through the sense of hearing,
+not only from the music of nature, but from the euphony of prosaic
+composition, the rhythm of poetic measure, the artistic composition
+of successive harmony in simple melody, and the combined harmony of
+counterpoint in the more complex works of that art. We know, also,
+that the mind is similarly gratified through the sense of seeing, not
+only by the visible beauties of nature, but by those of art, whether
+in symmetrical or picturesque compositions of forms, or in harmonious
+arrangements of gay or sombre colouring.
+
+Now, in respect to the first of these modes of sensation, we know,
+that from the time of Pythagoras, the fact has been established, that
+in whatever manner nature or art may address the ear, the degree of
+obedience paid to the fundamental law of harmony will determine the
+presence and degree of that beauty with which a perfect organ can impress
+a well-constituted mind; and it is my object in this, as it has been
+in former attempts, to prove it consistent with scientific truth, that
+that beauty which is addressed to the mind by objects of nature and art,
+through the eye, is similarly governed. In short, to shew that, as in
+compositions of sounds, there can be no true beauty in the absence of a
+strict obedience to this great law of nature, neither can there exist,
+in compositions of forms or colours, that principle of unity in variety
+which constitutes beauty, unless such compositions are governed by the
+same law.
+
+Although in the songs of birds, the gurgling of brooks, the sighing of
+the gentle summer winds, and all the other beautiful music of nature,
+no analysis might be able to detect the operation of any precise system
+of harmony, yet the pleasure thus afforded to the human mind we know to
+arise from its responding to every development of an obedience to this
+law. When, in like manner, we find even in those compositions of forms
+and colours which constitute the wildest and most rugged of Nature’s
+scenery, a species of picturesque grandeur and beauty to which the mind
+as readily responds as to her more mild and pleasing aspects, or to her
+sweetest music, we may rest assured that this beauty is simply another
+development of, and response to, the same harmonic law, although the
+precise nature of its operation may be too subtle to be easily detected.
+
+The _résumé_ of the various works I have already published upon the
+subject, along with the additional illustrations I am about to lay before
+my readers, will, I trust, point out a system of harmony, which, in
+formative art, as well as in that of colouring, will rise superior to the
+idiosyncracies of different artists, and bring back to one common type
+the sensations of the eye and the ear, thereby improving that knowledge
+of the laws of the universe which it is as much the business of science
+to combine with the ornamental as with the useful arts.
+
+In attempting this, however, I beg it may be understood, that I do not
+believe any system, based even upon the laws of nature, capable of
+forming a royal road to the perfection of art, or of “mapping the mighty
+maze of a creative mind.” At the same time, however, I must continue to
+reiterate the fact, that the diffusion of a general knowledge of the
+science of visible beauty will afford latent artistic genius just such a
+vantage ground as that which the general knowledge of philology diffused
+throughout this country affords its latent literary genius. Although
+_mere learning_ and _true genius_ differ as much in the practice of art
+as they do in the practice of literature, yet a precise and systematic
+education in the true science of beauty must certainly be as useful in
+promoting the practice and appreciation of the one, as a precise and
+systematic education in the science of philology is in promoting the
+practice and appreciation of the other.
+
+As all beauty is the result of harmony, it will be requisite here to
+remark, that harmony is not a simple quality, but, as Aristotle defines
+it, “the union of contrary principles having a ratio to each other.”
+Harmony thus operates in the production of all that is beautiful in
+nature, whether in the combinations, in the motions, or in the affinities
+of the elements of matter.
+
+The contrary principles to which Aristotle alludes, are those of
+uniformity and variety; for, according to the predominance of the one or
+the other of these principles, every kind of beauty is characterised.
+Hence the difference between symmetrical and picturesque beauty:—the
+first allied to the principle of uniformity, in being based upon precise
+laws that may be taught so as to enable men of ordinary capacity to
+produce it in their works—the second allied to the principle of variety
+often to so great a degree that they yield an obedience to the precise
+principles of harmony so subtilely, that they cannot be detected in its
+constitution, but are only felt in the response by which true genius
+acknowledges their presence. The generality of mankind may be capable
+of perceiving this latter kind of beauty, and of feeling its effects
+upon the mind, but men of genius, only, can impart it to works of art,
+whether addressed to the eye or the ear. Throughout the sounds, forms,
+and colours of nature, these two kinds of beauty are found not only in
+distinct developments, but in every degree of amalgamation. We find in
+the songs of some birds, such as those of the chaffinch, thrush, &c.,
+a rhythmical division, resembling in some measure the symmetrically
+precise arrangements of parts which characterises all artistic musical
+composition; while in the songs of other birds, and in the other numerous
+melodies with which nature charms and soothes the mind, there is no
+distinct regularity in the division of their parts. In the forms of
+nature, too, we find amongst the innumerable flowers with which the
+surface of the earth is so profusely decorated, an almost endless variety
+of systematic arrangements of beautiful figures, often so perfectly
+symmetrical in their combination, that the most careful application
+of the angleometer could scarcely detect the slightest deviation from
+geometrical precision; while, amongst the masses of foliage by which
+the forms of many trees are divided and subdivided into parts, as also
+amongst the hills and valleys, the mountains and ravines, which divide
+the earth’s surface, we find in every possible variety of aspect the
+beauty produced by that irregular species of symmetry which characterises
+the picturesque.
+
+In like manner, we find in wild as well as cultivated flowers the
+most symmetrical distributions of colours accompanying an equally
+precise species of harmony in their various kinds of contrasts, often
+as mathematically regular as the geometric diagrams by which writers
+upon colour sometimes illustrate their works; while in the general
+colouring of the picturesque beauties of nature, there is an endless
+variety in its distributions, its blendings, and its modifications. In
+the forms and colouring of animals, too, the same endless variety of
+regular and irregular symmetry is to be found. But the highest degree of
+beauty in nature is the result of an equal balance of uniformity with
+variety. Of this the human figure is an example; because, when it is
+of those proportions universally acknowledged to be the most perfect,
+its uniformity bears to its variety an apparently equal ratio. The
+harmony of combination in the normal proportions of its parts, and the
+beautifully simple harmony of succession in the normal melody of its
+softly undulating outline, are the perfection of symmetrical beauty,
+while the innumerable changes upon the contour which arise from the
+actions and attitudes occasioned by the various emotions of the mind,
+are calculated to produce every species of picturesque beauty, from the
+softest and most pleasing to the grandest and most sublime.
+
+Amongst the purely picturesque objects of inanimate nature, I may, as in
+a former work, instance an ancient oak tree, for its beauty is enhanced
+by want of apparent symmetry. Thus, the more fantastically crooked its
+branches, and the greater the dissimilarity and variety it exhibits in
+its masses of foliage, the more beautiful it appears to the artist and
+the amateur; and, as in the human figure, any attempt to produce variety
+in the proportions of its lateral halves would be destructive of its
+symmetrical beauty, so in the oak tree any attempt to produce palpable
+similarity between any of its opposite sides would equally deteriorate
+its picturesque beauty. But picturesque beauty is not the result of the
+total absence of symmetry; for, as none of the irregularly constructed
+music of nature could be pleasing to the ear unless there existed in
+the arrangement of its notes an obedience, however subtle, to the great
+harmonic law of Nature, so neither could any object be picturesquely
+beautiful, unless the arrangement of its parts yields, although it may be
+obscurely, an obedience to the same law.
+
+However symmetrically beautiful any architectural structure may be, when
+in a complete and perfect state, it must, as it proceeds towards ruin,
+blend the picturesque with the symmetrical; but the type of its beauty
+will continue to be the latter, so long as a sufficient portion of it
+remains to convey an idea of its original perfection. It is the same with
+the human form and countenance; for age does not destroy their original
+beauty, but in both only lessens that which is symmetrical, while it
+increases that which is picturesque.
+
+In short, as a variety of simultaneously produced sounds, which do not
+relate to each other agreeably to this law, can only convey to the mind
+a feeling of mere noise; so a variety of forms or colours simultaneously
+exposed to the eye under similar circumstances, can only convey to the
+mind a feeling of chaotic confusion, or what may be termed _visible_
+discord. As, therefore, the two principles of uniformity and variety,
+or similarity and dissimilarity, are in operation in every harmonious
+combination of the elements of sound, of form, and of colour, we must
+first have recourse to numbers in the abstract before we can form a
+proper basis for a universal science of beauty.
+
+
+
+
+THE SCIENCE OF BEAUTY EVOLVED FROM THE HARMONIC LAW OF NATURE, AGREEABLY
+TO THE PYTHAGOREAN SYSTEM OF NUMERICAL RATIO.
+
+
+The scientific principles of beauty appear to have been well known to the
+ancient Greeks; and it must have been by the practical application of
+that knowledge to the arts of Design, that that people continued for a
+period of upwards of three hundred years to execute, in every department
+of these arts, works surpassing in chaste beauty any that had ever before
+appeared, and which have not been equalled during the two thousand years
+which have since elapsed.
+
+Æsthetic science, as the science of beauty is now termed, is based
+upon that great harmonic law of nature which pervades and governs the
+universe. It is in its nature neither absolutely physical nor absolutely
+metaphysical, but of an intermediate nature, assimilating in various
+degrees, more or less, to one or other of those opposite kinds of
+science. It specially embodies the inherent principles which govern
+impressions made upon the mind through the senses of hearing and seeing.
+Thus, the æsthetic pleasure derived from listening to the beautiful in
+musical composition, and from contemplating the beautiful in works of
+formative art, is in both cases simply a response in the human mind to
+artistic developments of the great harmonic law upon which the science
+is based.
+
+Although the eye and the ear are two different senses, and, consequently,
+various in their modes of receiving impressions; yet the sensorium is
+but one, and the mind by which these impressions are perceived and
+appreciated is also characterised by unity. There appears, likewise,
+a striking analogy between the natural constitution of the two kinds
+of beauty, which is this, that the more physically æsthetic elements
+of the highest works of musical composition are melody, harmony, and
+tone, whilst those of the highest works of formative art are contour,
+proportion, and colour. The melody or theme of a musical composition
+and its harmony are respectively analogous,—1st, To the outline of
+an artistic work of formative art; and 2d, To the proportion which
+exists amongst its parts. To the careful investigator these analogies
+become identities in their effect upon the mind, like those of the more
+metaphysically æsthetic emotions produced by expression in either of
+these arts.
+
+Agreeably to the first analogy, the outline and contour of an object,
+suppose that of a building in shade when viewed against a light
+background, has a similar effect upon the mind with that of the simple
+melody of a musical composition when addressed to the ear unaccompanied
+by the combined harmony of counterpoint. Agreeably to the second analogy,
+the various parts into which the surface of the supposed elevation is
+divided being simultaneously presented to the eye, will, if arranged
+agreeably to the same great law, affect the mind like that of an equally
+harmonious arrangement of musical notes accompanying the supposed melody.
+
+There is, however, a difference between the construction of these two
+organs of sense, viz., that the ear must in a great degree receive its
+impressions involuntarily; while the eye, on the other hand, is provided
+by nature with the power of either dwelling upon, or instantly shutting
+out or withdrawing itself from an object. The impression of a sound,
+whether simple or complex, when made upon the ear, is instantaneously
+conveyed to the mind; but when the sound ceases, the power of observation
+also ceases. But the eye can dwell upon objects presented to it so long
+as they are allowed to remain pictured on the retina; and the mind has
+thereby the power of leisurely examining and comparing them. Hence the
+ear guides more as a mere sense, at once and without reflection; whilst
+the eye, receiving its impressions gradually, and part by part, is more
+directly under the influence of mental analysis, consequently producing
+a more metaphysically æsthetic emotion. Hence, also, the acquired power
+of the mind in appreciating impressions made upon it through the organ of
+sight under circumstances, such as perspective, &c., which to those who
+take a hasty view of the subject appear impossible.
+
+Dealing as this science therefore does, alike with the sources and the
+resulting principles of beauty, it is scarcely less dependent on the
+accuracy of the senses than on the power of the understanding, inasmuch
+as the effect which it produces is as essential a property of objects,
+as are its laws inherent in the human mind. It necessarily comprehends a
+knowledge of those first principles in art, by which certain combinations
+of sounds, forms, and colours produce an effect upon the mind, connected,
+in the first instance, with sensation, and in the second with the
+reasoning faculty. It is, therefore, not only the basis of all true
+practice in art, but of all sound judgment on questions of artistic
+criticism, and necessarily includes those laws whereon a correct taste
+must be based. Doubtless many eloquent and ingenious treatises have been
+written upon beauty and taste; but in nearly every case, with no other
+effect than that of involving the subject in still greater uncertainty.
+Even when restricted to the arts of design, they have failed to exhibit
+any definite principles whereby the true may be distinguished from
+the false, and some natural and recognised laws of beauty reduced to
+demonstration. This may be attributed, in a great degree, to the neglect
+of a just discrimination between what is merely agreeable, or capable
+of exciting pleasurable sensations, and what is essentially beautiful;
+but still more to the confounding of the operations of the understanding
+with those of the imagination. Very slight reflection, however, will
+suffice to shew how essentially distinct these two faculties of the mind
+are; the former being regulated, in matters of taste, by irrefragable
+principles existing in nature, and responded to by an inherent principle
+existing in the human mind; while the latter operates in the production
+of ideal combinations of its own creation, altogether independent of any
+immediate impression made upon the senses. The beauty of a flower, for
+example, or of a dew-drop, depends on certain combinations of form and
+colour, manifestly referable to definite and systematic, though it may
+be unrecognised, laws; but when Oberon, in “Midsummer Night’s Dream,” is
+made to exclaim—
+
+    “And that same dew, which sometimes on the buds
+    Was wont to swell, like round and orient pearls,
+    Stood now within the pretty floweret’s eyes,
+    Like tears that did their own disgrace bewail,”—
+
+the poet introduces a new element of beauty equally legitimate, yet
+altogether distinct from, although accompanying that which constitutes
+the more precise science of æsthetics as here defined. The composition
+of the rhythm is an operation of the understanding, but the beauty of the
+poetic fancy is an operation of the imagination.
+
+Our physical and mental powers, æsthetically considered, may therefore be
+classed under three heads, in their relation to the fine arts, viz., the
+receptive, the perceptive, and the conceptive.
+
+The senses of hearing and seeing are respectively, in the degree of
+their physical power, receptive of impressions made upon them, and of
+these impressions the sensorium, in the degree of its mental power, is
+perceptive. This perception enables the mind to form a judgment whereby
+it appreciates the nature and quality of the impression originally made
+on the receptive organ. The mode of this operation is intuitive, and
+the quickness and accuracy with which the nature and quality of the
+impression is apprehended, will be in the degree of the intellectual
+vigour of the mind by which it is perceived. Thus we are, by the
+cultivation of these intuitive faculties, enabled to decide with accuracy
+as to harmony or discord, proportion or deformity, and assign sound
+reasons for our judgment in matters of taste. But mental conception is
+the intuitive power of constructing original ideas from these materials;
+for after the receptive power has acted, the perception operates in
+establishing facts, and then the judgment is formed upon these operations
+by the reasoning powers, which lead, in their turn, to the creations of
+the imagination.
+
+The power of forming these creations is the true characteristic
+of genius, and determines the point at which art is placed beyond
+all determinable canons,—at which, indeed, æsthetics give place to
+metaphysics.
+
+In the science of beauty, therefore, the human mind is the subject, and
+the effect of external nature, as well as of works of art, the object.
+The external world, and the individual mind, with all that lies within
+the scope of its powers, may be considered as two separate existences,
+having a distinct relation to each other. The subject is affected by the
+object, through that inherent faculty by which it is enabled to respond
+to every development of the all-governing harmonic law of nature; and the
+media of communication are the sensorium and its inlets—the organs of
+sense.
+
+This harmonic law of nature was either originally discovered by that
+illustrious philosopher Pythagoras, upwards of five hundred years before
+Christ, or a knowledge of it obtained by him about that period, from
+the Egyptian or Chaldean priests. For after having been initiated into
+all the Grecian and barbarian sacred mysteries, he went to Egypt, where
+he remained upwards of twenty years, studying in the colleges of its
+priests; and from Egypt he went into the East, and visited the Persian
+and Chaldean magi.[3]
+
+By the generality of the biographers of Pythagoras, it is said to be
+difficult to give a clear idea of his philosophy, as it is almost certain
+he never committed it to writing, and that it has been disfigured by the
+fantastic dreams and chimeras of later Pythagoreans. Diogenes Laërtius,
+however, whose “Lives of the Philosophers” was supposed to be written
+about the end of the second century of our era, says “there are three
+volumes extant written by Pythagoras. One on education, one on politics,
+and one on natural philosophy.” And adds, that there were several other
+books extant, attributed to Pythagoras, but which were not written by
+him. Also, in his “Life of Philolaus,” that Plato wrote to Dion to take
+care and purchase the books of Pythagoras.[4] But whether this great
+philosopher committed his discoveries to writing or not, his doctrines
+regarding the philosophy of beauty are well-known to be, that he
+considered numbers as the essence and the principle of all things, and
+attributed to them a real and distinct existence; so that, in his view,
+they were the elements out of which the universe was constructed, and to
+which it owed its beauty. Diogenes Laërtius gives the following account
+of this law:—“That the monad was the beginning of everything. From the
+monad proceeds an indefinite duad, which is subordinate to the monad as
+to its cause. That from the monad and indefinite duad proceeds numbers.
+That the part of science to which Pythagoras applied himself above all
+others, was arithmetic; and that he taught ‘that from numbers proceed
+signs, and from these latter, lines, of which plane figures consist;
+that from plane figures are derived solid bodies; that of all plane
+figures the most beautiful was the circle, and of all solid bodies the
+most beautiful was the sphere.’ He discovered the numerical relations of
+sounds on a single string; and taught that everything owes its existence
+and consistency to harmony. In so far as I know, the most condensed
+account of all that is known of the Pythagorian system of numbers is the
+following:—‘The monad or unity is that quantity, which, being deprived of
+all number, remains fixed. It is the fountain of all number. The duad is
+imperfect and passive, and the cause of increase and division. The triad,
+composed of the monad and duad, partakes of the nature of both. The
+tetrad, tetractys, or quaternion number is most perfect. The decad, which
+is the sum of the four former, comprehends all arithmetical and musical
+proportions.’”[5]
+
+These short quotations, I believe, comprise all that is known, for
+certain, of the manner in which Pythagoras systematised the law of
+numbers. Yet, from the teachings of this great philosopher and his
+disciples, the harmonic law of nature, in which the fundamental
+principles of beauty are embodied, became so generally understood and
+universally applied in practice throughout all Greece, that the fragments
+of their works, which have reached us through a period of two thousand
+years, are still held to be examples of the highest artistic excellence
+ever attained by mankind. In the present state of art, therefore, a
+knowledge of this law, and of the manner in which it may again be applied
+in the production of beauty in all works of form and colour, must be
+of singular advantage; and the object of this work is to assist in the
+attainment of such a knowledge.
+
+It has been remarked, with equal comprehensiveness and truth, by a
+writer[6] in the _British and Foreign Medical Review_, that “there
+is harmony of numbers in all nature—in the force of gravity—in the
+planetary movements—in the laws of heat, light, electricity, and chemical
+affinity—in the forms of animals and plants—in the perceptions of the
+mind. The direction, indeed, of modern natural and physical science is
+towards a generalization which shall express the fundamental laws of all
+by one simple numerical ratio. And we think modern science will soon shew
+that the mysticism of Pythagoras was mystical only to the unlettered,
+and that it was a system of philosophy founded on the then existing
+mathematics, which latter seem to have comprised more of the philosophy
+of numbers than our present.” Many years of careful investigation have
+convinced me of the truth of this remark, and of the great advantage
+derivable from an application of the Pythagorean system in the arts
+of design. For so simple is its nature, that any one of an ordinary
+capacity of mind, and having a knowledge of the most simple rules of
+arithmetic, may, in a very short period, easily comprehend its nature,
+and be able to apply it in practice.
+
+The elements of the Pythagorean system of harmonic number, so far as can
+be gathered from the quotations I have given above, seem to be simply
+the indivisible monad (1); the duad (2), arising from the union of one
+monad with another; the triad (3), arising from the union of the monad
+with the duad; and the tetrad (4), arising from the union of one duad
+with another, which tetrad is considered a perfect number. From the
+union of these four elements arises the decad (10), the number, which,
+agreeably to the Pythagorean system, comprehends all arithmetical and
+harmonic proportions. If, therefore, we take these elements and unite
+them progressively in the following order, we shall find the series of
+harmonic numbers (2), (3), (5), and (7), which, with their multiples, are
+the complete numerical elements of all harmony, thus:—
+
+    1 + 1 = 2
+    1 + 2 = 3
+    2 + 3 = 5
+    3 + 4 = 7
+
+In order to render an extended series of harmonic numbers useful, it
+must be divided into scales; and it is a rule in the formation of these
+scales, that the first must begin with the monad (1) and end with the
+duad (2), the second begin with the duad (2) and end with the tetrad (4),
+and that the beginning and end of all other scales must be continued in
+the same arithmetical progression. These primary elements will then form
+the foundation of a series of such scales.
+
+      I. (1)                                    (2)
+     II. (2)                (3)                 (4)
+    III. (4)      (5)       (6)       (7)       (8)
+     IV. (8) (9) (10) (  ) (12) (  ) (14) (15) (16)
+
+The first of these scales has in (1) and (2) a beginning and an end; but
+the second has in (2), (3), and (4) the essential requisites demanded
+by Aristotle in every composition, viz., “a beginning, a middle, and
+an end;” while the third has not only these essential requisites, but
+two intermediate parts (5) and (7), by which the beginning, the middle,
+and the end are united. In the fourth scale, however, the arithmetical
+progression is interrupted by the omission of numbers 11 and 13, which,
+not being multiples of either (2), (3), (5), or (7), are inadmissible.
+
+Such is the nature of the harmonic law which governs the progressive
+scales of numbers by the simple multiplication of the monad.
+
+I shall now use these numbers as divisors in the formation of a series
+of four such scales of parts, which has for its primary element, instead
+of the indivisible monad, a quantity which may be indefinitely divided,
+but which cannot be added to or multiplied. Like the monad, however, this
+quantity is represented by (1). The following is this series of four
+scales of harmonic parts:—
+
+      I.  (1)                                                  (¹⁄₂)
+     II. (¹⁄₂)                     (¹⁄₃)                       (¹⁄₄)
+    III. (¹⁄₄)       (¹⁄₅)         (¹⁄₆)         (¹⁄₇)         (¹⁄₈)
+     IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) (    ) (¹⁄₁₂) (    ) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)
+
+The scales I., II., and III. may now be rendered as complete as scale
+IV., simply by multiplying upwards by 2 from (¹⁄₉), (¹⁄₅), (¹⁄₃), (¹⁄₇),
+and (¹⁄₁₅), thus:—
+
+      I.  (1)  (⁸⁄₉) (⁴⁄₅)         (²⁄₃)         (⁴⁄₇)  (⁸⁄₁₅) (¹⁄₂)
+     II. (¹⁄₂) (⁴⁄₉) (²⁄₅)         (¹⁄₃)         (²⁄₇)  (⁴⁄₁₅) (¹⁄₄)
+    III. (¹⁄₄) (²⁄₉) (¹⁄₅)         (¹⁄₆)         (¹⁄₇)  (²⁄₁₅) (¹⁄₈)
+     IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) (    ) (¹⁄₁₂) (    ) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)
+
+We now find between the beginning and the end of scale I. the quantities
+(⁸⁄₉), (⁴⁄₅), (²⁄₃), (⁴⁄₇), and (⁸⁄₁₅).
+
+The three first of these quantities we find to be the remainders of the
+whole indefinite quantity contained in (1), after subtracting from it
+the primary harmonic quantities (¹⁄₉), (¹⁄₅), and (¹⁄₃); we, however,
+find also amongst these harmonic quantities that of (¹⁄₄), which being
+subtracted from (1) leaves (³⁄₄), a quantity the most suitable whereby
+to fill up the hiatus between (⁴⁄₅) and (²⁄₃) in scale I., which arises
+from the omission of (¹⁄₁₁) in scale IV. In like manner we find the two
+last of these quantities, (⁴⁄₇) and (⁸⁄₁₅), are respectively the largest
+of the two parts into which 7 and 15 are susceptible of being divided.
+Finding the number 5 to be divisible into parts more unequal than (2)
+to (3) and less unequal than (4) to (7), (³⁄₅) naturally fills up the
+hiatus between these quantities in scale I., which hiatus arises from the
+omission of (¹⁄₁₃) in scale IV. Thus:—
+
+      I.  (1)  (⁸⁄₉) (⁴⁄₅)  (³⁄₄)  (²⁄₃)  (³⁄₅)  (⁴⁄₇)  (⁸⁄₁₅) (¹⁄₂)
+     II. (¹⁄₂) (⁴⁄₉) (²⁄₅)  (    ) (¹⁄₃)  (    ) (²⁄₇)  (⁴⁄₁₅) (¹⁄₄)
+    III. (¹⁄₄) (²⁄₉) (¹⁄₅)  (    ) (¹⁄₆)  (    ) (¹⁄₇)  (²⁄₁₅) (¹⁄₈)
+     IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) (    ) (¹⁄₁₂) (    ) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)
+
+Scale I. being now complete, we have only to divide these latter
+quantities by (2) downwards in order to complete the other three. Thus:—
+
+      I.  (1)  (⁸⁄₉) (⁴⁄₅)  (³⁄₄)  (²⁄₃)  (³⁄₅)  (⁴⁄₇)  (⁸⁄₁₅) (¹⁄₂)
+     II. (¹⁄₂) (⁴⁄₉) (²⁄₅)  (³⁄₈)  (¹⁄₃)  (³⁄₁₀) (²⁄₇)  (⁴⁄₁₅) (¹⁄₄)
+    III. (¹⁄₄) (²⁄₉) (¹⁄₅)  (³⁄₁₆) (¹⁄₆)  (³⁄₂₀) (¹⁄₇)  (²⁄₁₅) (¹⁄₈)
+     IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) (³⁄₃₂) (¹⁄₁₂) (³⁄₄₀) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)
+
+The harmony existing amongst these numbers or quantities consists of the
+numerical relations which the parts bear to the whole and to each other;
+and the more simple these relations are, the more perfect is the harmony.
+The following are the numerical harmonic ratios which the parts bear to
+the whole:—
+
+      I. (1:1) (8:9) (4: 5) (3: 4) (2: 3) (3: 5) (4: 7) (8:15) (1: 2)
+     II. (1:2) (4:9) (2: 5) (3: 8) (1: 3) (3:10) (2: 7) (4:15) (1: 4)
+    III. (1:4) (2:9) (1: 5) (3:16) (1: 6) (3:20) (1: 7) (2:15) (1: 8)
+     IV. (1:8) (1:9) (1:10) (3:32) (1:12) (3:40) (1:14) (1:15) (1:16)
+
+The following are the principal numerical relations which the parts in
+each scale bear to one another:—
+
+     (¹⁄₂):(⁴⁄₇)  =  (7:8)
+     (⁴⁄₅):(⁸⁄₉)  =  (9:10)
+     (²⁄₃):(⁴⁄₅)  =  (5:6)
+     (⁴⁄₇):(²⁄₃)  =  (6:7)
+    (⁸⁄₁₅):(⁴⁄₇)  = (14:15)
+     (¹⁄₂):(⁸⁄₁₅) = (15:16)
+
+Although these relations are exemplified by parts of scale I., the same
+ratios exist between the relative parts of scales II., III., and IV.,
+and would exist between the parts of any other scales that might be added
+to that series.
+
+These are the simple elements of the science of that harmony which
+pervades the universe, and by which the various kinds of beauty
+æsthetically impressed upon the senses of hearing and seeing are
+governed.
+
+
+
+
+THE SCIENCE OF BEAUTY AS APPLIED TO SOUNDS.
+
+
+It is well-known that all sounds arise from a peculiar action of the
+air, and that this action may be excited by the concussion resulting
+from the sudden displacement of a portion of the atmosphere itself, or
+by the rapid motions of bodies, or of confined columns of air; in all
+which cases, when the motions are irregular, and the force great, the
+sound conveyed to the sensorium is called a noise. But that musical
+sounds are the result of equal and regular vibratory motions, either
+of an elastic body, or of a column of air in a tube, exciting in the
+surrounding atmosphere a regular and equal pulsation. The ear is the
+medium of communication between those varieties of atmospheric action and
+the seat of consciousness. To describe fully the beautiful arrangement
+of the various parts of this organ, and their adaptation to the purpose
+of collecting and conveying these undulatory motions of the atmosphere,
+is as much beyond the scope of my present attempt as it is beyond my
+anatomical knowledge; but I may simply remark, that within the ear, and
+most carefully protected in the construction of that organ, there is a
+small cavity containing a pellucid fluid, in which the minute extremities
+of the auditory nerve float; and that this fluid is the last of the media
+through which the action producing the sensation of sound is conveyed
+to the nerve, and thence to the sensorium, where its nature becomes
+perceptible to the mind.
+
+The impulses which produce musical notes must arrive at a certain
+frequency before the ear loses the intervals of silence between them,
+and is impressed by only one continued sound; and as they increase
+in frequency the sound becomes more acute upon the ear. The pitch of
+a musical note is, therefore, determined by the frequency of these
+impulses; but, on the other hand, its intensity or loudness will depend
+upon the violence and the quality of its tone on the material employed in
+producing them. All such sounds, therefore, whatever be their loudness
+or the quality of their tone in which the impulses occur with the same
+frequency are in perfect unison, having the same pitch. Upon this the
+whole doctrine of harmonies is founded, and by this the laws of numerical
+ratio are found to operate in the production of harmony, and the theory
+of music rendered susceptible of exact reasoning.
+
+The mechanical means by which such sounds can be produced are extremely
+various; but, as it is my purpose simply to shew the nature of harmony
+of sound as related to, or as evolving numerical harmonic ratio, I shall
+confine myself to the most simple mode of illustration—namely, that of
+the monochord. This is an instrument consisting of a string of a given
+length stretched between two bridges standing upon a graduated scale.
+Suppose this string to be stretched until its tension is such that,
+when drawn a little to a side and suddenly let go, it would vibrate at
+the rate of 64 vibrations in a second of time, producing to a certain
+distance in the surrounding atmosphere a series of pulsations of the same
+frequency.
+
+These pulsations will communicate through the ear a musical note which
+would, therefore, be the fundamental note of such a string. Now, the
+phenomenon said to be discovered by Pythagoras is well known to those
+acquainted with the science of acoustics, namely, that immediately after
+the string is thus put into vibratory motion, it spontaneously divides
+itself, by a node, into two equal parts, the vibrations of each of which
+occur with a double frequency—namely, 128 in a second of time, and,
+consequently, produce a note doubly acute in pitch, although much weaker
+as to intensity or loudness; that it then, while performing these two
+series of vibrations, divides itself, by two nodes, into three parts,
+each of which vibrates with a frequency triple that of the whole string;
+that is, performs 192 vibrations in a second of time, and produces a
+note corresponding in increase of acuteness, but still less intense than
+the former, and that this continues to take place in the arithmetical
+progression of 2, 3, 4, &c. Simultaneous vibrations, agreeably to the
+same law of progression, which, however, seem to admit of no other primes
+than the numbers 2, 3, 5, and 7, are easily excited upon any stringed
+instrument, even by the lightest possible touch of any of its strings
+while in a state of vibratory motion, and the notes thus produced are
+distinguished by the name of harmonics. It follows, then, that one-half
+of a musical string, when divided from the whole by the pressure of the
+finger, or any other means, and put into vibratory motion, produces a
+note doubly acute to that produced by the vibratory motion of the whole
+string; the third part, similarly separated, a note trebly acute; and
+the same with every part into which any musical string may be divided.
+This is the fundamental principle by which all stringed instruments are
+made to produce harmony. It is the same with wind instruments, the sounds
+of which are produced by the frequency of the pulsations occasioned in
+the surrounding atmosphere by agitating a column of air confined within
+a tube as in an organ, in which the frequency of pulsation becomes
+greater in an inverse ratio to the length of the pipes. But the following
+series of four successive scales of musical notes will give the reader
+a more comprehensive view of the manner in which they follow the law of
+numerical ratio just explained than any more lengthened exposition.
+
+It is here requisite to mention, that in the construction of these
+scales, I have not only adopted the old German or literal mode of
+indicating the notes, but have included, as the Germans do, the note
+termed by us B flat as B natural, and the note we term B natural as
+H. Now, although this arrangement differs from that followed in the
+construction of our modern Diatonic scale, yet as the ratio of 4:7
+is more closely related to that of 1:2 than that of 8:15, and as it
+is offered by nature in the spontaneous division of the monochord,
+I considered it quite admissible. The figures give the parts of the
+monochord which would produce the notes.
+
+   I. {  (1)   (⁸⁄₉)  (⁴⁄₅)   (³⁄₄)  (²⁄₃)   (³⁄₅)  (⁴⁄₇)   (⁸⁄₁₅)  (¹⁄₂)*
+      {   C      D      E       F      G       A      B       H      _c_
+
+  II. { (¹⁄₂)* (⁴⁄₉)  (²⁄₅)   (³⁄₈)  (¹⁄₃)*  (³⁄₁₀) (²⁄₇)   (²⁄₁₅)  (¹⁄₄)*
+      {  _c_    _d_    _e_     _f_    _g_     _a_    _b_     _h_     _c′_
+
+ III. { (¹⁄₄)* (²⁄₉)  (¹⁄₅)*  (³⁄₁₆) (¹⁄₆)*  (³⁄₂₀) (¹⁄₇)*  (²⁄₁₅)  (¹⁄₈)*
+      {  _c′_   _d′_   _e′_    _f′_   _g′_    _a′_   _b′_    _h′_    _c′′_
+
+  IV. { (¹⁄₈)* (¹⁄₉)* (¹⁄₁₀)* (³⁄₃₂) (¹⁄₁₂)* (³⁄₄₀) (¹⁄₁₄)* (¹⁄₁₅)* (¹⁄₁₆)*
+      {  _c′′_  _d′′_  _e′′_   _f′′_  _g′′_   _a′′_  _b′′_   _h′′_   _c′′′_
+
+The notes marked (*) are the harmonics which naturally arise from the
+division of the string by 2, 3, 5, and 7, and the multiples of these
+primes.
+
+Thus every musical sound is composed of a certain number of parts called
+pulsations, and these parts must in every scale relate harmonically
+to some fundamental number. When these parts are multiples of the
+fundamental number by 2, 4, 8, &c., like the pulsations of the sounds
+indicated by _c_, _c′_, _c′′_, _c′′′_, they are called tonic notes, being
+the most consonant; when the pulsations are similar multiples by 3, 6,
+12, &c., like those of the sounds indicated by _g_, _g′_, _g′′_, they are
+called dominant notes, being the next most consonant; and multiples by
+5, 10, &c., like those of the sounds indicated by _e_, _e′_, _e′′_, they
+are called mediant notes, from a similar cause. In harmonic combinations
+of musical sounds, the æsthetic feeling produced by their agreement
+depends upon the relations they bear to each other with reference to the
+number of pulsations produced in a given time by the fundamental note of
+the scale to which they belong; and it will be observed, that the more
+simple the numerical ratios are amongst the pulsations of any number of
+notes simultaneously produced, the more perfect their agreement. Hence
+the origin of the common chord or fundamental concord in the united
+sounds of the tonic, the dominant, and the mediant notes, the ratios and
+coincidences of whose pulsations 2:1, 3:2, 5:4, may thus be exemplified:—
+
+[Illustration]
+
+In musical composition, the law of number also governs its division
+into parts, in order to produce upon the ear, along with the beauty of
+harmony, that of rhythm. Thus a piece of music is divided into parts
+each of which contains a certain number of other parts called bars,
+which may be divided and subdivided into any number of notes, and the
+performance of each bar is understood to occupy the same portion of time,
+however numerous the notes it contains may be; so that the music of art
+is regularly symmetrical in its structure; while that of nature is in
+general as irregular and indefinite in its rhythm as it is in its harmony.
+
+Thus I have endeavoured briefly to explain the manner in which the law of
+numerical ratio operates in that species of beauty perceived through the
+ear.
+
+The definite principles of the art of music founded upon this law have
+been for ages so systematised that those who are instructed in them
+advance steadily in proportion to their natural endowments, while those
+who refuse this instruction rarely attain to any excellence. In the
+sister arts of form and colour, however, a system of tuition, founded
+upon this law, is still a desideratum, and a knowledge of the scientific
+principles by which these arts are governed is confined to a very few,
+and scarcely acknowledged amongst those whose professions most require
+their practical application.
+
+
+
+
+THE SCIENCE OF BEAUTY AS APPLIED TO FORMS.
+
+
+It is justly remarked, in the “Illustrated Record of the New York
+Exhibition of 1853,” that “it is a question worthy of consideration how
+far the mediocrity of the present day is attributable to an overweening
+reliance on natural powers and a neglect of the lights of science;”
+and there is expressed a thorough conviction of the fact that, besides
+the evils of the copying system, “much genius is now wasted in the
+acquirement of rudimentary knowledge in the slow school of practical
+experiment, and that the excellence of the ancient Greek school of
+design arose from a thoroughly digested canon of form, and the use
+of geometrical formulas, which make the works even of the second and
+third-rate genius of that period the wonder and admiration of the present
+day.”
+
+That such a canon of form, and that the use of such geometrical formula,
+entered into the education, and thereby facilitated the practice of
+ancient Greek art, I have in a former work expressed my firm belief,
+which is founded on the remarkable fact, that for a period of nearly
+three centuries, and throughout a whole country politically divided into
+states often at war with each other, works of sculpture, architecture,
+and ornamental design were executed, which surpass in symmetrical
+beauty any works of the kind produced during the two thousand years that
+have since elapsed. So decided is this superiority, that the artistic
+remains of the extraordinary period I alluded to are, in all civilised
+nations, still held to be the most perfect specimens of formative art in
+the world; and even when so fragmentary as to be denuded of everything
+that can convey an idea of expression, they still excite admiration and
+wonder by the purity of their geometric beauty. And so universal was
+this excellence, that it seems to have characterised every production of
+formative art, however humble the use to which it was applied.
+
+The common supposition, that this excellence was the result of an
+extraordinary amount of genius existing among the Greek people during
+that particular period, is not consistent with what we know of the
+progress of mankind in any other direction, and is, in the present state
+of art, calculated to retard its progress, inasmuch as such an idea
+would suggest that, instead of making any exertion to arrive at a like
+general excellence, the world must wait for it until a similar supposed
+psychological phenomenon shall occur.
+
+But history tends to prove that the long period of universal artistic
+excellence throughout Greece could only be the result of an early
+inculcation of some well-digested system of correct elementary
+principles, by which the ordinary amount of genius allotted to mankind
+in every age was properly nurtured and cultivated; and by which, also, a
+correct knowledge and appreciation of art were disseminated amongst the
+people generally. Indeed, Müller, in his “Ancient Art and its Remains,”
+shews clearly that some certain fixed principles, constituting a science
+of proportions, were known in Greece, and that they formed the basis of
+all artists’ education and practice during the period referred to; also,
+that art began to decline, and its brightest period to close, as this
+science fell into disuse, and the Greek artists, instead of working for
+an enlightened community, who understood the nature of the principles
+which guided them, were called upon to gratify the impatient whims of
+pampered and tyrannical rulers.
+
+By being instructed in this science of proportion, the Greek artists
+were enabled to impart to their representations of the human figure
+a mathematically correct species of symmetrical beauty; whether
+accompanying the slender and delicately undulated form of the Venus,—its
+opposite, the massive and powerful mould of the Hercules,—or the
+characteristic representation of any other deity in the heathen
+mythology. And this seems to have been done with equal ease in the minute
+figure cut on a precious gem, and in the most colossal statue. The same
+instruction likewise enabled the architects of Greece to institute those
+varieties of proportions in structure called the Classical Orders of
+Architecture; which are so perfect that, since the science which gave
+them birth has been buried in oblivion, classical architecture has been
+little more than an imitative art; for all who have since written upon
+the subject, from Vitruvius downwards, have arrived at nothing, in
+so far as the great elementary principles in question are concerned,
+beyond the most vague and unsatisfactory conjectures. For a more clear
+understanding of the nature of this application of the Pythagorean law of
+number to the harmony of form, it will be requisite to repeat the fact,
+that modern science has shewn that the cause of the impression, produced
+by external nature upon the sensorium, called light, may be traced to a
+molecular or ethereal action. This action is excited naturally by the
+sun, artificially by the combustion of various substances, and sometimes
+physically within the eye. Like the atmospheric pulsations which produce
+sound, the action which produces light is capable, within a limited
+sphere, of being reflected from some bodies and transmitted through
+others; and by this reflection and transmission the visible nature of
+forms and figures is communicated to the sensorium. The eye is the
+medium of this communication; and its structural beauty, and perfect
+adaptation to the purpose of conveying this action, must, like those of
+the ear, be left to the anatomist fully to describe. It is here only
+necessary to remark, that the optic nerve, like the auditory nerve,
+ends in a carefully protected fluid, which is the last of the media
+interposed between this peculiarly subtle action and the nerve upon
+which it impresses the presence of the object from which it is reflected
+or through which it is transmitted, and the nature of such object made
+perceptible to the mind. The eye and the ear are thus, in one essential
+point, similar in their physiology, relatively to the means provided
+for receiving impressions from external nature; it is, therefore, but
+reasonable to believe that the eye is capable of appreciating the exact
+subdivision of spaces, just as the ear is capable of appreciating the
+exact subdivision of intervals of time; so that the division of space
+into exact numbers of equal parts will æsthetically affect the mind
+through the medium of the eye.
+
+We assume, therefore, that the standard of symmetry, so estimated, is
+deduced from the simplest law that could have been conceived—the law
+that the angles of direction must all bear to some fixed angle the same
+simple relations which the different notes in a chord of music bear to
+the fundamental note; that is, relations expressed arithmetically by the
+smallest natural numbers. Thus the eye, being guided in its estimate by
+direction rather than by distance, just as the ear is guided by number
+of vibrations rather than by magnitude, both it and the ear convey
+simplicity and harmony to the mind without effort, and the mind with
+equal facility receives and appreciates them.
+
+
+_On the Rectilinear Forms and Proportions of Architecture._
+
+As we are accustomed in all cases to refer direction to the horizontal
+and vertical lines, and as the meeting of these lines makes the right
+angle, it naturally constitutes the fundamental angle, by the harmonic
+division of which a system of proportion may be established, and the
+theory of symmetrical beauty, like that of music, rendered susceptible of
+exact reasoning.
+
+Let therefore the right angle be the fundamental angle, and let it be
+divided upon the quadrant of a circle into the harmonic parts already
+explained, thus:—
+
+            Super-           Sub-            Sub-    Sub-  Semi-sub-
+     Right  tonic  Mediant dominant Dominant mediant tonic  tonic   Tonic
+     Angle. Angles. Angles. Angles. Angles. Angles. Angles. Angles. Angles.
+   I.  (1)   (⁸⁄₉)   (⁴⁄₅)   (³⁄₄)   (²⁄₃)   (³⁄₅)   (⁴⁄₇)   (⁸⁄₁₅)  (¹⁄₂)
+  II. (¹⁄₂)  (⁴⁄₉)   (²⁄₅)   (³⁄₈)   (¹⁄₃)   (³⁄₁₀)  (²⁄₇)   (⁴⁄₁₅)  (¹⁄₄)
+ III. (¹⁄₄)  (²⁄₉)   (¹⁄₅)   (³⁄₁₆)  (¹⁄₆)   (³⁄₂₀)  (¹⁄₇)   (²⁄₁₅)  (¹⁄₈)
+  IV. (¹⁄₈)  (¹⁄₉)   (¹⁄₁₀)  (³⁄₃₂)  (¹⁄₁₂)  (³⁄₄₀)  (¹⁄₁₄)  (¹⁄₁₅)  (¹⁄₁₆)
+
+In order that the analogy may be kept in view, I have given to the parts
+of each of these four scales the appropriate nomenclature of the notes
+which form the diatonic scale in music.
+
+When a right angled triangle is constructed so that its two smallest
+angles are equal, I term it simply the triangle of (¹⁄₂), because the
+smaller angles are each one-half of the right angle. But when the two
+angles are unequal, the triangle may be named after the smallest. For
+instance, when the smaller angle, which we shall here suppose to be
+one-third of the right angle, is made with the vertical line, the
+triangle may be called the vertical scalene triangle of (¹⁄₃); and when
+made with the horizontal line, the horizontal scalene triangle of (¹⁄₃).
+As every rectangle is made up of two of these right angled triangles,
+the same terminology may also be applied to these figures. Thus, the
+equilateral rectangle or perfect square is simply the rectangle of (¹⁄₂),
+being composed of two similar right angled triangles of (¹⁄₂); and when
+two vertical scalene triangles of (¹⁄₃), and of similar dimensions, are
+united by their hypothenuses, they form the vertical rectangle of (¹⁄₃),
+and in like manner the horizontal triangles of (¹⁄₃) similarly united
+would form the horizontal rectangle of (¹⁄₃). As the isosceles triangle
+is in like manner composed of two right angled scalene triangles joined
+by one of their sides, the same terminology may be applied to every
+variety of that figure. All the angles of the first of the above scales,
+except that of (¹⁄₂), give rectangles whose longest sides are in the
+horizontal line, while the other three give rectangles whose longest
+sides are in the vertical line. I have illustrated in Plate I. the manner
+in which this harmonic law acts upon these elementary rectilinear figures
+by constructing a series agreeably to the angles of scales II., III.,
+IV. Throughout this series _a b c_ is the primary scalene triangle, of
+which the rectangle _a b c e_ is composed; _d c e_ the vertical isosceles
+triangle; and when the plate is turned, _d e a_ the horizontal isosceles
+triangle, both of which are composed of the same primary scalene triangle.
+
+[Sidenote: Plate I.]
+
+Thus the most simple elements of symmetry in rectilinear forms are the
+three following figures:—
+
+    The equilateral rectangle or perfect square,
+    The oblong rectangle, and
+    The isosceles triangle.
+
+It has been shewn that in harmonic combinations of musical sounds, the
+æsthetic feeling produced by their agreement depends upon the relation
+they bear to each other with reference to the number of pulsations
+produced in a given time by the fundamental note of the scale to which
+they belong; and that the more simply they relate to each other in this
+way the more perfect the harmony, as in the common chord of the first
+scale, the relations of whose parts are in the simple ratios of 2:1, 3:2,
+and 5:4. It is equally consistent with this law, that when applied to
+form in the composition of an assortment of figures of any kind, their
+respective proportions should bear a very simple ratio to each other in
+order that a definite and pleasing harmony may be produced amongst the
+various parts. Now, this is as effectually done by forming them upon the
+harmonic divisions of the right angle as musical harmony is produced by
+sounds resulting from harmonic divisions of a vibratory body.
+
+Having in previous works[7] given the requisite illustrations of this
+fact in full detail, I shall here confine myself to the most simple kind,
+taking for my first example one of the finest specimens of classical
+architecture in the world—the front portico of the Parthenon of Athens.
+
+The angles which govern the proportions of this beautiful elevation are
+the following harmonic parts of the right angle—
+
+    Tonic   Dominant  Mediant  Subtonic  Supertonic
+    Angles.  Angles.  Angles.   Angle.    Angles.
+    (¹⁄₂)    (¹⁄₃)   (¹⁄₅)      (¹⁄₇)     (¹⁄₉)
+    (¹⁄₄)    (¹⁄₆)   (¹⁄₁₀)               (¹⁄₁₈)
+    (¹⁄₈)
+    (¹⁄₁₆)
+
+[Sidenote: Plate II.]
+
+In Plate II. I give a diagram of its rectilinear orthography, which is
+simply constructed by lines drawn, either horizontally, vertically, or
+obliquely, which latter make with either of the former lines one or other
+of the harmonic angles in the above series. For example, the horizontal
+line AB represents the length of the base or surface of the upper step of
+the substructure of the building. The line AE, which makes an angle of
+(¹⁄₅) with the horizontal, determines the height of the colonnade. The
+line AD, which makes an angle of (¹⁄₄) with the horizontal, determines
+the height of the portico, exclusive of the pediment. The line AC, which
+makes an angle of (¹⁄₃) with the horizontal, determines the height of the
+portico, including the pediment. The line GD, which makes an angle of
+(¹⁄₇) with the horizontal, determines the form of the pediment. The lines
+EZ and LY, which respectively make angles of (¹⁄₁₆) and (¹⁄₁₈) with the
+horizontal, determine the breadth of the architrave, frieze, and cornice.
+The line _v n u_, which makes an angle of (¹⁄₃) with the vertical,
+determines the breadth of the triglyphs. The line _t d_, which makes an
+angle of (¹⁄₂), determines the breadth of the metops. The lines _c b
+r f_, and _a i_, which make each an angle of (¹⁄₆) with the vertical,
+determine the width of the five centre intercolumniations. The line _z
+k_, which makes an angle of (¹⁄₈) with the vertical, determines the width
+of the two remaining intercolumniations. The lines _c s_, _q x_, and _y
+h_, each of which makes an angle of (¹⁄₁₀) with the vertical, determine
+the diameters of the three columns on each side of the centre. The line
+_w l_, which makes an angle of (¹⁄₉) with the vertical, determines the
+diameter of the two remaining or corner columns.
+
+In all this, the length and breadth of the parts are determined by
+horizontal and vertical lines, which are necessarily at right angles with
+each other, and the position of which are determined by one or other of
+the lines making the harmonic angles above enumerated.
+
+Now, the lengths and breadths thus so simply determined by these few
+angles, have been proved to be correct by their agreement with the most
+careful measurements which could possibly be made of this exquisite
+specimen of formative art. These measurements were obtained by the
+“Society of Dilettanti,” London, who, expressly for that purpose, sent Mr
+F. C. Penrose, a highly educated architect, to Athens, where he remained
+for about five months, engaged in the execution of this interesting
+commission, the results of which are now published in a magnificent
+volume by the Society.[8] The agreement was so striking, that Mr Penrose
+has been publicly thanked by an eminent man of science for bearing
+testimony to the truth of my theory, who in doing so observes, “The
+dimensions which he (Mr Penrose) gives are to me the surest verification
+of the theory I could have desired. The minute discrepancies form that
+very element of practical incertitude, both as to execution and direct
+measurement, which always prevails in materialising a mathematical
+calculation made under such conditions.”[9]
+
+Although the measurements taken by Mr Penrose are undeniably correct, as
+all who examine the great work just referred to must acknowledge, and
+although they have afforded me the best possible means of testing the
+accuracy of my theory as applied to the Parthenon, yet the ideas of Mr
+Penrose as to the principles they evolve are founded upon the fallacious
+doctrine which has so long prevailed, and still prevails, in the
+æsthetics of architecture, viz., that harmony may be imparted by ratios
+between the lengths and breadths of parts.
+
+I have taken for my second example an elevation which, although of
+smaller dimensions, is no less celebrated for the beauty of its
+proportions than the Parthenon itself, viz., the front portico of the
+temple of Theseus, which has also been measured by Mr Penrose.
+
+The angles which govern the proportions of this elevation are the
+following harmonic parts of the right angle:—
+
+    Tonic   Dominant  Mediant
+    Angles.  Angles.  Angles.
+     (¹⁄₂)    (¹⁄₃)    (²⁄₅)
+     (¹⁄₄)    (¹⁄₆)    (¹⁄₅)
+              (¹⁄₁₂)
+
+[Sidenote: Plate III.]
+
+A diagram of the rectilinear orthography of this portico is given in
+Plate III. Its construction is similar to that of the Parthenon in
+respect to the harmonic parts of the right angle, and I have therefore
+only to observe, that the line A E makes an angle of (¹⁄₄); the line A D
+an angle of (¹⁄₃); the line A C an angle of (²⁄₅); the line G D an angle
+of (¹⁄₆); and the lines E Z and L Y angles of (¹⁄₁₂) with the horizontal.
+
+As to the colonnade or vertical part, the line _a b_, which determines
+the three middle intercolumniations, makes an angle of (¹⁄₅); the line
+_c d_, which determines the two outer intercolumniations, makes an angle
+of (¹⁄₆); and the line _e f_, which determines the lesser diameter of
+the columns, makes an angle of (¹⁄₁₂) with the vertical. I need give no
+further details here, as my intention is to shew the simplicity of the
+method by which this theory may be reduced to practice, and because I
+have given in my other works ample details, in full illustration of the
+orthography of these two structures, especially the first.[10]
+
+The foregoing examples being both horizontal rectangular compositions,
+the proportions of their principal parts have necessarily been determined
+by lines drawn from the extremities of the base, making angles with
+the horizontal line, and forming thereby the diagonals of the various
+rectangles into which, in their leading features, they are necessarily
+resolved. But the example I am now about to give is of another character,
+being a vertical pyramidal composition, and consequently the proportions
+of its principal parts are determined by the angles which the oblique
+lines make with the vertical line representing the height of the
+elevation, and forming a series of isosceles triangles; for the isosceles
+triangle is the type of all pyramidal composition.
+
+This third example is the east end of Lincoln Cathedral, a Gothic
+structure, which is acknowledged to be one of the finest specimens of
+that style of architecture existing in this country.
+
+The angles which govern the proportions of this elevation are the
+following harmonic parts of the right angle:—
+
+    Tonic.  Dominant.  Mediant.  Subtonic.  Supertonic.
+    (¹⁄₂)     (¹⁄₃)     (¹⁄₅)      (¹⁄₇)      (²⁄₉)
+    (¹⁄₄)     (¹⁄₆)     (¹⁄₁₀)                (¹⁄₉)
+              (¹⁄₁₂)
+
+[Sidenote: Plate IV.]
+
+In Plate IV. I give a diagram of the vertical, horizontal, and oblique
+lines, which compose the orthography of this beautiful elevation.
+
+The line A B represents the full height of this structure. The line A C,
+which makes an angle of (²⁄₉) with the vertical, determines the width of
+the design, the tops of the aisle windows, and the bases of the pediments
+on the inner buttresses; A G, (¹⁄₅) with the vertical, that of the outer
+buttress; A F, (¹⁄₉) with the vertical, that of the space between the
+outer and inner buttresses and the width of the great centre window; and
+A E, (¹⁄₁₂) with vertical, that of both the inner buttresses and the
+space between these. A H, which makes (¹⁄₄) with the vertical, determines
+the form of the pediment of the centre, and the full height of the base
+and surbase. A I, which makes (¹⁄₃) with the vertical, determines the
+form of the pediment of the smaller gables, the base of the pediment on
+the outer buttress, the base of the ornamental recess between the outer
+and inner buttresses, the spring of the arch of the centre window, the
+tops of the pediments on the inner buttresses, and the spring of the
+arch of the upper window. A K, which makes (¹⁄₂), determines the height
+of the outer buttress; and A Z, which makes (¹⁄₆) with the horizontal,
+determines that of the inner buttresses. For the reasons already given,
+I need not here go into further detail.[11] It is, however, worthy of
+remark in this place, that notwithstanding the great difference which
+exists between the style of composition in this Gothic design, and in
+that of the east end of the Parthenon, the harmonic elements upon which
+the orthographic beauty of the one depends, are almost identical with
+those of the other.
+
+
+_On the Curvilinear Forms and Proportions of Architecture._
+
+Each regular rectilinear figure has a curvilinear figure that exclusively
+belongs to it, and to which may be applied a corresponding terminology.
+For instance, the circle belongs to the equilateral rectangle; that
+is, the rectangle of (¹⁄₂), an ellipse to every other rectangle, and
+a composite ellipse to every isosceles triangle. Thus the most simple
+elements of beauty in the curvilinear forms of architectural design are
+the following three figures:—
+
+    The circle,
+    The ellipse, and
+    The composite ellipse.
+
+I find it necessary in this place to go into some details regarding
+the specific character of the two latter figures, because the proper
+mode of describing these beautiful curves, and their high value in the
+practice of the architectural draughtsman and ornamental designer, seem
+as yet unknown. In proof of this assertion, I must again refer to Mr
+Penrose’s great work published by the “Society of Dilettanti.” At page
+52 of that work it is observed, that “by whatever means an ellipse is to
+be constructed mechanically, it is a work of time (if not of absolute
+difficulty) so to arrange the foci, &c., as to produce an ellipse of any
+exact length and breadth which may be desired.” Now, this is far from
+being the case, for the method of arranging the foci of an ellipse of any
+given length and breadth is extremely simple, being as follows:—
+
+Let A B C (figure 1) be the length, and D B E the breadth of the desired
+ellipse.
+
+[Illustration: Fig. 1.]
