,,the ,,science ,,of ,,beauty4 ,,edinburgh3 ,,printed ,,by ,,ballantyne ,,and ,,company1 ,,paul',s ,,work4 ,,the ,,science ,,of ,,beauty1 ,,as ,,developed ,,in ,,nature ,,and ,,applied ,,in ,,art4 ,,by ,d4 ,r4 ,,hay1 ,f4,r4,s4,e4 8,the irregular combinations of fanciful invention may delight awhile1 by that novelty of which the common satiety of life sends us all in quest2 the pleasures of sudden wonder are soon exhausted1 and the mind can only repose on the stability of truth40 ,,dr ,,johnson4 ,,william ,,blackwood ,,and ,,sons1 ,,edinburgh ,,and ,,london4 ,,mdccclvi4 ,,to ,,john ,,goodsir1 ,,esq41 ,f4,r4,s ,s4 ,l4 ,e41 ,,professor ,,of ,,anatomy ,,in ,,the ,,university ,,of ,,edinburgh1 ,,as ,,an ,,expression ,,of ,,gratitude ,,for ,,valuable ,,assistance1 ,,as ,,also ,,of ,,high ,,esteem ,,and ,,sincere ,,regard1 ,,this ,,volume ,,is ,,dedicated1 ,,by ,,the ,,author4 ,,preface4 ,my theory of beauty in form and colour being now admitted by the best authorities to be based on truth1 ,i have of late been often asked1 by those who wished to become acquainted with its nature1 and the manner of its being applied in art1 which of my publications ,i would recommend for their perusal4 ,this question ,i have always found difficulty in answering2 for although the law upon which my theory is based is characterised by unity1 yet the subjects in which it is applied1 and the modes of its application1 are equally characterised by variety1 and consequently occupy several volumes4 ,under these circumstances1 ,i consulted a highly respected friend1 whose mathematical talents and good taste are well known1 and to whom ,i have been greatly indebted for much valuable assistance during the course of my investigations4 ,the advice ,i received on this occasion1 was to publish a rsum of my former works1 of such a character as not only to explain the nature of my theory1 but to exhibit to the general reader1 by the most simple modes of illustration and description1 how it is developed in nature1 and how it may be extensively and with ease applied in those arts in which beauty forms an essential element4 ,the following pages1 with their illustrations1 are the results of an attempt to accomplish this object4 ,to those who are already acquainted1 through my former works1 with the nature1 scope1 and tendency of my theory1 ,i have the satisfaction to intimate that ,i have been enabled to include in this rsum much original matter1 with reference both to form and colour4 ,d4 ,r4 ,,hay4 ,,contents4 ,,page ,,introduction #a ,,the ,,science ,,of ,,beauty1 ,,evolved ,,from ,,the ,,harmonic ,,law ,,of ,,nature1 ,,agreeably ,,to ,,the ,,pythagorean ,,system ,,of ,,numerical ,,ratio #ae ,,the ,,science ,,of ,,beauty1 ,,as ,,applied ,,to ,,sounds #bh ,,the ,,science ,,of ,,beauty1 ,,as ,,applied ,,to ,,forms #cd ,,the ,,science ,,of ,,beauty1 ,,as ,,developed ,,in ,,the ,,form ,,of ,,the ,,human ,,head ,,and ,,countenance #ed ,,the ,,science ,,of ,,beauty1 ,,as ,,developed ,,in ,,the ,,form ,,of ,,the ,,human ,,figure #fa ,,the ,,science ,,of ,,beauty1 ,,as ,,developed ,,in ,,colours #fg ,,the ,,science ,,of ,,beauty ,,applied ,,to ,,the ,,forms ,,and ,,proportions ,,of ,,ancient ,,grecian ,,vases ,,and ,,ornaments #hb ,,appendix1 ,,no4 ,i4 #ia ,,appendix1 ,,no4 ,,ii4 #ii ,,appendix1 ,,no4 ,,iii4 #ajj ,,appendix1 ,,no4 ,,iv4 #ajj ,,appendix1 ,,no4 ,v4 #ajd ,,appendix1 ,,no4 ,,vi4 #aje ,,illustrations4 ,,plates ,illustration3 ,i4 ,three ,scales of the ,elementary ,rectilinear ,figures1 viz41 the ,scalene ,triangle1 the ,isosceles ,triangle1 and the ,rectangle1 comprising twenty-seven varieties of each1 according to the harmonic parts of the ,right ,angle from to 4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,ii4 ,diagram of the ,rectilinear ,orthography of the ,principal ,front of the ,parthenon of ,athens1 in which its ,proportions are determined by harmonic parts of the ,right ,angle4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,iii4 ,diagram of the ,rectilinear ,orthography of the ,portico of the ,temple of ,theseus at ,athens1 in which its ,proportions are determined by harmonic parts of the ,right ,angle4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,iv4 ,diagram of the ,rectilinear ,orthography of the ,east ,end of ,lincoln ,cathedral1 in which its ,proportions are determined by harmonic parts of the ,right ,angle4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,v4 ,four ,ellipses described from ,foci1 determined by harmonic parts of the ,right ,angle1 shewing in each the ,scalene ,triangle1 the ,isosceles ,triangle1 and the ,rectangle to which it belongs4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,vi4 ,the ,composite ,ellipse of and of the ,right ,angle1 shewing its greater and lesser ,axis1 its various ,foci1 and the ,isosceles ,triangle in which they are placed4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,vii4 ,the ,composite ,ellipse of and of the ,right ,angle1 shewing how it forms the ,entasis of the ,columns of the ,parthenon of ,athens4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,viii4 ,sectional ,outlines of two ,mouldings of the ,parthenon of ,athens1 full size1 shewing the harmonic nature of their ,curves1 and the simple manner of their ,construction4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,ix4 ,three ,diagrams1 giving a ,vertical1 a ,front1 and a ,side ,aspect of the ,geometrical ,construction of the ,human ,head and ,countenance1 in which the ,proportions are determined by harmonic parts of the ,right ,angle4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,x4 ,diagram in which the ,symmetrical ,proportions of the ,human ,figure are determined by harmonic parts of the ,right ,angle4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,xi4 ,the ,contour of the ,human ,figure as viewed in ,front and in ,profile1 its ,curves being determined by ,ellipses1 whose ,foci are determined by harmonic parts of the ,right ,angle4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,xii4 ,rectilinear ,diagram1 shewing the ,proportions of the ,portland ,vase1 as determined by harmonic parts of the ,right ,angle1 and the outline of its form by an ,elliptic ,curve harmonically described4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,xiii4 ,rectilinear ,diagram of the ,proportions and ,curvilinear ,outline of the form of an ancient ,grecian ,vase1 the proportions determined by harmonic parts of the ,right ,angle1 and the melody of the form by ,curves of two ,ellipses4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,xiv4 ,rectilinear ,diagram of the ,proportions and ,curvilinear ,outline of the form an ancient ,grecian ,vase1 the proportions determined by harmonic parts of the ,right ,angle1 and the melody of the form by an ,elliptic ,curve4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,xv4 ,two ,diagrams of ,etruscan ,vases1 the harmony of ,proportions and melody of the ,contour determined1 respectively1 by parts of the ,right ,angle and an ,elliptic ,curve4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,xvi4 ,two ,diagrams of ,etruscan ,vases1 whose harmony of ,proportion and melody of ,contour are determined as above4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,xvii4 ,diagram shewing the ,geometric ,construction of an ,ornament belonging to the ,parthenon at ,athens4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,xviii4 ,diagram of the ,geometrical ,construction of the ancient ,grecian ,ornament called the ,honeysuckle4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,xix4 ,an additional ,illustration of the ,contour of the ,human ,figure1 as viewed in ,front and in ,profile4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,xx4 ,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 ,front4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,xxi4 ,diagram of a variation on the ,form of the ,portland ,vase4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,xxii4 ,diagram of a second variation on the ,form of the ,portland ,vase4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,illustration3 ,,xxiii4 ,diagram of a third variation on the ,form of the ,portland ,vase4 ,d4 ,r4 ,hay del4 ,g4 ,aikman sc4 ,,introduction4 ,twelve years ago1 one of our most eminent philosophers1#a through the medium of the ,edinburgh ,review1#b gave the following account of what was then the state of the fine arts as connected with science3--8,the disposition to introduce into the intellectual community the principles of free intercourse1 is by no means general2 but we are confident that ,art will not sufficiently develop her powers1 nor ,science attain her most commanding position1 till the practical knowledge of the one is taken in return for the sound deductions of the other4444 ,it is in the fine arts1 principally1 and in the speculations with which they are associated1 that the controlling power of scientific truth has not exercised its legitimate influence4 ,in discussing the principles of painting1 sculpture1 architecture1 and landscape gardening1 philosophers have renounced science as a guide1 and even as an auxiliary2 and a school has arisen whose speculations will brook no restraint1 and whose decisions stand in opposition to the strongest convictions of our senses4 ,that the external world1 in its gay colours and lovely forms1 is exhibited to the mind only as a tinted mass1 neither within nor without the eye1 neither touching it nor distant from it--an ubiquitous chaos1 which experience only can analyse and transform into the realities which compose it2 that the beautiful and sublime in nature and in art derive their power over the mind from association alone1 are among the philosophical doctrines of the present day1 which1 if it be safe1 it is scarcely prudent to question4 ,nor are these opinions the emanations of poetical or ill-trained minds1 which ingenuity has elaborated1 and which fashion sustains4 ,they are conclusions at which most of our distinguished philosophers have arrived4 ,they have been given to the world with all the authority of demonstrated truth2 and in proportion to the hold which they have taken of the public mind1 have they operated as a check upon the progress of knowledge40 ,such1 then1 was the state of art as connected with science twelve years ago4 ,but although the causes which then placed science and the fine arts at variance have since been gradually diminishing1 yet they are still far from being removed4 ,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 ,design1 in #ahed1 the pupils in which amount to upwards of two hundred4 ,the meeting on that occasion included1 besides the pupils1 a numerous and highly respectable assemblage of artists and men of science4 ,the chairman1 a ,professor in our ,university1 and editor of one of the most voluminous works on art1 science1 and literature ever produced in this country1 after extolling the general progress of the pupils1 so far as evinced by the drawings exhibited on the occasion1 drew the attention of the meeting to a discovery made by the head master of the architectural and ornamental department of the school1 viz4--,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 ages1 subdivided by the mythic numbers #c and #g1 and their intermediate odd number #e4 ,now1 it may be remarked1 that the figure thus referred to is not an ovoid1 neither is it in any way of a mysterious nature1 being produced simply by two equal circles cutting each other in their centres4 ,neither can it be shewn that the numbers #c and #g are in any way more mythic than other numbers4 ,in fact1 the terms mysterious and mythic so applied1 can only be regarded as a remnant of an ancient terminology1 calculated to obscure the simplicity of scientific truth1 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 design4 ,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 generally4 ,in the same metropolis there has likewise existed for upwards of ten years a ,philosophical ,institution of great importance and utility1 whose members amount to nearly three thousand1 embracing a large proportion of the higher classes of society1 both in respect to talent and wealth4 ,at the close of the session of this ,institution1 in #ahed1 a learned and eloquent philologus1 who occasionally lectures upon beauty1 was appointed to deliver the closing address1 and touching upon the subject of the beautiful1 he thus concluded-- 8,in the worship of the beautiful1 and in that alone1 we are inferior to the ,greeks4 ,let us therefore be glad to borrow from them2 not slavishly1 but with a wise adaptation--not exclusively1 but with a cunning selection2 in art1 as in religion1 let us learn to prove all things1 and hold fast that which is good--not merely one thing which is good1 but all good things--,classicalism1 ,medivalism1 ,modernism--let us have and hold them all in one wide and lusty embrace4 ,why should the world of art be more narrow1 more monotonous1 than the world of nature8 ,did ,god make all the flowers of one pattern1 to please the devotees of the rose or the lily2 and did ,he make all the hills1 with the green folds of their queenly mantles1 all at one slope1 to suit the angleometer of the most mathematical of decorators8 ,i trow not4 ,let us go and do likewise40 ,i here take for granted1 that what the lecturer meant by 8the worship of the beautiful10 is the production and appreciation of works of art in which beauty should be a primary element2 and judging from the remains which we possess of such works as were produced by the ancient ,grecians1 our inferiority to them in these respects cannot certainly be denied4 ,but ,i must reiterate what ,i have often before asserted1 that it is not by borrowing from them1 however cunning our selection1 or however wise our adaptations1 that this inferiority is to be removed1 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 years4 ,and ,i hold1 that as in religion1 so in art1 there is only one truth1 a grain of which is worth any amount of philological eloquence4 ,i also take for granted1 that what is meant by ,classicalism in the above quotation1 is the ancient ,grecian style of art2 by ,medivalism1 the semi-barbaric style of the middle ages2 and by ,modernism1 that chaotic jumble of all previous styles and fashions of art1 which is the peculiar characteristic of our present school1 and which is1 doubtless1 the result of a system of education based upon plagiarism and mere imitation4 ,therefore a recommendation to embrace with equal fervour 8as good things10 these very opposite articisms 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 times4 ,now1 8the world of nature0 and 8the world of art0 have not that equality of scope which this lecturer on beauty ascribes to them1 but differ very decidedly in that particular4 ,neither will it be difficult to shew why 8the world of art should be more narrow than the world of nature0--that it should be thereby rendered more monotonous does not follow4 ,it is well known1 that the 8world of nature0 consists of productions1 including objects of every degree of beauty from the very lowest to the highest1 and calculated to suit not only the tastes arising from various degrees of intellect1 but those arising from the natural instincts of the lower animals4 ,on the other hand1 8the world of art10 being devoted to the gratification and improvement of intelligent minds only1 is therefore narrowed in its scope by the exclusion from its productions of the lower degrees of beauty--even mediocrity is inadmissible2 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 beauty4 ,consequently1 all objects in nature are not equally well adapted for artistic study1 and it therefore requires1 on the part of the artist1 besides true genius1 much experience and care to enable him to choose proper subjects from nature2 and it is in the choice of such subjects1 and not in plagiarism from the ancients1 that he should select with knowledge and adapt with wisdom4 ,hence1 all such latitudinarian doctrines as those ,i have quoted must act as a check upon the progress of knowledge in the scientific truth of art4 ,i have observed in some of my works1 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 investigations2 for our ideas of visible beauty are still undefined1 and our attempts to produce it in the various branches of art are left dependant1 in a great measure1 upon chance4 ,our schools are conducted without reference to any first principles or definite laws of beauty1 and from the drawing of a simple architectural moulding to the intricate combinations of form in the human figure1 the pupils trust to their hands and eyes alone1 servilely and mechanically copying the works of the ancients1 instead of being instructed in the unerring principles of science1 upon which the beauty of those works normally depends4 ,the instruction they receive is imparted without reference to the judgment or understanding1 and they are thereby led to imitate effects without investigating causes4 ,doubtless1 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 based2 but it should be kept in mind1 that true genius includes an intuitive perception of those principles along with its creative power4 ,it is1 therefore1 to the generality of mankind that instruction in the definite laws of beauty will be of most service1 not only in improving the practice of those who follow the arts professionally1 but in enabling all of us to distinguish the true from the false1 and to exercise a sound and discriminating taste in forming our judgment upon artistic productions4 sthetic culture should consequently supersede servile copying1 as the basis of instruction in our schools of art4 ,many teachers of drawing1 however1 still assert1 that1 by copying the great works of the ancients1 the mind of the pupil will become imbued with ideas similar to theirs--that he will imbibe their feeling for the beautiful1 and thereby become inspired with their genius1 and think as they thought4 ,to study carefully and to investigate the principles which constitute the excellence of the works of the ancients1 is no doubt of much benefit to the student2 but it would be as unreasonable to suppose that he should become inspired with artistic genius by merely copying them1 as it would be to imagine1 that1 in literature1 poetic inspiration could be created by making boys transcribe or repeat the works of the ancient poets4 ,sir ,joshua ,reynolds considered copying as a delusive kind of industry1 and has observed1 that 8,nature herself is not to be too closely copied10 asserting that 8there are excellences in the art of painting beyond what is commonly called the imitation of nature10 and that 8a mere copier of nature can never produce any thing great40 ,proclus1 an eminent philosopher and mathematician of the later ,platonist school 7,a4,d4 #dhe71 says1 that 8he who takes for his model such forms as nature produces1 and confines himself to an exact imitation of these1 will never attain to what is perfectly beautiful4 ,for the works of nature are full of disproportion1 and fall very short of the true standard of beauty40 ,it is remarked by ,mr4 ,j4 ,c4 ,daniel1 in the introduction to his translation of ,m4 ,victor ,cousin's 8,philosophy of the ,beautiful10 that 8the ,english writers have advocated no theory which allows the beautiful to be universal and absolute2 nor have they professedly founded their views on original and ultimate principles4 ,thus the doctrine of the ,english school has for the most part been1 that beauty is mutable and special1 and the inference that has been drawn from this teaching is1 that all tastes are equally just1 provided that each man speaks of what he feels40 ,he then observes1 that the ,german1 and some of the ,french writers1 have thought far differently2 for with them the beautiful is 8simple1 immutable1 absolute1 though its forms are manifold40 ,so far back as the year #agbe1 the same truths advanced by the modern ,german and ,french writers1 and so eloquently illustrated by ,m4 ,cousin1 were given to the world by ,hutchison in his 8,inquiry into the ,original of our ,ideas of ,beauty and ,virtue40 ,this author says--8,we1 by absolute beauty1 understand only that beauty which we perceive in objects1 without comparison to any thing external1 of which the object is supposed an imitation or picture1 such as the beauty perceived from the works of nature1 artificial forms1 figures1 theorems4 ,comparative or relative beauty is that which we perceive in objects commonly considered as imitations or resemblances of something else40 ,dr4 ,reid also1 in his 8,intellectual ,powers of ,man10 says--8,that taste1 which we may call rational1 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 kind1 the pleasure being annexed to this judgment1 and regulated by it4 ,this taste may be true or false1 according as it is founded on a true or false judgment4 ,and if it may be true or false1 it must have first principles40 ,m4 ,victor ,cousin's opinion upon this subject is1 however1 still more conclusive4 ,he observes--8,if the idea of the beautiful is not absolute1 like the idea of the true--if it is nothing more than the expression of individual sentiment1 the rebound of a changing sensation1 or the result of each person's fancy--then the discussions on the fine arts waver without support1 and will never end4 ,for a theory of the fine arts to be possible1 there must be something absolute in beauty1 just as there must be something absolute in the idea of goodness1 to render morals a possible science40 ,the basis of the science of beauty must thus be founded upon fixed principles1 and when these principles are evolved with the same care which has characterised the labours of investigators in natural science1 and are applied in the fine arts as the natural sciences have been in the useful arts1 a solid foundation will be laid1 not only for correct practice1 but also for a just appreciation of productions in every branch of the arts of design4 ,we know that the mind receives pleasure through the sense of hearing1 not only from the music of nature1 but from the euphony of prosaic composition1 the rhythm of poetic measure1 the artistic composition of successive harmony in simple melody1 and the combined harmony of counterpoint in the more complex works of that art4 ,we know1 also1 that the mind is similarly gratified through the sense of seeing1 not only by the visible beauties of nature1 but by those of art1 whether in symmetrical or picturesque compositions of forms1 or in harmonious arrangements of gay or sombre colouring4 ,now1 in respect to the first of these modes of sensation1 we know1 that from the time of ,pythagoras1 the fact has been established1 that in whatever manner nature or art may address the ear1 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 mind2 and it is my object in this1 as it has been in former attempts1 to prove it consistent with scientific truth1 that that beauty which is addressed to the mind by objects of nature and art1 through the eye1 is similarly governed4 ,in short1 to shew that1 as in compositions of sounds1 there can be no true beauty in the absence of a strict obedience to this great law of nature1 neither can there exist1 in compositions of forms or colours1 that principle of unity in variety which constitutes beauty1 unless such compositions are governed by the same law4 ,although in the songs of birds1 the gurgling of brooks1 the sighing of the gentle summer winds1 and all the other beautiful music of nature1 no analysis might be able to detect the operation of any precise system of harmony1 yet the pleasure thus afforded to the human mind we know to arise from its responding to every development of an obedience to this law4 ,when1 in like manner1 we find even in those compositions of forms and colours which constitute the wildest and most rugged of ,nature's scenery1 a species of picturesque grandeur and beauty to which the mind as readily responds as to her more mild and pleasing aspects1 or to her sweetest music1 we may rest assured that this beauty is simply another development of1 and response to1 the same harmonic law1 although the precise nature of its operation may be too subtle to be easily detected4 ,the rsum of the various works ,i have already published upon the subject1 along with the additional illustrations ,i am about to lay before my readers1 will1 ,i trust1 point out a system of harmony1 which1 in formative art1 as well as in that of colouring1 will rise superior to the idiosyncracies of different artists1 and bring back to one common type the sensations of the eye and the ear1 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 arts4 ,in attempting this1 however1 ,i beg it may be understood1 that ,i do not believe any system1 based even upon the laws of nature1 capable of forming a royal road to the perfection of art1 or of 8mapping the mighty maze of a creative mind40 ,at the same time1 however1 ,i must continue to reiterate the fact1 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 genius4 ,although mere learning and true genius differ as much in the practice of art as they do in the practice of literature1 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 one1 as a precise and systematic education in the science of philology is in promoting the practice and appreciation of the other4 ,as all beauty is the result of harmony1 it will be requisite here to remark1 that harmony is not a simple quality1 but1 as ,aristotle defines it1 8the union of contrary principles having a ratio to each other40 ,harmony thus operates in the production of all that is beautiful in nature1 whether in the combinations1 in the motions1 or in the affinities of the elements of matter4 ,the contrary principles to which ,aristotle alludes1 are those of uniformity and variety2 for1 according to the predominance of the one or the other of these principles1 every kind of beauty is characterised4 ,hence the difference between symmetrical and picturesque beauty3--the first allied to the principle of uniformity1 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 subtilely1 that they cannot be detected in its constitution1 but are only felt in the response by which true genius acknowledges their presence4 ,the generality of mankind may be capable of perceiving this latter kind of beauty1 and of feeling its effects upon the mind1 but men of genius1 only1 can impart it to works of art1 whether addressed to the eye or the ear4 ,throughout the sounds1 forms1 and colours of nature1 these two kinds of beauty are found