J 
 
 Ulrich Middeldorf 
 
PROPORTION, 
 
PROPORTION, 
 OR THE GEOMETRIC PRINCIPLE 
 BEAUTY, ANALYSED. 
 
 OF 
 
 # 
 
 D. R. HAY, 
 
 DECORATIVE PAINTER TO THE QUEEN, EDINBURGH; 
 
 AUTHOR OF 
 
 " THE LAWS OF HARMONIOUS COLOURING ADAPTED TO INTERIOR DECORATIONS," 
 " THE NATURAL PRINCIPLES AND ANALOGY OF THE 
 HARMONY OF FORM," ETC. 
 
 WILLIAM BLACKWOOD AND SONS, EDINBURGH, 
 AND 22 PALL-MALL, LONDON. 
 
 M.DCCC.XLIII. 
 
Digitized by the Internet Archive 
 
 in 2013 
 
 http://archive.org/details/proportionorgeomOOhayd 
 
INTRODUCTION. 
 
 To attempt to reduce to fixed principles what is called Taste in the 
 combination of forms, is a difficult and perhaps a presumptuous task, 
 although such principles are universally felt to exist. Yet it is 
 not a hopeless labour, as it seems almost certain that the Grecians, 
 at the period of their highest refinement, had a positive geometric 
 principle of beauty systematically developed and applied in all their 
 works. 
 
 It is recorded in a curious treatise called the " Beau Ideal," quoted 
 by Hogarth in the preface to his " Analysis of Beauty," that a key 
 for finding all harmonious proportions in Painting, Sculpture, Archi- 
 tecture, Music, &c, was brought to Greece by Pythagoras, on his 
 return from travelling through Phoenicia, Egypt, and Chaldea, (about 
 a.m. 3484,) and that its application to the arts raised them to their 
 
 A 
 
ii 
 
 INTRODUCTION. 
 
 highest pitch of excellence. Flaxman, the celebrated sculptor, in 
 remarking, in his lectures, that the lines of Grecian composition 
 enchant the beholder by the harmony and perfection of their geo- 
 metric beauty, adds — 44 This portion of study seems to have been 
 highly esteemed by Pamphilus,* the learned Macedonian painter, who 
 denied that any one could succeed in the study of painting without 
 arithmetic and geometry. But this magic bond of arrangement was 
 utterly lost when the other perfections of Grecian genius were over- 
 whelmed in barbarism." 
 
 Vitruvius, the well-known Roman writer on architecture, seems 
 also to have been impressed with a conviction of the fact, that the 
 ancient Grecians possessed geometric rules of beauty, but at the same 
 time appears in considerable doubt as to what these rules were. At 
 the commencement of his work on Civil Architecture, he observes — 
 44 Symmetry results from proportion, which, in the Greek language, 
 is termed Analogia. Proportion is the commensuration of the various 
 constituent parts with the whole, in the existence of which symmetry 
 
 * Pamphilus flourished a.m. 3641. He was the master of Apelles, and is said to have applied 
 the key of harmony, or analogy, of the Grecians to the art of Painting. 
 
INTRODUCTION. 
 
 iii 
 
 is found to consist." He then remarks, that the proportions of a per- 
 fect building ought to resemble those of a well-formed human body, 
 of the proportions of which he gives various details, adding, that 
 the general standard by which all measurements were made was 
 deduced from its members. 
 
 He says, also, that the perfect number amongst the ancients was 
 thought to be ten, but that mathematicians, on the contrary, con- 
 tended that it was six ; and upon this principle the cities of Greece 
 appeared to have divided the drachma into six parts. The Romans, 
 however, preferred the ancient number ten ; but, in process of time, 
 being led to consider both ten and six as perfect numbers, they formed 
 what they thought a still more perfect one by combining the two. 
 
 That doubt existed as to the proper mode of harmonious division 
 at the period he writes, is clear from what follows : — " If it be true," 
 he adds, " that the decennary notation was suggested by the members 
 (the digits, for example) of man, and that the laws of proportion 
 arose from the relative measures existing between certain parts of 
 each member and the whole body, it will follow, that those are entitled 
 to our commendation, who, in building temples to their deities, pro- 
 
iv 
 
 INTRODUCTION. 
 
 portioned the edifices so that the several parts might be commensu- 
 rate with the whole." 
 
 This knowledge of the principles of geometric beauty which the 
 ancient Grecians possessed, and applied systematically to the arts of 
 design, seems, therefore, to have been lost before the time of Vitru- 
 vius.* 
 
 Although my former essay was almost exclusively directed to 
 proving the analogy that exists in the harmonies of sound, colour, and 
 form, and although I find that in a very extraordinary work,f lately 
 published, it has been clearly proved that such an analogy not only exists 
 as regards the elements of harmony, but also pervades nature, science, 
 and art generally, I must support my present attempt by again refer- 
 ring to the same subject, because the effects of harmony in regard to 
 sound have been thoroughly investigated as a branch of natural phi- 
 losophy, and fixed principles established for their regulation. 
 
 In Painting and Sculpture, form addresses itself to the eye in two 
 ways. The first quality required in works of these arts, and that 
 
 * In the era of Julius Ca?sar. 
 
 f Field's Outlines of Analogical Philosophy. London : 1839. 
 
INTRODUCTION. 
 
 v 
 
 which is most easily appreciated by the generality of mankind, is a 
 faithful imitation of the configuration of any natural object or objects 
 that may be represented in such works. These arts are consequently 
 called the imitative arts, in contradistinction to those in which no 
 natural object is imitated. The other quality of form required to 
 constitute excellence in works of art, lies in what is technically called 
 composition. This, in Painting and Sculpture, is the disposition or 
 arrangement of such natural objects, or their parts, as may be brought 
 together in forming a subject, independently of their individual merits. 
 This disposition may be made so as greatly to enhance the beauty or 
 grandeur of the subject, by the scientific combination of similarity and 
 dissimilarity, and of various modes of simplicity and variety. No 
 production in Painting or Sculpture can be reckoned a great work of 
 art which depends exclusively on the first of those qualities of form ; 
 but many have been reckoned so from the science displayed in their 
 mode of composition, although defective in accuracy of imitation. 
 
 These qualities are very different in their effect upon the mind. 
 Imitation in works of art is appreciated by the degree of deception 
 to which the judgment, through the limited capacity of the visual 
 
Vi 
 
 INTRODUCTION. 
 
 organ, is subjected; while scientific composition of form seems to be 
 appreciated by an inherent feeling responsive to certain mathematical 
 principles of propriety and harmony existing in nature, and conveying 
 an impression to the mind through the medium of the senses. 
 
 The beauty of all original architectural compositions depends upon 
 mathematical harmony alone, because in such there is no imitation ; 
 and it can scarcely be doubted that the Five Orders owe their origin 
 and the perfection of their proportions to some systematic mode of 
 applying those principles practically in this art. Upon this subject 
 Field makes the following excellent observation: — " In proportion as 
 we acquire rules and principles of our own, we shall be released from 
 the servile necessity of continuing mere imitators of those ancients, 
 with the philosophy of whose practice we are little acquainted, and 
 who certainly did not work without principles themselves." * 
 
 So far as I know, Burke is the only writer on taste who does not 
 admit that proportion is one of the constituents of beauty. He says 
 that " proportion relates almost wholly to convenience ;"f and that it 
 
 * Outlines of Analogical Philosophy, 
 f Essay on the Sublime and Beautiful. 
 
INTRODUCTION. 
 
 vii 
 
 should therefore be considered as a creature of the understanding, 
 rather than a primary cause acting on the senses and imagina- 
 tion. It is very evident that proportion and fitness are here, as 
 well as throughout this part of the Essay, considered as the same 
 quality ; but they appear to be qualities of a very different nature. 
 Fitness certainly is a creature of the understanding relating to use 
 and convenience, and cannot therefore be considered a necessary 
 concomitant of beauty. But proportion is a quality of a very 
 different kind; it is the essence of symmetry, and symmetry is 
 the first principle of harmony to the eye. Proportion must surely, 
 therefore, be regarded as a constituent of beauty in form and 
 figures. 
 
 In treating of this property in forms, apart from fitness, Burke 
 defines it thus : — " Proportion is the measure of relative quantity 
 Since all quantity is divisible, it is evident that every distinct part 
 into which any quantity is divided must bear some relation to the 
 other parts, or to the whole. These relations give an origin to the 
 idea of proportion. They are discovered by mensuration, and they 
 are the objects of mathematical enquiry. But whether any part of 
 
vm 
 
 INTRODUCTION. 
 
 any determinate quantity be a fourth, or a fifth, or a sixth, or a moiety 
 of the whole ; or whether it be of equal length with any other part, or 
 double its length, or but one half, is a matter merely indifferent to 
 the mind." He then proceeds to state, that all proportions and 
 arrangements of quantity are alike to the understanding ; that beauty 
 has nothing to do with calculation and geometry ; and concludes, " If 
 it had, we might then point out some certain measures which we 
 could demonstrate to be beautiful, either as simply considered or as 
 related to others." 
 
 In despite of so great an authority to the contrary, an attempt 
 shall be made, in the present Essay, to prove that beauty does depend 
 upon calculation and geometry ; and that, therefore, we can point out 
 and demonstrate certain measures to be beautiful, either as simply 
 considered, or as related to others. 
 
 The beauty of forms must consist in the production of a pleasing 
 effect upon the visual perceptions ;* the physical effect of forms, when 
 
 * This fact is admitted by this eminent author in another part of the same Essay, where he 
 observes, — " Another principle of beautiful objects is, that the line of their parts is continually vary- 
 ing its direction ; but it varies it by a very insensible deviation — it never varies it so quickly as to 
 .surprise, or, by the sharpness of its angle, to cause any twitching or convulsion of the optic nerve." 
 
INTRODUCTION. ix 
 
 pleasing to that sense, is appropriately called harmony, and when dis- 
 pleasing it is called discord. Now, when we find that combined or 
 successive sounds are harmonious or discordant to the sense of hear- 
 ing, agreeably to the arithmetical proportion they bear to one another 
 in the number of pulsations they occasion in the surrounding atmo- 
 sphere in a given period, and that these pulsations have an equally 
 exact geometrical proportion relatively to the size of the vibratory 
 body producing them, it appears extraordinary that so acute a reasoner 
 as Burke should advance a doctrine so much at variance with the 
 general tenor of his excellent Essay, especially as he admits, in another 
 part, that " if we can comprehend clearly how things operate upon 
 one of our senses, there can be very little difficulty in conceiving in 
 what manner they affect the rest." 
 
 It is with much reluctance that I venture to differ so decidedly 
 on this point from the opinions of so great an author, particu- 
 larly as I cordially concur in the general principles developed in 
 his admirable Essay on the Sublime and Beautiful ; as well as in 
 the various observations by which they are introduced, and in 
 
 B 
 
x INTRODUCTION. 
 
 none more than in that with which 1 shall conclude this Intro- 
 duction : — 
 
 " By looking into physical causes, our minds are opened and 
 enlarged ; and in this pursuit, whether we take or whether we 
 lose our game, the chase is certainly of service." 
 