+
+Take A B upon the compasses, and place the point of one leg upon E and
+the point of the other upon the line A B, it will meet it at F, which is
+one focus: keeping the point of the one leg upon E, remove the point of
+the other to the line B C, and it will meet it at G, which is the other
+focus. But, when the proportions of an ellipse are to be imparted by
+means of one of the harmonic angles, suppose the angle of (¹⁄₃), then the
+following is the process:—
+
+Let A B C (figure 2) represent the length of the intended ellipse.
+Through B draw B _e_ indefinitely, at right angles with A B C; through C
+draw the line C _f_ indefinitely, making, with B C, an angle of (¹⁄₃).
+
+Take B C upon the compasses, and place the point of one leg upon D where
+C f intersects B _e_, and the point of the other upon the line A B, it
+will meet it at F, which is one focus. Keeping the point of one leg still
+upon D, remove the point of the other to the line B C, and it will meet
+it at G, which is the other focus.
+
+[Illustration: Fig. 2.]
+
+The foci being in either case thus simply ascertained, the method of
+describing the curve on a small scale is equally simple.
+
+[Sidenote: Plate V.]
+
+A pin is fixed into each of the two foci, and another into the point D.
+Around these three pins a waxed thread, flexible but not elastic, is
+tied, care being taken that the knot be of a kind that will not slip.
+The pin at D is now removed, and a hard black lead pencil introduced
+within the thread band. The pencil is then moved around the pins fixed
+in the foci, keeping the thread band at a full and equal tension; thus
+simply the ellipse is described. When, however, the governing angle is
+acute, say less than (¹⁄₆), it is requisite to adopt a more accurate
+method of description,[12] as the architectural examples which follow
+will shew. But architectural draughtsmen and ornamental designers would
+do well to supply themselves, for ordinary practice, with half a dozen
+series of ellipses, varying in the proportions of their axes from (⁴⁄₉)
+to (¹⁄₆) of the scale, and the length of their major axes from 1 to 6
+inches. These should be described by the above simple process, upon
+very strong drawing paper, and carefully cut out, the edge of the paper
+being kept smooth, and each ellipse having its greater and lesser axes,
+its foci, and the hypothenuse of its scalene triangle drawn upon it. To
+exemplify this, I give Plate V., which exhibits the ellipses of (¹⁄₃),
+(¹⁄₄), (¹⁄₅), and (¹⁄₆), inscribed in their rectangles, on which _a b_
+and _c d_ are respectively the greater and lesser axes, _o o_ the foci,
+and _d b_ the angle of each. Such a series of these beautiful figures
+would be found particularly useful in drawing the mouldings of Grecian
+architecture; for, to describe the curvilinear contour of such mouldings
+from single points, as has been done with those which embellish even our
+most pretending attempts at the restoration of that classical style of
+architecture, is to give the resemblance of an external form without the
+harmony which constitutes its real beauty.
+
+Mr Penrose, owing to the supposed difficulty regarding the description of
+ellipses just alluded to, endeavours to shew that the curves of all the
+mouldings throughout the Parthenon were either parabolic or hyperbolic;
+but I believe such curves can have no connexion with the elementary forms
+of architecture, for they are curves which represent motion, and do not,
+by continued production, form closed figures.
+
+But I have shewn, in a former work,[13] that the contours of these
+mouldings are composed of curves of the composite ellipse,—a figure
+which I so name because it is composed simply of arcs of various
+ellipses harmonically flowing into each other. The composite ellipse,
+when drawn systematically upon the isosceles triangle, resembles closely
+parabolic and hyperbolic curves—only differing from these inasmuch as it
+possesses the essential quality of circumscribing harmonically one of the
+elementary rectilinear figures employed in architecture, while those of
+the parabola and hyperbola, as I have just observed, are merely curves
+of motion, and, consequently, never can harmonically circumscribe or be
+resolved into any regular figure.
+
+The composite ellipse may be thus described.
+
+[Sidenote: Plate VI.]
+
+Let A B C (Plate VI.) be a vertical isosceles triangle of (¹⁄₆), bisect A
+B in D, and through D draw indefinitely D _f_ perpendicular to A B, and
+through B draw indefinitely B _g_, making the angle D B _g_ (¹⁄₈), D _f_
+and B _g_ intersecting each other in M. Take B D and D M as semi-axes of
+an ellipse, the foci of which will be at _p_ and _q_, in each of these,
+and in each of the foci _h t_ and _k r_ in the lines A C and B C, fix
+a pin, and one also in the point M, tie a thread around these pins,
+withdraw the pin from M, and trace the composite ellipse in the manner
+already described with respect to the simple ellipse.
+
+In some of my earlier works I described this figure by taking the angles
+of the isosceles triangle as foci; but the above method is much more
+correct. As the elementary angle of the triangle is (¹⁄₆), and that of
+the elliptic curve described around it (¹⁄₈), I call it the composite
+ellipse of (¹⁄₆) and (¹⁄₈), their harmonic ratio being 4:3; and so on of
+all others, according to the difference that may thus exist between the
+elementary angles.
+
+The visible curves which soften and beautify the melody of the outline
+of the front of the Parthenon, as given in Mr Penrose’s great work, I
+have carefully analysed, and have found them in as perfect agreement
+with this system, as its rectilinear harmony has been shewn to be. This
+I demonstrated in the work just referred to[14] by a series of twelve
+plates, shewing that the entasis of the columns (a subject upon which
+there has been much speculation) is simply an arc of an ellipse of
+(¹⁄₄₈), whose greater axis makes with the vertical an angle of (¹⁄₆₄);
+or simply, the form of one of these columns is the frustrum of an
+elliptic-sided or prolate-spheroidal cone, whose section is a composite
+ellipse of (¹⁄₄₈) and (¹⁄₆₄), the harmonic ratio of these two angles
+being 4:3, the same as that of the angles of the composite ellipse just
+exemplified.
+
+[Sidenote: Plate VII.]
+
+[Sidenote: Plate VIII.]
+
+In Plate VII. is represented the section of such a cone, of which A B C
+is the isosceles triangle of (¹⁄₄₈), and B D and D M the semi-axes of an
+ellipse of (¹⁄₆₄). M N and O P are the entases of the column, and _d e f_
+the normal construction of the capital. All these are fully illustrated
+in the work above referred to,[15] in which I have also shewn that the
+curve of the neck of the column is that of an ellipse of (¹⁄₆); the curve
+of the capital or echinus, that of an ellipse of (¹⁄₁₄); the curve of
+the moulding under the cymatium of the pediment, that of an ellipse of
+(¹⁄₃); and the curve of the bed-moulding of the cornice of the pediment,
+that of an ellipse of (¹⁄₃). The curve of the cavetto of the soffit of
+the corona is composed of ellipses of (¹⁄₆) and (¹⁄₁₄); the curve of the
+cymatium which surmounts the corona, is that of an ellipse of (¹⁄₃); the
+curve of the moulding of the capital of the antæ of the posticum, that of
+an ellipse of (¹⁄₃); the curves of the lower moulding of the same capital
+are composed of those of an ellipse of (¹⁄₃) and of the circle (¹⁄₂); the
+curve of the moulding which is placed between the two latter is that of
+an ellipse of (¹⁄₃); the curve of the upper moulding of the band under
+the beams of the ceiling of the peristyle, that of an ellipse of (¹⁄₃);
+the curve of the lower moulding of the same band, that of an ellipse of
+(¹⁄₄); and the curves of the moulding at the bottom of the small step
+or podium between the columns, are those of the circle (¹⁄₂) and of an
+ellipse of (¹⁄₃). I have also shewn the curve of the fluting of the
+columns to be that of (¹⁄₁₄). The greater axis of each of these ellipses,
+when not in the vertical or horizontal lines, makes an harmonic angle
+with one or other of them. In Plate VIII., sections of the two last-named
+mouldings are represented full size, which will give the reader an idea
+of the simple manner in which the ellipses are employed in the production
+of those harmonic curves.
+
+Thus we find that the system here adopted for applying this law of
+nature to the production of beauty in the abstract forms employed
+in architectural composition, so far from involving us in anything
+complicated, is characterised by extreme simplicity.
+
+In concluding this part of my treatise, I may here repeat what I have
+advanced in a late work,[16] viz., my conviction of the probability that
+a system of applying this law of nature in architectural construction
+was the only great practical secret of the Freemasons, all their other
+secrets being connected, not with their art, but with the social
+constitution of their society. This valuable secret, however, seems
+to have been lost, as its practical application fell into disuse; but,
+as that ancient society consisted of speculative as well as practical
+masons, the secrets connected with their social union have still been
+preserved, along with the excellent laws by which the brotherhood is
+governed. It can scarcely be doubted that there was some such practically
+useful secret amongst the Freemasons or early Gothic architects; for
+we find in all the venerable remains of their art which exist in this
+country, symmetrical elegance of form pervading the general design,
+harmonious proportion amongst all the parts, beautiful geometrical
+arrangements throughout all the tracery, as well as in the elegantly
+symmetrised foliated decorations which belong to that style of
+architecture. But it is at the same time worthy of remark, that whenever
+they diverged from architecture to sculpture and painting, and attempted
+to represent the human figure, or even any of the lower animals, their
+productions are such as to convince us that in this country these arts
+were in a very degraded state of barbarism—the figures are often much
+disproportioned in their parts and distorted in their attitudes, while
+their representations of animals and chimeras are whimsically absurd.
+It would, therefore, appear that architecture, as a fine art, must have
+been preserved by some peculiar influence from partaking of the barbarism
+so apparent in the sister arts of that period. Although its practical
+secrets have been long lost, the Freemasons of the present day trace the
+original possession of them to Moses, who, they say, “modelled masonry
+into a perfect system, and circumscribed its mysteries by _land-marks_
+significant and unalterable.” Now, as Moses received his education in
+Egypt, where Pythagoras is said to have acquired his first knowledge of
+the harmonic law of numbers, it is highly probable that this perfect
+system of the great Jewish legislator was based upon the same law of
+nature which constituted the foundation of the Pythagorean philosophy,
+and ultimately led to that excellence in art which is still the
+admiration of the world.
+
+Pythagoras, it would appear, formed a system much more perfect and
+comprehensive than that practised by the Freemasons in the middle ages of
+Christianity; for it was as applicable to sculpture, painting, and music,
+as it was to architecture. This perfection in architecture is strikingly
+exemplified in the Parthenon, as compared with the Gothic structures
+of the middle ages; for it will be found that the whole six elementary
+figures I have enumerated as belonging to architecture, are required in
+completing the orthographic beauty of that noble structure. And amongst
+these, none conduce more to that beauty than the simple and composite
+ellipses. Now, in the architecture of the best periods of Gothic, or,
+indeed, in that of any after period (Roman architecture included), these
+beautiful curves seem to have been ignored, and that of the circle alone
+employed.
+
+Be those matters as they may, however, the great law of numerical
+harmonic ratio remains unalterable, and a proper application of it in
+the science of art will never fail to be as productive of effect, as its
+operation in nature is universal, certain, and continual.
+
+
+
+
+THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE HUMAN HEAD AND COUNTENANCE.
+
+
+The most remarkable characteristics of the human head and countenance
+are the globular form of the cranium, united as it is with the prolate
+spheroidal form produced by the parts which constitute the face, and the
+approximation of the profile to the vertical; for in none of the lower
+animals does the skull present so near a resemblance to a combination
+of these geometric forms, nor the plane of the face to this direction.
+We also find that although these peculiar characteristics are variously
+modified among the numerous races of mankind, yet one law appears to
+govern the beauty of the whole. The highest and most cultivated of these
+races, however, present only an approximation to the perfect development
+of those distinguishing marks of humanity; and therefore the beauty of
+form and proportion which in nature characterises the human head and
+countenance, exhibits only a partial development of the harmonic law of
+visible beauty. On the other hand, we find that, in their sculpture, the
+ancient Greeks surpassed ordinary nature, and produced in their beau
+ideal a species of beauty free from the imperfections and peculiarities
+that constitute the individuality by which the countenances of men are
+distinguished from each other. It may be requisite here to remark, that
+this species of beauty is independent of the more intellectual quality
+of expression. For as Sir Charles Bell has said, “Beauty of countenance
+may be defined in words, as well as demonstrated in art. A face may
+be beautiful in sleep, and a statue without expression may be highly
+beautiful. But it will be said there is expression in the sleeping figure
+or in the statue. Is it not rather that we see in these the capacity
+for expression?—that our minds are active in imagining what may be the
+motions of these features when awake or animated? Thus, we speak of an
+expressive face before we have seen a movement grave or cheerful, or any
+indication in the features of what prevails in the heart.”
+
+This capacity for expression certainly enhances our admiration of the
+human countenance; but it is more a concomitant of the primary cause of
+its beauty than the cause itself. This cause rests on that simple and
+secure basis—the harmonic law of nature; for the nearer the countenance
+approximates to an harmonious combination of the most perfect figures in
+geometry, or rather the more its general form and the relation of its
+individual parts are arranged in obedience to that law, the higher its
+degree of beauty, and the greater its capacity for the expression of the
+passions.
+
+Various attempts have been made to define geometrically the difference
+between the ordinary and the ideal beauty of the human head and
+countenance, the most prominent of which is that of Camper. He traced,
+upon a profile of the skull, a line in a horizontal direction, passing
+through the foramen of the ear and the exterior margin of the sockets of
+the front teeth of the upper jaw, upon which he raised an oblique line,
+tangential to the margin of these sockets, and to the most prominent part
+of the forehead. Agreeably to the obliquity of this line, he determined
+the relative proportion of the areas occupied by the brain and by the
+face, and hence inferred the degree of intellect. When he applied this
+measurement to the heads of the antique statues, he found the angle much
+greater than in ordinary nature; but that this simple fact afforded no
+rule for the reproduction of the ideal beauty of ancient Greek art, is
+very evident from the heads and countenances by which his treatise is
+illustrated. Sir Charles Bell justly remarks, that although, by Camper’s
+method, the forehead may be thrown forward, yet, while the features
+of common nature are preserved, we refuse to acknowledge a similarity
+to the beautiful forms of the antique marbles. “It is true,” he says,
+“that, by advancing the forehead, it is raised, the face is shortened,
+and the eye brought to the centre of the head. But with all this, there
+is much wanting—that which measurement, or a mere line, will not shew
+us.”—“The truth is, that we are more moved by the features than by the
+form of the whole head. Unless there be a conformity in every feature to
+the general shape of the head, throwing the forehead forward on the face
+produces deformity; and the question returns with full force—How is it
+that we are led to concede that the antique head of the Apollo, or of the
+Jupiter, is beautiful when the facial line makes a hundred degrees with
+the horizontal line? In other words—How do we admit that to be beautiful
+which is not natural? Simply for the same reason that, if we discover a
+broken portion of an antique, a nose, or a chin of marble, we can say,
+without deliberation—This must have belonged to a work of antiquity;
+which proves that the character is distinguishable in every part—in each
+feature, as well as in the whole head.”
+
+Dr Oken says upon this subject:[17]—“The face is beautiful whose nose is
+parallel to the spine. No human face has grown into this estate; but
+every nose makes an acute angle with the spine. The facial angle is, as
+is well known, 80°. What, as yet, no man has remarked, and what is not to
+be remarked, either, without our view of the cranial signification, the
+old masters have felt through inspiration. They have not only made the
+facial angle a right angle, but have even stepped beyond this—the Romans
+going up to 96°, the Greeks even to 100°. Whence comes it that this
+unnatural face of the Grecian works of art is still more beautiful than
+that of the Roman, when the latter comes nearer to nature? The reason
+thereof resides in the fact of the Grecian artistic face representing
+nature’s design more than that of the Roman; for, in the former, the nose
+is placed quite perpendicular, or parallel to the spinal cord, and thus
+returns whither it has been derived.”
+
+Other various and conflicting opinions upon this subject have been
+given to the world; but we find that the principle from which arose
+the ideal beauty of the head and countenance, as represented in works
+of ancient Greek art, is still a matter of dispute. When, however, we
+examine carefully a fine specimen, we find its beauty and grandeur to
+depend more upon the degree of harmony amongst its parts, as to their
+relative proportions and mode of arrangement, than upon their excellence
+taken individually. It is, therefore, clear that those (and they are
+many) who attribute the beauty of ancient Greek sculpture merely to a
+selection of parts from various models, must be in error. No assemblage
+of parts from ordinary nature could have produced its principal
+characteristic, the excess in the angle of the facial line, much less
+could it have led to that exquisite harmony of parts by which it is so
+eminently distinguished; neither can we reasonably agree with Dr Oken and
+others, who assert that it was produced by an exclusive degree of the
+inspiration of genius amongst the Greek people during a certain period.
+
+That the inspiration of genius, combined with a careful study of nature,
+were essential elements in the production of the great works which have
+been handed down to us, no one will deny; but these elements have existed
+in all ages, whilst the ideal head belongs exclusively to the Greeks
+during the period in which the schools of Pythagoras and Plato were open.
+Is it not, therefore, reasonable to suppose, that, besides genius and the
+study of nature, another element was employed in the production of this
+excellence, and that this element arose from the precise mathematical
+doctrines taught in the schools of these philosophers?
+
+An application of the great harmonic law seems to prove that there is no
+object in nature in which the science of beauty is more clearly developed
+than in the human head and countenance, nor to the representations of
+which the same science is more easily applied; and it is to the mode
+in which this is done that the varieties of sex and character may be
+imparted to works of art. Having gone into full detail, and given ample
+illustrations in a former work,[18] it is unnecessary for me to enter
+upon that part of the subject in this _résumé_; but only to shew the
+typical structure of beauty by which this noble work of creation is
+distinguished.
+
+The angles which govern the form and proportions of the human head and
+countenance are, with the right angle, a series of seven, which, from the
+simplicity of their ratios to each other, are calculated to produce the
+most perfect concord. It consists of the right angle and its following
+parts—
+
+    Tonic.   Dominant.   Mediant.   Subtonic.
+    (¹⁄₂)     (¹⁄₃)       (¹⁄₅)       (¹⁄₇)
+    (¹⁄₄)     (¹⁄₆)
+
+These angles, and the figures which belong to them, are thus arranged:—
+
+[Sidenote: Plate IX.]
+
+The vertical line A B (Plate IX. fig. 2) represents the full length of
+the head and face. Taking this line as the greater axis of an ellipse of
+(¹⁄₃), such an ellipse is described around it. Through A the lines A G,
+A K, A L, A M, and A N, are drawn on each side of the line A B, making,
+with the vertical, respectively the angles of (¹⁄₃), (¹⁄₄), (¹⁄₅), (¹⁄₆),
+and (¹⁄₇). Through the points G, K, L, M, and N, where these straight
+lines meet the curved line of the ellipse, horizontal lines are drawn
+by which the following isosceles triangles are formed, A G G, A K K, A
+L L, A M M, and A N N. From the centre X of the equilateral triangle
+A G G the curvilinear figure of (¹⁄₂), viz., the circle, is described
+circumscribing that triangle.
+
+The curvilinear plane figures of (¹⁄₂) and (¹⁄₃), respectively, represent
+the solid bodies of which they are sections, viz., a sphere and a prolate
+spheroid. These bodies, from the manner in which they are here placed,
+are partially amalgamated, as shewn in figures 1 and 3 of the same plate,
+thus representing the form of the human head and countenance, both in
+their external appearance and osseous structure, more correctly than they
+could be represented by any other geometrical figures. Thus, the angles
+of (¹⁄₂) and (¹⁄₃) determine the typical form.
+
+From each of the points _u_ and _n_, where A M cuts G G on both sides of
+A B, a circle is described through the points _p_ and _q_, where A K cuts
+G G on both sides of A B, and with the same radius a circle is described
+from the point _a_, where K K cuts A B.
+
+The circles _u_ and _n_ determine the position and size of the eyeballs,
+and the circle _a_ the width of the nose, as also the horizontal width of
+the mouth.
+
+The lines G G and K K also determine the length of the joinings of the
+ear to the head. The lines L L and M M determine the vertical width of
+the mouth and lips when at perfect repose, and the line N N the superior
+edge of the chin. Thus simply are the features arranged and proportioned
+on the facial surface.
+
+It must, however, be borne in mind, that in treating simply of the
+æsthetic beauty of the human head and countenance, we have only to do
+with the external appearance. In this research, therefore, the system
+of Dr Camper, Dr Owen, and others, whose investigations were more of
+a physiological than an æsthetic character, can be of little service;
+because, according to that system, the facial angle is determined by
+drawing a line tangential to the exterior margin of the sockets of
+the front teeth of the upper jaw, and the most prominent part of the
+forehead. Now, as these sockets are, when the skull is naturally clothed,
+and the features in repose, entirely concealed by the upper lip, we must
+take the prominent part of it, instead of the sockets under it, in order
+to determine properly this distinguishing mark of humanity. And I believe
+it will be found, that when the head is properly poised, the nearer the
+angle which this line makes with the horizontal approaches 90°, the more
+symmetrically beautiful will be the general arrangement of the parts (see
+line _y z_, figure 3, Plate IX.).
+
+
+
+
+THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE FORM OF THE HUMAN FIGURE.
+
+
+The manner in which this science is developed in the symmetrical
+proportions of the entire human figure, is as remarkable for its
+simplicity as it has been shewn to be in those of the head and
+countenance. Having gone into very full details, and given ample
+illustration in two former works[19] upon this subject, I may here
+confine myself to the illustration of one description of figure, and to a
+reiteration of some facts stated in these works. These facts are, _1st_,
+That on a given line the human figure is developed, as to its principal
+points, entirely by lines drawn either from the extremities of this line,
+or from some obvious or determined localities. _2d_, That the angles
+which these lines make with the given line, are all simple sub-multiples
+of some given fundamental angle, or bear to it a proportion expressible
+under the most simple relations, such as those which constitute the scale
+of music. _3d_, That the contour is resolved into a series of ellipses
+of the same simple angles. And, _4th_, That these ellipses, like the
+lines, are inclined to the first given line by angles which are simple
+sub-multiples of the given fundamental angle. From which four facts,
+and agreeably to the hypothesis I have adopted, it results as a natural
+consequence that the only effort which the mind exercises through the
+eye, in order to put itself in possession of the data for forming its
+judgment, is this, that it compares the angles about a point, and thereby
+appreciates the simplicity of their relations. In selecting the prominent
+features of a figure, the eye is not seeking to compare their relative
+distances—it is occupied solely with their relative positions. In tracing
+the contour, in like manner, it is not left in vague uncertainty as
+to what is the curve which is presented to it; unconsciously it feels
+the complete ellipse developed before it; and if that ellipse and its
+position are both formed by angles of the same simple relative value as
+those which aided its determination of the positions of the prominent
+features, it is satisfied, and finds the symmetry perfect.
+
+Müller, and other investigators into the archæology of art, refer to the
+great difficulty which exists in discovering the principles which the
+ancients followed in regard to the proportions of the human figure, from
+the different sexes and characters to which they require to be applied.
+But in the system thus founded upon the harmonic law of nature, no such
+difficulty is felt, for it is as applicable to the massive proportions
+which characterise the ancient representations of the Hercules, as to the
+delicate and perfectly symmetrical beauty of the Venus. This change is
+effected simply by an increase in the fundamental angle. For instance,
+in the construction of a figure of the exact proportions of the Venus,
+the right angle is adopted. But in the construction of a figure of the
+massive proportions of the Hercules, it is requisite to adopt an angle
+which bears to the right angle the ratio of 6:5. The adoption of this
+angle I have shewn in another work[20] to produce in the Hercules those
+proportions which are so characteristic of physical power. The ellipses
+which govern the outline, being also formed upon the same larger class
+of angles, give the contour of the muscles a more massive character. In
+comparing the male and female forms thus geometrically constructed, it
+will be found that that of the female is more harmoniously symmetrical,
+because the right angle is the fundamental angle for the trunk and the
+limbs as well as for the head and countenance; while in that of the male,
+the right angle is the fundamental angle for the head only. It may also
+be observed, that, from the greater proportional width of the pelvis
+of the female, the centres of that motion which the heads of the thigh
+bones perform in the cotyloid cavities, and the centres of that still
+more extensive range of motion which the arm is capable of performing at
+the shoulder joints, are nearly in the same line which determines the
+central motion of the vertebral column, while those of the male are not;
+consequently all the motions of the female are more graceful than those
+of the male.
+
+This difference between the fundamental angles, which impart to the
+human figure, on the one hand, the beauty of feminine proportion and
+contour, and on the other, the grandeur of masculine strength, being in
+the ratio of 5:6, allows ample latitude for those intermediate classes of
+proportions which the ancients imparted to their various other deities
+in which these two qualities were blended. I therefore confine myself to
+an illustration of the external contour of the form, and the relative
+proportions of all the parts of a female figure, such as those of the
+statues of the Venus of Melos and Venus of Medici.
+
+The angles which govern the form and proportions of such a figure are,
+with the right angle, a series of twelve, as follows:—
+
+    Tonic.   Dominant.   Mediant.   Subtonic.   Supertonic.
+    (¹⁄₂)      (¹⁄₃)       (¹⁄₅)       (¹⁄₇)       (¹⁄₉)
+    (¹⁄₄)      (¹⁄₆)       (¹⁄₁₀)      (¹⁄₁₄)
+    (¹⁄₈)      (¹⁄₁₂)
+
+These angles are employed in the construction of a diagram, which
+determines the proportions of the parts throughout the whole figure.
+Thus:—
+
+[Sidenote: Plate X.]
+
+Let the line A B (fig. 1, plate X.) represent the height of the figure
+to be constructed. At the point A, make the angles of C A D (¹⁄₃), F A G
+(¹⁄₄), H A I (¹⁄₅), K A L (¹⁄₆), and M A N (¹⁄₇). At the point B, make
+the angles K B L (¹⁄₈), U B A (¹⁄₁₂), and O B A (¹⁄₁₄).
+
+Through the point K, in which the lines A K and B K intersect one
+another, draw P K O parallel to A B, and through C F H and M, where
+this line meets A C, A F, A H, and A M, draw C D, F G, H I, and M N,
+perpendicular to A B; draw also K L perpendicular to A B; join B F and
+B H, and through C draw C E, making with A B the angle (¹⁄₂), which
+completes the arrangement of the eleven angles upon A B; for F B G is
+very nearly (¹⁄₁₀), and H B I very nearly (¹⁄₉).
+
+At the point _f_, where A C intersects O B, draw _f a_ perpendicular to
+A B; and through the point _i_, where B O intersects M N, draw S _i_ T
+parallel to A C.
+
+Through _m_, where S _i_ T intersects F B, draw _m n_; through _β_, where
+S _i_ T intersects K B, draw _β w_; through T draw T _g_, making an angle
+of (¹⁄₃) with O P. Join N P, M B, and _g_ P, and where N P intersects K
+B, draw Q R perpendicular to A B.
+
+On A E as a diameter, describe a circle cutting A C in _r_, and draw _r
+o_ perpendicular to A B.
+
+With A _o_ and _o r_ as semi-axes, describe the ellipse A _r e_, cutting
+A H in _t_; and draw _t u_ perpendicular to A B. With A _u_ and _t u_, as
+semi-axes describe the ellipse A _t d_. On _a_ L, as major axis, describe
+the ellipse of (¹⁄₃).
+
+For the side aspect or profile of the figure the diagram is thus
+constructed—
+
+On one side of a line A B (fig. 2, Plate X.) construct the rectilinear
+portion of a diagram the same as fig. 1. Through _i_ draw W Y parallel to
+A B, and draw A _z_ perpendicular to A B. Make W _a_ equal to A _a_ (fig.
+1), and on _a l_, as major axis, describe the ellipse of (¹⁄₄). Through
+_a_ draw _a p_ parallel to A F, and through _p_ draw _p t_ perpendicular
+to W Y. Through _a_ draw _f a u_ perpendicular to W Y.
+
+Upon a diameter equal to A E describe a circle whose circumference shall
+touch A B and A _z_. With semi-axes equal to A _o_ and _o r_ (fig.
+1), describe an ellipse with its major axis parallel to A B, and its
+circumference touching O P and _z_ A.
+
+[Sidenote: Plate XI.]
+
+Thus simply are the diagrams of the general proportions of the human
+figure, as viewed in front and in profile, constructed; and Plate XI.
+gives the contour in both points of view, as composed entirely of the
+curvilinear figures of (¹⁄₂), (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆).
+
+Further detail here would be out of place, and I shall therefore refer
+those who require it to the Appendix, or the more elaborate works to
+which I have already referred.
+
+The beauty derived from proportion, imparted by the system here pointed
+out, and from a contour of curves derived from the same harmonic angles,
+is not confined to the human figure, but is found in various minor
+degrees of perfection in all the organic forms of nature, whether animate
+or inanimate, of which I have in other works given many examples.[21]
+
+
+
+
+THE SCIENCE OF BEAUTY, AS DEVELOPED IN COLOURS.
+
+
+There is not amongst the various phenomena of nature one that more
+readily excites our admiration, or makes on the mind a more vivid
+impression of the order, variety, and harmonious beauty of the creation,
+than that of colour. On the general landscape this phenomenon is
+displayed in the production of that species of harmony in which colours
+are so variously blended, and in which they are by light, shade, and
+distance modified in such an infinity of gradation and hue. Although
+genius is continually struggling, with but partial success, to imitate
+those effects, yet, through the Divine beneficence, all whose organs of
+sight are in an ordinary degree of perfection can appreciate and enjoy
+them. In winter this pleasure is often to a certain extent withdrawn,
+when the colourless snow alone clothes the surface of the earth. But
+this is only a pause in the general harmony, which, as the spring
+returns, addresses itself the more pleasingly to our perception in its
+vernal melody, which, gradually resolving itself into the full rich hues
+of luxuriant beauty exhibited in the foliage and flowers of summer,
+subsequently rises into the more vivid and powerful harmonies of autumn’s
+colouring. Thus the eye is prepared again to enjoy that rest which such
+exciting causes may be said to have rendered necessary.
+
+When we pass from the general colouring of nature to that of particular
+objects, we are again wrapt in wonder and admiration by the beauty and
+harmony which so constantly, and in such infinite variety, present
+themselves to our view, and which are so often found combined in the
+most minute objects. And the systematic order and uniformity perceptible
+amidst this endless variety in the colouring of animate and inanimate
+nature is thus another characteristic of beauty equally prevalent
+throughout creation.
+
+By this uniformity in colour, various species of animals are often
+distinguished; and in each individual of most of these species, how much
+is this beauty enhanced when the uniformity prevails in the resemblance
+of their lateral halves! The human countenance exemplifies this in a
+striking manner; the slightest variety of colour between one and another
+of the double parts is at once destructive of its symmetrical beauty.
+Many of the lower animals, whether inhabitants of the earth, the air, or
+the water, owe much of their beauty to this kind of uniformity in the
+colour of the furs, feathers, scales, or shells, with which they are
+clothed.
+
+In the vegetable kingdom, we find a great degree of uniformity of colour
+in the leaves, flowers, and fruit of the same plant, combined with all
+the harmonious beauty of variety which a little careful examination
+develops.
+
+In the colours of minerals, too, the same may be observed. In short, in
+the beauty of colouring, as in every other species of beauty, uniformity
+and variety are found to combine.
+
+An appreciation of colour depends, in the first place, as much upon the
+physical powers of the eye in conveying a proper impression to the mind,
+as that of music on those of the ear. But an ear for music, or an eye for
+colour, are, in so far as beauty is concerned, erroneous expressions;
+because they are merely applicable to the impression made upon the
+senses, and do not refer to the æsthetical principles of harmony, by
+which beauty can alone be understood.
+
+A good eye, combined with experience, may enable us to form a correct
+idea as to the purity of an individual colour, or of the relative
+difference existing between two separate hues; but this sort of
+discrimination does not constitute that kind of appreciation of the
+harmony of colour by which we admire and enjoy its development in nature
+and art. The power of perceiving and appreciating beauty of any kind,
+is a principle inherent in the human mind, which may be improved by
+cultivation in the degree of the perfection of the art senses. Great
+pains have been bestowed on the education of the ear, in assisting it to
+appreciate the melody and harmony of sound; but still much remains to be
+done in regard to the cultivation of the eye, in appreciating colour as
+well as form.
+
+It is true, that there are individuals whose powers of vision are
+perfect, in so far as regards the appreciation of light, shade, and
+configuration, but who are totally incapable of perceiving effects
+produced by the intermediate phenomenon of colour, every object appearing
+to them either white, black, or neutral gray; others, who are equally
+blind as to the effect of one of the three primary colours, but see the
+other two perfectly, either singly or combined; while there are many
+who, having the full physical power of perceiving all the varieties of
+the phenomenon, and who are even capable of making nice distinctions
+amongst a variety of various colours, are yet incapable of appreciating
+the æsthetic quality of harmony which exists in their proper combination.
+It is the same with respect to the effects of sounds upon the ear—some
+have organs so constituted, that notes above or below a certain pitch
+are to them inaudible; while others, with physical powers otherwise
+perfect, are incapable of appreciating either melody or harmony in
+musical composition. But perceptions so imperfectly constituted are,
+by the goodness of the Creator, of very rare occurrence; therefore all
+attempts at improvement in the science of æsthetics must be suited to the
+capacities of the generality of mankind, amongst whom the perception of
+colour exists in a variety as great as that by which their countenances
+are distinguished. Artists now and then appear who have this intuitive
+perception in such perfection, that they are capable of transferring to
+their works the most beautiful harmonies and most delicate gradations of
+colours, in a manner that no acquired knowledge could have enabled them
+to impart. To those who possess such a gift, as well as to those to whom
+the ordinary powers of perception are denied, it would be equally useless
+to offer an explanation of the various modes in which the harmony of
+colour develops itself, or to attempt a definition of the many various
+colours, hues, tints, and shades, arising out of the simple elements of
+this phenomenon. But to those whose powers lie between these extremes,
+being neither above nor below cultivation, such an explanation and
+definition must form a step towards the improvement of that inherent
+principle which constitutes the basis of æsthetical science.
+
+Although the variety and harmony of colour which nature is continually
+presenting to our view, are apparent to all whose visual organs are in a
+natural state, and thus to the generality of mankind; yet a knowledge of
+the simplicity by which this variety and beauty are produced, is, after
+ages of philosophic research and experimental inquiry, only beginning to
+be properly understood.
+
+Light may be considered as an active, and darkness a passive principle in
+the economy of Nature, and colour an intermediate phenomenon arising from
+their joint influence; and it is in the ratios in which these primary
+principles act upon each other, by which I here intend to explain the
+science of beauty as evolved in colour. It has been usual to consider
+colour as an inherent quality in light, and to suppose that coloured
+bodies absorb certain classes of its rays, and reflect or transmit the
+remainder; but it appears to me that colour is more probably the result
+of certain modes in which the opposite principles of motion and rest,
+or force and resistance, operate in the production, refraction, and
+reflection of light, and that each colour is mutually related, although
+in different degrees, to these active and passive principles.
+
+White and black are the representatives of light and darkness, or
+activity and rest, and are therefore calculated as pigments to reduce
+colours and hues to tints and shades.
+
+Having, however, fully illustrated the nature of tints and shades in
+a former work,[22] I shall here confine myself to colours in their
+full intensity—shewing the various modifications which their union
+with each other produce, along with the harmonic relations which these
+modifications bear to the primaries, and to each other in respect to
+warmth and coolness of tone, as well as to light and shade.
+
+The primary colours are red, yellow, and blue. Of these, yellow is
+most allied to light, and blue to shade, while red is neutral in these
+respects, being equally allied to both. In respect to tone, that of red
+is warm, and that of blue cool, while the tone of yellow is neutral. The
+ratios of their relations to each other in these respects will appear in
+the harmonic scales to which, for the first time, I am about to subject
+colours, and to systematise their various simple and compound relations,
+which are as follow:—
+
+From the binary union of the primary colours, the secondary colours arise—
+
+Orange colour, from the union of yellow and red.
+
+Green, from the union of yellow and blue.
+
+Purple, from the union of red and blue.
+
+From the binary union of the secondary colours, the primary hues arise—
+
+Yellow-hue, from the union of orange and green.
+
+Red-hue, from the union of orange and purple.
+
+Blue-hue, from the union of purple and green.
+
+From the binary union of the primary hues, the secondary hues arise—
+
+Orange-hue, from the union of yellow-hue and red-hue.
+
+Green-hue, from the union of yellow-hue and blue-hue.
+
+Purple-hue, from the union of red-hue and blue-hue.
+
+Each hue owes its characteristic distinction to the proportionate
+predominance or subordination of one or other of the three primary
+colours in its composition.
+
+It follows, that in every hue of _red_, yellow and blue are subordinate;
+in every hue of _yellow_, red and blue are subordinate; and in every
+hue of _blue_, red and yellow are subordinate. In like manner, in every
+hue of _green_, red is subordinate; in every hue of _orange_, blue is
+subordinate; and in every hue of _purple_, yellow is subordinate.
+
+By the union of two primary colours, in the production of a secondary
+colour, the nature of both primaries is altered; and as there are only
+three primary or simple colours in the scale, the two that are united
+harmonically in a compound colour, form the natural contrast to the
+remaining simple colour.
+
+Notwithstanding all the variety that extends beyond the six positive
+colours, it may be said that there are only three proper contrasts of
+colour in nature, and that all others are simply modifications of these.
+
+Pure red is the most perfect contrast to pure green; because it is
+characterised amongst the primary colours by warmth of tone, while
+amongst the secondary colours green is distinguished by coolness of tone,
+both being equally related to the primary elements of light and shade.
+
+Pure yellow is the most perfect contrast to pure purple; because it is
+characterised amongst the primary colours as most allied to light, whilst
+pure purple is characterised amongst the secondaries as most allied to
+shade, both being equally neutral as to tone.
+
+Pure blue is the most perfect contrast to pure orange; because it is
+characterised amongst the primary colours as not only the most allied
+to shade, but as being the coolest in tone, whilst pure orange is
+characterised amongst the secondaries as being the most allied to light
+and the warmest in tone. The same principle operates throughout all the
+modifications of these primary and secondary colours.
+
+Such is the simple nature of contrast upon which the beauty of colouring
+mainly depends.
+
+It being now established as a scientific fact, that the effect of
+light upon the eye is the result of an ethereal action, similar to the
+atmospheric action by which the effect of sound is produced upon the
+ear; also, that the various colours which light assumes are the effect
+of certain modifications in this ethereal action;—just as the various
+sounds, which constitute the scale of musical notes, are known to be the
+effect of certain modifications in the atmospheric action by which sounds
+in general are produced:
+
+Therefore, as harmony may thus be impressed upon the mind through
+either of these two art senses—hearing and seeing—the principles which
+govern the modifications in the ethereal action of light, so as to
+produce through the eye the effect of harmony, cannot be supposed to
+differ from those principles which we know govern the modifications of
+the atmospheric action of sound, in producing through the ear a like
+effect. I shall therefore endeavour to illustrate the science of beauty
+as evolved in colours, by forming scales of their various modifications
+agreeably to the same Pythagorean system of numerical ratio from which
+the harmonic elements of beauty in sounds were originally evolved, and by
+which I have endeavoured in this, as in previous works, to systematise
+the harmonic beauty of forms.
+
+[Illustration]
+
+It will be observed, that with a view to avoid complexity as much as
+possible, I have, in the arrangement of the above series of scales, not
+only confined myself to the merely elementary parts of the Pythagorean
+system, but have left out the harmonic modifications upon (¹⁄₁₁)
+and (¹⁄₁₃), in order that the arithmetical progression might not be
+interrupted.[23]
+
+The above elementary process will, I trust, be found sufficient to
+explain the progress, by harmonic union, of a primary colour to a
+toned gray, and how the simple and compound colours naturally arrange
+themselves into the elements of five scales, the parts of which continue
+from primary to secondary colour; from secondary colour to primary hue;
+from primary hue to secondary hue; from secondary hue to primary-toned
+gray; and from primary-toned gray to secondary-toned gray in the simple
+ratio of 2:1; thereby producing a series of the most beautiful and
+perfect contrasts.
+
+The natural arrangement of the primary colours upon the solar spectrum is
+red, yellow, blue, and I have therefore adopted the same arrangement on
+the present occasion. Red being, consequently, the first tonic, and blue
+the second, the divisions express the numerical ratios which the colours
+bear to one another, in respect to that colourific power for which red
+is pre-eminent. Thus, yellow is to red, as 2:3; blue to yellow, as 3:4;
+purple to orange, as 5:6; and green to purple, as 6:7.
+
+The following series of completed scales are arranged upon the
+foregoing principle, with the natural connecting links of red-orange,
+yellow-orange, yellow-green, and blue-green, introduced in their proper
+places.
+
+The appropriate terminology of musical notes has been adopted, and the
+scales are composed as follows:—
+
+    Scale I. consists of primary and secondary colours;
+    Scale II. of secondary colours and primary hues;
+    Scale III. of primary and secondary hues;
+    Scale IV. of secondary hues and primary-toned grays; and
+    Scale V. of primary and secondary-toned grays;
+
+All the parts in each of these scales, from the first tonic to the
+second, relate to the same parts of the scale below them in the simple
+ratio of 2:1; and serially to the first tonic in the following ratios:—
+
+                 8:9, 4:5, 3:4, 2:3, 3:5, 4:7, 8:15, 1:2.
+
+
+_First Series of Scales._
+
+ ----+------+-------+-------+-------+-------+-------+------+-------+-------
+     |Tonic.|       |       |       |       |       |      |       |
+     |      |Supertonic.    |       |       |       |      |       |
+     |      |       |Mediant.       |       |       |      |       |
+     |      |       |       |Subdominant.   |       |      |       |
+     |      |       |       |       |Dominant.      |      |       |
+     |      |       |       |       |       |Submediant.   |       |
+     |      |       |       |       |       |       |Subtonic.     |
+     |      |       |       |       |       |       |      |Semi-subtonic.
+     |      |       |       |       |       |       |      |       |Tonic.
+ ----+------+-------+-------+-------+-------+-------+------+-------+-------
+ I.  |(¹⁄₂) |(⁴⁄₉)  |(²⁄₅)  |(³⁄₈)  |(¹⁄₃)  |(³⁄₁₀) |(²⁄₇) |(⁴⁄₁₅) |(¹⁄₄)
+     |Red.  |Red-   |Orange.|Yellow-|Yellow.|Yellow-|Green.|Blue-  |Blue.
+     |      |orange.|       |orange.|       |green. |      |green. |
+ ----+------+-------+-------+-------+-------+-------+------+-------+-------
+ II. |(¹⁄₄) |(²⁄₉)  |(¹⁄₅)  |(³⁄₁₆) |(¹⁄₆)  |(³⁄₂₀) |(¹⁄₇) |(²⁄₁₅) |(¹⁄₈)
+     |Green.|Blue-  |Blue   |Blue-  |Purple |Red-   |Red   |Red-   |Orange.
+     |      |green  |hue.   |purple |hue.   |purple |hue.  |orange |
+     |      |hue.   |       |hue.   |       |hue.   |      |hue.   |
+ ----+------+-------+-------+-------+-------+-------+------+-------+-------
+ III.|(¹⁄₈) |(¹⁄₉)  |(¹⁄₁₀) |(³⁄₃₂) |(¹⁄₁₂) |(³⁄₄₀) |(¹⁄₁₄)|(¹⁄₁₅) |(¹⁄₁₆)
+     |Red   |Red-   |Orange |Yellow-|Yellow |Yellow-|Green |Blue-  |Blue
+     |hue.  |orange |hue.   |orange |hue.   |green  |hue.  |green  |hue.
+     |      |hue.   |       |hue.   |       |hue.   |      |hue.   |
+ ----+------+-------+-------+-------+-------+-------+------+-------+-------
+ IV. |(¹⁄₁₆)|(¹⁄₁₈) |(¹⁄₂₀) |(³⁄₆₄) |(¹⁄₂₄) |(³⁄₈₀) |(¹⁄₂₈)|(¹⁄₃₀) |(¹⁄₃₂)
+     |Green |Blue-  |Blue-  |Blue-  |Purple |Red-   |Red-  |Red-   |Orange
+     |hue.  |green- |toned  |purple-|hue.   |purple-|toned |orange-|hue.
+     |      |toned  |gray.  |toned  |       |toned  |gray. |toned  |
+     |      |gray.  |       |gray.  |       |gray.  |      |gray.  |
+ ----+------+-------+-------+-------+-------+-------+------+-------+-------
+ V.  |(¹⁄₃₂)|(¹⁄₃₆) |(¹⁄₄₀) |(³⁄₁₂₈)|(¹⁄₄₈) |(³⁄₁₆₀)|(¹⁄₅₆)|(¹⁄₆₀) |(¹⁄₆₄)
+     |Red-  |Red-   |Orange-|Yellow-|Yellow-|Yellow-|Green-| Blue- |Blue-
+     |toned |orange-|toned  |orange-|toned  |green- |toned | green-|toned
+     |gray. |toned  |gray.  |toned  |gray.  |toned  |gray. | toned |gray.
+     |      |gray.  |       |gray.  |       | gray. |      | gray. |
+ ----+------+-------+-------+-------+-------+-------+------+-------+-------
+
+To the scales of chromatic power I add another series of scales, in
+which yellow, being the first tonic, and blue the second, the numerical
+divisions express the ratios which the colours in each scale bear to one
+another in respect to light and shade. Thus red is to yellow, in respect
+to light, as 2:3; blue to red, as 3:4; green to orange, as 5:6, and
+purple to green, as 6:7.
+
+These scales may therefore be termed scales for the colour-blind,
+because, in comparing colours, those whose sight is thus defective,
+naturally compare the ratios of the light and shade of which different
+colours are primarily constituted.
+
+[Illustration]
+
+The following is a series of five complete scales of the harmonic parts
+into which the light and shade in colours may be divided in each scale
+according to the above arrangement:—
+
+
+_Second Series of Scales._
+
+ ----+-------+-------+-------+-------+------+-------+-------+-------+------
+     |Tonic. |       |       |       |      |       |       |       |
+     |       |Supertonic.    |       |      |       |       |       |
+     |       |       |Mediant.       |      |       |       |       |
+     |       |       |       |Subdominant.  |       |       |       |
+     |       |       |       |       |Dominant.     |       |       |
+     |       |       |       |       |      |Submediant.    |       |
+     |       |       |       |       |      |       |Subtonic.      |
+     |       |       |       |       |      |       |       |Semi-subtonic.
+     |       |       |       |       |      |       |       |       |Tonic.
+ ----+-------+-------+-------+-------+------+-------+-------+-------+------
+ I.  |(¹⁄₂)  |(⁴⁄₉)  |(²⁄₅)  |(³⁄₈)  |(¹⁄₃) |(³⁄₁₀) |(²⁄₇)  |(⁴⁄₁₅) |(¹⁄₄)
+     |Yellow.|Yellow-|Orange.|Red-   |Red.  |Red-   |Purple.|Blue-  |Blue.
+     |       |orange.|       |orange.|      |purple.|       |purple.|
+ ----+-------+-------+-------+-------+------+-------+-------+-------+------
+ II. |(¹⁄₄)  |(²⁄₉)  |(¹⁄₅)  |(³⁄₁₆) |(¹⁄₆) |(³⁄₂₀) |(¹⁄₇)  |(²⁄₁₅) |(¹⁄₈)
+     |Purple.|Blue-  |Blue   |Blue-  |Green.|Yellow-|Yellow |Yellow-|Orange
+     |       |purple |hue.   |green  |      |green  |hue.   |orange |
+     |       |hue.   |       |hue.   |      |hue.   |       |hue.   |
+ ----+-------+-------+-------+-------+------+-------+-------+-------+------
+ III.|(¹⁄₈)  |(¹⁄₉)  |(¹⁄₁₀) |(³⁄₃₂) |(¹⁄₁₂)|(³⁄₄₀) |(¹⁄₁₄) |(¹⁄₁₅) |(¹⁄₁₆)
+     |Yellow |Yellow-|Orange |Red-   |Red   |Red-   |Purple |Blue-  |Blue
+     |hue.   |orange |hue.   |orange |hue.  |purple |hue.   |purple |hue.
+     |       |hue.   |       |hue.   |      |hue.   |       |hue.   |
+ ----+-------+-------+-------+-------+------+-------+-------+-------+------
+ IV. |(¹⁄₁₆) |(¹⁄₁₈) |(¹⁄₂₀) |(³⁄₆₄) |(¹⁄₂₄)|(³⁄₈₀) |(¹⁄₂₈) |(¹⁄₃₀) |(¹⁄₃₂)
+     |Purple |Blue-  |Blue-  |Blue-  |Green |Yellow-|Yellow-|Yellow-|Orange
+     |hue.   |purple-|toned  |green- |hue.  |green- |toned  |orange-|hue.
+     |       |toned  |gray.  |toned  |      |toned  |gray.  |toned  |
+     |       |gray.  |       |gray.  |      |gray.  |       |gray.  |
+ ----+-------+-------+-------+-------+------+-------+-------+-------+------
+ V.  |(¹⁄₃₂) |(¹⁄₃₆) |(¹⁄₄₀) |(³⁄₁₂₈)|(¹⁄₄₈)|(³⁄₁₆₀)|(¹⁄₅₆) |(¹⁄₆₀) |(¹⁄₆₄)
+     |Yellow-|Yellow-|Orange-|Red-   |Red-  |Red-   |Purple-|Blue-  |Blue-
+     |toned  |orange-|toned  |orange-|toned |purple-|toned  |green- |toned
+     |gray.  |toned  |gray.  |toned  |gray. |toned  |gray.  |toned  |gray.
+     |       |gray.  |       |gray.  |      |gray.  |       |gray.  |
+ ----+-------+-------+-------+-------+------+-------+-------+-------+------
+
+Should I be correct in arranging colours upon scales identical with those
+upon which musical notes have been arranged, and in assuming that colours
+have the same ratios to each other, in respect to their harmonic power
+upon the eye, which musical notes have in respect to their harmonic power
+upon the ear, the colourist may yet be enabled to impart harmonic beauty
+to his works with as much certainty and ease, as the musician imparts the
+same quality to his compositions: for the colourist has no more right to
+trust exclusively to his eye in the arrangement of colours, than the
+musician has to trust exclusively to his ear in the arrangement of sounds.
+
+We find, in comparing the dominant parts in the first and second scales
+of the second series, that they are equal as to light and shade, so that
+their relative powers of contrast depend entirely upon colour. Hence it
+is that red and green are the two colours, the difference between which
+the colour-blind are least able to appreciate. Professor George Wilson,
+in his excellent work, “Researches on Colour-Blindness,” mentions the
+case of an engraver, which proves the power of the eye in being able to
+appreciate these original constituents of colour, irrespective of the
+intermediate phenomenon of tone. This engraver, instead of expressing
+regret on account of his being colour-blind, observed to the professor,
+“My defective vision is, to a certain extent, a useful and valuable
+quality. Thus, an engraver has two negatives to deal with, _i.e._, white
+and black. Now, when I look at a picture, I see it only in white and
+black, or light and shade, or, as artists term it, the effect. I find
+at times many of my brother engravers in doubt how to translate certain
+colours of pictures, which to me are matters of decided certainty and
+ease. Thus to me it is valuable.”
+
+The colour-blind are therefore as incapable of receiving pleasure from
+the harmonious union of various colours, as those who, to use a common
+term, have no ear for music, are of being gratified by the “melody of
+sweet sounds.”
+
+The generality of mankind are, however, capable of appreciating the
+harmony of colour which, like that of both sound and form, arises from
+the simultaneous exhibition of opposite principles having a ratio to each
+other. These principles are in continual operation throughout nature,
+and from them we often derive pleasure without being conscious of the
+cause. All who are not colour-blind must have felt themselves struck
+with the harmonic beauty of a cloudless sky, although in it there is no
+configuration, and at first sight apparently but one colour. Now, as
+we know that there can be no more impression of harmony made upon the
+mind by looking upon a single colour, than there could be by listening
+to a single continued musical note, however sweet its tone, we are apt
+at first to imagine that the organ of vision has, in some measure,
+conveyed a false impression to the mind. But it has not done so; for
+light, when reflected from the atmosphere, produces those cool tones of
+blue, gray, and purple, which seem to clothe the distant mountains; but,
+when transmitted through the same atmosphere, it produces those numerous
+warm tints, the most intense of which give the gorgeous effects which
+so often accompany the setting sun. We have, therefore, in the upper
+part of a clear sky, where the atmosphere may be said to be illuminated
+principally by reflection from the surface of the earth, a comparatively
+cool tone of blue, the result of reflection, which gradually blends into
+the warm tints, the result of transmission through the same atmosphere.
+Such a composition of harmonious colouring is to the eye what the voice
+of the soft breath of summer amongst the trees, the hum of insects on
+a sultry day, or the simple harmony of the Æolian harp, is to the ear.
+To such a composition of chromatic harmony must also be referred the
+universal concurrence of mankind in appreciating the peculiar beauty of
+white marble statuary. That the principal constituent of beauty in such
+works ought to be harmony of form, no one will deny; but this is not the
+only element, as appears from the fact, that a cast in plaster of Paris,
+of a fine white marble statue, although identical in form, is far less
+beautiful than the original. Now this undoubtedly must be the consequence
+of its having been changed from a semi-translucent substance, which,
+like the atmosphere, can transmit as well as reflect light, to an opaque
+substance, which can only reflect it. Thus the opposite principles of
+chromatic warmth and coolness are equally balanced in white marble—the
+one being the natural result of the partial transmission of light, and
+the other that of its reflection.
+
+As a series of coloured illustrations would be beyond the scope of this
+_résumé_, I may refer those who wish to prosecute the inquiry, with the
+assistance of such a series, to my published works upon the subject.[24]
+
+
+
+
+THE SCIENCE OF BEAUTY, APPLIED TO THE FORMS AND PROPORTIONS OF ANCIENT
+GRECIAN VASES AND ORNAMENTS.
+
+
+In examining the remains of the ornamental works of the ancient Greek
+artists, it appears highly probable that the harmony of their proportions
+and melody of their contour are equally the result of a systematised
+application of the same harmonic law. This probability not being fully
+elucidated in any of my former works, I will require to go into some
+detail on the present occasion. I take for my first illustration an
+unexceptionable example, viz.:—
+
+
+_The Portland Vase._
+
+Although this beautiful specimen of ancient art was found about the
+middle of the sixteenth century, inclosed in a marble sarcophagus within
+a sepulchral chamber under the Monte del Grano, near Rome, and although
+the date of its production is unknown, yet its being a work of ancient
+Grecian art is undoubted; and the exquisite beauty of its form has been
+universally acknowledged, both during the time it remained in the palace
+of the Barberini family at Rome, and since it was added to the treasures
+of the British Museum. The forms and proportions of this gem of art
+appear to me to yield an obedience to the great harmonic law of nature,
+similar to that which I have instanced in the proportions and contour of
+the best specimens of ancient Grecian architecture.
+
+[Sidenote: Plate XII.]
+
+Let the line A B (Plate XII.) represent the full height of the vase.
+Through A draw A _a_, and through B draw B _b_ indefinitely, A _a_ making
+an angle of (¹⁄₂), and B _b_ an angle of (¹⁄₃), with the vertical.
+Through the point C, where A _a_ and B _b_ intersect one another, draw
+D C E vertical. Through A C and B respectively, draw A D, C F, and B
+E horizontal. Draw similar lines on the other side of A B, and the
+rectilinear portion of the diagram is complete.
+
+The curvilinear contour may be thus added:—
+
+Take a cut-out ellipse of (¹⁄₄), whose greater axis is equal to the line
+A B, and
+
+_1st._ Place it upon the diagram, so that its circumference may be
+tangential to the lines C E and C F, and its greater axis _m n_ may make
+an angle of (¹⁄₅) with the vertical, and trace its circumference.
+
+_2d._ Place it with its circumference tangential to that of the first at
+the point m, while its greater axis (of which _o p_ is a part) is in the
+horizontal, and trace the portion of its circumference _q o r_.
+
+_3d._ Place it with its circumference tangential to that of the above at
+_v_, while its greater axis (of which _u v_ is a part) makes an angle of
+(³⁄₁₀) with the vertical, and trace the portion of its circumference _s v
+t_.
+
+Thus the curvilinear contour of the body and neck are harmonically
+determined.
+
+The curve of the handle may be determined by the same ellipse placed so
+that its greater axis (of which _i k_ is a part) makes an angle of (¹⁄₆)
+with the vertical.
+
+Make similar tracings on the other side of A B, and the diagram is
+complete. The inscribing rectangle D G E K is that of (²⁄₅).
+
+The outline resulting from this diagram, not only is in perfect agreement
+with my recollection of the form, but with the measurements of the
+original given in the “Penny Cyclopædia;” of the accuracy of which there
+can be no doubt. They are stated thus:—“It is about ten inches in height,
+and beautifully curved from the top downwards; the diameter at the top
+being about three inches and a-half; at the neck or smallest part, two
+inches; at the largest (mid-height), seven inches; and at the bottom,
+five inches.”