not only in distinct developments1 but in every degree of amalgamation4 ,we find in the songs of some birds1 such as those of the chaffinch1 thrush1 c41 a rhythmical division1 resembling in some measure the symmetrically precise arrangements of parts which characterises all artistic musical composition2 while in the songs of other birds1 and in the other numerous melodies with which nature charms and soothes the mind1 there is no distinct regularity in the division of their parts4 ,in the forms of nature1 too1 we find amongst the innumerable flowers with which the surface of the earth is so profusely decorated1 an almost endless variety of systematic arrangements of beautiful figures1 often so perfectly symmetrical in their combination1 that the most careful application of the angleometer could scarcely detect the slightest deviation from geometrical precision2 while1 amongst the masses of foliage by which the forms of many trees are divided and subdivided into parts1 as also amongst the hills and valleys1 the mountains and ravines1 which divide the earth's surface1 we find in every possible variety of aspect the beauty produced by that irregular species of symmetry which characterises the picturesque4 ,in like manner1 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 contrasts1 often as mathematically regular as the geometric diagrams by which writers upon colour sometimes illustrate their works2 while in the general colouring of the picturesque beauties of nature1 there is an endless variety in its distributions1 its blendings1 and its modifications4 ,in the forms and colouring of animals1 too1 the same endless variety of regular and irregular symmetry is to be found4 ,but the highest degree of beauty in nature is the result of an equal balance of uniformity with variety4 ,of this the human figure is an example2 because1 when it is of those proportions universally acknowledged to be the most perfect1 its uniformity bears to its variety an apparently equal ratio4 ,the harmony of combination in the normal proportions of its parts1 and the beautifully simple harmony of succession in the normal melody of its softly undulating outline1 are the perfection of symmetrical beauty1 while the innumerable changes upon the contour which arise from the actions and attitudes occasioned by the various emotions of the mind1 are calculated to produce every species of picturesque beauty1 from the softest and most pleasing to the grandest and most sublime4 ,amongst the purely picturesque objects of inanimate nature1 ,i may1 as in a former work1 instance an ancient oak tree1 for its beauty is enhanced by want of apparent symmetry4 ,thus1 the more fantastically crooked its branches1 and the greater the dissimilarity and variety it exhibits in its masses of foliage1 the more beautiful it appears to the artist and the amateur2 and1 as in the human figure1 any attempt to produce variety in the proportions of its lateral halves would be destructive of its symmetrical beauty1 so in the oak tree any attempt to produce palpable similarity between any of its opposite sides would equally deteriorate its picturesque beauty4 ,but picturesque beauty is not the result of the total absence of symmetry2 for1 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 obedience1 however subtle1 to the great harmonic law of ,nature1 so neither could any object be picturesquely beautiful1 unless the arrangement of its parts yields1 although it may be obscurely1 an obedience to the same law4 ,however symmetrically beautiful any architectural structure may be1 when in a complete and perfect state1 it must1 as it proceeds towards ruin1 blend the picturesque with the symmetrical2 but the type of its beauty will continue to be the latter1 so long as a sufficient portion of it remains to convey an idea of its original perfection4 ,it is the same with the human form and countenance2 for age does not destroy their original beauty1 but in both only lessens that which is symmetrical1 while it increases that which is picturesque4 ,in short1 as a variety of simultaneously produced sounds1 which do not relate to each other agreeably to this law1 can only convey to the mind a feeling of mere noise2 so a variety of forms or colours simultaneously exposed to the eye under similar circumstances1 can only convey to the mind a feeling of chaotic confusion1 or what may be termed visible discord4 ,as1 therefore1 the two principles of uniformity and variety1 or similarity and dissimilarity1 are in operation in every harmonious combination of the elements of sound1 of form1 and of colour1 we must first have recourse to numbers in the abstract before we can form a proper basis for a universal science of beauty4 ,,the ,,science ,,of ,,beauty ,,evolved ,,from ,,the ,,harmonic ,,law ,,of ,,nature1 ,,agreeably ,,to ,,the ,,pythagorean ,,system ,,of ,,numerical ,,ratio4 ,the scientific principles of beauty appear to have been well known to the ancient ,greeks2 and it must have been by the practical application of that knowledge to the arts of ,design1 that that people continued for a period of upwards of three hundred years to execute1 in every department of these arts1 works surpassing in chaste beauty any that had ever before appeared1 and which have not been equalled during the two thousand years which have since elapsed4 sthetic science1 as the science of beauty is now termed1 is based upon that great harmonic law of nature which pervades and governs the universe4 ,it is in its nature neither absolutely physical nor absolutely metaphysical1 but of an intermediate nature1 assimilating in various degrees1 more or less1 to one or other of those opposite kinds of science4 ,it specially embodies the inherent principles which govern impressions made upon the mind through the senses of hearing and seeing4 ,thus1 the sthetic pleasure derived from listening to the beautiful in musical composition1 and from contemplating the beautiful in works of formative art1 is in both cases simply a response in the human mind to artistic developments of the great harmonic law upon which the science is based4 ,although the eye and the ear are two different senses1 and1 consequently1 various in their modes of receiving impressions2 yet the sensorium is but one1 and the mind by which these impressions are perceived and appreciated is also characterised by unity4 ,there appears1 likewise1 a striking analogy between the natural constitution of the two kinds of beauty1 which is this1 that the more physically sthetic elements of the highest works of musical composition are melody1 harmony1 and tone1 whilst those of the highest works of formative art are contour1 proportion1 and colour4 ,the melody or theme of a musical composition and its harmony are respectively analogous1--#ast1 ,to the outline of an artistic work of formative art2 and #b;d1 ,to the proportion which exists amongst its parts4 ,to the careful investigator these analogies become identities in their effect upon the mind1 like those of the more metaphysically sthetic emotions produced by expression in either of these arts4 ,agreeably to the first analogy1 the outline and contour of an object1 suppose that of a building in shade when viewed against a light background1 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 counterpoint4 ,agreeably to the second analogy1 the various parts into which the surface of the supposed elevation is divided being simultaneously presented to the eye1 will1 if arranged agreeably to the same great law1 affect the mind like that of an equally harmonious arrangement of musical notes accompanying the supposed melody4 ,there is1 however1 a difference between the construction of these two organs of sense1 viz41 that the ear must in a great degree receive its impressions involuntarily2 while the eye1 on the other hand1 is provided by nature with the power of either dwelling upon1 or instantly shutting out or withdrawing itself from an object4 ,the impression of a sound1 whether simple or complex1 when made upon the ear1 is instantaneously conveyed to the mind2 but when the sound ceases1 the power of observation also ceases4 ,but the eye can dwell upon objects presented to it so long as they are allowed to remain pictured on the retina2 and the mind has thereby the power of leisurely examining and comparing them4 ,hence the ear guides more as a mere sense1 at once and without reflection2 whilst the eye1 receiving its impressions gradually1 and part by part1 is more directly under the influence of mental analysis1 consequently producing a more metaphysically sthetic emotion4 ,hence1 also1 the acquired power of the mind in appreciating impressions made upon it through the organ of sight under circumstances1 such as perspective1 c41 which to those who take a hasty view of the subject appear impossible4 ,dealing as this science therefore does1 alike with the sources and the resulting principles of beauty1 it is scarcely less dependent on the accuracy of the senses than on the power of the understanding1 inasmuch as the effect which it produces is as essential a property of objects1 as are its laws inherent in the human mind4 ,it necessarily comprehends a knowledge of those first principles in art1 by which certain combinations of sounds1 forms1 and colours produce an effect upon the mind1 connected1 in the first instance1 with sensation1 and in the second with the reasoning faculty4 ,it is1 therefore1 not only the basis of all true practice in art1 but of all sound judgment on questions of artistic criticism1 and necessarily includes those laws whereon a correct taste must be based4 ,doubtless many eloquent and ingenious treatises have been written upon beauty and taste2 but in nearly every case1 with no other effect than that of involving the subject in still greater uncertainty4 ,even when restricted to the arts of design1 they have failed to exhibit any definite principles whereby the true may be distinguished from the false1 and some natural and recognised laws of beauty reduced to demonstration4 ,this may be attributed1 in a great degree1 to the neglect of a just discrimination between what is merely agreeable1 or capable of exciting pleasurable sensations1 and what is essentially beautiful2 but still more to the confounding of the operations of the understanding with those of the imagination4 ,very slight reflection1 however1 will suffice to shew how essentially distinct these two faculties of the mind are2 the former being regulated1 in matters of taste1 by irrefragable principles existing in nature1 and responded to by an inherent principle existing in the human mind2 while the latter operates in the production of ideal combinations of its own creation1 altogether independent of any immediate impression made upon the senses4 ,the beauty of a flower1 for example1 or of a dew-drop1 depends on certain combinations of form and colour1 manifestly referable to definite and systematic1 though it may be unrecognised1 laws2 but when ,oberon1 in 8,midsummer ,night's ,dream10 is made to exclaim-- 8,and that same dew1 which sometimes on the buds ,was wont to swell1 like round and orient pearls1 ,stood now within the pretty floweret's eyes1 ,like tears that did their own disgrace bewail10-- the poet introduces a new element of beauty equally legitimate1 yet altogether distinct from1 although accompanying that which constitutes the more precise science of sthetics as here defined4 ,the composition of the rhythm is an operation of the understanding1 but the beauty of the poetic fancy is an operation of the imagination4 ,our physical and mental powers1 sthetically considered1 may therefore be classed under three heads1 in their relation to the fine arts1 viz41 the receptive1 the perceptive1 and the conceptive4 ,the senses of hearing and seeing are respectively1 in the degree of their physical power1 receptive of impressions made upon them1 and of these impressions the sensorium1 in the degree of its mental power1 is perceptive4 ,this perception enables the mind to form a judgment whereby it appreciates the nature and quality of the impression originally made on the receptive organ4 ,the mode of this operation is intuitive1 and the quickness and accuracy with which the nature and quality of the impression is apprehended1 will be in the degree of the intellectual vigour of the mind by which it is perceived4 ,thus we are1 by the cultivation of these intuitive faculties1 enabled to decide with accuracy as to harmony or discord1 proportion or deformity1 and assign sound reasons for our judgment in matters of taste4 ,but mental conception is the intuitive power of constructing original ideas from these materials2 for after the receptive power has acted1 the perception operates in establishing facts1 and then the judgment is formed upon these operations by the reasoning powers1 which lead1 in their turn1 to the creations of the imagination4 ,the power of forming these creations is the true characteristic of genius1 and determines the point at which art is placed beyond all determinable canons1--at which1 indeed1 sthetics give place to metaphysics4 ,in the science of beauty1 therefore1 the human mind is the subject1 and the effect of external nature1 as well as of works of art1 the object4 ,the external world1 and the individual mind1 with all that lies within the scope of its powers1 may be considered as two separate existences1 having a distinct relation to each other4 ,the subject is affected by the object1 through that inherent faculty by which it is enabled to respond to every development of the all-governing harmonic law of nature2 and the media of communication are the sensorium and its inlets--the organs of sense4 ,this harmonic law of nature was either originally discovered by that illustrious philosopher ,pythagoras1 upwards of five hundred years before ,christ1 or a knowledge of it obtained by him about that period1 from the ,egyptian or ,chaldean priests4 ,for after having been initiated into all the ,grecian and barbarian sacred mysteries1 he went to ,egypt1 where he remained upwards of twenty years1 studying in the colleges of its priests2 and from ,egypt he went into the ,east1 and visited the ,persian and ,chaldean magi4#c ,by the generality of the biographers of ,pythagoras1 it is said to be difficult to give a clear idea of his philosophy1 as it is almost certain he never committed it to writing1 and that it has been disfigured by the fantastic dreams and chimeras of later ,pythagoreans4 ,diogenes ,lartius1 however1 whose 8,lives of the ,philosophers0 was supposed to be written about the end of the second century of our era1 says 8there are three volumes extant written by ,pythagoras4 ,one on education1 one on politics1 and one on natural philosophy40 ,and adds1 that there were several other books extant1 attributed to ,pythagoras1 but which were not written by him4 ,also1 in his 8,life of ,philolaus10 that ,plato wrote to ,dion to take care and purchase the books of ,pythagoras4#d ,but whether this great philosopher committed his discoveries to writing or not1 his doctrines regarding the philosophy of beauty are well-known to be1 that he considered numbers as the essence and the principle of all things1 and attributed to them a real and distinct existence2 so that1 in his view1 they were the elements out of which the universe was constructed1 and to which it owed its beauty4 ,diogenes ,lartius gives the following account of this law3--8,that the monad was the beginning of everything4 ,from the monad proceeds an indefinite duad1 which is subordinate to the monad as to its cause4 ,that from the monad and indefinite duad proceeds numbers4 ,that the part of science to which ,pythagoras applied himself above all others1 was arithmetic2 and that he taught #8that from numbers proceed signs1 and from these latter1 lines1 of which plane figures consist2 that from plane figures are derived solid bodies2 that of all plane figures the most beautiful was the circle1 and of all solid bodies the most beautiful was the sphere4' ,he discovered the numerical relations of sounds on a single string2 and taught that everything owes its existence and consistency to harmony4 ,in so far as ,i know1 the most condensed account of all that is known of the ,pythagorian system of numbers is the following3--#8,the monad or unity is that quantity1 which1 being deprived of all number1 remains fixed4 ,it is the fountain of all number4 ,the duad is imperfect and passive1 and the cause of increase and division4 ,the triad1 composed of the monad and duad1 partakes of the nature of both4 ,the tetrad1 tetractys1 or quaternion number is most perfect4 ,the decad1 which is the sum of the four former1 comprehends all arithmetical and musical proportions4'0#e ,these short quotations1 ,i believe1 comprise all that is known1 for certain1 of the manner in which ,pythagoras systematised the law of numbers4 ,yet1 from the teachings of this great philosopher and his disciples1 the harmonic law of nature1 in which the fundamental principles of beauty are embodied1 became so generally understood and universally applied in practice throughout all ,greece1 that the fragments of their works1 which have reached us through a period of two thousand years1 are still held to be examples of the highest artistic excellence ever attained by mankind4 ,in the present state of art1 therefore1 a knowledge of this law1 and of the manner in which it may again be applied in the production of beauty in all works of form and colour1 must be of singular advantage2 and the object of this work is to assist in the attainment of such a knowledge4 ,it has been remarked1 with equal comprehensiveness and truth1 by a writer#f in the ,british and ,foreign ,medical ,review1 that 8there is harmony of numbers in all nature--in the force of gravity--in the planetary movements--in the laws of heat1 light1 electricity1 and chemical affinity--in the forms of animals and plants--in the perceptions of the mind4 ,the direction1 indeed1 of modern natural and physical science is towards a generalization which shall express the fundamental laws of all by one simple numerical ratio4 ,and we think modern science will soon shew that the mysticism of ,pythagoras was mystical only to the unlettered1 and that it was a system of philosophy founded on the then existing mathematics1 which latter seem to have comprised more of the philosophy of numbers than our present40 ,many years of careful investigation have convinced me of the truth of this remark1 and of the great advantage derivable from an application of the ,pythagorean system in the arts of design4 ,for so simple is its nature1 that any one of an ordinary capacity of mind1 and having a knowledge of the most simple rules of arithmetic1 may1 in a very short period1 easily comprehend its nature1 and be able to apply it in practice4 ,the elements of the ,pythagorean system of harmonic number1 so far as can be gathered from the quotations ,i have given above1 seem to be simply the indivisible monad 7#a72 the duad 7#b71 arising from the union of one monad with another2 the triad 7#c71 arising from the union of the monad with the duad2 and the tetrad 7#d71 arising from the union of one duad with another1 which tetrad is considered a perfect number4 ,from the union of these four elements arises the decad 7#aj71 the number1 which1 agreeably to the ,pythagorean system1 comprehends all arithmetical and harmonic proportions4 ,if1 therefore1 we take these elements and unite them progressively in the following order1 we shall find the series of harmonic numbers 7#b71 7#c71 7#e71 and 7#g71 which1 with their multiples1 are the complete numerical elements of all harmony1 thus3-- #a #a #b #a #b #c #b #c #e #c #d #g ,in order to render an extended series of harmonic numbers useful1 it must be divided into scales2 and it is a rule in the formation of these scales1 that the first must begin with the monad 7#a7 and end with the duad 7#b71 the second begin with the duad 7#b7 and end with the tetrad 7#d71 and that the beginning and end of all other scales must be continued in the same arithmetical progression4 ,these primary elements will then form the foundation of a series of such scales4 ,i4 7#a7 7#b7 ,,ii4 7#b7 7#c7 7#d7 ,,iii4 7#d7 7#e7 7#f7 7#g7 7#h7 ,,iv4 7#h7 7#i7 7#aj7 7 7 7#ab7 7 7 7#ad7 7#ae7 7#af7 ,the first of these scales has in 7#a7 and 7#b7 a beginning and an end2 but the second has in 7#b71 7#c71 and 7#d7 the essential requisites demanded by ,aristotle in every composition1 viz41 8a beginning1 a middle1 and an end20 while the third has not only these essential requisites1 but two intermediate parts 7#e7 and 7#g71 by which the beginning1 the middle1 and the end are united4 ,in the fourth scale1 however1 the arithmetical progression is interrupted by the omission of numbers #aa and #ac1 which1 not being multiples of either 7#b71 7#c71 7#e71 or 7#g71 are inadmissible4 ,such is the nature of the harmonic law which governs the progressive scales of numbers by the simple multiplication of the monad4 ,i shall now use these numbers as divisors in the formation of a series of four such scales of parts1 which has for its primary element1 instead of the indivisible monad1 a quantity which may be indefinitely divided1 but which cannot be added to or multiplied4 ,like the monad1 however1 this quantity is represented by 7#a74 ,the following is this series of four scales of harmonic parts3-- ,i4 7#a7 77 ,,ii4 77 77 77 ,,iii4 77 77 77 77 77 ,,iv4 77 77 77 7 7 77 7 7 77 77 77 ,the scales ,i41 ,,ii41 and ,,iii4 may now be rendered as complete as scale ,,iv41 simply by multiplying upwards by #b from 771 771 771 771 and 771 thus3-- ,i4 7#a7 77 77 77 77 77 77 ,,ii4 77 77 77 77 77 77 77 ,,iii4 77 77 77 77 77 77 77 ,,iv4 77 77 77 7 7 77 7 7 77 77 77 ,we now find between the beginning and the end of scale ,i4 the quantities 771 771 771 771 and 774 ,the three first of these quantities we find to be the remainders of the whole indefinite quantity contained in 7#a71 after subtracting from it the primary harmonic quantities 771 771 and 772 we1 however1 find also amongst these harmonic quantities that of 771 which being subtracted from 7#a7 leaves 771 a quantity the most suitable whereby to fill up the hiatus between 77 and 77 in scale ,i41 which arises from the omission of 77 in scale ,,iv4 ,in like manner we find the two last of these quantities1 77 and 771 are respectively the largest of the two parts into which #g and #ae are susceptible of being divided4 ,finding the number #e to be divisible into parts more unequal than 7#b7 to 7#c7 and less unequal than 7#d7 to 7#g71 77 naturally fills up the hiatus between these quantities in scale ,i41 which hiatus arises from the omission of 77 in scale ,,iv4 ,thus3-- ,i4 7#a7 77 77 77 77 77 77 77 77 ,,ii4 77 77 77 7 7 77 7 7 77 77 77 ,,iii4 77 77 77 7 7 77 7 7 77 77 77 ,,iv4 77 77 77 7 7 77 7 7 77 77 77 ,scale ,i4 being now complete1 we have only to divide these latter quantities by 7#b7 downwards in order to complete the other three4 ,thus3-- ,i4 7#a7 77 77 77 77 77 77 77 77 ,,ii4 77 77 77 77 77 77 77 77 77 ,,iii4 77 77 77 77 77 77 77 77 77 ,,iv4 77 77 77 77 77 77 77 77 77 ,the harmony existing amongst these numbers or quantities consists of the numerical relations which the parts bear to the whole and to each other2 and the more simple these relations are1 the more perfect is the harmony4 ,the following are the numerical harmonic ratios which the parts bear to the whole3-- ,i4 7#a3a7 7#h3i7 7#d3 #e7 7#c3 #d7 7#b3 #c7 7#c3 #e7 7#d3 #g7 7#h3ae7 7#a3 #b7 ,,ii4 7#a3b7 7#d3i7 7#b3 #e7 7#c3 #h7 7#a3 #c7 7#c3aj7 7#b3 #g7 7#d3ae7 7#a3 #d7 ,,iii4 7#a3d7 7#b3i7 7#a3 #e7 7#c3af7 7#a3 #f7 7#c3bj7 7#a3 #g7 7#b3ae7 7#a3 #h7 ,,iv4 7#a3h7 7#a3i7 7#a3aj7 7#c3cb7 7#a3ab7 7#c3dj7 7#a3ad7 7#a3ae7 7#a3af7 ,the following are the principal numerical relations which the parts in each scale bear to one another3-- 77377 7#g3h7 77377 7#i3aj7 77377 7#e3f7 77377 7#f3g7 77377 7#ad3ae7 77377 7#ae3af7 ,although these relations are exemplified by parts of scale ,i41 the same ratios exist between the relative parts of scales ,,ii41 ,,iii41 and ,,iv41 and would exist between the parts of any other scales that might be added to that series4 ,these are the simple elements of the science of that harmony which pervades the universe1 and by which the various kinds of beauty sthetically impressed upon the senses of hearing and seeing are governed4 ,,the ,,science ,,of ,,beauty ,,as ,,applied ,,to ,,sounds4 ,it is well-known that all sounds arise from a peculiar action of the air1 and that this action may be excited by the concussion resulting from the sudden displacement of a portion of the atmosphere itself1 or by the rapid motions of bodies1 or of confined columns of air2 in all which cases1 when the motions are irregular1 and the force great1 the sound conveyed to the sensorium is called a noise4 ,but that musical sounds are the result of equal and regular vibratory motions1 either of an elastic body1 or of a column of air in a tube1 exciting in the surrounding atmosphere a regular and equal pulsation4 ,the ear is the medium of communication between those varieties of atmospheric action and the seat of consciousness4 ,to describe fully the beautiful arrangement of the various parts of this organ1 and their adaptation to the purpose of collecting and conveying these undulatory motions of the atmosphere1 is as much beyond the scope of my present attempt as it is beyond my anatomical knowledge2 but ,i may simply remark1 that within the ear1 and most carefully protected in the construction of that organ1 there is a small cavity containing a pellucid fluid1 in which the minute extremities of the auditory nerve float2 and that this fluid is the last of the media through which the action producing the sensation of sound is conveyed to the nerve1 and thence to the sensorium1 where its nature becomes perceptible to the mind4 ,the impulses which produce musical notes must arrive at a certain frequency before the ear loses the intervals of silence between them1 and is impressed by only one continued sound2 and as they increase in frequency the sound becomes more acute upon the ear4 ,the pitch of a musical note is1 therefore1 determined by the frequency of these impulses2 but1 on the other hand1 its intensity or loudness will depend upon the violence and the quality of its tone on the material employed in producing them4 ,all such sounds1 therefore1 whatever be their loudness or the quality of their tone in which the impulses occur with the same frequency are in