PROPORTION, 
 
 OR THE 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 Proportion in its simplest mode is to form what time is to music, 
 or measure to poetry;* but in its more complex mode, it is to form 
 what grammar is to language, or harmony to music. Proportion 
 may lie in the relative sizes of two or more objects — the relative dimen- 
 sions that the length bears to the breadth of an object — the relative 
 obtusity or acuteness of various angles — the relative degrees of cur- 
 
 * Poetry is composed of two things, — of the natural perception of the beautiful, and of the artis- 
 tic development of this perception. In the former sense we are all poets ; in the latter sense only 
 a few possess the divine gift, and merit the distinguished name. We are all poets ; for we are all 
 capable of seizing, among the aspects of the actual, that harmony of proportions which constitutes 
 beauty, and of finding in the field of the possible and the spiritual, that image of perfection of which 
 external grace and sublimity are simply the embodiments. The meanest event, the most insignifi- 
 cant object, if suggestive to us of brighter thoughts and deeper feelings than those that people the 
 range of our ordinary musings, become for us a poetical event — a poetical object. — MaccalVs Agents 
 of Civilization. 
 
2 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 vature in various objects, or in the parts of one object, or, it may be, 
 in the general relation that various forms bear to one another in ren- 
 dering their combinations harmonious. Proportion is, in short, that 
 geometrical quality in forms and figures by which they are rendered 
 pleasing to the sense of sight, independently of their use or any 
 other consideration. A short description of the structure of the 
 organ of this sense, and the manner in which geometrical configura- 
 tion acts upon it, will therefore be an appropriate commencement to 
 this essay. 
 
 Figure 1 represents the external ap- 
 pearance of the human eye, the visible 
 parts of which are the sclerotic coat, or 
 white of the eye, to which the muscles 
 for giving it motion are attached ; the 
 iris, or that part which denotes the colour of the eye, being a fiat 
 membrane of a circular form, having in its centre the pupil, apparently 
 a dark circular spot, but in reality a simple aperture in the opaque 
 membrane that surrounds it, its seeming depth of colour being derived 
 from the darkness of the inner chamber of the eye. When this organ 
 
 Fig. 1. 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 of sense is examined anatomically, it is found to be of a globular form, 
 as shown in figure 2, on which A is the white or scle- Fig. 2. 
 rotic coat, B the iris, and C the pupil. There is upon 
 this globe a slight projection in front of the iris. The 
 front covering of this projection is the cornea, a trans- 
 parent circular membrane embracing the iris and pupil, and through 
 which they are seen, as in figure 1. It closes in the anterior chamber 
 of the eye, which contains a transparent liquid called the aqueous 
 humour. Behind the iris is the posterior chamber of the eye, which 
 is also filled with a transparent liquid, called the vitreous humour. 
 Attached to the back of the iris, and close to the pupil, is that por- 
 tion of the eye called the crystalline lens, which consists of concen- 
 tric coats composed of fibres, and is contained within a transparent 
 capsule or bag. Upon the inner surface of the posterior chamber of 
 the eye, and covering two-thirds of it, lies the retina, a tender reticu- 
 lar membrane consisting of an expansion of the optic nerve. Exactly 
 in the centre of the retina, and consequently opposite to the pupil, 
 there is a small hole, or transparent spot , called the foramen centrale, 
 over which the membrane forming the retina does not pass. Between 
 
4 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 the retina and the sclerotic coat, or white portion of the outer surface, 
 intervenes a delicate membrane called the choroid coat, which, on the 
 side next the retina, is covered with a black pigment. Figure 3 is a 
 Fl S- a vertical section of the eye, on which 
 
 A is the white or sclerotic coat, B 
 the iris, C the pupil, D the cornea, 
 E the anterior chamber, F the pos- 
 terior chamber, G the crystalline 
 lens, H the retina, I the choroid coat, 
 and K the foramen centrale. 
 
 A 
 
 These parts perform the functions of seeing in the following man- 
 ner : — When rays of light, either natural or artificial, direct or reflex, 
 fall upon opaque objects, they are reflected in all directions from every 
 illuminated part of such objects. But they pass through every trans- 
 parent body, either in a direct or refracted line, according to the 
 nature of the substance. Therefore such rays as fall upon the white 
 of the eye, or upon the iris, are reflected back again, or partially 
 absorbed ; while those that fall within the range of the pupil are 
 transmitted through the aqueous humour, crystalline lens, and 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 5 
 
 vitreous humour, all being transparent media, until they reach the 
 back of the posterior chamber, where they act upon the retina, and 
 are absorbed by the black pigment behind it. This action upon the 
 retina is the depicting of the configuration of any object upon its sur- 
 face, in the same manner that objects are depicted within a camera 
 obscura with a concave table. The three transparent media have 
 various powers of refraction, by which the rays that enter the pupil 
 converge and are spread upon the retina. From the structure of the 
 human eye, it will be observed that it is limited in its range of percep- 
 tion — that is, that the rays which necessarily traverse each other in 
 passing through the crystalline lens, or one of the other two trans- 
 parent media, must enter the posterior chamber at a certain angle to 
 reach the retina. And as that membrane covers more than a half of 
 this chamber, the rays, as they approach its extremities, must, from 
 their obliquity, form a gradually less distinct impression. Hence, the 
 dying away of objects around the focal range of sight. The greatest 
 angle at which an object can be clearly and distinctly appreciated by 
 the human eye is 60° — that is, when the rays of light are reflected 
 from every angle in the perimeter of a rectilinear figure, or from 
 
6 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 every portion of the circumference of a curvilinear one, in such a 
 direction as to form with one another, in passing through the crys- 
 talline lens or other media, angles of that number of degrees ; which 
 figure will thus occupy about one-third of the posterior surface of the 
 inner chamber of the eye. But we know that objects are seen con- 
 siderably beyond this range, although in a less distinct manner, and 
 therefore other rays besides those must necessarily reach the retina ; 
 and we also know that the eye has the power of concentrating or 
 condensing upon the retina the rays which enter it, in order that 
 minute objects may be more perfectly depicted upon its surface. 
 This is partly performed by the contraction of the pupil. But how 
 rays reach the outer portion of the retina to form the indistinct kind 
 of vision that surrounds the more perfect kind, has not yet been 
 satisfactorily explained. This is, however, of little importance to 
 the present enquiry. The mode in which rays of light may be sup- 
 posed to enter the eye is shown in figure 4, on which the line A to B 
 is the optic axis. It will be observed that these rays traverse each 
 other at three different points upon this axis ; but as to which of 
 these points is that upon which the rays really do so, writers on .the 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 7 
 
 physiology of the eye dif- 4 - 
 fer : however this may be, 
 we know from our own 
 sensations, that the rays ^ 
 must reach the surface of 
 the retina somewhat in 
 the manner here explained, 
 being at the same time affected by the refractive powers of the vari- 
 ous transparent media through which they pass. Eyes differ also 
 in regard to the distance at which objects are most distinctly seen, 
 and this difference no doubt arises from variety in the construction 
 of the transparent media, and also from variety in the degree of con- 
 vexity of the cornea. 
 
 In the essay lately published by me* on this subject, I showed that a 
 representative of every variety of rectilinear figure occurred within 
 a diagram produced by the division of the circumference of a circle 
 into twelve parts, and by drawing from each of those parts two 
 straight lines. In investigating the structure of the human eye, I find 
 
 * Natural Principles and Analogy of the Harmony of Form : Edinburgh, 1842. 
 
 c 
 
8 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 that the same diagram points out the manner of its division into the 
 various parts hy which its extraordinary functions are performed. 
 
 Figure 5 is this diagram, on which A 
 is the centre point of the posterior sur- 
 face of the crystalline lens ; B the cen- 
 tre point of the cornea ; C the centre 
 point of the eyeball, and of the ante- 
 r u r j or surface of the crystalline lens ; E 
 D F is that portion of the surface of 
 the retina upon which vision is most perfect ; GDH that portion 
 of the surface of the retina which bounds perfect vision ; O P the 
 points at which vision almost ceases, and I D K is the extent of the 
 retina ; L to M is that portion of the eye which is flat, containing the 
 ciliary process, the iris, and the pupil. The exact relative sizes of the 
 different parts of this organ, are variously stated by writers on its 
 physiology, and the coincidences upon this diagram correspond to 
 those generally given. 
 
 Although I have hitherto referred to the effects of forms upon one 
 eye only, in order to be more explicit, these effects are much modi- 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 fied by the rays entering both our eyes simultaneously ; hence the 
 mild and pleasing influence of horizontal composition, and the more 
 powerful and grand impression made by that which is vertical. These 
 are the sensible effects of figure upon the organs of vision, and it 
 is only of such that I mean to treat. My observations can there- 
 fore have no reference to any geometrical property in figures be- 
 yond what can be superficially depicted, as they are reflected upon 
 the retina ; for it is well known that we only find out by experience 
 that bodies possess other dimensions than what may be thus appre- 
 ciated. 
 
 The effects of geometrical configuration on the eye are, in the first 
 instance, regulated by the relation they bear to the conformation of 
 that organ itself ; hence the soft influence of those of the curved kind, 
 and the acute and more powerful effect of those whose outlines are 
 composed of angles. On the mode of proportioning these elements of 
 form in the combinations of various figures, their effect upon the eye 
 depends — when a proper mode is adopted, geometric beauty is the 
 result, while the adoption of an improper mode results in deformity. 
 
 The first principle and most simple element of geometric beauty is 
 
10 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 proportion, and this quality cannot exist without variety ; for there 
 can be no proportion in one simple homogeneous part, nor in the 
 repetition of such a part. The smallest number of parts by which 
 this element is attainable are two, and the greatest number three. 
 The first is the relation that the length bears to the breadth, and the 
 second is the relative quantity of the three kinds of configuration that 
 are produced by the straight, the angled, and the curved line ; the 
 various combinations of which elements lead to infinite variety. Those 
 dimensions must relate to one another agreeably to mathematical 
 laws, which are responded to by an inherent principle in the human 
 mind, correlative to them and regulating every effect of external 
 nature upon our senses. This may be called the first principle of 
 taste in regard to figure, and is possessed by mankind in every phase 
 of variety, and when it develops itself in any high degree, it consti- 
 tutes genius. 
 
 This mathematical proportion of parts is that particular element 
 of beauty called harmony, which, in a general sense, means the adap- 
 tation of the parts of any thing one to another, or the just proportion 
 of sounds, colours, forms, words, or sentiments. 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 1 1 
 
 Poetry, as already observed, owes its beauty, in the first instance, 
 to a geometric principle of proportion, upon which rules are estab- 
 lished for its construction ; and that, when these rules are followed, it 
 produces a sensible effect of pleasure upon the ear, independently of 
 the meaning conveyed in the words of which the measure is com- 
 posed. But in music this geometric principle is much more generally 
 understood, and its rules philosophically determined by the science 
 of acoustics. 
 
 Agreeably to the experimental enquiries of natural philosophers into 
 this science, it has been determined that the elements of the art of 
 music are, in the first instance, three in number ; and these three 
 primary parts are technically called the tonic, the mediant, and the 
 dominant, expressive of their relative effects in musical composition. 
 These sounds or notes are in the general or diatonic scale ; the 1st, 
 3d, and 5th. And there are, therefore, other intermediate notes that 
 act as connecting links to these primaries, and they are called the 2d, 
 4th, 6th, and 7th degrees of the scale, for the 8th degree is a repetition 
 of the tonic ; and it has been established, that agreeably to the mode in 
 which these sounds are arranged, whether in succession or combination, 
 
12 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 will their effects be harmonious or discordant, and that this depends 
 upon a geometric principle. The art of music consists in a thorough 
 knowledge of the effects that these elementary parts have naturally 
 upon one another ; and the object of musical composition is, in the 
 first instance, to make them pleasing to the ear. Thus far the 
 effects of combined sounds upon the ear are merely sensual, and please 
 or displease in so far as the established rules have been attended to or 
 violated in the mode of their combination, and the degree of these 
 effects will be in the ratio of the sensibility of this organ. Musical 
 composition has no doubt a much higher aim, and produces effects of 
 a more exalted kind, but it would be beyond the object of this essay 
 to treat of them. 
 