+
+The harmonic elements of this beautiful form, therefore, appear to be the
+following parts of the right angle:—
+
+    Tonic.  Dominant.  Mediant.  Submediant.
+    (¹⁄₂)    (¹⁄₃)      (¹⁄₅)      (³⁄₁₀)
+    (¹⁄₄)    (¹⁄₆)
+
+When we reflect upon the variety of harmonic ellipses that may
+be described, and the innumerable positions in which they may be
+harmonically placed with respect to the horizontal and vertical lines,
+as well as upon the various modes in which their circumferences may be
+combined, the variety which may be introduced amongst such forms as the
+foregoing appears almost endless. My second example is that of—
+
+
+_An Ancient Grecian Marble Vase of a Vertical Composition._
+
+I shall now proceed to another class of the ancient Greek vase, the form
+of which is of a more complex character. The specimen I have chosen for
+the first example of this class is one of those so correctly measured and
+beautifully delineated by Tatham in his unequalled work.[25] This vase is
+a work of ancient Grecian art in Parian marble, which he met with in the
+collection at the Villa Albani, near Rome. Its height is 4 ft. 4¹⁄₂ in.
+
+[Sidenote: Plate XIII.]
+
+The following is the formula by which I endeavour to develop its harmonic
+elements:—
+
+Let A B (Plate XIII.) represent the full height of this vase. Through B
+draw B D, making an angle of (¹⁄₅) with the vertical. Through D draw D O
+vertical, through A draw A C, making an angle of (²⁄₅); through B draw
+B L, making an angle of (¹⁄₂), and B S, making an angle of (³⁄₁₀), each
+with the vertical. Through A draw A D, through B draw B O, through L draw
+L N, through C draw C F, and through S draw S P, all horizontal. Through
+A draw A H, making an angle of (¹⁄₁₀) with the vertical, and through
+H draw H M vertical. Draw similar lines on the other side of A B, and
+the rectilinear portion of the diagram is complete, and its inscribing
+rectangle that of (³⁄₈).
+
+The curvilinear portion may thus be added—
+
+Take a cut-out ellipse of (¹⁄₃), whose greater axis is about the length
+of the body of the intended vase, place it with its lesser axis upon the
+line S P, and its greater axis upon the line D O, and trace the part _a
+b_ of its circumference upon the diagram. Place the same ellipse with
+one of its foci upon C, and its greater axis upon C F, and trace its
+circumference upon the diagram. Take a cut-out ellipse of (¹⁄₅), whose
+greater axis is nearly equal to that of the ellipse already used; place
+it with its greater axis upon M H, and its lesser axis upon L N, and
+trace its circumference upon the diagram. Make similar tracings upon the
+other side of A B, and the diagram is complete. In this, as in the other
+diagrams, the strong portions of the lines give the contour of the vase.
+The harmonic elements of this classical form, therefore, appear to be the
+right angle and its following parts:—
+
+    Tonic.  Dominant.  Mediant. Submediant.
+    (¹⁄₂)     (¹⁄₃)     (²⁄₅)     (³⁄₁₀)
+                        (¹⁄₅)
+                        (¹⁄₁₀)
+
+My third example is that of—
+
+
+_An Ancient Grecian Vase of a Horizontal Composition._
+
+This example belongs to the same class as the last, but it is of a
+horizontal composition. It was carefully drawn from the original in the
+museum of the Vatican by Tatham, in whose etchings it will be found with
+its ornamental decorations. The diagram of its harmonic elements may be
+constructed as follows:—
+
+[Sidenote: Plate XIV.]
+
+Let A B (Plate XIV.) represent the full height of the vase. Through B
+draw B D, making an angle of (²⁄₅) with the vertical. Through A draw A H,
+A L, and A C, making respectively the following angles, (¹⁄₅) with the
+vertical, (⁴⁄₉) with the vertical, and (³⁄₁₀) with the horizontal. These
+angles determine the horizontal lines H B, L N, and C F, which divide
+the vase into its parts, and the inscribing rectangle D G K O is (³⁄₈).
+This completes the rectilinear portion of the diagram. The ellipse by
+which the curvilinear portion is added is one of (¹⁄₅), the greater axis
+of which, at _a b_, as also at _c d_, makes an angle of (¹⁄₁₂) with
+the vertical, and the same axis at _e f_ an angle of (¹⁄₁₂) with the
+horizontal.
+
+The harmonic elements of this vase, therefore, appear to be:—
+
+     Tonic.    Dominant.  Mediant. Submediant.  Supertonic.
+    The Right    (¹⁄₁₂)    (²⁄₅)     (³⁄₁₀)       (⁴⁄₉)
+     Angle.                (¹⁄₅)
+
+My remaining examples are those of—
+
+
+_Etruscan Vases._
+
+Of these vases I give four examples, by which the simplicity of the
+method employed in applying the harmonic law will be apparent.
+
+[Sidenote: Plate XV.]
+
+The inscribing rectangle D G E K of fig. 1, Plate XV., is one of (³⁄₈),
+within which are arranged tracings from an ellipse of (³⁄₁₀), whose
+greater axis at _a b_ makes an angle of (¹⁄₁₂), at _c d_ an angle of
+(³⁄₁₀), and at _e f_ an angle of (³⁄₄), with the vertical. The harmonic
+elements of the contour of this vase, therefore, appear to be:—
+
+     Tonic.    Dominant.  Subdominants.  Submediant.
+    The Right   (¹⁄₁₂)        (³⁄₄)        (³⁄₁₀)
+     Angle.                   (³⁄₈)
+
+The inscribing rectangle L M N O of fig. 2 is that of (¹⁄₂), within which
+are arranged tracings from an ellipse of (¹⁄₃), whose greater axis, at
+_a b_ and _c d_ respectively, makes angles of (¹⁄₂) and (⁴⁄₉) with the
+horizontal, while that at _e f_ is in the horizontal line. The harmonic
+elements of the contour of this vase, therefore, appear to be:—
+
+    Tonic.  Dominant.  Subtonic.
+    (¹⁄₂)     (¹⁄₃)      (⁴⁄₉)
+
+[Sidenote: Plate XVI.]
+
+The inscribing rectangle P Q R S of fig. 1, Plate XVI., is one of (⁴⁄₉),
+within which are arranged tracings from an ellipse of (³⁄₈), whose
+greater axis, at _a b_, _c d_, and _e f_, makes respectively angles
+of (¹⁄₆) with the horizontal, (³⁄₅) and (⁴⁄₅) with the vertical. Its
+harmonic elements, therefore, appear to be:—
+
+    Tonic.    Dominant.  Mediant.  Supertonic.  Subdominant.  Submediant.
+    The Right   (¹⁄₆)     (⁴⁄₅)       (⁴⁄₉)        (³⁄₈)         (³⁄₅)
+    Angle.
+
+The inscribing rectangle T U V X of fig. 2 is one of (⁴⁄₉), within which
+are arranged tracings from an ellipse of (³⁄₈) whose greater axis at _a
+b_ is in the vertical line, and at _c d_ makes an angle of (¹⁄₂). The
+harmonic elements of the contour of this vase, therefore, appear to be:—
+
+    Tonic.  Submediant.  Supertonic.
+    (¹⁄₂)     (³⁄₈)        (⁴⁄₉)
+
+These four Etruscan vases, the contours of which are thus reduced to the
+harmonic law of nature, are in the British Museum, and engravings of
+them are to be found in the well-known work of Mr Henry Moses, Plates
+4, 6, 14, and 7, respectively, where they are represented with their
+appropriate decorations and colours.
+
+To these, I add two examples of—
+
+
+_Ancient Grecian Ornament._
+
+I have elsewhere shewn[26] that the elliptic curve pervades the Parthenon
+from the entases of the column to the smallest moulding, and we need not,
+therefore, be surprised to find it employed in the construction of the
+only two ornaments belonging to that great work.
+
+[Sidenote: Plate XVII.]
+
+In the diagram (Plate XVII.), I endeavour to exhibit the geometric
+construction of the upper part of one of the ornamental apices, termed
+antefixæ, which surmounted the cornice of the Parthenon.
+
+The first ellipse employed is that of (¹⁄₃), whose greater axis _a b_ is
+in the vertical line; the second is also that of (¹⁄₃), whose greater
+axis _c d_ makes, with the vertical, an angle of (¹⁄₁₂); the third
+ellipse is the same with its major axis _e f_ in the vertical line.
+Through one of the foci of this ellipse at A the line A C is drawn, and
+upon the part of the circumference C _e_, the number of parts, 1, 2, 3,
+4, 5, 6, 7, of which the surmounting part of this ornament is to consist,
+are set off. That part of the circumference of the ellipse whose larger
+axis is _c d_ is divided from _g_ to _c_ into a like number of parts. The
+third ellipse employed is one of (¹⁄₄).
+
+Take a cut-out ellipse of this kind, whose larger axis is equal in length
+to the inscribing rectangle. Place it with its vertex upon the same
+ellipse at _g_, so that its circumference will pass through C, and trace
+it; remove its apix first to _p_, then to _q_, and proceed in the same
+way to _q_, _r_, _s_, _t_, _u_, and _v_, so that its circumference will
+pass through the seven divisions on _c g_ and _e_ C: _v o_, _u n_, _t m_,
+_s i_, _r k_, _q j_, _p l_, and _g x_, are parts of the larger axes of
+the ellipses from which the curves are traced. The small ellipse of which
+the ends of the parts are formed is that of (¹⁄₃).
+
+[Sidenote: Plate XVIII.]
+
+In the diagram (Plate XVIII.), I endeavour to exhibit the geometric
+construction of the ancient Grecian ornament, commonly called the
+_Honeysuckle_, from its resemblance to the flower of that name. The first
+part of the process is similar to that just explained with reference to
+the antefixæ of the Parthenon, although the angles in some parts differ.
+The contour is determined by the circumference of an ellipse of (¹⁄₃),
+whose major axis A B makes an angle of (¹⁄₉) with the vertical, and
+the leaves or petals are arranged upon a portion of the perimeter of a
+similar ellipse whose larger axis E F is in the vertical line, and these
+parts are again arranged upon a similar ellipse whose larger axis C D
+makes an angle of (¹⁄₁₂) with the vertical. The first series of curved
+lines proceeding from 1, 2, 3, 4, 5, 6, 7, and 8, are between K E and H
+C, part of the circumference of an ellipse of (¹⁄₃); and those between C
+H and A G are parts of the circumference of four ellipses, each of (¹⁄₃),
+but varying as to the lengths of their larger axes from 5 to 3 inches.
+The change from the convex to the concave, which produces the ogie forms
+of which this ornament is composed, takes place upon the line C H, and
+the lines _a b_, _c d_, _e f_, _g h_, _i k_, _l m_, _n o_, and _p q_, are
+parts of the larger axis of the four ellipses the circumference of which
+give the upper parts of the petals or leaves.
+
+This peculiar Grecian ornament is often, like the antefixæ of the
+Parthenon, combined with the curve of the spiral scroll. But the volute
+is so well understood that I have not rendered my diagrams more complex
+by adding that figure. Many varieties of this union are to be found in
+Tatham’s etchings, already referred to. The antefixæ of the Parthenon,
+and its only other ornament the honeysuckle, as represented on the soffit
+of the cornice, are to be found in Stewart’s “Athens.”
+
+
+
+
+APPENDIX.
+
+
+No. I.
+
+In pages 34, 35, and 58, I have reiterated an opinion advanced in several
+of my former works, viz., that, besides genius, and the study of nature,
+an additional cause must be assigned for the general excellence which
+characterises such works of Grecian art as were executed during a period
+commencing about 500 B.C., and ending about 200 B.C. And that this cause
+most probably was, that the artists of that period were instructed in
+a system of fixed principles, based upon the doctrines of Pythagoras
+and Plato. This opinion has not been objected to by the generality of
+those critics who have reviewed my works; but has, however, met with an
+opponent, whose recondite researches and learned observations are worthy
+of particular attention. These are given in an essay by Mr C. Knight
+Watson, “On the Classical Authorities for Ancient Art,” which appeared in
+the _Cambridge Journal of Classical and Sacred Philology_ in June 1854.
+As this essay is not otherwise likely to meet the eyes of the generality
+of my readers, and as the objections he raises to my opinion only occupy
+two out of the sixteen ample paragraphs which constitute the first part
+of the essay, I shall quote them fully:—
+
+    “The next name on our list is that of the famous Euphranor
+    (B.C. 362). For the fact that to the practice of sculpture
+    and of painting he added an exposition of the theory, we are
+    indebted to Pliny, who says (xxxv. 11, 40), ‘Volumina quoque
+    composuit de symmetria et coloribus.’ When we reflect on the
+    _critical_ position occupied by Euphranor in the history
+    of Greek art, as a connecting link between the idealism of
+    Pheidias and the naturalism of Lysippus, we can scarcely
+    overestimate the value of a treatise on art proceeding from
+    such a quarter. This is especially the case with the first
+    of the two works here assigned to Euphranor. The inquiries
+    which of late years have been instituted by Mr D. R. Hay of
+    Edinburgh, on the proportions of the human figure, and on the
+    natural principles of beauty as illustrated by works of Greek
+    art, constitute an epoch in the study of æsthetics and the
+    philosophy of form. Now, in the presence of these inquiries,
+    or of such less solid results as Mr Hay’s predecessors in
+    the same field have elicited, it naturally becomes an object
+    of considerable interest to ascertain how far these laws of
+    form and principles of beauty were consciously developed in
+    the mind, and by the chisel, of the sculptor: how far any
+    such system of curves and proportions as Mr Hay’s was used
+    by the Greek as a practical manual of his craft. Without in
+    the least wishing to impugn the accuracy of that gentleman’s
+    results—a piece of presumption I should do well to avoid—I must
+    be permitted to doubt whether the ‘Symmetria’ of Euphranor
+    contained anything analogous to them in kind, or indeed equal
+    in value. It must not be forgotten that the truth of Mr Hay’s
+    theory is perfectly compatible with the fact, that of such
+    theory the Greek may have been utterly ignorant. It is on this
+    fact I insist: it is here that I join issue with Mr Hay, and
+    with his reviewer in a recent number of _Blackwood’s Magazine_.
+    Or, to speak more accurately,—while I am quite prepared to find
+    that the Elgin marbles will best of all stand the test which Mr
+    Hay has hitherto applied, I believe, to works of a later age, I
+    am none the less convinced that it is precisely that golden age
+    of Hellenic art to which they belong, precisely that first and
+    chief of Hellenic artists by whom they were executed, to which
+    and to whom any such line of research on the laws of form would
+    have been pre-eminently alien. Pheidias, remember, by the right
+    of primogeniture, is the ruling spirit of idealism in art. Of
+    spontaneity was that idealism begotten and nurtured: by any
+    such system as Mr Hay’s, that spontaneity would be smothered
+    and paralysed. Pheidias copied an idea in his own mind—‘Ipsius
+    in mente insidebat species pulchritudinis eximia quædam’
+    (_Cic._);—later ages copied _him_. He created: they criticised.
+    He was the author of Iliads: they the authors of Poetics.
+    Doubtless, if you unsphere the _spirit_ of Mr Hay’s theories,
+    you will find nothing discordant with what I have here said.
+    That is a sound view of Beauty which makes it consist in that
+    due subordination of the parts to the whole, that due relation
+    of the parts to each other, which Mendelssohn had in his
+    mind when he said that the essence of beauty was ‘unity in
+    variety’—variety beguiling the imagination, the perception of
+    unity exercising the thewes and sinews of the intellect. On
+    such a view of beauty, Mr Hay’s theory may, _in spirit_, be
+    said to rest. But here, as in higher things, it is the letter
+    that killeth, while the spirit giveth life. And accordingly I
+    must enter a protest against any endeavour to foist upon the
+    palmy days of Hellenic art systems of geometrical proportions
+    incompatible, as I believe, with those higher and broader
+    principles by which the progress of ancient sculpture was
+    ordered and governed—systems which will bear nothing of that
+    ‘felicity and chance by which’—and not by rule—‘Lord Bacon
+    believed that a painter may make a better face than ever was:’
+    systems which take no account of that fundamental distinction
+    between the schools of Athens and of Argos, and their
+    respective disciples and descendants, without which you will
+    make nonsense of the pages of Pliny, and—what is worse—sense
+    of the pages of his commentators;—systems, in short, which may
+    have their value as instruments for the education of the eye,
+    and for instructions in the arts of design, but must be cast
+    aside as matters of learned trifling and curious disputation,
+    where they profess to be royal roads to art, and to map the
+    mighty maze of a creative mind. And even as regards the
+    application of such a system of proportions to those works of
+    sculpture which are posterior to the Pheidian age, only partial
+    can have been the prevalence which it or any other _one_ system
+    can have obtained. The discrepancies of different artists
+    in the treatment of what was called, technically called,
+    _Symmetria_ (as in the title of Euphranor’s work) were, by the
+    concurrent testimony of all ancient writers, far too salient
+    and important to warrant the supposition of any uniform scale
+    of proportions, as advocated by Mr Hay. Even in Egypt, where
+    one might surely have expected that such uniformity would have
+    been observed with far greater rigour than in Greece, the
+    discoveries of Dr Lepsius (_Vorläufige Nachricht_, Berlin,
+    1849) have elicited three totally different κανόνες, one of
+    which is identical with the system of proportions of the human
+    figure detailed in Diodorus. While we thus venture to differ
+    from Mr Hay on the historical data he has mixed up with his
+    inquiries, we feel bound to pay him a large and glad tribute of
+    praise for having devised a system of proportions which rises
+    superior to the idiosyncracies of different artists, which
+    brings back to one common type the sensations of eye and ear,
+    and so makes a giant stride towards that _codification_, if
+    I may so speak, of the laws of the universe which it is the
+    business of the science to effect. I have no hesitation in
+    saying, that, for scientific precision of method and importance
+    of results, Albert Durer, Da Vinci, and Hogarth, not to mention
+    less noteworthy writers, must all yield the palm to Mr Hay.
+
+    “I am quite aware that in the digression I have here allowed
+    myself, on systems of proportions prevalent among ancient
+    artists, and on the probable contents of such treatises as that
+    of Euphranor, _De Symmetria_, I have laid myself open to the
+    charge of treating an intricate question in a very perfunctory
+    way. At present the exigencies of the subject more immediately
+    in hand allow me only to urge in reply, that, as regards the
+    point at issue—I mean the ‘solidarité’ between theories such as
+    Mr Hay’s and the practice of Pheidias—the _onus probandi_ rests
+    with my adversaries.”
+
+I am bound, in the first place, gratefully to acknowledge the kind and
+complimentary notice which, notwithstanding our difference of opinion,
+this author has been pleased to take of my works; and, in the second, to
+assure him that if any of them profess to be “royal roads to art,” or
+to “map the mighty maze of a creative mind,” they certainly profess to
+do more than I ever meant they should; for I never entertained the idea
+that a system of æsthetic culture, even when based upon a law of nature,
+was capable of effecting any such object. But I doubt not that this too
+common misapprehension of the scope and tendency of my works must arise
+from a want of perspicuity in my style.
+
+I have certainly, on one occasion,[27] gone the length of stating
+that as poetic genius must yield obedience to the rules of rhythmical
+measure, even in the highest flights of her inspirations; and musical
+genius must, in like manner, be subject to the strictly defined laws of
+harmony in the most delicate, as well as in the most powerfully grand
+of her compositions; so must genius, in the formative arts, either
+consciously or unconsciously have clothed her creations of ideal beauty
+with proportions strictly in accordance with the laws which nature has
+set up as inflexible standards. If, therefore, the laws of proportion, in
+their relation to the arts of design, constitute the harmony of geometry,
+as definitely as those that are applicable to poetry and music produce
+the harmony of acoustics; the former ought, certainly, to hold the same
+relative position in those arts which are addressed to the eye, that is
+accorded to the latter in those which are addressed to the ear. Until
+so much science be brought to bear upon the arts of design, the student
+must continue to copy from individual and imperfect objects in nature, or
+from the few existing remains of ancient Greek art, in total ignorance of
+the laws by which their proportions are produced, and, what is equally
+detrimental to art, the accuracy of all criticism must continue to rest
+upon the indefinite and variable basis of mere opinion.
+
+It cannot be denied that men of great artistic genius are possessed of
+an intuitive feeling of appreciation for what is beautiful, by means
+of which they impart to their works the most perfect proportions,
+independently of any knowledge of the definite laws which govern that
+species of beauty. But they often do so at the expense of much labour,
+making many trials before they can satisfy themselves in imparting to
+them the true proportions which their minds can conceive, and which,
+along with those other qualities of expression, action, or attitude,
+which belong more exclusively to the province of genius. In such
+cases, an acquaintance with the rules which constitute the science of
+proportion, instead of proving fetters to genius, would doubtless afford
+her such a vantage ground as would promote the more free exercise of
+her powers, and give confidence and precision in the embodiment of her
+inspirations; qualities which, although quite compatible with genius, are
+not always intuitively developed along with that gift.
+
+It is also true that the operations of the conceptive faculty of the mind
+are uncontrolled by definite laws, and that, therefore, there cannot
+exist any rules by the inculcation of which an ordinary mind can be
+imbued with genius sufficient to produce works of high art. Nevertheless,
+such a mind may be improved in its perceptive faculty by instruction in
+the science of proportion, so as to be enabled to exercise as correct and
+just an appreciation of the conceptions of others, in works of plastic
+art, as that manifested by the educated portion of mankind in respect
+to poetry and music. In short, it appears that, in those arts which are
+addressed to the ear, men of genius communicate the original conceptions
+of their minds under the control of certain scientific laws, by means
+of which the educated easily distinguish the true from the false, and
+by which the works of the poet and musical composer may be placed above
+mere imitations of nature, or of the works of others; while, in those
+arts that are addressed to the eye in their own peculiar language, such
+as sculpture, architecture, painting, and ornamental design, no such laws
+are as yet acknowledged.
+
+Although I am, and ever have been, far from endeavouring “to foist upon
+the palmy days of Hellenic art” any system incompatible with those higher
+and more intellectual qualities which genius alone can impart; yet, from
+what has been handed down to us by writers on the subject, meagre as it
+is, I cannot help continuing to believe that, along with the physical and
+metaphysical sciences, æsthetic science was taught in the early schools
+of Greece.
+
+I shall here take the liberty to repeat the proofs I advanced in a former
+work as the ground of this belief, and to which the author, from whose
+essay I have quoted, undoubtedly refers. It is well known that, in the
+time of Pythagoras, the treasures of science were veiled in mystery to
+all but the properly initiated, and the results of its various branches
+only given to the world in the works of those who had acquired this
+knowledge. So strictly was this secresy maintained amongst the disciples
+and pupils of Pythagoras, that any one divulging the sacred doctrines
+to the profane, was expelled the community, and none of his former
+associates allowed to hold further intercourse with him; it is even
+said, that one of his pupils incurred the displeasure of the philosopher
+for having published the solution of a problem in geometry.[28] The
+difficulty, therefore, which is expressed by writers, shortly after the
+period in which Pythagoras lived, regarding a precise knowledge of his
+theories, is not to be wondered at, more especially when it is considered
+that he never committed them to writing. It would appear, however, that
+he proceeded upon the principle, that the order and beauty so apparent
+throughout the whole universe, must compel men to believe in, and refer
+them to, an intelligible cause. Pythagoras and his disciples sought for
+properties in the science of numbers, by the knowledge of which they
+might attain to that of nature; and they conceived those properties to
+be indicated in the phenomena of sonorous bodies. Observing that Nature
+herself had thus irrevocably fixed the numerical value of the intervals
+of musical tones, they justly concluded that, as she is always uniform
+in her works, the same laws must regulate the general system of the
+universe.[29] Pythagoras, therefore, considered numerical proportion as
+the great principle inherent in all things, and traced the various forms
+and phenomena of the world to numbers as their basis and essence.
+
+How the principles of numbers were applied in the arts is not recorded,
+farther than what transpires in the works of Plato, whose doctrines were
+from the school of Pythagoras. In explaining the principle of beauty,
+as developed in the elements of the material world, he commences in the
+following words:—“But when the Artificer began to adorn the universe,
+he first of all figured with forms and numbers, fire and earth, water
+and air—which possessed, indeed, certain traces of the true elements,
+but were in every respect so constituted as it becomes anything to be
+from which Deity is absent. But we should always persevere in asserting
+that Divinity rendered them, as much as possible, the most beautiful
+and the best, when they were in a state of existence opposite to such
+a condition.” Plato goes on further to say, that these elementary
+bodies must have forms; and as it is necessary that every depth should
+comprehend the nature of a plane, and as of plane figures the triangle
+is the most elementary, he adopts two triangles as the originals or
+representatives of the isosceles and the scalene kinds. The first
+triangle of Plato is that which forms the half of the square, and is
+regulated by the number, 2; and the second, that which forms the half
+of the equilateral triangle, which is regulated by the number, 3;
+from various combinations of these, he formed the bodies of which he
+considered the elements to be composed. To these elementary figures I
+have already referred.
+
+Vitruvius, who studied architecture ages after the arts of Greece had
+been buried in the oblivion which succeeded her conquest, gives the
+measurements of various details of monuments of Greek art then existing.
+But he seems to have had but a vague traditionary knowledge of the
+principle of harmony and proportion from which these measurements
+resulted. He says—“The several parts which constitute a temple ought
+to be subject to the laws of symmetry; the principles of which should
+be familiar to all who profess the science of architecture. Symmetry
+results from proportion, which, in the Greek language, is termed
+analogy. Proportion is the commensuration of the various constituent
+parts with the whole; in the existence of which symmetry is found to
+consist. For no building can possess the attributes of composition
+in which symmetry and proportion are disregarded; nor unless there
+exist that perfect conformation of parts which may be observed in a
+well-formed human being.” After going at some length into details, he
+adds—“Since, therefore, the human figure appears to have been formed
+with such propriety, that the several members are commensurate with
+the whole, the artists of antiquity (meaning those of Greece at the
+period of her highest refinement) must be allowed to have followed the
+dictates of a judgment the most rational, when, transferring to works of
+art principles derived from nature, every part was so regulated as to
+bear a just proportion to the whole. Now, although the principles were
+universally acted upon, yet they were more particularly attended to in
+the construction of temples and sacred edifices, the beauties or defects
+of which were destined to remain as a perpetual testimony of their skill
+or of their inability.”
+
+Vitruvius, however, gives no explanation of this ancient principle
+of proportion, as derived from the human form; but plainly shews his
+uncertainty upon the subject, by concluding this part of his essay in the
+following words: “If it be true, therefore, that the decenary notation
+was suggested by the members of man, and that the laws of proportion
+arose from the relative measures existing between certain parts of each
+member and the whole body, it will follow, that those are entitled to our
+commendation who, in building temples to their deities, proportioned the
+edifices, so that the several parts of them might be commensurate with
+the whole.” It thus appears certain that the Grecians, at the period of
+their highest excellence, had arrived at a knowledge of some definite
+mathematical law of proportion, which formed a standard of perfectly
+symmetrical beauty, not only in the representation of the human figure
+in sculpture and painting, but in architectural design, and indeed in
+all works where beauty of form and harmony of proportion constituted
+excellence. That this law was not deduced from the proportions of
+the human figure, as supposed by Vitruvius, but had its origin in
+mathematical science, seems equally certain; for in no other way can we
+satisfactorily account for the proportions of the beau ideal forms of the
+ancient Greek deities, or of those of their architectural structures,
+such as the Parthenon, the temple of Theseus, &c., or for the beauty that
+pervades all the other formative art of the period.
+
+This system of geometrical harmony, founded, as I have shewn it to be,
+upon numerical relations, must consequently have formed part of the Greek
+philosophy of the period, by means of which the arts began to progress
+towards that great excellence which they soon after attained. A little
+further investigation will shew, that immediately after this period a
+theory connected with art was acknowledged and taught, and also that
+there existed a Science of Proportion.
+
+Pamphilus, the celebrated painter, who flourished about four hundred
+years before the Christian era, from whom Apelles received the
+rudiments of his art, and whose school was distinguished for scientific
+cultivation, artistic knowledge, and the greatest accuracy in drawing,
+would admit no pupil unacquainted with geometry.[30] The terms upon which
+he engaged with his students were, that each should pay him one talent
+(£225 sterling) previous to receiving his instructions; for this he
+engaged “to give them, _for ten years_, lessons founded on an excellent
+theory.”[31]
+
+It was by the advice of Pamphilus that the magistrates of Sicyon ordained
+that the study of drawing should constitute part of the education of the
+citizens—“a law,” says the Abbé Barthélémie, “which rescued the fine arts
+from servile hands.”
+
+It is stated of Parrhasius, the rival of Zeuxis, who flourished about
+the same period as Pamphilus, that he accelerated the progress of art by
+purity and correctness of design; “for he was acquainted with the science
+of Proportions. Those he gave his gods and heroes were so happy, that
+artists did not hesitate to adopt them.” Parrhasius, it is also stated,
+was so admired by his contemporaries, that they decreed him the name of
+Legislator.[32] The whole history of the arts in Egypt and Greece concurs
+to prove that they were based on geometric precision, and were perfected
+by a continued application of the same science; while in all other
+countries we find them originating in rude and misshapen imitations of
+nature.
+
+In the earliest stages of Greek art, the gods—then the only statues—were
+represented in a tranquil and fixed posture, with the features exhibiting
+a stiff inflexible earnestness, their only claim to excellence being
+symmetrical proportion; and this attention to geometric precision
+continued as art advanced towards its culminating point, and was
+thereafter still exhibited in the neatly and regularly folded drapery,
+and in the curiously braided and symmetrically arranged hair.[33]
+
+These researches, imperfect as they are, cannot fail to exhibit the
+great contrast that exists between the system of elementary education
+in art practised in ancient Greece, and that adopted in this country at
+the present period. But it would be of very little service to point out
+this contrast, were it not accompanied by some attempt to develop the
+principles which seem to have formed the basis of this excellence amongst
+the Greeks.
+
+But in making such an attempt, I cannot accuse myself of assuming
+anything incompatible with the free exercise of that spontaneity of
+genius which the learned essayist says is the parent and nurse of
+idealism. For it is in no way more incompatible with the free exercise of
+artistic genius, that those who are so gifted should have the advantage
+of an elementary education in the science of æsthetics, than it is
+incompatible with the free exercise of literary or poetic genius, that
+those who possess it should have the advantage of such an elementary
+education in the science of philology as our literary schools and
+colleges so amply afford.
+
+
+No. II.
+
+The letter from which I have made a quotation at page 42, arose out of
+the following circumstance:—In order that my theory, as applied to the
+orthographic beauty of the Parthenon, might be brought before the highest
+tribunal which this country afforded, I sent a paper upon the subject,
+accompanied by ample illustrations, to the Royal Institute of British
+Architects, and it was read at a meeting of that learned body on the 7th
+of February 1853; at the conclusion of which, Mr Penrose kindly undertook
+to examine my theoretical views, in connexion with the measurements he
+had taken of that beautiful structure by order of the Dilettanti Society,
+and report upon the subject to the Royal Institute. This report was
+published by Mr Penrose, vol. xi., No. 539 of _The Builder_, and the
+letter from which I have quoted appeared in No. 542 of the same journal.
+It was as follows:—
+
+              “GEOMETRICAL RELATIONS IN ARCHITECTURE.
+
+    “Will you allow me, through the medium of your columns, to
+    thank Mr Penrose for his testimony to the truth of Mr Hay’s
+    revival of Pythagoras? The dimensions which he gives are to
+    me the surest verification of the theory that I could have
+    desired. The minute discrepancies form that very element
+    of practical incertitude, both as to execution and direct
+    measurement, which always prevails in materialising a
+    mathematical calculation under such conditions.
+
+    “It is time that the scattered computations by which
+    architecture has been analysed—more than enough—be synthetised
+    into those formulæ which, as Mrs Somerville tells us, ‘are
+    emblematic of omniscience.’ The young architects of our day
+    feel trembling beneath their feet the ground whence men are
+    about to evoke the great and slumbering corpse of art. Sir, it
+    is food of this kind a reviving poetry demands.
+
+                ——‘Give us truths,
+        For we are weary of the surfaces,
+        And die of inanition.’
+
+    “I, for one, as I listen to such demonstrations, whose scope
+    extends with every research into them, feel as if listening to
+    those words of Pythagoras, which sowed in the mind of Greece
+    the poetry whose manifestation in beauty has enchained the
+    world in worship ever since its birth. And I am sure that in
+    such a quarter, and in such thoughts, _we_ must look for the
+    first shining of that lamp of art, which even now is prepared
+    to burn.
+
+    “I know that this all sounds rhapsodical; but I know also that
+    until the architect becomes a poet, and not a tradesman, we
+    may look in vain for architecture: and I know that valuable
+    as isolated and detailed investigations are in their proper
+    bearings, yet that such purposes and bearings are to be found
+    in the enunciation of principles sublime as the generalities of
+    ‘mathematical beauty.’
+
+                                                        “AUTOCTHON.”
+
+
+No. III.
+
+Of the work alluded to at page 58 I was favoured with two opinions—the
+one referring to the theory it propounds, and the other to its anatomical
+accuracy—both of which I have been kindly permitted to publish.
+
+The first is from Sir WILLIAM HAMILTON, Bart., professor of logic and
+metaphysics in the University of Edinburgh, and is as follows:—
+
+    “Your very elegant volume is to me extremely interesting, as
+    affording an able contribution to what is the ancient, and,
+    I conceive, the true theory of the Beautiful. But though
+    your doctrine coincides with the one prevalent through all
+    antiquity, it appears to me quite independent and original
+    in you; and I esteem it the more, that it stands opposed to
+    the hundred one-sided and exclusive views prevalent in modern
+    times.—_16 Great King Street, March 5, 1849._”
+
+The second is from JOHN GOODSIR, Esq., professor of anatomy in the
+University of Edinburgh, and is as follows:—
+
+    “I have examined the plates in your work on the proportions
+    of the human head and countenance, and find the head you have
+    given as typical of human beauty to be anatomically correct
+    in its structure, only differing from ordinary nature in its
+    proportions being more mathematically precise, and consequently
+    more symmetrically beautiful.—_College, Edinburgh, 17th April
+    1849._”
+
+
+No. IV.
+
+I shall here shew, as I have done in a former work, how the curvilinear
+outline of the figure is traced upon the rectilinear diagrams by portions
+of the ellipse of (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆).
+
+[Sidenote: Plate XIX.]
+
+The outline of the head and face, from points (1) to (3) (fig. 1, Plate
+XIX.), takes the direction of the two first curves of the diagram. From
+point (3), the outline of the sterno-mastoid muscle continues to (4),
+where, joining the outline of the trapezius muscle, at first concave, it
+becomes convex after passing through (5), reaches the point (6), where
+the convex outline of the deltoid muscle commences, and, passing through
+(7), takes the outline of the arm as far as (8). The outline of the
+muscles on the side, the latissimus dorsi and serratus magnus, commences
+under the arm at the point (9), and joins the outline of the oblique
+muscle of the abdomen by a concave curve at (10), which, rising into
+convexity as it passes through the points (11) and (12), ends at (13),
+where it joins the outline of the gluteus medius muscle. The outline of
+this latter muscle passes convexly through the point (14), and ends at
+(15), where the outline of the tensor vaginæ femoris and vastus externus
+muscle of the thigh commences. This convex outline joins the concave
+outline of the biceps of the thigh at (16), which ends in that of the
+slight convexity of the condyles of the thigh-bone at (17). From this
+point, the outline of the outer surface of the leg, which includes the
+biceps, peroneus longus, and soleus muscles, after passing through the
+point (18), continues convexly to (19), where the concave outline of the
+tendons of the peroneus longus continues to (20), whence the outline of
+the outer ankle and foot commences.
+
+The outline of the mamma and fold of the arm-pit commences at (21),
+and passes through the points (22) and (23). The circle at (24) is the
+outline of the areola, in the centre of which the nipple is placed.
+
+The outline of the pubes commences at (25), and ends at the point (26),
+from which the outline of the inner surface of the thigh proceeds over
+the gracilis, the sartorius, and vastus internus muscles, until it meets
+the internal face of the knee-joint at (27), the outline of which ends
+at (28). The outline of the inside of the leg commences by proceeding
+over the gastrocnemius muscle as far as (29), where it meets that of the
+soleus muscle, and continues over the tendons of the heel until it meets
+the outline of the inner ankle and foot at (30).
+
+The outline of the outer surface of the arm, as viewed in front, proceeds
+from (8) over the remainder of the deltoid, in which there is here a
+slight concavity, and, next, from (31) over the biceps muscle till (32),
+where it takes the line of the long supinator, and passing concavely, and
+almost imperceptibly, into the long and short radial extensor muscles,
+reaches the wrist at (33). The outline of the inner surface of the arm
+from opposite (9) commences by passing over the triceps extensor, which
+ends at (34), then over the gentle convexity of the condyles of the bone
+of the arm at (35), and, lastly, over the flexor sublimis which ends at
+the wrist-joint (36).
+
+The outline of the front of the figure commences at the point (1), (fig.
+2, Plate II.), and, passing almost vertically over the platzsma-myoidis
+muscles, reaches the point (2), where the neck ends by a concave curve.
+From (2) the outline rises convexly over the ends of the clavicles,
+and continues so over the pectoral muscle till it reaches (3), where
+the mamma swells out convexly to (4), and returns convexly towards
+(5), where the curve becomes concave. From (5) the outline follows the
+undulations of the rectus muscle of the abdomen, concave at the points
+(6) and (7), and having its greatest convexity at (8). This series
+of curves ends with a slight concavity at the point (9), where the
+horizontal branch of the pubes is situated, over which the outline is
+convex and ends at (10).
+
+The outline of the thigh commences at the point (11) with a slight
+concave curve, and then swells out convexly over the extensors of the
+leg, and, reaching (12), becomes gently concave, and, passing over the
+patella at (13), becomes again convex until it reaches the ligament of
+that bone, where it becomes gently concave towards the point (14), whence
+it follows the slightly convex curve of the shin-bone, and then, becoming
+as slightly concave, ends with the muscles in front of the leg at (15).
+
+The outline of the back commences at the point (16), and, following with
+a concave curve the muscles of the neck as far as (17), swells into a
+convex curve over the trapezius muscle towards the point (18); passing
+through which point, it continues to swell outward until it reaches
+(19), half way between (18) and (20); whence the convexity, becoming
+less and less, falls into the concave curve of the muscles of the loins
+at (21), and passing through the point (22), it rises into convexity. It
+then passes through the point (23), follows the outline of the gluteus
+maximus, the convex curve of which rises to the point (24), and then
+returns inwards to that of (25), where it ends in the fold of the hip.
+From this point the outline follows the curve of the hamstring muscles
+by a slight concavity as far as (26), and then, becoming gently convex,
+it reaches (27); whence it becomes again gently concave, with a slight
+indication of the condyle of the thigh-bone at (28), and, reaching (29),
+follows the convex curve of the gastrocnemius muscle through the point
+(30), and falling into the convex curve of the tendo Achilles at (31),
+ends in the concavity over the heel at (32).
+
+The outline of the front of the arm commences at the point (33), by a
+gentle concavity at the arm-pit, and then swells out in a convex curve
+over the biceps, reaching (34), where it becomes concave, and passing
+through (35), again becomes convex in passing over the long supinator,
+and, becoming gently concave as it passes the radial extensors, rises
+slightly at (36), and ends at (37), where the outline of the wrist
+commences. The outline of the back of the arm commences with a concave
+curve at (38), which becomes convex as it passes from the deltoid to the
+long extensor and ends at the elbow (39), from below which the outline
+follows the convex curve of the extensor ulnaris, reaching the wrist at
+the point (40).
+
+It will be seen that the various undulations of the outline are regulated
+by points which are determined generally by the intersections and
+sometimes by directions and extensions of the lines of the diagram, in
+the same manner in which I shewed proportion to be imparted, in a late
+work, to the osseous structure. The mode in which the curves of (¹⁄₂),
+(¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆) are thus so harmoniously blended in the
+outline of the female figure, only remains to be explained.
+
+The curves which compose the outline of the female form are therefore
+simply those of (¹⁄₂), (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆).
+
+Manner in which these curves are disposed in the lateral outline (figure
+1, Plate XIX.):—
+
+                                         Points.       Curves.
+    Head                              from  1 to  2     (¹⁄₂)
+    Face                                ”   2  ”  3     (¹⁄₃)
+    Neck                                ”   3  ”  4     (¹⁄₅)
+    Shoulder                            ”   4  ”  6     (¹⁄₆)
+       ”                                ”   6  ”  8     (¹⁄₄)
+    Trunk                               ”   9  ” 15     (¹⁄₄)
+      ”                                 ”  21  ” 24     (¹⁄₂)
+    Outer surface of thigh and leg      ”  15  ” 20     (¹⁄₆)
+    Inner surface of thigh and leg      ”  25  ” 30     (¹⁄₆)
+    Outer surface of the arm            ”   8  ” 33     (¹⁄₆)
+    Inner surface of the arm            ”   9  ” 36     (¹⁄₆)
+
+Manner in which they are disposed in the outline (figure 2, Plate XIX.):—
+
+                                         Points.       Curves.
+    Front of neck                     from  1 to  2     (¹⁄₆)
+        ”    trunk                      ”   2  ” 10     (¹⁄₄)
+    Back of  neck                       ”  16  ” 18     (¹⁄₆)
+        ”    trunk                      ”  18  ” 23     (¹⁄₄)
+        ”      ”                        ”  23  ” 25     (¹⁄₃)
+    Front of thigh and leg              ”  11  ” 13     (¹⁄₄)
+      ”        ”        ”               ”  13  ” 15     (¹⁄₆)
+    Back of thigh and leg               ”  25  ” 32     (¹⁄₆)
+    Front of the arm                    ”  33  ” 37     (¹⁄₆)
+    Back of the arm                     ”  38  ” 40     (¹⁄₆)
+    Foot                                ”   0  ”  0     (¹⁄₆)
+
+[Sidenote: Plate XX.]
+
+In order to exemplify more clearly the manner in which these various
+curves appear in the outline of the figure, I give in Plate XX. the whole
+curvilinear figures, complete, to which these portions belong that form
+the outline of the sides of the head, neck, and trunk, and of the outer
+surface of the thighs and legs.
+
+The various angles which the axes of these ellipses form with the
+vertical, will be found amongst other details in the works I have just
+referred to.
+
+
+No. V.
+
+At page 85 I have remarked upon the variety that may be introduced into
+any particular form of vase; and, in order to give the reader an idea of
+the ease with which this may be done without violating the harmonic law,
+I shall here give three examples:—
+
+[Sidenote: Plate XXI.]
+
+The first of these (Plate XXI.) differs from the Portland vase, in the
+concave curve of the neck flowing more gradually into the convex curve of
+the body.
+
+[Sidenote: Plate XXII.]
+
+The second (Plate XXII.) differs from the same vase in the same change
+of contour, as also in being of a smaller diameter at the top and at the
+bottom.
+
+[Sidenote: Plate XXIII.]
+
+The third (Plate XXIII.) is the most simple arrangement of the elliptic
+curve by which this kind of form may be produced; and it differs from the
+Portland vase in the relative proportions of height and diameter, and in
+having a fuller curve of contour.
+
+The following comparison of the angles employed in these examples, with
+the angles employed in the original, will shew that the changes of
+contour in these forms, arise more from the mode in which the angles are
+arranged than in a change of the angles themselves:—
+
+                  Line        Line        Line         Line
+    Plate VIII.  _A C_ (¹⁄₂) _B C_  (¹⁄₃) _o p_  (H)  _v u_  (³⁄₁₀)
+    Plate XXI.         (¹⁄₂)        (¹⁄₃)       (²⁄₉)        (¹⁄₄)
+    Plate XXII.        (¹⁄₂)        (¹⁄₃)       (¹⁄₈)        (⁴⁄₉)
+    Plate XXIII.       (¹⁄₂)        (¹⁄₄)        (H)          (-)
+
+                  Line         Line
+    Plate VIII.  _m n_ (¹⁄₃)  _i k_ (¹⁄₅)  ellipse   (¹⁄₄)  rectangle (²⁄₅)
+    Plate XXI.         (²⁄₉)        (¹⁄₅)            (¹⁄₄)            (²⁄₅)
+    Plate XXII.        (¹⁄₃)        (¹⁄₅)            (¹⁄₄)            (²⁄₅)
+    Plate XXIII.       (¹⁄₅)        (¹⁄₅)  ellipses {(¹⁄₃)}           (¹⁄₃)
+                                                    {(¹⁄₄)}
+
+The harmonic elements of each are therefore simply the following parts of
+the right angle:—
+
+                  Tonic.  Dominant.  Mediant.  Submediant.
+    Plate VIII.   (¹⁄₂)     (¹⁄₃)     (¹⁄₅)      (³⁄₁₀)
+                  (¹⁄₄)
+
+                  Tonic.  Dominant.  Mediant.  Supertonic.
+    Plate XXI.    (¹⁄₂)     (¹⁄₃)     (¹⁄₅)       (²⁄₉)
+                  (¹⁄₄)
+
+                  Tonic.  Dominant.  Mediant.  Supertonic.
+    Plate XXII.   (¹⁄₂)     (¹⁄₃)     (¹⁄₅)       (⁴⁄₉)
+                  (¹⁄₄)
+                  (¹⁄₈)
+
+                  Tonic.  Dominant.  Mediant.
+    Plate XXIII.  (¹⁄₂)     (¹⁄₃)     (¹⁄₅)
+                  (¹⁄₄)
+
+
+No. VI.
+
+So far as I know, there has been only one attempt in modern times,
+besides my own, to establish a universal system of proportion, based on a
+law of nature, and applicable to art. This attempt consists of a work of
+457 pages, with 166 engraved illustrations, by Dr Zeising, a professor in
+Leipzic, where it was published in 1854.
+
+One of the most learned and talented professors in our Edinburgh
+University has reviewed that work as follows:—
+
+“It has been rather cleverly said that the intellectual distinction
+between an Englishman and a Scotchman is this—‘Give an Englishman two
+facts, and he looks out for a third; give a Scotchman two facts, and he
+looks out for a theory.’ Neither of these tests distinguishes the German;
+he is as likely to seek for a third fact as for a theory, and as likely
+to build a theory on two facts as to look abroad for further information.
+But once let him have a theory in his mind, and he will ransack heaven
+and earth until he almost buries it under the weight of accumulated
+facts. This remark applies with more than common force to a treatise
+published last year by Dr Zeising, a professor in Leipsic, ‘On a law of
+proportion which rules all nature.’ The ingenious author, after proving
+from the writings of ancient and modern philosophers that there always
+existed the belief (whence derived it is difficult to say), that some
+law does bind into one formula all the visible works of God, proceeds to
+criticise the opinions of individual writers respecting that connecting
+law. It is not our purpose to follow him through his lengthy examination.
+Suffice it to say that he believes he has found the lost treasure in the
+_Timæus_ of Plato, c. 31. The passage is confessedly an obscure one, and
+will not bear a literal translation. The interpretation which Dr Zeising
+puts on it is certainly a little strained, but we are disposed to admit
+that he does it with considerable reason. Agreeably to him, the passage
+runs thus:—‘That bond is the most beautiful which binds the things as
+much as possible into one; and proportion effects this most perfectly
+when three things are so united that the greater bears to the middle the
+same ratio that the middle bears to the less.’
+
+“We must do Dr Zeising the justice to say that he has not made more
+than a legitimate use of the materials which were presented to him in
+the writings of the ancients, in his endeavour to establish the fact
+of the existence of this law amongst them. The canon of Polycletes,
+the tradition of Varro mentioned by Pliny relative to that canon, the
+writings of Galen and others, are all brought to bear on the same point
+with more or less force. The sum of this portion of the argument is
+fairly this,—that the ancient sculptors had _some_ law of proportion—some
+authorised examplar to which they referred as their work proceeded. That
+it was the law here attributed to Plato is by no means made out; but,
+considering the incidental manner in which that law is referred to, and
+the obscurity of the passages as they exist, it is, perhaps, too much to
+expect more than this broad feature of coincidence—the fact that some
+law was known and appealed to. Dr Zeising now proceeds to examine modern
+theories, and it is fair to state that he appears generally to take a
+very just view of them.
+
+“Let us now turn to Dr Zeising’s own theory. It is this—that in every
+beautiful form lines are divided in extreme and mean ratio; or, that
+any line considered as a whole, bears to its larger part the same
+proportion that the larger bears to the smaller—thus, a line of 5
+inches will be divided into parts which are very nearly 2 and 3 inches
+respectively (1·9 and 3·1 inches). This is a well-known division of a
+line, and has been called the GOLDEN rule, but when or why, it is not
+easy to ascertain. With this rule in his hand, Dr Zeising proceeds to
+examine all nature and art; nay, he even ventures beyond the threshold
+of nature to scan Deity. We will not follow him so far. Let us turn over
+the pages of his carefully illustrated work, and see how he applies his
+line. We meet first with the Apollo Belvidere—the golden line divides
+him happily. We cannot say the same of the division of a handsome face
+which occurs a little further on. Our preconceived notions have made the
+face terminate with the chin, and not with the centre of the throat.
+It is evident that, with such a rule as this, a little latitude as to
+the extreme point of the object to be measured, relieves its inventor
+from a world of perplexities. This remark is equally applicable to the
+_arm_ which follows, to which the rule appears to apply admirably, yet
+we have tried it on sundry plates of arms, both fleshy and bony, without
+a shadow of success. Whether the rule was made for the arm or the arm
+for the rule, we do not pretend to decide. But let us pass hastily on
+to page 284, where the Venus de Medicis and Raphael’s Eve are presented
+to us. They bear the application of the line right well. It might,
+perhaps, be objected that it is remarkable that the same rule applies
+so exactly to the existing position of the figures, such as the Apollo
+and the Venus, the one of which is upright, and the other crouching.
+But let that pass. We find Dr Zeising next endeavouring to square his
+theory with the distances of the planets, with wofully scanty success.
+Descending from his lofty position, he spans the earth from corner to
+corner, at which occupation we will leave him for a moment, whilst we
+offer a suggestion which is equally applicable to poets, painters,
+novelists, and theorisers. Never err in excess—defect is the safe side—it
+is seldom a fault, often a real merit. Leave something for the student
+of your works to do—don’t chew the cud for him. Be assured he will not
+omit to pay you for every little thing which you have enabled him to
+discover. Poor Professor Zeising! he is far too German to leave any
+field of discovery open for his readers. But let us return to him; we
+left him on his back, lost for a time in a hopeless attempt to double
+Cape Horn. We will be kind to him, as the child is to his man in the
+Noah’s ark, and set him on his legs amongst his toys again. He is now in
+the vegetable kingdom, amidst oak leaves and sections of the stems of
+divers plants. He is in his element once more, and it were ungenerous
+not to admit the merit of his endeavours, and the success which now and
+then attends it. We will pass over his horses and their riders, together
+with that portly personage, the Durham ox, for we have caught a glimpse
+of a form familiar to our eyes, the ever-to-be-admired Parthenon. This
+is the true test of a theory. Unlike the Durham ox just passed before
+us, the Parthenon will stand still to be measured. It has so stood for
+twenty centuries, and every one that has scanned its proportions has
+pronounced them exquisite. Beauty is not an adaptation to the acquired
+taste of a single nation, or the conventionality of a single generation.
+It emanates from a deep-rooted principle in nature, and appeals to the
+verdict of our whole humanity. We don’t find fault with the Durham ox—his
+proportions are probably good, though they be the result of breeding
+and cross-breeding; still we are not sure whether, in the march of
+agriculture, our grandchildren may not think him a very wretched beast.
+But there is no mistake about the Parthenon; as a type of proportion it
+stands, has stood, and shall stand. Well, then, let us see how Dr Zeising
+succeeds with his rule here. Alas! not a single point comes right. The
+Parthenon is condemned, or its condemnation condemns the theory. Choose
+your part. We choose the latter alternative; and now, our choice being
+made, we need proceed no further. But a question or two have presented
+themselves as we went along, which demand an answer. It may be asked—How
+do you account for the esteem in which this law of the section in
+extreme and mean ratio was held? We reply—That it was esteemed just in
+the same way that a tree is esteemed for its fruit. To divide a right
+angle into two or three, four or six, equal parts was easy enough. But
+to divide it into five or ten such parts was a real difficulty. And how
+was the difficulty got over? It was effected by means of this golden
+rule. This is its great, its ruling application; and if we adopt the
+notion that the ancients were possessed with the idea of the existence
+of angular symmetry, we shall have no difficulty in accounting for their
+appreciation of this problem. Nay, we may even go further, and admit,
+with Dr Zeising, the interpretation of the passage of Plato,—only with
+this limitation, that Plato, as a geometer, was carried away by the
+geometry of æsthetics from the thing itself. It may be asked again—Is it
+not probable that _some_ proportionality does exist amongst the parts of
+natural objects? We reply—That, _à priori_, we expect some such system
+to exist, but that it is inconsistent with the scheme of _least effort_,
+which pervades and characterises all natural succession in space or in
+time, that that system should be a complicated one. Whatever it is, its
+essence must be simplicity. And no system of simple linear proportion
+is found in nature; quite the contrary. We are, therefore, driven to
+another hypothesis, viz.—that the simplicity is one of angles, not of
+lines; that the eye estimates by search round a point, not by ascending
+and descending, going to the right and to the left,—a theory which we
+conceive all nature conspires to prove. Beauty was not created for the
+eye of man, but the eye of man and his mental eye were created for the
+appreciation of beauty. Examine the forms of animals and plants so minute
+that nothing short of the most recent improvements in the microscope can
+succeed in detecting their symmetry; or examine the forms of those little
+silicious creations which grew thousands of years before Man was placed
+on the earth, and, with forms of marvellous and varied beauty, they all
+point to its source in angular symmetry. This is the keystone of formal
+beauty, alike in the minutest animalcule, and in the noblest of God’s
+works, his own image—Man.”
+
+
+THE END.
+
+               BALLANTYNE AND COMPANY, PRINTERS, EDINBURGH.
+
+
+
+
+FOOTNOTES
+
+
+[1] Sir David Brewster.
+
+[2] No. CLVIII., October 1843.
+
+[3] Diogenes Laërtius’s “Lives of the Philosophers,” literally
+translated. Bohn: London.
+
+[4] Ibid.
+
+[5] Rose’s “Biographical Dictionary.”
+
+[6] Professor Laycock, now of the University of Edinburgh.
+
+[7] “The Geometric Beauty of the Human Figure Defined,” &c.
+
+[8] Longman and Co., London.
+
+[9] See Appendix.
+
+[10] “The Orthographic Beauty of the Parthenon,” &c., and “The Harmonic
+Law of Nature applied to Architectural Design.”
+
+[11] For further details, see “Harmonic Law of Nature,” &c.
+
+[12] By a very simple machine, which I have lately invented, an ellipse
+of any given proportions, even to those of (¹⁄₆₄), which is the curve of
+the entases of the columns of the Parthenon (see Plate VII.), and of any
+length, from half an inch to fifty feet or upwards, may be easily and
+correctly described; the length and angle of the required ellipse being
+all that need be given.
+
+[13] “The Orthographic Beauty of the Parthenon,” &c.
+
+[14] “The Orthographic Beauty of the Parthenon,” &c.
+
+[15] Ibid.
+
+[16] “The Harmonic Law of Nature applied to Architectural Design.”
+
+[17] “Physio-philosophy.” By Dr Oken. Translated by Talk; and published
+by the Ray Society. London, 1848.
+
+[18] “The Science of those Proportions by which the Human Head and
+Countenance, as represented in Works of ancient Greek Art, are
+distinguished from those of ordinary Nature.”
+
+[19] “The Geometric Beauty of the Human Figure Defined,” &c., and “The
+Natural Principles of Beauty Developed in the Human Figure.”
+
+[20] “The Geometric Beauty of the Human Figure Defined,” &c.
+
+[21] “Essay on Ornamental Design,” &c., and “The Geometric Beauty of the
+Human Figure,” &c.
+
+[22] “A Nomenclature of Colours, applicable to the Arts and Natural
+Sciences,” &c., &c.
+
+[23] See pp. 24 and 25.
+
+[24] “The Principles of Beauty in Colouring Systematised,” Fourteen
+Diagrams, each containing Six Colours and Hues.
+
+“A Nomenclature of Colours,” &c., Forty Diagrams, each containing Twelve
+Examples of Colours, Hues, Tints, and Shades.
+
+“The Laws of Harmonious Colouring,” &c., One Diagram, containing Eighteen
+Colours and Hues.