perfect unison1 having the same pitch4 ,upon this the whole doctrine of harmonies is founded1 and by this the laws of numerical ratio are found to operate in the production of harmony1 and the theory of music rendered susceptible of exact reasoning4 ,the mechanical means by which such sounds can be produced are extremely various2 but1 as it is my purpose simply to shew the nature of harmony of sound as related to1 or as evolving numerical harmonic ratio1 ,i shall confine myself to the most simple mode of illustration--namely1 that of the monochord4 ,this is an instrument consisting of a string of a given length stretched between two bridges standing upon a graduated scale4 ,suppose this string to be stretched until its tension is such that1 when drawn a little to a side and suddenly let go1 it would vibrate at the rate of #fd vibrations in a second of time1 producing to a certain distance in the surrounding atmosphere a series of pulsations of the same frequency4 ,these pulsations will communicate through the ear a musical note which would1 therefore1 be the fundamental note of such a string4 ,now1 the phenomenon said to be discovered by ,pythagoras is well known to those acquainted with the science of acoustics1 namely1 that immediately after the string is thus put into vibratory motion1 it spontaneously divides itself1 by a node1 into two equal parts1 the vibrations of each of which occur with a double frequency--namely1 #abh in a second of time1 and1 consequently1 produce a note doubly acute in pitch1 although much weaker as to intensity or loudness2 that it then1 while performing these two series of vibrations1 divides itself1 by two nodes1 into three parts1 each of which vibrates with a frequency triple that of the whole string2 that is1 performs #aib vibrations in a second of time1 and produces a note corresponding in increase of acuteness1 but still less intense than the former1 and that this continues to take place in the arithmetical progression of #b1 #c1 #d1 c4 ,simultaneous vibrations1 agreeably to the same law of progression1 which1 however1 seem to admit of no other primes than the numbers #b1 #c1 #e1 and #g1 are easily excited upon any stringed instrument1 even by the lightest possible touch of any of its strings while in a state of vibratory motion1 and the notes thus produced are distinguished by the name of harmonics4 ,it follows1 then1 that one-half of a musical string1 when divided from the whole by the pressure of the finger1 or any other means1 and put into vibratory motion1 produces a note doubly acute to that produced by the vibratory motion of the whole string2 the third part1 similarly separated1 a note trebly acute2 and the same with every part into which any musical string may be divided4 ,this is the fundamental principle by which all stringed instruments are made to produce harmony4 ,it is the same with wind instruments1 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 organ1 in which the frequency of pulsation becomes greater in an inverse ratio to the length of the pipes4 ,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 exposition4 ,it is here requisite to mention1 that in the construction of these scales1 ,i have not only adopted the old ,german or literal mode of indicating the notes1 but have included1 as the ,germans do1 the note termed by us ,b flat as ,b natural1 and the note we term ,b natural as ,h4 ,now1 although this arrangement differs from that followed in the construction of our modern ,diatonic scale1 yet as the ratio of #d3g is more closely related to that of #a3b than that of #h3ae1 and as it is offered by nature in the spontaneous division of the monochord1 ,i considered it quite admissible4 ,the figures give the parts of the monochord which would produce the notes4 ,i4 7#a7 77 77 77 77 77 77 77 7799 ,c ,d ,e ,f ,g ,a ,b ,h c ,,ii4 7799 77 77 77 7799 77 77 77 7799 c d e f g a b h c ,,iii4 7799 77 7799 77 7799 77 7799 77 7799 c d e f g a b h c ,,iv4 7799 7799 7799 77 7799 77 7799 7799 7799 c d e f g a b h c ,the notes marked 7997 are the harmonics which naturally arise from the division of the string by #b1 #c1 #e1 and #g1 and the multiples of these primes4 ,thus every musical sound is composed of a certain number of parts called pulsations1 and these parts must in every scale relate harmonically to some fundamental number4 ,when these parts are multiples of the fundamental number by #b1 #d1 #h1 c41 like the pulsations of the sounds indicated by c1 c1 c1 c1 they are called tonic notes1 being the most consonant2 when the pulsations are similar multiples by #c1 #f1 #ab1 c41 like those of the sounds indicated by g1 g1 g1 they are called dominant notes1 being the next most consonant2 and multiples by #e1 #aj1 c41 like those of the sounds indicated by e1 e1 e1 they are called mediant notes1 from a similar cause4 ,in harmonic combinations of musical sounds1 the sthetic 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 belong2 and it will be observed1 that the more simple the numerical ratios are amongst the pulsations of any number of notes simultaneously produced1 the more perfect their agreement4 ,hence the origin of the common chord or fundamental concord in the united sounds of the tonic1 the dominant1 and the mediant notes1 the ratios and coincidences of whose pulsations #b3a1 #c3b1 #e3d1 may thus be exemplified3-- ,illustration ,in musical composition1 the law of number also governs its division into parts1 in order to produce upon the ear1 along with the beauty of harmony1 that of rhythm4 ,thus a piece of music is divided into parts each of which contains a certain number of other parts called bars1 which may be divided and subdivided into any number of notes1 and the performance of each bar is understood to occupy the same portion of time1 however numerous the notes it contains may be2 so that the music of art is regularly symmetrical in its structure2 while that of nature is in general as irregular and indefinite in its rhythm as it is in its harmony4 ,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 ear4 ,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 endowments1 while those who refuse this instruction rarely attain to any excellence4 ,in the sister arts of form and colour1 however1 a system of tuition1 founded upon this law1 is still a desideratum1 and a knowledge of the scientific principles by which these arts are governed is confined to a very few1 and scarcely acknowledged amongst those whose professions most require their practical application4 ,,the ,,science ,,of ,,beauty ,,as ,,applied ,,to ,,forms4 ,it is justly remarked1 in the 8,illustrated ,record of the ,new ,york ,exhibition of #ahec10 that 8it 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 science20 and there is expressed a thorough conviction of the fact that1 besides the evils of the copying system1 8much genius is now wasted in the acquirement of rudimentary knowledge in the slow school of practical experiment1 and that the excellence of the ancient ,greek school of design arose from a thoroughly digested canon of form1 and the use of geometrical formulas1 which make the works even of the second and third-rate genius of that period the wonder and admiration of the present day40 ,that such a canon of form1 and that the use of such geometrical formula1 entered into the education1 and thereby facilitated the practice of ancient ,greek art1 ,i have in a former work expressed my firm belief1 which is founded on the remarkable fact1 that for a period of nearly three centuries1 and throughout a whole country politically divided into states often at war with each other1 works of sculpture1 architecture1 and ornamental design were executed1 which surpass in symmetrical beauty any works of the kind produced during the two thousand years that have since elapsed4 ,so decided is this superiority1 that the artistic remains of the extraordinary period ,i alluded to are1 in all civilised nations1 still held to be the most perfect specimens of formative art in the world2 and even when so fragmentary as to be denuded of everything that can convey an idea of expression1 they still excite admiration and wonder by the purity of their geometric beauty4 ,and so universal was this excellence1 that it seems to have characterised every production of formative art1 however humble the use to which it was applied4 ,the common supposition1 that this excellence was the result of an extraordinary amount of genius existing among the ,greek people during that particular period1 is not consistent with what we know of the progress of mankind in any other direction1 and is1 in the present state of art1 calculated to retard its progress1 inasmuch as such an idea would suggest that1 instead of making any exertion to arrive at a like general excellence1 the world must wait for it until a similar supposed psychological phenomenon shall occur4 ,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 principles1 by which the ordinary amount of genius allotted to mankind in every age was properly nurtured and cultivated2 and by which1 also1 a correct knowledge and appreciation of art were disseminated amongst the people generally4 ,indeed1 ,mller1 in his 8,ancient ,art and its ,remains10 shews clearly that some certain fixed principles1 constituting a science of proportions1 were known in ,greece1 and that they formed the basis of all artists' education and practice during the period referred to2 also1 that art began to decline1 and its brightest period to close1 as this science fell into disuse1 and the ,greek artists1 instead of working for an enlightened community1 who understood the nature of the principles which guided them1 were called upon to gratify the impatient whims of pampered and tyrannical rulers4 ,by being instructed in this science of proportion1 the ,greek artists were enabled to impart to their representations of the human figure a mathematically correct species of symmetrical beauty2 whether accompanying the slender and delicately undulated form of the ,venus1--its opposite1 the massive and powerful mould of the ,hercules1--or the characteristic representation of any other deity in the heathen mythology4 ,and this seems to have been done with equal ease in the minute figure cut on a precious gem1 and in the most colossal statue4 ,the same instruction likewise enabled the architects of ,greece to institute those varieties of proportions in structure called the ,classical ,orders of ,architecture2 which are so perfect that1 since the science which gave them birth has been buried in oblivion1 classical architecture has been little more than an imitative art2 for all who have since written upon the subject1 from ,vitruvius downwards1 have arrived at nothing1 in so far as the great elementary principles in question are concerned1 beyond the most vague and unsatisfactory conjectures4 ,for a more clear understanding of the nature of this application of the ,pythagorean law of number to the harmony of form1 it will be requisite to repeat the fact1 that modern science has shewn that the cause of the impression1 produced by external nature upon the sensorium1 called light1 may be traced to a molecular or ethereal action4 ,this action is excited naturally by the sun1 artificially by the combustion of various substances1 and sometimes physically within the eye4 ,like the atmospheric pulsations which produce sound1 the action which produces light is capable1 within a limited sphere1 of being reflected from some bodies and transmitted through others2 and by this reflection and transmission the visible nature of forms and figures is communicated to the sensorium4 ,the eye is the medium of this communication2 and its structural beauty1 and perfect adaptation to the purpose of conveying this action1 must1 like those of the ear1 be left to the anatomist fully to describe4 ,it is here only necessary to remark1 that the optic nerve1 like the auditory nerve1 ends in a carefully protected fluid1 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 transmitted1 and the nature of such object made perceptible to the mind4 ,the eye and the ear are thus1 in one essential point1 similar in their physiology1 relatively to the means provided for receiving impressions from external nature2 it is1 therefore1 but reasonable to believe that the eye is capable of appreciating the exact subdivision of spaces1 just as the ear is capable of appreciating the exact subdivision of intervals of time2 so that the division of space into exact numbers of equal parts will sthetically affect the mind through the medium of the eye4 ,we assume1 therefore1 that the standard of symmetry1 so estimated1 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 note2 that is1 relations expressed arithmetically by the smallest natural numbers4 ,thus the eye1 being guided in its estimate by direction rather than by distance1 just as the ear is guided by number of vibrations rather than by magnitude1 both it and the ear convey simplicity and harmony to the mind without effort1 and the mind with equal facility receives and appreciates them4 ,on the ,rectilinear ,forms and ,proportions of ,architecture4 ,as we are accustomed in all cases to refer direction to the horizontal and vertical lines1 and as the meeting of these lines makes the right angle1 it naturally constitutes the fundamental angle1 by the harmonic division of which a system of proportion may be established1 and the theory of symmetrical beauty1 like that of music1 rendered susceptible of exact reasoning4 ,let therefore the right angle be the fundamental angle1 and let it be divided upon the quadrant of a circle into the harmonic parts already explained1 thus3-- ,super- ,sub- ,sub- ,sub- ,semi-sub- ,right tonic ,mediant dominant ,dominant mediant tonic tonic ,tonic ,angle4 ,angles4 ,angles4 ,angles4 ,angles4 ,angles4 ,angles4 ,angles4 ,angles4 ,i4 7#a7 77 77 77 77 77 77 77 77 ,,ii4 77 77 77 77 77 77 77 77 77 ,,iii4 77 77 77 77 77 77 77 77 77 ,,iv4 77 77 77 77 77 77 77 77 77 ,in order that the analogy may be kept in view1 ,i have given to the parts of each of these four scales the appropriate nomenclature of the notes which form the diatonic scale in music4 ,when a right angled triangle is constructed so that its two smallest angles are equal1 ,i term it simply the triangle of 771 because the smaller angles are each one-half of the right angle4 ,but when the two angles are unequal1 the triangle may be named after the smallest4 ,for instance1 when the smaller angle1 which we shall here suppose to be one-third of the right angle1 is made with the vertical line1 the triangle may be called the vertical scalene triangle of 772 and when made with the horizontal line1 the horizontal scalene triangle of 774 ,as every rectangle is made up of two of these right angled triangles1 the same terminology may also be applied to these figures4 ,thus1 the equilateral rectangle or perfect square is simply the rectangle of 771 being composed of two similar right angled triangles of 772 and when two vertical scalene triangles of 771 and of similar dimensions1 are united by their hypothenuses1 they form the vertical rectangle of 771 and in like manner the horizontal triangles of 77 similarly united would form the horizontal rectangle of 774 ,as the isosceles triangle is in like manner composed of two right angled scalene triangles joined by one of their sides1 the same terminology may be applied to every variety of that figure4 ,all the angles of the first of the above scales1 except that of 771 give rectangles whose longest sides are in the horizontal line1 while the other three give rectangles whose longest sides are in the vertical line4 ,i have illustrated in ,plate ,i4 the manner in which this harmonic law acts upon these elementary rectilinear figures by constructing a series agreeably to the angles of scales ,,ii41 ,,iii41 ,,iv4 ,throughout this series a b c is the primary scalene triangle1 of which the rectangle a b c e is composed2 d c e the vertical isosceles triangle2 and when the plate is turned1 d e a the horizontal isosceles triangle1 both of which are composed of the same primary scalene triangle4 ,sidenote3 ,plate ,i4 ,thus the most simple elements of symmetry in rectilinear forms are the three following figures3-- ,the equilateral rectangle or perfect square1 ,the oblong rectangle1 and ,the isosceles triangle4 ,it has been shewn that in harmonic combinations of musical sounds1 the sthetic 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 belong2 and that the more simply they relate to each other in this way the more perfect the harmony1 as in the common chord of the first scale1 the relations of whose parts are in the simple ratios of #b3a1 #c3b1 and #e3d4 ,it is equally consistent with this law1 that when applied to form in the composition of an assortment of figures of any kind1 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 parts4 ,now1 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 body4 ,having in previous works#g given the requisite illustrations of this fact in full detail1 ,i shall here confine myself to the most simple kind1 taking for my first example one of the finest specimens of classical architecture in the world--the front portico of the ,parthenon of ,athens4 ,the angles which govern the proportions of this beautiful elevation are the following harmonic parts of the right angle-- ,tonic ,dominant ,mediant ,subtonic ,supertonic ,angles4 ,angles4 ,angles4 ,angle4 ,angles4 77 77 77 77 77 77 77 77 77 77 77 ,sidenote3 ,plate ,,ii4 ,in ,plate ,,ii4 ,i give a diagram of its rectilinear orthography1 which is simply constructed by lines drawn1 either horizontally1 vertically1 or obliquely1 which latter make with either of the former lines one or other of the harmonic angles in the above series4 ,for example1 the horizontal line ,,ab represents the length of the base or surface of the upper step of the substructure of the building4 ,the line ,,ae1 which makes an angle of 77 with the horizontal1 determines the height of the colonnade4 ,the line ,,ad1 which makes an angle of 77 with the horizontal1 determines the height of the portico1 exclusive of the pediment4 ,the line ,,ac1 which makes an angle of 77 with the horizontal1 determines the height of the portico1 including the pediment4 ,the line ,,gd1 which makes an angle of 77 with the horizontal1 determines the form of the pediment4 ,the lines ,,ez and ,,ly1 which respectively make angles of 77 and 77 with the horizontal1 determine the breadth of the architrave1 frieze1 and cornice4 ,the line v n u1 which makes an angle of 77 with the vertical1 determines the breadth of the triglyphs4 ,the line t d1 which makes an angle of 771 determines the breadth of the metops4 ,the lines c b r f1 and a i1 which make each an angle of 77 with the vertical1 determine the width of the five centre intercolumniations4 ,the line z k1 which makes an angle of 77 with the vertical1 determines the width of the two remaining intercolumniations4 ,the lines c s1 q x1 and y h1 each of which makes an angle of 77 with the vertical1 determine the diameters of the three columns on each side of the centre4 ,the line w l1 which makes an angle of 77 with the vertical1 determines the diameter of the two remaining or corner columns4 ,in all this1 the length and breadth of the parts are determined by horizontal and vertical lines1 which are necessarily at right angles with each other1 and the position of which are determined by one or other of the lines making the harmonic angles above enumerated4 ,now1 the lengths and breadths thus so simply determined by these few angles1 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 art4 ,these measurements were obtained by the 8,society of ,dilettanti10 ,london1 who1 expressly for that purpose1 sent ,mr ,f4 ,c4 ,penrose1 a highly educated architect1 to ,athens1 where he remained for about five months1 engaged in the execution of this interesting commission1 the results of which are now published in a magnificent volume by the ,society4#h ,the agreement was so striking1 that ,mr ,penrose has been publicly thanked by an eminent man of science for bearing testimony to the truth of my theory1 who in doing so observes1 8,the dimensions which he 7,mr ,penrose7 gives are to me the surest verification of the theory ,i could have desired4 ,the minute discrepancies form that very element of practical incertitude1 both as to execution and direct measurement1 which always prevails in materialising a mathematical calculation made under such conditions40#i ,although the measurements taken by ,mr ,penrose are undeniably correct1 as all who examine the great work just referred to must acknowledge1 and although they have afforded me the best possible means of testing the accuracy of my theory as applied to the ,parthenon1 yet the ideas of ,mr ,penrose as to the principles they evolve are founded upon the fallacious doctrine which has so long prevailed1 and still prevails1 in the sthetics of architecture1 viz41 that harmony may be imparted by ratios between the lengths and breadths of parts4 ,i have taken for my second example an elevation which1 although of smaller dimensions1 is no less celebrated for the beauty of its proportions than the ,parthenon itself1 viz41 the front portico of the temple of ,theseus1 which has also been measured by ,mr ,penrose4 ,the angles which govern the proportions of this elevation are the following harmonic parts of the right angle3-- ,tonic ,dominant ,mediant ,angles4 ,angles4 ,angles4 77 77 77 77 77 77 77 ,sidenote3 ,plate ,,iii4 ,a diagram of the rectilinear orthography of this portico is given in ,plate ,,iii4 ,its construction is similar to that of the ,parthenon in respect to the harmonic parts of the right angle1 and ,i have therefore only to observe1 that the line ,a ,e makes an angle of 772 the line ,a ,d an angle of 772 the line ,a ,c an angle of 772 the line ,g ,d an angle of 772 and the lines ,e ,z and ,l ,y angles of 77 with the horizontal4 ,as to the colonnade or vertical part1 the line a b1 which determines the three middle intercolumniations1 makes an angle of 772 the line c d1 which determines the two outer intercolumniations1 makes an angle of 772 and the line e f1 which determines the lesser diameter of the columns1 makes an angle of 77 with the vertical4 ,i need give no further details here1 as my intention is to shew the simplicity of the method by which this theory may be reduced to practice1 and because ,i have given in my other works ample details1 in full illustration of the orthography of these two structures1 especially the first4#aj ,the foregoing examples being both horizontal rectangular compositions1 the proportions of their principal parts have necessarily been determined by lines drawn from the extremities of the base1 making angles with the horizontal line1 and forming thereby the diagonals of the various rectangles into which1 in their leading features1 they are necessarily resolved4 ,but the example ,i am now about to give is of another character1 being a vertical pyramidal composition1 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 elevation1 and forming a series of isosceles triangles2 for the isosceles triangle is the type of all pyramidal composition4 ,this third example is the east end of ,lincoln ,cathedral1 a ,gothic structure1 which is acknowledged to be one of the finest specimens of that style of architecture existing in this country4 ,the angles which govern the proportions of this elevation are the following harmonic parts of the right angle3-- ,tonic4 ,dominant4 ,mediant4 ,subtonic4 ,supertonic4 77 77 77 77 77 77 77 77 77 77 ,sidenote3 ,plate ,,iv4 ,in ,plate ,,iv4 ,i give a diagram of the vertical1 horizontal1 and oblique lines1 which compose the orthography of this beautiful elevation4 ,the line ,a ,b represents the full height of this structure4 ,the line ,a ,c1 which makes an angle of 77 with the vertical1 determines the width of the design1 the tops of the aisle windows1 and the bases of the pediments on the inner buttresses2 ,a ,g1 77 with the vertical1 that of the outer buttress2 ,a ,f1 77 with the vertical1 that of the space between the outer and inner buttresses and the width of the great centre window2 and ,a ,e1 77 with vertical1 that of both the inner buttresses and the space between these4 ,a ,h1 which makes 77 with the vertical1 determines the form of the pediment of the centre1 and the full height of the base and surbase4 ,a ,i1 which makes 77 with the vertical1 determines the form of the pediment of the smaller gables1 the base of the pediment on the outer buttress1 the base of the ornamental recess between the outer and inner buttresses1 the spring of the arch of the centre window1 the tops of the pediments on the inner buttresses1 and the spring of the arch of the upper window4 ,a ,k1 which makes 771 determines the height of the outer buttress2 and ,a ,z1 which makes 77 with the horizontal1 determines that of the inner buttresses4 ,for the reasons already given1 ,i need not here go into further detail4#aa ,it is1 however1 worthy of remark in this place1 that notwithstanding the great difference which exists between the style of composition in this ,gothic design1 and in that of the east end of the ,parthenon1 the harmonic elements upon which the orthographic beauty of the one depends1 are almost identical with those of the other4 ,on the ,curvilinear ,forms and ,proportions of ,architecture4 ,each regular rectilinear figure has a curvilinear figure that exclusively belongs to it1 and to which may be applied a corresponding terminology4 ,for instance1 the circle belongs to the equilateral rectangle2 that is1 the rectangle of 771 an ellipse to every other rectangle1 and a composite ellipse to every isosceles triangle4 ,thus the most simple elements of beauty in the curvilinear forms of architectural design are the following three figures3-- ,the circle1 ,the ellipse1 and ,the composite ellipse4 ,i find it necessary in this place to go into some details regarding the specific character of the two latter figures1 because the proper mode of describing these beautiful curves1 and their high value in the practice of the architectural draughtsman and ornamental designer1 seem as yet unknown4 ,in proof of this assertion1 ,i must again refer to ,mr ,penrose's great work published by the 8,society of ,dilettanti40 ,at page #eb of that work it is observed1 that 8by whatever means an ellipse is to be constructed mechanically1 it is a work of time 7if not of absolute difficulty7 so to arrange the foci1 