 By the science of chromatics it has been shown that in colouring 
 also there are three primary elements — blue, red, and yellow; and that 
 the complete scale of the colourist has other four secondary or inter- 
 mediate hues — purple, orange, green, and neutral. The art of paint- 
 ing teaches the proper use of these, the harmony arising from which 
 is in its simplest elementary kind merely sensual, although, like other 
 harmonies, it can be made, in the hands of men of genius, and com- 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 13 
 
 bined with subjects of an exalted kind, to produce powerful effects 
 upon the mind through the sense by which it is appreciated. 
 
 But there has, as yet, been no systematic arrangement, or geome- 
 tric principles of proportion, applied to form, by which harmony may 
 with certainty be produced ; although it has been universally acknow- 
 ledged that there is a harmony and discord in the modes of combining 
 forms, as certainly as harmony and discord can be produced by various 
 modes of combining either sounds or colours. 
 
 In a former essay already alluded to, I gave a detail of the kinds of 
 lines by which forms are produced, and it will only be necessary here 
 to observe, that in geometry they are understood to be three — the 
 straight line, the crooked line, and the curved line. As the crooked line 
 is, however, but a combination of two or more straight lines, it might 
 be argued that there are only two kinds of lines, the straight and the 
 curved, and that they are correlative to silence and noise in sound, 
 and to light and darkness in colour, especially as a single straight line 
 can enclose no figure ; while the whole series are produced within an 
 octave of the circle — a single curved line. But such an enquiry could 
 not result in any practical benefit. 
 
14 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 A line and a point, mathematically considered, are individually 
 length without breadth, and position without magnitude ; but in the 
 arts they are understood to have a physical existence, and are conse- 
 quently such as may be drawn with a pen, or any similar instrument, 
 and are therefore, in this essay, so considered. 
 
 In attempting to establish a system of linear harmony, applicable 
 to form, of an intelligible and practical kind, geometrical figures must 
 be employed ; and in order that such a system may correspond to 
 those that regulate other harmonies, three simple elementary homo- 
 geneous parts must be shown to exist. 
 
 The distinctions of figures do not seem to rest so much upon the 
 number of lines that compose their circumference, or perimeter, as upon 
 other peculiarities in their configuration. Regular curvilinear figures 
 must have either one or two points as a centre, and have no angles ; 
 and regular rectilinear figures must be composed of acute angles, right 
 angles, obtuse angles, or an equal mixture of the acute and obtuse 
 kinds. Amongst these geometrical figures there are three which are 
 perfectly simple and homogeneous in the nature of their configura- 
 tion, and are, in every other respect, quite analogous to the tonic, 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 15 
 
 mediant, and dominant notes of the dia- 
 
 Fig. 6. 
 
 tonic scale of the musician, and to the three 
 primaries, blue, red, and yellow, of the 
 colourist. These figures are the circle, 
 the triangle, and the square, in the rela- 
 tive proportions in which they are given in 
 figures 6, 7, 8. 
 
 I have endeavoured to show in my former essay, that these 
 figures, in such proportions, bear an analogy to the three primary 
 parts of sound in the quantity of their circumference and perimeter. 
 I have also demonstrated, that if two circles be produced from one 
 centre point, having in their circumference the relative proportions 
 
 D 
 
16 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 of 1 to 2, as in figure 6, those two rectilinear figures, in the pro- 
 portions in which I have given them, can be placed harmonically 
 between two such circles, as shown by the dotted lines in figures 
 7 and 8. 
 
 The homogeneous simplicity of these figures consists, first, in the 
 circle being the most perfect curve, and composed of one line drawn 
 round one point, from which every portion is equidistant ; secondly, 
 in the equilateral triangle being composed of three sides, the smallest 
 number possessed by any rectilinear figure, which sides are equal, 
 and each of which, as well as each of its angles, are equidistant 
 from one point ; and thirdly, in the square being composed of four 
 equal sides and four right angles, each side and each angle being 
 also equidistant from one point, and the right angle itself being 
 homogeneous. 
 
 Without referring to the analogy of sound, it might be shown that 
 from their configuration, compared to the conformation of the eye, the 
 effects of those particular forms upon that organ entitle them to hold 
 the situation amongst other forms in which I have placed them. The 
 
GEOMETRIC PRINCIPLE OE BEAUTY ANALYSED. 
 
 17 
 
 pupil of the eye is circular ; hence the rays, or pencils of light, which 
 pass from external objects to the back of the inner chamber, or retina, 
 are most easily transmitted when the object is circular, as already 
 explained. The circle is, therefore, not only geometrically the most 
 simple of the homogeneous forms, but naturally so in reference to 
 the organ by which it is perceived. The square is the next most 
 consonant form to the eye, as its angles, although more in num- 
 ber, are less acute than those of the triangle, and are the exact 
 mean between acuteness and obtusity. The triangle, of the three, 
 is the figure which, from its being composed of acute angles and 
 oblique lines, exercises the most powerful influence on that delicate 
 organ. 
 
 In this respect it corresponds to the note E in the diatonic scale 
 in music ; for compositions having that note for their key exercise the 
 same relative influence on the ear. Indeed, round and acute are 
 terms as often employed to express qualities of sound, as they are to 
 express the particular configuration of objects presented to the visual 
 organs. 
 
18 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 It is well known in chromatics, that the primary colour, blue, exer- 
 cises a softer influence on the eye than either of the other two, red 
 and yellow ; and this no doubt occurs from its being the most allied 
 to darkness or black of the three, and hence associating more inti- 
 mately with the colour of the retina itself. The colour that stands 
 next to it as a primary in the solar spectrum, is red, which conse- 
 quently holds the situation that the triangle does in my series of 
 forms ; and this colour is well known to affect the eye more forcibly 
 than the yellow, which, in the natural series, is furthest removed from 
 the blue ; so that the more acute effect of the triangle upon the eye, 
 although holding a medial situation, is quite in accordance with the 
 analogy of acoustics and chromatics. 
 
 The scales by which harmony in sound and colour is produced, as 
 already noticed, have, besides the three primary parts, other four of a 
 secondary kind by which these are connected. So, to complete the 
 scale of forms, it was necessary to adopt figures corresponding to 
 these, as the 2d, 4th, 6th, and 7th notes in music do to the 1st, 3d, 
 and 5th ; and as the secondary colours, orange, green, purple, and 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 19 
 
 neutral, do to the primaries, blue, red, and yellow. The figures 
 I now adopt for this purpose are the rectangle, figure 9, the rhom- 
 bus, figure 10, the hexagon, figure 11, and the dodecagon, figure 12. 
 Fig. 9. Fig. 10. 
 
 These figures, it will be shown, occur naturally from the intersec- 
 tions of two lines produced between two circles of the relative pro- 
 portions already stated, and form, with the three primaries, a 
 
20 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 series embracing a representative of every variety of geometric 
 figures, whether right-angled, acutely angled, obtusely angled, or 
 curved. 
 
 Forms composed of straight lines are, in geometrical language, 
 called rectilinear figures, and are all necessarily angular, and are named 
 according to the kind and number of angles they contain. These 
 angles are of three kinds, viz. the right angle, the acute angle, and 
 the obtuse angle. The first of these is composed of the two positive 
 directions of the straight line — the horizontal and vertical ; and this 
 is the only homogeneous angle, as it admits of no modification in any 
 degree, for whatever direction one of its sides may take, the other 
 must be perpendicular to it. All forms having this angle are called 
 rectangular figures ; even a triangle having one of its three angles 
 of this kind, is called a right-angled triangle. The other two — the 
 acute and the obtuse angles — admit of any change in the relative 
 direction of the lines which produce them, without changing their 
 denomination or character. These angles are geometrically regu- 
 lated by the circle in this way : — Its circumference is divided into 360 
 equal parts, called degrees, figure 13, which degrees are again divided 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 2 J 
 
 into 60th parts, called minutes ; these, again, into 60 seconds, and 
 these seconds into 60 thirds ; and this subdivision may be carried on 
 to any imaginable extent. The circle is divided into two equal parts 
 by a line drawn through its centre, and cutting the circumference at 
 
 each end, figure 14. This line is in geometry called a diameter, and 
 Fig. 13. Fig. 14. 
 
 when horizontally placed, is the base or groundwork from which all 
 angles arise. The half of the circle is called a semicircle, and the half 
 of the semicircle a quadrant. Any line drawn from the centre of this 
 diameter to the circumference of the semicircle is a radius, and will 
 divide it into two portions called arcs. If it cut the circumference in 
 the centre, these arcs will be equal, and the angles formed with the 
 diameter on each side of the radius will be right angles ; and as 
 
22 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 these arcs contain 90 degrees, right angles are called angles of 90°, 
 Fig. 15. figure 15. But if the radius touch the 
 
 circumference at any other point than 
 the centre, two different angles are form- 
 ed; one an acute angle, and the other 
 an obtuse angle, because the one arc 
 contains fewer, and the other more than 
 
 Fig. 16. 
 
 90°, as shown in figures 16 — 17. Upon 
 
 Fig. 17. 
 
 this division of the circumference of the circle depends the cha- 
 racter and direction of every straight line, as well as every rec- 
 tilinear figure. This simple account of the mode in which 
 angles are reckoned, will make the following description of geo- 
 metric figures more easily understood to those who have not studied 
 geometry. 
 
GEOMETRIC PRINCIPLE OF BEAUTY 
 
 ANALYSED. 
 
 23 
 
 OF THE CIRCLE AND OTHER CURVILINEAR FIGURES. 
 
 The circle itself, as already shown, is in the parts that compose it 
 the most simple and homogeneous of all forms. Its secondary is the 
 ellipse, also a perfect curve, because it is a line in all its parts equi- 
 distant from one or other of two points, and necessarily uniting its 
 beginning and end at the same point. It is heterogeneous, for these 
 points may be placed near or apart ; but the figure described around 
 them will still be an ellipse, however much it may resemble a circle 
 from the closeness of its two centre points, or, on the other hand, a 
 straight line from their separation. These points are called its foci ; 
 and if they remain in the same position while the line which forms 
 the circumference is increased, the ellipses thus formed will continue 
 in appearance to approach the proportions of the circle, but, 
 although produced to infinity, can never form that particular figure. 
 It has consequently two diameters, the longest of which is called the 
 transverse diameter, and the shortest the conjugate diameter, as 
 
24 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 Fig. 18. shown in figure 18, on which A B are 
 
 "\ % the foci, C D the transverse diameter, 
 
 E 
 
 and E F the conjugate diameter. It will 
 be shown in another part, that although 
 this figure can be produced in every vari- 
 ety of proportion, from the straight line 
 to the circle ; yet, as it possesses the most 
 simple' kind of variety of parts which constitute the first elements of 
 proportion, there are certain rules for its formation which agree mathe- 
 matically with the principles of geometric beauty, and which entitle 
 the one here given to be termed the ellipse, in contradistinction to 
 other varieties of the same figure. In this proportion it is also 
 entitled to hold the place of tonic in the secondary series of geome- 
 trical figures, and to produce within its octave the rectilinear figures 
 of that class. The only other curve that can produce a form or figure 
 useful in the arts of design is the spiral : this curve, as well as that 
 producing the circle and ellipse, is a real unmixed curve ; and although 
 it can form no figure of itself, is of much importance in the arts of 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 25 
 
 ornamental design in producing the volute. It is a curved line, receding 
 gradually from a focus or centre, figure 19. 
 Its centre may be a point, a circular figure, 
 or an ellipse, and these may be large or 
 small, or its aberration may be in any de- 
 gree ; its direction still forms a spiral line, 
 and the figure it produces when closed by 
 another line, a volute. There are various 
 other curved lines, which, although inapplicable in the arts of design, 
 have been reckoned of much importance in science. They are the 
 cycloid, the parabola, the hyperbola, and others. But all these I look 
 upon as compounds. For instance, the cycloid is the mixture of a 
 straight line and a circle ; for during the formation of the circle, by 
 the revolution of a radius around its centre point circularly, that 
 point is traversing a straight line. If the revolution were stopped, 
 and the progress of the radius continued, the ends of the radius 
 would produce two parallel lines. On the other hand, if the pro- 
 gress were stopped, and the revolution allowed to proceed, one end 
 
26 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 of the radius would describe a circle. But this curve cannot of itself 
 enclose a space or produce a figure. 
 