+
+[25] “Etchings Representing the Best Examples of Grecian and Roman
+Architectural Ornament, drawn from the Originals,” &c. By Charles
+Heathcote Tatham, Architect. London: Priestly and Weale. 1826.
+
+[26] “The Orthographic Beauty of the Parthenon,” &c.
+
+[27] “Science of those Proportions,” &c.
+
+[28] Abbé Barthélémie’s “Travels of Anacharsis in Greece,” vol iv., pp.
+193, 195.
+
+[29] Abbé Barthélémie (vol. ii., pp. 168, 169), who cites as his
+authorities, Cicer. De Nat. Deor., lib. i., cap. ii., t. 2, p. 405;
+Justin Mart., Ovat. ad Gent., p. 10; Aristot. Metaph., lib. i., cap. v.,
+t. 2, p. 845.
+
+[30] Müller’s “Ancient Art and its Remains.”
+
+[31] “Anacharsis’ Travels in Greece.” By the Abbé Barthélémie, vol. ii.,
+p. 325.
+
+[32] “Anacharsis’ Travels in Greece.” By the Abbé Barthélémie, vol. vi.,
+p. 225. The authorities the Abbé quotes are—Quintil., lib. xii., cap. x.,
+p. 744; Plin., lib. xxxv., cap. ix., p. 691.
+
+[33] Müller’s “Archæology of Art,” &c.
+
+
+
+
+Works by the Same Author.
+
+
+I.
+
+In royal 8vo, with Copperplate Illustrations, price 2s. 6d.,
+
+THE HARMONIC LAW OF NATURE APPLIED TO ARCHITECTURAL DESIGN.
+
+_From the Athenæum._
+
+The beauty of the theory is its universality, and its simplicity. In
+nature, the Creator accomplished his purposes by the simplest means—the
+harmony of nature is indestructible and self-restoring. Mr Hay’s book on
+the “Parthenon,” on the “Natural Principles of Beauty as developed in the
+Human Figure,” his “Principles of Symmetrical Beauty,” his “Principles of
+Colouring, and Nomenclature of Colours,” his “Science of Proportion,” and
+“Essay on Ornamental Design,” we have already noticed with praise as the
+results of philosophical and original thought.
+
+_From the Daily News._
+
+This essay is a new application to Lincoln cathedral in Gothic
+architecture, and to the Temple of Theseus in Greek architecture, of the
+principles of symmetrical beauty already so profusely illustrated and
+demonstrated by Mr Hay. The theory which Mr Hay has propounded in so many
+volumes is not only a splendid contribution towards a science of æsthetic
+proportions, but, for the first time in the history of art, proves the
+possibility, and lays the foundations of such a science. To those who are
+not acquainted with the facts, these expressions will sound hyperbolical,
+but they are most true.
+
+
+II.
+
+In royal 8vo, with Copperplate Illustrations, price 5s.,
+
+THE NATURAL PRINCIPLES OF BEAUTY, AS DEVELOPED IN THE HUMAN FIGURE.
+
+_From the Spectator._
+
+We cannot refuse to entertain Mr Hay’s system as of singular intrinsic
+excellence. The simplicity of his law and its generality impress
+themselves more deeply on the conviction with each time of enforcement.
+His theory proceeds from the idea, that in nature every thing is effected
+by means more simple than any other that could have been conceived,—an
+idea certainly consistent with whatever we can trace out or imagine of
+the all-wise framing of the universe.
+
+_From the Sun._
+
+By founding (if we may so phrase it) this noble theory, Mr Hay has
+covered his name with distinction, and has laid the basis, we conceive,
+of no ephemeral reputation. By illustrating it anew, through the
+medium of this graceful treatise, he has conferred a real boon upon
+the community, for he has afforded the public another opportunity of
+following the golden rule of the poet—by looking through the holy and
+awful mystery of creation to the holier and yet more awful mystery of
+Omnipotence.
+
+_From the Cambridge Journal of Classical and Sacred Philology._
+
+The inquiries which of late years have been instituted by Mr D. R. Hay
+of Edinburgh, on the proportions of the human figure, and on the natural
+principles of beauty, as illustrated by works of Greek art, constitute an
+epoch in the study of æsthetics and the philosophy of form.
+
+
+III.
+
+In royal 8vo, with Copperplate Illustrations, price 5s.,
+
+THE ORTHOGRAPHIC BEAUTY OF THE PARTHENON REFERRED TO A LAW OF NATURE.
+
+To which are prefixed, a few Observations on the Importance of Æsthetic
+Science as an Element in Architectural Education.
+
+_From the Scottish Literary Gazette._
+
+We think this work will satisfy every impartial mind that Mr Hay has
+developed the true theory of the Parthenon—that he has, in fact, to use
+a kindred phraseology, both _parsed_ and _scanned_ this exquisitely
+beautiful piece of architectural composition, and that, in doing so, he
+has provided the true key by which the treasures of Greek art may be
+further unlocked, and rendered the means of correcting, improving, and
+elevating modern practice.
+
+_From the Edinburgh Guardian._
+
+Again and again the attempt has been made to detect harmonic ratios in
+the measurement of Athenian architecture, but ever without reward. Mr Hay
+has, however, made the discovery, and to an extent of which no one had
+previously dreamt.
+
+
+IV.
+
+In 8vo, 100 Plates, price 6s.,
+
+FIRST PRINCIPLES OF SYMMETRICAL BEAUTY.
+
+_From the Spectator._
+
+This is a grammar of pure form, in which the elements of symmetrical, as
+distinguished from picturesque beauty, are demonstrated, by reducing the
+outlines or planes of curvilinear and rectilinear forms to their origin
+in the principles of geometrical proportion. In thus analysing symmetry
+of outline in natural and artificial objects, Mr Hay determines the fixed
+principles of beauty in positive shape, and shews how beautiful forms may
+be reproduced and infinitely varied with mathematical precision. Hitherto
+the originating and copying of beautiful forms have been alike empirical;
+the production of a new design for a vase or a jug has been a matter of
+chance between the eye and the hand; and the copying of a Greek moulding
+or ornament, a merely mechanical process. By a series of problems, Mr
+Hay places both the invention and imitation of beautiful forms on a sure
+basis of science, giving to the fancy of original minds a clue to the
+evolving of new and elegant shapes, in which the infinite resources of
+nature are made subservient to the uses of art.
+
+The volume is illustrated by one hundred diagrams beautifully executed,
+that serve to explain the text, and suggest new ideas of beauty of
+contour in common objects. To designers of pottery, hardware, and
+architectural ornaments, this work is particularly valuable; but artists
+of every kind, and workmen of intelligence, will find it of great utility.
+
+_From the Athenæum._
+
+The volume before us is the seventh of Mr Hay’s works. It is the most
+practical and systematic, and likely to be one of the most useful.
+It is, in short, a grammar of form, or a spelling-book of beauty.
+This is beginning at the right end of the matter; and the necessity
+for this kind of knowledge will inevitably, though gradually, be
+felt. The work will, therefore, be ultimately appreciated and
+adopted as an introduction to the study of beautiful forms.
+
+The third part of the work treats of the Greek oval or composite ellipse,
+as Mr Hay calls it. It is an ellipse of three foci, and gives practical
+forms for vases and architectural mouldings, which are curious to the
+architect, and will be very useful both to the potter, the moulder,
+and the pattern-drawer. A fourth part contains applications of this to
+practice. Of the details worked out with so much judgment and ingenuity
+by Mr Hay, we should in vain attempt to communicate just notions without
+the engravings of which his book is full. We must, therefore, refer to
+the work itself. The forms there given are full of beauty, and so far
+tend to prove the system.
+
+
+V.
+
+In 8vo, 14 Coloured Diagrams, Second Edition, price 15s.,
+
+THE PRINCIPLES OF BEAUTY IN COLOURING SYSTEMATISED.
+
+_From the Spectator._
+
+In this new analysis of the harmonies of colour, Mr Hay has performed the
+useful service of tracing to the operation of certain fixed principles
+the sources of beauty in particular combinations of hues and tints; so
+that artists may, by the aid of this book, produce, with mathematical
+certainty, the richest effects, hitherto attainable by genius alone. Mr
+Hay has reduced this branch of art to a perfect system, and proved that
+an offence against good taste in respect to combinations of colour is, in
+effect, a violation of natural laws.
+
+
+VI.
+
+In 8vo, 228 Examples of Colours, Hues, Tints, and Shades, price 63s.,
+
+A NOMENCLATURE OF COLOURS, APPLICABLE TO THE ARTS AND NATURAL SCIENCES.
+
+_From the Daily News._
+
+In this work Mr Hay has brought a larger amount of practical knowledge
+to bear on the subject of colour than any other writer with whom we are
+acquainted, and in proportion to this practical knowledge is, as might be
+expected, the excellence of his treatise. There is much in this volume
+which we would most earnestly recommend to the notice of artists, house
+decorators, and, indeed, to all whose business or profession requires a
+knowledge of the management of colour. The work is replete with hints
+which they might turn to profitable account, and which they will find
+nowhere else.
+
+_From the Athenæum._
+
+We have formerly stated the high opinion we entertain of Mr Hay’s
+previous exertions for the improvement of decorative art in this country.
+We have already awarded him the merit of invention and creation of the
+new and the beautiful in form. In his former treatises he furnished a
+theory of definite proportions for the creation of the beautiful in form.
+In the present work he proposes to supply a scale of definite proportions
+for chromatic beauty. For this purpose he sets out very properly with a
+precise nomenclature of colour. In this he has constructed a vocabulary
+for the artist—an alphabet for the artizan. He has gone further—he
+constructs words for three syllables. From this time, it will be possible
+to write a letter in Edinburgh about a coloured composition, which shall
+be read off in London, Paris, St Petersburg, or Pekin, and shall so
+express its nature that it can be reproduced in perfect identity. This Mr
+Hay has done, or at least so nearly, as to deserve our thanks on behalf
+of art, and artists of all grades, even to the decorative artizan—not one
+of whom, be he house-painter, china pattern-drawer, or calico printer,
+should be without the simple manual of “words for colours.”
+
+
+VII.
+
+In post 8vo, with a Coloured Diagram, Sixth Edition, price 7s. 6d.,
+
+THE LAWS OF HARMONIOUS COLOURING ADAPTED TO INTERIOR DECORATIONS.
+
+_From the Atlas._
+
+Every line of this useful book shews that the author has contrived to
+intellectualise his subject in a very interesting manner. The principles
+of harmony in colour as applied to decorative purposes, are explained and
+enforced in a lucid and practical style, and the relations of the various
+tints and shades to each other, so as to produce a harmonious result, are
+descanted upon most satisfactorily and originally.
+
+_From the Edinburgh Review._
+
+In so far as we know, Mr Hay is the first and only modern artist who
+has entered upon the study of these subjects without the trammels of
+prejudice and authority. Setting aside the ordinances of fashion, as
+well as the dicta of speculation, he has sought the foundation of
+his profession in the properties of light, and in the laws of visual
+sensation, by which these properties are recognised and modified. The
+truths to which he has appealed are fundamental and irrefragable.
+
+_From the Athenæum._
+
+We have regarded, and do still regard, the production of Mr Hay’s works
+as a remarkable psychological phenomenon—one which is instructive both
+for the philosopher and the critic to study with care and interest, not
+unmingled with respect. We see how his mind has been gradually guided
+by Nature herself out of one track, and into another, and ever and anon
+leading him to some vein of the beautiful and true, hitherto unworked.
+
+
+VIII.
+
+In 4to, 25 Plates, price 36s.,
+
+ON THE SCIENCE OF THOSE PROPORTIONS BY WHICH THE HUMAN HEAD AND
+COUNTENANCE, AS REPRESENTED IN ANCIENT GREEK ART, ARE DISTINGUISHED FROM
+THOSE OF ORDINARY NATURE.
+
+(PRINTED BY PERMISSION.)
+
+_From a Letter to the Author by Sir William Hamilton, Bart., Professor of
+Logic and Metaphysics in the Edinburgh University._
+
+Your very elegant volume, “Science of those Proportions,” &c., is to me
+extremely interesting, as affording an able contribution to what is the
+ancient, and, I conceive, the true theory of the beautiful. But though
+your doctrine coincides with the one prevalent through all antiquity, it
+appears to me quite independent and original in you; and I esteem it the
+more that it stands opposed to the hundred one-sided and exclusive views
+prevalent in modern times.
+
+_From Chambers’s Edinburgh Journal._
+
+We now come to another, and much more remarkable corroboration, which
+calls upon us to introduce to our readers one of the most valuable and
+original contributions that have ever been made to the Philosophy of Art,
+viz., Mr Hay’s work “On the Science of those Proportions,” &c. Mr Hay’s
+plan is simply to form a scale composed of the well-known vibrations of
+the monochord, which are the alphabet of music, and then to draw upon the
+quadrant of a circle angles _answering to these vibrations_. With the
+series of triangles thus obtained he combines a circle and an ellipse,
+the proportions of which are derived from the triangles themselves; and
+thus he obtains an infallible rule for the composition of the head of
+ideal beauty.
+
+
+IX.
+
+In 4to, 16 Plates, price 30s.,
+
+THE GEOMETRIC BEAUTY OF THE HUMAN FIGURE DEFINED.
+
+To which is prefixed, a SYSTEM of ÆSTHETIC PROPORTION applicable to
+ARCHITECTURE and the other FORMATIVE ARTS.
+
+_From the Cambridge Journal of Classical and Sacred Philology._
+
+We feel bound to pay Mr Hay a large and glad tribute of praise for
+having devised a system of proportions which rises superior to the
+idiosyncrasies of different artists, which brings back to one common type
+the sensations of Eye and Ear, and so makes a giant stride towards that
+_codification_ of the laws of the universe which it is the business of
+science to effect. We have no hesitation in saying that, for scientific
+precision of method and importance of results, Albert Durer, Da Vinci,
+and Hogarth—not to mention less noteworthy writers—must all yield the
+palm to Mr Hay.
+
+
+X.
+
+In oblong folio, 57 Plates and numerous Woodcuts, price 42s.,
+
+AN ESSAY ON ORNAMENTAL DESIGN, IN WHICH ITS TRUE PRINCIPLES ARE DEVELOPED
+AND ELUCIDATED, &c.
+
+_From the Athenæum._
+
+In conclusion, Mr Hay’s book goes forth with our best wishes. It must be
+good. It must be prolific of thought—stimulant of invention. It is to be
+acknowledged as a benefit of an unusual character conferred on the arts
+of ornamental design.
+
+_From the Spectator._
+
+Mr Hay has studied the subject deeply and scientifically. In this
+treatise on ornamental design, the student will find a clue to the
+discovery of the source of an endless variety of beautiful forms and
+combinations of lines, in the application of certain fixed laws of
+harmonious proportion to the purposes of art. Mr Hay also exemplifies
+the application of his theory of linear harmony to the production of
+beautiful forms generally, testing its soundness by applying it to the
+human figure, and the purest creations of Greek art.
+
+_From Fraser’s Magazine._
+
+Each part of this work is enriched by diagrams of great beauty, direct
+emanations of principle, and, consequently, presenting entirely new
+combinations of form. Had our space permitted, we should have made some
+extracts from this “Essay on Ornamental Design;” and we would have done
+so, because of the discriminating taste by which it is pervaded, and the
+forcible observations which it contains; but we cannot venture on the
+indulgence.
+
+
+XI.
+
+In 4to, 17 Plates and 38 Woodcuts, price 25s.,
+
+PROPORTION, OR THE GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED.
+
+
+XII.
+
+In 4to, 18 Plates and numerous Woodcuts, price 15s.,
+
+THE NATURAL PRINCIPLES AND ANALOGY OF THE HARMONY OF FORM.
+
+_From the Edinburgh Review._
+
+Notwithstanding some trivial points of difference between Mr Hay’s views
+and our own, we have derived the greatest pleasure from the perusal of
+these works. They are all composed with accuracy and even elegance. His
+opinions and views are distinctly brought before the reader, and stated
+with that modesty which characterises genius, and that firmness which
+indicates truth.
+
+_From Blackwood’s Magazine._
+
+We have no doubt that when Mr Hay’s Art-discovery is duly developed and
+taught, as it should be, in our schools, it will do more to improve the
+general taste than anything which has yet been devised.
+
+
+
+*** END OF THE PROJECT GUTENBERG EBOOK 75399 ***
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+<div style='text-align:center'>*** START OF THE PROJECT GUTENBERG EBOOK 75399 ***</div>
+
+<p><span class="pagenum"><a id="Page_i"></a>[i]</span></p>
+
+<h1>THE SCIENCE OF BEAUTY.</h1>
+
+<hr class="chap x-ebookmaker-drop">
+
+<p><span class="pagenum"><a id="Page_ii"></a>[ii]</span></p>
+
+<p class="center">EDINBURGH:<br>
+PRINTED BY BALLANTYNE AND COMPANY,<br>
+PAUL’S WORK.</p>
+
+<hr class="chap x-ebookmaker-drop">
+
+<p><span class="pagenum"><a id="Page_iii"></a>[iii]</span></p>
+
+<p class="titlepage larger"><span class="smaller">THE</span><br>
+<br>
+<span class="larger">SCIENCE OF BEAUTY,</span><br>
+<br>
+AS DEVELOPED IN NATURE AND<br>
+APPLIED IN ART.</p>
+
+<p class="titlepage"><span class="smaller">BY</span><br>
+D. R. HAY, F.R.S.E.</p>
+
+<div class="blockquote">
+
+<p>“The irregular combinations of fanciful invention may delight awhile, by that
+novelty of which the common satiety of life sends us all in quest; the pleasures
+of sudden wonder are soon exhausted, and the mind can only repose on the stability
+of truth.”</p>
+
+<p class="right"><span class="smcap">Dr Johnson.</span></p>
+
+</div>
+
+<p class="titlepage">WILLIAM BLACKWOOD AND SONS,<br>
+<span class="smaller">EDINBURGH AND LONDON.<br>
+MDCCCLVI.</span></p>
+
+<p><span class="pagenum"><a id="Page_iv"></a>[iv]</span></p>
+
+<hr class="chap x-ebookmaker-drop">
+
+<p><span class="pagenum"><a id="Page_v"></a>[v]</span></p>
+
+<p class="dedication"><span class="smaller">TO</span><br>
+JOHN GOODSIR, ESQ., F.R.S S. L. &amp; E.,<br>
+<span class="smaller"><span class="allsmcap">PROFESSOR OF ANATOMY IN THE UNIVERSITY OF EDINBURGH,</span><br>
+AS AN EXPRESSION OF GRATITUDE FOR VALUABLE ASSISTANCE,<br>
+AS ALSO OF HIGH ESTEEM AND SINCERE REGARD,<br>
+THIS VOLUME IS DEDICATED,<br>
+BY</span><br>
+<span class="spacer">THE AUTHOR.</span></p>
+
+<p><span class="pagenum"><a id="Page_vi"></a>[vi]</span></p>
+
+<hr class="chap x-ebookmaker-drop">
+
+<div class="chapter">
+
+<p><span class="pagenum"><a id="Page_vii"></a>[vii]</span></p>
+
+<h2 class="nobreak" id="PREFACE">PREFACE.</h2>
+
+</div>
+
+<p>My theory of beauty in form and colour being now
+admitted by the best authorities to be based on truth,
+I have of late been often asked, by those who wished
+to become acquainted with its nature, and the manner
+of its being applied in art, which of my publications I
+would recommend for their perusal. This question
+I have always found difficulty in answering; for
+although the law upon which my theory is based is
+characterised by unity, yet the subjects in which it is
+applied, and the modes of its application, are equally
+characterised by variety, and consequently occupy
+several volumes.</p>
+
+<p>Under these circumstances, I consulted a highly
+respected friend, whose mathematical talents and
+good taste are well known, and to whom I have
+been greatly indebted for much valuable assistance
+during the course of my investigations. The advice
+I received on this occasion, was to publish a <i>résumé</i>
+of my former works, of such a character as not only<span class="pagenum"><a id="Page_viii"></a>[viii]</span>
+to explain the nature of my theory, but to exhibit to
+the general reader, by the most simple modes of illustration
+and description, how it is developed in nature,
+and how it may be extensively and with ease applied
+in those arts in which beauty forms an essential
+element.</p>
+
+<p>The following pages, with their illustrations, are
+the results of an attempt to accomplish this object.</p>
+
+<p>To those who are already acquainted, through
+my former works, with the nature, scope, and tendency
+of my theory, I have the satisfaction to intimate
+that I have been enabled to include in this <i>résumé</i>
+much original matter, with reference both to form
+and colour.</p>
+
+<p class="right">D. R. HAY.</p>
+
+<hr class="chap x-ebookmaker-drop">
+
+<div class="chapter">
+
+<p><span class="pagenum"><a id="Page_ix"></a>[ix]</span></p>
+
+<h2 class="nobreak" id="CONTENTS">CONTENTS.</h2>
+
+</div>
+
+<table id="contents">
+  <tr>
+    <td></td>
+    <td class="tdpg smaller">PAGE</td>
+  </tr>
+  <tr>
+    <td><span class="smcap">Introduction</span></td>
+    <td class="tdpg"><a href="#INTRODUCTION">1</a></td>
+  </tr>
+  <tr>
+    <td><span class="smcap">The Science of Beauty, evolved from the
+    Harmonic Law of Nature, agreeably to the Pythagorean System of
+    Numerical Ratio</span></td>
+    <td class="tdpg"><a href="#EVOLVED_FROM_THE_HARMONIC_LAW_OF_NATURE">15</a></td>
+  </tr>
+  <tr>
+    <td><span class="smcap">The Science of Beauty, as applied to
+    Sounds</span></td>
+    <td class="tdpg"><a href="#AS_APPLIED_TO_SOUNDS">28</a></td>
+  </tr>
+  <tr>
+    <td><span class="smcap">The Science of Beauty, as applied to
+    Forms</span></td>
+    <td class="tdpg"><a href="#AS_APPLIED_TO_FORMS">34</a></td>
+  </tr>
+  <tr>
+    <td><span class="smcap">The Science of Beauty, as developed in
+    the Form of the Human Head and Countenance</span></td>
+    <td class="tdpg"><a href="#AS_DEVELOPED_IN_THE_HUMAN_HEAD_AND_COUNTENANCE">54</a></td>
+  </tr>
+  <tr>
+    <td><span class="smcap">The Science of Beauty, as developed in
+    the Form of the Human Figure</span></td>
+    <td class="tdpg"><a href="#AS_DEVELOPED_IN_THE_FORM_OF_THE_HUMAN_FIGURE">61</a></td>
+  </tr>
+  <tr>
+    <td><span class="smcap">The Science of Beauty, as developed in
+    Colours</span></td>
+    <td class="tdpg"><a href="#AS_DEVELOPED_IN_COLOURS">67</a></td>
+  </tr>
+  <tr>
+    <td><span class="smcap">The Science of Beauty applied to the
+    Forms and Proportions of Ancient Grecian Vases and Ornaments</span></td>
+    <td class="tdpg"><a href="#APPLIED_TO_THE_FORMS_AND_PROPORTIONS">82</a></td>
+  </tr>
+  <tr>
+    <td><span class="smcap">Appendix, No. I.</span></td>
+    <td class="tdpg"><a href="#APPENDIX_I">91</a></td>
+  </tr>
+  <tr>
+    <td><span class="smcap">Appendix, No. II.</span></td>
+    <td class="tdpg"><a href="#APPENDIX_II">99</a></td>
+  </tr>
+  <tr>
+    <td><span class="smcap">Appendix, No. III.</span></td>
+    <td class="tdpg"><a href="#APPENDIX_III">100</a></td>
+  </tr>
+  <tr>
+    <td><span class="smcap">Appendix, No. IV.</span></td>
+    <td class="tdpg"><a href="#APPENDIX_IV">100</a></td>
+  </tr>
+  <tr>
+    <td><span class="smcap">Appendix, No. V.</span></td>
+    <td class="tdpg"><a href="#APPENDIX_V">104</a></td>
+  </tr>
+  <tr>
+    <td><span class="smcap">Appendix, No. VI.</span></td>
+    <td class="tdpg"><a href="#APPENDIX_VI">105</a></td>
+  </tr>
+</table>
+
+<hr class="chap x-ebookmaker-drop">
+
+<div class="chapter">
+
+<p><span class="pagenum"><a id="Page_x"></a>[x]</span></p>
+
+<h2 class="nobreak" id="ILLUSTRATIONS">ILLUSTRATIONS.</h2>
+
+</div>
+
+<h3>PLATES</h3>
+
+<figure class="figcenter illowp100" id="plate01" style="max-width: 62.5em;">
+  <img class="w100" src="images/plate01.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">I. Three Scales of the
+  Elementary Rectilinear Figures, viz., the Scalene Triangle, the Isosceles
+  Triangle, and the Rectangle, comprising twenty-seven varieties of each,
+  according to the harmonic parts of the Right Angle from ¹⁄₂ to ¹⁄₁₆.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp100" id="plate02" style="max-width: 62.5em;">
+  <img class="w100" src="images/plate02.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">II. Diagram of the
+  Rectilinear Orthography of the Principal Front of the Parthenon of
+  Athens, in which its Proportions are determined by harmonic parts of
+  the Right Angle.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp100" id="plate03" style="max-width: 62.5em;">
+  <img class="w100" src="images/plate03.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">III. Diagram of the
+  Rectilinear Orthography of the Portico of the Temple of Theseus at Athens,
+  in which its Proportions are determined by harmonic parts of the Right Angle.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp50" id="plate04" style="max-width: 40.625em;">
+  <img class="w100" src="images/plate04.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">IV. Diagram of the
+  Rectilinear Orthography of the East End of Lincoln Cathedral, in which
+  its Proportions are determined by harmonic parts of the Right Angle.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp100" id="plate05" style="max-width: 62.5em;">
+  <img class="w100" src="images/plate05.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">V. Four Ellipses described
+  from Foci, determined by harmonic parts of the Right Angle, shewing in each
+  the Scalene Triangle, the Isosceles Triangle, and the Rectangle to which
+  it belongs.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp50" id="plate06" style="max-width: 39.0625em;">
+  <img class="w100" src="images/plate06.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">VI. The Composite Ellipse
+  of ¹⁄₆ and ¹⁄₈ of the Right Angle, shewing its greater and lesser Axis,
+  its various Foci, and the Isosceles Triangle in which they are placed.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp50" id="plate07" style="max-width: 35.9375em;">
+  <img class="w100" src="images/plate07.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">VII. The Composite Ellipse
+  of ¹⁄₄₈ and ¹⁄₆₄ of the Right Angle, shewing how it forms the Entasis of
+  the Columns of the Parthenon of Athens.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp50" id="plate08" style="max-width: 39.0625em;">
+  <img class="w100" src="images/plate08.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">VIII. Sectional Outlines of
+  two Mouldings of the Parthenon of Athens, full size, shewing the harmonic
+  nature of their Curves, and the simple manner of their Construction.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp100" id="plate09" style="max-width: 62.5em;">
+  <img class="w100" src="images/plate09.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">IX. Three Diagrams, giving
+  a Vertical, a Front, and a Side Aspect of the Geometrical Construction of
+  the Human Head and Countenance, in which the Proportions are determined
+  by harmonic parts of the Right Angle.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<p><span class="pagenum"><a id="Page_xi"></a>[xi]</span></p>
+
+<figure class="figcenter illowp50" id="plate10" style="max-width: 35.9375em;">
+  <img class="w100" src="images/plate10.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">X. Diagram in which the
+  Symmetrical Proportions of the Human Figure are determined by harmonic
+  parts of the Right Angle.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp50" id="plate11" style="max-width: 37.5em;">
+  <img class="w100" src="images/plate11.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">XI. The Contour of the
+  Human Figure as viewed in Front and in Profile, its Curves being determined
+  by Ellipses, whose Foci are determined by harmonic parts of the Right Angle.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp50" id="plate12" style="max-width: 35.9375em;">
+  <img class="w100" src="images/plate12.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">XII. Rectilinear Diagram,
+  shewing the Proportions of the Portland Vase, as determined by harmonic
+  parts of the Right Angle, and the outline of its form by an Elliptic Curve
+  harmonically described.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp50" id="plate13" style="max-width: 43.75em;">
+  <img class="w100" src="images/plate13.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">XIII. Rectilinear Diagram
+  of the Proportions and Curvilinear Outline of the form of an ancient
+  Grecian Vase, the proportions determined by harmonic parts of the Right
+  Angle, and the melody of the form by Curves of two Ellipses.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp100" id="plate14" style="max-width: 62.5em;">
+  <img class="w100" src="images/plate14.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">XIV. Rectilinear Diagram
+  of the Proportions and Curvilinear Outline of the form an ancient Grecian
+  Vase, the proportions determined by harmonic parts of the Right Angle,
+  and the melody of the form by an Elliptic Curve.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp100" id="plate15" style="max-width: 62.5em;">
+  <img class="w100" src="images/plate15.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">XV. Two Diagrams of Etruscan
+  Vases, the harmony of Proportions and melody of the Contour determined,
+  respectively, by parts of the Right Angle and an Elliptic Curve.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp100" id="plate16" style="max-width: 62.5em;">
+  <img class="w100" src="images/plate16.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">XVI. Two Diagrams of Etruscan
+  Vases, whose harmony of Proportion and melody of Contour are determined as
+  above.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp50" id="plate17" style="max-width: 40.625em;">
+  <img class="w100" src="images/plate17.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">XVII. Diagram shewing the
+  Geometric Construction of an Ornament belonging to the Parthenon at Athens.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp50" id="plate18" style="max-width: 37.5em;">
+  <img class="w100" src="images/plate18.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">XVIII. Diagram of the
+  Geometrical Construction of the ancient Grecian Ornament called the
+  Honeysuckle.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp50" id="plate19" style="max-width: 39.0625em;">
+  <img class="w100" src="images/plate19.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">XIX. An additional Illustration
+  of the Contour of the Human Figure, as viewed in Front and in Profile.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp50" id="plate20" style="max-width: 35.9375em;">
+  <img class="w100" src="images/plate20.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">XX. Diagram shewing the manner
+  in which the Elliptic Curves are arranged in order to produce an Outline of
+  the Form of the Human Figure as viewed in Front.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp50" id="plate21" style="max-width: 39.0625em;">
+  <img class="w100" src="images/plate21.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">XXI. Diagram of a variation
+  on the Form of the Portland Vase.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp50" id="plate22" style="max-width: 42.1875em;">
+  <img class="w100" src="images/plate22.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">XXII. Diagram of a second
+  variation on the Form of the Portland Vase.</p>   <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<figure class="figcenter illowp50" id="plate23" style="max-width: 42.1875em;">
+  <img class="w100" src="images/plate23.jpg" alt="">
+  <figcaption class="caption"><p class="hanging">XXIII. Diagram of a third
+  variation on the Form of the Portland Vase.</p>
+  <p><i>D. R. Hay delᵗ.</i> <i class="spacer">G. Aikman sc.</i></p></figcaption>
+</figure>
+
+<hr class="chap x-ebookmaker-drop">
+
+<div class="chapter">
+
+<p><span class="pagenum"><a id="Page_1"></a>[1]</span></p>
+
+<h2 class="nobreak" id="INTRODUCTION">INTRODUCTION.</h2>
+
+</div>
+
+<p>Twelve years ago, one of our most eminent philosophers,<a id="FNanchor_1" href="#Footnote_1" class="fnanchor">[1]</a>
+through the medium of the <i>Edinburgh Review</i>,<a id="FNanchor_2" href="#Footnote_2" class="fnanchor">[2]</a> gave the
+following account of what was then the state of the fine arts
+as connected with science:—“The disposition to introduce
+into the intellectual community the principles of free intercourse,
+is by no means general; but we are confident that
+Art will not sufficiently develop her powers, nor Science attain
+her most commanding position, till the practical knowledge of
+the one is taken in return for the sound deductions of the
+other.... It is in the fine arts, principally, and in the
+speculations with which they are associated, that the controlling
+power of scientific truth has not exercised its legitimate
+influence. In discussing the principles of painting, sculpture,
+architecture, and landscape gardening, philosophers have renounced
+science as a guide, and even as an auxiliary; and a
+school has arisen whose speculations will brook no restraint,
+and whose decisions stand in opposition to the strongest convictions
+of our senses. That the external world, in its gay
+colours and lovely forms, is exhibited to the mind only as a
+tinted mass, neither within nor without the eye, neither touching
+it nor distant from it—an ubiquitous chaos, which experience<span class="pagenum"><a id="Page_2"></a>[2]</span>
+only can analyse and transform into the realities which
+compose it; that the beautiful and sublime in nature and in
+art derive their power over the mind from association alone,
+are among the philosophical doctrines of the present day,
+which, if it be safe, it is scarcely prudent to question. Nor
+are these opinions the emanations of poetical or ill-trained
+minds, which ingenuity has elaborated, and which fashion sustains.
+They are conclusions at which most of our distinguished
+philosophers have arrived. They have been given to the
+world with all the authority of demonstrated truth; and in
+proportion to the hold which they have taken of the public
+mind, have they operated as a check upon the progress of
+knowledge.”</p>
+
+<p>Such, then, was the state of art as connected with science
+twelve years ago. But although the causes which then placed
+science and the fine arts at variance have since been gradually
+diminishing, yet they are still far from being removed. In
+proof of this I may refer to what took place at the annual
+distribution of the prizes to the students attending our Scottish
+Metropolitan School of Design, in 1854, the pupils in which
+amount to upwards of two hundred. The meeting on
+that occasion included, besides the pupils, a numerous and
+highly respectable assemblage of artists and men of science.
+The chairman, a Professor in our University, and editor of
+one of the most voluminous works on art, science, and
+literature ever produced in this country, after extolling the
+general progress of the pupils, so far as evinced by the drawings
+exhibited on the occasion, drew the attention of the
+meeting to a discovery made by the head master of the architectural
+and ornamental department of the school, viz.—That
+the ground-plan of the Parthenon at Athens had been constructed
+by the application of the <i>mysterious</i> ovoid or <i>Vesica<span class="pagenum"><a id="Page_3"></a>[3]</span>
+Piscis</i> of the middle ages, subdivided by the <i>mythic</i> numbers 3
+and 7, and their intermediate odd number 5. Now, it may be
+remarked, that the figure thus referred to is not an ovoid, neither
+is it in any way of a mysterious nature, being produced simply
+by two equal circles cutting each other in their centres. Neither
+can it be shewn that the numbers 3 and 7 are in any way more
+mythic than other numbers. In fact, the terms <i>mysterious</i>
+and <i>mythic</i> so applied, can only be regarded as a remnant of
+an ancient terminology, calculated to obscure the simplicity of
+scientific truth, and when used by those employed to teach—for
+doubtless the chairman only gave the description he received—must
+tend to retard the connexion of that truth with
+the arts of design. I shall now give a specimen of the manner
+in which a knowledge of the philosophy of the fine arts is at
+present inculcated upon the public mind generally. In the same
+metropolis there has likewise existed for upwards of ten years
+a Philosophical Institution of great importance and utility,
+whose members amount to nearly three thousand, embracing a
+large proportion of the higher classes of society, both in respect
+to talent and wealth. At the close of the session of this
+Institution, in 1854, a learned and eloquent philologus, who
+occasionally lectures upon beauty, was appointed to deliver
+the closing address, and touching upon the subject of the
+beautiful, he thus concluded—</p>
+
+<p>“In the worship of the beautiful, and in that alone, we are
+inferior to the Greeks. Let us therefore be glad to borrow
+from them; not slavishly, but with a wise adaptation—not
+exclusively, but with a cunning selection; in art, as in religion,
+let us learn to prove all things, and hold fast that which is
+good—not merely one thing which is good, but all good
+things—Classicalism, Mediævalism, Modernism—let us have
+and hold them all in one wide and lusty embrace. Why<span class="pagenum"><a id="Page_4"></a>[4]</span>
+should the world of art be more narrow, more monotonous,
+than the world of nature? Did God make all the flowers of
+one pattern, to please the devotees of the rose or the lily; and
+did He make all the hills, with the green folds of their queenly
+mantles, all at one slope, to suit the angleometer of the most
+mathematical of decorators? I trow not. Let us go and do
+likewise.”</p>
+
+<p>I here take for granted, that what the lecturer meant by
+“the worship of the beautiful,” is the production and appreciation
+of works of art in which beauty should be a primary
+element; and judging from the remains which we possess of
+such works as were produced by the ancient Grecians, our inferiority
+to them in these respects cannot certainly be denied.
+But I must reiterate what I have often before asserted, that it
+is not by borrowing from them, however cunning our selection,
+or however wise our adaptations, that this inferiority is to be
+removed, but by a re-discovery of the science which these
+ancient artists must have employed in the production of that
+symmetrical beauty and chaste elegance which pervaded all
+their works for a period of nearly three hundred years. And
+I hold, that as in religion, so in art, there is only one truth, a
+grain of which is worth any amount of philological eloquence.</p>
+
+<p>I also take for granted, that what is meant by Classicalism
+in the above quotation, is the ancient Grecian style of art; by
+Mediævalism, the semi-barbaric style of the middle ages; and
+by Modernism, that chaotic jumble of all previous styles and
+fashions of art, which is the peculiar characteristic of our present
+school, and which is, doubtless, the result of a system of
+education based upon plagiarism and mere imitation. Therefore
+a recommendation to embrace with equal fervour “as
+good things,” these very opposite artic<i>isms</i> must be a doctrine
+as mischievous in art as it would be in religion to recommend<span class="pagenum"><a id="Page_5"></a>[5]</span>
+as equally good things the various <i>isms</i> into which it has also
+been split in modern times.</p>
+
+<p>Now, “the world of nature” and “the world of art” have
+not that equality of scope which this lecturer on beauty ascribes
+to them, but differ very decidedly in that particular. Neither
+will it be difficult to shew why “the world of art <i>should</i> be
+more narrow than the world of nature”—that it should be
+thereby rendered more monotonous does not follow.</p>
+
+<p>It is well known, that the “world of nature” consists of
+productions, including objects of every degree of beauty from
+the very lowest to the highest, and calculated to suit not only
+the tastes arising from various degrees of intellect, but those
+arising from the natural instincts of the lower animals. On
+the other hand, “the world of art,” being devoted to the
+gratification and improvement of intelligent minds only, is
+therefore narrowed in its scope by the exclusion from its productions
+of the lower degrees of beauty—even mediocrity is
+inadmissible; and we know that the science of the ancient
+Greek artists enabled them to excel the highest individual
+productions of nature in the perfection of symmetrical beauty.
+Consequently, all objects in nature are not equally well adapted
+for artistic study, and it therefore requires, on the part of the
+artist, besides true genius, much experience and care to enable
+him to choose proper subjects from nature; and it is in the
+choice of such subjects, and not in plagiarism from the ancients,
+that he should select with knowledge and adapt with wisdom.
+Hence, all such latitudinarian doctrines as those I have
+quoted must act as a check upon the progress of knowledge
+in the scientific truth of art. I have observed in some of my
+works, that in this country a course had been followed in our
+search for the true science of beauty not differing from that
+by which the alchymists of the middle ages conducted their<span class="pagenum"><a id="Page_6"></a>[6]</span>
+investigations; for our ideas of visible beauty are still undefined,
+and our attempts to produce it in the various branches
+of art are left dependant, in a great measure, upon chance.
+Our schools are conducted without reference to any first principles
+or definite laws of beauty, and from the drawing of a
+simple architectural moulding to the intricate combinations of
+form in the human figure, the pupils trust to their hands
+and eyes alone, servilely and mechanically copying the works
+of the ancients, instead of being instructed in the unerring
+principles of science, upon which the beauty of those works
+normally depends. The instruction they receive is imparted
+without reference to the judgment or understanding, and they
+are thereby led to imitate effects without investigating causes.
+Doubtless, men of great genius sometimes arrive at excellence
+in the arts of design without a knowledge of the principles
+upon which beauty of form is based; but it should be kept
+in mind, that true genius includes an intuitive perception of
+those principles along with its creative power. It is, therefore,
+to the generality of mankind that instruction in the definite
+laws of beauty will be of most service, not only in improving
+the practice of those who follow the arts professionally, but
+in enabling all of us to distinguish the true from the false,
+and to exercise a sound and discriminating taste in forming
+our judgment upon artistic productions. Æsthetic culture
+should consequently supersede servile copying, as the basis
+of instruction in our schools of art. Many teachers of drawing,
+however, still assert, that, by copying the great works
+of the ancients, the mind of the pupil will become imbued
+with ideas similar to theirs—that he will imbibe their feeling
+for the beautiful, and thereby become inspired with their
+genius, and think as they thought. To study carefully and
+to investigate the principles which constitute the excellence<span class="pagenum"><a id="Page_7"></a>[7]</span>
+of the works of the ancients, is no doubt of much benefit
+to the student; but it would be as unreasonable to suppose
+that he should become inspired with artistic genius by merely
+copying them, as it would be to imagine, that, in literature,
+poetic inspiration could be created by making boys
+transcribe or repeat the works of the ancient poets. Sir
+Joshua Reynolds considered copying as a delusive kind of
+industry, and has observed, that “Nature herself is not to be
+too closely copied,” asserting that “there are excellences in
+the art of painting beyond what is commonly called the
+imitation of nature,” and that “a mere copier of nature can
+never produce any thing great.” Proclus, an eminent philosopher
+and mathematician of the later Platonist school (<span class="allsmcap">A.D.</span>
+485), says, that “he who takes for his model such forms as
+nature produces, and confines himself to an exact imitation
+of these, will never attain to what is perfectly beautiful. For
+the works of nature are full of disproportion, and fall very
+short of the true standard of beauty.”</p>
+
+<p>It is remarked by Mr. J. C. Daniel, in the introduction
+to his translation of M. Victor Cousin’s “Philosophy of the
+Beautiful,” that “the English writers have advocated no
+theory which allows the beautiful to be universal and absolute;
+nor have they professedly founded their views on original
+and ultimate principles. Thus the doctrine of the English
+school has for the most part been, that beauty is mutable and
+special, and the inference that has been drawn from this
+teaching is, that all tastes are equally just, provided that each
+man speaks of what he feels.” He then observes, that the
+German, and some of the French writers, have thought far
+differently; for with them the beautiful is “simple, immutable,
+absolute, though its <i>forms</i> are manifold.”</p>
+
+<p>So far back as the year 1725, the same truths advanced by<span class="pagenum"><a id="Page_8"></a>[8]</span>
+the modern German and French writers, and so eloquently
+illustrated by M. Cousin, were given to the world by Hutchison
+in his “Inquiry into the Original of our Ideas of Beauty
+and Virtue.” This author says—“We, by absolute beauty,
+understand only that beauty which we perceive in objects,
+without comparison to any thing external, of which the object
+is supposed an imitation or picture, such as the beauty perceived
+from the works of nature, artificial forms, figures,
+theorems. Comparative or relative beauty is that which we
+perceive in objects commonly considered as imitations or
+resemblances of something else.”</p>
+
+<p>Dr. Reid also, in his “Intellectual Powers of Man,” says—“That
+taste, which we may call rational, is that part of our
+constitution by which we are made to receive pleasure from
+the contemplation of what we conceive to be excellent in its
+kind, the pleasure being annexed to this judgment, and regulated
+by it. This taste may be true or false, according as it
+is founded on a true or false judgment. And if it may be
+true or false, it must have first principles.”</p>
+
+<p>M. Victor Cousin’s opinion upon this subject is, however,
+still more conclusive. He observes—“If the idea of the beautiful
+is not absolute, like the idea of the true—if it is nothing
+more than the expression of individual sentiment, the rebound
+of a changing sensation, or the result of each person’s fancy—then
+the discussions on the fine arts waver without support,
+and will never end. For a theory of the fine arts to be possible,
+there must be something absolute in beauty, just as there must
+be something absolute in the idea of goodness, to render morals
+a possible science.”</p>
+
+<p>The basis of the science of beauty must thus be founded
+upon fixed principles, and when these principles are evolved
+with the same care which has characterised the labours of investigators<span class="pagenum"><a id="Page_9"></a>[9]</span>
+in natural science, and are applied in the fine arts
+as the natural sciences have been in the useful arts, a solid
+foundation will be laid, not only for correct practice, but also
+for a just appreciation of productions in every branch of the
+arts of design.</p>
+
+<p>We know that the mind receives pleasure through the sense
+of hearing, not only from the music of nature, but from the
+euphony of prosaic composition, the rhythm of poetic measure,
+the artistic composition of successive harmony in simple melody,
+and the combined harmony of counterpoint in the more complex
+works of that art. We know, also, that the mind is
+similarly gratified through the sense of seeing, not only by the
+visible beauties of nature, but by those of art, whether in symmetrical
+or picturesque compositions of forms, or in harmonious
+arrangements of gay or sombre colouring.</p>
+
+<p>Now, in respect to the first of these modes of sensation, we
+know, that from the time of Pythagoras, the fact has been
+established, that in whatever manner nature or art may address
+the ear, the degree of obedience paid to the fundamental law
+of harmony will determine the presence and degree of that
+beauty with which a perfect organ can impress a well-constituted
+mind; and it is my object in this, as it has been in
+former attempts, to prove it consistent with scientific truth,
+that that beauty which is addressed to the mind by objects of
+nature and art, through the eye, is similarly governed. In
+short, to shew that, as in compositions of sounds, there can be
+no true beauty in the absence of a strict obedience to this great
+law of nature, neither can there exist, in compositions of forms
+or colours, that principle of unity in variety which constitutes
+beauty, unless such compositions are governed by the same law.</p>
+
+<p>Although in the songs of birds, the gurgling of brooks, the
+sighing of the gentle summer winds, and all the other beautiful<span class="pagenum"><a id="Page_10"></a>[10]</span>
+music of nature, no analysis might be able to detect the operation
+of any precise system of harmony, yet the pleasure thus
+afforded to the human mind we know to arise from its responding
+to every development of an obedience to this law. When,
+in like manner, we find even in those compositions of forms
+and colours which constitute the wildest and most rugged of
+Nature’s scenery, a species of picturesque grandeur and beauty
+to which the mind as readily responds as to her more mild
+and pleasing aspects, or to her sweetest music, we may rest
+assured that this beauty is simply another development of,
+and response to, the same harmonic law, although the precise
+nature of its operation may be too subtle to be easily detected.</p>
+
+<p>The <i>résumé</i> of the various works I have already published
+upon the subject, along with the additional illustrations I am
+about to lay before my readers, will, I trust, point out a system
+of harmony, which, in formative art, as well as in that of
+colouring, will rise superior to the idiosyncracies of different
+artists, and bring back to one common type the sensations of
+the eye and the ear, thereby improving that knowledge of
+the laws of the universe which it is as much the business of
+science to combine with the ornamental as with the useful arts.</p>
+
+<p>In attempting this, however, I beg it may be understood,
+that I do not believe any system, based even upon the laws of
+nature, capable of forming a royal road to the perfection of
+art, or of “mapping the mighty maze of a creative mind.”
+At the same time, however, I must continue to reiterate the
+fact, that the diffusion of a general knowledge of the science
+of visible beauty will afford latent artistic genius just such
+a vantage ground as that which the general knowledge of
+philology diffused throughout this country affords its latent
+literary genius. Although <i>mere learning</i> and <i>true genius</i> differ
+as much in the practice of art as they do in the practice of<span class="pagenum"><a id="Page_11"></a>[11]</span>
+literature, yet a precise and systematic education in the true
+science of beauty must certainly be as useful in promoting
+the practice and appreciation of the one, as a precise and systematic
+education in the science of philology is in promoting
+the practice and appreciation of the other.</p>
+
+<p>As all beauty is the result of harmony, it will be requisite
+here to remark, that harmony is not a simple quality, but, as
+Aristotle defines it, “the union of contrary principles having
+a ratio to each other.” Harmony thus operates in the production
+of all that is beautiful in nature, whether in the combinations,
+in the motions, or in the affinities of the elements
+of matter.</p>
+
+<p>The contrary principles to which Aristotle alludes, are those
+of uniformity and variety; for, according to the predominance
+of the one or the other of these principles, every kind of beauty
+is characterised. Hence the difference between symmetrical
+and picturesque beauty:—the first allied to the principle of
+uniformity, in being based upon precise laws that may be
+taught so as to enable men of ordinary capacity to produce it
+in their works—the second allied to the principle of variety
+often to so great a degree that they yield an obedience to the
+precise principles of harmony so subtilely, that they cannot be
+detected in its constitution, but are only felt in the response
+by which true genius acknowledges their presence. The
+generality of mankind may be capable of perceiving this latter
+kind of beauty, and of feeling its effects upon the mind, but
+men of genius, only, can impart it to works of art, whether
+addressed to the eye or the ear. Throughout the sounds,
+forms, and colours of nature, these two kinds of beauty are
+found not only in distinct developments, but in every degree
+of amalgamation. We find in the songs of some birds, such
+as those of the chaffinch, thrush, &amp;c., a rhythmical division,<span class="pagenum"><a id="Page_12"></a>[12]</span>
+resembling in some measure the symmetrically precise arrangements
+of parts which characterises all artistic musical composition;
+while in the songs of other birds, and in the other
+numerous melodies with which nature charms and soothes the
+mind, there is no distinct regularity in the division of their
+parts. In the forms of nature, too, we find amongst the
+innumerable flowers with which the surface of the earth is so
+profusely decorated, an almost endless variety of systematic
+arrangements of beautiful figures, often so perfectly symmetrical
+in their combination, that the most careful application
+of the angleometer could scarcely detect the slightest deviation
+from geometrical precision; while, amongst the masses of
+foliage by which the forms of many trees are divided and subdivided
+into parts, as also amongst the hills and valleys, the
+mountains and ravines, which divide the earth’s surface, we
+find in every possible variety of aspect the beauty produced
+by that irregular species of symmetry which characterises the
+picturesque.</p>
+
+<p>In like manner, we find in wild as well as cultivated flowers
+the most symmetrical distributions of colours accompanying
+an equally precise species of harmony in their various kinds of
+contrasts, often as mathematically regular as the geometric
+diagrams by which writers upon colour sometimes illustrate
+their works; while in the general colouring of the picturesque
+beauties of nature, there is an endless variety in its distributions,
+its blendings, and its modifications. In the forms and
+colouring of animals, too, the same endless variety of regular
+and irregular symmetry is to be found. But the highest
+degree of beauty in nature is the result of an equal balance
+of uniformity with variety. Of this the human figure is an
+example; because, when it is of those proportions universally
+acknowledged to be the most perfect, its uniformity bears to<span class="pagenum"><a id="Page_13"></a>[13]</span>
+its variety an apparently equal ratio. The harmony of combination
+in the normal proportions of its parts, and the beautifully
+simple harmony of succession in the normal melody of
+its softly undulating outline, are the perfection of symmetrical
+beauty, while the innumerable changes upon the contour which
+arise from the actions and attitudes occasioned by the various
+emotions of the mind, are calculated to produce every species
+of picturesque beauty, from the softest and most pleasing to
+the grandest and most sublime.</p>
+
+<p>Amongst the purely picturesque objects of inanimate nature,
+I may, as in a former work, instance an ancient oak tree, for
+its beauty is enhanced by want of apparent symmetry. Thus,
+the more fantastically crooked its branches, and the greater
+the dissimilarity and variety it exhibits in its masses of foliage,
+the more beautiful it appears to the artist and the amateur;
+and, as in the human figure, any attempt to produce variety
+in the proportions of its lateral halves would be destructive of
+its symmetrical beauty, so in the oak tree any attempt to
+produce palpable similarity between any of its opposite sides
+would equally deteriorate its picturesque beauty. But picturesque
+beauty is not the result of the total absence of
+symmetry; for, as none of the irregularly constructed music
+of nature could be pleasing to the ear unless there existed in
+the arrangement of its notes an obedience, however subtle, to
+the great harmonic law of Nature, so neither could any object
+be picturesquely beautiful, unless the arrangement of its parts
+yields, although it may be obscurely, an obedience to the
+same law.</p>
+
+<p>However symmetrically beautiful any architectural structure
+may be, when in a complete and perfect state, it must, as it
+proceeds towards ruin, blend the picturesque with the symmetrical;
+but the type of its beauty will continue to be the<span class="pagenum"><a id="Page_14"></a>[14]</span>
+latter, so long as a sufficient portion of it remains to convey
+an idea of its original perfection. It is the same with the
+human form and countenance; for age does not destroy their
+original beauty, but in both only lessens that which is symmetrical,
+while it increases that which is picturesque.</p>
+
+<p>In short, as a variety of simultaneously produced sounds,
+which do not relate to each other agreeably to this law, can
+only convey to the mind a feeling of mere noise; so a variety
+of forms or colours simultaneously exposed to the eye under
+similar circumstances, can only convey to the mind a feeling
+of chaotic confusion, or what may be termed <i>visible</i> discord.
+As, therefore, the two principles of uniformity and variety, or
+similarity and dissimilarity, are in operation in every harmonious
+combination of the elements of sound, of form, and of colour,
+we must first have recourse to numbers in the abstract before
+we can form a proper basis for a universal science of beauty.</p>
+
+<hr class="chap x-ebookmaker-drop">
+
+<div class="chapter">
+
+<p><span class="pagenum"><a id="Page_15"></a>[15]</span></p>
+
+<h2 class="nobreak" id="EVOLVED_FROM_THE_HARMONIC_LAW_OF_NATURE">THE SCIENCE OF BEAUTY EVOLVED FROM THE HARMONIC LAW OF NATURE, AGREEABLY
+TO THE PYTHAGOREAN SYSTEM OF NUMERICAL RATIO.</h2>
+
+</div>
+
+<p>The scientific principles of beauty appear to have been well
+known to the ancient Greeks; and it must have been by the
+practical application of that knowledge to the arts of Design,
+that that people continued for a period of upwards of three
+hundred years to execute, in every department of these arts,
+works surpassing in chaste beauty any that had ever before
+appeared, and which have not been equalled during the two
+thousand years which have since elapsed.</p>
+
+<p>Æsthetic science, as the science of beauty is now termed, is
+based upon that great harmonic law of nature which pervades
+and governs the universe. It is in its nature neither absolutely
+physical nor absolutely metaphysical, but of an intermediate
+nature, assimilating in various degrees, more or less,
+to one or other of those opposite kinds of science. It
+specially embodies the inherent principles which govern impressions
+made upon the mind through the senses of hearing
+and seeing. Thus, the æsthetic pleasure derived from
+listening to the beautiful in musical composition, and from
+contemplating the beautiful in works of formative art, is in
+both cases simply a response in the human mind to artistic<span class="pagenum"><a id="Page_16"></a>[16]</span>
+developments of the great harmonic law upon which the
+science is based.</p>
+
+<p>Although the eye and the ear are two different senses,
+and, consequently, various in their modes of receiving impressions;
+yet the sensorium is but one, and the mind by
+which these impressions are perceived and appreciated is also
+characterised by unity. There appears, likewise, a striking
+analogy between the natural constitution of the two kinds of
+beauty, which is this, that the more physically æsthetic
+elements of the highest works of musical composition are
+melody, harmony, and tone, whilst those of the highest works
+of formative art are contour, proportion, and colour. The
+melody or theme of a musical composition and its harmony
+are respectively analogous,—1st, To the outline of an artistic
+work of formative art; and 2d, To the proportion which
+exists amongst its parts. To the careful investigator these
+analogies become identities in their effect upon the mind, like
+those of the more metaphysically æsthetic emotions produced
+by expression in either of these arts.</p>
+
+<p>Agreeably to the first analogy, the outline and contour of
+an object, suppose that of a building in shade when viewed
+against a light background, has a similar effect upon the mind
+with that of the simple melody of a musical composition when
+addressed to the ear unaccompanied by the combined harmony
+of counterpoint. Agreeably to the second analogy, the
+various parts into which the surface of the supposed elevation
+is divided being simultaneously presented to the eye, will, if
+arranged agreeably to the same great law, affect the mind
+like that of an equally harmonious arrangement of musical
+notes accompanying the supposed melody.</p>
+
+<p>There is, however, a difference between the construction of
+these two organs of sense, viz., that the ear must in a great<span class="pagenum"><a id="Page_17"></a>[17]</span>
+degree receive its impressions involuntarily; while the eye, on
+the other hand, is provided by nature with the power of
+either dwelling upon, or instantly shutting out or withdrawing
+itself from an object. The impression of a sound, whether
+simple or complex, when made upon the ear, is instantaneously
+conveyed to the mind; but when the sound ceases,
+the power of observation also ceases. But the eye can dwell
+upon objects presented to it so long as they are allowed to
+remain pictured on the retina; and the mind has thereby the
+power of leisurely examining and comparing them. Hence
+the ear guides more as a mere sense, at once and without
+reflection; whilst the eye, receiving its impressions gradually,
+and part by part, is more directly under the influence of
+mental analysis, consequently producing a more metaphysically
+æsthetic emotion. Hence, also, the acquired power of
+the mind in appreciating impressions made upon it through
+the organ of sight under circumstances, such as perspective,
+&amp;c., which to those who take a hasty view of the subject
+appear impossible.</p>
+
+<p>Dealing as this science therefore does, alike with the sources
+and the resulting principles of beauty, it is scarcely less dependent
+on the accuracy of the senses than on the power of
+the understanding, inasmuch as the effect which it produces
+is as essential a property of objects, as are its laws inherent in
+the human mind. It necessarily comprehends a knowledge
+of those first principles in art, by which certain combinations
+of sounds, forms, and colours produce an effect upon the
+mind, connected, in the first instance, with sensation, and in
+the second with the reasoning faculty. It is, therefore, not
+only the basis of all true practice in art, but of all sound
+judgment on questions of artistic criticism, and necessarily
+includes those laws whereon a correct taste must be based.<span class="pagenum"><a id="Page_18"></a>[18]</span>
+Doubtless many eloquent and ingenious treatises have been
+written upon beauty and taste; but in nearly every case, with
+no other effect than that of involving the subject in still
+greater uncertainty. Even when restricted to the arts of
+design, they have failed to exhibit any definite principles
+whereby the true may be distinguished from the false, and
+some natural and recognised laws of beauty reduced to
+demonstration. This may be attributed, in a great degree,
+to the neglect of a just discrimination between what is merely
+agreeable, or capable of exciting pleasurable sensations, and
+what is essentially beautiful; but still more to the confounding
+of the operations of the understanding with those of the
+imagination. Very slight reflection, however, will suffice to
+shew how essentially distinct these two faculties of the mind
+are; the former being regulated, in matters of taste, by irrefragable
+principles existing in nature, and responded to by
+an inherent principle existing in the human mind; while the
+latter operates in the production of ideal combinations of its
+own creation, altogether independent of any immediate impression
+made upon the senses. The beauty of a flower, for
+example, or of a dew-drop, depends on certain combinations
+of form and colour, manifestly referable to definite and systematic,
+though it may be unrecognised, laws; but when
+Oberon, in “Midsummer Night’s Dream,” is made to exclaim—</p>
+
+<div class="poetry-container">
+<div class="poetry">
+  <div class="stanza">
+    <div class="verse indent0">“And that same dew, which sometimes on the buds</div>
+    <div class="verse indent0">Was wont to swell, like round and orient pearls,</div>
+    <div class="verse indent0">Stood now within the pretty floweret’s eyes,</div>
+    <div class="verse indent0">Like tears that did their own disgrace bewail,”—</div>
+  </div>
+</div>
+</div>
+
+<p class="noindent">the poet introduces a new element of beauty equally legitimate,
+yet altogether distinct from, although accompanying
+that which constitutes the more precise science of æsthetics<span class="pagenum"><a id="Page_19"></a>[19]</span>
+as here defined. The composition of the rhythm is an operation
+of the understanding, but the beauty of the poetic
+fancy is an operation of the imagination.</p>
+
+<p>Our physical and mental powers, æsthetically considered,
+may therefore be classed under three heads, in their relation
+to the fine arts, viz., the receptive, the perceptive, and the
+conceptive.</p>
+
+<p>The senses of hearing and seeing are respectively, in the
+degree of their physical power, receptive of impressions made
+upon them, and of these impressions the sensorium, in the
+degree of its mental power, is perceptive. This perception
+enables the mind to form a judgment whereby it appreciates
+the nature and quality of the impression originally made on
+the receptive organ. The mode of this operation is intuitive,
+and the quickness and accuracy with which the nature and
+quality of the impression is apprehended, will be in the degree
+of the intellectual vigour of the mind by which it is perceived.