c41 as to produce an ellipse of any exact length and breadth which may be desired40 ,now1 this is far from being the case1 for the method of arranging the foci of an ellipse of any given length and breadth is extremely simple1 being as follows3-- ,let ,a ,b ,c 7figure #a7 be the length1 and ,d ,b ,e the breadth of the desired ellipse4 ,illustration3 ,fig4 #a4 ,take ,a ,b upon the compasses1 and place the point of one leg upon ,e and the point of the other upon the line ,a ,b1 it will meet it at ,f1 which is one focus3 keeping the point of the one leg upon ,e1 remove the point of the other to the line ,b ,c1 and it will meet it at ,g1 which is the other focus4 ,but1 when the proportions of an ellipse are to be imparted by means of one of the harmonic angles1 suppose the angle of 771 then the following is the process3-- ,let ,a ,b ,c 7figure #b7 represent the length of the intended ellipse4 ,through ,b draw ,b e indefinitely1 at right angles with ,a ,b ,c2 through ,c draw the line ,c f indefinitely1 making1 with ,b ,c1 an angle of 774 ,take ,b ,c upon the compasses1 and place the point of one leg upon ,d where ,c f intersects ,b e1 and the point of the other upon the line ,a ,b1 it will meet it at ,f1 which is one focus4 ,keeping the point of one leg still upon ,d1 remove the point of the other to the line ,b ,c1 and it will meet it at ,g1 which is the other focus4 ,illustration3 ,fig4 #b4 ,the foci being in either case thus simply ascertained1 the method of describing the curve on a small scale is equally simple4 ,sidenote3 ,plate ,v4 ,a pin is fixed into each of the two foci1 and another into the point ,d4 ,around these three pins a waxed thread1 flexible but not elastic1 is tied1 care being taken that the knot be of a kind that will not slip4 ,the pin at ,d is now removed1 and a hard black lead pencil introduced within the thread band4 ,the pencil is then moved around the pins fixed in the foci1 keeping the thread band at a full and equal tension2 thus simply the ellipse is described4 ,when1 however1 the governing angle is acute1 say less than 771 it is requisite to adopt a more accurate method of description1#ab as the architectural examples which follow will shew4 ,but architectural draughtsmen and ornamental designers would do well to supply themselves1 for ordinary practice1 with half a dozen series of ellipses1 varying in the proportions of their axes from 77 to 77 of the scale1 and the length of their major axes from #a to #f inches4 ,these should be described by the above simple process1 upon very strong drawing paper1 and carefully cut out1 the edge of the paper being kept smooth1 and each ellipse having its greater and lesser axes1 its foci1 and the hypothenuse of its scalene triangle drawn upon it4 ,to exemplify this1 ,i give ,plate ,v41 which exhibits the ellipses of 771 771 771 and 771 inscribed in their rectangles1 on which a b and c d are respectively the greater and lesser axes1 o o the foci1 and d b the angle of each4 ,such a series of these beautiful figures would be found particularly useful in drawing the mouldings of ,grecian architecture2 for1 to describe the curvilinear contour of such mouldings from single points1 as has been done with those which embellish even our most pretending attempts at the restoration of that classical style of architecture1 is to give the resemblance of an external form without the harmony which constitutes its real beauty4 ,mr ,penrose1 owing to the supposed difficulty regarding the description of ellipses just alluded to1 endeavours to shew that the curves of all the mouldings throughout the ,parthenon were either parabolic or hyperbolic2 but ,i believe such curves can have no connexion with the elementary forms of architecture1 for they are curves which represent motion1 and do not1 by continued production1 form closed figures4 ,but ,i have shewn1 in a former work1#ac that the contours of these mouldings are composed of curves of the composite ellipse1--a figure which ,i so name because it is composed simply of arcs of various ellipses harmonically flowing into each other4 ,the composite ellipse1 when drawn systematically upon the isosceles triangle1 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 architecture1 while those of the parabola and hyperbola1 as ,i have just observed1 are merely curves of motion1 and1 consequently1 never can harmonically circumscribe or be resolved into any regular figure4 ,the composite ellipse may be thus described4 ,sidenote3 ,plate ,,vi4 ,let ,a ,b ,c 7,plate ,,vi47 be a vertical isosceles triangle of 771 bisect ,a ,b in ,d1 and through ,d draw indefinitely ,d f perpendicular to ,a ,b1 and through ,b draw indefinitely ,b g1 making the angle ,d ,b g 771 ,d f and ,b g intersecting each other in ,m4 ,take ,b ,d and ,d ,m as semi-axes of an ellipse1 the foci of which will be at p and q1 in each of these1 and in each of the foci h t and k r in the lines ,a ,c and ,b ,c1 fix a pin1 and one also in the point ,m1 tie a thread around these pins1 withdraw the pin from ,m1 and trace the composite ellipse in the manner already described with respect to the simple ellipse4 ,in some of my earlier works ,i described this figure by taking the angles of the isosceles triangle as foci2 but the above method is much more correct4 ,as the elementary angle of the triangle is 771 and that of the elliptic curve described around it 771 ,i call it the composite ellipse of 77 and 771 their harmonic ratio being #d3c2 and so on of all others1 according to the difference that may thus exist between the elementary angles4 ,the visible curves which soften and beautify the melody of the outline of the front of the ,parthenon1 as given in ,mr ,penrose's great work1 ,i have carefully analysed1 and have found them in as perfect agreement with this system1 as its rectilinear harmony has been shewn to be4 ,this ,i demonstrated in the work just referred to#ad by a series of twelve plates1 shewing that the entasis of the columns 7a subject upon which there has been much speculation7 is simply an arc of an ellipse of 771 whose greater axis makes with the vertical an angle of 772 or simply1 the form of one of these columns is the frustrum of an elliptic-sided or prolate-spheroidal cone1 whose section is a composite ellipse of 77 and 771 the harmonic ratio of these two angles being #d3c1 the same as that of the angles of the composite ellipse just exemplified4 ,sidenote3 ,plate ,,vii4 ,sidenote3 ,plate ,,viii4 ,in ,plate ,,vii4 is represented the section of such a cone1 of which ,a ,b ,c is the isosceles triangle of 771 and ,b ,d and ,d ,m the semi-axes of an ellipse of 774 ,m ,n and ,o ,p are the entases of the column1 and d e f the normal construction of the capital4 ,all these are fully illustrated in the work above referred to1#ae in which ,i have also shewn that the curve of the neck of the column is that of an ellipse of 772 the curve of the capital or echinus1 that of an ellipse of 772 the curve of the moulding under the cymatium of the pediment1 that of an ellipse of 772 and the curve of the bed-moulding of the cornice of the pediment1 that of an ellipse of 774 ,the curve of the cavetto of the soffit of the corona is composed of ellipses of 77 and 772 the curve of the cymatium which surmounts the corona1 is that of an ellipse of 772 the curve of the moulding of the capital of the ant of the posticum1 that of an ellipse of 772 the curves of the lower moulding of the same capital are composed of those of an ellipse of 77 and of the circle 772 the curve of the moulding which is placed between the two latter is that of an ellipse of 772 the curve of the upper moulding of the band under the beams of the ceiling of the peristyle1 that of an ellipse of 772 the curve of the lower moulding of the same band1 that of an ellipse of 772 and the curves of the moulding at the bottom of the small step or podium between the columns1 are those of the circle 77 and of an ellipse of 774 ,i have also shewn the curve of the fluting of the columns to be that of 774 ,the greater axis of each of these ellipses1 when not in the vertical or horizontal lines1 makes an harmonic angle with one or other of them4 ,in ,plate ,,viii41 sections of the two last-named mouldings are represented full size1 which will give the reader an idea of the simple manner in which the ellipses are employed in the production of those harmonic curves4 ,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 composition1 so far from involving us in anything complicated1 is characterised by extreme simplicity4 ,in concluding this part of my treatise1 ,i may here repeat what ,i have advanced in a late work1#af viz41 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 ,freemasons1 all their other secrets being connected1 not with their art1 but with the social constitution of their society4 ,this valuable secret1 however1 seems to have been lost1 as its practical application fell into disuse2 but1 as that ancient society consisted of speculative as well as practical masons1 the secrets connected with their social union have still been preserved1 along with the excellent laws by which the brotherhood is governed4 ,it can scarcely be doubted that there was some such practically useful secret amongst the ,freemasons or early ,gothic architects2 for we find in all the venerable remains of their art which exist in this country1 symmetrical elegance of form pervading the general design1 harmonious proportion amongst all the parts1 beautiful geometrical arrangements throughout all the tracery1 as well as in the elegantly symmetrised foliated decorations which belong to that style of architecture4 ,but it is at the same time worthy of remark1 that whenever they diverged from architecture to sculpture and painting1 and attempted to represent the human figure1 or even any of the lower animals1 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 attitudes1 while their representations of animals and chimeras are whimsically absurd4 ,it would1 therefore1 appear that architecture1 as a fine art1 must have been preserved by some peculiar influence from partaking of the barbarism so apparent in the sister arts of that period4 ,although its practical secrets have been long lost1 the ,freemasons of the present day trace the original possession of them to ,moses1 who1 they say1 8modelled masonry into a perfect system1 and circumscribed its mysteries by land-marks significant and unalterable40 ,now1 as ,moses received his education in ,egypt1 where ,pythagoras is said to have acquired his first knowledge of the harmonic law of numbers1 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 philosophy1 and ultimately led to that excellence in art which is still the admiration of the world4 ,pythagoras1 it would appear1 formed a system much more perfect and comprehensive than that practised by the ,freemasons in the middle ages of ,christianity2 for it was as applicable to sculpture1 painting1 and music1 as it was to architecture4 ,this perfection in architecture is strikingly exemplified in the ,parthenon1 as compared with the ,gothic structures of the middle ages2 for it will be found that the whole six elementary figures ,i have enumerated as belonging to architecture1 are required in completing the orthographic beauty of that noble structure4 ,and amongst these1 none conduce more to that beauty than the simple and composite ellipses4 ,now1 in the architecture of the best periods of ,gothic1 or1 indeed1 in that of any after period 7,roman architecture included71 these beautiful curves seem to have been ignored1 and that of the circle alone employed4 ,be those matters as they may1 however1 the great law of numerical harmonic ratio remains unalterable1 and a proper application of it in the science of art will never fail to be as productive of effect1 as its operation in nature is universal1 certain1 and continual4 ,,the ,,science ,,of ,,beauty1 ,,as ,,developed ,,in ,,the ,,human ,,head ,,and ,,countenance4 ,the most remarkable characteristics of the human head and countenance are the globular form of the cranium1 united as it is with the prolate spheroidal form produced by the parts which constitute the face1 and the approximation of the profile to the vertical2 for in none of the lower animals does the skull present so near a resemblance to a combination of these geometric forms1 nor the plane of the face to this direction4 ,we also find that although these peculiar characteristics are variously modified among the numerous races of mankind1 yet one law appears to govern the beauty of the whole4 ,the highest and most cultivated of these races1 however1 present only an approximation to the perfect development of those distinguishing marks of humanity2 and therefore the beauty of form and proportion which in nature characterises the human head and countenance1 exhibits only a partial development of the harmonic law of visible beauty4 ,on the other hand1 we find that1 in their sculpture1 the ancient ,greeks surpassed ordinary nature1 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 other4 ,it may be requisite here to remark1 that this species of beauty is independent of the more intellectual quality of expression4 ,for as ,sir ,charles ,bell has said1 8,beauty of countenance may be defined in words1 as well as demonstrated in art4 ,a face may be beautiful in sleep1 and a statue without expression may be highly beautiful4 ,but it will be said there is expression in the sleeping figure or in the statue4 ,is it not rather that we see in these the capacity for expression8--that our minds are active in imagining what may be the motions of these features when awake or animated8 ,thus1 we speak of an expressive face before we have seen a movement grave or cheerful1 or any indication in the features of what prevails in the heart40 ,this capacity for expression certainly enhances our admiration of the human countenance2 but it is more a concomitant of the primary cause of its beauty than the cause itself4 ,this cause rests on that simple and secure basis--the harmonic law of nature2 for the nearer the countenance approximates to an harmonious combination of the most perfect figures in geometry1 or rather the more its general form and the relation of its individual parts are arranged in obedience to that law1 the higher its degree of beauty1 and the greater its capacity for the expression of the passions4 ,various attempts have been made to define geometrically the difference between the ordinary and the ideal beauty of the human head and countenance1 the most prominent of which is that of ,camper4 ,he traced1 upon a profile of the skull1 a line in a horizontal direction1 passing through the foramen of the ear and the exterior margin of the sockets of the front teeth of the upper jaw1 upon which he raised an oblique line1 tangential to the margin of these sockets1 and to the most prominent part of the forehead4 ,agreeably to the obliquity of this line1 he determined the relative proportion of the areas occupied by the brain and by the face1 and hence inferred the degree of intellect4 ,when he applied this measurement to the heads of the antique statues1 he found the angle much greater than in ordinary nature2 but that this simple fact afforded no rule for the reproduction of the ideal beauty of ancient ,greek art1 is very evident from the heads and countenances by which his treatise is illustrated4 ,sir ,charles ,bell justly remarks1 that although1 by ,camper's method1 the forehead may be thrown forward1 yet1 while the features of common nature are preserved1 we refuse to acknowledge a similarity to the beautiful forms of the antique marbles4 8,it is true10 he says1 8that1 by advancing the forehead1 it is raised1 the face is shortened1 and the eye brought to the centre of the head4 ,but with all this1 there is much wanting--that which measurement1 or a mere line1 will not shew us40--8,the truth is1 that we are more moved by the features than by the form of the whole head4 ,unless there be a conformity in every feature to the general shape of the head1 throwing the forehead forward on the face produces deformity2 and the question returns with full force--,how is it that we are led to concede that the antique head of the ,apollo1 or of the ,jupiter1 is beautiful when the facial line makes a hundred degrees with the horizontal line8 ,in other words--,how do we admit that to be beautiful which is not natural8 ,simply for the same reason that1 if we discover a broken portion of an antique1 a nose1 or a chin of marble1 we can say1 without deliberation--,this must have belonged to a work of antiquity2 which proves that the character is distinguishable in every part--in each feature1 as well as in the whole head40 ,dr ,oken says upon this subject3#ag--8,the face is beautiful whose nose is parallel to the spine4 ,no human face has grown into this estate2 but every nose makes an acute angle with the spine4 ,the facial angle is1 as is well known1 #hj4 ,what1 as yet1 no man has remarked1 and what is not to be remarked1 either1 without our view of the cranial signification1 the old masters have felt through inspiration4 ,they have not only made the facial angle a right angle1 but have even stepped beyond this--the ,romans going up to #if1 the ,greeks even to #ajj4 ,whence comes it that this unnatural face of the ,grecian works of art is still more beautiful than that of the ,roman1 when the latter comes nearer to nature8 ,the reason thereof resides in the fact of the ,grecian artistic face representing nature's design more than that of the ,roman2 for1 in the former1 the nose is placed quite perpendicular1 or parallel to the spinal cord1 and thus returns whither it has been derived40 ,other various and conflicting opinions upon this subject have been given to the world2 but we find that the principle from which arose the ideal beauty of the head and countenance1 as represented in works of ancient ,greek art1 is still a matter of dispute4 ,when1 however1 we examine carefully a fine specimen1 we find its beauty and grandeur to depend more upon the degree of harmony amongst its parts1 as to their relative proportions and mode of arrangement1 than upon their excellence taken individually4 ,it is1 therefore1 clear that those 7and they are many7 who attribute the beauty of ancient ,greek sculpture merely to a selection of parts from various models1 must be in error4 ,no assemblage of parts from ordinary nature could have produced its principal characteristic1 the excess in the angle of the facial line1 much less could it have led to that exquisite harmony of parts by which it is so eminently distinguished2 neither can we reasonably agree with ,dr ,oken and others1 who assert that it was produced by an exclusive degree of the inspiration of genius amongst the ,greek people during a certain period4 ,that the inspiration of genius1 combined with a careful study of nature1 were essential elements in the production of the great works which have been handed down to us1 no one will deny2 but these elements have existed in all ages1 whilst the ideal head belongs exclusively to the ,greeks during the period in which the schools of ,pythagoras and ,plato were open4 ,is it not1 therefore1 reasonable to suppose1 that1 besides genius and the study of nature1 another element was employed in the production of this excellence1 and that this element arose from the precise mathematical doctrines taught in the schools of these philosophers8 ,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 countenance1 nor to the representations of which the same science is more easily applied2 and it is to the mode in which this is done that the varieties of sex and character may be imparted to works of art4 ,having gone into full detail1 and given ample illustrations in a former work1#ah it is unnecessary for me to enter upon that part of the subject in this rsum2 but only to shew the typical structure of beauty by which this noble work of creation is distinguished4 ,the angles which govern the form and proportions of the human head and countenance are1 with the right angle1 a series of seven1 which1 from the simplicity of their ratios to each other1 are calculated to produce the most perfect concord4 ,it consists of the right angle and its following parts-- ,tonic4 ,dominant4 ,mediant4 ,subtonic4 77 77 77 77 77 77 ,these angles1 and the figures which belong to them1 are thus arranged3-- ,sidenote3 ,plate ,,ix4 ,the vertical line ,a ,b 7,plate ,,ix4 fig4 #b7 represents the full length of the head and face4 ,taking this line as the greater axis of an ellipse of 771 such an ellipse is described around it4 ,through ,a the lines ,a ,g1 ,a ,k1 ,a ,l1 ,a ,m1 and ,a ,n1 are drawn on each side of the line ,a ,b1 making1 with the vertical1 respectively the angles of 771 771 771 771 and 774 ,through the points ,g1 ,k1 ,l1 ,m1 and ,n1 where these straight lines meet the curved line of the ellipse1 horizontal lines are drawn by which the following isosceles triangles are formed1 ,a ,g ,g1 ,a ,k ,k1 ,a ,l ,l1 ,a ,m ,m1 and ,a ,n ,n4 ,from the centre ,x of the equilateral triangle ,a ,g ,g the curvilinear figure of 771 viz41 the circle1 is described circumscribing that triangle4 ,the curvilinear plane figures of 77 and 771 respectively1 represent the solid bodies of which they are sections1 viz41 a sphere and a prolate spheroid4 ,these bodies1 from the manner in which they are here placed1 are partially amalgamated1 as shewn in figures #a and #c of the same plate1 thus representing the form of the human head and countenance1 both in their external appearance and osseous structure1 more correctly than they could be represented by any other geometrical figures4 ,thus1 the angles of 77 and 77 determine the typical form4 ,from each of the points u and n1 where ,a ,m cuts ,g ,g on both sides of ,a ,b1 a circle is described through the points p and q1 where ,a ,k cuts ,g ,g on both sides of ,a ,b1 and with the same radius a circle is described from the point a1 where ,k ,k cuts ,a ,b4 ,the circles u and n determine the position and size of the eyeballs1 and the circle a the width of the nose1 as also the horizontal width of the mouth4 ,the lines ,g ,g and ,k ,k also determine the length of the joinings of the ear to the head4 ,the lines ,l ,l and ,m ,m determine the vertical width of the mouth and lips when at perfect repose1 and the line ,n ,n the superior edge of the chin4 ,thus simply are the features arranged and proportioned on the facial surface4 ,it must1 however1 be borne in mind1 that in treating simply of the sthetic beauty of the human head and countenance1 we have only to do with the external appearance4 ,in this research1 therefore1 the system of ,dr ,camper1 ,dr ,owen1 and others1 whose investigations were more of a physiological than an sthetic character1 can be of little service2 because1 according to that system1 the facial angle is determined by drawing a line tangential to the exterior margin of the sockets of the front teeth of the upper jaw1 and the most prominent part of the forehead4 ,now1 as these sockets are1 when the skull is naturally clothed1 and the features in repose1 entirely concealed by the upper lip1 we must take the prominent part of it1 instead of the sockets under it1 in order to determine properly this distinguishing mark of humanity4 ,and ,i believe it will be found1 that when the head is properly poised1 the nearer the angle which this line makes with the horizontal approaches #ij1 the more symmetrically beautiful will be the general arrangement of the parts 7see line y z1 figure #c1 ,plate ,,ix474 ,,the ,,science ,,of ,,beauty1 ,,as ,,developed ,,in ,,the ,,form ,,of ,,the ,,human ,,figure4 ,the manner in which this science is developed in the symmetrical proportions of the entire human figure1 is as remarkable for its simplicity as it has been shewn to be in those of the head and countenance4 ,having gone into very full details1 and given ample illustration in two former works#ai upon this subject1 ,i may here confine myself to the illustration of one description of figure1 and to a reiteration of some facts stated in these works4 ,these facts are1 #ast1 ,that on a given line the human figure is developed1 as to its principal points1 entirely by lines drawn either from the extremities of this line1 or from some obvious or determined localities4 #b;d1 ,that the angles which these lines make with the given line1 are all simple sub-multiples of some given fundamental angle1 or bear to it a proportion expressible under the most simple relations1 such as those which constitute the scale of music4 #c;d1 ,that the contour is resolved into a series of ellipses of the same simple angles4 ,and1 #dth1 ,that these ellipses1 like the lines1 are inclined to the first given line by angles which are simple sub-multiples of the given fundamental angle4 ,from which four facts1 and agreeably to the hypothesis ,i have adopted1 it results as a natural consequence that the only effort which the mind exercises through the eye1 in order to put itself in possession of the data for forming its judgment1 is this1 that it compares the angles about a point1 and thereby appreciates the simplicity of their relations4 ,in selecting the prominent features of a figure1 the eye is not seeking to compare their relative distances--it is occupied solely with their relative positions4 ,in tracing the contour1 in like manner1 it is not left in vague uncertainty as to what is the curve which is presented to it2 unconsciously it feels the complete ellipse developed before it2 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 features1 it is satisfied1 and finds the symmetry perfect4 ,mller1 and other investigators into the archology of art1 refer to the great difficulty which exists in discovering the principles which the ancients followed in regard to the proportions of the human figure1 from the different sexes and characters to which they require to be applied4 ,but in the system thus founded upon the harmonic law of nature1 no such difficulty is felt1 for it is as applicable to the massive proportions which characterise the ancient representations of the ,hercules1 as to the delicate and perfectly symmetrical beauty of the ,venus4 ,this change is effected simply by an increase in the fundamental angle4 ,for instance1 in the construction of a figure of the exact proportions of the ,venus1 the right angle is adopted4 ,but in the construction of a figure of the massive proportions of the ,hercules1 it is requisite to adopt an angle which bears to the right angle the