 The parabola, in the same way, is the mixture of an elliptic 
 curve and an angle ; or, it may be termed, an ellipse formed upon 
 a definite and an indefinite point, both acting at the same time upon 
 the formation of the curve. 
 
 The hyperbola appears to be a mixture of an angle and a circular 
 curve, which, like the ellipse in its production round an elongated 
 centre, will continue to infinity to approach the straight line with 
 which it is associated, as the ellipse has been shown to recede from it. 
 
 Every curvilinear figure has a rectilinear figure as its basis, as will 
 afterwards be shown. 
 
 OF THE SQUARE, AND OTHER RECTANGULAR FIGURES. 
 
 The square is homogeneous in its parts, none of which can be 
 altered without destroying its form. The parts are, as already 
 stated, four straight lines of equal length equidistant from a centre, 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 27 
 
 and uniting at their extremities in four right angles, which are like- 
 wise equidistant from the same centre, and being of 90" each, make 
 up the full number 360 contained in the circumference of the circle. 
 When a quadrilateral rectangle has two of its opposite sides longer 
 than the other two, it is called an oblong or right-angled parallelo- 
 gram ; and every rectangle of this kind, from a perfect square to 
 a straight line, is so, whatever may be the proportion between its 
 length and breadth. In the proportion of this figure, therefore, 
 there is the same latitude that exists in regard to the ellipse ; and as 
 geometricians have given no rules by which to distinguish the paral- 
 lelogram or oblong from the innumerable series that lie between the 
 square and the straight line, it is of importance to fix some rule for 
 the formation of one whose proportions may be mathematically regu- 
 lated agreeably to the principles of taste. This I shall attempt 
 in its proper place, and here proceed with a definition of the other 
 primary, and the figures that are allied to it. 
 
28 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 m 
 
 OF THE TRIANGLE. 
 
 % 
 
 The equilateral triangle is the proper triangle ; it is, like the other 
 primary figures, homogeneous in its parts, being formed of three 
 straight lines of equal lengths, equidistant from one point or centre, 
 and by their union producing three acute angles, also equidistant 
 from the same point. Like the square, it cannot be altered in any 
 of its parts, without destroying its form and altering its character. 
 Each of its angles are 60°, which together make 180°, being half the 
 number contained in the circumference of the circle. 
 
 There are various other triangles, some of which have one right 
 angle and two acute angles ; others, one obtuse angle and two 
 acute angles. But they cannot have less than two acute angles, or 
 more than one right or obtuse angle ; and, whatever their varieties 
 in other respects may be, their three angles make up 180°. If two 
 straight lines of the same length meet at an angle of 60°, whatever 
 their length may be, a third straight line joining the other two ends 
 will produce an equilateral triangle. 
 
 The equilateral triangle has for its secondary the rhombus, which 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 29 
 
 may be termed a perfect mixture of the triangle and square. The 
 ellipse is the secondary to the circle, by having two foci, while the 
 circle has only one. The secondary to the square is removed from 
 that figure by having two of its sides shortened, while the number of 
 its angles and the direction of its sides are the same. The rhombus 
 is removed from the equilateral triangle by being two figures of the 
 same kind placed together, which two triangles produce a quadrila- 
 teral figure. Two of its opposite angles are 60°, and the other two 
 are 120°. It has, therefore, two acute and two obtuse angles, which, 
 put together, are equal to four right angles. It is the only figure 
 that can occur within the circle, having an equal number of acute 
 and obtuse angles. It may be shortened until its angles be nearly 
 right angles, or it may be lengthened until it approaches the straight 
 line so closely that its figure cannot be distinguished, and it there- 
 fore possesses the peculiarities of the other second figures. 
 
30 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 OF POLYGONS. 
 
 Although this term applies to all figures having more sides than 
 one, yet those whose sides exceed four, are generally denominated 
 polygons ; and all regular rectilinear polygons of this kind are con- 
 sequently obtusely angled. The hexagon has been adopted in the 
 series as the representative of this class, from its being the first regu- 
 lar figure of the kind that occurs from the intersections of the domi- 
 nant lines, as shall presently be shown. It is the figure that approaches 
 nearest to the configuration of the circle of any rectilinear plane figure 
 that can be joined together in any number by its sides. As poly- 
 gons increase in the number of their sides, they can scarcely be dis- 
 tinguished from the circle. If two circles be described round one 
 centre point, of the relative proportion in circumference of 1 to 2, and 
 if the outer one be divided into 12 parts, and four lines drawn from 
 each division, these lines will produce between the circle a series of 
 rectilinear figures, three of which will touch the outer circle by their 
 angles, other three will touch the inner circle by their sides, and 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 31 
 
 the centre figure will do both, as appears in figures 20, 21, 22, 
 and 23. 
 
 Forms must have other distinguishing qualities besides their confi- 
 guration. We know, that although the diatonic scale of the musi- 
 cian has only seven distinctive characters of notes, this scale may be 
 
 F 
 
32 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 repeated at least nine times in musical composition between the 
 gravest and the most acute tones appreciable by the human ear : 
 and we also know, that in colours there are many gradations 
 from the deepest and most mellow to the most intensely brilliant 
 of each individual hue. The gradations in both cases are inde- 
 pendent of quantity ; for a note in music may be long or short, or 
 even strong or weak, without affecting its gravity or acuteness, as 
 a colour may cover a large or a small surface without affecting its 
 peculiar degree of intensity. So it is with form ; the three pri- 
 mary figures, with their attendant secondaries, can be produced 
 within the range of ocular perception in an endless variety of com- 
 bination, and in various degrees of modification in regard to their 
 proportions. To this modification the laws of harmonic ratios shall 
 now be applied. 
 
 The simple manner in which the rectilinear figures are produced 
 between two circles, has been described above, and that the primary 
 rectangle is formed upon the circumference of the inner circle. It 
 therefore follows, that the curvilinear figure to be adopted as a 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 33 
 
 secondary to the circle, should be an ellipse proportioned to the 
 secondaries of the other two primary figures ; that is, it ought to be 
 capable of inscribing the rhombus in such a manner, that the vertices 
 of the angles of that figure should cut Fig. 24. 
 
 its circumference into four equal parts, 
 and of such proportions as may be re- 
 gularly inscribed by the perimeter of the 
 oblong rectangle, having likewise the 
 relative configuration to the circle shown 
 in figure 24. The other peculiarities 
 of this figure, as well as those of the other secondaries, will be 
 noticed hereafter. 
 
 Fig. 25. 
 
 When the circumference of such an ellipse is 
 divided into twelve parts, and lines produced 
 within it as shown in figure 25, it will be found 
 that a secondary series of geometrical figures is 
 formed by their intersections. 
 
 Amongst these, upon the transversed diameter 
 
34 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 Fig. 26. of the ellipse will be produced an oblong rectangle, 
 which gives the proportions to a third class of figures, 
 as shown in figure 26. The rectangles inscribing those 
 three curvilinear figures, 24, 25, and 26, will be shown to 
 have an harmonious proportion to one another strictly in 
 conformity with the principles of harmonic ratios in form, 
 the development of which is attempted in this essay. 
 
 OF THE GEOMETRY OF HARMONY. 
 
 We must now turn to the geometry of harmony, in order to find, 
 and more easily to comprehend and systematize, the harmony of geo- 
 metry. This geometry of harmony, as I have before observed, has 
 formed an interesting branch of natural philosophy, and its laws have 
 been fully demonstrated and established. They are as follow : — The 
 perfect absence of sound is silence, from which point sound commences 
 by the most sonorous or the gravest kind, and ends at the other 
 extreme — the most acute. The one is called the lowest, and the other 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 :55 
 
 the highest sound ; but as each loses its distinction in silence, these 
 extremes may be said to meet. Natural philosophers have proved 
 that sound is produced by the air being put into a state of undulation, 
 either by the vibration of an elastic body, or of a column of the air it- 
 self ; and as this undulation does not produce a tone appreciable by 
 the human ear until the vibrations that produce it reach the rapidity 
 of 32 in a second, and that the extreme of appreciable acuteness is 
 almost attained when those vibrations have reached the rapidity of 
 16,384 in the same period, the range of human perception in regard 
 to sound may be said to be divided into 16,384 parts. This branch 
 of natural philosophy is called acoustics, and by it harmony of sound 
 has been ascertained to depend upon a certain arithmetical division 
 of those vibrations, and that this division bears an inverse ratio to the 
 geometrical division of the body by which the vibrations are produced. 
 For instance, suppose the string of a harp, or any other musical 
 string, to be put into a state of vibration, and if it were of the proper 
 length, thickness, and tension to give 512 vibrations in a second, the 
 sound thus produced would be middle C of the musical scale, or that 
 
36 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 note which stands intermediately in the range of human perception 
 between the gravest and most acute sounds. Figure 27 represents 
 this string and the note which it produces — the dotted line showing 
 the motion of vibration. 
 
 Fig. 27. 
 
 — o — 
 
 If this string were divided into two parts, and one of those parts 
 rendered quiescent, the rate of vibration would be doubly rapid, and it 
 would consequently give 1024 vibrations in a second ; 
 
 Fig. 28. 
 
 thus producing the next C above it in the scale, which is called an 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 37 
 
 octave to the tonic ; because, between it and the original note, the 
 other six, D, E, F, G, A, B, can be produced, being in all eight. This 
 sound is the same note, though doubly acute ; and therefore the most 
 consonant sound that can accompany that produced by the whole 
 string or monochord. And it is the case with all sounds within the 
 range of human perception, that, when the number of vibrations in a 
 second are as 2 to 1, the octave or scale is complete. This is the first 
 and most simple division of the string. The second division of the 
 string is by three ; and when one of those third parts is rendered qui- 
 escent, the relative number of vibrations to those of the original 
 string, or monochord, will be found to be as 3 to 2, and the note thus 
 produced is G, the dominant or fifth degree of the scale, and next 
 most consonant to the tonic. 
 
 Fig. 29. 
 
 i „... • i 1 
 
 
 
 
 i 
 
 
 
 
 
 
 The third division of the string is by 5, and on one of those fifth 
 
88 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 parts being rendered quiescent, the vibrations will then be found to be 
 in point of rapidity relatively to the tonic as 5 to 4, producing the 
 
 next consonant note E, the third degree or mediant. 
 
 Fig. 30. 
 