+Thus we are, by the cultivation of these intuitive faculties,
+enabled to decide with accuracy as to harmony or discord,
+proportion or deformity, and assign sound reasons for our
+judgment in matters of taste. But mental conception is the
+intuitive power of constructing original ideas from these
+materials; for after the receptive power has acted, the perception
+operates in establishing facts, and then the judgment
+is formed upon these operations by the reasoning
+powers, which lead, in their turn, to the creations of the
+imagination.</p>
+
+<p>The power of forming these creations is the true characteristic
+of genius, and determines the point at which art is
+placed beyond all determinable canons,—at which, indeed,
+æsthetics give place to metaphysics.</p>
+
+<p>In the science of beauty, therefore, the human mind is the<span class="pagenum"><a id="Page_20"></a>[20]</span>
+subject, and the effect of external nature, as well as of works
+of art, the object. The external world, and the individual
+mind, with all that lies within the scope of its powers, may be
+considered as two separate existences, having a distinct relation
+to each other. The subject is affected by the object, through
+that inherent faculty by which it is enabled to respond to every
+development of the all-governing harmonic law of nature; and
+the media of communication are the sensorium and its inlets—the
+organs of sense.</p>
+
+<p>This harmonic law of nature was either originally discovered
+by that illustrious philosopher Pythagoras, upwards of five
+hundred years before Christ, or a knowledge of it obtained by
+him about that period, from the Egyptian or Chaldean priests.
+For after having been initiated into all the Grecian and barbarian
+sacred mysteries, he went to Egypt, where he remained
+upwards of twenty years, studying in the colleges of its priests;
+and from Egypt he went into the East, and visited the Persian
+and Chaldean magi.<a id="FNanchor_3" href="#Footnote_3" class="fnanchor">[3]</a></p>
+
+<p>By the generality of the biographers of Pythagoras, it is
+said to be difficult to give a clear idea of his philosophy, as it
+is almost certain he never committed it to writing, and that it
+has been disfigured by the fantastic dreams and chimeras of
+later Pythagoreans. Diogenes Laërtius, however, whose “Lives
+of the Philosophers” was supposed to be written about the
+end of the second century of our era, says “there are three
+volumes extant written by Pythagoras. One on education,
+one on politics, and one on natural philosophy.” And adds,
+that there were several other books extant, attributed to
+Pythagoras, but which were not written by him. Also, in his
+“Life of Philolaus,” that Plato wrote to Dion to take care and
+purchase the books of Pythagoras.<a id="FNanchor_4" href="#Footnote_4" class="fnanchor">[4]</a> But whether this great<span class="pagenum"><a id="Page_21"></a>[21]</span>
+philosopher committed his discoveries to writing or not, his
+doctrines regarding the philosophy of beauty are well-known
+to be, that he considered numbers as the essence and the
+principle of all things, and attributed to them a real and distinct
+existence; so that, in his view, they were the elements
+out of which the universe was constructed, and to which it
+owed its beauty. Diogenes Laërtius gives the following account
+of this law:—“That the monad was the beginning of
+everything. From the monad proceeds an indefinite duad,
+which is subordinate to the monad as to its cause. That from
+the monad and indefinite duad proceeds numbers. That the
+part of science to which Pythagoras applied himself above all
+others, was arithmetic; and that he taught ‘that from numbers
+proceed signs, and from these latter, lines, of which plane
+figures consist; that from plane figures are derived solid
+bodies; that of all plane figures the most beautiful was the
+circle, and of all solid bodies the most beautiful was the
+sphere.’ He discovered the numerical relations of sounds on
+a single string; and taught that everything owes its existence
+and consistency to harmony. In so far as I know, the most
+condensed account of all that is known of the Pythagorian
+system of numbers is the following:—‘The monad or unity is
+that quantity, which, being deprived of all number, remains
+fixed. It is the fountain of all number. The duad is imperfect
+and passive, and the cause of increase and division.
+The triad, composed of the monad and duad, partakes of
+the nature of both. The tetrad, tetractys, or quaternion
+number is most perfect. The decad, which is the sum of
+the four former, comprehends all arithmetical and musical
+proportions.’”<a id="FNanchor_5" href="#Footnote_5" class="fnanchor">[5]</a></p>
+
+<p>These short quotations, I believe, comprise all that is known,<span class="pagenum"><a id="Page_22"></a>[22]</span>
+for certain, of the manner in which Pythagoras systematised
+the law of numbers. Yet, from the teachings of this great
+philosopher and his disciples, the harmonic law of nature, in
+which the fundamental principles of beauty are embodied,
+became so generally understood and universally applied in
+practice throughout all Greece, that the fragments of their
+works, which have reached us through a period of two thousand
+years, are still held to be examples of the highest artistic
+excellence ever attained by mankind. In the present state of
+art, therefore, a knowledge of this law, and of the manner in
+which it may again be applied in the production of beauty in
+all works of form and colour, must be of singular advantage;
+and the object of this work is to assist in the attainment of
+such a knowledge.</p>
+
+<p>It has been remarked, with equal comprehensiveness and
+truth, by a writer<a id="FNanchor_6" href="#Footnote_6" class="fnanchor">[6]</a> in the <i>British and Foreign Medical Review</i>,
+that “there is harmony of numbers in all nature—in
+the force of gravity—in the planetary movements—in the
+laws of heat, light, electricity, and chemical affinity—in the
+forms of animals and plants—in the perceptions of the mind.
+The direction, indeed, of modern natural and physical science
+is towards a generalization which shall express the fundamental
+laws of all by one simple numerical ratio. And we
+think modern science will soon shew that the mysticism of
+Pythagoras was mystical only to the unlettered, and that it
+was a system of philosophy founded on the then existing
+mathematics, which latter seem to have comprised more of the
+philosophy of numbers than our present.” Many years of
+careful investigation have convinced me of the truth of this
+remark, and of the great advantage derivable from an application
+of the Pythagorean system in the arts of design. For so<span class="pagenum"><a id="Page_23"></a>[23]</span>
+simple is its nature, that any one of an ordinary capacity of
+mind, and having a knowledge of the most simple rules of
+arithmetic, may, in a very short period, easily comprehend its
+nature, and be able to apply it in practice.</p>
+
+<p>The elements of the Pythagorean system of harmonic
+number, so far as can be gathered from the quotations I
+have given above, seem to be simply the indivisible monad
+(1); the duad (2), arising from the union of one monad with
+another; the triad (3), arising from the union of the monad
+with the duad; and the tetrad (4), arising from the union of
+one duad with another, which tetrad is considered a perfect
+number. From the union of these four elements arises the
+decad (10), the number, which, agreeably to the Pythagorean
+system, comprehends all arithmetical and harmonic proportions.
+If, therefore, we take these elements and unite them
+progressively in the following order, we shall find the series
+of harmonic numbers (2), (3), (5), and (7), which, with their
+multiples, are the complete numerical elements of all harmony,
+thus:—</p>
+
+<table>
+  <tr>
+    <td>1</td>
+    <td>+</td>
+    <td>1</td>
+    <td>=</td>
+    <td>2</td>
+  </tr>
+  <tr>
+    <td>1</td>
+    <td>+</td>
+    <td>2</td>
+    <td>=</td>
+    <td>3</td>
+  </tr>
+  <tr>
+    <td>2</td>
+    <td>+</td>
+    <td>3</td>
+    <td>=</td>
+    <td>5</td>
+  </tr>
+  <tr>
+    <td>3</td>
+    <td>+</td>
+    <td>4</td>
+    <td>=</td>
+    <td>7</td>
+  </tr>
+</table>
+
+<p>In order to render an extended series of harmonic numbers
+useful, it must be divided into scales; and it is a rule in the
+formation of these scales, that the first must begin with the
+monad (1) and end with the duad (2), the second begin with
+the duad (2) and end with the tetrad (4), and that the beginning
+and end of all other scales must be continued in the same
+arithmetical progression. These primary elements will then
+form the foundation of a series of such scales.</p>
+
+<p><span class="pagenum"><a id="Page_24"></a>[24]</span></p>
+
+<table>
+  <tr>
+    <td class="tdr">I.</td>
+    <td>(1)</td>
+    <td></td>
+    <td></td>
+    <td></td>
+    <td></td>
+    <td></td>
+    <td></td>
+    <td></td>
+    <td>(2)</td>
+  </tr>
+  <tr>
+    <td class="tdr">II.</td>
+    <td>(2)</td>
+    <td></td>
+    <td></td>
+    <td></td>
+    <td>(3)</td>
+    <td></td>
+    <td></td>
+    <td></td>
+    <td>(4)</td>
+  </tr>
+  <tr>
+    <td class="tdr">III.</td>
+    <td>(4)</td>
+    <td></td>
+    <td>(5)</td>
+    <td></td>
+    <td>(6)</td>
+    <td></td>
+    <td>(7)</td>
+    <td></td>
+    <td>(8)</td>
+  </tr>
+  <tr>
+    <td class="tdr">IV.</td>
+    <td>(8)</td>
+    <td>(9)</td>
+    <td>(10)</td>
+    <td>(&nbsp;&nbsp;)</td>
+    <td>(12)</td>
+    <td>(&nbsp;&nbsp;)</td>
+    <td>(14)</td>
+    <td>(15)</td>
+    <td>(16)</td>
+  </tr>
+</table>
+
+<p>The first of these scales has in (1) and (2) a beginning and
+an end; but the second has in (2), (3), and (4) the essential
+requisites demanded by Aristotle in every composition, viz.,
+“a beginning, a middle, and an end;” while the third has not
+only these essential requisites, but two intermediate parts (5)
+and (7), by which the beginning, the middle, and the end are
+united. In the fourth scale, however, the arithmetical progression
+is interrupted by the omission of numbers 11 and 13,
+which, not being multiples of either (2), (3), (5), or (7), are
+inadmissible.</p>
+
+<p>Such is the nature of the harmonic law which governs the
+progressive scales of numbers by the simple multiplication of
+the monad.</p>
+
+<p>I shall now use these numbers as divisors in the formation
+of a series of four such scales of parts, which has for its
+primary element, instead of the indivisible monad, a quantity
+which may be indefinitely divided, but which cannot be added
+to or multiplied. Like the monad, however, this quantity is
+represented by (1). The following is this series of four scales
+of harmonic parts:—</p>
+
+<table>
+  <tr>
+    <td class="tdr">I.</td>
+    <td>(1)</td>
+    <td></td>
+    <td></td>
+    <td></td>
+    <td></td>
+    <td></td>
+    <td></td>
+    <td></td>
+    <td>(¹⁄₂)</td>
+  </tr>
+  <tr>
+    <td class="tdr">II.</td>
+    <td>(¹⁄₂)</td>
+    <td></td>
+    <td></td>
+    <td></td>
+    <td>(¹⁄₃)</td>
+    <td></td>
+    <td></td>
+    <td></td>
+    <td>(¹⁄₄)</td>
+  </tr>
+  <tr>
+    <td class="tdr">III.</td>
+    <td>(¹⁄₄)</td>
+    <td></td>
+    <td>(¹⁄₅)</td>
+    <td></td>
+    <td>(¹⁄₆)</td>
+    <td></td>
+    <td>(¹⁄₇)</td>
+    <td></td>
+    <td>(¹⁄₈)</td>
+  </tr>
+  <tr>
+    <td class="tdr">IV.</td>
+    <td>(¹⁄₈)</td>
+    <td>(¹⁄₉)</td>
+    <td>(¹⁄₁₀)</td>
+    <td>(&nbsp;&nbsp;&nbsp;)</td>
+    <td>(¹⁄₁₂)</td>
+    <td>(&nbsp;&nbsp;&nbsp;)</td>
+    <td>(¹⁄₁₄)</td>
+    <td>(¹⁄₁₅)</td>
+    <td>(¹⁄₁₆)</td>
+  </tr>
+</table>
+
+<p>The scales I., II., and III. may now be rendered as complete<span class="pagenum"><a id="Page_25"></a>[25]</span>
+as scale IV., simply by multiplying upwards by 2 from
+(¹⁄₉), (¹⁄₅), (¹⁄₃), (¹⁄₇), and (¹⁄₁₅), thus:—</p>
+
+<table>
+  <tr>
+    <td class="tdr">I.</td>
+    <td>(1)</td>
+    <td>(⁸⁄₉)</td>
+    <td>(⁴⁄₅)</td>
+    <td></td>
+    <td>(²⁄₃)</td>
+    <td></td>
+    <td>(⁴⁄₇)</td>
+    <td>(⁸⁄₁₅)</td>
+    <td>(¹⁄₂)</td>
+  </tr>
+  <tr>
+    <td class="tdr">II.</td>
+    <td>(¹⁄₂)</td>
+    <td>(⁴⁄₉)</td>
+    <td>(²⁄₅)</td>
+    <td></td>
+    <td>(¹⁄₃)</td>
+    <td></td>
+    <td>(²⁄₇)</td>
+    <td>(⁴⁄₁₅)</td>
+    <td>(¹⁄₄)</td>
+  </tr>
+  <tr>
+    <td class="tdr">III.</td>
+    <td>(¹⁄₄)</td>
+    <td>(²⁄₉)</td>
+    <td>(¹⁄₅)</td>
+    <td></td>
+    <td>(¹⁄₆)</td>
+    <td></td>
+    <td>(¹⁄₇)</td>
+    <td>(²⁄₁₅)</td>
+    <td>(¹⁄₈)</td>
+  </tr>
+  <tr>
+    <td class="tdr">IV.</td>
+    <td>(¹⁄₈)</td>
+    <td>(¹⁄₉)</td>
+    <td>(¹⁄₁₀)</td>
+    <td>(&nbsp;&nbsp;&nbsp;)</td>
+    <td>(¹⁄₁₂)</td>
+    <td>(&nbsp;&nbsp;&nbsp;)</td>
+    <td>(¹⁄₁₄)</td>
+    <td>(¹⁄₁₅)</td>
+    <td>(¹⁄₁₆)</td>
+  </tr>
+</table>
+
+<p>We now find between the beginning and the end of scale I.
+the quantities (⁸⁄₉), (⁴⁄₅), (²⁄₃), (⁴⁄₇), and (⁸⁄₁₅).</p>
+
+<p>The three first of these quantities we find to be the remainders
+of the whole indefinite quantity contained in (1), after
+subtracting from it the primary harmonic quantities (¹⁄₉), (¹⁄₅),
+and (¹⁄₃); we, however, find also amongst these harmonic
+quantities that of (¹⁄₄), which being subtracted from (1) leaves
+(³⁄₄), a quantity the most suitable whereby to fill up the hiatus
+between (⁴⁄₅) and (²⁄₃) in scale I., which arises from the omission
+of (¹⁄₁₁) in scale IV. In like manner we find the two last of
+these quantities, (⁴⁄₇) and (⁸⁄₁₅), are respectively the largest of the
+two parts into which 7 and 15 are susceptible of being divided.
+Finding the number 5 to be divisible into parts more unequal
+than (2) to (3) and less unequal than (4) to (7), (³⁄₅) naturally
+fills up the hiatus between these quantities in scale I., which
+hiatus arises from the omission of (¹⁄₁₃) in scale IV. Thus:—</p>
+
+<table>
+  <tr>
+    <td class="tdr">I.</td>
+    <td>(1)</td>
+    <td>(⁸⁄₉)</td>
+    <td>(⁴⁄₅)</td>
+    <td>(³⁄₄)</td>
+    <td>(²⁄₃)</td>
+    <td>(³⁄₅)</td>
+    <td>(⁴⁄₇)</td>
+    <td>(⁸⁄₁₅)</td>
+    <td>(¹⁄₂)</td>
+  </tr>
+  <tr>
+    <td class="tdr">II.</td>
+    <td>(¹⁄₂)</td>
+    <td>(⁴⁄₉)</td>
+    <td>(²⁄₅)</td>
+    <td>(&nbsp;&nbsp;&nbsp;)</td>
+    <td>(¹⁄₃)</td>
+    <td>(&nbsp;&nbsp;&nbsp;)</td>
+    <td>(²⁄₇)</td>
+    <td>(⁴⁄₁₅)</td>
+    <td>(¹⁄₄)</td>
+  </tr>
+  <tr>
+    <td class="tdr">III.</td>
+    <td>(¹⁄₄)</td>
+    <td>(²⁄₉)</td>
+    <td>(¹⁄₅)</td>
+    <td>(&nbsp;&nbsp;&nbsp;)</td>
+    <td>(¹⁄₆)</td>
+    <td>(&nbsp;&nbsp;&nbsp;)</td>
+    <td>(¹⁄₇)</td>
+    <td>(²⁄₁₅)</td>
+    <td>(¹⁄₈)</td>
+  </tr>
+  <tr>
+    <td class="tdr">IV.</td>
+    <td>(¹⁄₈)</td>
+    <td>(¹⁄₉)</td>
+    <td>(¹⁄₁₀)</td>
+    <td>(&nbsp;&nbsp;&nbsp;)</td>
+    <td>(¹⁄₁₂)</td>
+    <td>(&nbsp;&nbsp;&nbsp;)</td>
+    <td>(¹⁄₁₄)</td>
+    <td>(¹⁄₁₅)</td>
+    <td>(¹⁄₁₆)</td>
+  </tr>
+</table>
+
+<p>Scale I. being now complete, we have only to divide these<span class="pagenum"><a id="Page_26"></a>[26]</span>
+latter quantities by (2) downwards in order to complete the
+other three. Thus:—</p>
+
+<table>
+  <tr>
+    <td class="tdr">I.</td>
+    <td>(1)</td>
+    <td>(⁸⁄₉)</td>
+    <td>(⁴⁄₅)</td>
+    <td>(³⁄₄)</td>
+    <td>(²⁄₃)</td>
+    <td>(³⁄₅)</td>
+    <td>(⁴⁄₇)</td>
+    <td>(⁸⁄₁₅)</td>
+    <td>(¹⁄₂)</td>
+  </tr>
+  <tr>
+    <td class="tdr">II.</td>
+    <td>(¹⁄₂)</td>
+    <td>(⁴⁄₉)</td>
+    <td>(²⁄₅)</td>
+    <td>(³⁄₈)</td>
+    <td>(¹⁄₃)</td>
+    <td>(³⁄₁₀)</td>
+    <td>(²⁄₇)</td>
+    <td>(⁴⁄₁₅)</td>
+    <td>(¹⁄₄)</td>
+  </tr>
+  <tr>
+    <td class="tdr">III.</td>
+    <td>(¹⁄₄)</td>
+    <td>(²⁄₉)</td>
+    <td>(¹⁄₅)</td>
+    <td>(³⁄₁₆)</td>
+    <td>(¹⁄₆)</td>
+    <td>(³⁄₂₀)</td>
+    <td>(¹⁄₇)</td>
+    <td>(²⁄₁₅)</td>
+    <td>(¹⁄₈)</td>
+  </tr>
+  <tr>
+    <td class="tdr">IV.</td>
+    <td>(¹⁄₈)</td>
+    <td>(¹⁄₉)</td>
+    <td>(¹⁄₁₀)</td>
+    <td>(³⁄₃₂)</td>
+    <td>(¹⁄₁₂)</td>
+    <td>(³⁄₄₀)</td>
+    <td>(¹⁄₁₄)</td>
+    <td>(¹⁄₁₅)</td>
+    <td>(¹⁄₁₆)</td>
+  </tr>
+</table>
+
+<p>The harmony existing amongst these numbers or quantities
+consists of the numerical relations which the parts bear to the
+whole and to each other; and the more simple these relations
+are, the more perfect is the harmony. The following are
+the numerical harmonic ratios which the parts bear to the
+whole:—</p>
+
+<table>
+  <tr>
+    <td class="tdr">I.</td>
+    <td>(1:1)</td>
+    <td>(8:9)</td>
+    <td>(4:5)</td>
+    <td>(3:4)</td>
+    <td>(2:3)</td>
+    <td>(3:5)</td>
+    <td>(4:7)</td>
+    <td>(8:15)</td>
+    <td>(1:2)</td>
+  </tr>
+  <tr>
+    <td class="tdr">II.</td>
+    <td>(1:2)</td>
+    <td>(4:9)</td>
+    <td>(2:5)</td>
+    <td>(3:8)</td>
+    <td>(1:3)</td>
+    <td>(3:10)</td>
+    <td>(2:7)</td>
+    <td>(4:15)</td>
+    <td>(1:4)</td>
+  </tr>
+  <tr>
+    <td class="tdr">III.</td>
+    <td>(1:4)</td>
+    <td>(2:9)</td>
+    <td>(1:5)</td>
+    <td>(3:16)</td>
+    <td>(1:6)</td>
+    <td>(3:20)</td>
+    <td>(1:7)</td>
+    <td>(2:15)</td>
+    <td>(1:8)</td>
+  </tr>
+  <tr>
+    <td class="tdr">IV.</td>
+    <td>(1:8)</td>
+    <td>(1:9)</td>
+    <td>(1:10)</td>
+    <td>(3:32)</td>
+    <td>(1:12)</td>
+    <td>(3:40)</td>
+    <td>(1:14)</td>
+    <td>(1:15)</td>
+    <td>(1:16)</td>
+  </tr>
+</table>
+
+<p>The following are the principal numerical relations which
+the parts in each scale bear to one another:—</p>
+
+<table>
+  <tr>
+    <td>(¹⁄₂):(⁴⁄₇)</td>
+    <td>=</td>
+    <td>(7:8)</td>
+  </tr>
+  <tr>
+    <td>(⁴⁄₅):(⁸⁄₉)</td>
+    <td>=</td>
+    <td>(9:10)</td>
+  </tr>
+  <tr>
+    <td>(²⁄₃):(⁴⁄₅)</td>
+    <td>=</td>
+    <td>(5:6)</td>
+  </tr>
+  <tr>
+    <td>(⁴⁄₇):(²⁄₃)</td>
+    <td>=</td>
+    <td>(6:7)</td>
+  </tr>
+  <tr>
+    <td>(⁸⁄₁₅):(⁴⁄₇)</td>
+    <td>=</td>
+    <td>(14:15)</td>
+  </tr>
+  <tr>
+    <td>(¹⁄₂):(⁸⁄₁₅)</td>
+    <td>=</td>
+    <td>(15:16)</td>
+  </tr>
+</table>
+
+<p>Although these relations are exemplified by parts of scale I.,
+the same ratios exist between the relative parts of scales II.,<span class="pagenum"><a id="Page_27"></a>[27]</span>
+III., and IV., and would exist between the parts of any other
+scales that might be added to that series.</p>
+
+<p>These are the simple elements of the science of that
+harmony which pervades the universe, and by which the
+various kinds of beauty æsthetically impressed upon the senses
+of hearing and seeing are governed.</p>
+
+<hr class="chap x-ebookmaker-drop">
+
+<div class="chapter">
+
+<p><span class="pagenum"><a id="Page_28"></a>[28]</span></p>
+
+<h2 class="nobreak" id="AS_APPLIED_TO_SOUNDS">THE SCIENCE OF BEAUTY AS APPLIED TO SOUNDS.</h2>
+
+</div>
+
+<p>It is well-known that all sounds arise from a peculiar action
+of the air, and that this action may be excited by the concussion
+resulting from the sudden displacement of a portion of
+the atmosphere itself, or by the rapid motions of bodies, or of
+confined columns of air; in all which cases, when the motions
+are irregular, and the force great, the sound conveyed to the
+sensorium is called a noise. But that musical sounds are the
+result of equal and regular vibratory motions, either of an
+elastic body, or of a column of air in a tube, exciting in the
+surrounding atmosphere a regular and equal pulsation. The
+ear is the medium of communication between those varieties
+of atmospheric action and the seat of consciousness. To describe
+fully the beautiful arrangement of the various parts of
+this organ, and their adaptation to the purpose of collecting
+and conveying these undulatory motions of the atmosphere, is
+as much beyond the scope of my present attempt as it is beyond
+my anatomical knowledge; but I may simply remark,
+that within the ear, and most carefully protected in the construction
+of that organ, there is a small cavity containing a
+pellucid fluid, in which the minute extremities of the auditory
+nerve float; and that this fluid is the last of the media
+through which the action producing the sensation of sound is<span class="pagenum"><a id="Page_29"></a>[29]</span>
+conveyed to the nerve, and thence to the sensorium, where its
+nature becomes perceptible to the mind.</p>
+
+<p>The impulses which produce musical notes must arrive at a
+certain frequency before the ear loses the intervals of silence
+between them, and is impressed by only one continued sound;
+and as they increase in frequency the sound becomes more
+acute upon the ear. The pitch of a musical note is, therefore,
+determined by the frequency of these impulses; but, on the
+other hand, its intensity or loudness will depend upon the
+violence and the quality of its tone on the material employed
+in producing them. All such sounds, therefore, whatever be
+their loudness or the quality of their tone in which the impulses
+occur with the same frequency are in perfect unison,
+having the same pitch. Upon this the whole doctrine of
+harmonies is founded, and by this the laws of numerical ratio
+are found to operate in the production of harmony, and the
+theory of music rendered susceptible of exact reasoning.</p>
+
+<p>The mechanical means by which such sounds can be produced
+are extremely various; but, as it is my purpose simply
+to shew the nature of harmony of sound as related to, or as
+evolving numerical harmonic ratio, I shall confine myself to
+the most simple mode of illustration—namely, that of the
+monochord. This is an instrument consisting of a string of a
+given length stretched between two bridges standing upon a
+graduated scale. Suppose this string to be stretched until its
+tension is such that, when drawn a little to a side and suddenly
+let go, it would vibrate at the rate of 64 vibrations in a
+second of time, producing to a certain distance in the surrounding
+atmosphere a series of pulsations of the same frequency.</p>
+
+<p>These pulsations will communicate through the ear a musical
+note which would, therefore, be the fundamental note of such<span class="pagenum"><a id="Page_30"></a>[30]</span>
+a string. Now, the phenomenon said to be discovered by
+Pythagoras is well known to those acquainted with the science
+of acoustics, namely, that immediately after the string is thus
+put into vibratory motion, it spontaneously divides itself, by
+a node, into two equal parts, the vibrations of each of which
+occur with a double frequency—namely, 128 in a second
+of time, and, consequently, produce a note doubly acute in
+pitch, although much weaker as to intensity or loudness;
+that it then, while performing these two series of vibrations,
+divides itself, by two nodes, into three parts, each of which
+vibrates with a frequency triple that of the whole string;
+that is, performs 192 vibrations in a second of time, and
+produces a note corresponding in increase of acuteness, but
+still less intense than the former, and that this continues to
+take place in the arithmetical progression of 2, 3, 4, &amp;c.
+Simultaneous vibrations, agreeably to the same law of progression,
+which, however, seem to admit of no other primes
+than the numbers 2, 3, 5, and 7, are easily excited upon any
+stringed instrument, even by the lightest possible touch of
+any of its strings while in a state of vibratory motion, and the
+notes thus produced are distinguished by the name of harmonics.
+It follows, then, that one-half of a musical string,
+when divided from the whole by the pressure of the finger, or
+any other means, and put into vibratory motion, produces a
+note doubly acute to that produced by the vibratory motion
+of the whole string; the third part, similarly separated, a note
+trebly acute; and the same with every part into which any
+musical string may be divided. This is the fundamental principle
+by which all stringed instruments are made to produce
+harmony. It is the same with wind instruments, the sounds
+of which are produced by the frequency of the pulsations
+occasioned in the surrounding atmosphere by agitating a<span class="pagenum"><a id="Page_31"></a>[31]</span>
+column of air confined within a tube as in an organ, in which
+the frequency of pulsation becomes greater in an inverse ratio
+to the length of the pipes. But the following series of four
+successive scales of musical notes will give the reader a more
+comprehensive view of the manner in which they follow the
+law of numerical ratio just explained than any more lengthened
+exposition.</p>
+
+<p>It is here requisite to mention, that in the construction of
+these scales, I have not only adopted the old German or literal
+mode of indicating the notes, but have included, as the Germans
+do, the note termed by us B flat as B natural, and the
+note we term B natural as H. Now, although this arrangement
+differs from that followed in the construction of our
+modern Diatonic scale, yet as the ratio of 4:7 is more closely
+related to that of 1:2 than that of 8:15, and as it is offered
+by nature in the spontaneous division of the monochord, I
+considered it quite admissible. The figures give the parts of
+the monochord which would produce the notes.</p>
+
+<table>
+  <tr>
+    <td rowspan="2" class="tdr valign">I.</td>
+    <td>{</td>
+    <td>(1)</td>
+    <td>(⁸⁄₉)</td>
+    <td>(⁴⁄₅)</td>
+    <td>(³⁄₄)</td>
+    <td>(²⁄₃)</td>
+    <td>(³⁄₅)</td>
+    <td>(⁴⁄₇)</td>
+    <td>(⁸⁄₁₅)</td>
+    <td>(¹⁄₂)*</td>
+  </tr>
+  <tr>
+    <td>{</td>
+    <td>C</td>
+    <td>D</td>
+    <td>E</td>
+    <td>F</td>
+    <td>G</td>
+    <td>A</td>
+    <td>B</td>
+    <td>H</td>
+    <td><i>c</i></td>
+  </tr>
+  <tr>
+    <td rowspan="2" class="tdr valign">II.</td>
+    <td>{</td>
+    <td>(¹⁄₂)*</td>
+    <td>(⁴⁄₉)</td>
+    <td>(²⁄₅)</td>
+    <td>(³⁄₈)</td>
+    <td>(¹⁄₃)*</td>
+    <td>(³⁄₁₀)</td>
+    <td>(²⁄₇)</td>
+    <td>(²⁄₁₅)</td>
+    <td>(¹⁄₄)*</td>
+  </tr>
+  <tr>
+    <td>{</td>
+    <td><i>c</i></td>
+    <td><i>d</i></td>
+    <td><i>e</i></td>
+    <td><i>f</i></td>
+    <td><i>g</i></td>
+    <td><i>a</i></td>
+    <td><i>b</i></td>
+    <td><i>h</i></td>
+    <td><i>c′</i></td>
+  </tr>
+  <tr>
+    <td rowspan="2" class="tdr valign">III.</td>
+    <td>{</td>
+    <td>(¹⁄₄)*</td>
+    <td>(²⁄₉)</td>
+    <td>(¹⁄₅)*</td>
+    <td>(³⁄₁₆)</td>
+    <td>(¹⁄₆)*</td>
+    <td>(³⁄₂₀)</td>
+    <td>(¹⁄₇)*</td>
+    <td>(²⁄₁₅)</td>
+    <td>(¹⁄₈)*</td>
+  </tr>
+  <tr>
+    <td>{</td>
+    <td><i>c′</i></td>
+    <td><i>d′</i></td>
+    <td><i>e′</i></td>
+    <td><i>f′</i></td>
+    <td><i>g′</i></td>
+    <td><i>a′</i></td>
+    <td><i>b′</i></td>
+    <td><i>h′</i></td>
+    <td><i>c′′</i></td>
+  </tr>
+  <tr>
+    <td rowspan="2" class="tdr valign">IV.</td>
+    <td>{</td>
+    <td>(¹⁄₈)*</td>
+    <td>(¹⁄₉)*</td>
+    <td>(¹⁄₁₀)*</td>
+    <td>(³⁄₃₂)</td>
+    <td>(¹⁄₁₂)*</td>
+    <td>(³⁄₄₀)</td>
+    <td>(¹⁄₁₄)*</td>
+    <td>(¹⁄₁₅)*</td>
+    <td>(¹⁄₁₆)*</td>
+  </tr>
+  <tr>
+    <td>{</td>
+    <td><i>c′′</i></td>
+    <td><i>d′′</i></td>
+    <td><i>e′′</i></td>
+    <td><i>f′′</i></td>
+    <td><i>g′′</i></td>
+    <td><i>a′′</i></td>
+    <td><i>b′′</i></td>
+    <td><i>h′′</i></td>
+    <td><i>c′′′</i></td>
+  </tr>
+</table>
+
+<p><span class="pagenum"><a id="Page_32"></a>[32]</span></p>
+
+<p>The notes marked (*) are the harmonics which naturally
+arise from the division of the string by 2, 3, 5, and 7, and the
+multiples of these primes.</p>
+
+<p>Thus every musical sound is composed of a certain number
+of parts called pulsations, and these parts must in every scale
+relate harmonically to some fundamental number. When
+these parts are multiples of the fundamental number by 2, 4,
+8, &amp;c., like the pulsations of the sounds indicated by <i>c</i>, <i>c′</i>, <i>c′′</i>, <i>c′′′</i>,
+they are called tonic notes, being the most consonant; when
+the pulsations are similar multiples by 3, 6, 12, &amp;c., like those
+of the sounds indicated by <i>g</i>, <i>g′</i>, <i>g′′</i>, they are called dominant
+notes, being the next most consonant; and multiples by 5,
+10, &amp;c., like those of the sounds indicated by <i>e</i>, <i>e′</i>, <i>e′′</i>, they are
+called mediant notes, from a similar cause. In harmonic combinations
+of musical sounds, the æsthetic feeling produced by
+their agreement depends upon the relations they bear to each
+other with reference to the number of pulsations produced in
+a given time by the fundamental note of the scale to which
+they belong; and it will be observed, that the more simple
+the numerical ratios are amongst the pulsations of any number
+of notes simultaneously produced, the more perfect their agreement.
+Hence the origin of the common chord or fundamental
+concord in the united sounds of the tonic, the dominant, and
+the mediant notes, the ratios and coincidences of whose pulsations
+2:1, 3:2, 5:4, may thus be exemplified:—</p>
+
+<figure class="figcenter illowp100" id="illus1" style="max-width: 37.5em;">
+  <img class="w100" src="images/illus1.jpg" alt="">
+</figure>
+
+<p><span class="pagenum"><a id="Page_33"></a>[33]</span></p>
+
+<p>In musical composition, the law of number also governs its
+division into parts, in order to produce upon the ear, along
+with the beauty of harmony, that of rhythm. Thus a piece of
+music is divided into parts each of which contains a certain
+number of other parts called bars, which may be divided
+and subdivided into any number of notes, and the performance
+of each bar is understood to occupy the same portion
+of time, however numerous the notes it contains may be; so
+that the music of art is regularly symmetrical in its structure;
+while that of nature is in general as irregular and indefinite in
+its rhythm as it is in its harmony.</p>
+
+<p>Thus I have endeavoured briefly to explain the manner in
+which the law of numerical ratio operates in that species of
+beauty perceived through the ear.</p>
+
+<p>The definite principles of the art of music founded upon
+this law have been for ages so systematised that those who are
+instructed in them advance steadily in proportion to their
+natural endowments, while those who refuse this instruction
+rarely attain to any excellence. In the sister arts of form
+and colour, however, a system of tuition, founded upon this
+law, is still a desideratum, and a knowledge of the scientific
+principles by which these arts are governed is confined to a
+very few, and scarcely acknowledged amongst those whose
+professions most require their practical application.</p>
+
+<hr class="chap x-ebookmaker-drop">
+
+<div class="chapter">
+
+<p><span class="pagenum"><a id="Page_34"></a>[34]</span></p>
+
+<h2 class="nobreak" id="AS_APPLIED_TO_FORMS">THE SCIENCE OF BEAUTY AS APPLIED TO FORMS.</h2>
+
+</div>
+
+<p>It is justly remarked, in the “Illustrated Record of the
+New York Exhibition of 1853,” that “it is a question worthy
+of consideration how far the mediocrity of the present day is
+attributable to an overweening reliance on natural powers
+and a neglect of the lights of science;” and there is expressed
+a thorough conviction of the fact that, besides the evils of the
+copying system, “much genius is now wasted in the acquirement
+of rudimentary knowledge in the slow school of practical
+experiment, and that the excellence of the ancient Greek school
+of design arose from a thoroughly digested canon of form, and
+the use of geometrical formulas, which make the works even
+of the second and third-rate genius of that period the wonder
+and admiration of the present day.”</p>
+
+<p>That such a canon of form, and that the use of such geometrical
+formula, entered into the education, and thereby facilitated
+the practice of ancient Greek art, I have in a former
+work expressed my firm belief, which is founded on the remarkable
+fact, that for a period of nearly three centuries, and
+throughout a whole country politically divided into states
+often at war with each other, works of sculpture, architecture,
+and ornamental design were executed, which surpass in symmetrical<span class="pagenum"><a id="Page_35"></a>[35]</span>
+beauty any works of the kind produced during the
+two thousand years that have since elapsed. So decided is
+this superiority, that the artistic remains of the extraordinary
+period I alluded to are, in all civilised nations, still held to be
+the most perfect specimens of formative art in the world; and
+even when so fragmentary as to be denuded of everything
+that can convey an idea of expression, they still excite admiration
+and wonder by the purity of their geometric beauty.
+And so universal was this excellence, that it seems to have
+characterised every production of formative art, however
+humble the use to which it was applied.</p>
+
+<p>The common supposition, that this excellence was the result
+of an extraordinary amount of genius existing among the
+Greek people during that particular period, is not consistent
+with what we know of the progress of mankind in any other
+direction, and is, in the present state of art, calculated to
+retard its progress, inasmuch as such an idea would suggest
+that, instead of making any exertion to arrive at a like general
+excellence, the world must wait for it until a similar supposed
+psychological phenomenon shall occur.</p>
+
+<p>But history tends to prove that the long period of universal
+artistic excellence throughout Greece could only be the result
+of an early inculcation of some well-digested system of correct
+elementary principles, by which the ordinary amount of genius
+allotted to mankind in every age was properly nurtured and
+cultivated; and by which, also, a correct knowledge and appreciation
+of art were disseminated amongst the people generally.
+Indeed, Müller, in his “Ancient Art and its Remains,” shews
+clearly that some certain fixed principles, constituting a science
+of proportions, were known in Greece, and that they formed
+the basis of all artists’ education and practice during the
+period referred to; also, that art began to decline, and its<span class="pagenum"><a id="Page_36"></a>[36]</span>
+brightest period to close, as this science fell into disuse, and
+the Greek artists, instead of working for an enlightened community,
+who understood the nature of the principles which
+guided them, were called upon to gratify the impatient whims
+of pampered and tyrannical rulers.</p>
+
+<p>By being instructed in this science of proportion, the Greek
+artists were enabled to impart to their representations of the
+human figure a mathematically correct species of symmetrical
+beauty; whether accompanying the slender and delicately
+undulated form of the Venus,—its opposite, the massive and
+powerful mould of the Hercules,—or the characteristic representation
+of any other deity in the heathen mythology. And
+this seems to have been done with equal ease in the minute figure
+cut on a precious gem, and in the most colossal statue. The
+same instruction likewise enabled the architects of Greece to
+institute those varieties of proportions in structure called the
+Classical Orders of Architecture; which are so perfect that, since
+the science which gave them birth has been buried in oblivion,
+classical architecture has been little more than an imitative art;
+for all who have since written upon the subject, from Vitruvius
+downwards, have arrived at nothing, in so far as the great
+elementary principles in question are concerned, beyond the
+most vague and unsatisfactory conjectures. For a more clear
+understanding of the nature of this application of the Pythagorean
+law of number to the harmony of form, it will be requisite
+to repeat the fact, that modern science has shewn that
+the cause of the impression, produced by external nature upon
+the sensorium, called light, may be traced to a molecular or
+ethereal action. This action is excited naturally by the sun,
+artificially by the combustion of various substances, and sometimes
+physically within the eye. Like the atmospheric pulsations
+which produce sound, the action which produces light is<span class="pagenum"><a id="Page_37"></a>[37]</span>
+capable, within a limited sphere, of being reflected from some
+bodies and transmitted through others; and by this reflection
+and transmission the visible nature of forms and figures is
+communicated to the sensorium. The eye is the medium of
+this communication; and its structural beauty, and perfect
+adaptation to the purpose of conveying this action, must, like
+those of the ear, be left to the anatomist fully to describe. It
+is here only necessary to remark, that the optic nerve, like
+the auditory nerve, ends in a carefully protected fluid, which
+is the last of the media interposed between this peculiarly
+subtle action and the nerve upon which it impresses the
+presence of the object from which it is reflected or through
+which it is transmitted, and the nature of such object made
+perceptible to the mind. The eye and the ear are thus, in
+one essential point, similar in their physiology, relatively to
+the means provided for receiving impressions from external
+nature; it is, therefore, but reasonable to believe that the eye
+is capable of appreciating the exact subdivision of spaces, just
+as the ear is capable of appreciating the exact subdivision of
+intervals of time; so that the division of space into exact
+numbers of equal parts will æsthetically affect the mind
+through the medium of the eye.</p>
+
+<p>We assume, therefore, that the standard of symmetry, so
+estimated, is deduced from the simplest law that could have
+been conceived—the law that the angles of direction must all
+bear to some fixed angle the same simple relations which the
+different notes in a chord of music bear to the fundamental
+note; that is, relations expressed arithmetically by the smallest
+natural numbers. Thus the eye, being guided in its estimate
+by direction rather than by distance, just as the ear is guided
+by number of vibrations rather than by magnitude, both it
+and the ear convey simplicity and harmony to the mind without<span class="pagenum"><a id="Page_38"></a>[38]</span>
+effort, and the mind with equal facility receives and appreciates
+them.</p>
+
+<h3><i>On the Rectilinear Forms and Proportions of Architecture.</i></h3>
+
+<p>As we are accustomed in all cases to refer direction to the
+horizontal and vertical lines, and as the meeting of these
+lines makes the right angle, it naturally constitutes the
+fundamental angle, by the harmonic division of which a
+system of proportion may be established, and the theory of
+symmetrical beauty, like that of music, rendered susceptible of
+exact reasoning.</p>
+
+<p>Let therefore the right angle be the fundamental angle,
+and let it be divided upon the quadrant of a circle into the
+harmonic parts already explained, thus:—</p>
+
+<table>
+  <tr>
+    <th></th>
+    <th>Right Angle.</th>
+    <th>Supertonic Angles.</th>
+    <th>Mediant Angles.</th>
+    <th>Subdominant Angles.</th>
+    <th>Dominant Angles.</th>
+    <th>Submediant Angles.</th>
+    <th>Subtonic Angles.</th>
+    <th>Semi-subtonic Angles.</th>
+    <th>Tonic Angles.</th>
+  </tr>
+  <tr>
+    <td class="tdr">I.</td>
+    <td>(1)</td>
+    <td>(⁸⁄₉)</td>
+    <td>(⁴⁄₅)</td>
+    <td>(³⁄₄)</td>
+    <td>(²⁄₃)</td>
+    <td>(³⁄₅)</td>
+    <td>(⁴⁄₇)</td>
+    <td>(⁸⁄₁₅)</td>
+    <td>(¹⁄₂)</td>
+  </tr>
+  <tr>
+    <td class="tdr">II.</td>
+    <td>(¹⁄₂)</td>
+    <td>(⁴⁄₉)</td>
+    <td>(²⁄₅)</td>
+    <td>(³⁄₈)</td>
+    <td>(¹⁄₃)</td>
+    <td>(³⁄₁₀)</td>
+    <td>(²⁄₇)</td>
+    <td>(⁴⁄₁₅)</td>
+    <td>(¹⁄₄)</td>
+  </tr>
+  <tr>
+    <td class="tdr">III.</td>
+    <td>(¹⁄₄)</td>
+    <td>(²⁄₉)</td>
+    <td>(¹⁄₅)</td>
+    <td>(³⁄₁₆)</td>
+    <td>(¹⁄₆)</td>
+    <td>(³⁄₂₀)</td>
+    <td>(¹⁄₇)</td>
+    <td>(²⁄₁₅)</td>
+    <td>(¹⁄₈)</td>
+  </tr>
+  <tr>
+    <td class="tdr">IV.</td>
+    <td>(¹⁄₈)</td>
+    <td>(¹⁄₉)</td>
+    <td>(¹⁄₁₀)</td>
+    <td>(³⁄₃₂)</td>
+    <td>(¹⁄₁₂)</td>
+    <td>(³⁄₄₀)</td>
+    <td>(¹⁄₁₄)</td>
+    <td>(¹⁄₁₅)</td>
+    <td>(¹⁄₁₆)</td>
+  </tr>
+</table>
+
+<p>In order that the analogy may be kept in view, I have
+given to the parts of each of these four scales the appropriate
+nomenclature of the notes which form the diatonic scale in
+music.</p>
+
+<p>When a right angled triangle is constructed so that its
+two smallest angles are equal, I term it simply the triangle
+of (¹⁄₂), because the smaller angles are each one-half of the
+right angle. But when the two angles are unequal, the
+triangle may be named after the smallest. For instance, when
+the smaller angle, which we shall here suppose to be one-third<span class="pagenum"><a id="Page_39"></a>[39]</span>
+of the right angle, is made with the vertical line, the triangle
+may be called the vertical scalene triangle of (¹⁄₃); and when
+made with the horizontal line, the horizontal scalene triangle
+of (¹⁄₃). As every rectangle is made up of two of these right
+angled triangles, the same terminology may also be applied to
+these figures. Thus, the equilateral rectangle or perfect
+square is simply the rectangle of (¹⁄₂), being composed of two
+similar right angled triangles of (¹⁄₂); and when two vertical
+scalene triangles of (¹⁄₃), and of similar dimensions, are united
+by their hypothenuses, they form the vertical rectangle of (¹⁄₃),
+and in like manner the horizontal triangles of (¹⁄₃) similarly
+united would form the horizontal rectangle of (¹⁄₃). As the
+isosceles triangle is in like manner composed of two right
+angled scalene triangles joined by one of their sides, the same
+terminology may be applied to every variety of that figure.
+All the angles of the first of the above scales, except that
+of (¹⁄₂), give rectangles whose longest sides are in the horizontal
+line, while the other three give rectangles whose longest sides
+are in the vertical line. I have illustrated in <a href="#plate01">Plate I.</a> the
+manner in which this harmonic law acts upon these elementary
+rectilinear figures by constructing a series agreeably to the
+angles of scales II., III., IV. Throughout this series <i>a b c</i> is
+the primary scalene triangle, of which the rectangle <i>a b c e</i>
+is composed; <i>d c e</i> the vertical isosceles triangle; and when
+the plate is turned, <i>d e a</i> the horizontal isosceles triangle, both
+of which are composed of the same primary scalene triangle.</p>
+
+<div class="sidenote"><a href="#plate01">Plate I.</a></div>
+
+<p>Thus the most simple elements of symmetry in rectilinear
+forms are the three following figures:—</p>
+
+<ul>
+<li>The equilateral rectangle or perfect square,</li>
+<li>The oblong rectangle, and</li>
+<li>The isosceles triangle.</li>
+</ul>
+
+<p>It has been shewn that in harmonic combinations of<span class="pagenum"><a id="Page_40"></a>[40]</span>
+musical sounds, the æsthetic feeling produced by their agreement
+depends upon the relation they bear to each other with
+reference to the number of pulsations produced in a given
+time by the fundamental note of the scale to which they
+belong; and that the more simply they relate to each other
+in this way the more perfect the harmony, as in the common
+chord of the first scale, the relations of whose parts are in the
+simple ratios of 2:1, 3:2, and 5:4. It is equally consistent
+with this law, that when applied to form in the composition of
+an assortment of figures of any kind, their respective proportions
+should bear a very simple ratio to each other in order
+that a definite and pleasing harmony may be produced
+amongst the various parts. Now, this is as effectually done
+by forming them upon the harmonic divisions of the right
+angle as musical harmony is produced by sounds resulting
+from harmonic divisions of a vibratory body.</p>
+
+<p>Having in previous works<a id="FNanchor_7" href="#Footnote_7" class="fnanchor">[7]</a> given the requisite illustrations of
+this fact in full detail, I shall here confine myself to the most
+simple kind, taking for my first example one of the finest
+specimens of classical architecture in the world—the front
+portico of the Parthenon of Athens.</p>
+
+<p>The angles which govern the proportions of this beautiful
+elevation are the following harmonic parts of the right angle—</p>
+
+<table>
+  <tr>
+    <th>Tonic Angles.</th>
+    <th>Dominant Angles.</th>
+    <th>Mediant Angles.</th>
+    <th>Subtonic Angle.</th>
+    <th>Supertonic Angles.</th>
+  </tr>
+  <tr>
+    <td>(¹⁄₂)</td>
+    <td>(¹⁄₃)</td>
+    <td>(¹⁄₅)</td>
+    <td>(¹⁄₇)</td>
+    <td>(¹⁄₉)</td>
+  </tr>
+  <tr>
+    <td>(¹⁄₄)</td>
+    <td>(¹⁄₆)</td>
+    <td>(¹⁄₁₀)</td>
+    <td></td>
+    <td>(¹⁄₁₈)</td>
+  </tr>
+  <tr>
+    <td>(¹⁄₈)</td>
+    <td></td>
+    <td></td>
+    <td></td>
+    <td></td>
+  </tr>
+  <tr>
+    <td>(¹⁄₁₆)</td>
+    <td></td>
+    <td></td>
+    <td></td>
+    <td></td>
+  </tr>
+</table>
+
+<div class="sidenote"><a href="#plate02">Plate II.</a></div>
+
+<p>In <a href="#plate02">Plate II.</a> I give a diagram of its rectilinear orthography,
+which is simply constructed by lines drawn, either horizontally,<span class="pagenum"><a id="Page_41"></a>[41]</span>
+vertically, or obliquely, which latter make with either of the
+former lines one or other of the harmonic angles in the above
+series. For example, the horizontal line AB represents the
+length of the base or surface of the upper step of the substructure
+of the building. The line AE, which makes an
+angle of (¹⁄₅) with the horizontal, determines the height of the
+colonnade. The line AD, which makes an angle of (¹⁄₄) with
+the horizontal, determines the height of the portico, exclusive
+of the pediment. The line AC, which makes an angle of (¹⁄₃)
+with the horizontal, determines the height of the portico, including
+the pediment. The line GD, which makes an angle
+of (¹⁄₇) with the horizontal, determines the form of the pediment.
+The lines EZ and LY, which respectively make angles
+of (¹⁄₁₆) and (¹⁄₁₈) with the horizontal, determine the breadth of
+the architrave, frieze, and cornice. The line <i>v n u</i>, which makes
+an angle of (¹⁄₃) with the vertical, determines the breadth of
+the triglyphs. The line <i>t d</i>, which makes an angle of (¹⁄₂),
+determines the breadth of the metops. The lines <i>c b r f</i>, and
+<i>a i</i>, which make each an angle of (¹⁄₆) with the vertical, determine
+the width of the five centre intercolumniations. The
+line <i>z k</i>, which makes an angle of (¹⁄₈) with the vertical, determines
+the width of the two remaining intercolumniations. The
+lines <i>c s</i>, <i>q x</i>, and <i>y h</i>, each of which makes an angle of (¹⁄₁₀)
+with the vertical, determine the diameters of the three columns
+on each side of the centre. The line <i>w l</i>, which makes an
+angle of (¹⁄₉) with the vertical, determines the diameter of
+the two remaining or corner columns.</p>
+
+<p>In all this, the length and breadth of the parts are determined
+by horizontal and vertical lines, which are necessarily
+at right angles with each other, and the position of which are
+determined by one or other of the lines making the harmonic
+angles above enumerated.</p>
+
+<p><span class="pagenum"><a id="Page_42"></a>[42]</span></p>
+
+<p>Now, the lengths and breadths thus so simply determined
+by these few angles, have been proved to be correct by their
+agreement with the most careful measurements which could
+possibly be made of this exquisite specimen of formative art.