ratio of #f3e4 ,the adoption of this angle ,i have shewn in another work#bj to produce in the ,hercules those proportions which are so characteristic of physical power4 ,the ellipses which govern the outline1 being also formed upon the same larger class of angles1 give the contour of the muscles a more massive character4 ,in comparing the male and female forms thus geometrically constructed1 it will be found that that of the female is more harmoniously symmetrical1 because the right angle is the fundamental angle for the trunk and the limbs as well as for the head and countenance2 while in that of the male1 the right angle is the fundamental angle for the head only4 ,it may also be observed1 that1 from the greater proportional width of the pelvis of the female1 the centres of that motion which the heads of the thigh bones perform in the cotyloid cavities1 and the centres of that still more extensive range of motion which the arm is capable of performing at the shoulder joints1 are nearly in the same line which determines the central motion of the vertebral column1 while those of the male are not2 consequently all the motions of the female are more graceful than those of the male4 ,this difference between the fundamental angles1 which impart to the human figure1 on the one hand1 the beauty of feminine proportion and contour1 and on the other1 the grandeur of masculine strength1 being in the ratio of #e3f1 allows ample latitude for those intermediate classes of proportions which the ancients imparted to their various other deities in which these two qualities were blended4 ,i therefore confine myself to an illustration of the external contour of the form1 and the relative proportions of all the parts of a female figure1 such as those of the statues of the ,venus of ,melos and ,venus of ,medici4 ,the angles which govern the form and proportions of such a figure are1 with the right angle1 a series of twelve1 as follows3-- ,tonic4 ,dominant4 ,mediant4 ,subtonic4 ,supertonic4 77 77 77 77 77 77 77 77 77 77 77 ,these angles are employed in the construction of a diagram1 which determines the proportions of the parts throughout the whole figure4 ,thus3-- ,sidenote3 ,plate ,x4 ,let the line ,a ,b 7fig4 #a1 plate ,x47 represent the height of the figure to be constructed4 ,at the point ,a1 make the angles of ,c ,a ,d 771 ,f ,a ,g 771 ,h ,a ,i 771 ,k ,a ,l 771 and ,m ,a ,n 774 ,at the point ,b1 make the angles ,k ,b ,l 771 ,u ,b ,a 771 and ,o ,b ,a 774 ,through the point ,k1 in which the lines ,a ,k and ,b ,k intersect one another1 draw ,p ,k ,o parallel to ,a ,b1 and through ,c ,f ,h and ,m1 where this line meets ,a ,c1 ,a ,f1 ,a ,h1 and ,a ,m1 draw ,c ,d1 ,f ,g1 ,h ,i1 and ,m ,n1 perpendicular to ,a ,b2 draw also ,k ,l perpendicular to ,a ,b2 join ,b ,f and ,b ,h1 and through ,c draw ,c ,e1 making with ,a ,b the angle 771 which completes the arrangement of the eleven angles upon ,a ,b2 for ,f ,b ,g is very nearly 771 and ,h ,b ,i very nearly 774 ,at the point f1 where ,a ,c intersects ,o ,b1 draw f a perpendicular to ,a ,b2 and through the point i1 where ,b ,o intersects ,m ,n1 draw ,s i ,t parallel to ,a ,c4 ,through m1 where ,s i ,t intersects ,f ,b1 draw m n2 through 1 where ,s i ,t intersects ,k ,b1 draw w2 through ,t draw ,t g1 making an angle of 77 with ,o ,p4 ,join ,n ,p1 ,m ,b1 and g ,p1 and where ,n ,p intersects ,k ,b1 draw ,q ,r perpendicular to ,a ,b4 ,on ,a ,e as a diameter1 describe a circle cutting ,a ,c in r1 and draw r o perpendicular to ,a ,b4 ,with ,a o and o r as semi-axes1 describe the ellipse ,a r e1 cutting ,a ,h in t2 and draw t u perpendicular to ,a ,b4 ,with ,a u and t u1 as semi-axes describe the ellipse ,a t d4 ,on a ,l1 as major axis1 describe the ellipse of 774 ,for the side aspect or profile of the figure the diagram is thus constructed-- ,on one side of a line ,a ,b 7fig4 #b1 ,plate ,x47 construct the rectilinear portion of a diagram the same as fig4 #a4 ,through i draw ,w ,y parallel to ,a ,b1 and draw ,a z perpendicular to ,a ,b4 ,make ,w a equal to ,a a 7fig4 #a71 and on a l1 as major axis1 describe the ellipse of 774 ,through a draw a p parallel to ,a ,f1 and through p draw p t perpendicular to ,w ,y4 ,through a draw f a u perpendicular to ,w ,y4 ,upon a diameter equal to ,a ,e describe a circle whose circumference shall touch ,a ,b and ,a z4 ,with semi-axes equal to ,a o and o r 7fig4 #a71 describe an ellipse with its major axis parallel to ,a ,b1 and its circumference touching ,o ,p and z ,a4 ,sidenote3 ,plate ,,xi4 ,thus simply are the diagrams of the general proportions of the human figure1 as viewed in front and in profile1 constructed2 and ,plate ,,xi4 gives the contour in both points of view1 as composed entirely of the curvilinear figures of 771 771 771 771 and 774 ,further detail here would be out of place1 and ,i shall therefore refer those who require it to the ,appendix1 or the more elaborate works to which ,i have already referred4 ,the beauty derived from proportion1 imparted by the system here pointed out1 and from a contour of curves derived from the same harmonic angles1 is not confined to the human figure1 but is found in various minor degrees of perfection in all the organic forms of nature1 whether animate or inanimate1 of which ,i have in other works given many examples4#ba ,,the ,,science ,,of ,,beauty1 ,,as ,,developed ,,in ,,colours4 ,there is not amongst the various phenomena of nature one that more readily excites our admiration1 or makes on the mind a more vivid impression of the order1 variety1 and harmonious beauty of the creation1 than that of colour4 ,on the general landscape this phenomenon is displayed in the production of that species of harmony in which colours are so variously blended1 and in which they are by light1 shade1 and distance modified in such an infinity of gradation and hue4 ,although genius is continually struggling1 with but partial success1 to imitate those effects1 yet1 through the ,divine beneficence1 all whose organs of sight are in an ordinary degree of perfection can appreciate and enjoy them4 ,in winter this pleasure is often to a certain extent withdrawn1 when the colourless snow alone clothes the surface of the earth4 ,but this is only a pause in the general harmony1 which1 as the spring returns1 addresses itself the more pleasingly to our perception in its vernal melody1 which1 gradually resolving itself into the full rich hues of luxuriant beauty exhibited in the foliage and flowers of summer1 subsequently rises into the more vivid and powerful harmonies of autumn's colouring4 ,thus the eye is prepared again to enjoy that rest which such exciting causes may be said to have rendered necessary4 ,when we pass from the general colouring of nature to that of particular objects1 we are again wrapt in wonder and admiration by the beauty and harmony which so constantly1 and in such infinite variety1 present themselves to our view1 and which are so often found combined in the most minute objects4 ,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 creation4 ,by this uniformity in colour1 various species of animals are often distinguished2 and in each individual of most of these species1 how much is this beauty enhanced when the uniformity prevails in the resemblance of their lateral halves6 ,the human countenance exemplifies this in a striking manner2 the slightest variety of colour between one and another of the double parts is at once destructive of its symmetrical beauty4 ,many of the lower animals1 whether inhabitants of the earth1 the air1 or the water1 owe much of their beauty to this kind of uniformity in the colour of the furs1 feathers1 scales1 or shells1 with which they are clothed4 ,in the vegetable kingdom1 we find a great degree of uniformity of colour in the leaves1 flowers1 and fruit of the same plant1 combined with all the harmonious beauty of variety which a little careful examination develops4 ,in the colours of minerals1 too1 the same may be observed4 ,in short1 in the beauty of colouring1 as in every other species of beauty1 uniformity and variety are found to combine4 ,an appreciation of colour depends1 in the first place1 as much upon the physical powers of the eye in conveying a proper impression to the mind1 as that of music on those of the ear4 ,but an ear for music1 or an eye for colour1 are1 in so far as beauty is concerned1 erroneous expressions2 because they are merely applicable to the impression made upon the senses1 and do not refer to the sthetical principles of harmony1 by which beauty can alone be understood4 ,a good eye1 combined with experience1 may enable us to form a correct idea as to the purity of an individual colour1 or of the relative difference existing between two separate hues2 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 art4 ,the power of perceiving and appreciating beauty of any kind1 is a principle inherent in the human mind1 which may be improved by cultivation in the degree of the perfection of the art senses4 ,great pains have been bestowed on the education of the ear1 in assisting it to appreciate the melody and harmony of sound2 but still much remains to be done in regard to the cultivation of the eye1 in appreciating colour as well as form4 ,it is true1 that there are individuals whose powers of vision are perfect1 in so far as regards the appreciation of light1 shade1 and configuration1 but who are totally incapable of perceiving effects produced by the intermediate phenomenon of colour1 every object appearing to them either white1 black1 or neutral gray2 others1 who are equally blind as to the effect of one of the three primary colours1 but see the other two perfectly1 either singly or combined2 while there are many who1 having the full physical power of perceiving all the varieties of the phenomenon1 and who are even capable of making nice distinctions amongst a variety of various colours1 are yet incapable of appreciating the sthetic quality of harmony which exists in their proper combination4 ,it is the same with respect to the effects of sounds upon the ear--some have organs so constituted1 that notes above or below a certain pitch are to them inaudible2 while others1 with physical powers otherwise perfect1 are incapable of appreciating either melody or harmony in musical composition4 ,but perceptions so imperfectly constituted are1 by the goodness of the ,creator1 of very rare occurrence2 therefore all attempts at improvement in the science of sthetics must be suited to the capacities of the generality of mankind1 amongst whom the perception of colour exists in a variety as great as that by which their countenances are distinguished4 ,artists now and then appear who have this intuitive perception in such perfection1 that they are capable of transferring to their works the most beautiful harmonies and most delicate gradations of colours1 in a manner that no acquired knowledge could have enabled them to impart4 ,to those who possess such a gift1 as well as to those to whom the ordinary powers of perception are denied1 it would be equally useless to offer an explanation of the various modes in which the harmony of colour develops itself1 or to attempt a definition of the many various colours1 hues1 tints1 and shades1 arising out of the simple elements of this phenomenon4 ,but to those whose powers lie between these extremes1 being neither above nor below cultivation1 such an explanation and definition must form a step towards the improvement of that inherent principle which constitutes the basis of sthetical science4 ,although the variety and harmony of colour which nature is continually presenting to our view1 are apparent to all whose visual organs are in a natural state1 and thus to the generality of mankind2 yet a knowledge of the simplicity by which this variety and beauty are produced1 is1 after ages of philosophic research and experimental inquiry1 only beginning to be properly understood4 ,light may be considered as an active1 and darkness a passive principle in the economy of ,nature1 and colour an intermediate phenomenon arising from their joint influence2 and it is in the ratios in which these primary principles act upon each other1 by which ,i here intend to explain the science of beauty as evolved in colour4 ,it has been usual to consider colour as an inherent quality in light1 and to suppose that coloured bodies absorb certain classes of its rays1 and reflect or transmit the remainder2 but it appears to me that colour is more probably the result of certain modes in which the opposite principles of motion and rest1 or force and resistance1 operate in the production1 refraction1 and reflection of light1 and that each colour is mutually related1 although in different degrees1 to these active and passive principles4 ,white and black are the representatives of light and darkness1 or activity and rest1 and are therefore calculated as pigments to reduce colours and hues to tints and shades4 ,having1 however1 fully illustrated the nature of tints and shades in a former work1#bb ,i shall here confine myself to colours in their full intensity--shewing the various modifications which their union with each other produce1 along with the harmonic relations which these modifications bear to the primaries1 and to each other in respect to warmth and coolness of tone1 as well as to light and shade4 ,the primary colours are red1 yellow1 and blue4 ,of these1 yellow is most allied to light1 and blue to shade1 while red is neutral in these respects1 being equally allied to both4 ,in respect to tone1 that of red is warm1 and that of blue cool1 while the tone of yellow is neutral4 ,the ratios of their relations to each other in these respects will appear in the harmonic scales to which1 for the first time1 ,i am about to subject colours1 and to systematise their various simple and compound relations1 which are as follow3-- ,from the binary union of the primary colours1 the secondary colours arise-- ,orange colour1 from the union of yellow and red4 ,green1 from the union of yellow and blue4 ,purple1 from the union of red and blue4 ,from the binary union of the secondary colours1 the primary hues arise-- ,yellow-hue1 from the union of orange and green4 ,red-hue1 from the union of orange and purple4 ,blue-hue1 from the union of purple and green4 ,from the binary union of the primary hues1 the secondary hues arise-- ,orange-hue1 from the union of yellow-hue and red-hue4 ,green-hue1 from the union of yellow-hue and blue-hue4 ,purple-hue1 from the union of red-hue and blue-hue4 ,each hue owes its characteristic distinction to the proportionate predominance or subordination of one or other of the three primary colours in its composition4 ,it follows1 that in every hue of red1 yellow and blue are subordinate2 in every hue of yellow1 red and blue are subordinate2 and in every hue of blue1 red and yellow are subordinate4 ,in like manner1 in every hue of green1 red is subordinate2 in every hue of orange1 blue is subordinate2 and in every hue of purple1 yellow is subordinate4 ,by the union of two primary colours1 in the production of a secondary colour1 the nature of both primaries is altered2 and as there are only three primary or simple colours in the scale1 the two that are united harmonically in a compound colour1 form the natural contrast to the remaining simple colour4 ,notwithstanding all the variety that extends beyond the six positive colours1 it may be said that there are only three proper contrasts of colour in nature1 and that all others are simply modifications of these4 ,pure red is the most perfect contrast to pure green2 because it is characterised amongst the primary colours by warmth of tone1 while amongst the secondary colours green is distinguished by coolness of tone1 both being equally related to the primary elements of light and shade4 ,pure yellow is the most perfect contrast to pure purple2 because it is characterised amongst the primary colours as most allied to light1 whilst pure purple is characterised amongst the secondaries as most allied to shade1 both being equally neutral as to tone4 ,pure blue is the most perfect contrast to pure orange2 because it is characterised amongst the primary colours as not only the most allied to shade1 but as being the coolest in tone1 whilst pure orange is characterised amongst the secondaries as being the most allied to light and the warmest in tone4 ,the same principle operates throughout all the modifications of these primary and secondary colours4 ,such is the simple nature of contrast upon which the beauty of colouring mainly depends4 ,it being now established as a scientific fact1 that the effect of light upon the eye is the result of an ethereal action1 similar to the atmospheric action by which the effect of sound is produced upon the ear2 also1 that the various colours which light assumes are the effect of certain modifications in this ethereal action2--just as the various sounds1 which constitute the scale of musical notes1 are known to be the effect of certain modifications in the atmospheric action by which sounds in general are produced3 ,therefore1 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 light1 so as to produce through the eye the effect of harmony1 cannot be supposed to differ from those principles which we know govern the modifications of the atmospheric action of sound1 in producing through the ear a like effect4 ,i shall therefore endeavour to illustrate the science of beauty as evolved in colours1 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 evolved1 and by which ,i have endeavoured in this1 as in previous works1 to systematise the harmonic beauty of forms4 ,illustration ,it will be observed1 that with a view to avoid complexity as much as possible1 ,i have1 in the arrangement of the above series of scales1 not only confined myself to the merely elementary parts of the ,pythagorean system1 but have left out the harmonic modifications upon 77 and 771 in order that the arithmetical progression might not be interrupted4#bc ,the above elementary process will1 ,i trust1 be found sufficient to explain the progress1 by harmonic union1 of a primary colour to a toned gray1 and how the simple and compound colours naturally arrange themselves into the elements of five scales1 the parts of which continue from primary to secondary colour2 from secondary colour to primary hue2 from primary hue to secondary hue2 from secondary hue to primary-toned gray2 and from primary-toned gray to secondary-toned gray in the simple ratio of #b3a2 thereby producing a series of the most beautiful and perfect contrasts4 ,the natural arrangement of the primary colours upon the solar spectrum is red1 yellow1 blue1 and ,i have therefore adopted the same arrangement on the present occasion4 ,red being1 consequently1 the first tonic1 and blue the second1 the divisions express the numerical ratios which the colours bear to one another1 in respect to that colourific power for which red is pre-eminent4 ,thus1 yellow is to red1 as #b3c2 blue to yellow1 as #c3d2 purple to orange1 as #e3f2 and green to purple1 as #f3g4 ,the following series of completed scales are arranged upon the foregoing principle1 with the natural connecting links of red-orange1 yellow-orange1 yellow-green1 and blue-green1 introduced in their proper places4 ,the appropriate terminology of musical notes has been adopted1 and the scales are composed as follows3-- ,scale ,i4 consists of primary and secondary colours2 ,scale ,,ii4 of secondary colours and primary hues2 ,scale ,,iii4 of primary and secondary hues2 ,scale ,,iv4 of secondary hues and primary-toned grays2 and ,scale ,v4 of primary and secondary-toned grays2 ,all the parts in each of these scales1 from the first tonic to the second1 relate to the same parts of the scale below them in the simple ratio of #b3a2 and serially to the first tonic in the following ratios3-- #h3i1 #d3e1 #c3d1 #b3c1 #c3e1 #d3g1 #h3ae1 #a3b4 ,first ,series of ,scales4 ---------------------------------------- ------------------------- ,tonic4 ,supertonic4 ,mediant4 ,subdominant4 ,dominant4 ,submediant4 ,subtonic4 ,semi-subtonic4 ,tonic4 ---------------------------------------- ------------------------- ,i4 77 77 77 77 77 77 77 77 77 ,red4 ,red- ,orange4,yellow-,yellow4,yellow-,green4, blue- ,blue4 orange4 orange4 green4 green4 ---------------------------------------- ------------------------- ,,ii4 77 77 77 77 77 77 77 77 77 ,green4,blue- ,blue ,blue- ,purple ,red- ,red ,red- ,orange4 green hue4 purple hue4 purple hue4 orange hue4 hue4 hue4 hue4 ---------------------------------------- ------------------------- ,,iii477 77 77 77 77 77 7777 77 ,red ,red- ,orange ,yellow-,yellow ,yellow-,green ,blue- ,blue hue4 orange hue4 orange hue4 green hue4 green hue4 hue4 hue4 hue4 hue4 ---------------------------------------- ------------------------- ,,iv4 7777 77 77 77 77 7777 77 ,green ,blue- ,blue- ,blue- ,purple ,red- ,red- ,red- ,orange hue4 green- toned purple-hue4 purple-toned orange-hue4 toned gray4 toned toned gray4 toned gray4 gray4 gray4 gray4 ---------------------------------------- ------------------------- ,v4 7777 77 7777 777777 77 ,red- ,red- ,orange-,yellow-,yellow-,yellow-,green- ,blue- ,blue- toned orange-toned orange-toned green- toned green-toned gray4 toned gray4 toned gray4 toned gray4 toned gray4 gray4 gray4 gray4 gray4 ---------------------------------------- ------------------------- ,to the scales of chromatic power ,i add another series of scales1 in which yellow1 being the first tonic1 and blue the second1 the numerical divisions express the ratios which the colours in each scale bear to one another in respect to light and shade4 ,thus red is to yellow1 in respect to light1 as #b3c2 blue to red1 as #c3d2 green to orange1 as #e3f1 and purple to green1 as #f3g4 ,these scales may therefore be termed scales for the colour-blind1 because1 in comparing colours1 those whose sight is thus defective1 naturally compare the ratios of the light and shade of which different colours are primarily constituted4 ,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 arrangement3-- ,second ,series of ,scales4 ---------------------------------------- ------------------------- ,tonic4 ,supertonic4 ,mediant4 ,subdominant4 ,dominant4 ,submediant4 ,subtonic4 ,semi-subtonic4 ,tonic4 ---------------------------------------- ------------------------- ,i4 77 77 77 77 77 77 77 77 77 ,yellow4,yellow-,orange4,red- ,red4 ,red- ,purple4,blue- ,blue4 orange4 orange4 purple4 purple4 ---------------------------------------- ------------------------- ,,ii4 77 77 77 77 77 77 77 77 77 ,purple4,blue- ,blue ,blue- ,green4,yellow-,yellow ,yellow-,orange purple hue4 green green hue4 orange hue4 hue4 hue4 hue4 ---------------------------------------- ------------------------- ,,iii477 77 77 77 7777 77 77 77 ,yellow ,yellow-,orange ,red- ,red ,red- ,purple ,blue- ,blue hue4 orange hue4 orange hue4 purple hue4 purple hue4 hue4 hue4 hue4 hue4 ---------------------------------------- ------------------------- ,,iv4 77 77 77 77 7777 77 77 77 ,purple ,blue- ,blue- ,blue- ,green ,yellow-,yellow-,yellow-,orange hue4 purple-toned green- hue4 green- toned orange-hue4 toned gray4 toned toned gray4 toned gray4 gray4 gray4 gray4 ---------------------------------------- ------------------------- ,v4 77 77 77 77777777 77 77 ,yellow-,yellow-,orange-,red- ,red- ,red- ,purple-,blue- ,blue- toned orange-toned orange-toned purple-toned green- toned gray4 toned gray4 toned gray4 toned gray4 toned gray4 gray4 gray4 gray4 gray4 ---------------------------------------- ------------------------- ,should ,i be correct in arranging colours upon scales identical with those upon which musical notes have been arranged1 and in assuming that colours have the same ratios to each other1 in respect to their harmonic power upon the eye1 which musical notes have in respect to their harmonic power upon the ear1 the colourist may yet be enabled to impart harmonic beauty to his works with as much certainty and ease1 as the musician imparts the same quality to his compositions3 for the colourist has no more right to trust exclusively to his eye in the arrangement of colours1 than the musician has to trust exclusively to his ear in the arrangement of sounds4 ,we find1 in comparing the dominant parts in the first and second scales of the second series1 that they are equal as to light and shade1 so that their relative powers of contrast depend entirely upon colour4 ,hence it is that red and green are the two colours1 the difference between which the colour-blind are least able to appreciate4 ,professor ,george ,wilson1 in his excellent work1 8,researches on ,colour-,blindness10 mentions the case of an engraver1 which proves the power of the eye in being able to appreciate these original constituents of colour1 irrespective of the intermediate phenomenon of tone4 ,this engraver1 instead of expressing regret on account of his being colour-blind1 observed to the professor1 8,my defective vision is1 to a certain extent1 a useful and valuable quality4 ,thus1 an engraver has two negatives to deal with1 i4e41 white and black4 ,now1 when ,i look at a picture1 ,i see it only in white and black1 or light and shade1 or1 as artists term it1 the effect4 ,i find at times many of my brother engravers in doubt how to translate certain colours of pictures1 which to me are matters of decided certainty and ease4 ,thus to me it is valuable40 ,the colour-blind are therefore as incapable of receiving pleasure from the harmonious union of various colours1 as those who1 to use a common term1 have no ear for music1 are of being gratified by the 8melody of sweet sounds40 ,the generality of mankind are1 however1 capable of appreciating the harmony of colour which1 like that of both sound and form1 arises from the simultaneous exhibition of opposite principles having a ratio to each other4 ,these principles are in continual operation throughout nature1 and from them we often derive pleasure without being conscious of the cause4 ,all who are not colour-blind must have felt themselves struck with the harmonic beauty of a cloudless sky1 although in it there is no configuration1 and at first sight apparently but one colour4 ,now1 as we know that there can be no more impression of harmony made upon the mind by looking upon a single colour1 than there could be by listening to a single continued musical note1 however sweet its tone1 we are apt at first to imagine that the organ of vision has1 in some measure1 conveyed a false impression to the mind4 ,but it has not done so2 for light1 when reflected