 If, when the string is divided by three, one-third part instead of 
 two was made to vibrate, the note produced would be the harmonic, 
 called a twelfth, and its vibrations would be to those of the original 
 note as 3 to 1. And if, when the string is divided by five, one in- 
 stead of four of those divisions was made to vibrate, the note produced 
 would be the harmonic, called a seventeenth, giving vibrations whose 
 rapidity would be relatively to the original note as 5 to 1. These are 
 leading harmonies in the musical scale.* 
 
 * A more comprehensive mode of producing the harmonies is as follows : — Taking the same 
 string — one-half gives an octave — one-third part a 12th — one-fourth part a double octave, or 15th — 
 one-fifth part a 17th — one-sixth part a 19th — one-seventh part a 21st, flattened a semitone — and 
 one-eighth part a 22d or treble octave. 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 39 
 
 It appears, therefore, that harmony in music has a geometrical pro- 
 portion, and that the division of the vibratory body, and consequently 
 the vibrations it produces, is the most simple, and at the same time 
 the most comprehensive, that could be adopted, viz. by 2, by 3, and 
 by 5 ; because any other mode of division must be made either by 
 multiples or mixtures of these. Those divisions thus produce the 
 leading tones in all musical composition, and the intermediate notes 
 by which they are connected in the musical scale, are produced by 
 other relative divisions, as shall presently be shown. Now, suppose a 
 string capable of producing, when extended, 32 vibrations in a second 
 of time, which gives the lowest appreciable sound, and that this string 
 or monochord is divided into 16,384 parts, its first division by two 
 will produce an octave above it, giving 64 vibrations in a second, and 
 having 8192 parts ; and proceeding to subdivide into octaves the 
 length of string and numbers of vibrations, would stand relatively to 
 one another thus, which is the first and most comprehensive geometry 
 of harmony : — 
 
 String. 16-384 8192 4096 2048 1024 512 256 128 64 32 
 Vibrations. 32 64 128 256 512 1024 2048 4096 8192 16384 
 
 G 
 
40 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 The more minute kind may be thus detailed within an octave : — 
 
 Names of notes in 
 the diatonic scale 
 of music. 
 
 Number of vibra- 
 tions in a second 
 of time. 
 
 Relative pro- 
 portion to 
 key-note. 
 
 Number of 
 parts of 
 string. 
 
 Relative pro* 
 portion to 
 monochord, 
 
 C, do, or 1st degree 
 
 512 
 
 1 
 
 1024 
 
 2 
 
 D, re, — 2d — 
 
 576 
 
 9 to 8 
 
 910| 
 
 8 to 9 
 
 E, mi, — 3d — 
 
 640 
 
 5 - 4 
 
 818} 
 
 4-5 
 
 F, fa, — 4th — 
 
 682f 
 
 4-3 
 
 768 
 
 3 - 4 
 
 G, sol, — 5th — 
 
 768 
 
 3 - 2 
 
 68<2§ 
 
 2 - 3 
 
 A, la, — 6th — 
 
 853^ 
 
 5-3 
 
 614§ 
 
 3 - 5 
 
 B, si, — 7th — 
 
 960 
 
 15-8 
 
 546 T ^ 
 
 8 - 15 
 
 C, do, — 8th — 
 
 1024 
 
 2 
 
 512 
 
 1 
 
 The relative proportions that these notes bear to one another in 
 number of vibrations and parts of monochord are as follows : — 
 
 
 
 Relative proportion 
 
 
 
 
 Relative proportii 
 
 
 
 in number of 
 
 
 
 
 in parts 
 
 
 
 vibrations. 
 
 
 
 
 of monochord. 
 
 c 
 
 to D 
 
 8 to 9 
 
 called a tone 
 
 C to 
 
 D 
 
 9 to 8 
 
 D 
 
 — E 
 
 9 — 10 
 
 tone 
 
 D — 
 
 E 
 
 10—9 
 
 E 
 
 — F 
 
 15 — 16 
 
 semitone 
 
 E — 
 
 F 
 
 16 — 15 
 
 F 
 
 — G 
 
 8—9 
 
 tone 
 
 F — 
 
 G 
 
 9—8 
 
 G 
 
 — A 
 
 9 — 10 
 
 tone 
 
 G — 
 
 A 
 
 10—9 
 
 A 
 
 — B 
 
 8—9 
 
 tone 
 
 A — 
 
 B 
 
 9—8 
 
 B 
 
 — C 
 
 15 — 16 
 
 semitone 
 
 B — 
 
 C 
 
 16 — 15 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 41 
 
 The following is a Table, showing the number of vibrations in a 
 second of time in each note of the diatonic scale of music, within the 
 range of human appreciation : — 
 
 DO 
 
 32 
 
 RE 
 
 36 
 
 MI 
 
 40 
 
 FA 
 
 42f 
 
 SOL 
 
 48 
 
 LA 
 
 5H 
 
 SI 
 
 60 
 
 DO 
 
 64 
 
 64 
 
 72 
 
 80 
 
 85^ 
 
 96 
 
 106f 
 
 120 
 
 128 
 
 128 
 
 144 
 
 160 
 
 I70f 
 
 192 
 
 2131 
 
 240 
 
 256 
 
 256 
 
 288 
 
 320 
 
 341^ 
 
 384 
 
 426f 
 
 480 
 
 512 
 
 512 
 
 576 
 
 640 
 
 682| 
 
 768 
 
 8531 
 
 960 
 
 1024 
 
 1024 
 
 1152 
 
 1280 
 
 13651 
 
 1536 
 
 I706f 
 
 1920 
 
 2048 
 
 2048 
 
 2304 
 
 2560 
 
 2730f 
 
 3072 
 
 3413^- 
 
 3840 
 
 4096 
 
 4096 
 
 4608 
 
 5120 
 
 5461A 
 
 6144 
 
 6826f 
 
 7680 
 
 8192 
 
 8192 
 
 9216 
 
 10240 
 
 10922§ 
 
 12288 
 
 13653£ 
 
 15360 
 
 16384 
 
 From the foregoing Table, it will be seen that this scale or series 
 of notes is repeated nine times within the range of human percep- 
 tion. It will, however, be sufficient for our present purpose to take 
 two of those octaves, and arrange them upon the bass, tenor, and 
 treble clefs, which are written thus : — 
 
42 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 Fier. 31 
 
 
 
 Relation to 
 
 Relation to 
 
 Relation to 
 
 
 
 Degrees of Scale. 
 
 Vibration. 
 
 Tonic. 
 
 one another. 
 
 Monocliord. 
 
 String. 
 
 Degrees of Scale. 
 
 15th cc 
 
 1024 
 
 4 to 1 . 
 
 
 
 ( 1 to 4 
 
 512 
 
 cc 15th 
 
 
 
 
 • 15 to 10 
 
 
 
 
 14th b 
 
 960 
 
 30 ... 8 ■ 
 
 8. 
 
 
 j 8. ..30 
 
 546& 
 
 b 14th 
 
 13th a 
 
 8531 
 
 10... 3 j 
 
 
 . 9 
 
 | 3... 10 
 
 614f 
 
 a 13th 
 
 
 
 
 9 .. 
 
 . 10 
 
 
 
 
 12th g 
 
 768 
 
 6... 2 j 
 
 8.. 
 
 
 2... 6 
 
 682f 
 
 g 12th 
 
 11th f 
 
 682f 
 
 8... 3 j 
 
 
 . 9 
 
 [3... 8 
 
 768 
 
 filth 
 
 
 
 
 15 .. 
 
 . 16 
 
 
 
 
 10th g 
 
 640 
 
 10... 4 j 
 
 
 
 4... 10 
 
 819^ 
 
 e 10th 
 
 
 
 
 9.. 
 
 . 10 
 
 
 
 9th d 
 
 576 
 
 18... 8 j 
 
 9 .. 
 
 . 8 
 
 \ 8... 18 
 
 910f 
 
 d9th 
 
 8th C 
 
 512 
 
 2... 1 j 
 
 
 
 [ 1 ... 2 
 
 1024 
 
 C 8th 
 
 
 
 
 15.. 
 
 . 10 
 
 
 
 
 7th B 
 
 480 
 
 15... 8 | 
 
 
 
 ! 8... 15 
 
 1092 T 4 3 
 
 B 7th 
 
 
 
 
 8.. 
 
 . 9 . 
 
 
 
 6th A 
 
 42Gf 
 
 5... 3 | 
 
 
 
 3... 5 
 
 1228| 
 
 A 6th 
 
 
 
 9.. 
 
 .10 . 
 
 
 
 
 5th G 
 
 384 
 
 3... 2 | 
 
 8 .. 
 
 
 ; 2... 3 
 
 1365£ 
 
 G 5th 
 
 4th F 
 
 3411 
 
 4... 3 j 
 
 
 . 9 . 
 
 ! 3... 4 
 
 1536 
 
 F 4th 
 
 
 
 
 15.. 
 
 16 . 
 
 
 
 
 3d E 
 
 320 
 
 5... 4 j 
 
 
 
 4... 5 
 
 1038| 
 
 E3d 
 
 
 
 
 9... 
 
 10 j 
 
 
 
 2d D 
 
 288 
 
 9... 8 j 
 
 
 
 8... 9 
 
 1820$ 
 
 D 2d 
 
 
 
 
 8... 
 
 9 i 
 
 
 
 1st C 
 
 256 
 
 1 * 
 
 
 
 1 
 
 2048 
 
 C 1st 
 
GEOMETRIC PRINCIPLE OE BEAUTY ANALYSED. 
 
 43 
 
 The arithmetical relations that these notes bear to the tonic, and 
 to one another, in ascending and descending ; as also, the correspond- 
 ing geometrical relations in the portions of the monochord by which 
 they are produced, are exhibited in the above Table. 
 
 Thus the only kind of harmony, with the philosophy of which we 
 are thoroughly acquainted, and which constitutes the beautiful in 
 sound, owes its excellence to an adherence to certain geometrical 
 rules, which act mechanically upon the ear ; the atmosphere being the 
 medium of communication between that organ and the body which, 
 by its vibrations, produces the sound. This body may be composed 
 of any substance, from a column of the air itself to the hardest stone. 
 
 Now it is, as before observed, universally admitted that the eye is 
 as susceptible of what is proportioned or deformed, or, in other words, 
 beautiful or ugly in form, as the ear is to what is harmonious or dis- 
 cordant in sound. Indeed, to such an extent, that the terms which 
 express those sensations are synonymous in both cases. It has already 
 been shown that an analogy exists in the primary and secondary parts 
 employed to produce an effect in either case, and it shall now be 
 attempted to be proved that those parts are equally productive of 
 
44 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 infinite variety, and equally under the control of certain geometrical 
 principles, in the one case acting mechanically on the eye, as they have 
 been shown in the other to act upon the ear. 
 
 OF THE HARMONY OF GEOMETRY. 
 
 The circle, as already shown, is geometrically divided into 360 
 degrees, &c. ; and I shall endeavour to prove, that, in the division of 
 those degrees by the harmonic ratios, the principle of geometric beauty 
 or proportion lies. 
 
 In the first division by two, which determines the octaves in sound, 
 the diameter of the circle or horizontal line — the base of all geome- 
 trical figures — is produced. The second octave gives a radius per- 
 Fig- 32- pendicular to it, producing the right 
 
 angle of 90°; and the third, the angle 
 of 45°, which is the diagonal of the 
 primary square. We have thus the first 
 elements of figure ; and this division 
 by two gives the first harmonic ratio. 
 Figure 32. 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 45 
 
 The next harmonic division is by Fig. 33. 
 
 three ; and when produced upon each of 
 those parts it falls upon the numbers 
 240°, 120°, and 60°, (Figure 33;) and 
 the lines produced between the first and 
 the second, and between the second and 
 the third of these dominant divisions, 
 are those which, being repeated from each twelfth division of the 
 circle, will produce, by their intersections within it, the diagram 
 which contains the whole series of rectilinear figures, and which has 
 been already so often referred to. The remainder of those divisions 
 will be better understood by what follows regarding rectangles. 
 