+These measurements were obtained by the “Society of Dilettanti,”
+London, who, expressly for that purpose, sent Mr
+F. C. Penrose, a highly educated architect, to Athens, where
+he remained for about five months, engaged in the execution
+of this interesting commission, the results of which are now
+published in a magnificent volume by the Society.<a id="FNanchor_8" href="#Footnote_8" class="fnanchor">[8]</a> The
+agreement was so striking, that Mr Penrose has been publicly
+thanked by an eminent man of science for bearing testimony
+to the truth of my theory, who in doing so observes, “The
+dimensions which he (Mr Penrose) gives are to me the surest
+verification of the theory I could have desired. The minute
+discrepancies form that very element of practical incertitude,
+both as to execution and direct measurement, which always
+prevails in materialising a mathematical calculation made
+under such conditions.”<a id="FNanchor_9" href="#Footnote_9" class="fnanchor">[9]</a></p>
+
+<p>Although the measurements taken by Mr Penrose are
+undeniably correct, as all who examine the great work
+just referred to must acknowledge, and although they have
+afforded me the best possible means of testing the accuracy of
+my theory as applied to the Parthenon, yet the ideas of Mr
+Penrose as to the principles they evolve are founded upon the
+fallacious doctrine which has so long prevailed, and still prevails,
+in the æsthetics of architecture, viz., that harmony may be
+imparted by ratios between the lengths and breadths of parts.</p>
+
+<p>I have taken for my second example an elevation which,
+although of smaller dimensions, is no less celebrated for the
+beauty of its proportions than the Parthenon itself, viz., the<span class="pagenum"><a id="Page_43"></a>[43]</span>
+front portico of the temple of Theseus, which has also been
+measured by Mr Penrose.</p>
+
+<p>The angles which govern the proportions of this elevation
+are the following harmonic parts of the right angle:—</p>
+
+<table>
+  <tr>
+    <th>Tonic Angles.</th>
+    <th>Dominant Angles.</th>
+    <th>Mediant Angles.</th>
+  </tr>
+  <tr>
+    <td>(¹⁄₂)</td>
+    <td>(¹⁄₃)</td>
+    <td>(²⁄₅)</td>
+  </tr>
+  <tr>
+    <td>(¹⁄₄)</td>
+    <td>(¹⁄₆)</td>
+    <td>(¹⁄₅)</td>
+  </tr>
+  <tr>
+    <td></td>
+    <td>(¹⁄₁₂)</td>
+    <td></td>
+  </tr>
+</table>
+
+<div class="sidenote"><a href="#plate03">Plate III.</a></div>
+
+<p>A diagram of the rectilinear orthography of this portico is
+given in <a href="#plate03">Plate III.</a> Its construction is similar to that of
+the Parthenon in respect to the harmonic parts of the right
+angle, and I have therefore only to observe, that the line A E
+makes an angle of (¹⁄₄); the line A D an angle of (¹⁄₃); the
+line A C an angle of (²⁄₅); the line G D an angle of (¹⁄₆); and
+the lines E Z and L Y angles of (¹⁄₁₂) with the horizontal.</p>
+
+<p>As to the colonnade or vertical part, the line <i>a b</i>, which
+determines the three middle intercolumniations, makes an
+angle of (¹⁄₅); the line <i>c d</i>, which determines the two outer
+intercolumniations, makes an angle of (¹⁄₆); and the line <i>e f</i>,
+which determines the lesser diameter of the columns, makes an
+angle of (¹⁄₁₂) with the vertical. I need give no further details
+here, as my intention is to shew the simplicity of the method
+by which this theory may be reduced to practice, and because I
+have given in my other works ample details, in full illustration
+of the orthography of these two structures, especially the first.<a id="FNanchor_10" href="#Footnote_10" class="fnanchor">[10]</a></p>
+
+<p>The foregoing examples being both horizontal rectangular
+compositions, the proportions of their principal parts have
+necessarily been determined by lines drawn from the extremities
+of the base, making angles with the horizontal line, and forming<span class="pagenum"><a id="Page_44"></a>[44]</span>
+thereby the diagonals of the various rectangles into which, in
+their leading features, they are necessarily resolved. But the
+example I am now about to give is of another character,
+being a vertical pyramidal composition, and consequently the
+proportions of its principal parts are determined by the angles
+which the oblique lines make with the vertical line representing
+the height of the elevation, and forming a series of isosceles
+triangles; for the isosceles triangle is the type of all pyramidal
+composition.</p>
+
+<p>This third example is the east end of Lincoln Cathedral, a
+Gothic structure, which is acknowledged to be one of the
+finest specimens of that style of architecture existing in this
+country.</p>
+
+<p>The angles which govern the proportions of this elevation
+are the following harmonic parts of the right angle:—</p>
+
+<table>
+  <tr>
+    <th>Tonic.</th>
+    <th>Dominant.</th>
+    <th>Mediant.</th>
+    <th>Subtonic.</th>
+    <th>Supertonic.</th>
+  </tr>
+  <tr>
+    <td>(¹⁄₂)</td>
+    <td>(¹⁄₃)</td>
+    <td>(¹⁄₅)</td>
+    <td>(¹⁄₇)</td>
+    <td>(²⁄₉)</td>
+  </tr>
+  <tr>
+    <td>(¹⁄₄)</td>
+    <td>(¹⁄₆)</td>
+    <td>(¹⁄₁₀)</td>
+    <td></td>
+    <td>(¹⁄₉)</td>
+  </tr>
+  <tr>
+    <td></td>
+    <td>(¹⁄₁₂)</td>
+    <td></td>
+    <td></td>
+    <td></td>
+  </tr>
+</table>
+
+<div class="sidenote"><a href="#plate04">Plate IV.</a></div>
+
+<p>In <a href="#plate04">Plate IV.</a> I give a diagram of the vertical, horizontal, and
+oblique lines, which compose the orthography of this beautiful
+elevation.</p>
+
+<p>The line A B represents the full height of this structure.
+The line A C, which makes an angle of (²⁄₉) with the vertical,
+determines the width of the design, the tops of the aisle windows,
+and the bases of the pediments on the inner buttresses;
+A G, (¹⁄₅) with the vertical, that of the outer buttress; A F,
+(¹⁄₉) with the vertical, that of the space between the outer and
+inner buttresses and the width of the great centre window;
+and A E, (¹⁄₁₂) with vertical, that of both the inner buttresses
+and the space between these. A H, which makes (¹⁄₄) with the
+vertical, determines the form of the pediment of the centre,<span class="pagenum"><a id="Page_45"></a>[45]</span>
+and the full height of the base and surbase. A I, which makes
+(¹⁄₃) with the vertical, determines the form of the pediment of
+the smaller gables, the base of the pediment on the outer
+buttress, the base of the ornamental recess between the outer
+and inner buttresses, the spring of the arch of the centre
+window, the tops of the pediments on the inner buttresses,
+and the spring of the arch of the upper window. A K, which
+makes (¹⁄₂), determines the height of the outer buttress; and
+A Z, which makes (¹⁄₆) with the horizontal, determines that of
+the inner buttresses. For the reasons already given, I need not
+here go into further detail.<a id="FNanchor_11" href="#Footnote_11" class="fnanchor">[11]</a> It is, however, worthy of remark
+in this place, that notwithstanding the great difference which
+exists between the style of composition in this Gothic design,
+and in that of the east end of the Parthenon, the harmonic
+elements upon which the orthographic beauty of the one
+depends, are almost identical with those of the other.</p>
+
+<h3><i>On the Curvilinear Forms and Proportions of Architecture.</i></h3>
+
+<p>Each regular rectilinear figure has a curvilinear figure that
+exclusively belongs to it, and to which may be applied a corresponding
+terminology. For instance, the circle belongs to
+the equilateral rectangle; that is, the rectangle of (¹⁄₂), an
+ellipse to every other rectangle, and a composite ellipse to
+every isosceles triangle. Thus the most simple elements of
+beauty in the curvilinear forms of architectural design are the
+following three figures:—</p>
+
+<ul>
+<li>The circle,</li>
+<li>The ellipse, and</li>
+<li>The composite ellipse.</li>
+</ul>
+
+<p>I find it necessary in this place to go into some details<span class="pagenum"><a id="Page_46"></a>[46]</span>
+regarding the specific character of the two latter figures,
+because the proper mode of describing these beautiful curves,
+and their high value in the practice of the architectural
+draughtsman and ornamental designer, seem as yet unknown.
+In proof of this assertion, I must again refer to Mr Penrose’s
+great work published by the “Society of Dilettanti.” At page
+52 of that work it is observed, that “by whatever means an
+ellipse is to be constructed mechanically, it is a work of time
+(if not of absolute difficulty) so to arrange the foci, &amp;c., as to
+produce an ellipse of any exact length and breadth which may
+be desired.” Now, this is far from being the case, for the
+method of arranging the foci of an ellipse of any given length
+and breadth is extremely simple, being as follows:—</p>
+
+<p>Let A B C (figure 1) be the length, and D B E the breadth
+of the desired ellipse.</p>
+
+<figure class="figcenter illowp100" id="figure1" style="max-width: 37.5em;">
+  <img class="w100" src="images/figure1.jpg" alt="">
+  <figcaption class="caption"><p>Fig. 1.</p></figcaption>
+</figure>
+
+<p>Take A B upon the compasses, and place the point of one
+leg upon E and the point of the other upon the line A B, it
+will meet it at F, which is one focus: keeping the point of
+the one leg upon E, remove the point of the other to the line
+B C, and it will meet it at G, which is the other focus.
+But, when the proportions of an ellipse are to be imparted<span class="pagenum"><a id="Page_47"></a>[47]</span>
+by means of one of the harmonic angles, suppose the angle
+of (¹⁄₃), then the following is the process:—</p>
+
+<p>Let A B C (figure 2) represent the length of the intended
+ellipse. Through B draw B <i>e</i> indefinitely, at right angles with
+A B C; through C draw the line C <i>f</i> indefinitely, making, with
+B C, an angle of (¹⁄₃).</p>
+
+<p>Take B C upon the compasses, and place the point of one
+leg upon D where C f intersects B <i>e</i>, and the point of the
+other upon the line A B, it will meet it at F, which is one
+focus. Keeping the point of one leg still upon D, remove
+the point of the other to the line B C, and it will meet it at
+G, which is the other focus.</p>
+
+<figure class="figcenter illowp100" id="figure2" style="max-width: 34.375em;">
+  <img class="w100" src="images/figure2.jpg" alt="">
+  <figcaption class="caption"><p>Fig. 2.</p></figcaption>
+</figure>
+
+<p>The foci being in either case thus simply ascertained, the
+method of describing the curve on a small scale is equally
+simple.</p>
+
+<div class="sidenote"><a href="#plate05">Plate V.</a></div>
+
+<p>A pin is fixed into each of the two foci, and another into
+the point D. Around these three pins a waxed thread,
+flexible but not elastic, is tied, care being taken that the knot
+be of a kind that will not slip. The pin at D is now removed,
+and a hard black lead pencil introduced within the thread
+band. The pencil is then moved around the pins fixed in the
+foci, keeping the thread band at a full and equal tension;<span class="pagenum"><a id="Page_48"></a>[48]</span>
+thus simply the ellipse is described. When, however, the
+governing angle is acute, say less than (¹⁄₆), it is requisite to
+adopt a more accurate method of description,<a id="FNanchor_12" href="#Footnote_12" class="fnanchor">[12]</a> as the architectural
+examples which follow will shew. But architectural
+draughtsmen and ornamental designers would do well to
+supply themselves, for ordinary practice, with half a dozen
+series of ellipses, varying in the proportions of their axes from
+(⁴⁄₉) to (¹⁄₆) of the scale, and the length of their major axes
+from 1 to 6 inches. These should be described by the above
+simple process, upon very strong drawing paper, and carefully
+cut out, the edge of the paper being kept smooth,
+and each ellipse having its greater and lesser axes, its foci,
+and the hypothenuse of its scalene triangle drawn upon
+it. To exemplify this, I give <a href="#plate05">Plate V.</a>, which exhibits the
+ellipses of (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆), inscribed in their rectangles,
+on which <i>a b</i> and <i>c d</i> are respectively the greater and lesser
+axes, <i>o o</i> the foci, and <i>d b</i> the angle of each. Such a series
+of these beautiful figures would be found particularly useful
+in drawing the mouldings of Grecian architecture; for, to
+describe the curvilinear contour of such mouldings from
+single points, as has been done with those which embellish
+even our most pretending attempts at the restoration of
+that classical style of architecture, is to give the resemblance
+of an external form without the harmony which constitutes
+its real beauty.</p>
+
+<p>Mr Penrose, owing to the supposed difficulty regarding the
+description of ellipses just alluded to, endeavours to shew
+that the curves of all the mouldings throughout the Parthenon<span class="pagenum"><a id="Page_49"></a>[49]</span>
+were either parabolic or hyperbolic; but I believe such curves
+can have no connexion with the elementary forms of architecture,
+for they are curves which represent motion, and do
+not, by continued production, form closed figures.</p>
+
+<p>But I have shewn, in a former work,<a id="FNanchor_13" href="#Footnote_13" class="fnanchor">[13]</a> that the contours of
+these mouldings are composed of curves of the composite
+ellipse,—a figure which I so name because it is composed simply
+of arcs of various ellipses harmonically flowing into each
+other. The composite ellipse, when drawn systematically upon
+the isosceles triangle, resembles closely parabolic and hyperbolic
+curves—only differing from these inasmuch as it possesses
+the essential quality of circumscribing harmonically one of the
+elementary rectilinear figures employed in architecture, while
+those of the parabola and hyperbola, as I have just observed,
+are merely curves of motion, and, consequently, never can harmonically
+circumscribe or be resolved into any regular figure.</p>
+
+<p>The composite ellipse may be thus described.</p>
+
+<div class="sidenote"><a href="#plate06">Plate VI.</a></div>
+
+<p>Let A B C (<a href="#plate06">Plate VI.</a>) be a vertical isosceles triangle of
+(¹⁄₆), bisect A B in D, and through D draw indefinitely D <i>f</i>
+perpendicular to A B, and through B draw indefinitely B <i>g</i>,
+making the angle D B <i>g</i> (¹⁄₈), D <i>f</i> and B <i>g</i> intersecting each
+other in M. Take B D and D M as semi-axes of an ellipse,
+the foci of which will be at <i>p</i> and <i>q</i>, in each of these, and in
+each of the foci <i>h t</i> and <i>k r</i> in the lines A C and B C, fix
+a pin, and one also in the point M, tie a thread around these
+pins, withdraw the pin from M, and trace the composite ellipse
+in the manner already described with respect to the simple
+ellipse.</p>
+
+<p>In some of my earlier works I described this figure by taking
+the angles of the isosceles triangle as foci; but the above
+method is much more correct. As the elementary angle of<span class="pagenum"><a id="Page_50"></a>[50]</span>
+the triangle is (¹⁄₆), and that of the elliptic curve described
+around it (¹⁄₈), I call it the composite ellipse of (¹⁄₆) and (¹⁄₈),
+their harmonic ratio being 4:3; and so on of all others, according
+to the difference that may thus exist between the elementary
+angles.</p>
+
+<p>The visible curves which soften and beautify the melody of
+the outline of the front of the Parthenon, as given in Mr
+Penrose’s great work, I have carefully analysed, and have
+found them in as perfect agreement with this system, as its
+rectilinear harmony has been shewn to be. This I demonstrated
+in the work just referred to<a id="FNanchor_14" href="#Footnote_14" class="fnanchor">[14]</a> by a series of twelve plates,
+shewing that the entasis of the columns (a subject upon which
+there has been much speculation) is simply an arc of an ellipse
+of (¹⁄₄₈), whose greater axis makes with the vertical an angle of
+(¹⁄₆₄); or simply, the form of one of these columns is the frustrum
+of an elliptic-sided or prolate-spheroidal cone, whose
+section is a composite ellipse of (¹⁄₄₈) and (¹⁄₆₄), the harmonic
+ratio of these two angles being 4:3, the same as that of the
+angles of the composite ellipse just exemplified.</p>
+
+<div class="sidenote"><a href="#plate07">Plate VII.</a></div>
+
+<div class="sidenote"><a href="#plate08">Plate VIII.</a></div>
+
+<p>In <a href="#plate07">Plate VII.</a> is represented the section of such a cone, of
+which A B C is the isosceles triangle of (¹⁄₄₈), and B D and
+D M the semi-axes of an ellipse of (¹⁄₆₄). M N and O P are
+the entases of the column, and <i>d e f</i> the normal construction of
+the capital. All these are fully illustrated in the work above
+referred to,<a id="FNanchor_15" href="#Footnote_15" class="fnanchor">[15]</a> in which I have also shewn that the curve of the
+neck of the column is that of an ellipse of (¹⁄₆); the curve of
+the capital or echinus, that of an ellipse of (¹⁄₁₄); the curve of
+the moulding under the cymatium of the pediment, that of an
+ellipse of (¹⁄₃); and the curve of the bed-moulding of the cornice
+of the pediment, that of an ellipse of (¹⁄₃). The curve of the
+cavetto of the soffit of the corona is composed of ellipses of (¹⁄₆)<span class="pagenum"><a id="Page_51"></a>[51]</span>
+and (¹⁄₁₄); the curve of the cymatium which surmounts the
+corona, is that of an ellipse of (¹⁄₃); the curve of the moulding
+of the capital of the antæ of the posticum, that of an
+ellipse of (¹⁄₃); the curves of the lower moulding of the same
+capital are composed of those of an ellipse of (¹⁄₃) and of the
+circle (¹⁄₂); the curve of the moulding which is placed between
+the two latter is that of an ellipse of (¹⁄₃); the curve of the upper
+moulding of the band under the beams of the ceiling of the
+peristyle, that of an ellipse of (¹⁄₃); the curve of the lower
+moulding of the same band, that of an ellipse of (¹⁄₄); and the
+curves of the moulding at the bottom of the small step or
+podium between the columns, are those of the circle (¹⁄₂) and of
+an ellipse of (¹⁄₃). I have also shewn the curve of the fluting
+of the columns to be that of (¹⁄₁₄). The greater axis of each of
+these ellipses, when not in the vertical or horizontal lines,
+makes an harmonic angle with one or other of them. In
+<a href="#plate08">Plate VIII.</a>, sections of the two last-named mouldings are
+represented full size, which will give the reader an idea of the
+simple manner in which the ellipses are employed in the
+production of those harmonic curves.</p>
+
+<p>Thus we find that the system here adopted for applying
+this law of nature to the production of beauty in the abstract
+forms employed in architectural composition, so far from
+involving us in anything complicated, is characterised by
+extreme simplicity.</p>
+
+<p>In concluding this part of my treatise, I may here repeat
+what I have advanced in a late work,<a id="FNanchor_16" href="#Footnote_16" class="fnanchor">[16]</a> viz., my conviction of
+the probability that a system of applying this law of nature in
+architectural construction was the only great practical secret
+of the Freemasons, all their other secrets being connected, not
+with their art, but with the social constitution of their society.<span class="pagenum"><a id="Page_52"></a>[52]</span>
+This valuable secret, however, seems to have been lost, as its
+practical application fell into disuse; but, as that ancient
+society consisted of speculative as well as practical masons,
+the secrets connected with their social union have still been
+preserved, along with the excellent laws by which the brotherhood
+is governed. It can scarcely be doubted that there was
+some such practically useful secret amongst the Freemasons
+or early Gothic architects; for we find in all the venerable
+remains of their art which exist in this country, symmetrical
+elegance of form pervading the general design, harmonious proportion
+amongst all the parts, beautiful geometrical arrangements
+throughout all the tracery, as well as in the elegantly
+symmetrised foliated decorations which belong to that style of
+architecture. But it is at the same time worthy of remark,
+that whenever they diverged from architecture to sculpture
+and painting, and attempted to represent the human figure,
+or even any of the lower animals, their productions are such
+as to convince us that in this country these arts were in a
+very degraded state of barbarism—the figures are often much
+disproportioned in their parts and distorted in their attitudes,
+while their representations of animals and chimeras are whimsically
+absurd. It would, therefore, appear that architecture,
+as a fine art, must have been preserved by some peculiar
+influence from partaking of the barbarism so apparent in the
+sister arts of that period. Although its practical secrets have
+been long lost, the Freemasons of the present day trace the
+original possession of them to Moses, who, they say, “modelled
+masonry into a perfect system, and circumscribed its mysteries
+by <i>land-marks</i> significant and unalterable.” Now, as Moses
+received his education in Egypt, where Pythagoras is said to
+have acquired his first knowledge of the harmonic law of
+numbers, it is highly probable that this perfect system of the<span class="pagenum"><a id="Page_53"></a>[53]</span>
+great Jewish legislator was based upon the same law of nature
+which constituted the foundation of the Pythagorean philosophy,
+and ultimately led to that excellence in art which is
+still the admiration of the world.</p>
+
+<p>Pythagoras, it would appear, formed a system much more
+perfect and comprehensive than that practised by the Freemasons
+in the middle ages of Christianity; for it was as
+applicable to sculpture, painting, and music, as it was to
+architecture. This perfection in architecture is strikingly
+exemplified in the Parthenon, as compared with the Gothic
+structures of the middle ages; for it will be found that the
+whole six elementary figures I have enumerated as belonging
+to architecture, are required in completing the orthographic
+beauty of that noble structure. And amongst these, none
+conduce more to that beauty than the simple and composite
+ellipses. Now, in the architecture of the best periods of
+Gothic, or, indeed, in that of any after period (Roman architecture
+included), these beautiful curves seem to have been
+ignored, and that of the circle alone employed.</p>
+
+<p>Be those matters as they may, however, the great law of
+numerical harmonic ratio remains unalterable, and a proper
+application of it in the science of art will never fail to be as
+productive of effect, as its operation in nature is universal, certain,
+and continual.</p>
+
+<hr class="chap x-ebookmaker-drop">
+
+<div class="chapter">
+
+<p><span class="pagenum"><a id="Page_54"></a>[54]</span></p>
+
+<h2 class="nobreak" id="AS_DEVELOPED_IN_THE_HUMAN_HEAD_AND_COUNTENANCE">THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE HUMAN HEAD AND COUNTENANCE.</h2>
+
+</div>
+
+<p>The most remarkable characteristics of the human head and
+countenance are the globular form of the cranium, united as
+it is with the prolate spheroidal form produced by the parts
+which constitute the face, and the approximation of the profile
+to the vertical; for in none of the lower animals does the skull
+present so near a resemblance to a combination of these
+geometric forms, nor the plane of the face to this direction.
+We also find that although these peculiar characteristics are
+variously modified among the numerous races of mankind, yet
+one law appears to govern the beauty of the whole. The
+highest and most cultivated of these races, however, present
+only an approximation to the perfect development of those
+distinguishing marks of humanity; and therefore the beauty
+of form and proportion which in nature characterises the human
+head and countenance, exhibits only a partial development of
+the harmonic law of visible beauty. On the other hand, we
+find that, in their sculpture, the ancient Greeks surpassed
+ordinary nature, and produced in their beau ideal a species of
+beauty free from the imperfections and peculiarities that constitute
+the individuality by which the countenances of men are
+distinguished from each other. It may be requisite here to<span class="pagenum"><a id="Page_55"></a>[55]</span>
+remark, that this species of beauty is independent of the more
+intellectual quality of expression. For as Sir Charles Bell has
+said, “Beauty of countenance may be defined in words, as well
+as demonstrated in art. A face may be beautiful in sleep, and
+a statue without expression may be highly beautiful. But it
+will be said there is expression in the sleeping figure or in the
+statue. Is it not rather that we see in these the capacity for
+expression?—that our minds are active in imagining what may
+be the motions of these features when awake or animated?
+Thus, we speak of an expressive face before we have seen a
+movement grave or cheerful, or any indication in the features
+of what prevails in the heart.”</p>
+
+<p>This capacity for expression certainly enhances our admiration
+of the human countenance; but it is more a concomitant
+of the primary cause of its beauty than the cause itself. This
+cause rests on that simple and secure basis—the harmonic law
+of nature; for the nearer the countenance approximates to
+an harmonious combination of the most perfect figures in
+geometry, or rather the more its general form and the relation
+of its individual parts are arranged in obedience to that law,
+the higher its degree of beauty, and the greater its capacity
+for the expression of the passions.</p>
+
+<p>Various attempts have been made to define geometrically
+the difference between the ordinary and the ideal beauty of the
+human head and countenance, the most prominent of which is
+that of Camper. He traced, upon a profile of the skull, a line
+in a horizontal direction, passing through the foramen of the
+ear and the exterior margin of the sockets of the front teeth of
+the upper jaw, upon which he raised an oblique line, tangential
+to the margin of these sockets, and to the most prominent part
+of the forehead. Agreeably to the obliquity of this line, he
+determined the relative proportion of the areas occupied by the<span class="pagenum"><a id="Page_56"></a>[56]</span>
+brain and by the face, and hence inferred the degree of intellect.
+When he applied this measurement to the heads of the
+antique statues, he found the angle much greater than in
+ordinary nature; but that this simple fact afforded no rule for
+the reproduction of the ideal beauty of ancient Greek art, is
+very evident from the heads and countenances by which his
+treatise is illustrated. Sir Charles Bell justly remarks, that
+although, by Camper’s method, the forehead may be thrown
+forward, yet, while the features of common nature are preserved,
+we refuse to acknowledge a similarity to the beautiful forms of
+the antique marbles. “It is true,” he says, “that, by advancing
+the forehead, it is raised, the face is shortened, and the eye
+brought to the centre of the head. But with all this, there is
+much wanting—that which measurement, or a mere line, will
+not shew us.”—“The truth is, that we are more moved by the
+features than by the form of the whole head. Unless there be
+a conformity in every feature to the general shape of the head,
+throwing the forehead forward on the face produces deformity;
+and the question returns with full force—How is it that we are
+led to concede that the antique head of the Apollo, or of the
+Jupiter, is beautiful when the facial line makes a hundred degrees
+with the horizontal line? In other words—How do we
+admit that to be beautiful which is not natural? Simply for
+the same reason that, if we discover a broken portion of an
+antique, a nose, or a chin of marble, we can say, without deliberation—This
+must have belonged to a work of antiquity;
+which proves that the character is distinguishable in every part—in
+each feature, as well as in the whole head.”</p>
+
+<p>Dr Oken says upon this subject:<a id="FNanchor_17" href="#Footnote_17" class="fnanchor">[17]</a>—“The face is beautiful
+whose nose is parallel to the spine. No human face has grown<span class="pagenum"><a id="Page_57"></a>[57]</span>
+into this estate; but every nose makes an acute angle with the
+spine. The facial angle is, as is well known, 80°. What, as
+yet, no man has remarked, and what is not to be remarked,
+either, without our view of the cranial signification, the old
+masters have felt through inspiration. They have not only
+made the facial angle a right angle, but have even stepped
+beyond this—the Romans going up to 96°, the Greeks even to
+100°. Whence comes it that this unnatural face of the Grecian
+works of art is still more beautiful than that of the Roman,
+when the latter comes nearer to nature? The reason thereof
+resides in the fact of the Grecian artistic face representing
+nature’s design more than that of the Roman; for, in the
+former, the nose is placed quite perpendicular, or parallel to
+the spinal cord, and thus returns whither it has been derived.”</p>
+
+<p>Other various and conflicting opinions upon this subject have
+been given to the world; but we find that the principle from
+which arose the ideal beauty of the head and countenance, as
+represented in works of ancient Greek art, is still a matter of
+dispute. When, however, we examine carefully a fine specimen,
+we find its beauty and grandeur to depend more upon the
+degree of harmony amongst its parts, as to their relative proportions
+and mode of arrangement, than upon their excellence
+taken individually. It is, therefore, clear that those (and they
+are many) who attribute the beauty of ancient Greek sculpture
+merely to a selection of parts from various models, must be in
+error. No assemblage of parts from ordinary nature could
+have produced its principal characteristic, the excess in the
+angle of the facial line, much less could it have led to that
+exquisite harmony of parts by which it is so eminently distinguished;
+neither can we reasonably agree with Dr Oken
+and others, who assert that it was produced by an exclusive<span class="pagenum"><a id="Page_58"></a>[58]</span>
+degree of the inspiration of genius amongst the Greek people
+during a certain period.</p>
+
+<p>That the inspiration of genius, combined with a careful
+study of nature, were essential elements in the production of
+the great works which have been handed down to us, no one
+will deny; but these elements have existed in all ages, whilst
+the ideal head belongs exclusively to the Greeks during the
+period in which the schools of Pythagoras and Plato were
+open. Is it not, therefore, reasonable to suppose, that, besides
+genius and the study of nature, another element was employed
+in the production of this excellence, and that this
+element arose from the precise mathematical doctrines taught
+in the schools of these philosophers?</p>
+
+<p>An application of the great harmonic law seems to prove
+that there is no object in nature in which the science of
+beauty is more clearly developed than in the human head
+and countenance, nor to the representations of which the
+same science is more easily applied; and it is to the mode in
+which this is done that the varieties of sex and character
+may be imparted to works of art. Having gone into full
+detail, and given ample illustrations in a former work,<a id="FNanchor_18" href="#Footnote_18" class="fnanchor">[18]</a> it is
+unnecessary for me to enter upon that part of the subject in
+this <i>résumé</i>; but only to shew the typical structure of beauty
+by which this noble work of creation is distinguished.</p>
+
+<p>The angles which govern the form and proportions of the
+human head and countenance are, with the right angle, a
+series of seven, which, from the simplicity of their ratios to
+each other, are calculated to produce the most perfect concord.
+It consists of the right angle and its following parts—</p>
+
+<p><span class="pagenum"><a id="Page_59"></a>[59]</span></p>
+
+<table>
+  <tr>
+    <th>Tonic.</th>
+    <th>Dominant.</th>
+    <th>Mediant.</th>
+    <th>Subtonic.</th>
+  </tr>
+  <tr>
+    <td>(¹⁄₂)</td>
+    <td>(¹⁄₃)</td>
+    <td>(¹⁄₅)</td>
+    <td>(¹⁄₇)</td>
+  </tr>
+  <tr>
+    <td>(¹⁄₄)</td>
+    <td>(¹⁄₆)</td>
+    <td></td>
+    <td></td>
+  </tr>
+</table>
+
+<p>These angles, and the figures which belong to them, are
+thus arranged:—</p>
+
+<div class="sidenote"><a href="#plate09">Plate IX.</a></div>
+
+<p>The vertical line A B (<a href="#plate09">Plate IX.</a> fig. 2) represents the
+full length of the head and face. Taking this line as the
+greater axis of an ellipse of (¹⁄₃), such an ellipse is described
+around it. Through A the lines A G, A K, A L, A M, and
+A N, are drawn on each side of the line A B, making, with
+the vertical, respectively the angles of (¹⁄₃), (¹⁄₄), (¹⁄₅), (¹⁄₆), and
+(¹⁄₇). Through the points G, K, L, M, and N, where these
+straight lines meet the curved line of the ellipse, horizontal
+lines are drawn by which the following isosceles triangles are
+formed, A G G, A K K, A L L, A M M, and A N N. From
+the centre X of the equilateral triangle A G G the curvilinear
+figure of (¹⁄₂), viz., the circle, is described circumscribing that
+triangle.</p>
+
+<p>The curvilinear plane figures of (¹⁄₂) and (¹⁄₃), respectively,
+represent the solid bodies of which they are sections, viz., a
+sphere and a prolate spheroid. These bodies, from the manner
+in which they are here placed, are partially amalgamated,
+as shewn in figures 1 and 3 of the same plate, thus representing
+the form of the human head and countenance, both in
+their external appearance and osseous structure, more correctly
+than they could be represented by any other geometrical
+figures. Thus, the angles of (¹⁄₂) and (¹⁄₃) determine the
+typical form.</p>
+
+<p>From each of the points <i>u</i> and <i>n</i>, where A M cuts G G on
+both sides of A B, a circle is described through the points <i>p</i>
+and <i>q</i>, where A K cuts G G on both sides of A B, and with<span class="pagenum"><a id="Page_60"></a>[60]</span>
+the same radius a circle is described from the point <i>a</i>, where
+K K cuts A B.</p>
+
+<p>The circles <i>u</i> and <i>n</i> determine the position and size of the
+eyeballs, and the circle <i>a</i> the width of the nose, as also the
+horizontal width of the mouth.</p>
+
+<p>The lines G G and K K also determine the length of the
+joinings of the ear to the head. The lines L L and M M determine
+the vertical width of the mouth and lips when at
+perfect repose, and the line N N the superior edge of the chin.
+Thus simply are the features arranged and proportioned on
+the facial surface.</p>
+
+<p>It must, however, be borne in mind, that in treating simply
+of the æsthetic beauty of the human head and countenance,
+we have only to do with the external appearance. In
+this research, therefore, the system of Dr Camper, Dr Owen,
+and others, whose investigations were more of a physiological
+than an æsthetic character, can be of little service; because,
+according to that system, the facial angle is determined by
+drawing a line tangential to the exterior margin of the sockets
+of the front teeth of the upper jaw, and the most prominent
+part of the forehead. Now, as these sockets are, when the
+skull is naturally clothed, and the features in repose, entirely
+concealed by the upper lip, we must take the prominent part
+of it, instead of the sockets under it, in order to determine
+properly this distinguishing mark of humanity. And I believe
+it will be found, that when the head is properly poised,
+the nearer the angle which this line makes with the horizontal
+approaches 90°, the more symmetrically beautiful will be the
+general arrangement of the parts (see line <i>y z</i>, figure 3,
+<a href="#plate09">Plate IX.</a>).</p>
+
+<hr class="chap x-ebookmaker-drop">
+
+<div class="chapter">
+
+<p><span class="pagenum"><a id="Page_61"></a>[61]</span></p>
+
+<h2 class="nobreak" id="AS_DEVELOPED_IN_THE_FORM_OF_THE_HUMAN_FIGURE">THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE FORM OF THE HUMAN FIGURE.</h2>
+
+</div>
+
+<p>The manner in which this science is developed in the symmetrical
+proportions of the entire human figure, is as remarkable
+for its simplicity as it has been shewn to be in those of the
+head and countenance. Having gone into very full details,
+and given ample illustration in two former works<a id="FNanchor_19" href="#Footnote_19" class="fnanchor">[19]</a> upon this
+subject, I may here confine myself to the illustration of one
+description of figure, and to a reiteration of some facts stated
+in these works. These facts are, <i>1st</i>, That on a given line the
+human figure is developed, as to its principal points, entirely
+by lines drawn either from the extremities of this line, or
+from some obvious or determined localities. <i>2d</i>, That the
+angles which these lines make with the given line, are all
+simple sub-multiples of some given fundamental angle, or
+bear to it a proportion expressible under the most simple relations,
+such as those which constitute the scale of music.
+<i>3d</i>, That the contour is resolved into a series of ellipses of
+the same simple angles. And, <i>4th</i>, That these ellipses, like
+the lines, are inclined to the first given line by angles which
+are simple sub-multiples of the given fundamental angle.<span class="pagenum"><a id="Page_62"></a>[62]</span>
+From which four facts, and agreeably to the hypothesis I
+have adopted, it results as a natural consequence that the
+only effort which the mind exercises through the eye, in
+order to put itself in possession of the data for forming its
+judgment, is this, that it compares the angles about a point,
+and thereby appreciates the simplicity of their relations. In
+selecting the prominent features of a figure, the eye is not
+seeking to compare their relative distances—it is occupied
+solely with their relative positions. In tracing the contour,
+in like manner, it is not left in vague uncertainty as to what
+is the curve which is presented to it; unconsciously it feels
+the complete ellipse developed before it; and if that ellipse
+and its position are both formed by angles of the same
+simple relative value as those which aided its determination
+of the positions of the prominent features, it is satisfied, and
+finds the symmetry perfect.</p>
+
+<p>Müller, and other investigators into the archæology of art,
+refer to the great difficulty which exists in discovering the
+principles which the ancients followed in regard to the proportions
+of the human figure, from the different sexes and
+characters to which they require to be applied. But in the
+system thus founded upon the harmonic law of nature, no
+such difficulty is felt, for it is as applicable to the massive
+proportions which characterise the ancient representations of
+the Hercules, as to the delicate and perfectly symmetrical
+beauty of the Venus. This change is effected simply by an
+increase in the fundamental angle. For instance, in the
+construction of a figure of the exact proportions of the Venus,
+the right angle is adopted. But in the construction of a
+figure of the massive proportions of the Hercules, it is requisite
+to adopt an angle which bears to the right angle the ratio
+of 6:5. The adoption of this angle I have shewn in another<span class="pagenum"><a id="Page_63"></a>[63]</span>
+work<a id="FNanchor_20" href="#Footnote_20" class="fnanchor">[20]</a> to produce in the Hercules those proportions which
+are so characteristic of physical power. The ellipses which
+govern the outline, being also formed upon the same larger
+class of angles, give the contour of the muscles a more massive
+character. In comparing the male and female forms thus
+geometrically constructed, it will be found that that of the
+female is more harmoniously symmetrical, because the right
+angle is the fundamental angle for the trunk and the limbs
+as well as for the head and countenance; while in that of the
+male, the right angle is the fundamental angle for the head
+only. It may also be observed, that, from the greater proportional
+width of the pelvis of the female, the centres
+of that motion which the heads of the thigh bones perform
+in the cotyloid cavities, and the centres of that still
+more extensive range of motion which the arm is capable
+of performing at the shoulder joints, are nearly in the same
+line which determines the central motion of the vertebral
+column, while those of the male are not; consequently all
+the motions of the female are more graceful than those of
+the male.</p>
+
+<p>This difference between the fundamental angles, which
+impart to the human figure, on the one hand, the beauty of
+feminine proportion and contour, and on the other, the grandeur
+of masculine strength, being in the ratio of 5:6, allows
+ample latitude for those intermediate classes of proportions
+which the ancients imparted to their various other deities in
+which these two qualities were blended. I therefore confine
+myself to an illustration of the external contour of the form,
+and the relative proportions of all the parts of a female figure,
+such as those of the statues of the Venus of Melos and Venus
+of Medici.</p>
+
+<p><span class="pagenum"><a id="Page_64"></a>[64]</span></p>
+
+<p>The angles which govern the form and proportions of such
+a figure are, with the right angle, a series of twelve, as
+follows:—</p>
+
+<table>
+  <tr>
+    <th>Tonic.</th>
+    <th>Dominant.</th>
+    <th>Mediant.</th>
+    <th>Subtonic.</th>
+    <th>Supertonic.</th>
+  </tr>
+  <tr>
+    <td>(¹⁄₂)</td>
+    <td>(¹⁄₃)</td>
+    <td>(¹⁄₅)</td>
+    <td>(¹⁄₇)</td>
+    <td>(¹⁄₉)</td>
+  </tr>
+  <tr>
+    <td>(¹⁄₄)</td>
+    <td>(¹⁄₆)</td>
+    <td>(¹⁄₁₀)</td>
+    <td>(¹⁄₁₄)</td>
+    <td></td>
+  </tr>
+  <tr>
+    <td>(¹⁄₈)</td>
+    <td>(¹⁄₁₂)</td>
+    <td></td>
+    <td></td>
+    <td></td>
+  </tr>
+</table>
+
+<p>These angles are employed in the construction of a diagram,
+which determines the proportions of the parts throughout the
+whole figure. Thus:—</p>
+
+<div class="sidenote"><a href="#plate10">Plate X.</a></div>
+
+<p>Let the line A B (fig. 1, <a href="#plate10">plate X.</a>) represent the height of
+the figure to be constructed. At the point A, make the
+angles of C A D (¹⁄₃), F A G (¹⁄₄), H A I (¹⁄₅), K A L (¹⁄₆), and
+M A N (¹⁄₇). At the point B, make the angles K B L (¹⁄₈),
+U B A (¹⁄₁₂), and O B A (¹⁄₁₄).</p>
+
+<p>Through the point K, in which the lines A K and B K
+intersect one another, draw P K O parallel to A B, and
+through C F H and M, where this line meets A C, A F, A H,
+and A M, draw C D, F G, H I, and M N, perpendicular to
+A B; draw also K L perpendicular to A B; join B F and B H,
+and through C draw C E, making with A B the angle (¹⁄₂),
+which completes the arrangement of the eleven angles upon
+A B; for F B G is very nearly (¹⁄₁₀), and H B I very
+nearly (¹⁄₉).</p>
+
+<p>At the point <i>f</i>, where A C intersects O B, draw <i>f a</i>
+perpendicular to A B; and through the point <i>i</i>, where B O
+intersects M N, draw S <i>i</i> T parallel to A C.</p>
+
+<p>Through <i>m</i>, where S <i>i</i> T intersects F B, draw <i>m n</i>; through
+<i>β</i>, where S <i>i</i> T intersects K B, draw <i>β w</i>; through T draw T <i>g</i>,
+making an angle of (¹⁄₃) with O P. Join N P, M B, and <i>g</i> P,<span class="pagenum"><a id="Page_65"></a>[65]</span>
+and where N P intersects K B, draw Q R perpendicular to
+A B.</p>
+
+<p>On A E as a diameter, describe a circle cutting A C in <i>r</i>,
+and draw <i>r o</i> perpendicular to A B.</p>
+
+<p>With A <i>o</i> and <i>o r</i> as semi-axes, describe the ellipse A <i>r e</i>,
+cutting A H in <i>t</i>; and draw <i>t u</i> perpendicular to A B.
+With A <i>u</i> and <i>t u</i>, as semi-axes describe the ellipse A <i>t d</i>.
+On <i>a</i> L, as major axis, describe the ellipse of (¹⁄₃).</p>
+
+<p>For the side aspect or profile of the figure the diagram is
+thus constructed—</p>
+
+<p>On one side of a line A B (fig. 2, <a href="#plate10">Plate X.</a>) construct the
+rectilinear portion of a diagram the same as fig. 1. Through <i>i</i>
+draw W Y parallel to A B, and draw A <i>z</i> perpendicular to A B.
+Make W <i>a</i> equal to A <i>a</i> (fig. 1), and on <i>a l</i>, as major axis,
+describe the ellipse of (¹⁄₄). Through <i>a</i> draw <i>a p</i> parallel to
+A F, and through <i>p</i> draw <i>p t</i> perpendicular to W Y. Through
+<i>a</i> draw <i>f a u</i> perpendicular to W Y.</p>
+
+<p>Upon a diameter equal to A E describe a circle whose
+circumference shall touch A B and A <i>z</i>. With semi-axes
+equal to A <i>o</i> and <i>o r</i> (fig. 1), describe an ellipse with its major
+axis parallel to A B, and its circumference touching O P and
+<i>z</i> A.</p>
+
+<div class="sidenote"><a href="#plate11">Plate XI.</a></div>
+
+<p>Thus simply are the diagrams of the general proportions of
+the human figure, as viewed in front and in profile, constructed;
+and <a href="#plate11">Plate XI.</a> gives the contour in both points of view, as
+composed entirely of the curvilinear figures of (¹⁄₂), (¹⁄₃), (¹⁄₄),
+(¹⁄₅), and (¹⁄₆).</p>
+
+<p>Further detail here would be out of place, and I shall
+therefore refer those who require it to the Appendix, or the
+more elaborate works to which I have already referred.</p>
+
+<p>The beauty derived from proportion, imparted by the
+system here pointed out, and from a contour of curves derived<span class="pagenum"><a id="Page_66"></a>[66]</span>
+from the same harmonic angles, is not confined to the human
+figure, but is found in various minor degrees of perfection in
+all the organic forms of nature, whether animate or inanimate,
+of which I have in other works given many examples.<a id="FNanchor_21" href="#Footnote_21" class="fnanchor">[21]</a></p>
+
+<hr class="chap x-ebookmaker-drop">
+
+<div class="chapter">
+
+<p><span class="pagenum"><a id="Page_67"></a>[67]</span></p>
+
+<h2 class="nobreak" id="AS_DEVELOPED_IN_COLOURS">THE SCIENCE OF BEAUTY, AS DEVELOPED IN COLOURS.</h2>
+
+</div>
+
+<p>There is not amongst the various phenomena of nature one that
+more readily excites our admiration, or makes on the mind a
+more vivid impression of the order, variety, and harmonious
+beauty of the creation, than that of colour. On the general
+landscape this phenomenon is displayed in the production of
+that species of harmony in which colours are so variously blended,
+and in which they are by light, shade, and distance modified
+in such an infinity of gradation and hue. Although genius
+is continually struggling, with but partial success, to imitate
+those effects, yet, through the Divine beneficence, all whose
+organs of sight are in an ordinary degree of perfection can
+appreciate and enjoy them. In winter this pleasure is often
+to a certain extent withdrawn, when the colourless snow alone
+clothes the surface of the earth. But this is only a pause in
+the general harmony, which, as the spring returns, addresses
+itself the more pleasingly to our perception in its vernal
+melody, which, gradually resolving itself into the full rich hues
+of luxuriant beauty exhibited in the foliage and flowers of
+summer, subsequently rises into the more vivid and powerful
+harmonies of autumn’s colouring. Thus the eye is prepared
+again to enjoy that rest which such exciting causes may be
+said to have rendered necessary.</p>
+
+<p><span class="pagenum"><a id="Page_68"></a>[68]</span></p>
+
+<p>When we pass from the general colouring of nature to that
+of particular objects, we are again wrapt in wonder and admiration
+by the beauty and harmony which so constantly, and in
+such infinite variety, present themselves to our view, and
+which are so often found combined in the most minute objects.
+And the systematic order and uniformity perceptible amidst
+this endless variety in the colouring of animate and inanimate
+nature is thus another characteristic of beauty equally prevalent
+throughout creation.</p>
+
+<p>By this uniformity in colour, various species of animals are
+often distinguished; and in each individual of most of these
+species, how much is this beauty enhanced when the uniformity
+prevails in the resemblance of their lateral halves! The
+human countenance exemplifies this in a striking manner; the
+slightest variety of colour between one and another of the
+double parts is at once destructive of its symmetrical beauty.
+Many of the lower animals, whether inhabitants of the earth,
+the air, or the water, owe much of their beauty to this kind of
+uniformity in the colour of the furs, feathers, scales, or shells,
+with which they are clothed.</p>
+
+<p>In the vegetable kingdom, we find a great degree of uniformity
+of colour in the leaves, flowers, and fruit of the same
+plant, combined with all the harmonious beauty of variety
+which a little careful examination develops.</p>
+
+<p>In the colours of minerals, too, the same may be observed.
+In short, in the beauty of colouring, as in every other species
+of beauty, uniformity and variety are found to combine.</p>
+
+<p>An appreciation of colour depends, in the first place, as
+much upon the physical powers of the eye in conveying a proper
+impression to the mind, as that of music on those of the
+ear. But an ear for music, or an eye for colour, are, in so far
+as beauty is concerned, erroneous expressions; because they<span class="pagenum"><a id="Page_69"></a>[69]</span>
+are merely applicable to the impression made upon the senses,
+and do not refer to the æsthetical principles of harmony, by
+which beauty can alone be understood.</p>
+
+<p>A good eye, combined with experience, may enable us to
+form a correct idea as to the purity of an individual colour, or
+of the relative difference existing between two separate hues;
+but this sort of discrimination does not constitute that kind of
+appreciation of the harmony of colour by which we admire
+and enjoy its development in nature and art. The power of
+perceiving and appreciating beauty of any kind, is a principle
+inherent in the human mind, which may be improved by cultivation
+in the degree of the perfection of the art senses.
+Great pains have been bestowed on the education of the ear,
+in assisting it to appreciate the melody and harmony of sound;
+but still much remains to be done in regard to the cultivation
+of the eye, in appreciating colour as well as form.</p>
+
+<p>It is true, that there are individuals whose powers of vision
+are perfect, in so far as regards the appreciation of light, shade,
+and configuration, but who are totally incapable of perceiving
+effects produced by the intermediate phenomenon of colour,
+every object appearing to them either white, black, or neutral
+gray; others, who are equally blind as to the effect of one of
+the three primary colours, but see the other two perfectly, either
+singly or combined; while there are many who, having the full
+physical power of perceiving all the varieties of the phenomenon,
+and who are even capable of making nice distinctions
+amongst a variety of various colours, are yet incapable of appreciating
+the æsthetic quality of harmony which exists in their
+proper combination. It is the same with respect to the effects
+of sounds upon the ear—some have organs so constituted, that
+notes above or below a certain pitch are to them inaudible; while
+others, with physical powers otherwise perfect, are incapable<span class="pagenum"><a id="Page_70"></a>[70]</span>
+of appreciating either melody or harmony in musical composition.