from the atmosphere1 produces those cool tones of blue1 gray1 and purple1 which seem to clothe the distant mountains2 but1 when transmitted through the same atmosphere1 it produces those numerous warm tints1 the most intense of which give the gorgeous effects which so often accompany the setting sun4 ,we have1 therefore1 in the upper part of a clear sky1 where the atmosphere may be said to be illuminated principally by reflection from the surface of the earth1 a comparatively cool tone of blue1 the result of reflection1 which gradually blends into the warm tints1 the result of transmission through the same atmosphere4 ,such a composition of harmonious colouring is to the eye what the voice of the soft breath of summer amongst the trees1 the hum of insects on a sultry day1 or the simple harmony of the olian harp1 is to the ear4 ,to such a composition of chromatic harmony must also be referred the universal concurrence of mankind in appreciating the peculiar beauty of white marble statuary4 ,that the principal constituent of beauty in such works ought to be harmony of form1 no one will deny2 but this is not the only element1 as appears from the fact1 that a cast in plaster of ,paris1 of a fine white marble statue1 although identical in form1 is far less beautiful than the original4 ,now this undoubtedly must be the consequence of its having been changed from a semi-translucent substance1 which1 like the atmosphere1 can transmit as well as reflect light1 to an opaque substance1 which can only reflect it4 ,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 light1 and the other that of its reflection4 ,as a series of coloured illustrations would be beyond the scope of this rsum1 ,i may refer those who wish to prosecute the inquiry1 with the assistance of such a series1 to my published works upon the subject4#bd ,,the ,,science ,,of ,,beauty1 ,,applied ,,to ,,the ,,forms ,,and ,,proportions ,,of ,,ancient ,,grecian ,,vases ,,and ,,ornaments4 ,in examining the remains of the ornamental works of the ancient ,greek artists1 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 law4 ,this probability not being fully elucidated in any of my former works1 ,i will require to go into some detail on the present occasion4 ,i take for my first illustration an unexceptionable example1 viz43-- ,the ,portland ,vase4 ,although this beautiful specimen of ancient art was found about the middle of the sixteenth century1 inclosed in a marble sarcophagus within a sepulchral chamber under the ,monte del ,grano1 near ,rome1 and although the date of its production is unknown1 yet its being a work of ancient ,grecian art is undoubted2 and the exquisite beauty of its form has been universally acknowledged1 both during the time it remained in the palace of the ,barberini family at ,rome1 and since it was added to the treasures of the ,british ,museum4 ,the forms and proportions of this gem of art appear to me to yield an obedience to the great harmonic law of nature1 similar to that which ,i have instanced in the proportions and contour of the best specimens of ancient ,grecian architecture4 ,sidenote3 ,plate ,,xii4 ,let the line ,a ,b 7,plate ,,xii47 represent the full height of the vase4 ,through ,a draw ,a a1 and through ,b draw ,b b indefinitely1 ,a a making an angle of 771 and ,b b an angle of 771 with the vertical4 ,through the point ,c1 where ,a a and ,b b intersect one another1 draw ,d ,c ,e vertical4 ,through ,a ,c and ,b respectively1 draw ,a ,d1 ,c ,f1 and ,b ,e horizontal4 ,draw similar lines on the other side of ,a ,b1 and the rectilinear portion of the diagram is complete4 ,the curvilinear contour may be thus added3-- ,take a cut-out ellipse of 771 whose greater axis is equal to the line ,a ,b1 and #ast4 ,place it upon the diagram1 so that its circumference may be tangential to the lines ,c ,e and ,c ,f1 and its greater axis m n may make an angle of 77 with the vertical1 and trace its circumference4 #b;d4 ,place it with its circumference tangential to that of the first at the point m1 while its greater axis 7of which o p is a part7 is in the horizontal1 and trace the portion of its circumference q o r4 #c;d4 ,place it with its circumference tangential to that of the above at v1 while its greater axis 7of which u v is a part7 makes an angle of 77 with the vertical1 and trace the portion of its circumference s v t4 ,thus the curvilinear contour of the body and neck are harmonically determined4 ,the curve of the handle may be determined by the same ellipse placed so that its greater axis 7of which i k is a part7 makes an angle of 77 with the vertical4 ,make similar tracings on the other side of ,a ,b1 and the diagram is complete4 ,the inscribing rectangle ,d ,g ,e ,k is that of 774 ,the outline resulting from this diagram1 not only is in perfect agreement with my recollection of the form1 but with the measurements of the original given in the 8,penny ,cyclopdia20 of the accuracy of which there can be no doubt4 ,they are stated thus3--8,it is about ten inches in height1 and beautifully curved from the top downwards2 the diameter at the top being about three inches and a-half2 at the neck or smallest part1 two inches2 at the largest 7mid-height71 seven inches2 and at the bottom1 five inches40 ,the harmonic elements of this beautiful form1 therefore1 appear to be the following parts of the right angle3-- ,tonic4 ,dominant4 ,mediant4 ,submediant4 77 77 77 77 77 77 ,when we reflect upon the variety of harmonic ellipses that may be described1 and the innumerable positions in which they may be harmonically placed with respect to the horizontal and vertical lines1 as well as upon the various modes in which their circumferences may be combined1 the variety which may be introduced amongst such forms as the foregoing appears almost endless4 ,my second example is that of-- ,an ,ancient ,grecian ,marble ,vase of a ,vertical ,composition4 ,i shall now proceed to another class of the ancient ,greek vase1 the form of which is of a more complex character4 ,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 work4#be ,this vase is a work of ancient ,grecian art in ,parian marble1 which he met with in the collection at the ,villa ,albani1 near ,rome4 ,its height is #d ft4 #d in4 ,sidenote3 ,plate ,,xiii4 ,the following is the formula by which ,i endeavour to develop its harmonic elements3-- ,let ,a ,b 7,plate ,,xiii47 represent the full height of this vase4 ,through ,b draw ,b ,d1 making an angle of 77 with the vertical4 ,through ,d draw ,d ,o vertical1 through ,a draw ,a ,c1 making an angle of 772 through ,b draw ,b ,l1 making an angle of 771 and ,b ,s1 making an angle of 771 each with the vertical4 ,through ,a draw ,a ,d1 through ,b draw ,b ,o1 through ,l draw ,l ,n1 through ,c draw ,c ,f1 and through ,s draw ,s ,p1 all horizontal4 ,through ,a draw ,a ,h1 making an angle of 77 with the vertical1 and through ,h draw ,h ,m vertical4 ,draw similar lines on the other side of ,a ,b1 and the rectilinear portion of the diagram is complete1 and its inscribing rectangle that of 774 ,the curvilinear portion may thus be added-- ,take a cut-out ellipse of 771 whose greater axis is about the length of the body of the intended vase1 place it with its lesser axis upon the line ,s ,p1 and its greater axis upon the line ,d ,o1 and trace the part a b of its circumference upon the diagram4 ,place the same ellipse with one of its foci upon ,c1 and its greater axis upon ,c ,f1 and trace its circumference upon the diagram4 ,take a cut-out ellipse of 771 whose greater axis is nearly equal to that of the ellipse already used2 place it with its greater axis upon ,m ,h1 and its lesser axis upon ,l ,n1 and trace its circumference upon the diagram4 ,make similar tracings upon the other side of ,a ,b1 and the diagram is complete4 ,in this1 as in the other diagrams1 the strong portions of the lines give the contour of the vase4 ,the harmonic elements of this classical form1 therefore1 appear to be the right angle and its following parts3-- ,tonic4 ,dominant4 ,mediant4 ,submediant4 77 77 77 77 77 77 ,my third example is that of-- ,an ,ancient ,grecian ,vase of a ,horizontal ,composition4 ,this example belongs to the same class as the last1 but it is of a horizontal composition4 ,it was carefully drawn from the original in the museum of the ,vatican by ,tatham1 in whose etchings it will be found with its ornamental decorations4 ,the diagram of its harmonic elements may be constructed as follows3-- ,sidenote3 ,plate ,,xiv4 ,let ,a ,b 7,plate ,,xiv47 represent the full height of the vase4 ,through ,b draw ,b ,d1 making an angle of 77 with the vertical4 ,through ,a draw ,a ,h1 ,a ,l1 and ,a ,c1 making respectively the following angles1 77 with the vertical1 77 with the vertical1 and 77 with the horizontal4 ,these angles determine the horizontal lines ,h ,b1 ,l ,n1 and ,c ,f1 which divide the vase into its parts1 and the inscribing rectangle ,d ,g ,k ,o is 774 ,this completes the rectilinear portion of the diagram4 ,the ellipse by which the curvilinear portion is added is one of 771 the greater axis of which1 at a b1 as also at c d1 makes an angle of 77 with the vertical1 and the same axis at e f an angle of 77 with the horizontal4 ,the harmonic elements of this vase1 therefore1 appear to be3-- ,tonic4 ,dominant4 ,mediant4 ,submediant4 ,supertonic4 ,the ,right 77 77 77 77 ,angle4 77 ,my remaining examples are those of-- ,etruscan ,vases4 ,of these vases ,i give four examples1 by which the simplicity of the method employed in applying the harmonic law will be apparent4 ,sidenote3 ,plate ,,xv4 ,the inscribing rectangle ,d ,g ,e ,k of fig4 #a1 ,plate ,,xv41 is one of 771 within which are arranged tracings from an ellipse of 771 whose greater axis at a b makes an angle of 771 at c d an angle of 771 and at e f an angle of 771 with the vertical4 ,the harmonic elements of the contour of this vase1 therefore1 appear to be3-- ,tonic4 ,dominant4 ,subdominants4 ,submediant4 ,the ,right 77 77 77 ,angle4 77 ,the inscribing rectangle ,l ,m ,n ,o of fig4 #b is that of 771 within which are arranged tracings from an ellipse of 771 whose greater axis1 at a b and c d respectively1 makes angles of 77 and 77 with the horizontal1 while that at e f is in the horizontal line4 ,the harmonic elements of the contour of this vase1 therefore1 appear to be3-- ,tonic4 ,dominant4 ,subtonic4 77 77 77 ,sidenote3 ,plate ,,xvi4 ,the inscribing rectangle ,p ,q ,r ,s of fig4 #a1 ,plate ,,xvi41 is one of 771 within which are arranged tracings from an ellipse of 771 whose greater axis1 at a b1 c d1 and e f1 makes respectively angles of 77 with the horizontal1 77 and 77 with the vertical4 ,its harmonic elements1 therefore1 appear to be3-- ,tonic4 ,dominant4 ,mediant4 ,supertonic4 ,subdominant4 ,submediant4 ,the ,right 77 77 77 77 77 ,angle4 ,the inscribing rectangle ,t ,u ,v ,x of fig4 #b is one of 771 within which are arranged tracings from an ellipse of 77 whose greater axis at a b is in the vertical line1 and at c d makes an angle of 774 ,the harmonic elements of the contour of this vase1 therefore1 appear to be3-- ,tonic4 ,submediant4 ,supertonic4 77 77 77 ,these four ,etruscan vases1 the contours of which are thus reduced to the harmonic law of nature1 are in the ,british ,museum1 and engravings of them are to be found in the well-known work of ,mr ,henry ,moses1 ,plates #d1 #f1 #ad1 and #g1 respectively1 where they are represented with their appropriate decorations and colours4 ,to these1 ,i add two examples of-- ,ancient ,grecian ,ornament4 ,i have elsewhere shewn#bf that the elliptic curve pervades the ,parthenon from the entases of the column to the smallest moulding1 and we need not1 therefore1 be surprised to find it employed in the construction of the only two ornaments belonging to that great work4 ,sidenote3 ,plate ,,xvii4 ,in the diagram 7,plate ,,xvii471 ,i endeavour to exhibit the geometric construction of the upper part of one of the ornamental apices1 termed antefix1 which surmounted the cornice of the ,parthenon4 ,the first ellipse employed is that of 771 whose greater axis a b is in the vertical line2 the second is also that of 771 whose greater axis c d makes1 with the vertical1 an angle of 772 the third ellipse is the same with its major axis e f in the vertical line4 ,through one of the foci of this ellipse at ,a the line ,a ,c is drawn1 and upon the part of the circumference ,c e1 the number of parts1 #a1 #b1 #c1 #d1 #e1 #f1 #g1 of which the surmounting part of this ornament is to consist1 are set off4 ,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 parts4 ,the third ellipse employed is one of 774 ,take a cut-out ellipse of this kind1 whose larger axis is equal in length to the inscribing rectangle4 ,place it with its vertex upon the same ellipse at g1 so that its circumference will pass through ,c1 and trace it2 remove its apix first to p1 then to q1 and proceed in the same way to q1 r1 s1 t1 u1 and v1 so that its circumference will pass through the seven divisions on c g and e ,c3 v o1 u n1 t m1 s i1 r k1 q j1 p l1 and g x1 are parts of the larger axes of the ellipses from which the curves are traced4 ,the small ellipse of which the ends of the parts are formed is that of 774 ,sidenote3 ,plate ,,xviii4 ,in the diagram 7,plate ,,xviii471 ,i endeavour to exhibit the geometric construction of the ancient ,grecian ornament1 commonly called the ,honeysuckle1 from its resemblance to the flower of that name4 ,the first part of the process is similar to that just explained with reference to the antefix of the ,parthenon1 although the angles in some parts differ4 ,the contour is determined by the circumference of an ellipse of 771 whose major axis ,a ,b makes an angle of 77 with the vertical1 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 line1 and these parts are again arranged upon a similar ellipse whose larger axis ,c ,d makes an angle of 77 with the vertical4 ,the first series of curved lines proceeding from #a1 #b1 #c1 #d1 #e1 #f1 #g1 and #h1 are between ,k ,e and ,h ,c1 part of the circumference of an ellipse of 772 and those between ,c ,h and ,a ,g are parts of the circumference of four ellipses1 each of 771 but varying as to the lengths of their larger axes from #e to #c inches4 ,the change from the convex to the concave1 which produces the ogie forms of which this ornament is composed1 takes place upon the line ,c ,h1 and the lines a b1 c d1 e f1 g h1 i k1 l m1 n o1 and p q1 are parts of the larger axis of the four ellipses the circumference of which give the upper parts of the petals or leaves4 ,this peculiar ,grecian ornament is often1 like the antefix of the ,parthenon1 combined with the curve of the spiral scroll4 ,but the volute is so well understood that ,i have not rendered my diagrams more complex by adding that figure4 ,many varieties of this union are to be found in ,tatham's etchings1 already referred to4 ,the antefix of the ,parthenon1 and its only other ornament the honeysuckle1 as represented on the soffit of the cornice1 are to be found in ,stewart's 8,athens40 ,,appendix4 ,no4 ,i4 ,in pages #cd1 #ce1 and #eh1 ,i have reiterated an opinion advanced in several of my former works1 viz41 that1 besides genius1 and the study of nature1 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 #ejj ,b4,c41 and ending about #bjj ,b4,c4 ,and that this cause most probably was1 that the artists of that period were instructed in a system of fixed principles1 based upon the doctrines of ,pythagoras and ,plato4 ,this opinion has not been objected to by the generality of those critics who have reviewed my works2 but has1 however1 met with an opponent1 whose recondite researches and learned observations are worthy of particular attention4 ,these are given in an essay by ,mr ,c4 ,knight ,watson1 8,on the ,classical ,authorities for ,ancient ,art10 which appeared in the ,cambridge ,journal of ,classical and ,sacred ,philology in ,june #ahed4 ,as this essay is not otherwise likely to meet the eyes of the generality of my readers1 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 essay1 ,i shall quote them fully3-- 8,the next name on our list is that of the famous ,euphranor 7,b4,c4 #cfb74 ,for the fact that to the practice of sculpture and of painting he added an exposition of the theory1 we are indebted to ,pliny1 who says 7xxxv4 #aa1 #dj71 #8,volumina quoque composuit de symmetria et coloribus4' ,when we reflect on the critical position occupied by ,euphranor in the history of ,greek art1 as a connecting link between the idealism of ,pheidias and the naturalism of ,lysippus1 we can scarcely overestimate the value of a treatise on art proceeding from such a quarter4 ,this is especially the case with the first of the two works here assigned to ,euphranor4 ,the inquiries which of late years have been instituted by ,mr ,d4 ,r4 ,hay of ,edinburgh1 on the proportions of the human figure1 and on the natural principles of beauty as illustrated by works of ,greek art1 constitute an epoch in the study of sthetics and the philosophy of form4 ,now1 in the presence of these inquiries1 or of such less solid results as ,mr ,hay's predecessors in the same field have elicited1 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 mind1 and by the chisel1 of the sculptor3 how far any such system of curves and proportions as ,mr ,hay's was used by the ,greek as a practical manual of his craft4 ,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 #8,symmetria' of ,euphranor contained anything analogous to them in kind1 or indeed equal in value4 ,it must not be forgotten that the truth of ,mr ,hay's theory is perfectly compatible with the fact1 that of such theory the ,greek may have been utterly ignorant4 ,it is on this fact ,i insist3 it is here that ,i join issue with ,mr ,hay1 and with his reviewer in a recent number of ,blackwood's ,magazine4 ,or1 to speak more accurately1--while ,i am quite prepared to find that the ,elgin marbles will best of all stand the test which ,mr ,hay has hitherto applied1 ,i believe1 to works of a later age1 ,i am none the less convinced that it is precisely that golden age of ,hellenic art to which they belong1 precisely that first and chief of ,hellenic artists by whom they were executed1 to which and to whom any such line of research on the laws of form would have been pre-eminently alien4 ,pheidias1 remember1 by the right of primogeniture1 is the ruling spirit of idealism in art4 ,of spontaneity was that idealism begotten and nurtured3 by any such system as ,mr ,hay's1 that spontaneity would be smothered and paralysed4 ,pheidias copied an idea in his own mind--#8,ipsius in mente insidebat species pulchritudinis eximia qudam' 7,cic472--later ages copied him4 ,he created3 they criticised4 ,he was the author of ,iliads3 they the authors of ,poetics4 ,doubtless1 if you unsphere the spirit of ,mr ,hay's theories1 you will find nothing discordant with what ,i have here said4 ,that is a sound view of ,beauty which makes it consist in that due subordination of the parts to the whole1 that due relation of the parts to each other1 which ,mendelssohn had in his mind when he said that the essence of beauty was #8unity in variety'--variety beguiling the imagination1 the perception of unity exercising the thewes and sinews of the intellect4 ,on such a view of beauty1 ,mr ,hay's theory may1 in spirit1 be said to rest4 ,but here1 as in higher things1 it is the letter that killeth1 while the spirit giveth life4 ,and accordingly ,i must enter a protest against any endeavour to foist upon the palmy days of ,hellenic art systems of geometrical proportions incompatible1 as ,i believe1 with those higher and broader principles by which the progress of ancient sculpture was ordered and governed--systems which will bear nothing of that #8felicity and chance by which'--and not by rule--#8,lord ,bacon believed that a painter may make a better face than ever was3' systems which take no account of that fundamental distinction between the schools of ,athens and of ,argos1 and their respective disciples and descendants1 without which you will make nonsense of the pages of ,pliny1 and--what is worse--sense of the pages of his commentators2--systems1 in short1 which may have their value as instruments for the education of the eye1 and for instructions in the arts of design1 but must be cast aside as matters of learned trifling and curious disputation1 where they profess to be royal roads to art1 and to map the mighty maze of a creative mind4 ,and even as regards the application of such a system of proportions to those works of sculpture which are posterior to the ,pheidian age1 only partial can have been the prevalence which it or any other one system can have obtained4 ,the discrepancies of different artists in the treatment of what was called1 technically called1 ,symmetria 7as in the title of ,euphranor's work7 were1 by the concurrent testimony of all ancient writers1 far too salient and important to warrant the supposition of any uniform scale of proportions1 as advocated by ,mr ,hay4 ,even in ,egypt1 where one might surely have expected that such uniformity would have been observed with far greater rigour than in ,greece1 the discoveries of ,dr ,lepsius 7,vorlufige ,nachricht1 ,berlin1 #ahdi7 have elicited three totally different 1 one of which is identical with the system of proportions of the human figure detailed in ,diodorus4 ,while we thus venture to differ from ,mr ,hay on the historical data he has mixed up with his inquiries1 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 artists1 which brings back to one common type the sensations of eye and ear1 and so makes a giant stride towards that codification1 if ,i may so speak1 of the laws of the universe which it is the business of the science to effect4 ,i have no hesitation in saying1 that1 for scientific precision of method and importance of results1 ,albert ,durer1 ,da ,vinci1 and ,hogarth1 not to mention less noteworthy writers1 must all yield the palm to ,mr ,hay4 8,i am quite aware that in the digression ,i have here allowed myself1 on systems of proportions prevalent among ancient artists1 and on the probable contents of such treatises as that of ,euphranor1 ,de ,symmetria1 ,i have laid myself open to the charge of treating an intricate question in a very perfunctory way4 ,at present the exigencies of the subject more immediately in hand allow me only to urge in reply1 that1 as regards the point at issue--,i mean the #8solidarit' between theories such as ,mr ,hay's and the practice of ,pheidias--the onus probandi rests with my adversaries40 ,i am bound1 in the first place1 gratefully to acknowledge the kind and complimentary notice which1 notwithstanding our difference of opinion1 this author has been pleased to take of my works2 and1 in the second1 to assure him that if any of them profess to be 8royal roads to art10 or to 8map the mighty maze of a creative mind10 they certainly profess to do more than ,i ever meant they should2 for ,i never entertained the idea that a system of sthetic culture1 even when based upon a law of nature1 was capable of effecting any such object4 ,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 style4 ,i have certainly1 on one occasion1#bg gone the length of stating that as poetic genius must yield obedience to the rules of rhythmical measure1 even in the highest flights of her inspirations2 and musical genius must1 in like manner1 be subject to the strictly defined laws of harmony in the most delicate1 as well as in the most powerfully grand of her compositions2 so must genius1 in the formative arts1 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 standards4 ,if1 therefore1 the laws of proportion1 in their relation to the arts of design1 constitute the harmony of geometry1 as definitely as those that are applicable to poetry and music produce the harmony of acoustics2 the former ought1 certainly1 to hold the same relative position in those arts which are addressed to the eye1 that is accorded to the latter in those which are addressed to the ear4 ,until so much science be brought to bear upon the arts of design1 the student must continue to copy from individual and imperfect objects in nature1 or from the few existing remains of ancient ,greek art1 in total ignorance of the laws by which their proportions are produced1 and1 what is equally detrimental to art1 the accuracy of all criticism must continue to rest upon the indefinite and variable basis of mere opinion4 ,it cannot be denied that men of great artistic genius are possessed of an intuitive feeling of appreciation for what is beautiful1 by means of which they impart to their works the most perfect proportions1 independently of any knowledge of the definite laws which govern that species of beauty4 ,but they often do so at the expense of much labour1 making many trials before they can satisfy themselves in imparting to them the true proportions which their minds can conceive1 and which1 along with those other qualities of expression1 action1 or attitude1 which belong more exclusively to the province of genius4 ,in such cases1 an acquaintance with the rules which constitute the science of proportion1 instead of proving fetters to genius1 would doubtless afford her such a vantage ground as would promote the more free exercise of her powers1 and give confidence and precision in the embodiment of her inspirations2 qualities which1 although quite compatible with genius1 are not always intuitively developed along with that gift4 ,it is also true that the operations of the conceptive faculty of the mind are uncontrolled by definite laws1 and that1 therefore1 there cannot exist any rules by the inculcation of which an ordinary mind can be imbued with genius sufficient to produce works of high art4 ,nevertheless1 such a mind may be improved in its perceptive faculty by instruction in the science of proportion1 so as to be enabled to exercise as correct and just an appreciation of the conceptions of others1 in works of plastic art1 as that manifested by the educated portion of mankind in respect to poetry and music4 ,in short1 it appears that1 in those arts which are addressed to the ear1 men of genius communicate the original conceptions of their minds under the control of certain scientific laws1 by means of which the educated easily distinguish the true from the false1 and by which the works of the poet and musical composer may be placed above mere imitations of nature1 or of the works of others2 while1 in those arts that are addressed to the eye in their own peculiar language1 such as sculpture1 