 Rectangles only differ from one another in their proportion — that 
 is, the ratio that their length bears to their breadth. This pro- 
 portion is determined by one measurement, which is the diagonal. 
 The difference between the various kinds of angles has been already 
 explained. It has been shown that the angle of 90° is the right 
 angle ; and that all angles having more degrees are obtuse, and 
 that those having less are acute. The diagonal, by which the pro- 
 
46 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 portion of a rectangle is determined, is a line drawn from the vertex 
 of one of its angles to that which is opposite, and every such diagonal 
 must form two acute angles — the square being the only figure of this 
 kind in which these angles are the same, 45°. This angle is there- 
 fore to the angle of 90°, as 1 to 2, and they consequently are in the 
 harmonic ratio of an octave. As no diagonal of a rectangle can be 
 an obtuse angle, all variety of this peculiar line must be found within 
 the quadrant. 
 
 The oblong is simply a modification of the square, and this modi- 
 fication is regulated by the number of degrees in the angle of the 
 diagonal, which, when the oblong is placed vertically, must exceed 
 45° ; and, when horizontally placed, must be under that number. 
 If, therefore, a series of these diagonals be produced by an har- 
 monic division of the degrees that occur upon a quadrant — that is, 
 by 2, by 3, and by 5 — the rectangles formed upon them must bear 
 an harmonious relation to one another. 
 
 The three dominant rectangles already described, were in my 
 former essay proportioned to one another by a different process, 
 but they also agree with this more correct mode — the first being 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 47 
 
 produced by the division of the quadrant by 2, which gives 45°, the 
 diagonal of the primary square, and relates to the right angle as 1 to 
 2. The second division by 3, gives the vertical diagonal of the first 
 oblong 60°, and relates to the right angle as 2 to 3 ; and the third 
 division by 5, gives the vertical diagonal of the second oblong 72°, 
 which is in the relative proportion to the right angle of 4 to 5. 
 
 The square being homogeneous, has only one proper position, in 
 which its diagonal is always 45° ; the other two rectangles being 
 heterogeneous, have two, a vertical and an horizontal. Their vertical 
 diagonals, as shown above, are respectively 60° and 72°. Their hori- 
 zontal diagonals are therefore 30° and 18°. The first of these latter 
 numbers, 30°, is relatively to its vertical number, 60°, as 1 to 2, and 
 is consequently in the ratio of an octave ; and being relatively to 45° 
 
 H 
 
48 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 as 2 to 3, it is in the harmonic ratio of a fifth or dominant to that 
 diagonal. Its relation to the tonic, 90°, is that of a musical twelfth, 
 being as 1 to 3. 
 
 The second number, 18°, is relatively to its own vertical number, 
 72°, as a fifteenth or double octave, being as 1 to 4. It is to the tonic, 
 90°, as a seventeenth, being relatively to it as 1 to 5 ; and is the third 
 degree to the diagonal 22° 30', to which it relates as 4 to 5. These 
 and all other harmonic ratios will be found in the following General 
 Scale of geometric degrees, which is divided into nine series or 
 octaves, agreeably to the acoustical division of vibrations : — 
 
 First. 
 
 Second. 
 
 Third. 
 
 Fourth. 
 
 Fifth. 
 
 Sixth. 
 
 Seventh. 
 
 Eighth. 
 
 360° 
 
 320° 
 
 288° 
 
 270° 
 
 240° 
 
 216° 
 
 192° 
 
 180° 
 
 180 
 
 160 
 
 144 
 
 135 
 
 120 
 
 108 
 
 96 
 
 90 
 
 90 
 
 80 
 
 72 
 
 67 30' 
 
 60 
 
 54 
 
 48 
 
 45 
 
 45 
 
 40 
 
 36 
 
 33 45 
 
 30 
 
 27 
 
 24 
 
 22 30' 
 
 22 30' 
 
 20 
 
 18 
 
 10 52 30" 
 
 15 
 
 13 30' 
 
 12 
 
 11 15 
 
 11 15 
 
 10 
 
 9 
 
 8 2615 
 
 7 30' 
 
 6 45 
 
 6 
 
 5 37 30' 
 
 5 37 30" 
 
 5 
 
 4 30' 
 
 4 13 7 30"' 
 
 3 45 
 
 3 22 30" 
 
 3 
 
 2 48 45 
 
 2 48 45 
 
 2 30 
 
 ' 2 15 
 
 2 6 33 45 
 
 1 52 
 
 30" 1 41 15 
 
 1 30' 
 
 1 24 22 
 
 1 24 22 30"' 1 15 1 7 30" 1 316 52 30"" 5615 50 37 30"' 45 4211 15 
 
 It has been shown (pages 36, 37, and 38) that the three leading 
 harmonies, agreeably to the established laws of acoustics, are produced 
 by portions of the monochord relating proportionally to one another, 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 49 
 
 in the first instance, or within an octave, as 1 to 2, 2 to 3, and 4 to 5, 
 and are called the 8th, 5th, and 3d degrees of the diatonic scale. But 
 that, when the portions of the monochord are as 1 to 2, 1 to 3, and 
 1 to 5, the harmonies of an 8th, a 12th, and a 17th, are produced. 
 This is, therefore, precisely the case in regard to the formation of 
 these three rectangles. The angles of their diagonals, in the first in- 
 stance, relate to the right angle as 1 to 2, 2 to 3, and 4 to 5. But 
 when the horizontal, instead of the vertical diagonal, is employed in 
 the construction of the two latter figures, the three will be found to 
 relate to the right angle as 1 to 2, 1 to 3, and 1 to 5. The harmonic 
 ratios are therefore in this instance quite analogous. 
 
 These and the other harmonious relations of the diagonals are 
 apparent in the following comparative scale, in which those from 90° 
 to 11° 15" are compared with the acoustical and geometrical construc- 
 tion of three octaves of the musical scale. The degrees are numbered 
 from the bottom upwards, in order that they may correspond to the 
 manner in which the notes are arranged upon the clefs, and the 
 degrees in the diagonals are arranged in the same order, that thev 
 may correspond to the parts of the monochord. 
 
50 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 COMPARATIVE SCALE. 
 
 Musical Sounds. 
 
 n , v -? a !-°° f . Number of 
 
 Degrees of Vibrations to vibrations 
 the Scale. the Tonic. inaSecond . 
 
 Relations 
 to one 
 another. 
 
 Number of 
 degrees in 
 diagonals. 
 
 Geometric Figures. 
 Ratio of dia- 
 gonals to 
 the right 
 angle. 
 
 Nos. of 
 
 rect- 
 angles. 
 
 22d 
 
 8 to 1 
 
 1024 s 
 
 
 ( 
 
 15' 
 
 1 to 8 
 
 22d 
 
 
 
 
 15 to 16 
 
 
 
 
 
 21st 
 
 60 ... 8 
 
 960 < 
 
 
 c 
 
 
 8 ... 60 
 
 21st 
 
 
 
 
 o a 
 
 8 ... 9 
 
 
 
 
 
 20th 
 
 20 ... 3 
 
 853 1 I 
 
 
 > 13 
 
 OA 
 
 30 
 
 3 ... 20 
 
 20th 
 
 
 
 
 A T A 
 
 9 ... 10 
 
 
 
 
 
 19th 
 
 12 ... 2 
 
 *7 OO I 
 
 768 i 
 
 
 ( 
 
 
 2 ... 12 
 
 1 Ail 
 
 19th 
 
 
 
 
 O A 
 
 8 ... 9 
 
 
 
 
 
 18th 
 
 16 ... 3 
 
 682§ j 
 
 
 | 16 
 
 52 30" 
 
 3 ... 16 
 
 18th 
 
 
 
 
 15... 16 
 
 
 
 
 
 T Til 
 
 17th 
 
 A A A 
 
 20 ... 4 
 
 O A A J 
 
 
 l TO 
 ( 
 
 
 A OA 
 
 4 ... 20 
 
 17th 
 
 
 
 
 A 1A 
 
 9 ... 10 
 
 
 
 
 
 luth 
 
 c\o o 
 
 3b ... 8 
 
 570 < 
 
 
 f OA 
 ( 
 
 
 8 ... 3b 
 
 lbth 
 
 
 
 
 O A 
 
 8 ... 9 
 
 
 
 
 
 loth 
 
 A 1 
 
 4 ... 1 
 
 512 j 
 
 
 ' 00 
 
 OA 
 JO 
 
 1 A 
 
 1 ... 4 
 
 loth 
 
 
 
 
 1 e? to 
 
 15 ... lb 
 
 
 
 
 
 14rtll 
 
 OA O 
 
 oO ... o 
 
 y( OA J 
 
 O A 
 
 f OA 
 ( 
 
 
 O OA 
 
 8 ... 30 
 
 14th 
 
 loth 
 
 10 ... 6 
 
 42b§ 
 
 o ... y 
 
 I 07 
 ( 
 
 
 O 1 A 
 
 o ... 10 
 
 13th 
 
 
 
 
 A 1 A 
 
 9 ... 10 
 
 
 
 
 
 J til 
 
 o ... J 
 
 OO 1 ) 
 
 
 ' 9.O. 
 > OU 
 
 
 o P. 
 
 Z ... b 
 
 lJth 
 
 
 
 
 8... 9 
 
 
 
 
 
 11th 
 
 8... 3 
 
 341i | 
 
 
 | 33 
 
 45 
 
 3... 8 
 
 11th 
 
 
 
 
 15... 16 
 
 
 
 
 
 10th 
 
 10 ... 4 
 
 320 j 
 
 
 | 36 
 
 
 4 ... 10 
 
 10th 
 
 
 
 
 9... 10 
 
 
 
 
 
 9th 
 
 18... 8 
 
 288 j 
 
 
 \ 40 
 
 
 8... 18 
 
 9th 
 
 
 
 
 8... 9 
 
 ) -~> 
 
 
 
 
 8th 
 
 2 ... 1 
 
 256 | 
 
 
 | 45 
 
 
 1 ... 2 
 
 8th 
 
 
 
 
 15 ... 16 
 
 
 
 
 
 7th 
 
 15... 8 
 
 240 j 
 
 
 ] 48 
 
 
 8 ... 15 
 
 7th 
 
 6th 
 
 5... 3 
 
 213^ | 
 
 8... 9 
 
 | 54 
 
 
 3... 5 
 
 6th 
 
 
 
 
 9... 10 
 
 
 
 
 
 5th 
 
 3... 2 
 
 192 j 
 
 
 | 60 
 
 
 2... 3 
 
 5th 
 
 
 
 
 8... 9 
 
 
 
 
 
 4th 
 
 4.. .3 
 
 170f | 
 
 
 | 67 
 
 30 
 
 3... 4 
 
 4th 
 
 
 
 15 ... 16 
 
 
 
 
 
 3d 
 
 5... 4 
 
 160 | 
 
 
 | 72 
 
 
 4... 5 
 
 3d 
 
 
 
 
 9... 10 
 
 
 
 
 
 2d 
 
 9... 8 
 
 144 j 
 
 
 J 80 
 
 
 8... 9 
 
 2d 
 
 
 
 
 8... 9 
 
 
 
 
 
 1st 
 
 1 
 
 128 > 
 
 
 < 90 
 
 
 1 
 
 1st 
 
PLATE I. 
 