+But perceptions so imperfectly constituted are, by the
+goodness of the Creator, of very rare occurrence; therefore all
+attempts at improvement in the science of æsthetics must be
+suited to the capacities of the generality of mankind, amongst
+whom the perception of colour exists in a variety as great as
+that by which their countenances are distinguished. Artists
+now and then appear who have this intuitive perception in
+such perfection, that they are capable of transferring to their
+works the most beautiful harmonies and most delicate gradations
+of colours, in a manner that no acquired knowledge
+could have enabled them to impart. To those who possess
+such a gift, as well as to those to whom the ordinary powers
+of perception are denied, it would be equally useless to offer
+an explanation of the various modes in which the harmony of
+colour develops itself, or to attempt a definition of the
+many various colours, hues, tints, and shades, arising out of
+the simple elements of this phenomenon. But to those whose
+powers lie between these extremes, being neither above nor
+below cultivation, such an explanation and definition must
+form a step towards the improvement of that inherent principle
+which constitutes the basis of æsthetical science.</p>
+
+<p>Although the variety and harmony of colour which nature
+is continually presenting to our view, are apparent to all whose
+visual organs are in a natural state, and thus to the generality
+of mankind; yet a knowledge of the simplicity by which this
+variety and beauty are produced, is, after ages of philosophic
+research and experimental inquiry, only beginning to be properly
+understood.</p>
+
+<p>Light may be considered as an active, and darkness a
+passive principle in the economy of Nature, and colour an
+intermediate phenomenon arising from their joint influence;<span class="pagenum"><a id="Page_71"></a>[71]</span>
+and it is in the ratios in which these primary principles act
+upon each other, by which I here intend to explain the science
+of beauty as evolved in colour. It has been usual to consider
+colour as an inherent quality in light, and to suppose that
+coloured bodies absorb certain classes of its rays, and reflect or
+transmit the remainder; but it appears to me that colour is
+more probably the result of certain modes in which the opposite
+principles of motion and rest, or force and resistance, operate
+in the production, refraction, and reflection of light, and that
+each colour is mutually related, although in different degrees,
+to these active and passive principles.</p>
+
+<p>White and black are the representatives of light and darkness,
+or activity and rest, and are therefore calculated as
+pigments to reduce colours and hues to tints and shades.</p>
+
+<p>Having, however, fully illustrated the nature of tints and
+shades in a former work,<a id="FNanchor_22" href="#Footnote_22" class="fnanchor">[22]</a> I shall here confine myself to
+colours in their full intensity—shewing the various modifications
+which their union with each other produce, along with
+the harmonic relations which these modifications bear to the
+primaries, and to each other in respect to warmth and coolness
+of tone, as well as to light and shade.</p>
+
+<p>The primary colours are red, yellow, and blue. Of these,
+yellow is most allied to light, and blue to shade, while red is
+neutral in these respects, being equally allied to both. In
+respect to tone, that of red is warm, and that of blue cool,
+while the tone of yellow is neutral. The ratios of their relations
+to each other in these respects will appear in the harmonic
+scales to which, for the first time, I am about to subject colours,
+and to systematise their various simple and compound relations,
+which are as follow:—</p>
+
+<p><span class="pagenum"><a id="Page_72"></a>[72]</span></p>
+
+<p>From the binary union of the primary colours, the
+secondary colours arise—</p>
+
+<p>Orange colour, from the union of yellow and red.</p>
+
+<p>Green, from the union of yellow and blue.</p>
+
+<p>Purple, from the union of red and blue.</p>
+
+<p>From the binary union of the secondary colours, the primary
+hues arise—</p>
+
+<p>Yellow-hue, from the union of orange and green.</p>
+
+<p>Red-hue, from the union of orange and purple.</p>
+
+<p>Blue-hue, from the union of purple and green.</p>
+
+<p>From the binary union of the primary hues, the secondary
+hues arise—</p>
+
+<p>Orange-hue, from the union of yellow-hue and red-hue.</p>
+
+<p>Green-hue, from the union of yellow-hue and blue-hue.</p>
+
+<p>Purple-hue, from the union of red-hue and blue-hue.</p>
+
+<p>Each hue owes its characteristic distinction to the proportionate
+predominance or subordination of one or other of the
+three primary colours in its composition.</p>
+
+<p>It follows, that in every hue of <i>red</i>, yellow and blue are subordinate;
+in every hue of <i>yellow</i>, red and blue are subordinate;
+and in every hue of <i>blue</i>, red and yellow are subordinate. In
+like manner, in every hue of <i>green</i>, red is subordinate; in
+every hue of <i>orange</i>, blue is subordinate; and in every hue of
+<i>purple</i>, yellow is subordinate.</p>
+
+<p>By the union of two primary colours, in the production of
+a secondary colour, the nature of both primaries is altered; and
+as there are only three primary or simple colours in the scale,
+the two that are united harmonically in a compound colour,
+form the natural contrast to the remaining simple colour.</p>
+
+<p>Notwithstanding all the variety that extends beyond the
+six positive colours, it may be said that there are only three<span class="pagenum"><a id="Page_73"></a>[73]</span>
+proper contrasts of colour in nature, and that all others are
+simply modifications of these.</p>
+
+<p>Pure red is the most perfect contrast to pure green; because
+it is characterised amongst the primary colours by warmth of
+tone, while amongst the secondary colours green is distinguished
+by coolness of tone, both being equally related to the
+primary elements of light and shade.</p>
+
+<p>Pure yellow is the most perfect contrast to pure purple;
+because it is characterised amongst the primary colours as most
+allied to light, whilst pure purple is characterised amongst
+the secondaries as most allied to shade, both being equally
+neutral as to tone.</p>
+
+<p>Pure blue is the most perfect contrast to pure orange;
+because it is characterised amongst the primary colours as
+not only the most allied to shade, but as being the coolest in
+tone, whilst pure orange is characterised amongst the secondaries
+as being the most allied to light and the warmest in
+tone. The same principle operates throughout all the modifications
+of these primary and secondary colours.</p>
+
+<p>Such is the simple nature of contrast upon which the beauty
+of colouring mainly depends.</p>
+
+<p>It being now established as a scientific fact, that the effect
+of light upon the eye is the result of an ethereal action, similar
+to the atmospheric action by which the effect of sound is
+produced upon the ear; also, that the various colours which
+light assumes are the effect of certain modifications in this
+ethereal action;—just as the various sounds, which constitute
+the scale of musical notes, are known to be the effect of certain
+modifications in the atmospheric action by which sounds in
+general are produced:</p>
+
+<p>Therefore, as harmony may thus be impressed upon the
+mind through either of these two art senses—hearing and<span class="pagenum"><a id="Page_74"></a>[74]</span>
+seeing—the principles which govern the modifications in the
+ethereal action of light, so as to produce through the eye the
+effect of harmony, cannot be supposed to differ from those
+principles which we know govern the modifications of the
+atmospheric action of sound, in producing through the ear a
+like effect. I shall therefore endeavour to illustrate the
+science of beauty as evolved in colours, by forming scales of
+their various modifications agreeably to the same Pythagorean
+system of numerical ratio from which the harmonic elements
+of beauty in sounds were originally evolved, and by which
+I have endeavoured in this, as in previous works, to systematise
+the harmonic beauty of forms.</p>
+
+<figure class="figcenter illowp80" id="illus2" style="max-width: 43.75em;">
+  <img class="w100" src="images/illus2.jpg" alt="">
+</figure>
+
+<p>It will be observed, that with a view to avoid complexity<span class="pagenum"><a id="Page_75"></a>[75]</span>
+as much as possible, I have, in the arrangement of the above
+series of scales, not only confined myself to the merely elementary
+parts of the Pythagorean system, but have left out
+the harmonic modifications upon (¹⁄₁₁) and (¹⁄₁₃), in order that
+the arithmetical progression might not be interrupted.<a id="FNanchor_23" href="#Footnote_23" class="fnanchor">[23]</a></p>
+
+<p>The above elementary process will, I trust, be found sufficient
+to explain the progress, by harmonic union, of a primary
+colour to a toned gray, and how the simple and compound
+colours naturally arrange themselves into the elements of five
+scales, the parts of which continue from primary to secondary
+colour; from secondary colour to primary hue; from primary
+hue to secondary hue; from secondary hue to primary-toned
+gray; and from primary-toned gray to secondary-toned gray
+in the simple ratio of 2:1; thereby producing a series of the
+most beautiful and perfect contrasts.</p>
+
+<p>The natural arrangement of the primary colours upon the
+solar spectrum is red, yellow, blue, and I have therefore
+adopted the same arrangement on the present occasion. Red
+being, consequently, the first tonic, and blue the second, the
+divisions express the numerical ratios which the colours bear
+to one another, in respect to that colourific power for which
+red is pre-eminent. Thus, yellow is to red, as 2:3; blue
+to yellow, as 3:4; purple to orange, as 5:6; and green to
+purple, as 6:7.</p>
+
+<p>The following series of completed scales are arranged upon
+the foregoing principle, with the natural connecting links of
+red-orange, yellow-orange, yellow-green, and blue-green, introduced
+in their proper places.</p>
+
+<p>The appropriate terminology of musical notes has been
+adopted, and the scales are composed as follows:—</p>
+
+<p><span class="pagenum"><a id="Page_76"></a>[76]</span></p>
+
+<ul>
+<li>Scale I. consists of primary and secondary colours;</li>
+<li>Scale II. of secondary colours and primary hues;</li>
+<li>Scale III. of primary and secondary hues;</li>
+<li>Scale IV. of secondary hues and primary-toned grays; and</li>
+<li>Scale V. of primary and secondary-toned grays;</li>
+</ul>
+
+<p>All the parts in each of these scales, from the first tonic to
+the second, relate to the same parts of the scale below them
+in the simple ratio of 2:1; and serially to the first tonic in
+the following ratios:—</p>
+
+<p class="center">8:9, 4:5, 3:4, 2:3, 3:5, 4:7, 8:15, 1:2.</p>
+
+<h3><i>First Series of Scales.</i></h3>
+
+<table class="borders">
+  <tr>
+   <th></th>
+   <th>Tonic.</th>
+   <th>Supertonic.</th>
+   <th>Mediant.</th>
+   <th>Subdominant.</th>
+   <th>Dominant.</th>
+   <th>Submediant.</th>
+   <th>Subtonic.</th>
+   <th>Semi-Subtonic.</th>
+   <th>Tonic.</th>
+  </tr>
+  <tr>
+    <td rowspan="2" class="tdr valign">I.</td>
+    <td>(¹⁄₂)</td>
+    <td>(⁴⁄₉)</td>
+    <td>(²⁄₅)</td>
+    <td>(³⁄₈)</td>
+    <td>(¹⁄₃)</td>
+    <td>(³⁄₁₀)</td>
+    <td>(²⁄₇)</td>
+    <td>(⁴⁄₁₅)</td>
+    <td>(¹⁄₄)</td>
+  </tr>
+  <tr>
+    <td>Red.</td>
+    <td>Red-orange.</td>
+    <td>Orange.</td>
+    <td>Yellow-orange.</td>
+    <td>Yellow.</td>
+    <td>Yellow-green.</td>
+    <td>Green.</td>
+    <td>Blue-green.</td>
+    <td>Blue.</td>
+  </tr>
+  <tr>
+    <td rowspan="2" class="tdr valign">II.</td>
+    <td>(¹⁄₄)</td>
+    <td>(²⁄₉)</td>
+    <td>(¹⁄₅)</td>
+    <td>(³⁄₁₆)</td>
+    <td>(¹⁄₆)</td>
+    <td>(³⁄₂₀)</td>
+    <td>(¹⁄₇)</td>
+    <td>(²⁄₁₅)</td>
+    <td>(¹⁄₈)</td>
+  </tr>
+  <tr>
+    <td>Green.</td>
+    <td>Blue-green hue.</td>
+    <td>Blue hue.</td>
+    <td>Blue-purple hue.</td>
+    <td>Purple hue.</td>
+    <td>Red-purple hue.</td>
+    <td>Red hue.</td>
+    <td>Red-orange hue.</td>
+    <td>Orange.</td>
+  </tr>
+  <tr>
+    <td rowspan="2" class="tdr valign">III.</td>
+    <td>(¹⁄₈)</td>
+    <td>(¹⁄₉)</td>
+    <td>(¹⁄₁₀)</td>
+    <td>(³⁄₃₂)</td>
+    <td>(¹⁄₁₂)</td>
+    <td>(³⁄₄₀)</td>
+    <td>(¹⁄₁₄)</td>
+    <td>(¹⁄₁₅)</td>
+    <td>(¹⁄₁₆)</td>
+  </tr>
+  <tr>
+    <td>Red hue.</td>
+    <td>Red-orange hue.</td>
+    <td>Orange hue.</td>
+    <td>Yellow-orange hue.</td>
+    <td>Yellow hue.</td>
+    <td>Yellow-green hue.</td>
+    <td>Green hue.</td>
+    <td>Blue-green hue.</td>
+    <td>Blue hue.</td>
+  </tr>
+  <tr>
+    <td rowspan="2" class="tdr valign">IV.</td>
+    <td>(¹⁄₁₆)</td>
+    <td>(¹⁄₁₈)</td>
+    <td>(¹⁄₂₀)</td>
+    <td>(³⁄₆₄)</td>
+    <td>(¹⁄₂₄)</td>
+    <td>(³⁄₈₀)</td>
+    <td>(¹⁄₂₈)</td>
+    <td>(¹⁄₃₀)</td>
+    <td>(¹⁄₃₂)</td>
+  </tr>
+  <tr>
+    <td>Green hue.</td>
+    <td>Blue-green-toned gray.</td>
+    <td>Blue-toned gray.</td>
+    <td>Blue-purple-toned gray.</td>
+    <td>Purple hue.</td>
+    <td>Red-purple-toned gray.</td>
+    <td>Red-toned gray.</td>
+    <td>Red-orange-toned gray.</td>
+    <td>Orange hue.</td>
+  </tr>
+  <tr>
+    <td rowspan="2" class="tdr valign">V.</td>
+    <td>(¹⁄₃₂)</td>
+    <td>(¹⁄₃₆)</td>
+    <td>(¹⁄₄₀)</td>
+    <td>(³⁄₁₂₈)</td>
+    <td>(¹⁄₄₈)</td>
+    <td>(³⁄₁₆₀)</td>
+    <td>(¹⁄₅₆)</td>
+    <td>(¹⁄₆₀)</td>
+    <td>(¹⁄₆₄)</td>
+  </tr>
+  <tr>
+    <td>Red-toned gray.</td>
+    <td>Red-orange-toned gray.</td>
+    <td>Orange-toned gray.</td>
+    <td>Yellow-orange-toned gray.</td>
+    <td>Yellow-toned gray.</td>
+    <td>Yellow-green-toned gray.</td>
+    <td>Green-toned gray.</td>
+    <td>Blue-green-toned gray.</td>
+    <td>Blue-toned gray.</td>
+  </tr>
+</table>
+
+<p><span class="pagenum"><a id="Page_77"></a>[77]</span></p>
+
+<p>To the scales of chromatic power I add another series of
+scales, in which yellow, being the first tonic, and blue the
+second, the numerical divisions express the ratios which the
+colours in each scale bear to one another in respect to light
+and shade. Thus red is to yellow, in respect to light, as 2:3;
+blue to red, as 3:4; green to orange, as 5:6, and purple to
+green, as 6:7.</p>
+
+<p>These scales may therefore be termed scales for the colour-blind,
+because, in comparing colours, those whose sight is thus
+defective, naturally compare the ratios of the light and shade
+of which different colours are primarily constituted.</p>
+
+<figure class="figcenter illowp80" id="illus3" style="max-width: 43.75em;">
+  <img class="w100" src="images/illus3.jpg" alt="">
+</figure>
+
+<p>The following is a series of five complete scales of the harmonic<span class="pagenum"><a id="Page_78"></a>[78]</span>
+parts into which the light and shade in colours may be
+divided in each scale according to the above arrangement:—</p>
+
+<h3><i>Second Series of Scales.</i></h3>
+
+<table class="borders">
+  <tr>
+   <th></th>
+   <th>Tonic.</th>
+   <th>Supertonic.</th>
+   <th>Mediant.</th>
+   <th>Subdominant.</th>
+   <th>Dominant.</th>
+   <th>Submediant.</th>
+   <th>Subtonic.</th>
+   <th>Semi-Subtonic.</th>
+   <th>Tonic.</th>
+  </tr>
+  <tr>
+    <td rowspan="2" class="tdr valign">I.</td>
+    <td>(¹⁄₂)</td>
+    <td>(⁴⁄₉)</td>
+    <td>(²⁄₅)</td>
+    <td>(³⁄₈)</td>
+    <td>(¹⁄₃)</td>
+    <td>(³⁄₁₀)</td>
+    <td>(²⁄₇)</td>
+    <td>(⁴⁄₁₅)</td>
+    <td>(¹⁄₄)</td>
+  </tr>
+  <tr>
+    <td>Yellow.</td>
+    <td>Yellow-orange.</td>
+    <td>Orange.</td>
+    <td>Red-orange.</td>
+    <td>Red.</td>
+    <td>Red-purple.</td>
+    <td>Purple.</td>
+    <td>Blue-purple.</td>
+    <td>Blue.</td>
+  </tr>
+  <tr>
+    <td rowspan="2" class="tdr valign">II.</td>
+    <td>(¹⁄₄)</td>
+    <td>(²⁄₉)</td>
+    <td>(¹⁄₅)</td>
+    <td>(³⁄₁₆)</td>
+    <td>(¹⁄₆)</td>
+    <td>(³⁄₂₀)</td>
+    <td>(¹⁄₇)</td>
+    <td>(²⁄₁₅)</td>
+    <td>(¹⁄₈)</td>
+  </tr>
+  <tr>
+    <td>Purple.</td>
+    <td>Blue-purple hue.</td>
+    <td>Blue hue.</td>
+    <td>Blue-green hue.</td>
+    <td>Green.</td>
+    <td>Yellow-green hue.</td>
+    <td>Yellow hue.</td>
+    <td>Yellow-orange hue.</td>
+    <td>Orange.</td>
+  </tr>
+  <tr>
+    <td rowspan="2" class="tdr valign">III.</td>
+    <td>(¹⁄₈)</td>
+    <td>(¹⁄₉)</td>
+    <td>(¹⁄₁₀)</td>
+    <td>(³⁄₃₂)</td>
+    <td>(¹⁄₁₂)</td>
+    <td>(³⁄₄₀)</td>
+    <td>(¹⁄₁₄)</td>
+    <td>(¹⁄₁₅)</td>
+    <td>(¹⁄₁₆)</td>
+  </tr>
+  <tr>
+    <td>Yellow hue.</td>
+    <td>Yellow-orange hue.</td>
+    <td>Orange hue.</td>
+    <td>Red-orange hue.</td>
+    <td>Red hue.</td>
+    <td>Red-purple hue.</td>
+    <td>Purple hue.</td>
+    <td>Blue-purple hue.</td>
+    <td>Blue hue.</td>
+  </tr>
+  <tr>
+    <td rowspan="2" class="tdr valign">IV.</td>
+    <td>(¹⁄₁₆)</td>
+    <td>(¹⁄₁₈)</td>
+    <td>(¹⁄₂₀)</td>
+    <td>(³⁄₆₄)</td>
+    <td>(¹⁄₂₄)</td>
+    <td>(³⁄₈₀)</td>
+    <td>(¹⁄₂₈)</td>
+    <td>(¹⁄₃₀)</td>
+    <td>(¹⁄₃₂)</td>
+  </tr>
+  <tr>
+    <td>Purple hue.</td>
+    <td>Blue-purple-toned gray.</td>
+    <td>Blue-toned gray.</td>
+    <td>Blue-green-toned gray.</td>
+    <td>Green hue.</td>
+    <td>Yellow-green-toned gray.</td>
+    <td>Yellow-toned gray.</td>
+    <td>Yellow-orange-toned gray.</td>
+    <td>Orange hue.</td>
+  </tr>
+  <tr>
+    <td rowspan="2" class="tdr valign">V.</td>
+    <td>(¹⁄₃₂)</td>
+    <td>(¹⁄₃₆)</td>
+    <td>(¹⁄₄₀)</td>
+    <td>(³⁄₁₂₈)</td>
+    <td>(¹⁄₄₈)</td>
+    <td>(³⁄₁₆₀)</td>
+    <td>(¹⁄₅₆)</td>
+    <td>(¹⁄₆₀)</td>
+    <td>(¹⁄₆₄)</td>
+  </tr>
+  <tr>
+    <td>Yellow-toned gray.</td>
+    <td>Yellow-orange-toned gray.</td>
+    <td>Orange-toned gray.</td>
+    <td>Red-orange-toned gray.</td>
+    <td>Red-toned gray.</td>
+    <td>Red-purple-toned gray.</td>
+    <td>Purple-toned gray.</td>
+    <td>Blue-green-toned gray.</td>
+    <td>Blue-toned gray.</td>
+  </tr>
+</table>
+
+<p>Should I be correct in arranging colours upon scales identical
+with those upon which musical notes have been arranged,
+and in assuming that colours have the same ratios to
+each other, in respect to their harmonic power upon the eye,
+which musical notes have in respect to their harmonic power
+upon the ear, the colourist may yet be enabled to impart
+harmonic beauty to his works with as much certainty and
+ease, as the musician imparts the same quality to his compositions:
+for the colourist has no more right to trust exclusively<span class="pagenum"><a id="Page_79"></a>[79]</span>
+to his eye in the arrangement of colours, than the
+musician has to trust exclusively to his ear in the arrangement
+of sounds.</p>
+
+<p>We find, in comparing the dominant parts in the first
+and second scales of the second series, that they are equal
+as to light and shade, so that their relative powers of contrast
+depend entirely upon colour. Hence it is that red
+and green are the two colours, the difference between which
+the colour-blind are least able to appreciate. Professor
+George Wilson, in his excellent work, “Researches on
+Colour-Blindness,” mentions the case of an engraver, which
+proves the power of the eye in being able to appreciate
+these original constituents of colour, irrespective of the intermediate
+phenomenon of tone. This engraver, instead of
+expressing regret on account of his being colour-blind,
+observed to the professor, “My defective vision is, to a certain
+extent, a useful and valuable quality. Thus, an engraver has
+two negatives to deal with, <i>i.e.</i>, white and black. Now,
+when I look at a picture, I see it only in white and black,
+or light and shade, or, as artists term it, the effect. I find
+at times many of my brother engravers in doubt how to
+translate certain colours of pictures, which to me are matters
+of decided certainty and ease. Thus to me it is valuable.”</p>
+
+<p>The colour-blind are therefore as incapable of receiving
+pleasure from the harmonious union of various colours, as
+those who, to use a common term, have no ear for music, are
+of being gratified by the “melody of sweet sounds.”</p>
+
+<p>The generality of mankind are, however, capable of appreciating
+the harmony of colour which, like that of both sound
+and form, arises from the simultaneous exhibition of opposite
+principles having a ratio to each other. These principles are in
+continual operation throughout nature, and from them we<span class="pagenum"><a id="Page_80"></a>[80]</span>
+often derive pleasure without being conscious of the cause.
+All who are not colour-blind must have felt themselves struck
+with the harmonic beauty of a cloudless sky, although in it
+there is no configuration, and at first sight apparently but one
+colour. Now, as we know that there can be no more impression
+of harmony made upon the mind by looking upon a
+single colour, than there could be by listening to a single
+continued musical note, however sweet its tone, we are apt
+at first to imagine that the organ of vision has, in some
+measure, conveyed a false impression to the mind. But it
+has not done so; for light, when reflected from the atmosphere,
+produces those cool tones of blue, gray, and purple,
+which seem to clothe the distant mountains; but, when
+transmitted through the same atmosphere, it produces those
+numerous warm tints, the most intense of which give the
+gorgeous effects which so often accompany the setting sun.
+We have, therefore, in the upper part of a clear sky, where
+the atmosphere may be said to be illuminated principally
+by reflection from the surface of the earth, a comparatively
+cool tone of blue, the result of reflection, which gradually
+blends into the warm tints, the result of transmission through
+the same atmosphere. Such a composition of harmonious
+colouring is to the eye what the voice of the soft breath of
+summer amongst the trees, the hum of insects on a sultry
+day, or the simple harmony of the Æolian harp, is to the ear.
+To such a composition of chromatic harmony must also be
+referred the universal concurrence of mankind in appreciating
+the peculiar beauty of white marble statuary. That the
+principal constituent of beauty in such works ought to be
+harmony of form, no one will deny; but this is not the only
+element, as appears from the fact, that a cast in plaster of
+Paris, of a fine white marble statue, although identical in<span class="pagenum"><a id="Page_81"></a>[81]</span>
+form, is far less beautiful than the original. Now this undoubtedly
+must be the consequence of its having been changed
+from a semi-translucent substance, which, like the atmosphere,
+can transmit as well as reflect light, to an opaque substance,
+which can only reflect it. Thus the opposite principles of
+chromatic warmth and coolness are equally balanced in white
+marble—the one being the natural result of the partial transmission
+of light, and the other that of its reflection.</p>
+
+<p>As a series of coloured illustrations would be beyond the
+scope of this <i>résumé</i>, I may refer those who wish to prosecute
+the inquiry, with the assistance of such a series, to my published
+works upon the subject.<a id="FNanchor_24" href="#Footnote_24" class="fnanchor">[24]</a></p>
+
+<hr class="chap x-ebookmaker-drop">
+
+<div class="chapter">
+
+<p><span class="pagenum"><a id="Page_82"></a>[82]</span></p>
+
+<h2 class="nobreak" id="APPLIED_TO_THE_FORMS_AND_PROPORTIONS">THE SCIENCE OF BEAUTY, APPLIED TO THE FORMS AND PROPORTIONS OF ANCIENT
+GRECIAN VASES AND ORNAMENTS.</h2>
+
+</div>
+
+<p>In examining the remains of the ornamental works of the
+ancient Greek artists, it appears highly probable that the harmony
+of their proportions and melody of their contour are
+equally the result of a systematised application of the same
+harmonic law. This probability not being fully elucidated in
+any of my former works, I will require to go into some detail
+on the present occasion. I take for my first illustration an
+unexceptionable example, viz.:—</p>
+
+<h3><i>The Portland Vase.</i></h3>
+
+<p>Although this beautiful specimen of ancient art was found
+about the middle of the sixteenth century, inclosed in a marble
+sarcophagus within a sepulchral chamber under the Monte del
+Grano, near Rome, and although the date of its production
+is unknown, yet its being a work of ancient Grecian art is
+undoubted; and the exquisite beauty of its form has been
+universally acknowledged, both during the time it remained
+in the palace of the Barberini family at Rome, and since it
+was added to the treasures of the British Museum. The<span class="pagenum"><a id="Page_83"></a>[83]</span>
+forms and proportions of this gem of art appear to me to
+yield an obedience to the great harmonic law of nature, similar
+to that which I have instanced in the proportions and contour
+of the best specimens of ancient Grecian architecture.</p>
+
+<div class="sidenote"><a href="#plate12">Plate XII.</a></div>
+
+<p>Let the line A B (<a href="#plate12">Plate XII.</a>) represent the full height
+of the vase. Through A draw A <i>a</i>, and through B draw B <i>b</i>
+indefinitely, A <i>a</i> making an angle of (¹⁄₂), and B <i>b</i> an angle of
+(¹⁄₃), with the vertical. Through the point C, where A <i>a</i> and
+B <i>b</i> intersect one another, draw D C E vertical. Through
+A C and B respectively, draw A D, C F, and B E horizontal.
+Draw similar lines on the other side of A B, and the rectilinear
+portion of the diagram is complete.</p>
+
+<p>The curvilinear contour may be thus added:—</p>
+
+<p>Take a cut-out ellipse of (¹⁄₄), whose greater axis is equal
+to the line A B, and</p>
+
+<p><i>1st.</i> Place it upon the diagram, so that its circumference
+may be tangential to the lines C E and C F, and its greater
+axis <i>m n</i> may make an angle of (¹⁄₅) with the vertical, and trace
+its circumference.</p>
+
+<p><i>2d.</i> Place it with its circumference tangential to that of the
+first at the point m, while its greater axis (of which <i>o p</i> is a
+part) is in the horizontal, and trace the portion of its circumference
+<i>q o r</i>.</p>
+
+<p><i>3d.</i> Place it with its circumference tangential to that of the
+above at <i>v</i>, while its greater axis (of which <i>u v</i> is a part) makes
+an angle of (³⁄₁₀) with the vertical, and trace the portion of its
+circumference <i>s v t</i>.</p>
+
+<p>Thus the curvilinear contour of the body and neck are
+harmonically determined.</p>
+
+<p>The curve of the handle may be determined by the same
+ellipse placed so that its greater axis (of which <i>i k</i> is a part)
+makes an angle of (¹⁄₆) with the vertical.</p>
+
+<p><span class="pagenum"><a id="Page_84"></a>[84]</span></p>
+
+<p>Make similar tracings on the other side of A B, and the
+diagram is complete. The inscribing rectangle D G E K is
+that of (²⁄₅).</p>
+
+<p>The outline resulting from this diagram, not only is in perfect
+agreement with my recollection of the form, but with
+the measurements of the original given in the “Penny Cyclopædia;”
+of the accuracy of which there can be no doubt.
+They are stated thus:—“It is about ten inches in height, and
+beautifully curved from the top downwards; the diameter at
+the top being about three inches and a-half; at the neck or
+smallest part, two inches; at the largest (mid-height), seven
+inches; and at the bottom, five inches.”</p>
+
+<p>The harmonic elements of this beautiful form, therefore,
+appear to be the following parts of the right angle:—</p>
+
+<table>
+  <tr>
+    <th>Tonic.</th>
+    <th>Dominant.</th>
+    <th>Mediant.</th>
+    <th>Submediant.</th>
+  </tr>
+  <tr>
+    <td>(¹⁄₂)</td>
+    <td>(¹⁄₃)</td>
+    <td>(¹⁄₅)</td>
+    <td>(³⁄₁₀)</td>
+  </tr>
+  <tr>
+    <td>(¹⁄₄)</td>
+    <td>(¹⁄₆)</td>
+    <td></td>
+    <td></td>
+  </tr>
+</table>
+
+<p>When we reflect upon the variety of harmonic ellipses that
+may be described, and the innumerable positions in which
+they may be harmonically placed with respect to the horizontal
+and vertical lines, as well as upon the various modes in which
+their circumferences may be combined, the variety which may
+be introduced amongst such forms as the foregoing appears
+almost endless. My second example is that of—</p>
+
+<h3><i>An Ancient Grecian Marble Vase of a Vertical Composition.</i></h3>
+
+<p>I shall now proceed to another class of the ancient Greek
+vase, the form of which is of a more complex character. The<span class="pagenum"><a id="Page_85"></a>[85]</span>
+specimen I have chosen for the first example of this class is
+one of those so correctly measured and beautifully delineated
+by Tatham in his unequalled work.<a id="FNanchor_25" href="#Footnote_25" class="fnanchor">[25]</a> This vase is a work of
+ancient Grecian art in Parian marble, which he met with in
+the collection at the Villa Albani, near Rome. Its height is
+4 ft. 4¹⁄₂ in.</p>
+
+<div class="sidenote"><a href="#plate13">Plate XIII.</a></div>
+
+<p>The following is the formula by which I endeavour to develop
+its harmonic elements:—</p>
+
+<p>Let A B (<a href="#plate13">Plate XIII.</a>) represent the full height of this vase.
+Through B draw B D, making an angle of (¹⁄₅) with the vertical.
+Through D draw D O vertical, through A draw A C,
+making an angle of (²⁄₅); through B draw B L, making an
+angle of (¹⁄₂), and B S, making an angle of (³⁄₁₀), each with the
+vertical. Through A draw A D, through B draw B O, through
+L draw L N, through C draw C F, and through S draw S P,
+all horizontal. Through A draw A H, making an angle of
+(¹⁄₁₀) with the vertical, and through H draw H M vertical.
+Draw similar lines on the other side of A B, and the rectilinear
+portion of the diagram is complete, and its inscribing
+rectangle that of (³⁄₈).</p>
+
+<p>The curvilinear portion may thus be added—</p>
+
+<p>Take a cut-out ellipse of (¹⁄₃), whose greater axis is about
+the length of the body of the intended vase, place it with its
+lesser axis upon the line S P, and its greater axis upon the
+line D O, and trace the part <i>a b</i> of its circumference upon the
+diagram. Place the same ellipse with one of its foci upon C,
+and its greater axis upon C F, and trace its circumference
+upon the diagram. Take a cut-out ellipse of (¹⁄₅), whose
+greater axis is nearly equal to that of the ellipse already used;<span class="pagenum"><a id="Page_86"></a>[86]</span>
+place it with its greater axis upon M H, and its lesser axis
+upon L N, and trace its circumference upon the diagram.
+Make similar tracings upon the other side of A B, and the
+diagram is complete. In this, as in the other diagrams,
+the strong portions of the lines give the contour of the vase.
+The harmonic elements of this classical form, therefore, appear
+to be the right angle and its following parts:—</p>
+
+<table>
+  <tr>
+    <th>Tonic.</th>
+    <th>Dominant.</th>
+    <th>Mediant.</th>
+    <th>Submediant.</th>
+  </tr>
+  <tr>
+    <td>(¹⁄₂)</td>
+    <td>(¹⁄₃)</td>
+    <td>(²⁄₅)</td>
+    <td>(³⁄₁₀)</td>
+  </tr>
+  <tr>
+    <td></td>
+    <td></td>
+    <td>(¹⁄₅)</td>
+    <td></td>
+  </tr>
+  <tr>
+    <td></td>
+    <td></td>
+    <td>(¹⁄₁₀)</td>
+    <td></td>
+  </tr>
+</table>
+
+<p>My third example is that of—</p>
+
+<h3><i>An Ancient Grecian Vase of a Horizontal Composition.</i></h3>
+
+<p>This example belongs to the same class as the last, but it
+is of a horizontal composition. It was carefully drawn from
+the original in the museum of the Vatican by Tatham, in
+whose etchings it will be found with its ornamental decorations.
+The diagram of its harmonic elements may be constructed
+as follows:—</p>
+
+<div class="sidenote"><a href="#plate14">Plate XIV.</a></div>
+
+<p>Let A B (<a href="#plate14">Plate XIV.</a>) represent the full height of the vase.
+Through B draw B D, making an angle of (²⁄₅) with the vertical.
+Through A draw A H, A L, and A C, making respectively
+the following angles, (¹⁄₅) with the vertical, (⁴⁄₉) with the
+vertical, and (³⁄₁₀) with the horizontal. These angles determine
+the horizontal lines H B, L N, and C F, which divide the vase
+into its parts, and the inscribing rectangle D G K O is (³⁄₈).
+This completes the rectilinear portion of the diagram. The
+ellipse by which the curvilinear portion is added is one of (¹⁄₅),
+the greater axis of which, at <i>a b</i>, as also at <i>c d</i>, makes an angle<span class="pagenum"><a id="Page_87"></a>[87]</span>
+of (¹⁄₁₂) with the vertical, and the same axis at <i>e f</i> an angle of
+(¹⁄₁₂) with the horizontal.</p>
+
+<p>The harmonic elements of this vase, therefore, appear to
+be:—</p>
+
+<table>
+  <tr>
+    <th>Tonic.</th>
+    <th>Dominant.</th>
+    <th>Mediant.</th>
+    <th>Submediant.</th>
+    <th>Supertonic.</th>
+  </tr>
+  <tr>
+    <td>The Right Angle.</td>
+    <td>(¹⁄₁₂)</td>
+    <td>(²⁄₅)</td>
+    <td>(³⁄₁₀)</td>
+    <td>(⁴⁄₉)</td>
+  </tr>
+  <tr>
+    <td></td>
+    <td></td>
+    <td>(¹⁄₅)</td>
+    <td></td>
+    <td></td>
+  </tr>
+</table>
+
+<p>My remaining examples are those of—</p>
+
+<h3><i>Etruscan Vases.</i></h3>
+
+<p>Of these vases I give four examples, by which the simplicity
+of the method employed in applying the harmonic law
+will be apparent.</p>
+
+<div class="sidenote"><a href="#plate15">Plate XV.</a></div>
+
+<p>The inscribing rectangle D G E K of fig. 1, <a href="#plate15">Plate XV.</a>,
+is one of (³⁄₈), within which are arranged tracings from an
+ellipse of (³⁄₁₀), whose greater axis at <i>a b</i> makes an angle of
+(¹⁄₁₂), at <i>c d</i> an angle of (³⁄₁₀), and at <i>e f</i> an angle of (³⁄₄), with
+the vertical. The harmonic elements of the contour of this
+vase, therefore, appear to be:—</p>
+
+<table>
+  <tr>
+    <th>Tonic.</th>
+    <th>Dominant.</th>
+    <th>Subdominants.</th>
+    <th>Submediant.</th>
+  </tr>
+  <tr>
+    <td>The Right Angle.</td>
+    <td>(¹⁄₁₂)</td>
+    <td>(³⁄₄)</td>
+    <td>(³⁄₁₀)</td>
+  </tr>
+  <tr>
+    <td></td>
+    <td></td>
+    <td>(³⁄₈)</td>
+    <td></td>
+  </tr>
+</table>
+
+<p>The inscribing rectangle L M N O of fig. 2 is that of (¹⁄₂),
+within which are arranged tracings from an ellipse of (¹⁄₃),
+whose greater axis, at <i>a b</i> and <i>c d</i> respectively, makes angles of
+(¹⁄₂) and (⁴⁄₉) with the horizontal, while that at <i>e f</i> is in the
+horizontal line. The harmonic elements of the contour of
+this vase, therefore, appear to be:—</p>
+
+<table>
+  <tr>
+    <th>Tonic.</th>
+    <th>Dominant.</th>
+    <th>Subtonic.</th>
+  </tr>
+  <tr>
+    <td>(¹⁄₂)</td>
+    <td>(¹⁄₃)</td>
+    <td>(⁴⁄₉)</td>
+  </tr>
+</table>
+
+<p><span class="pagenum"><a id="Page_88"></a>[88]</span></p>
+
+<div class="sidenote"><a href="#plate16">Plate XVI.</a></div>
+
+<p>The inscribing rectangle P Q R S of fig. 1, <a href="#plate16">Plate XVI.</a>,
+is one of (⁴⁄₉), within which are arranged tracings from an
+ellipse of (³⁄₈), whose greater axis, at <i>a b</i>, <i>c d</i>, and <i>e f</i>, makes respectively
+angles of (¹⁄₆) with the horizontal, (³⁄₅) and (⁴⁄₅) with
+the vertical. Its harmonic elements, therefore, appear to be:—</p>
+
+<table>
+  <tr>
+    <th>Tonic.</th>
+    <th>Dominant.</th>
+    <th>Mediant.</th>
+    <th>Supertonic.</th>
+    <th>Subdominant.</th>
+    <th>Submediant.</th>
+  </tr>
+  <tr>
+    <td>The Right Angle.</td>
+    <td>(¹⁄₆)</td>
+    <td>(⁴⁄₅)</td>
+    <td>(⁴⁄₉)</td>
+    <td>(³⁄₈)</td>
+    <td>(³⁄₅)</td>
+  </tr>
+</table>
+
+<p>The inscribing rectangle T U V X of fig. 2 is one of (⁴⁄₉),
+within which are arranged tracings from an ellipse of (³⁄₈)
+whose greater axis at <i>a b</i> is in the vertical line, and at <i>c d</i>
+makes an angle of (¹⁄₂). The harmonic elements of the contour
+of this vase, therefore, appear to be:—</p>
+
+<table>
+  <tr>
+    <th>Tonic.</th>
+    <th>Submediant.</th>
+    <th>Supertonic.</th>
+  </tr>
+  <tr>
+    <td>(¹⁄₂)</td>
+    <td>(³⁄₈)</td>
+    <td>(⁴⁄₉)</td>
+  </tr>
+</table>
+
+<p>These four Etruscan vases, the contours of which are thus
+reduced to the harmonic law of nature, are in the British
+Museum, and engravings of them are to be found in the well-known
+work of Mr Henry Moses, Plates 4, 6, 14, and 7, respectively,
+where they are represented with their appropriate
+decorations and colours.</p>
+
+<p>To these, I add two examples of—</p>
+
+<h3><i>Ancient Grecian Ornament.</i></h3>
+
+<p>I have elsewhere shewn<a id="FNanchor_26" href="#Footnote_26" class="fnanchor">[26]</a> that the elliptic curve pervades
+the Parthenon from the entases of the column to the smallest
+moulding, and we need not, therefore, be surprised to find it<span class="pagenum"><a id="Page_89"></a>[89]</span>
+employed in the construction of the only two ornaments
+belonging to that great work.</p>
+
+<div class="sidenote"><a href="#plate17">Plate XVII.</a></div>
+
+<p>In the diagram (<a href="#plate17">Plate XVII.</a>), I endeavour to exhibit the
+geometric construction of the upper part of one of the ornamental
+apices, termed antefixæ, which surmounted the cornice
+of the Parthenon.</p>
+
+<p>The first ellipse employed is that of (¹⁄₃), whose greater axis
+<i>a b</i> is in the vertical line; the second is also that of (¹⁄₃), whose
+greater axis <i>c d</i> makes, with the vertical, an angle of (¹⁄₁₂);
+the third ellipse is the same with its major axis <i>e f</i> in the
+vertical line. Through one of the foci of this ellipse at A the
+line A C is drawn, and upon the part of the circumference C <i>e</i>,
+the number of parts, 1, 2, 3, 4, 5, 6, 7, of which the surmounting
+part of this ornament is to consist, are set off.
+That part of the circumference of the ellipse whose larger axis
+is <i>c d</i> is divided from <i>g</i> to <i>c</i> into a like number of parts. The
+third ellipse employed is one of (¹⁄₄).</p>
+
+<p>Take a cut-out ellipse of this kind, whose larger axis is
+equal in length to the inscribing rectangle. Place it with its
+vertex upon the same ellipse at <i>g</i>, so that its circumference
+will pass through C, and trace it; remove its apix first to <i>p</i>,
+then to <i>q</i>, and proceed in the same way to <i>q</i>, <i>r</i>, <i>s</i>,
+<i>t</i>, <i>u</i>, and <i>v</i>,
+so that its circumference will pass through the seven divisions
+on <i>c g</i> and <i>e</i> C: <i>v o</i>, <i>u n</i>, <i>t m</i>,
+<i>s i</i>, <i>r k</i>, <i>q j</i>, <i>p l</i>, and <i>g x</i>, are parts
+of the larger axes of the ellipses from which the curves are
+traced. The small ellipse of which the ends of the parts are
+formed is that of (¹⁄₃).</p>
+
+<div class="sidenote"><a href="#plate18">Plate XVIII.</a></div>
+
+<p>In the diagram (<a href="#plate18">Plate XVIII.</a>), I endeavour to exhibit the
+geometric construction of the ancient Grecian ornament, commonly
+called the <i>Honeysuckle</i>, from its resemblance to the flower
+of that name. The first part of the process is similar to that
+just explained with reference to the antefixæ of the Parthenon,<span class="pagenum"><a id="Page_90"></a>[90]</span>
+although the angles in some parts differ. The contour is
+determined by the circumference of an ellipse of (¹⁄₃), whose
+major axis A B makes an angle of (¹⁄₉) with the vertical, and the
+leaves or petals are arranged upon a portion of the perimeter
+of a similar ellipse whose larger axis E F is in the vertical
+line, and these parts are again arranged upon a similar ellipse
+whose larger axis C D makes an angle of (¹⁄₁₂) with the vertical.
+The first series of curved lines proceeding from 1, 2, 3, 4, 5,
+6, 7, and 8, are between K E and H C, part of the circumference
+of an ellipse of (¹⁄₃); and those between C H and A G
+are parts of the circumference of four ellipses, each of (¹⁄₃), but
+varying as to the lengths of their larger axes from 5 to 3 inches.
+The change from the convex to the concave, which produces
+the ogie forms of which this ornament is composed, takes
+place upon the line C H, and the lines <i>a b</i>, <i>c d</i>,
+<i>e f</i>, <i>g h</i>, <i>i k</i>, <i>l m</i>,
+<i>n o</i>, and <i>p q</i>, are parts of the larger axis of the four ellipses
+the circumference of which give the upper parts of the petals
+or leaves.</p>
+
+<p>This peculiar Grecian ornament is often, like the antefixæ
+of the Parthenon, combined with the curve of the spiral scroll.
+But the volute is so well understood that I have not rendered
+my diagrams more complex by adding that figure. Many
+varieties of this union are to be found in Tatham’s etchings,
+already referred to. The antefixæ of the Parthenon, and its
+only other ornament the honeysuckle, as represented on the
+soffit of the cornice, are to be found in Stewart’s “Athens.”</p>
+
+<hr class="chap x-ebookmaker-drop">
+
+<div class="chapter">
+
+<p><span class="pagenum"><a id="Page_91"></a>[91]</span></p>
+
+<h2 class="nobreak" id="APPENDIX">APPENDIX.</h2>
+
+</div>
+
+<h3 id="APPENDIX_I">No. I.</h3>
+
+<p>In pages <a href="#Page_34">34</a>, <a href="#Page_35">35</a>, and
+<a href="#Page_58">58</a>, I have reiterated an opinion advanced in several
+of my former works, viz., that, besides genius, and the study of nature, an
+additional cause must be assigned for the general excellence which characterises
+such works of Grecian art as were executed during a period commencing
+about 500 <span class="allsmcap">B.C.</span>, and ending about 200 <span class="allsmcap">B.C.</span> And that this cause
+most probably was, that the artists of that period were instructed in a
+system of fixed principles, based upon the doctrines of Pythagoras and
+Plato. This opinion has not been objected to by the generality of those
+critics who have reviewed my works; but has, however, met with an
+opponent, whose recondite researches and learned observations are worthy
+of particular attention. These are given in an essay by Mr C. Knight
+Watson, “On the Classical Authorities for Ancient Art,” which appeared in
+the <i>Cambridge Journal of Classical and Sacred Philology</i> in June 1854. As
+this essay is not otherwise likely to meet the eyes of the generality of my
+readers, and as the objections he raises to my opinion only occupy two out
+of the sixteen ample paragraphs which constitute the first part of the essay,
+I shall quote them fully:—</p>
+
+<div class="blockquote">
+
+<p>“The next name on our list is that of the famous Euphranor (<span class="allsmcap">B.C.</span> 362).
+For the fact that to the practice of sculpture and of painting he added an
+exposition of the theory, we are indebted to Pliny, who says (xxxv. 11, 40),
+‘Volumina quoque composuit de symmetria et coloribus.’ When we reflect
+on the <i>critical</i> position occupied by Euphranor in the history of Greek art,
+as a connecting link between the idealism of Pheidias and the naturalism
+of Lysippus, we can scarcely overestimate the value of a treatise on art
+proceeding from such a quarter. This is especially the case with the first of
+the two works here assigned to Euphranor. The inquiries which of late
+years have been instituted by Mr D. R. Hay of Edinburgh, on the proportions
+of the human figure, and on the natural principles of beauty as illustrated<span class="pagenum"><a id="Page_92"></a>[92]</span>
+by works of Greek art, constitute an epoch in the study of æsthetics
+and the philosophy of form. Now, in the presence of these inquiries, or of
+such less solid results as Mr Hay’s predecessors in the same field have
+elicited, it naturally becomes an object of considerable interest to ascertain
+how far these laws of form and principles of beauty were consciously
+developed in the mind, and by the chisel, of the sculptor: how far any
+such system of curves and proportions as Mr Hay’s was used by the Greek
+as a practical manual of his craft. Without in the least wishing to impugn
+the accuracy of that gentleman’s results—a piece of presumption I should
+do well to avoid—I must be permitted to doubt whether the ‘Symmetria’
+of Euphranor contained anything analogous to them in kind, or indeed equal
+in value. It must not be forgotten that the truth of Mr Hay’s theory is
+perfectly compatible with the fact, that of such theory the Greek may have
+been utterly ignorant. It is on this fact I insist: it is here that I join
+issue with Mr Hay, and with his reviewer in a recent number of <i>Blackwood’s
+Magazine</i>. Or, to speak more accurately,—while I am quite prepared to
+find that the Elgin marbles will best of all stand the test which Mr Hay has
+hitherto applied, I believe, to works of a later age, I am none the less convinced
+that it is precisely that golden age of Hellenic art to which they
+belong, precisely that first and chief of Hellenic artists by whom they were
+executed, to which and to whom any such line of research on the laws of
+form would have been pre-eminently alien. Pheidias, remember, by the
+right of primogeniture, is the ruling spirit of idealism in art. Of spontaneity
+was that idealism begotten and nurtured: by any such system as
+Mr Hay’s, that spontaneity would be smothered and paralysed. Pheidias
+copied an idea in his own mind—‘Ipsius in mente insidebat species pulchritudinis
+eximia quædam’ (<i>Cic.</i>);—later ages copied <i>him</i>. He created: they
+criticised. He was the author of Iliads: they the authors of Poetics. Doubtless,
+if you unsphere the <i>spirit</i> of Mr Hay’s theories, you will find nothing
+discordant with what I have here said. That is a sound view of Beauty
+which makes it consist in that due subordination of the parts to the whole,
+that due relation of the parts to each other, which Mendelssohn had in his
+mind when he said that the essence of beauty was ‘unity in variety’—variety
+beguiling the imagination, the perception of unity exercising the
+thewes and sinews of the intellect. On such a view of beauty, Mr Hay’s
+theory may, <i>in spirit</i>, be said to rest. But here, as in higher things, it is the
+letter that killeth, while the spirit giveth life. And accordingly I must
+enter a protest against any endeavour to foist upon the palmy days of
+Hellenic art systems of geometrical proportions incompatible, as I believe,
+with those higher and broader principles by which the progress of ancient
+sculpture was ordered and governed—systems which will bear nothing of
+that ‘felicity and chance by which’—and not by rule—‘Lord Bacon believed
+that a painter may make a better face than ever was:’ systems which
+take no account of that fundamental distinction between the schools of<span class="pagenum"><a id="Page_93"></a>[93]</span>
+Athens and of Argos, and their respective disciples and descendants, without
+which you will make nonsense of the pages of Pliny, and—what is worse—sense
+of the pages of his commentators;—systems, in short, which may have
+their value as instruments for the education of the eye, and for instructions
+in the arts of design, but must be cast aside as matters of learned trifling
+and curious disputation, where they profess to be royal roads to art, and to
+map the mighty maze of a creative mind. And even as regards the application
+of such a system of proportions to those works of sculpture which are
+posterior to the Pheidian age, only partial can have been the prevalence
+which it or any other <i>one</i> system can have obtained. The discrepancies
+of different artists in the treatment of what was called, technically called,
+<i>Symmetria</i> (as in the title of Euphranor’s work) were, by the concurrent
+testimony of all ancient writers, far too salient and important to warrant the
+supposition of any uniform scale of proportions, as advocated by Mr Hay.
+Even in Egypt, where one might surely have expected that such uniformity
+would have been observed with far greater rigour than in Greece, the discoveries
+of Dr Lepsius (<i>Vorläufige Nachricht</i>, Berlin, 1849) have elicited
+three totally different κανόνες, one of which is identical with the system of
+proportions of the human figure detailed in Diodorus. While we thus venture
+to differ from Mr Hay on the historical data he has mixed up with his
+inquiries, we feel bound to pay him a large and glad tribute of praise for
+having devised a system of proportions which rises superior to the idiosyncracies
+of different artists, which brings back to one common type the
+sensations of eye and ear, and so makes a giant stride towards that <i>codification</i>,
+if I may so speak, of the laws of the universe which it is the business
+of the science to effect. I have no hesitation in saying, that, for scientific
+precision of method and importance of results, Albert Durer, Da Vinci, and
+Hogarth, not to mention less noteworthy writers, must all yield the palm
+to Mr Hay.</p>
+
+<p>“I am quite aware that in the digression I have here allowed myself, on
+systems of proportions prevalent among ancient artists, and on the probable
+contents of such treatises as that of Euphranor, <i>De Symmetria</i>, I have laid
+myself open to the charge of treating an intricate question in a very perfunctory
+way. At present the exigencies of the subject more immediately
+in hand allow me only to urge in reply, that, as regards the point at issue—I
+mean the ‘solidarité’ between theories such as Mr Hay’s and the practice
+of Pheidias—the <i>onus probandi</i> rests with my adversaries.”</p>
+</div>
+
+<p>I am bound, in the first place, gratefully to acknowledge the kind and
+complimentary notice which, notwithstanding our difference of opinion, this
+author has been pleased to take of my works; and, in the second, to assure
+him that if any of them profess to be “royal roads to art,” or to “map
+the mighty maze of a creative mind,” they certainly profess to do more
+than I ever meant they should; for I never entertained the idea that a<span class="pagenum"><a id="Page_94"></a>[94]</span>
+system of æsthetic culture, even when based upon a law of nature, was
+capable of effecting any such object. But I doubt not that this too common
+misapprehension of the scope and tendency of my works must arise from a
+want of perspicuity in my style.</p>
+
+<p>I have certainly, on one occasion,<a id="FNanchor_27" href="#Footnote_27" class="fnanchor">[27]</a> gone the length of stating that
+as poetic genius must yield obedience to the rules of rhythmical measure,
+even in the highest flights of her inspirations; and musical genius must, in
+like manner, be subject to the strictly defined laws of harmony in the most
+delicate, as well as in the most powerfully grand of her compositions; so
+must genius, in the formative arts, either consciously or unconsciously have
+clothed her creations of ideal beauty with proportions strictly in accordance
+with the laws which nature has set up as inflexible standards. If, therefore,
+the laws of proportion, in their relation to the arts of design, constitute
+the harmony of geometry, as definitely as those that are applicable to poetry
+and music produce the harmony of acoustics; the former ought, certainly,
+to hold the same relative position in those arts which are addressed to the
+eye, that is accorded to the latter in those which are addressed to the ear.
+Until so much science be brought to bear upon the arts of design, the
+student must continue to copy from individual and imperfect objects in
+nature, or from the few existing remains of ancient Greek art, in total ignorance
+of the laws by which their proportions are produced, and, what is equally
+detrimental to art, the accuracy of all criticism must continue to rest upon
+the indefinite and variable basis of mere opinion.</p>
+
+<p>It cannot be denied that men of great artistic genius are possessed of an
+intuitive feeling of appreciation for what is beautiful, by means of which
+they impart to their works the most perfect proportions, independently of
+any knowledge of the definite laws which govern that species of beauty.
+But they often do so at the expense of much labour, making many trials
+before they can satisfy themselves in imparting to them the true proportions
+which their minds can conceive, and which, along with those other qualities of
+expression, action, or attitude, which belong more exclusively to the province
+of genius. In such cases, an acquaintance with the rules which constitute
+the science of proportion, instead of proving fetters to genius, would doubtless
+afford her such a vantage ground as would promote the more free exercise of
+her powers, and give confidence and precision in the embodiment of her
+inspirations; qualities which, although quite compatible with genius, are not
+always intuitively developed along with that gift.</p>
+
+<p>It is also true that the operations of the conceptive faculty of the mind
+are uncontrolled by definite laws, and that, therefore, there cannot exist any
+rules by the inculcation of which an ordinary mind can be imbued with
+genius sufficient to produce works of high art. Nevertheless, such a mind may
+be improved in its perceptive faculty by instruction in the science of proportion,
+so as to be enabled to exercise as correct and just an appreciation of<span class="pagenum"><a id="Page_95"></a>[95]</span>
+the conceptions of others, in works of plastic art, as that manifested by the
+educated portion of mankind in respect to poetry and music. In short, it
+appears that, in those arts which are addressed to the ear, men of genius
+communicate the original conceptions of their minds under the control of
+certain scientific laws, by means of which the educated easily distinguish the
+true from the false, and by which the works of the poet and musical composer
+may be placed above mere imitations of nature, or of the works of
+others; while, in those arts that are addressed to the eye in their own peculiar
+language, such as sculpture, architecture, painting, and ornamental
+design, no such laws are as yet acknowledged.</p>
+
+<p>Although I am, and ever have been, far from endeavouring “to foist upon
+the palmy days of Hellenic art” any system incompatible with those higher
+and more intellectual qualities which genius alone can impart; yet, from
+what has been handed down to us by writers on the subject, meagre as it is,
+I cannot help continuing to believe that, along with the physical and metaphysical
+sciences, æsthetic science was taught in the early schools of Greece.</p>
+
+<p>I shall here take the liberty to repeat the proofs I advanced in a former
+work as the ground of this belief, and to which the author, from whose
+essay I have quoted, undoubtedly refers. It is well known that, in the time
+of Pythagoras, the treasures of science were veiled in mystery to all but the
+properly initiated, and the results of its various branches only given to the
+world in the works of those who had acquired this knowledge. So strictly
+was this secresy maintained amongst the disciples and pupils of Pythagoras,
+that any one divulging the sacred doctrines to the profane, was expelled the
+community, and none of his former associates allowed to hold further intercourse
+with him; it is even said, that one of his pupils incurred the displeasure
+of the philosopher for having published the solution of a problem
+in geometry.<a id="FNanchor_28" href="#Footnote_28" class="fnanchor">[28]</a> The difficulty, therefore, which is expressed by writers,
+shortly after the period in which Pythagoras lived, regarding a precise
+knowledge of his theories, is not to be wondered at, more especially when it
+is considered that he never committed them to writing. It would appear,
+however, that he proceeded upon the principle, that the order and beauty so
+apparent throughout the whole universe, must compel men to believe in,
+and refer them to, an intelligible cause. Pythagoras and his disciples sought
+for properties in the science of numbers, by the knowledge of which they
+might attain to that of nature; and they conceived those properties to be
+indicated in the phenomena of sonorous bodies. Observing that Nature
+herself had thus irrevocably fixed the numerical value of the intervals of
+musical tones, they justly concluded that, as she is always uniform in her
+works, the same laws must regulate the general system of the universe.<a id="FNanchor_29" href="#Footnote_29" class="fnanchor">[29]</a><span class="pagenum"><a id="Page_96"></a>[96]</span>
+Pythagoras, therefore, considered numerical proportion as the great principle
+inherent in all things, and traced the various forms and phenomena of the
+world to numbers as their basis and essence.</p>
+
+<p>How the principles of numbers were applied in the arts is not recorded, farther
+than what transpires in the works of Plato, whose doctrines were from the
+school of Pythagoras. In explaining the principle of beauty, as developed in
+the elements of the material world, he commences in the following words:—“But
+when the Artificer began to adorn the universe, he first of all figured
+with forms and numbers, fire and earth, water and air—which possessed,
+indeed, certain traces of the true elements, but were in every respect so constituted
+as it becomes anything to be from which Deity is absent. But we
+should always persevere in asserting that Divinity rendered them, as much
+as possible, the most beautiful and the best, when they were in a state of
+existence opposite to such a condition.” Plato goes on further to say, that
+these elementary bodies must have forms; and as it is necessary that every
+depth should comprehend the nature of a plane, and as of plane figures the
+triangle is the most elementary, he adopts two triangles as the originals or
+representatives of the isosceles and the scalene kinds. The first triangle of
+Plato is that which forms the half of the square, and is regulated by the
+number, 2; and the second, that which forms the half of the equilateral
+triangle, which is regulated by the number, 3; from various combinations
+of these, he formed the bodies of which he considered the elements to be
+composed. To these elementary figures I have already referred.</p>
+
+<p>Vitruvius, who studied architecture ages after the arts of Greece had been
+buried in the oblivion which succeeded her conquest, gives the measurements
+of various details of monuments of Greek art then existing. But he
+seems to have had but a vague traditionary knowledge of the principle of
+harmony and proportion from which these measurements resulted. He says—“The
+several parts which constitute a temple ought to be subject to the
+laws of symmetry; the principles of which should be familiar to all who
+profess the science of architecture. Symmetry results from proportion,
+which, in the Greek language, is termed analogy. Proportion is the commensuration
+of the various constituent parts with the whole; in the
+existence of which symmetry is found to consist. For no building can
+possess the attributes of composition in which symmetry and proportion are
+disregarded; nor unless there exist that perfect conformation of parts which
+may be observed in a well-formed human being.” After going at some
+length into details, he adds—“Since, therefore, the human figure appears to
+have been formed with such propriety, that the several members are commensurate
+with the whole, the artists of antiquity (meaning those of Greece
+at the period of her highest refinement) must be allowed to have followed
+the dictates of a judgment the most rational, when, transferring to works of
+art principles derived from nature, every part was so regulated as to bear
+a just proportion to the whole. Now, although the principles were universally<span class="pagenum"><a id="Page_97"></a>[97]</span>
+acted upon, yet they were more particularly attended to in the construction
+of temples and sacred edifices, the beauties or defects of which
+were destined to remain as a perpetual testimony of their skill or of their
+inability.”</p>
+
+<p>Vitruvius, however, gives no explanation of this ancient principle of proportion,
+as derived from the human form; but plainly shews his uncertainty
+upon the subject, by concluding this part of his essay in the following words:
+“If it be true, therefore, that the decenary notation was suggested by the
+members of man, and that the laws of proportion arose from the relative
+measures existing between certain parts of each member and the whole body,
+it will follow, that those are entitled to our commendation who, in building
+temples to their deities, proportioned the edifices, so that the several parts
+of them might be commensurate with the whole.”
+It thus appears certain that the Grecians, at the period of their highest
+excellence, had arrived at a knowledge of some definite mathematical law
+of proportion, which formed a standard of perfectly symmetrical beauty,
+not only in the representation of the human figure in sculpture and painting,
+but in architectural design, and indeed in all works where beauty of
+form and harmony of proportion constituted excellence. That this law was
+not deduced from the proportions of the human figure, as supposed by
+Vitruvius, but had its origin in mathematical science, seems equally certain;
+for in no other way can we satisfactorily account for the proportions of the
+beau ideal forms of the ancient Greek deities, or of those of their architectural
+structures, such as the Parthenon, the temple of Theseus, &amp;c., or for
+the beauty that pervades all the other formative art of the period.</p>
+
+<p>This system of geometrical harmony, founded, as I have shewn it to be,
+upon numerical relations, must consequently have formed part of the Greek
+philosophy of the period, by means of which the arts began to progress
+towards that great excellence which they soon after attained. A little
+further investigation will shew, that immediately after this period a theory
+connected with art was acknowledged and taught, and also that there existed
+a Science of Proportion.</p>
+
+<p>Pamphilus, the celebrated painter, who flourished about four hundred
+years before the Christian era, from whom Apelles received the rudiments
+of his art, and whose school was distinguished for scientific cultivation,
+artistic knowledge, and the greatest accuracy in drawing, would admit no
+pupil unacquainted with geometry.<a id="FNanchor_30" href="#Footnote_30" class="fnanchor">[30]</a> The terms upon which he engaged
+with his students were, that each should pay him one talent (£225 sterling)
+previous to receiving his instructions; for this he engaged “to give them,
+<i>for ten years</i>, lessons founded on an excellent theory.”<a id="FNanchor_31" href="#Footnote_31" class="fnanchor">[31]</a></p>
+
+<p>It was by the advice of Pamphilus that the magistrates of Sicyon ordained
+that the study of drawing should constitute part of the education of the<span class="pagenum"><a id="Page_98"></a>[98]</span>
+citizens—“a law,” says the Abbé Barthélémie, “which rescued the fine arts
+from servile hands.”</p>
+
+<p>It is stated of Parrhasius, the rival of Zeuxis, who flourished about the
+same period as Pamphilus, that he accelerated the progress of art by purity
+and correctness of design; “for he was acquainted with the science of
+Proportions. Those he gave his gods and heroes were so happy, that artists
+did not hesitate to adopt them.” Parrhasius, it is also stated, was so
+admired by his contemporaries, that they decreed him the name of Legislator.<a id="FNanchor_32" href="#Footnote_32" class="fnanchor">[32]</a>
+The whole history of the arts in Egypt and Greece concurs to prove
+that they were based on geometric precision, and were perfected by a continued
+application of the same science; while in all other countries we find
+them originating in rude and misshapen imitations of nature.</p>
+
+<p>In the earliest stages of Greek art, the gods—then the only statues—were
+represented in a tranquil and fixed posture, with the features exhibiting
+a stiff inflexible earnestness, their only claim to excellence being symmetrical
+proportion; and this attention to geometric precision continued as art
+advanced towards its culminating point, and was thereafter still exhibited
+in the neatly and regularly folded drapery, and in the curiously braided and
+symmetrically arranged hair.<a id="FNanchor_33" href="#Footnote_33" class="fnanchor">[33]</a></p>
+
+<p>These researches, imperfect as they are, cannot fail to exhibit the great
+contrast that exists between the system of elementary education in art practised
+in ancient Greece, and that adopted in this country at the present
+period. But it would be of very little service to point out this contrast,
+were it not accompanied by some attempt to develop the principles which
+seem to have formed the basis of this excellence amongst the Greeks.</p>
+
+<p>But in making such an attempt, I cannot accuse myself of assuming anything
+incompatible with the free exercise of that spontaneity of genius which
+the learned essayist says is the parent and nurse of idealism. For it is in
+no way more incompatible with the free exercise of artistic genius, that
+those who are so gifted should have the advantage of an elementary education
+in the science of æsthetics, than it is incompatible with the free exercise
+of literary or poetic genius, that those who possess it should have the
+advantage of such an elementary education in the science of philology as
+our literary schools and colleges so amply afford.</p>
+
+<p><span class="pagenum"><a id="Page_99"></a>[99]</span></p>
+
+<h3 id="APPENDIX_II">No. II.</h3>
+
+<p>The letter from which I have made a quotation at page <a href="#Page_42">42</a>, arose out of
+the following circumstance:—In order that my theory, as applied to the
+orthographic beauty of the Parthenon, might be brought before the highest
+tribunal which this country afforded, I sent a paper upon the subject,
+accompanied by ample illustrations, to the Royal Institute of British Architects,
+and it was read at a meeting of that learned body on the 7th of
+February 1853; at the conclusion of which, Mr Penrose kindly undertook
+to examine my theoretical views, in connexion with the measurements he
+had taken of that beautiful structure by order of the Dilettanti Society, and
+report upon the subject to the Royal Institute. This report was published
+by Mr Penrose, vol. xi., No. 539 of <i>The Builder</i>, and the letter from which
+I have quoted appeared in No. 542 of the same journal. It was as
+follows:—</p>
+
+<div class="blockquote">
+
+<p class="center">“GEOMETRICAL RELATIONS IN ARCHITECTURE.</p>
+
+<p>“Will you allow me, through the medium of your columns, to thank Mr
+Penrose for his testimony to the truth of Mr Hay’s revival of Pythagoras?