architecture1 painting1 and ornamental design1 no such laws are as yet acknowledged4 ,although ,i am1 and ever have been1 far from endeavouring 8to foist upon the palmy days of ,hellenic art0 any system incompatible with those higher and more intellectual qualities which genius alone can impart2 yet1 from what has been handed down to us by writers on the subject1 meagre as it is1 ,i cannot help continuing to believe that1 along with the physical and metaphysical sciences1 sthetic science was taught in the early schools of ,greece4 ,i shall here take the liberty to repeat the proofs ,i advanced in a former work as the ground of this belief1 and to which the author1 from whose essay ,i have quoted1 undoubtedly refers4 ,it is well known that1 in the time of ,pythagoras1 the treasures of science were veiled in mystery to all but the properly initiated1 and the results of its various branches only given to the world in the works of those who had acquired this knowledge4 ,so strictly was this secresy maintained amongst the disciples and pupils of ,pythagoras1 that any one divulging the sacred doctrines to the profane1 was expelled the community1 and none of his former associates allowed to hold further intercourse with him2 it is even said1 that one of his pupils incurred the displeasure of the philosopher for having published the solution of a problem in geometry4#bh ,the difficulty1 therefore1 which is expressed by writers1 shortly after the period in which ,pythagoras lived1 regarding a precise knowledge of his theories1 is not to be wondered at1 more especially when it is considered that he never committed them to writing4 ,it would appear1 however1 that he proceeded upon the principle1 that the order and beauty so apparent throughout the whole universe1 must compel men to believe in1 and refer them to1 an intelligible cause4 ,pythagoras and his disciples sought for properties in the science of numbers1 by the knowledge of which they might attain to that of nature2 and they conceived those properties to be indicated in the phenomena of sonorous bodies4 ,observing that ,nature herself had thus irrevocably fixed the numerical value of the intervals of musical tones1 they justly concluded that1 as she is always uniform in her works1 the same laws must regulate the general system of the universe4#bi ,pythagoras1 therefore1 considered numerical proportion as the great principle inherent in all things1 and traced the various forms and phenomena of the world to numbers as their basis and essence4 ,how the principles of numbers were applied in the arts is not recorded1 farther than what transpires in the works of ,plato1 whose doctrines were from the school of ,pythagoras4 ,in explaining the principle of beauty1 as developed in the elements of the material world1 he commences in the following words3--8,but when the ,artificer began to adorn the universe1 he first of all figured with forms and numbers1 fire and earth1 water and air--which possessed1 indeed1 certain traces of the true elements1 but were in every respect so constituted as it becomes anything to be from which ,deity is absent4 ,but we should always persevere in asserting that ,divinity rendered them1 as much as possible1 the most beautiful and the best1 when they were in a state of existence opposite to such a condition40 ,plato goes on further to say1 that these elementary bodies must have forms2 and as it is necessary that every depth should comprehend the nature of a plane1 and as of plane figures the triangle is the most elementary1 he adopts two triangles as the originals or representatives of the isosceles and the scalene kinds4 ,the first triangle of ,plato is that which forms the half of the square1 and is regulated by the number1 #b2 and the second1 that which forms the half of the equilateral triangle1 which is regulated by the number1 #c2 from various combinations of these1 he formed the bodies of which he considered the elements to be composed4 ,to these elementary figures ,i have already referred4 ,vitruvius1 who studied architecture ages after the arts of ,greece had been buried in the oblivion which succeeded her conquest1 gives the measurements of various details of monuments of ,greek art then existing4 ,but he seems to have had but a vague traditionary knowledge of the principle of harmony and proportion from which these measurements resulted4 ,he says--8,the several parts which constitute a temple ought to be subject to the laws of symmetry2 the principles of which should be familiar to all who profess the science of architecture4 ,symmetry results from proportion1 which1 in the ,greek language1 is termed analogy4 ,proportion is the commensuration of the various constituent parts with the whole2 in the existence of which symmetry is found to consist4 ,for no building can possess the attributes of composition in which symmetry and proportion are disregarded2 nor unless there exist that perfect conformation of parts which may be observed in a well-formed human being40 ,after going at some length into details1 he adds--8,since1 therefore1 the human figure appears to have been formed with such propriety1 that the several members are commensurate with the whole1 the artists of antiquity 7meaning those of ,greece at the period of her highest refinement7 must be allowed to have followed the dictates of a judgment the most rational1 when1 transferring to works of art principles derived from nature1 every part was so regulated as to bear a just proportion to the whole4 ,now1 although the principles were universally acted upon1 yet they were more particularly attended to in the construction of temples and sacred edifices1 the beauties or defects of which were destined to remain as a perpetual testimony of their skill or of their inability40 ,vitruvius1 however1 gives no explanation of this ancient principle of proportion1 as derived from the human form2 but plainly shews his uncertainty upon the subject1 by concluding this part of his essay in the following words3 8,if it be true1 therefore1 that the decenary notation was suggested by the members of man1 and that the laws of proportion arose from the relative measures existing between certain parts of each member and the whole body1 it will follow1 that those are entitled to our commendation who1 in building temples to their deities1 proportioned the edifices1 so that the several parts of them might be commensurate with the whole40 ,it thus appears certain that the ,grecians1 at the period of their highest excellence1 had arrived at a knowledge of some definite mathematical law of proportion1 which formed a standard of perfectly symmetrical beauty1 not only in the representation of the human figure in sculpture and painting1 but in architectural design1 and indeed in all works where beauty of form and harmony of proportion constituted excellence4 ,that this law was not deduced from the proportions of the human figure1 as supposed by ,vitruvius1 but had its origin in mathematical science1 seems equally certain2 for in no other way can we satisfactorily account for the proportions of the beau ideal forms of the ancient ,greek deities1 or of those of their architectural structures1 such as the ,parthenon1 the temple of ,theseus1 c41 or for the beauty that pervades all the other formative art of the period4 ,this system of geometrical harmony1 founded1 as ,i have shewn it to be1 upon numerical relations1 must consequently have formed part of the ,greek philosophy of the period1 by means of which the arts began to progress towards that great excellence which they soon after attained4 ,a little further investigation will shew1 that immediately after this period a theory connected with art was acknowledged and taught1 and also that there existed a ,science of ,proportion4 ,pamphilus1 the celebrated painter1 who flourished about four hundred years before the ,christian era1 from whom ,apelles received the rudiments of his art1 and whose school was distinguished for scientific cultivation1 artistic knowledge1 and the greatest accuracy in drawing1 would admit no pupil unacquainted with geometry4#cj ,the terms upon which he engaged with his students were1 that each should pay him one talent 7#bbe sterling7 previous to receiving his instructions2 for this he engaged 8to give them1 for ten years1 lessons founded on an excellent theory40#ca ,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--8a law10 says the ,abb ,barthlmie1 8which rescued the fine arts from servile hands40 ,it is stated of ,parrhasius1 the rival of ,zeuxis1 who flourished about the same period as ,pamphilus1 that he accelerated the progress of art by purity and correctness of design2 8for he was acquainted with the science of ,proportions4 ,those he gave his gods and heroes were so happy1 that artists did not hesitate to adopt them40 ,parrhasius1 it is also stated1 was so admired by his contemporaries1 that they decreed him the name of ,legislator4#cb ,the whole history of the arts in ,egypt and ,greece concurs to prove that they were based on geometric precision1 and were perfected by a continued application of the same science2 while in all other countries we find them originating in rude and misshapen imitations of nature4 ,in the earliest stages of ,greek art1 the gods--then the only statues--were represented in a tranquil and fixed posture1 with the features exhibiting a stiff inflexible earnestness1 their only claim to excellence being symmetrical proportion2 and this attention to geometric precision continued as art advanced towards its culminating point1 and was thereafter still exhibited in the neatly and regularly folded drapery1 and in the curiously braided and symmetrically arranged hair4#cc ,these researches1 imperfect as they are1 cannot fail to exhibit the great contrast that exists between the system of elementary education in art practised in ancient ,greece1 and that adopted in this country at the present period4 ,but it would be of very little service to point out this contrast1 were it not accompanied by some attempt to develop the principles which seem to have formed the basis of this excellence amongst the ,greeks4 ,but in making such an attempt1 ,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 idealism4 ,for it is in no way more incompatible with the free exercise of artistic genius1 that those who are so gifted should have the advantage of an elementary education in the science of sthetics1 than it is incompatible with the free exercise of literary or poetic genius1 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 afford4 ,no4 ,,ii4 ,the letter from which ,i have made a quotation at page #db1 arose out of the following circumstance3--,in order that my theory1 as applied to the orthographic beauty of the ,parthenon1 might be brought before the highest tribunal which this country afforded1 ,i sent a paper upon the subject1 accompanied by ample illustrations1 to the ,royal ,institute of ,british ,architects1 and it was read at a meeting of that learned body on the #gth of ,february #ahec2 at the conclusion of which1 ,mr ,penrose kindly undertook to examine my theoretical views1 in connexion with the measurements he had taken of that beautiful structure by order of the ,dilettanti ,society1 and report upon the subject to the ,royal ,institute4 ,this report was published by ,mr ,penrose1 vol4 xi41 ,no4 #eci of ,the ,builder1 and the letter from which ,i have quoted appeared in ,no4 #edb of the same journal4 ,it was as follows3-- 8,,geometrical ,,relations ,,in ,,architecture4 8,will you allow me1 through the medium of your columns1 to thank ,mr ,penrose for his testimony to the truth of ,mr ,hay's revival of ,pythagoras8 ,the dimensions which he gives are to me the surest verification of the theory that ,i could have desired4 ,the minute discrepancies form that very element of practical incertitude1 both as to execution and direct measurement1 which always prevails in materialising a mathematical calculation under such conditions4 8,it is time that the scattered computations by which architecture has been analysed--more than enough--be synthetised into those formul which1 as ,mrs ,somerville tells us1 #8are emblematic of omniscience4' ,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 art4 ,sir1 it is food of this kind a reviving poetry demands4 ----#8,give us truths1 ,for we are weary of the surfaces1 ,and die of inanition4' 8,i1 for one1 as ,i listen to such demonstrations1 whose scope extends with every research into them1 feel as if listening to those words of ,pythagoras1 which sowed in the mind of ,greece the poetry whose manifestation in beauty has enchained the world in worship ever since its birth4 ,and ,i am sure that in such a quarter1 and in such thoughts1 we must look for the first shining of that lamp of art1 which even now is prepared to burn4 8,i know that this all sounds rhapsodical2 but ,i know also that until the architect becomes a poet1 and not a tradesman1 we may look in vain for architecture3 and ,i know that valuable as isolated and detailed investigations are in their proper bearings1 yet that such purposes and bearings are to be found in the enunciation of principles sublime as the generalities of #8mathematical beauty4' 8,,autocthon40 ,no4 ,,iii4 ,of the work alluded to at page #eh ,i was favoured with two opinions--the one referring to the theory it propounds1 and the other to its anatomical accuracy--both of which ,i have been kindly permitted to publish4 ,the first is from ,sir ,,william ,,hamilton1 ,bart41 professor of logic and metaphysics in the ,university of ,edinburgh1 and is as follows3-- 8,your very elegant volume is to me extremely interesting1 as affording an able contribution to what is the ancient1 and1 ,i conceive1 the true theory of the ,beautiful4 ,but though your doctrine coincides with the one prevalent through all antiquity1 it appears to me quite independent and original in you2 and ,i esteem it the more1 that it stands opposed to the hundred one-sided and exclusive views prevalent in modern times4--#af ,great ,king ,street1 ,march #e1 #ahdi40 ,the second is from ,,john ,,goodsir1 ,esq41 professor of anatomy in the ,university of ,edinburgh1 and is as follows3-- 8,i have examined the plates in your work on the proportions of the human head and countenance1 and find the head you have given as typical of human beauty to be anatomically correct in its structure1 only differing from ordinary nature in its proportions being more mathematically precise1 and consequently more symmetrically beautiful4--,college1 ,edinburgh1 #agth ,april #ahdi40 ,no4 ,,iv4 ,i shall here shew1 as ,i have done in a former work1 how the curvilinear outline of the figure is traced upon the rectilinear diagrams by portions of the ellipse of 771 771 771 and 774 ,sidenote3 ,plate ,,xix4 ,the outline of the head and face1 from points 7#a7 to 7#c7 7fig4 #a1 ,plate ,,xix471 takes the direction of the two first curves of the diagram4 ,from point 7#c71 the outline of the sterno-mastoid muscle continues to 7#d71 where1 joining the outline of the trapezius muscle1 at first concave1 it becomes convex after passing through 7#e71 reaches the point 7#f71 where the convex outline of the deltoid muscle commences1 and1 passing through 7#g71 takes the outline of the arm as far as 7#h74 ,the outline of the muscles on the side1 the latissimus dorsi and serratus magnus1 commences under the arm at the point 7#i71 and joins the outline of the oblique muscle of the abdomen by a concave curve at 7#aj71 which1 rising into convexity as it passes through the points 7#aa7 and 7#ab71 ends at 7#ac71 where it joins the outline of the gluteus medius muscle4 ,the outline of this latter muscle passes convexly through the point 7#ad71 and ends at 7#ae71 where the outline of the tensor vagin femoris and vastus externus muscle of the thigh commences4 ,this convex outline joins the concave outline of the biceps of the thigh at 7#af71 which ends in that of the slight convexity of the condyles of the thigh-bone at 7#ag74 ,from this point1 the outline of the outer surface of the leg1 which includes the biceps1 peroneus longus1 and soleus muscles1 after passing through the point 7#ah71 continues convexly to 7#ai71 where the concave outline of the tendons of the peroneus longus continues to 7#bj71 whence the outline of the outer ankle and foot commences4 ,the outline of the mamma and fold of the arm-pit commences at 7#ba71 and passes through the points 7#bb7 and 7#bc74 ,the circle at 7#bd7 is the outline of the areola1 in the centre of which the nipple is placed4 ,the outline of the pubes commences at 7#be71 and ends at the point 7#bf71 from which the outline of the inner surface of the thigh proceeds over the gracilis1 the sartorius1 and vastus internus muscles1 until it meets the internal face of the knee-joint at 7#bg71 the outline of which ends at 7#bh74 ,the outline of the inside of the leg commences by proceeding over the gastrocnemius muscle as far as 7#bi71 where it meets that of the soleus muscle1 and continues over the tendons of the heel until it meets the outline of the inner ankle and foot at 7#cj74 ,the outline of the outer surface of the arm1 as viewed in front1 proceeds from 7#h7 over the remainder of the deltoid1 in which there is here a slight concavity1 and1 next1 from 7#ca7 over the biceps muscle till 7#cb71 where it takes the line of the long supinator1 and passing concavely1 and almost imperceptibly1 into the long and short radial extensor muscles1 reaches the wrist at 7#cc74 ,the outline of the inner surface of the arm from opposite 7#i7 commences by passing over the triceps extensor1 which ends at 7#cd71 then over the gentle convexity of the condyles of the bone of the arm at 7#ce71 and1 lastly1 over the flexor sublimis which ends at the wrist-joint 7#cf74 ,the outline of the front of the figure commences at the point 7#a71 7fig4 #b1 ,plate ,,ii471 and1 passing almost vertically over the platzsma-myoidis muscles1 reaches the point 7#b71 where the neck ends by a concave curve4 ,from 7#b7 the outline rises convexly over the ends of the clavicles1 and continues so over the pectoral muscle till it reaches 7#c71 where the mamma swells out convexly to 7#d71 and returns convexly towards 7#e71 where the curve becomes concave4 ,from 7#e7 the outline follows the undulations of the rectus muscle of the abdomen1 concave at the points 7#f7 and 7#g71 and having its greatest convexity at 7#h74 ,this series of curves ends with a slight concavity at the point 7#i71 where the horizontal branch of the pubes is situated1 over which the outline is convex and ends at 7#aj74 ,the outline of the thigh commences at the point 7#aa7 with a slight concave curve1 and then swells out convexly over the extensors of the leg1 and1 reaching 7#ab71 becomes gently concave1 and1 passing over the patella at 7#ac71 becomes again convex until it reaches the ligament of that bone1 where it becomes gently concave towards the point 7#ad71 whence it follows the slightly convex curve of the shin-bone1 and then1 becoming as slightly concave1 ends with the muscles in front of the leg at 7#ae74 ,the outline of the back commences at the point 7#af71 and1 following with a concave curve the muscles of the neck as far as 7#ag71 swells into a convex curve over the trapezius muscle towards the point 7#ah72 passing through which point1 it continues to swell outward until it reaches 7#ai71 half way between 7#ah7 and 7#bj72 whence the convexity1 becoming less and less1 falls into the concave curve of the muscles of the loins at 7#ba71 and passing through the point 7#bb71 it rises into convexity4 ,it then passes through the point 7#bc71 follows the outline of the gluteus maximus1 the convex curve of which rises to the point 7#bd71 and then returns inwards to that of 7#be71 where it ends in the fold of the hip4 ,from this point the outline follows the curve of the hamstring muscles by a slight concavity as far as 7#bf71 and then1 becoming gently convex1 it reaches 7#bg72 whence it becomes again gently concave1 with a slight indication of the condyle of the thigh-bone at 7#bh71 and1 reaching 7#bi71 follows the convex curve of the gastrocnemius muscle through the point 7#cj71 and falling into the convex curve of the tendo ,achilles at 7#ca71 ends in the concavity over the heel at 7#cb74 ,the outline of the front of the arm commences at the point 7#cc71 by a gentle concavity at the arm-pit1 and then swells out in a convex curve over the biceps1 reaching 7#cd71 where it becomes concave1 and passing through 7#ce71 again becomes convex in passing over the long supinator1 and1 becoming gently concave as it passes the radial extensors1 rises slightly at 7#cf71 and ends at 7#cg71 where the outline of the wrist commences4 ,the outline of the back of the arm commences with a concave curve at 7#ch71 which becomes convex as it passes from the deltoid to the long extensor and ends at the elbow 7#ci71 from below which the outline follows the convex curve of the extensor ulnaris1 reaching the wrist at the point 7#dj74 ,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 diagram1 in the same manner in which ,i shewed proportion to be imparted1 in a late work1 to the osseous structure4 ,the mode in which the curves of 771 771 771 771 and 77 are thus so harmoniously blended in the outline of the female figure1 only remains to be explained4 ,the curves which compose the outline of the female form are therefore simply those of 771 771 771 771 and 774 ,manner in which these curves are disposed in the lateral outline 7figure #a1 ,plate ,,xix473-- ,points4 ,curves4 ,head from #a to #b 77 ,face 0 #b 0 #c 77 ,neck 0 #c 0 #d 77 ,shoulder 0 #d 0 #f 77 0 0 #f 0 #h 77 ,trunk 0 #i 0 #ae 77 0 0 #ba 0 #bd 77 ,outer surface of thigh and leg 0 #ae 0 #bj 77 ,inner surface of thigh and leg 0 #be 0 #cj 77 ,outer surface of the arm 0 #h 0 #cc 77 ,inner surface of the arm 0 #i 0 #cf 77 ,manner in which they are disposed in the outline 7figure #b1 ,plate ,,xix473-- ,points4 ,curves4 ,front of neck from #a to #b 77 0 trunk 0 #b 0 #aj 77 ,back of neck 0 #af 0 #ah 77 0 trunk 0 #ah 0 #bc 77 0 0 0 #bc 0 #be 77 ,front of thigh and leg 0 #aa 0 #ac 77 0 0 0 0 #ac 0 #ae 77 ,back of thigh and leg 0 #be 0 #cb 77 ,front of the arm 0 #cc 0 #cg 77 ,back of the arm 0 #ch 0 #dj 77 ,foot 0 #j 0 #j 77 ,sidenote3 ,plate ,,xx4 ,in order to exemplify more clearly the manner in which these various curves appear in the outline of the figure1 ,i give in ,plate ,,xx4 the whole curvilinear figures1 complete1 to which these portions belong that form the outline of the sides of the head1 neck1 and trunk1 and of the outer surface of the thighs and legs4 ,the various angles which the axes of these ellipses form with the vertical1 will be found amongst other details in the works ,i have just referred to4 ,no4 ,v4 ,at page #he ,i have remarked upon the variety that may be introduced into any particular form of vase2 and1 in order to give the reader an idea of the ease with which this may be done without violating the harmonic law1 ,i shall here give three examples3-- ,sidenote3 ,plate ,,xxi4 ,the first of these 7,plate ,,xxi47 differs from the ,portland vase1 in the concave curve of the neck flowing more gradually into the convex curve of the body4 ,sidenote3 ,plate ,,xxii4 ,the second 7,plate ,,xxii47 differs from the same vase in the same change of contour1 as also in being of a smaller diameter at the top and at the bottom4 ,sidenote3 ,plate ,,xxiii4 ,the third 7,plate ,,xxiii47 is the most simple arrangement of the elliptic curve by which this kind of form may be produced2 and it differs from the ,portland vase in the relative proportions of height and diameter1 and in having a fuller curve of contour4 ,the following comparison of the angles employed in these examples1 with the angles employed in the original1 will shew that the changes of contour in these forms1 arise more from the mode in which the angles are arranged than in a change of the angles themselves3-- ,line ,line ,line ,line ,plate ,,viii4 ,a ,c 77 ,b ,c 77 o p 7,h7 v u 77 ,plate ,,xxi4 77 77 77 77 ,plate ,,xxii4 77 77 77 77 ,plate ,,xxiii4 77 77 7,h7 7-7 ,line ,line ,plate ,,viii4 m n 77 i k 77 ellipse 77 rectangle 77 ,plate ,,xxi4 77 77 77 77 ,plate ,,xxii4 77 77 77 77 ,plate ,,xxiii4 77 77 ellipses 77 77 77 ,the harmonic elements of each are therefore simply the following parts of the right angle3-- ,tonic4 ,dominant4 ,mediant4 ,submediant4 ,plate ,,viii4 77 77 77 77 77 ,tonic4 ,dominant4 ,mediant4 ,supertonic4 ,plate ,,xxi4 77 77 77 77 77 ,tonic4 ,dominant4 ,mediant4 ,supertonic4 ,plate ,,xxii4 77 77 77 77 77 77 ,tonic4 ,dominant4 ,mediant4 ,plate ,,xxiii4 77 77 77 77 ,no4 ,,vi4 ,so far as ,i know1 there has been only one attempt in modern times1 besides my own1 to establish a universal system of proportion1 based on a law of nature1 and applicable to art4 ,this attempt consists of a work of #deg pages1 with #aff engraved illustrations1 by ,dr ,zeising1 a professor in ,leipzic1 where it was published in #ahed4 ,one of the most learned and talented professors in our ,edinburgh ,university has reviewed that work as follows3-- 8,it has been rather cleverly said that the intellectual distinction between an ,englishman and a ,scotchman is this--#8,give an ,englishman two facts1 and he looks out for a third2 give a ,scotchman two facts1 and he looks out for a theory4' ,neither of these tests distinguishes the ,german2 he is as likely to seek for a third fact as for a theory1 and as likely to build a theory on two facts as to look abroad for further information4 ,but once let him have a theory in his mind1 and he will ransack heaven and earth until he almost buries it under the weight of accumulated facts4 ,this remark applies with more than common force to a treatise published last year by ,dr ,zeising1 a professor in ,leipsic1 #8,on a law of proportion which rules all nature4' ,the ingenious author1 after proving from the writings of ancient and modern philosophers that there always existed the belief 7whence derived it is difficult to say71 that some law does bind into one formula all the visible works of ,god1 proceeds to criticise the opinions of individual writers respecting that connecting law4 ,it is not our purpose to follow him through his lengthy examination4 ,suffice it to say that he believes he has found the lost treasure in the ,timus of ,plato1 c4 #ca4 ,the passage is confessedly an obscure one1 and will not bear a literal translation4 ,the interpretation which ,dr ,zeising puts on it is certainly a little strained1 but we