 Rt'( It tuples produced bp the diagonals m the torcaoi/iu comparative scale 
 
 m mz F>.3 
 
 s|i> 
 
 Quantities in diagonals us diet/ 
 proceed from tlic Vertical line 
 
 Vertical 
 
 °1 
 
 90° 
 
 
 80° 
 
 3 
 
 72° 
 
 4 
 
 671.30' 
 
 
 60° 
 
 6 
 
 54° 
 
 / 
 
 48° 
 
 o 
 
 
 M° 
 
 9 
 
 40° 
 
 10 
 
 36° 
 
 li 
 
 33145' 
 
 12 
 
 30° 
 
 13 
 
 27° 
 
 14 
 
 24° 
 
 15 
 
 22130' 
 
 W 
 
 20° 
 
 17 
 
 18° 
 
 18 
 
 16Z52L, 
 
 19 
 
 ...lb 
 
 20 
 
 13°. 30' 
 
 21 
 
 12° 
 
 22 
 
 11115' 
 
 Rectangles 
 Horizontal 
 
 30 
 
 Rectangles 
 
 1120 
 
 WZL 
 
 W22 
 
 Quantities in diagonals as they 
 proceed from tJve Horizontal line . 
 
 Horizontal 
 
 Kci tangles 
 Square 
 
 Vertical 
 
 US15' 
 
 
 
 OB 
 
 
 * 
 
 F°l. 
 
 0° 
 
 9 
 
 V 
 
 10° 
 
 3 
 
 18° 
 
 4 
 
 22° 30' 
 
 - 
 
 o 
 
 30° 
 
 6. 
 
 36° 
 
 7 
 
 42° 
 
 8 
 
 45° 
 
 9. 
 
 50° 
 
 10 
 
 54° 
 
 11 
 
 56° 15' 
 
 12 
 13 
 14 
 
 60° 
 63° 
 66° 
 
 15 
 
 67" 30' 
 
 16. 
 
 70° 
 
 17 
 
 .72° 
 
 18 
 
 ..73?. ... 7 « 
 
 19. 
 
 .75" 
 
 20 
 
 70° 30' 
 
 21 
 
 78° 
 
 22 
 
 78°. 4b! 
 
 30 
 
 lurtantile. 
 
 II K ITmi ,lrf 
 
I' LATE II 
 
 56*15' 
 
 Frogress of the Rectangles fivm ike vertical and horizontal 
 positions of the strain lit line agreeably to the harmonious 
 divisions of the quadrant . shewing //fir it may he 
 continued-, and die various other series that arise 
 twm the intersections of these two /nodes of progress. 
 
 W 6 
 
 70? 
 
 76. 
 
 Ob 
 
 tit!" 
 
 r ,F30 
 
 61 
 
 45°. 
 
 W30 
 
 no 
 
 80 
 
 67-30 
 
 1 2 3 I 
 
 ToffO"as 8-9 4-5 3-4 
 
 24' 
 
 20" 
 
 I"" I \jlr] \'.<rVt] 1 /,- 1 | 77' 
 
 1130' 
 
 2-3 
 
 
 3-5 
 
 i 
 
 8-15 
 
 10" 
 
 ■/■:>: 
 
 36 l 
 
 10 
 
 For the number of degrees 
 in the diagonals of die 
 rectangles that proceed be- 
 yond the limits of this 
 plate, see gen/ral scale . 
 
 1-2 
 
PLATE III. 
 
 Progress nl llwlintuiuflm Hum thr /itwiiH/r/imus swuur, Im'ntf a r/mlinuiilum ol'lJir smix mm/iimtrd inPLjte.II half size 
 
 Geo Ai)vnan.S£. 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 51 
 
 In Plate I., I give the rectangles produced upon the diagonals in 
 the comparative scale, with their vertical and horizontal quantities 
 marked upon them, by which it will be seen, that those figures, in 
 their natural progress from 90°, increase in quantity of superficies 
 inversely to the number of degrees in the diagonal upon which they 
 are formed, in the same manner that the notes in music increase in 
 the numbers of their vibrations inversely to the parts of the mono- 
 chord producing them, although the ratios differ. 
 
 In Plate II., I have arranged them in such a manner that they pro- 
 ceed from the horizontal and vertical lines, forming with one another 
 various intermediate series, which mode of progression may be conti- 
 nued to any imaginable extent. In this series, the right angle is the 
 tonic, or key. 
 
 In Plate III., which is a continuation of Plate II., 45° may be 
 considered the key of the diagonals, and consequently the square 
 is the key of the rectangles. It will be observed, that a different 
 mode of progression is adopted from that of the first series, which 
 may also be continued to the most minute division of the quad- 
 rant ; but these two series are sufficient to show the harmonious 
 
52 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALSYED. 
 
 progression of this ruling figure, and the endless, though systematic, 
 variety to which the harmonious division of the geometric quadrant 
 leads. 
 
 In my former work on this subject, I divided the musical scale into 
 twelve equal parts or semitones, giving two of those to each full tone, 
 and upon this basis I endeavoured to found an analogy. But this 
 mode of division of the musical intervals was not correct, and the 
 work was scarcely published before I detected this apparent mistake. 
 I say apparent, because it appears that this mode of division is nearly 
 correct in a geometrical point of view, and arises out of the true 
 mode of division in the following manner : — Let a quadrant be 
 divided into two parts, and let one of those parts be subdivided into 
 the correct divisions of the monochord already exemplified ; let a line 
 be drawn above the quadrant, and parallel to its base, and the dia- 
 gonals which cut the arc of the quadrant at the proper divisions, will, 
 when they meet this line, divide it very nearly into the equal tones 
 and semitones adopted in my former Essay. 
 
 Thus are the diagonals, in the degrees of their obliquity, equal to 
 the correct musical intervals, as also to the lengths of string by which 
 
PLATE IV. 
 
 \cnusticai and Geometric liarntoiuj compared . 
 
 Figure Li 
 
 l\'cliUi\"e leiujths of musical strings leguired 
 to produce llif diatonir scale of notes. 
 I 8 
 
 Figure c: 
 
 Notes 
 
 H A C. 
 
 V. I) 
 
 Relative numbers of geometric derives required m 
 dwqoiials to produce the first series of recta n gles . 
 
 exhibited on a slraig/i/ line 
 
 W. -US . *■ ttP. tUf' 
 
 SO 
 
 Numbers of rectangles 7"' 
 
 ~7T 
 
 Figure' 3. 
 
 Oroiiwtririd relations of the diagonals which produce the first, series of rectangles tod^Mznnonk 
 division ol die vioitor/uuil hg which the diatonic series id musical nates is produced 
 Difference between divisions on the. chard 
 
 uiiil I'll die urc/i ol die circle . 
 
 vm vu v, v 
 
 Approximation loan egual division bulZ . parts, 
 line divided into egual purls 
 
 72 11 10 9 8 7 
 A A 
 
 vm vn vi 
 
 12 ii\ 10 9 ■ 8 
 
 
 IV 
 
 V 
 
 III 
 
 ft Y 
 
 
 
 1 
 
 7 
 
 6 
 
 5 
 
 / 
 
 3 2 / 
 
 a 
 
 V 
 
 X 
 
 
 IV 
 
 X 
 
 in 
 
 I 
 
 11 
 
 A V 
 
 I 
 
 I 
 
 f 
 
 
 IV 
 
 111 
 
 II 
 
 1 
 
 7 
 
 a 
 
 b 
 
 
 
 a 
 
 aeOtjlik/Tutri so< 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 53 
 
 the intervals are produced ; while the rectangles formed upon those 
 diagonals, which differ from one another only in one dimension, have 
 that difference nearly in equal sixth and twelfth parts. Therefore, 
 although my former mode of division was upon a wrong system, the 
 results were correct in regard to the harmonic ratios of the rect- 
 angles, upon which class of figures, any attempt to systematize the 
 harmony of form must depend.* — See Plate IV. 
 
 As, agreeably to the laws of acoustics, it was shown that the various 
 divisions of the monochord produced an harmonious arrangement of 
 
 * In a review of my former work, which appeared in the Athenceum, June 10th and 24th, alike 
 distinguished for its candour and ability, the writer points out this error in my mode of dividing 
 the musical scale, which certainly affects seriously the analogy I attempted to establish in that very 
 imperfect production. And he is quite right in supposing that I was misled by the popular mode 
 of speech which represents the octave as composed of five equal intervals and two half intervals, 
 making up six full tones. His observations upon this subject are so clear and decisive, that, an 
 earlier acquaintance with them would have enabled me greatly to improve the preceding part of 
 this Essay ; for the existence of a perfect analogy between the perimeters of my seven figures and 
 the musical intervals is not at all essential to my present theory, and I might therefore have 
 dispensed with further allusion to it. But it is not so with the harmonic ratios adopted in this 
 Essay for regulating diagonals, for they appear to me essential in reducing any theory of this kind 
 to system, without which it can be of little practical use. — 26th June 18-43. 
 
54 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 sound which may be either employed in combination or succession, 
 according to certain rules ; so has the harmonious division of the 
 circle in the first instance produced the three kinds of right lines — 
 the horizontal, the vertical, and the oblique ; in the second, the lead- 
 ing figure in every variety of rectilinear form ; and, in the third, a 
 proper series of diagonals, by which every modification of those forms 
 may be regulated. Because the rectangles are, in every arrange- 
 ment of form, the dominant figures, in so far as they rule the pro- 
 portions of all others, in the same way that the curves of the circle 
 and ellipse have been shown to rule the configuration of all rectilinear 
 forms. 
 
 The circle and the square seem to have a reciprocal effect upon 
 one another in regard to this harmonious mode of division ; for if the 
 perimeter of the latter be divided into sixteen equal parts, a circle 
 drawn within it tangential to its sides, will of course be divided at 
 its points of contact into four equal divisions or quadrants. But if 
 parallel lines be drawn across the area of the square from the other 
 divisions of its sides, these quadrants of the circle will each be divided 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 53 
 
 into three equal parts — Figure 37 ; and if from the points at which 
 these parallel lines cut the periphery of the circle, other straight 
 lines are drawn to the points of contact, the diagram, containing a 
 complete series of the rectilinear figures, will again be produced. 
 Figure 38. 
 
 Fig. 37. Fig. 38. 
 
 These lines form right angles with the side of the exterior square, 
 angles of 30° with any radius that meets them from the centre of the 
 circle, and angles of 60° with the lines drawn to the point of contact, 
 and they inscribe the inner circle by a square half the perimeter of 
 the exterior one. 
 
 If the four sides of any rectangle be divided in this manner — that 
 
56 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 is, each into 4 — the ellipse which it inscribes will be harmonically 
 divided into four parts by the points of contact, each of which will be 
 subdivided by parallel lines passing through the area of the oblong 
 into three proportionate divisions ; and by uniting the points of con- 
 tact with the points of intersection, a series of rectilinear figures will 
 be produced, inscribed by the ellipse, and leaving an area in their centre 
 capable of containing another ellipse of precisely the same propor- 
 tions and half the circumference of the first. This same process may 
 be performed within every rectangle in the series ; and the proper 
 angles, obtuse and acute, as well as the proper curve belonging to 
 any rectangle, either vertically or horizontally, thus accurately deter- 
 mined, as shall afterwards be shown. 
 
 As within a semicircle, every acute angle formed between a radius 
 and its base, from 0° to 90°, forms on the other side an obtuse angle, 
 the number of degrees in which, added to the acute angle, is always 
 180° ; so within a quadrant every diagonal, from 0° towards 45°, will 
 give the proportions to a horizonal rectangle, and will have on the 
 other side a corresponding diagonal, the number of degrees in which, 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 :>7 
 
 added to the first, will always be 90°. This diagonal will give the 
 proportions to a vertical rectangle, having a certain harmonious rela- 
 tion to that formed upon the horizontal diagonal, and conversely. 
 These relations are worthy of particular attention, and occur in the 
 first series of rectangles as follows : — 
 
 Vortical. 
 