+The dimensions which he gives are to me the surest verification of the
+theory that I could have desired. The minute discrepancies form that very
+element of practical incertitude, both as to execution and direct measurement,
+which always prevails in materialising a mathematical calculation under
+such conditions.</p>
+
+<p>“It is time that the scattered computations by which architecture has
+been analysed—more than enough—be synthetised into those formulæ
+which, as Mrs Somerville tells us, ‘are emblematic of omniscience.’ The
+young architects of our day feel trembling beneath their feet the ground
+whence men are about to evoke the great and slumbering corpse of art.
+Sir, it is food of this kind a reviving poetry demands.</p>
+
+<div class="poetry-container">
+<div class="poetry">
+  <div class="stanza">
+    <div class="verse indent8">——‘Give us truths,</div>
+    <div class="verse indent0">For we are weary of the surfaces,</div>
+    <div class="verse indent0">And die of inanition.’</div>
+  </div>
+</div>
+</div>
+
+<p>“I, for one, as I listen to such demonstrations, whose scope extends with
+every research into them, feel as if listening to those words of Pythagoras,
+which sowed in the mind of Greece the poetry whose manifestation in beauty
+has enchained the world in worship ever since its birth. And I am sure
+that in such a quarter, and in such thoughts, <i>we</i> must look for the first
+shining of that lamp of art, which even now is prepared to burn.</p>
+
+<p>“I know that this all sounds rhapsodical; but I know also that until the
+architect becomes a poet, and not a tradesman, we may look in vain for<span class="pagenum"><a id="Page_100"></a>[100]</span>
+architecture: and I know that valuable as isolated and detailed investigations
+are in their proper bearings, yet that such purposes and bearings are to
+be found in the enunciation of principles sublime as the generalities of
+‘mathematical beauty.’</p>
+
+<p class="right">“<span class="smcap">Autocthon.</span>”</p>
+
+</div>
+
+<h3 id="APPENDIX_III">No. III.</h3>
+
+<p>Of the work alluded to at page <a href="#Page_58">58</a> I was favoured with two opinions—the
+one referring to the theory it propounds, and the other to its anatomical
+accuracy—both of which I have been kindly permitted to publish.</p>
+
+<p>The first is from Sir <span class="smcap">William Hamilton</span>, Bart., professor of logic and
+metaphysics in the University of Edinburgh, and is as follows:—</p>
+
+<div class="blockquote">
+
+<p>“Your very elegant volume is to me extremely interesting, as affording
+an able contribution to what is the ancient, and, I conceive, the true theory
+of the Beautiful. But though your doctrine coincides with the one prevalent
+through all antiquity, it appears to me quite independent and original
+in you; and I esteem it the more, that it stands opposed to the hundred
+one-sided and exclusive views prevalent in modern times.—<i>16 Great King
+Street, March 5, 1849.</i>”</p>
+</div>
+
+<p>The second is from <span class="smcap">John Goodsir</span>, Esq., professor of anatomy in the
+University of Edinburgh, and is as follows:—</p>
+
+<div class="blockquote">
+
+<p>“I have examined the plates in your work on the proportions of the
+human head and countenance, and find the head you have given as typical
+of human beauty to be anatomically correct in its structure, only differing
+from ordinary nature in its proportions being more mathematically precise,
+and consequently more symmetrically beautiful.—<i>College, Edinburgh, 17th
+April 1849.</i>”</p>
+</div>
+
+<h3 id="APPENDIX_IV">No. IV.</h3>
+
+<p>I shall here shew, as I have done in a former work, how the curvilinear
+outline of the figure is traced upon the rectilinear diagrams by portions of
+the ellipse of (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆).</p>
+
+<div class="sidenote"><a href="#plate19">Plate XIX.</a></div>
+
+<p>The outline of the head and face, from points (1) to (3) (fig. 1, <a href="#plate19">Plate XIX.</a>),
+takes the direction of the two first curves of the diagram. From point (3),<span class="pagenum"><a id="Page_101"></a>[101]</span>
+the outline of the sterno-mastoid muscle continues to (4), where, joining the
+outline of the trapezius muscle, at first concave, it becomes convex after
+passing through (5), reaches the point (6), where the convex outline of the
+deltoid muscle commences, and, passing through (7), takes the outline of the
+arm as far as (8). The outline of the muscles on the side, the latissimus
+dorsi and serratus magnus, commences under the arm at the point (9), and
+joins the outline of the oblique muscle of the abdomen by a concave curve
+at (10), which, rising into convexity as it passes through the points (11) and
+(12), ends at (13), where it joins the outline of the gluteus medius muscle.
+The outline of this latter muscle passes convexly through the point (14), and
+ends at (15), where the outline of the tensor vaginæ femoris and vastus
+externus muscle of the thigh commences. This convex outline joins the
+concave outline of the biceps of the thigh at (16), which ends in that of the
+slight convexity of the condyles of the thigh-bone at (17). From this point,
+the outline of the outer surface of the leg, which includes the biceps, peroneus
+longus, and soleus muscles, after passing through the point (18), continues
+convexly to (19), where the concave outline of the tendons of the peroneus
+longus continues to (20), whence the outline of the outer ankle and foot
+commences.</p>
+
+<p>The outline of the mamma and fold of the arm-pit commences at (21),
+and passes through the points (22) and (23). The circle at (24) is the outline
+of the areola, in the centre of which the nipple is placed.</p>
+
+<p>The outline of the pubes commences at (25), and ends at the point (26),
+from which the outline of the inner surface of the thigh proceeds over the
+gracilis, the sartorius, and vastus internus muscles, until it meets the internal
+face of the knee-joint at (27), the outline of which ends at (28). The outline
+of the inside of the leg commences by proceeding over the gastrocnemius
+muscle as far as (29), where it meets that of the soleus muscle, and continues
+over the tendons of the heel until it meets the outline of the inner
+ankle and foot at (30).</p>
+
+<p>The outline of the outer surface of the arm, as viewed in front, proceeds
+from (8) over the remainder of the deltoid, in which there is here a slight
+concavity, and, next, from (31) over the biceps muscle till (32), where it
+takes the line of the long supinator, and passing concavely, and almost
+imperceptibly, into the long and short radial extensor muscles, reaches the
+wrist at (33). The outline of the inner surface of the arm from opposite
+(9) commences by passing over the triceps extensor, which ends at (34),
+then over the gentle convexity of the condyles of the bone of the arm at
+(35), and, lastly, over the flexor sublimis which ends at the wrist-joint (36).</p>
+
+<p>The outline of the front of the figure commences at the point (1), (fig. 2,
+<a href="#plate02">Plate II.</a>), and, passing almost vertically over the platzsma-myoidis muscles,
+reaches the point (2), where the neck ends by a concave curve. From (2)
+the outline rises convexly over the ends of the clavicles, and continues so
+over the pectoral muscle till it reaches (3), where the mamma swells out<span class="pagenum"><a id="Page_102"></a>[102]</span>
+convexly to (4), and returns convexly towards (5), where the curve becomes
+concave. From (5) the outline follows the undulations of the rectus muscle
+of the abdomen, concave at the points (6) and (7), and having its greatest
+convexity at (8). This series of curves ends with a slight concavity at the
+point (9), where the horizontal branch of the pubes is situated, over which
+the outline is convex and ends at (10).</p>
+
+<p>The outline of the thigh commences at the point (11) with a slight concave
+curve, and then swells out convexly over the extensors of the leg, and,
+reaching (12), becomes gently concave, and, passing over the patella at (13),
+becomes again convex until it reaches the ligament of that bone, where it
+becomes gently concave towards the point (14), whence it follows the slightly
+convex curve of the shin-bone, and then, becoming as slightly concave, ends
+with the muscles in front of the leg at (15).</p>
+
+<p>The outline of the back commences at the point (16), and, following with
+a concave curve the muscles of the neck as far as (17), swells into a convex
+curve over the trapezius muscle towards the point (18); passing through
+which point, it continues to swell outward until it reaches (19), half way
+between (18) and (20); whence the convexity, becoming less and less, falls
+into the concave curve of the muscles of the loins at (21), and passing
+through the point (22), it rises into convexity. It then passes through the
+point (23), follows the outline of the gluteus maximus, the convex curve of
+which rises to the point (24), and then returns inwards to that of (25),
+where it ends in the fold of the hip. From this point the outline follows
+the curve of the hamstring muscles by a slight concavity as far as (26), and
+then, becoming gently convex, it reaches (27); whence it becomes again
+gently concave, with a slight indication of the condyle of the thigh-bone at
+(28), and, reaching (29), follows the convex curve of the gastrocnemius
+muscle through the point (30), and falling into the convex curve of the
+tendo Achilles at (31), ends in the concavity over the heel at (32).</p>
+
+<p>The outline of the front of the arm commences at the point (33), by a
+gentle concavity at the arm-pit, and then swells out in a convex curve
+over the biceps, reaching (34), where it becomes concave, and passing
+through (35), again becomes convex in passing over the long supinator, and,
+becoming gently concave as it passes the radial extensors, rises slightly at
+(36), and ends at (37), where the outline of the wrist commences. The
+outline of the back of the arm commences with a concave curve at (38),
+which becomes convex as it passes from the deltoid to the long extensor
+and ends at the elbow (39), from below which the outline follows the convex
+curve of the extensor ulnaris, reaching the wrist at the point (40).</p>
+
+<p>It will be seen that the various undulations of the outline are regulated
+by points which are determined generally by the intersections and sometimes
+by directions and extensions of the lines of the diagram, in the same
+manner in which I shewed proportion to be imparted, in a late work, to the
+osseous structure. The mode in which the curves of (¹⁄₂), (¹⁄₃), (¹⁄₄), (¹⁄₅), and<span class="pagenum"><a id="Page_103"></a>[103]</span>
+(¹⁄₆) are thus so harmoniously blended in the outline of the female figure,
+only remains to be explained.</p>
+
+<p>The curves which compose the outline of the female form are therefore
+simply those of (¹⁄₂), (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆).</p>
+
+<p>Manner in which these curves are disposed in the lateral outline (figure 1,
+<a href="#plate19">Plate XIX.</a>):—</p>
+
+<table>
+  <tr>
+    <th></th>
+    <th></th>
+    <th colspan="3">Points.</th>
+    <th>Curves.</th>
+  </tr>
+  <tr>
+    <td class="tdl">Head</td>
+    <td>from</td>
+    <td class="tdr">1</td>
+    <td>to</td>
+    <td class="tdr">2</td>
+    <td>(¹⁄₂)</td>
+  </tr>
+  <tr>
+    <td class="tdl">Face</td>
+    <td>”</td>
+    <td class="tdr">2</td>
+    <td>”</td>
+    <td class="tdr">3</td>
+    <td>(¹⁄₃)</td>
+  </tr>
+  <tr>
+    <td class="tdl">Neck</td>
+    <td>”</td>
+    <td class="tdr">3</td>
+    <td>”</td>
+    <td class="tdr">4</td>
+    <td>(¹⁄₅)</td>
+  </tr>
+  <tr>
+    <td class="tdl">Shoulder</td>
+    <td>”</td>
+    <td class="tdr">4</td>
+    <td>”</td>
+    <td class="tdr">6</td>
+    <td>(¹⁄₆)</td>
+  </tr>
+  <tr>
+    <td class="tdl"><span class="ditto">”</span></td>
+    <td>”</td>
+    <td class="tdr">6</td>
+    <td>”</td>
+    <td class="tdr">8</td>
+    <td>(¹⁄₄)</td>
+  </tr>
+  <tr>
+    <td class="tdl">Trunk</td>
+    <td>”</td>
+    <td class="tdr">9</td>
+    <td>”</td>
+    <td class="tdr">15</td>
+    <td>(¹⁄₄)</td>
+  </tr>
+  <tr>
+    <td class="tdl"><span class="ditto">”</span></td>
+    <td>”</td>
+    <td class="tdr">21</td>
+    <td>”</td>
+    <td class="tdr">24</td>
+    <td>(¹⁄₂)</td>
+  </tr>
+  <tr>
+    <td class="tdl">Outer surface of thigh and leg</td>
+    <td>”</td>
+    <td class="tdr">15</td>
+    <td>”</td>
+    <td class="tdr">20</td>
+    <td>(¹⁄₆)</td>
+  </tr>
+  <tr>
+    <td class="tdl">Inner surface of thigh and leg</td>
+    <td>”</td>
+    <td class="tdr">25</td>
+    <td>”</td>
+    <td class="tdr">30</td>
+    <td>(¹⁄₆)</td>
+  </tr>
+  <tr>
+    <td class="tdl">Outer surface of the arm</td>
+    <td>”</td>
+    <td class="tdr">8</td>
+    <td>”</td>
+    <td class="tdr">33</td>
+    <td>(¹⁄₆)</td>
+  </tr>
+  <tr>
+    <td class="tdl">Inner surface of the arm</td>
+    <td>”</td>
+    <td class="tdr">9</td>
+    <td>”</td>
+    <td class="tdr">36</td>
+    <td>(¹⁄₆)</td>
+  </tr>
+</table>
+
+<p>Manner in which they are disposed in the outline (figure 2, <a href="#plate19">Plate XIX.</a>):—</p>
+
+<table>
+  <tr>
+    <th></th>
+    <th></th>
+    <th colspan="3">Points.</th>
+    <th>Curves.</th>
+  </tr>
+  <tr>
+    <td class="tdl">Front of neck</td>
+    <td>from</td>
+    <td class="tdr">1</td>
+    <td>to</td>
+    <td class="tdr">2</td>
+    <td>(¹⁄₆)</td>
+  </tr>
+  <tr>
+    <td class="tdl"><span class="ditto">”</span> trunk</td>
+    <td>”</td>
+    <td class="tdr">2</td>
+    <td>”</td>
+    <td class="tdr">10</td>
+    <td>(¹⁄₄)</td>
+  </tr>
+  <tr>
+    <td class="tdl">Back of neck</td>
+    <td>”</td>
+    <td class="tdr">16</td>
+    <td>”</td>
+    <td class="tdr">18</td>
+    <td>(¹⁄₆)</td>
+  </tr>
+  <tr>
+    <td class="tdl"><span class="ditto">”</span> trunk</td>
+    <td>”</td>
+    <td class="tdr">18</td>
+    <td>”</td>
+    <td class="tdr">23</td>
+    <td>(¹⁄₄)</td>
+  </tr>
+  <tr>
+    <td class="tdl"><span class="ditto">”</span> ”</td>
+    <td>”</td>
+    <td class="tdr">23</td>
+    <td>”</td>
+    <td class="tdr">25</td>
+    <td>(¹⁄₃)</td>
+  </tr>
+  <tr>
+    <td class="tdl">Front of thigh and leg</td>
+    <td>”</td>
+    <td class="tdr">11</td>
+    <td>”</td>
+    <td class="tdr">13</td>
+    <td>(¹⁄₄)</td>
+  </tr>
+  <tr>
+    <td class="tdl"><span class="ditto">”</span> ” <span class="ditto">”</span></td>
+    <td>”</td>
+    <td class="tdr">13</td>
+    <td>”</td>
+    <td class="tdr">15</td>
+    <td>(¹⁄₆)</td>
+  </tr>
+  <tr>
+    <td class="tdl">Back of thigh and leg</td>
+    <td>”</td>
+    <td class="tdr">25</td>
+    <td>”</td>
+    <td class="tdr">32</td>
+    <td>(¹⁄₆)</td>
+  </tr>
+  <tr>
+    <td class="tdl">Front of the arm</td>
+    <td>”</td>
+    <td class="tdr">33</td>
+    <td>”</td>
+    <td class="tdr">37</td>
+    <td>(¹⁄₆)</td>
+  </tr>
+  <tr>
+    <td class="tdl">Back of the arm</td>
+    <td>”</td>
+    <td class="tdr">38</td>
+    <td>”</td>
+    <td class="tdr">40</td>
+    <td>(¹⁄₆)</td>
+  </tr>
+  <tr>
+    <td class="tdl">Foot</td>
+    <td>”</td>
+    <td class="tdr">0</td>
+    <td>”</td>
+    <td class="tdr">0</td>
+    <td>(¹⁄₆)</td>
+  </tr>
+</table>
+
+<div class="sidenote"><a href="#plate20">Plate XX.</a></div>
+
+<p>In order to exemplify more clearly the manner in which these various
+curves appear in the outline of the figure, I give in <a href="#plate20">Plate XX.</a> the whole
+curvilinear figures, complete, to which these portions belong that form the
+outline of the sides of the head, neck, and trunk, and of the outer surface of
+the thighs and legs.</p>
+
+<p>The various angles which the axes of these ellipses form with the vertical,
+will be found amongst other details in the works I have just referred to.</p>
+
+<p><span class="pagenum"><a id="Page_104"></a>[104]</span></p>
+
+<h3 id="APPENDIX_V">No. V.</h3>
+
+<p>At page <a href="#Page_85">85</a> I have remarked upon the variety that may be introduced
+into any particular form of vase; and, in order to give the reader an idea
+of the ease with which this may be done without violating the harmonic
+law, I shall here give three examples:—</p>
+
+<div class="sidenote"><a href="#plate21">Plate XXI.</a></div>
+
+<p>The first of these (<a href="#plate21">Plate XXI.</a>) differs from the Portland vase, in the concave
+curve of the neck flowing more gradually into the convex curve of the
+body.</p>
+
+<div class="sidenote"><a href="#plate22">Plate XXII.</a></div>
+
+<p>The second (<a href="#plate22">Plate XXII.</a>) differs from the same vase in the same change
+of contour, as also in being of a smaller diameter at the top and at the
+bottom.</p>
+
+<div class="sidenote"><a href="#plate23">Plate XXIII.</a></div>
+
+<p>The third (<a href="#plate23">Plate XXIII.</a>) is the most simple arrangement of the elliptic
+curve by which this kind of form may be produced; and it differs from the
+Portland vase in the relative proportions of height and diameter, and in
+having a fuller curve of contour.</p>
+
+<p>The following comparison of the angles employed in these examples, with
+the angles employed in the original, will shew that the changes of contour
+in these forms, arise more from the mode in which the angles are arranged
+than in a change of the angles themselves:—</p>
+
+<table>
+  <tr>
+    <th></th>
+    <th>Line</th>
+    <th></th>
+    <th>Line</th>
+    <th></th>
+    <th>Line</th>
+    <th></th>
+    <th>Line</th>
+    <th></th>
+    <th>Line</th>
+    <th></th>
+    <th>Line</th>
+    <th></th>
+    <th></th>
+    <th></th>
+    <th></th>
+    <th></th>
+  </tr>
+  <tr>
+    <td class="nw">Plate VIII.</td>
+    <td><i>A&nbsp;C</i></td>
+    <td>(¹⁄₂)</td>
+    <td><i>B&nbsp;C</i></td>
+    <td>(¹⁄₃)</td>
+    <td><i>o&nbsp;p</i></td>
+    <td>(H)</td>
+    <td><i>v&nbsp;u</i></td>
+    <td>(³⁄₁₀)</td>
+    <td><i>m&nbsp;n</i></td>
+    <td>(¹⁄₃)</td>
+    <td><i>i&nbsp;k</i></td>
+    <td>(¹⁄₅)</td>
+    <td>ellipse</td>
+    <td>(¹⁄₄)</td>
+    <td>rectangle</td>
+    <td>(²⁄₅)</td>
+  </tr>
+  <tr>
+    <td class="nw">Plate XXI.</td>
+    <td></td>
+    <td>(¹⁄₂)</td>
+    <td></td>
+    <td>(¹⁄₃)</td>
+    <td></td>
+    <td>(²⁄₉)</td>
+    <td></td>
+    <td>(¹⁄₄)</td>
+    <td></td>
+    <td>(²⁄₉)</td>
+    <td></td>
+    <td>(¹⁄₅)</td>
+    <td></td>
+    <td>(¹⁄₄)</td>
+    <td></td>
+    <td>(²⁄₅)</td>
+  </tr>
+  <tr>
+    <td class="nw">Plate XXII.</td>
+    <td></td>
+    <td>(¹⁄₂)</td>
+    <td></td>
+    <td>(¹⁄₃)</td>
+    <td></td>
+    <td>(¹⁄₈)</td>
+    <td></td>
+    <td>(⁴⁄₉)</td>
+    <td></td>
+    <td>(¹⁄₃)</td>
+    <td></td>
+    <td>(¹⁄₅)</td>
+    <td></td>
+    <td>(¹⁄₄)</td>
+    <td></td>
+    <td>(²⁄₅)</td>
+  </tr>
+  <tr>
+    <td rowspan="2" class="nw valign">Plate XXIII.</td>
+    <td rowspan="2" class="valign"></td>
+    <td rowspan="2" class="valign">(¹⁄₂)</td>
+    <td rowspan="2" class="valign"></td>
+    <td rowspan="2" class="valign">(¹⁄₄)</td>
+    <td rowspan="2" class="valign"></td>
+    <td rowspan="2" class="valign">(H)</td>
+    <td rowspan="2" class="valign"></td>
+    <td rowspan="2" class="valign">(-)</td>
+    <td rowspan="2" class="valign"></td>
+    <td rowspan="2" class="valign">(¹⁄₅)</td>
+    <td rowspan="2" class="valign"></td>
+    <td rowspan="2" class="valign">(¹⁄₅)</td>
+    <td rowspan="2" class="valign">ellipses</td>
+    <td>{&nbsp;(¹⁄₃)&nbsp;}</td>
+    <td rowspan="2" class="valign"></td>
+    <td rowspan="2" class="valign">(¹⁄₃)</td>
+  </tr>
+  <tr>
+    <td>{&nbsp;(¹⁄₄)&nbsp;}</td>
+  </tr>
+</table>
+
+<p>The harmonic elements of each are therefore simply the following parts
+of the right angle:—</p>
+
+<table>
+  <tr>
+    <th></th>
+    <th>Tonic.</th>
+    <th>Dominant.</th>
+    <th>Mediant.</th>
+    <th>Submediant.</th>
+  </tr>
+  <tr>
+    <td class="nw">Plate VIII.</td>
+    <td>(¹⁄₂)</td>
+    <td>(¹⁄₃)</td>
+    <td>(¹⁄₅)</td>
+    <td>(³⁄₁₀)</td>
+  </tr>
+  <tr>
+    <td></td>
+    <td>(¹⁄₄)</td>
+    <td></td>
+    <td></td>
+    <td></td>
+  </tr>
+  <tr>
+    <th></th>
+    <th>Tonic.</th>
+    <th>Dominant.</th>
+    <th>Mediant.</th>
+    <th>Supertonic.</th>
+  </tr>
+  <tr>
+    <td class="nw">Plate XXI.</td>
+    <td>(¹⁄₂)</td>
+    <td>(¹⁄₃)</td>
+    <td>(¹⁄₅)</td>
+    <td>(²⁄₉)</td>
+  </tr>
+  <tr>
+    <td></td>
+    <td>(¹⁄₄)</td>
+    <td></td>
+    <td></td>
+    <td><span class="pagenum"><a id="Page_105"></a>[105]</span></td>
+  </tr>
+  <tr>
+    <th></th>
+    <th>Tonic.</th>
+    <th>Dominant.</th>
+    <th>Mediant.</th>
+    <th>Supertonic.</th>
+  </tr>
+  <tr>
+    <td class="nw">Plate XXII.</td>
+    <td>(¹⁄₂)</td>
+    <td>(¹⁄₃)</td>
+    <td>(¹⁄₅)</td>
+    <td>(⁴⁄₉)</td>
+  </tr>
+  <tr>
+    <td></td>
+    <td>(¹⁄₄)</td>
+    <td></td>
+    <td></td>
+    <td></td>
+  </tr>
+  <tr>
+    <td></td>
+    <td>(¹⁄₈)</td>
+    <td></td>
+    <td></td>
+    <td></td>
+  </tr>
+  <tr>
+    <th></th>
+    <th>Tonic.</th>
+    <th>Dominant.</th>
+    <th>Mediant.</th>
+    <th></th>
+  </tr>
+  <tr>
+    <td class="nw">Plate XXIII.</td>
+    <td>(¹⁄₂)</td>
+    <td>(¹⁄₃)</td>
+    <td>(¹⁄₅)</td>
+    <td></td>
+  </tr>
+  <tr>
+    <td></td>
+    <td>(¹⁄₄)</td>
+    <td></td>
+    <td></td>
+    <td></td>
+  </tr>
+</table>
+
+<h3 id="APPENDIX_VI">No. VI.</h3>
+
+<p>So far as I know, there has been only one attempt in modern times,
+besides my own, to establish a universal system of proportion, based
+on a law of nature, and applicable to art. This attempt consists of a
+work of 457 pages, with 166 engraved illustrations, by Dr Zeising, a professor
+in Leipzic, where it was published in 1854.</p>
+
+<p>One of the most learned and talented professors in our Edinburgh University
+has reviewed that work as follows:—</p>
+
+<p>“It has been rather cleverly said that the intellectual distinction between an
+Englishman and a Scotchman is this—‘Give an Englishman two facts, and
+he looks out for a third; give a Scotchman two facts, and he looks out for a
+theory.’ Neither of these tests distinguishes the German; he is as likely to
+seek for a third fact as for a theory, and as likely to build a theory on two facts
+as to look abroad for further information. But once let him have a theory
+in his mind, and he will ransack heaven and earth until he almost buries it
+under the weight of accumulated facts. This remark applies with more than
+common force to a treatise published last year by Dr Zeising, a professor in
+Leipsic, ‘On a law of proportion which rules all nature.’ The ingenious
+author, after proving from the writings of ancient and modern philosophers
+that there always existed the belief (whence derived it is difficult to say), that
+some law does bind into one formula all the visible works of God, proceeds
+to criticise the opinions of individual writers respecting that connecting law.
+It is not our purpose to follow him through his lengthy examination.
+Suffice it to say that he believes he has found the lost treasure in the
+<i>Timæus</i> of Plato, c. 31. The passage is confessedly an obscure one, and will
+not bear a literal translation. The interpretation which Dr Zeising puts on
+it is certainly a little strained, but we are disposed to admit that he does it
+with considerable reason. Agreeably to him, the passage runs thus:—‘That
+bond is the most beautiful which binds the things as much as possible
+into one; and proportion effects this most perfectly when three things are<span class="pagenum"><a id="Page_106"></a>[106]</span>
+so united that the greater bears to the middle the same ratio that the middle
+bears to the less.’</p>
+
+<p>“We must do Dr Zeising the justice to say that he has not made more than
+a legitimate use of the materials which were presented to him in the writings
+of the ancients, in his endeavour to establish the fact of the existence of this
+law amongst them. The canon of Polycletes, the tradition of Varro mentioned
+by Pliny relative to that canon, the writings of Galen and others,
+are all brought to bear on the same point with more or less force. The sum
+of this portion of the argument is fairly this,—that the ancient sculptors had
+<i>some</i> law of proportion—some authorised examplar to which they referred as
+their work proceeded. That it was the law here attributed to Plato is by no
+means made out; but, considering the incidental manner in which that law
+is referred to, and the obscurity of the passages as they exist, it is, perhaps,
+too much to expect more than this broad feature of coincidence—the
+fact that some law was known and appealed to. Dr Zeising now proceeds to
+examine modern theories, and it is fair to state that he appears generally to
+take a very just view of them.</p>
+
+<p>“Let us now turn to Dr Zeising’s own theory. It is this—that in every
+beautiful form lines are divided in extreme and mean ratio; or, that any
+line considered as a whole, bears to its larger part the same proportion that
+the larger bears to the smaller—thus, a line of 5 inches will be divided into
+parts which are very nearly 2 and 3 inches respectively (1·9 and 3·1 inches).
+This is a well-known division of a line, and has been called the <span class="smcap">golden</span> rule,
+but when or why, it is not easy to ascertain. With this rule in his hand,
+Dr Zeising proceeds to examine all nature and art; nay, he even ventures
+beyond the threshold of nature to scan Deity. We will not follow him
+so far. Let us turn over the pages of his carefully illustrated work, and
+see how he applies his line. We meet first with the Apollo Belvidere—the
+golden line divides him happily. We cannot say the same of the division
+of a handsome face which occurs a little further on. Our preconceived
+notions have made the face terminate with the chin, and not with the centre
+of the throat. It is evident that, with such a rule as this, a little latitude
+as to the extreme point of the object to be measured, relieves its inventor from
+a world of perplexities. This remark is equally applicable to the <i>arm</i> which
+follows, to which the rule appears to apply admirably, yet we have tried it
+on sundry plates of arms, both fleshy and bony, without a shadow of success.
+Whether the rule was made for the arm or the arm for the rule, we do not
+pretend to decide. But let us pass hastily on to page 284, where the Venus
+de Medicis and Raphael’s Eve are presented to us. They bear the application
+of the line right well. It might, perhaps, be objected that it is remarkable
+that the same rule applies so exactly to the existing position
+of the figures, such as the Apollo and the Venus, the one of which is
+upright, and the other crouching. But let that pass. We find Dr
+Zeising next endeavouring to square his theory with the distances of the<span class="pagenum"><a id="Page_107"></a>[107]</span>
+planets, with wofully scanty success. Descending from his lofty position, he
+spans the earth from corner to corner, at which occupation we will leave him
+for a moment, whilst we offer a suggestion which is equally applicable to
+poets, painters, novelists, and theorisers. Never err in excess—defect is the
+safe side—it is seldom a fault, often a real merit. Leave something for the student
+of your works to do—don’t chew the cud for him. Be assured he will not
+omit to pay you for every little thing which you have enabled him to discover.
+Poor Professor Zeising! he is far too German to leave any field of discovery
+open for his readers. But let us return to him; we left him on his
+back, lost for a time in a hopeless attempt to double Cape Horn. We
+will be kind to him, as the child is to his man in the Noah’s ark, and
+set him on his legs amongst his toys again. He is now in the vegetable
+kingdom, amidst oak leaves and sections of the stems of divers plants.
+He is in his element once more, and it were ungenerous not to admit
+the merit of his endeavours, and the success which now and then attends
+it. We will pass over his horses and their riders, together with that
+portly personage, the Durham ox, for we have caught a glimpse of a form
+familiar to our eyes, the ever-to-be-admired Parthenon. This is the true
+test of a theory. Unlike the Durham ox just passed before us, the Parthenon
+will stand still to be measured. It has so stood for twenty centuries,
+and every one that has scanned its proportions has pronounced them exquisite.
+Beauty is not an adaptation to the acquired taste of a single nation,
+or the conventionality of a single generation. It emanates from a deep-rooted
+principle in nature, and appeals to the verdict of our whole humanity.
+We don’t find fault with the Durham ox—his proportions are probably
+good, though they be the result of breeding and cross-breeding; still we are
+not sure whether, in the march of agriculture, our grandchildren may not
+think him a very wretched beast. But there is no mistake about the Parthenon;
+as a type of proportion it stands, has stood, and shall stand. Well,
+then, let us see how Dr Zeising succeeds with his rule here. Alas! not a
+single point comes right. The Parthenon is condemned, or its condemnation
+condemns the theory. Choose your part. We choose the
+latter alternative; and now, our choice being made, we need proceed no
+further. But a question or two have presented themselves as we went
+along, which demand an answer. It may be asked—How do you account
+for the esteem in which this law of the section in extreme and mean ratio
+was held? We reply—That it was esteemed just in the same way that a tree
+is esteemed for its fruit. To divide a right angle into two or three, four or
+six, equal parts was easy enough. But to divide it into five or ten such
+parts was a real difficulty. And how was the difficulty got over? It was
+effected by means of this golden rule. This is its great, its ruling application;
+and if we adopt the notion that the ancients were possessed with the idea of
+the existence of angular symmetry, we shall have no difficulty in accounting
+for their appreciation of this problem. Nay, we may even go further, and<span class="pagenum"><a id="Page_108"></a>[108]</span>
+admit, with Dr Zeising, the interpretation of the passage of Plato,—only
+with this limitation, that Plato, as a geometer, was carried away by the
+geometry of æsthetics from the thing itself. It may be asked again—Is it
+not probable that <i>some</i> proportionality does exist amongst the parts of
+natural objects? We reply—That, <i>à priori</i>, we expect some such system to
+exist, but that it is inconsistent with the scheme of <i>least effort</i>, which pervades
+and characterises all natural succession in space or in time, that that
+system should be a complicated one. Whatever it is, its essence must be
+simplicity. And no system of simple linear proportion is found in nature;
+quite the contrary. We are, therefore, driven to another hypothesis, viz.—that
+the simplicity is one of angles, not of lines; that the eye estimates by
+search round a point, not by ascending and descending, going to the right
+and to the left,—a theory which we conceive all nature conspires to prove.
+Beauty was not created for the eye of man, but the eye of man and his
+mental eye were created for the appreciation of beauty. Examine the forms
+of animals and plants so minute that nothing short of the most recent
+improvements in the microscope can succeed in detecting their symmetry;
+or examine the forms of those little silicious creations which grew thousands
+of years before Man was placed on the earth, and, with forms of marvellous
+and varied beauty, they all point to its source in angular symmetry. This
+is the keystone of formal beauty, alike in the minutest animalcule, and in
+the noblest of God’s works, his own image—Man.”</p>
+
+<p class="titlepage">THE END.</p>
+
+<p class="center smaller">BALLANTYNE AND COMPANY, PRINTERS, EDINBURGH.</p>
+
+<hr class="chap x-ebookmaker-drop">
+
+<div class="footnotes">
+
+<div class="chapter">
+
+<h2 class="nobreak" id="FOOTNOTES">FOOTNOTES</h2>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_1" href="#FNanchor_1" class="label">[1]</a> Sir David Brewster.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_2" href="#FNanchor_2" class="label">[2]</a> No. CLVIII., October 1843.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_3" href="#FNanchor_3" class="label">[3]</a> Diogenes Laërtius’s “Lives of the Philosophers,” literally translated. Bohn:
+London.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_4" href="#FNanchor_4" class="label">[4]</a> Ibid.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_5" href="#FNanchor_5" class="label">[5]</a> Rose’s “Biographical Dictionary.”</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_6" href="#FNanchor_6" class="label">[6]</a> Professor Laycock, now of the University of Edinburgh.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_7" href="#FNanchor_7" class="label">[7]</a> “The Geometric Beauty of the Human Figure Defined,” &amp;c.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_8" href="#FNanchor_8" class="label">[8]</a> Longman and Co., London.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_9" href="#FNanchor_9" class="label">[9]</a> See <a href="#APPENDIX">Appendix</a>.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_10" href="#FNanchor_10" class="label">[10]</a> “The Orthographic Beauty of the Parthenon,” &amp;c., and “The Harmonic Law
+of Nature applied to Architectural Design.”</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_11" href="#FNanchor_11" class="label">[11]</a> For further details, see “Harmonic Law of Nature,” &amp;c.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_12" href="#FNanchor_12" class="label">[12]</a> By a very simple machine, which I have lately invented, an ellipse of any
+given proportions, even to those of (¹⁄₆₄), which is the curve of the entases of the
+columns of the Parthenon (see <a href="#plate07">Plate VII.</a>), and of any length, from half an inch
+to fifty feet or upwards, may be easily and correctly described; the length and
+angle of the required ellipse being all that need be given.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_13" href="#FNanchor_13" class="label">[13]</a> “The Orthographic Beauty of the Parthenon,” &amp;c.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_14" href="#FNanchor_14" class="label">[14]</a> “The Orthographic Beauty of the Parthenon,” &amp;c.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_15" href="#FNanchor_15" class="label">[15]</a> Ibid.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_16" href="#FNanchor_16" class="label">[16]</a> “The Harmonic Law of Nature applied to Architectural Design.”</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_17" href="#FNanchor_17" class="label">[17]</a> “Physio-philosophy.” By Dr Oken. Translated by Talk; and published by
+the Ray Society. London, 1848.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_18" href="#FNanchor_18" class="label">[18]</a> “The Science of those Proportions by which the Human Head and Countenance,
+as represented in Works of ancient Greek Art, are distinguished from those
+of ordinary Nature.”</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_19" href="#FNanchor_19" class="label">[19]</a> “The Geometric Beauty of the Human Figure Defined,” &amp;c., and “The
+Natural Principles of Beauty Developed in the Human Figure.”</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_20" href="#FNanchor_20" class="label">[20]</a> “The Geometric Beauty of the Human Figure Defined,” &amp;c.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_21" href="#FNanchor_21" class="label">[21]</a> “Essay on Ornamental Design,” &amp;c., and “The Geometric Beauty of the Human
+Figure,” &amp;c.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_22" href="#FNanchor_22" class="label">[22]</a> “A Nomenclature of Colours, applicable to the Arts and Natural Sciences,”
+&amp;c., &amp;c.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_23" href="#FNanchor_23" class="label">[23]</a> See <a href="#Page_24">pp. 24 and 25</a>.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_24" href="#FNanchor_24" class="label">[24]</a> “The Principles of Beauty in Colouring Systematised,” Fourteen Diagrams,
+each containing Six Colours and Hues.</p>
+
+<p>“A Nomenclature of Colours,” &amp;c., Forty Diagrams, each containing Twelve
+Examples of Colours, Hues, Tints, and Shades.</p>
+
+<p>“The Laws of Harmonious Colouring,” &amp;c., One Diagram, containing Eighteen
+Colours and Hues.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_25" href="#FNanchor_25" class="label">[25]</a> “Etchings Representing the Best Examples of Grecian and Roman Architectural
+Ornament, drawn from the Originals,” &amp;c. By Charles Heathcote Tatham,
+Architect. London: Priestly and Weale. 1826.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_26" href="#FNanchor_26" class="label">[26]</a> “The Orthographic Beauty of the Parthenon,” &amp;c.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_27" href="#FNanchor_27" class="label">[27]</a> “Science of those Proportions,” &amp;c.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_28" href="#FNanchor_28" class="label">[28]</a> Abbé Barthélémie’s “Travels of Anacharsis in Greece,” vol iv., pp. 193, 195.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_29" href="#FNanchor_29" class="label">[29]</a> Abbé Barthélémie (vol. ii., pp. 168, 169), who cites as his authorities, Cicer. De Nat.
+Deor., lib. i., cap. ii., t. 2, p. 405; Justin Mart., Ovat. ad Gent., p. 10; Aristot. Metaph.,
+lib. i., cap. v., t. 2, p. 845.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_30" href="#FNanchor_30" class="label">[30]</a> Müller’s “Ancient Art and its Remains.”</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_31" href="#FNanchor_31" class="label">[31]</a> “Anacharsis’ Travels in Greece.” By the Abbé Barthélémie, vol. ii., p. 325.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_32" href="#FNanchor_32" class="label">[32]</a> “Anacharsis’ Travels in Greece.” By the Abbé Barthélémie, vol. vi., p. 225. The
+authorities the Abbé quotes are—Quintil., lib. xii., cap. x., p. 744; Plin., lib. xxxv., cap. ix.,
+p. 691.</p>
+
+</div>
+
+<div class="footnote">
+
+<p><a id="Footnote_33" href="#FNanchor_33" class="label">[33]</a> Müller’s “Archæology of Art,” &amp;c.</p>
+
+</div>
+
+</div>
+
+<hr class="chap x-ebookmaker-drop">
+
+<div class="chapter">
+
+<h2 class="nobreak" id="Works_by_the_Same_Author">Works by the Same Author.</h2>
+
+</div>
+
+<h3>I.<br>
+<span class="smaller">In royal 8vo, with Copperplate Illustrations, price 2s. 6d.,</span><br>
+THE HARMONIC LAW OF NATURE APPLIED TO ARCHITECTURAL DESIGN.</h3>
+
+<p class="center"><i>From the Athenæum.</i></p>
+
+<p>The beauty of the theory is its universality, and its simplicity. In
+nature, the Creator accomplished his purposes by the simplest means—the
+harmony of nature is indestructible and self-restoring. Mr Hay’s book on
+the “Parthenon,” on the “Natural Principles of Beauty as developed in the
+Human Figure,” his “Principles of Symmetrical Beauty,” his “Principles of
+Colouring, and Nomenclature of Colours,” his “Science of Proportion,” and
+“Essay on Ornamental Design,” we have already noticed with praise as the
+results of philosophical and original thought.</p>
+
+<p class="center"><i>From the Daily News.</i></p>
+
+<p>This essay is a new application to Lincoln cathedral in Gothic
+architecture, and to the Temple of Theseus in Greek architecture, of the
+principles of symmetrical beauty already so profusely illustrated and
+demonstrated by Mr Hay. The theory which Mr Hay has propounded in so many
+volumes is not only a splendid contribution towards a science of æsthetic
+proportions, but, for the first time in the history of art, proves the
+possibility, and lays the foundations of such a science. To those who are
+not acquainted with the facts, these expressions will sound hyperbolical,
+but they are most true.</p>
+
+<h3>II.<br>
+<span class="smaller">In royal 8vo, with Copperplate Illustrations, price 5s.,</span><br>
+THE NATURAL PRINCIPLES OF BEAUTY, AS DEVELOPED IN THE HUMAN FIGURE.</h3>
+
+<p class="center"><i>From the Spectator.</i></p>
+
+<p>We cannot refuse to entertain Mr Hay’s system as of singular intrinsic
+excellence. The simplicity of his law and its generality impress
+themselves more deeply on the conviction with each time of enforcement.
+His theory proceeds from the idea, that in nature every thing is effected
+by means more simple than any other that could have been conceived,—an
+idea certainly consistent with whatever we can trace out or imagine of
+the all-wise framing of the universe.</p>
+
+<p class="center"><i>From the Sun.</i></p>
+
+<p>By founding (if we may so phrase it) this noble theory, Mr Hay has
+covered his name with distinction, and has laid the basis, we conceive,
+of no ephemeral reputation. By illustrating it anew, through the
+medium of this graceful treatise, he has conferred a real boon upon
+the community, for he has afforded the public another opportunity of
+following the golden rule of the poet—by looking through the holy and
+awful mystery of creation to the holier and yet more awful mystery of
+Omnipotence.</p>
+
+<p class="center"><i>From the Cambridge Journal of Classical and Sacred Philology.</i></p>
+
+<p>The inquiries which of late years have been instituted by Mr D. R. Hay
+of Edinburgh, on the proportions of the human figure, and on the natural
+principles of beauty, as illustrated by works of Greek art, constitute an
+epoch in the study of æsthetics and the philosophy of form.</p>
+
+<h3>III.<br>
+<span class="smaller">In royal 8vo, with Copperplate Illustrations, price 5s.,</span><br>
+THE ORTHOGRAPHIC BEAUTY OF THE PARTHENON REFERRED TO A LAW OF NATURE.</h3>
+
+<p class="center">To which are prefixed, a few Observations on the Importance of Æsthetic
+Science as an Element in Architectural Education.</p>
+
+<p class="center"><i>From the Scottish Literary Gazette.</i></p>
+
+<p>We think this work will satisfy every impartial mind that Mr Hay has
+developed the true theory of the Parthenon—that he has, in fact, to
+use a kindred phraseology, both <i>parsed</i> and <i>scanned</i> this
+exquisitely beautiful piece of architectural composition, and that, in
+doing so, he has provided the true key by which the treasures of Greek
+art may be further unlocked, and rendered the means of correcting,
+improving, and elevating modern practice.</p>
+
+<p class="center"><i>From the Edinburgh Guardian.</i></p>
+
+<p>Again and again the attempt has been made to detect harmonic ratios in
+the measurement of Athenian architecture, but ever without reward. Mr Hay
+has, however, made the discovery, and to an extent of which no one had
+previously dreamt.</p>
+
+<h3>IV.<br>
+<span class="smaller">In 8vo, 100 Plates, price 6s.,</span><br>
+FIRST PRINCIPLES OF SYMMETRICAL BEAUTY.</h3>
+
+<p class="center"><i>From the Spectator.</i></p>
+
+<p>This is a grammar of pure form, in which the elements of symmetrical, as
+distinguished from picturesque beauty, are demonstrated, by reducing the
+outlines or planes of curvilinear and rectilinear forms to their origin
+in the principles of geometrical proportion. In thus analysing symmetry
+of outline in natural and artificial objects, Mr Hay determines the fixed
+principles of beauty in positive shape, and shews how beautiful forms may
+be reproduced and infinitely varied with mathematical precision. Hitherto
+the originating and copying of beautiful forms have been alike empirical;
+the production of a new design for a vase or a jug has been a matter of
+chance between the eye and the hand; and the copying of a Greek moulding
+or ornament, a merely mechanical process. By a series of problems, Mr
+Hay places both the invention and imitation of beautiful forms on a sure
+basis of science, giving to the fancy of original minds a clue to the
+evolving of new and elegant shapes, in which the infinite resources of
+nature are made subservient to the uses of art.</p>
+
+<p>The volume is illustrated by one hundred diagrams beautifully executed,
+that serve to explain the text, and suggest new ideas of beauty of
+contour in common objects. To designers of pottery, hardware, and
+architectural ornaments, this work is particularly valuable; but artists
+of every kind, and workmen of intelligence, will find it of great utility.</p>
+
+<p class="center"><i>From the Athenæum.</i></p>
+
+<p>The volume before us is the seventh of Mr Hay’s works. It is the most
+practical and systematic, and likely to be one of the most useful. It
+is, in short, a grammar of form, or a spelling-book of beauty. This is
+beginning at the right end of the matter; and the necessity for this kind
+of knowledge will inevitably, though gradually, be felt. The work will,
+therefore, be ultimately appreciated and adopted as an introduction to
+the study of beautiful forms.</p>
+
+<p>The third part of the work treats of the Greek oval or composite ellipse,
+as Mr Hay calls it. It is an ellipse of three foci, and gives practical
+forms for vases and architectural mouldings, which are curious to the
+architect, and will be very useful both to the potter, the moulder,
+and the pattern-drawer. A fourth part contains applications of this to
+practice. Of the details worked out with so much judgment and ingenuity
+by Mr Hay, we should in vain attempt to communicate just notions without
+the engravings of which his book is full. We must, therefore, refer to
+the work itself. The forms there given are full of beauty, and so far
+tend to prove the system.</p>
+
+<h3>V.<br>
+<span class="smaller">In 8vo, 14 Coloured Diagrams, Second Edition, price 15s.,</span><br>
+THE PRINCIPLES OF BEAUTY IN COLOURING SYSTEMATISED.</h3>
+
+<p class="center"><i>From the Spectator.</i></p>
+
+<p>In this new analysis of the harmonies of colour, Mr Hay has performed the
+useful service of tracing to the operation of certain fixed principles
+the sources of beauty in particular combinations of hues and tints; so
+that artists may, by the aid of this book, produce, with mathematical
+certainty, the richest effects, hitherto attainable by genius alone. Mr
+Hay has reduced this branch of art to a perfect system, and proved that
+an offence against good taste in respect to combinations of colour is, in
+effect, a violation of natural laws.</p>
+
+<h3>VI.<br>
+<span class="smaller">In 8vo, 228 Examples of Colours, Hues, Tints, and Shades, price 63s.,</span><br>
+A NOMENCLATURE OF COLOURS, APPLICABLE TO THE ARTS AND NATURAL SCIENCES.</h3>
+
+<p class="center"><i>From the Daily News.</i></p>
+
+<p>In this work Mr Hay has brought a larger amount of practical knowledge
+to bear on the subject of colour than any other writer with whom we are
+acquainted, and in proportion to this practical knowledge is, as might be
+expected, the excellence of his treatise. There is much in this volume
+which we would most earnestly recommend to the notice of artists, house
+decorators, and, indeed, to all whose business or profession requires a
+knowledge of the management of colour. The work is replete with hints
+which they might turn to profitable account, and which they will find
+nowhere else.</p>
+
+<p class="center"><i>From the Athenæum.</i></p>
+
+<p>We have formerly stated the high opinion we entertain of Mr Hay’s
+previous exertions for the improvement of decorative art in this country.
+We have already awarded him the merit of invention and creation of the
+new and the beautiful in form. In his former treatises he furnished a
+theory of definite proportions for the creation of the beautiful in form.
+In the present work he proposes to supply a scale of definite proportions
+for chromatic beauty. For this purpose he sets out very properly with a
+precise nomenclature of colour. In this he has constructed a vocabulary
+for the artist—an alphabet for the artizan. He has gone further—he
+constructs words for three syllables. From this time, it will be possible
+to write a letter in Edinburgh about a coloured composition, which shall
+be read off in London, Paris, St Petersburg, or Pekin, and shall so
+express its nature that it can be reproduced in perfect identity. This Mr
+Hay has done, or at least so nearly, as to deserve our thanks on behalf
+of art, and artists of all grades, even to the decorative artizan—not
+one of whom, be he house-painter, china pattern-drawer, or calico
+printer, should be without the simple manual of “words for colours.”</p>
+
+<h3>VII.<br>
+<span class="smaller">In post 8vo, with a Coloured Diagram, Sixth Edition, price 7s. 6d.,</span><br>
+THE LAWS OF HARMONIOUS COLOURING ADAPTED TO INTERIOR DECORATIONS.</h3>
+
+<p class="center"><i>From the Atlas.</i></p>
+
+<p>Every line of this useful book shews that the author has contrived to
+intellectualise his subject in a very interesting manner. The principles
+of harmony in colour as applied to decorative purposes, are explained and
+enforced in a lucid and practical style, and the relations of the various
+tints and shades to each other, so as to produce a harmonious result, are
+descanted upon most satisfactorily and originally.</p>
+
+<p class="center"><i>From the Edinburgh Review.</i></p>
+
+<p>In so far as we know, Mr Hay is the first and only modern artist who
+has entered upon the study of these subjects without the trammels of
+prejudice and authority. Setting aside the ordinances of fashion, as
+well as the dicta of speculation, he has sought the foundation of
+his profession in the properties of light, and in the laws of visual
+sensation, by which these properties are recognised and modified. The
+truths to which he has appealed are fundamental and irrefragable.</p>
+
+<p class="center"><i>From the Athenæum.</i></p>
+
+<p>We have regarded, and do still regard, the production of Mr Hay’s works
+as a remarkable psychological phenomenon—one which is instructive both
+for the philosopher and the critic to study with care and interest, not
+unmingled with respect. We see how his mind has been gradually guided
+by Nature herself out of one track, and into another, and ever and anon
+leading him to some vein of the beautiful and true, hitherto unworked.</p>
+
+<h3>VIII.<br>
+<span class="smaller">In 4to, 25 Plates, price 36s.,</span><br>
+ON THE SCIENCE OF THOSE PROPORTIONS BY WHICH THE HUMAN HEAD AND
+COUNTENANCE, AS REPRESENTED IN ANCIENT GREEK ART, ARE DISTINGUISHED FROM
+THOSE OF ORDINARY NATURE.</h3>
+
+<p class="center smaller">(PRINTED BY PERMISSION.)</p>
+
+<p class="center"><i>From a Letter to the Author by Sir William Hamilton, Bart., Professor
+of Logic and Metaphysics in the Edinburgh University.</i></p>
+
+<p>Your very elegant volume, “Science of those Proportions,” &amp;c., is to me
+extremely interesting, as affording an able contribution to what is the
+ancient, and, I conceive, the true theory of the beautiful. But though
+your doctrine coincides with the one prevalent through all antiquity, it
+appears to me quite independent and original in you; and I esteem it the
+more that it stands opposed to the hundred one-sided and exclusive views
+prevalent in modern times.</p>
+
+<p class="center"><i>From Chambers’s Edinburgh Journal.</i></p>
+
+<p>We now come to another, and much more remarkable corroboration, which
+calls upon us to introduce to our readers one of the most valuable and
+original contributions that have ever been made to the Philosophy of Art,
+viz., Mr Hay’s work “On the Science of those Proportions,” &amp;c. Mr Hay’s
+plan is simply to form a scale composed of the well-known vibrations of
+the monochord, which are the alphabet of music, and then to draw upon
+the quadrant of a circle angles <i>answering to these vibrations</i>.
+With the series of triangles thus obtained he combines a circle and
+an ellipse, the proportions of which are derived from the triangles
+themselves; and thus he obtains an infallible rule for the composition of
+the head of ideal beauty.</p>
+
+<h3>IX.<br>
+<span class="smaller">In 4to, 16 Plates, price 30s.,</span><br>
+THE GEOMETRIC BEAUTY OF THE HUMAN FIGURE DEFINED.</h3>
+
+<p class="center">To which is prefixed, a SYSTEM of ÆSTHETIC PROPORTION applicable to
+ARCHITECTURE and the other FORMATIVE ARTS.</p>
+
+<p class="center"><i>From the Cambridge Journal of Classical and Sacred Philology.</i></p>
+
+<p>We feel bound to pay Mr Hay a large and glad tribute of praise for
+having devised a system of proportions which rises superior to the
+idiosyncrasies of different artists, which brings back to one common
+type the sensations of Eye and Ear, and so makes a giant stride towards
+that <i>codification</i> of the laws of the universe which it is the
+business of science to effect. We have no hesitation in saying that, for
+scientific precision of method and importance of results, Albert Durer,
+Da Vinci, and Hogarth—not to mention less noteworthy writers—must all
+yield the palm to Mr Hay.</p>
+
+<h3>X.<br>
+<span class="smaller">In oblong folio, 57 Plates and numerous Woodcuts, price 42s.,</span><br>
+AN ESSAY ON ORNAMENTAL DESIGN, IN WHICH ITS TRUE PRINCIPLES ARE DEVELOPED
+AND ELUCIDATED, &amp;c.</h3>
+
+<p class="center"><i>From the Athenæum.</i></p>
+
+<p>In conclusion, Mr Hay’s book goes forth with our best wishes. It must be
+good. It must be prolific of thought—stimulant of invention. It is to be
+acknowledged as a benefit of an unusual character conferred on the arts
+of ornamental design.</p>
+
+<p class="center"><i>From the Spectator.</i></p>
+
+<p>Mr Hay has studied the subject deeply and scientifically. In this
+treatise on ornamental design, the student will find a clue to the
+discovery of the source of an endless variety of beautiful forms and
+combinations of lines, in the application of certain fixed laws of
+harmonious proportion to the purposes of art. Mr Hay also exemplifies
+the application of his theory of linear harmony to the production of
+beautiful forms generally, testing its soundness by applying it to the
+human figure, and the purest creations of Greek art.</p>
+
+<p class="center"><i>From Fraser’s Magazine.</i></p>
+
+<p>Each part of this work is enriched by diagrams of great beauty, direct
+emanations of principle, and, consequently, presenting entirely new
+combinations of form. Had our space permitted, we should have made some
+extracts from this “Essay on Ornamental Design;” and we would have done
+so, because of the discriminating taste by which it is pervaded, and the
+forcible observations which it contains; but we cannot venture on the
+indulgence.</p>
+
+<h3>XI.<br>
+<span class="smaller">In 4to, 17 Plates and 38 Woodcuts, price 25s.,</span><br>
+PROPORTION, OR THE GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED.</h3>
+
+<h3>XII.<br>
+<span class="smaller">In 4to, 18 Plates and numerous Woodcuts, price 15s.,</span><br>
+THE NATURAL PRINCIPLES AND ANALOGY OF THE HARMONY OF FORM.</h3>
+
+<p class="center"><i>From the Edinburgh Review.</i></p>
+
+<p>Notwithstanding some trivial points of difference between Mr Hay’s views
+and our own, we have derived the greatest pleasure from the perusal of
+these works. They are all composed with accuracy and even elegance. His
+opinions and views are distinctly brought before the reader, and stated
+with that modesty which characterises genius, and that firmness which
+indicates truth.</p>
+
+<p class="center"><i>From Blackwood’s Magazine.</i></p>
+
+<p>We have no doubt that when Mr Hay’s Art-discovery is duly developed and
+taught, as it should be, in our schools, it will do more to improve the
+general taste than anything which has yet been devised.</p>
+
+<div style='text-align:center'>*** END OF THE PROJECT GUTENBERG EBOOK 75399 ***</div>
+</body>
+</html>
+
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diff --git a/LICENSE.txt b/LICENSE.txt
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+This eBook, including all associated images, markup, improvements,
+metadata, and any other content or labor, has been confirmed to be
+in the PUBLIC DOMAIN IN THE UNITED STATES.
+
+Procedures for determining public domain status are described in
+the "Copyright How-To" at https://www.gutenberg.org.
+
+No investigation has been made concerning possible copyrights in
+jurisdictions other than the United States. Anyone seeking to utilize
+this eBook outside of the United States should confirm copyright
+status under the laws that apply to them.
diff --git a/README.md b/README.md
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+Project Gutenberg (https://www.gutenberg.org) public repository for
+eBook #75399 (https://www.gutenberg.org/ebooks/75399)