are disposed to admit that he does it with considerable reason4 ,agreeably to him1 the passage runs thus3--#8,that bond is the most beautiful which binds the things as much as possible into one2 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 less4' 8,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 ancients1 in his endeavour to establish the fact of the existence of this law amongst them4 ,the canon of ,polycletes1 the tradition of ,varro mentioned by ,pliny relative to that canon1 the writings of ,galen and others1 are all brought to bear on the same point with more or less force4 ,the sum of this portion of the argument is fairly this1--that the ancient sculptors had some law of proportion--some authorised examplar to which they referred as their work proceeded4 ,that it was the law here attributed to ,plato is by no means made out2 but1 considering the incidental manner in which that law is referred to1 and the obscurity of the passages as they exist1 it is1 perhaps1 too much to expect more than this broad feature of coincidence--the fact that some law was known and appealed to4 ,dr ,zeising now proceeds to examine modern theories1 and it is fair to state that he appears generally to take a very just view of them4 8,let us now turn to ,dr ,zeising's own theory4 ,it is this--that in every beautiful form lines are divided in extreme and mean ratio2 or1 that any line considered as a whole1 bears to its larger part the same proportion that the larger bears to the smaller--thus1 a line of #e inches will be divided into parts which are very nearly #b and #c inches respectively 7#ai and #ca inches74 ,this is a well-known division of a line1 and has been called the ,,golden rule1 but when or why1 it is not easy to ascertain4 ,with this rule in his hand1 ,dr ,zeising proceeds to examine all nature and art2 nay1 he even ventures beyond the threshold of nature to scan ,deity4 ,we will not follow him so far4 ,let us turn over the pages of his carefully illustrated work1 and see how he applies his line4 ,we meet first with the ,apollo ,belvidere--the golden line divides him happily4 ,we cannot say the same of the division of a handsome face which occurs a little further on4 ,our preconceived notions have made the face terminate with the chin1 and not with the centre of the throat4 ,it is evident that1 with such a rule as this1 a little latitude as to the extreme point of the object to be measured1 relieves its inventor from a world of perplexities4 ,this remark is equally applicable to the arm which follows1 to which the rule appears to apply admirably1 yet we have tried it on sundry plates of arms1 both fleshy and bony1 without a shadow of success4 ,whether the rule was made for the arm or the arm for the rule1 we do not pretend to decide4 ,but let us pass hastily on to page #bhd1 where the ,venus de ,medicis and ,raphael's ,eve are presented to us4 ,they bear the application of the line right well4 ,it might1 perhaps1 be objected that it is remarkable that the same rule applies so exactly to the existing position of the figures1 such as the ,apollo and the ,venus1 the one of which is upright1 and the other crouching4 ,but let that pass4 ,we find ,dr ,zeising next endeavouring to square his theory with the distances of the planets1 with wofully scanty success4 ,descending from his lofty position1 he spans the earth from corner to corner1 at which occupation we will leave him for a moment1 whilst we offer a suggestion which is equally applicable to poets1 painters1 novelists1 and theorisers4 ,never err in excess--defect is the safe side--it is seldom a fault1 often a real merit4 ,leave something for the student of your works to do--don't chew the cud for him4 ,be assured he will not omit to pay you for every little thing which you have enabled him to discover4 ,poor ,professor ,zeising6 he is far too ,german to leave any field of discovery open for his readers4 ,but let us return to him2 we left him on his back1 lost for a time in a hopeless attempt to double ,cape ,horn4 ,we will be kind to him1 as the child is to his man in the ,noah's ark1 and set him on his legs amongst his toys again4 ,he is now in the vegetable kingdom1 amidst oak leaves and sections of the stems of divers plants4 ,he is in his element once more1 and it were ungenerous not to admit the merit of his endeavours1 and the success which now and then attends it4 ,we will pass over his horses and their riders1 together with that portly personage1 the ,durham ox1 for we have caught a glimpse of a form familiar to our eyes1 the ever-to-be-admired ,parthenon4 ,this is the true test of a theory4 ,unlike the ,durham ox just passed before us1 the ,parthenon will stand still to be measured4 ,it has so stood for twenty centuries1 and every one that has scanned its proportions has pronounced them exquisite4 ,beauty is not an adaptation to the acquired taste of a single nation1 or the conventionality of a single generation4 ,it emanates from a deep-rooted principle in nature1 and appeals to the verdict of our whole humanity4 ,we don't find fault with the ,durham ox--his proportions are probably good1 though they be the result of breeding and cross-breeding2 still we are not sure whether1 in the march of agriculture1 our grandchildren may not think him a very wretched beast4 ,but there is no mistake about the ,parthenon2 as a type of proportion it stands1 has stood1 and shall stand4 ,well1 then1 let us see how ,dr ,zeising succeeds with his rule here4 ,alas6 not a single point comes right4 ,the ,parthenon is condemned1 or its condemnation condemns the theory4 ,choose your part4 ,we choose the latter alternative2 and now1 our choice being made1 we need proceed no further4 ,but a question or two have presented themselves as we went along1 which demand an answer4 ,it may be asked--,how do you account for the esteem in which this law of the section in extreme and mean ratio was held8 ,we reply--,that it was esteemed just in the same way that a tree is esteemed for its fruit4 ,to divide a right angle into two or three1 four or six1 equal parts was easy enough4 ,but to divide it into five or ten such parts was a real difficulty4 ,and how was the difficulty got over8 ,it was effected by means of this golden rule4 ,this is its great1 its ruling application2 and if we adopt the notion that the ancients were possessed with the idea of the existence of angular symmetry1 we shall have no difficulty in accounting for their appreciation of this problem4 ,nay1 we may even go further1 and admit1 with ,dr ,zeising1 the interpretation of the passage of ,plato1--only with this limitation1 that ,plato1 as a geometer1 was carried away by the geometry of sthetics from the thing itself4 ,it may be asked again--,is it not probable that some proportionality does exist amongst the parts of natural objects8 ,we reply--,that1 priori1 we expect some such system to exist1 but that it is inconsistent with the scheme of least effort1 which pervades and characterises all natural succession in space or in time1 that that system should be a complicated one4 ,whatever it is1 its essence must be simplicity4 ,and no system of simple linear proportion is found in nature2 quite the contrary4 ,we are1 therefore1 driven to another hypothesis1 viz4--that the simplicity is one of angles1 not of lines2 that the eye estimates by search round a point1 not by ascending and descending1 going to the right and to the left1--a theory which we conceive all nature conspires to prove4 ,beauty was not created for the eye of man1 but the eye of man and his mental eye were created for the appreciation of beauty4 ,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 symmetry2 or examine the forms of those little silicious creations which grew thousands of years before ,man was placed on the earth1 and1 with forms of marvellous and varied beauty1 they all point to its source in angular symmetry4 ,this is the keystone of formal beauty1 alike in the minutest animalcule1 and in the noblest of ,god's works1 his own image--,man40 ,,the ,,end4 ,,ballantyne ,,and ,,company1 ,,printers1 ,,edinburgh4 ,,footnotes #a ,sir ,david ,brewster4 #b ,no4 ,,clviii41 ,october #ahdc4 #c ,diogenes ,lartius's 8,lives of the ,philosophers10 literally translated4 ,bohn3 ,london4 #d ,ibid4 #e ,rose's 8,biographical ,dictionary40 #f ,professor ,laycock1 now of the ,university of ,edinburgh4 #g 8,the ,geometric ,beauty of the ,human ,figure ,defined10 c4 #h ,longman and ,co41 ,london4 #i ,see ,appendix4 #aj 8,the ,orthographic ,beauty of the ,parthenon10 c41 and 8,the ,harmonic ,law of ,nature applied to ,architectural ,design40 #aa ,for further details1 see 8,harmonic ,law of ,nature10 c4 #ab ,by a very simple machine1 which ,i have lately invented1 an ellipse of any given proportions1 even to those of 771 which is the curve of the entases of the columns of the ,parthenon 7see ,plate ,,vii471 and of any length1 from half an inch to fifty feet or upwards1 may be easily and correctly described2 the length and angle of the required ellipse being all that need be given4 #ac 8,the ,orthographic ,beauty of the ,parthenon10 c4 #ad 8,the ,orthographic ,beauty of the ,parthenon10 c4 #ae ,ibid4 #af 8,the ,harmonic ,law of ,nature applied to ,architectural ,design40 #ag 8,physio-philosophy40 ,by ,dr ,oken4 ,translated by ,talk2 and published by the ,ray ,society4 ,london1 #ahdh4 #ah 8,the ,science of those ,proportions by which the ,human ,head and ,countenance1 as represented in ,works of ancient ,greek ,art1 are distinguished from those of ordinary ,nature40 #ai 8,the ,geometric ,beauty of the ,human ,figure ,defined10 c41 and 8,the ,natural ,principles of ,beauty ,developed in the ,human ,figure40 #bj 8,the ,geometric ,beauty of the ,human ,figure ,defined10 c4 #ba 8,essay on ,ornamental ,design10 c41 and 8,the ,geometric ,beauty of the ,human ,figure10 c4 #bb 8,a ,nomenclature of ,colours1 applicable to the ,arts and ,natural ,sciences10 c41 c4 #bc ,see pp4 #bd and #be4 #bd 8,the ,principles of ,beauty in ,colouring ,systematised10 ,fourteen ,diagrams1 each containing ,six ,colours and ,hues4 8,a ,nomenclature of ,colours10 c41 ,forty ,diagrams1 each containing ,twelve ,examples of ,colours1 ,hues1 ,tints1 and ,shades4 8,the ,laws of ,harmonious ,colouring10 c41 ,one ,diagram1 containing ,eighteen ,colours and ,hues4 #be 8,etchings ,representing the ,best ,examples of ,grecian and ,roman ,architectural ,ornament1 drawn from the ,originals10 c4 ,by ,charles ,heathcote ,tatham1 ,architect4 ,london3 ,priestly and ,weale4 #ahbf4 #bf 8,the ,orthographic ,beauty of the ,parthenon10 c4 #bg 8,science of those ,proportions10 c4 #bh ,abb ,barthlmie's 8,travels of ,anacharsis in ,greece10 vol iv41 pp4 #aic1 #aie4 #bi ,abb ,barthlmie 7vol4 ii41 pp4 #afh1 #afi71 who cites as his authorities1 ,cicer4 ,de ,nat4 ,deor41 lib4 i41 cap4 ii41 t4 #b1 p4 #dje2 ,justin ,mart41 ,ovat4 ad ,gent41 p4 #aj2 ,aristot4 ,metaph41 lib4 i41 cap4 v41 t4 #b1 p4 #hde4 #cj ,mller's 8,ancient ,art and its ,remains40 #ca 8,anacharsis' ,travels in ,greece40 ,by the ,abb ,barthlmie1 vol4 ii41 p4 #cbe4 #cb 8,anacharsis' ,travels in ,greece40 ,by the ,abb ,barthlmie1 vol4 vi41 p4 #bbe4 ,the authorities the ,abb quotes are--,quintil41 lib4 xii41 cap4 x41 p4 #gdd2 ,plin41 lib4 xxxv41 cap4 ix41 p4 #fia4 #cc ,mller's 8,archology of ,art10 c4 ,works by the ,same ,author4 ,i4 ,in royal #hvo1 with ,copperplate ,illustrations1 price #bs4 #f;d41 ,,the ,,harmonic ,,law ,,of ,,nature ,,applied ,,to ,,architectural ,,design4 ,from the ,athenum4 ,the beauty of the theory is its universality1 and its simplicity4 ,in nature1 the ,creator accomplished his purposes by the simplest means--the harmony of nature is indestructible and self-restoring4 ,mr ,hay's book on the 8,parthenon10 on the 8,natural ,principles of ,beauty as developed in the ,human ,figure10 his 8,principles of ,symmetrical ,beauty10 his 8,principles of ,colouring1 and ,nomenclature of ,colours10 his 8,science of ,proportion10 and 8,essay on ,ornamental ,design10 we have already noticed with praise as the results of philosophical and original thought4 ,from the ,daily ,news4 ,this essay is a new application to ,lincoln cathedral in ,gothic architecture1 and to the ,temple of ,theseus in ,greek architecture1 of the principles of symmetrical beauty already so profusely illustrated and demonstrated by ,mr ,hay4 ,the theory which ,mr ,hay has propounded in so many volumes is not only a splendid contribution towards a science of sthetic proportions1 but1 for the first time in the history of art1 proves the possibility1 and lays the foundations of such a science4 ,to those who are not acquainted with the facts1 these expressions will sound hyperbolical1 but they are most true4 ,,ii4 ,in royal #hvo1 with ,copperplate ,illustrations1 price #es41 ,,the ,,natural ,,principles ,,of ,,beauty1 ,,as ,,developed ,,in ,,the ,,human ,,figure4 ,from the ,spectator4 ,we cannot refuse to entertain ,mr ,hay's system as of singular intrinsic excellence4 ,the simplicity of his law and its generality impress themselves more deeply on the conviction with each time of enforcement4 ,his theory proceeds from the idea1 that in nature every thing is effected by means more simple than any other that could have been conceived1--an idea certainly consistent with whatever we can trace out or imagine of the all-wise framing of the universe4 ,from the ,sun4 ,by founding 7if we may so phrase it7 this noble theory1 ,mr ,hay has covered his name with distinction1 and has laid the basis1 we conceive1 of no ephemeral reputation4 ,by illustrating it anew1 through the medium of this graceful treatise1 he has conferred a real boon upon the community1 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 ,omnipotence4 ,from the ,cambridge ,journal of ,classical and ,sacred ,philology4 ,the inquiries which of late years have been instituted by ,mr ,d4 ,r4 ,hay of ,edinburgh1 on the proportions of the human figure1 and on the natural principles of beauty1 as illustrated by works of ,greek art1 constitute an epoch in the study of sthetics and the philosophy of form4 ,,iii4 ,in royal #hvo1 with ,copperplate ,illustrations1 price #es41 ,,the ,,orthographic ,,beauty ,,of ,,the ,,parthenon ,,referred ,,to ,a ,,law ,,of ,,nature4 ,to which are prefixed1 a few ,observations on the ,importance of sthetic ,science as an ,element in ,architectural ,education4 ,from the ,scottish ,literary ,gazette4 ,we think this work will satisfy every impartial mind that ,mr ,hay has developed the true theory of the ,parthenon--that he has1 in fact1 to use a kindred phraseology1 both parsed and scanned this exquisitely beautiful piece of architectural composition1 and that1 in doing so1 he has provided the true key by which the treasures of ,greek art may be further unlocked1 and rendered the means of correcting1 improving1 and elevating modern practice4 ,from the ,edinburgh ,guardian4 ,again and again the attempt has been made to detect harmonic ratios in the measurement of ,athenian architecture1 but ever without reward4 ,mr ,hay has1 however1 made the discovery1 and to an extent of which no one had previously dreamt4 ,,iv4 ,in #hvo1 #ajj ,plates1 price #fs41 ,,first ,,principles ,,of ,,symmetrical ,,beauty4 ,from the ,spectator4 ,this is a grammar of pure form1 in which the elements of symmetrical1 as distinguished from picturesque beauty1 are demonstrated1 by reducing the outlines or planes of curvilinear and rectilinear forms to their origin in the principles of geometrical proportion4 ,in thus analysing symmetry of outline in natural and artificial objects1 ,mr ,hay determines the fixed principles of beauty in positive shape1 and shews how beautiful forms may be reproduced and infinitely varied with mathematical precision4 ,hitherto the originating and copying of beautiful forms have been alike empirical2 the production of a new design for a vase or a jug has been a matter of chance between the eye and the hand2 and the copying of a ,greek moulding or ornament1 a merely mechanical process4 ,by a series of problems1 ,mr ,hay places both the invention and imitation of beautiful forms on a sure basis of science1 giving to the fancy of original minds a clue to the evolving of new and elegant shapes1 in which the infinite resources of nature are made subservient to the uses of art4 ,the volume is illustrated by one hundred diagrams beautifully executed1 that serve to explain the text1 and suggest new ideas of beauty of contour in common objects4 ,to designers of pottery1 hardware1 and architectural ornaments1 this work is particularly valuable2 but artists of every kind1 and workmen of intelligence1 will find it of great utility4 ,from the ,athenum4 ,the volume before us is the seventh of ,mr ,hay's works4 ,it is the most practical and systematic1 and likely to be one of the most useful4 ,it is1 in short1 a grammar of form1 or a spelling-book of beauty4 ,this is beginning at the right end of the matter2 and the necessity for this kind of knowledge will inevitably1 though gradually1 be felt4 ,the work will1 therefore1 be ultimately appreciated and adopted as an introduction to the study of beautiful forms4 ,the third part of the work treats of the ,greek oval or composite ellipse1 as ,mr ,hay calls it4 ,it is an ellipse of three foci1 and gives practical forms for vases and architectural mouldings1 which are curious to the architect1 and will be very useful both to the potter1 the moulder1 and the pattern-drawer4 ,a fourth part contains applications of this to practice4 ,of the details worked out with so much judgment and ingenuity by ,mr ,hay1 we should in vain attempt to communicate just notions without the engravings of which his book is full4 ,we must1 therefore1 refer to the work itself4 ,the forms there given are full of beauty1 and so far tend to prove the system4 ,v4 ,in #hvo1 #ad ,coloured ,diagrams1 ,second ,edition1 price #aes41 ,,the ,,principles ,,of ,,beauty ,,in ,,colouring ,,systematised4 ,from the ,spectator4 ,in this new analysis of the harmonies of colour1 ,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 tints2 so that artists may1 by the aid of this book1 produce1 with mathematical certainty1 the richest effects1 hitherto attainable by genius alone4 ,mr ,hay has reduced this branch of art to a perfect system1 and proved that an offence against good taste in respect to combinations of colour is1 in effect1 a violation of natural laws4 ,,vi4 ,in #hvo1 #bbh ,examples of ,colours1 ,hues1 ,tints1 and ,shades1 price #fcs41 ,a ,,nomenclature ,,of ,,colours1 ,,applicable ,,to ,,the ,,arts ,,and ,,natural ,,sciences4 ,from the ,daily ,news4 ,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 acquainted1 and in proportion to this practical knowledge is1 as might be expected1 the excellence of his treatise4 ,there is much in this volume which we would most earnestly recommend to the notice of artists1 house decorators1 and1 indeed1 to all whose business or profession requires a knowledge of the management of colour4 ,the work is replete with hints which they might turn to profitable account1 and which they will find nowhere else4 ,from the ,athenum4 ,we have formerly stated the high opinion we entertain of ,mr ,hay's previous exertions for the improvement of decorative art in this country4 ,we have already awarded him the merit of invention and creation of the new and the beautiful in form4 ,in his former treatises he furnished a theory of definite proportions for the creation of the beautiful in form4 ,in the present work he proposes to supply a scale of definite proportions for chromatic beauty4 ,for this purpose he sets out very properly with a precise nomenclature of colour4 ,in this he has constructed a vocabulary for the artist--an alphabet for the artizan4 ,he has gone further--he constructs words for three syllables4 ,from this time1 it will be possible to write a letter in ,edinburgh about a coloured composition1 which shall be read off in ,london1 ,paris1 ,st ,petersburg1 or ,pekin1 and shall so express its nature that it can be reproduced in perfect identity4 ,this ,mr ,hay has done1 or at least so nearly1 as to deserve our thanks on behalf of art1 and artists of all grades1 even to the decorative artizan--not one of whom1 be he house-painter1 china pattern-drawer1 or calico printer1 should be without the simple manual of 8words for colours40 ,,vii4 ,in post #hvo1 with a ,coloured ,diagram1 ,sixth ,edition1 price #gs4 #f;d41 ,,the ,,laws ,,of ,,harmonious ,,colouring ,,adapted ,,to ,,interior ,,decorations4 ,from the ,atlas4 ,every line of this useful book shews that the author has contrived to intellectualise his subject in a very interesting manner4 ,the principles of harmony in colour as applied to decorative purposes1 are explained and enforced in a lucid and practical style1 and the relations of the various tints and shades to each other1 so as to produce a harmonious result1 are descanted upon most satisfactorily and originally4 ,from the ,edinburgh ,review4 ,in so far as we know1 ,mr ,hay is the first and only modern artist who has entered upon the study of these subjects without the trammels of prejudice and authority4 ,setting aside the ordinances of fashion1 as well as the dicta of speculation1 he has sought the foundation of his profession in the properties of light1 and in the laws of visual sensation1 by which these properties are recognised and modified4 ,the truths to which he has appealed are fundamental and irrefragable4 ,from the ,athenum4 ,we have regarded1 and do still regard1 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 interest1 not unmingled with respect4 ,we see how his mind has been gradually guided by ,nature herself out of one track1 and into another1 and ever and anon leading him to some vein of the beautiful and true1 hitherto unworked4 ,,viii4 ,in #dto1 #be ,plates1 price #cfs41 ,,on ,,the ,,science ,,of ,,those ,,proportions ,,by ,,which ,,the ,,human ,,head ,,and ,,countenance1 ,,as ,,represented ,,in ,,ancient ,,greek ,,art1 ,,are ,,distinguished ,,from ,,those ,,of ,,ordinary ,,nature4 7,,printed ,,by ,,permission47 ,from a ,letter to the ,author by ,sir ,william ,hamilton1 ,bart41 ,professor of ,logic and ,metaphysics in the ,edinburgh ,university4 ,your very elegant volume1 8,science of those ,proportions10 c41 is to me extremely interesting1 as affording an able contribution to what is the ancient1 and1 ,i conceive1 the true theory of the beautiful4 ,but though your doctrine coincides with the one prevalent through all antiquity1 it appears to me quite independent and original in you2 and ,i esteem it the more that it stands opposed to the hundred one-sided and exclusive views prevalent in modern times4 ,from ,chambers's ,edinburgh ,journal4 ,we now come to another1 and much more remarkable corroboration1 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 ,art1 viz41 ,mr ,hay's work 8,on the ,science of those ,proportions10 c4 ,mr ,hay's plan is simply to form a scale composed of the well-known vibrations of the monochord1 which are the alphabet of music1 and then to draw upon the quadrant of a circle angles answering to these vibrations4 ,with the series of triangles thus obtained he combines a circle and an ellipse1 the proportions of which are derived from the triangles themselves2 and thus he obtains an infallible rule for the composition of the head of ideal beauty4 ,,ix4 ,in #dto1 #af ,plates1 price #cjs41 ,,the ,,geometric ,,beauty ,,of ,,the ,,human ,,figure ,,defined4 ,to which is prefixed1 a ,,system of ,,sthetic ,,proportion applicable to ,,architecture and the other ,,formative ,,arts4 ,from the ,cambridge ,journal of ,classical and ,sacred ,philology4 ,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 artists1 which brings back to one common type the sensations of ,eye and ,ear1 and so makes a giant stride towards that codification of the laws of the universe which it is the business of science to effect4 ,we have no hesitation in saying that1 for scientific precision of method and importance of results1 ,albert ,durer1 ,da ,vinci1 and ,hogarth--not to mention less noteworthy writers--must all yield the palm to ,mr ,hay4 ,x4 ,in oblong folio1 #eg ,plates and numerous ,woodcuts1 price #dbs41 ,,an ,,essay ,,on ,,ornamental ,,design1 ,,in ,,which ,,its ,,true ,,principles ,,are ,,developed ,,and ,,elucidated1 c4 ,from the ,athenum4 ,in conclusion1 ,mr ,hay's book goes forth with our best wishes4 ,it must be good4 ,it must be prolific of thought--stimulant of invention4 ,it is to be acknowledged as a benefit of an unusual character conferred on the arts of ornamental design4 ,from the ,spectator4 ,mr ,hay has studied the subject deeply and scientifically4 ,in this treatise on ornamental design1 the student will find a clue to the discovery of the source of an endless variety of beautiful forms and combinations of lines1 in the application of certain fixed laws of harmonious proportion to the purposes of art4 ,mr ,hay also exemplifies the application of his theory of linear harmony to the production of beautiful forms generally1 testing its soundness by applying it to the human figure1 and the purest creations of ,greek art4 ,from ,fraser's ,magazine4 ,each part of this work is enriched by diagrams of great beauty1 direct emanations of principle1 and1 consequently1 presenting entirely new combinations of form4 ,had our space permitted1 we should have made some extracts from this 8,essay on ,ornamental ,design20 and we would have done so1 because of the discriminating taste by which it is pervaded1 and the forcible observations which it contains2 but we cannot venture on the indulgence4 ,,xi4 ,in #dto1 #ag ,plates and #ch ,woodcuts1 price #bes41 ,,proportion1 ,,or ,,the ,,geometric ,,principle ,,of ,,beauty ,,analysed4 ,,xii4 ,in #dto1 #ah ,plates and numerous ,woodcuts1 price #aes41 ,,the ,,natural ,,principles ,,and ,,analogy ,,of ,,the ,,harmony ,,of ,,form4 ,from the ,edinburgh ,review4 ,notwithstanding some trivial points of difference between ,mr ,hay's views and our own1 we have derived the greatest pleasure from the perusal of these works4 ,they are all composed with accuracy and even elegance4 ,his opinions and views are distinctly brought before the reader1 and stated with that modesty which characterises genius1 and that firmness which indicates truth4 ,from ,blackwood's ,magazine4 ,we have no doubt that when ,mr ,hay's ,art-discovery is duly developed and taught1 as it should be1 in our schools1 it will do more to improve the general taste than anything which has yet been devised4