 Horizontal. 
 
 
 I. 90° 
 
 0° 
 
 Being a Tonic. 
 
 II. 80° 
 
 10° 
 
 22d, or Treble Octave. 
 
 III. 72° 
 
 18° 
 
 ■ 15th, or Double Octave. 
 
 IV. 67° 30' . 
 
 22° 30' 
 
 12th, or Compound Fifth. 
 
 V. 60° 
 
 30° 
 
 an 8th, or Single Octave. 
 
 VI. 54° 
 
 ... 36° 
 
 a 5th, or Dominant. 
 
 *VII. 50° 
 
 .. 40" 
 
 3d, or Mediant. 
 
 VIII. 45° 
 
 45° 
 
 an Octave to the Tonic. 
 
 The ratios of those numbers are to one another as follows : — 
 
 I. It III. IV. V. VI. VII. VIII. 
 
 8 to 1 4 to 1 3 to 1 2 to 1 3 to 2 5 to 4 1 to 1 
 
 * The seventh diagonal requires to be lengthened to 50°, in order to produce an harmonic ratio 
 with 40°, which is the corresponding number on the opposite side of 45°. 
 
58 
 
 GEOMETRIC PRINCIPLE OE BEAUTY ANALYSED. 
 
 It has been observed, in the Introduction, that forms and figures 
 as used in the arts require one or other of two qualities to render 
 them pleasing : the first of which is the imitation of natural objects, 
 and the second is harmony, produced by the proportion and arrange- 
 ment of the elements of abstract form. The one is like a simple 
 description in plain language, while the other is the geometric 
 poetry of the graphic art. When forms are combined merely to 
 produce a pleasing effect, as in ornamental design, or even in archi- 
 tecture, where the subjects are not in imitation of any natural 
 objects, such compositions may be assimilated to those of melody or 
 harmony in music, independently of the analogy of their elements. 
 It has, however, been shown, that the elements of abstract form are 
 in many respects similar to those of abstract sound ; and I shall now 
 endeavour to show how those elements may be combined in harmony, 
 and how an endless variety of harmonious combinations may be pro- 
 duced. And, in doing so, I shall endeavour to render the plates as 
 explicit as possible to the general reader, and thereby avoid the neces- 
 sity of any long explanations. 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 59 
 
 Plates V., VI., VII., and VIII., are combinations of this kind, and 
 show, that if a quadrant be placed upon any diameter of a circle, and 
 lines drawn through any of the harmonic divisions until they reach 
 the circumference of the circle, and another line drawn from this 
 perpendicular to the diameter or base until it again meet the cir- 
 cumference, the repetition of these two lines from every similar 
 division of the circumference will produce an harmonious arrange- 
 ment. 
 
 If the diagonal, or oblique line, be drawn through the quadrant at 
 45°, an arc of 90° will be cut off the circle, and the perpendicular to 
 the base that leaves at the point of contact will divide the cir- 
 cumference into two equal parts of 180° ; and the repetition of the 
 diagonal at each arc of 90°, will produce a square of exactly half the 
 superficial contents of that which inscribes the circle, corresponding 
 in this respect to the harmonic ratio of a musical octave. Plate V., 
 Figure 1. 
 
 If the diagonal be 60°, the vertical, or perpendicular to the base, 
 will cut an arc of 120° off the circle ; the repetition of which perpen- 
 
 » 
 
60 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 dicular, along with the diagonal, will produce the equilateral triangle, 
 the proper parallelogram, the proper rhombus, and the hexagon. 
 Plate V., Figure 3. Figure 4 shows the harmonious combinations of 
 those lines. 
 
 If the diagonal be 72°, the vertical, or perpendicular to the base, 
 will cut an arc of 72° off the circle ; the repetition of which perpen- 
 dicular, along with the diagonal, will produce the pentagon and deca- 
 gon. Plate VII., Figure 1. 
 
 The diagonal lines from the quadrant form the perimeter of the 
 rectilinear figures within the circle ; while the diameter of the circle, 
 forming the base of the quadrant, gives the diagonal ; indeed, those 
 diagrams are simply circular arrangements of the harmonious rect- 
 angles. 
 
 These are the three primary divisions, viz. by 2, by 3, and by 5 ; 
 and the others being merely multiples, or compounds of these, fur- 
 ther description than that given upon the plates is unnecessary. It 
 will be observed, however, that when the division of the circumference 
 of the circle is uniform, the intersections form various concentric 
 
Geo^ikman sc. 
 
PLATE VI . 
 
 Harmonic Dun/rams 
 
 Figure I . 
 
 Division ot' the quadrant hu Four producing the 
 mirnwTdc anglA?. 
 22 "3t?' 45" 67° 30'. 
 
 Figures 3. 
 
 Division of the giiadranJ/buSixproduana the 
 II ,/ rum nil- angles 
 15°, 30" 45° 60° 75" 
 
 Figure/ 2 . Figure/ 4. 
 
 Di mi mm produces] bp the above angles. Diagram pnnlured hi/ the above angles. 
 
tiro di forum sc 
 
PLATE VIII . 
 
 Harmonic Diagrams 
 
 Figure I . 
 
 Division of 'ike quadrant />// Twelve producing the>ffarmanic angles. 
 7?30;i5?22?30", 30? 3 7 "30! and 45? 
 
 Figure 3 . 
 
 Mixture <>/'////" Three primary Divisions />// 2. 3an£L3 producing 
 ///r Ffarmanii: ant/Irs 4~>" !>(>''.'W'! I.i'.'.'l'.'aiid 22 "3d' 
 
 Figure 2. Figure 4 . 
 
 I) Kui nun produced In/ (he above angles. Diagram produced oylheabove. angles. 
 
N 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 61 
 
 polygons, which approach the figure of the circle so nearly, that they 
 at first sight deceive the eye. As an instance, Figures 2 and 4, 
 Plate VIII., have not a single curved line in their composition. But 
 when the division is irregular, by including the third diagonal of 18°, 
 a different effect is produced, in which no particular figure is distin- 
 guishable ; and the apparent curves, instead of being circular, are 
 hyperbolic. — Plate VIII., Figure 4. Examples of this kind of har- 
 mony might be multiplied to any extent. 
 
 Such combinations, of lines may be termed capriccios in the har- 
 mony of form, which, although they display every variety of figure 
 harmoniously arranged, have no subject, and are consequently mere 
 exemplifications of dexterity. Yet are they suggestive of subjects, 
 especially in mere ornamental design, and I shall, therefore, add other 
 four compositions — or more properly combinations — of this kind. 
 They are of a totally opposite character, being angular arrangements 
 of the homogeneous curve, while the former may be termed circular 
 arrangements of the homogeneous angle — the diameters of the circle, 
 or diagonals of the rectangles being superfluous. Plates IX., X., 
 XI., XII. 
 
62 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 Such combinations may be multiplied to any extent, and the result 
 never be otherwise than pleasing while the harmonic ratios are 
 attended to. It will be observed in those diagrams, that, as the 
 straight line assumed the appearance of the curve in the circular 
 mode of combination, so does the curve assume the appearance of the 
 straight line in those combinations that are angular.* 
 
 Having thus endeavoured to develop my theory, and give a few 
 results of the most simple combinations of the straight and curved 
 lines, agreeably to its rules I shall now endeavour to explain the 
 harmonic ratios of numbers, and show some of the peculiarities of the 
 seven figures which form the basis of my theory. 
 
 OF THE HARMONIC RATIOS OF NUMBERS. 
 
 The unit has no power of multiplication or division, but the second 
 numeral 2 has both those powers, and they are productive in both 
 
 * The Author has in progress, and shortly intends to publish, a series of Diagrams upon those 
 principles, with the diaper designs resulting from them. 
 
Plate ix. 
 
 ft May del* 
 
 Fr. ■ \;/,<„ri /. t /A. AW„, . r 
 
PLATE X 
 
« 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 cases of uniformity. It is the first multiple of 1. It is a submultiple 
 of the numbers 4, 6, 8, &c., progressively, as 2, 3, 4 ; and it is the 
 first even number. The second numeral 3 is also simply a multiple 
 of 1, and is the first odd number ; it is a submultiple of 6, 9, 12, 
 &c, progressively, as 2, 3, 4, and is a compound of 1 and 2 added 
 together. 
 
 The next multiple of 1, having no other aliquot parts, is 5 — a 
 similar compound of the first even and first odd numbers, 2 and 3. It 
 is a submultiple of 10, &c. These three numerals 2, 3, and 5, are, 
 therefore, the first three multiples of 1 that are multiples of no other 
 number — consequently they are adapted to divide the elements of 
 proportion into the primary harmonic ratios, as already shown ; and 
 in this capacity equally regulate the effects produced by external nature 
 upon the senses of hearing and seeing. The number 5, although by 
 it the third mode of division is performed, is of an intermediate cha- 
 racter, and the rectangle produced by its division of the quadrant, 
 is the mediant in the series, as already shown : the middle may thus be 
 said to be produced by a number which combines the first and last. 
 
 K 
 
64 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 The primary harmonic ratios produced by those modes of division, 
 are, as already shown, 1 to 2, 2 to 3, 4 to 5, and the secondary ones 
 being 8 to 9, 3 to 4, 3 to 5, and 8 to 15, no new mode of division 
 is required, 4, 8, 9, and 15, being multiples of 2, 3, and 5.* To 
 arrange and proportion the combination of various forms by those, 
 in such a manner as to produce one or unity, ought on all occasions 
 to be the aim of the artist ; for every composition should manifest 
 in all its parts a definite relation to a whole. This is the first 
 condition of order, and consequently the primary cause of geometric 
 beauty. 
 
 When the parts of a figure are perfectly homogeneous or identical, 
 there can exist in it no principle of proportion productive of this 
 effect, because its parts are throughout as 1 to 1, producing sameness. 
 The circle, the square, and the equilateral triangle, are figures of 
 
 * The number 7 is not employed to produce the harmonic ratios ; but if the quadrant be 
 divided by 18, and the product multiplied by 7, it will give 35° ; and a rectangle formed upon this 
 diagonal will have the peculiar property of being divided into two of its own proportions by a line 
 bisecting it through its shortest diameter. 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 (5.5 
 
 this description, and consequently want within themselves individually 
 the first element of proportion — variety. It would appear that those 
 figures can only be proportioned individually in octaves in regard to 
 superficies, and in octaves and fifths in regard to periphery. There- 
 fore, two squares are in proportion, if one be either in area or peri- 
 meter to the other, as 1 to 2, as shown on Plate XIII., Figure 1, 
 where A to B, B to C, C to D, and D to E, will be found to be re- 
 spectively in the ratio of 1 to 2 in quantity of area or superficies, while 
 in perimeter they are as 2 to 3. But A to C, B to D, and C to E, 
 will be found in the lineal quantity of their perimeter in the ratio of 
 1 to 2. 
 
 The circles inscribing and inscribed by those squares will neces- 
 sarily have the same binary and ternary ratios, Plate XIII., Figure 
 1 ; but the simple mode of division shown by the strong lines in 
 Figure 3, seems the most harmonious. 
 
 The equilateral triangle cannot harmonically inscribe a larger 
 figure of its own kind than that which bears to it the superficial 
 ratio of 1 to 4, a double octave. A is to B, and B to C, in 
 
66 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 that ratio, and are in perimeter in that of 1 to 2, as shown on 
 Plate XIII., Figure 2. This seems to be the only kind of proportion 
 that can be applied to those figures while they retain their homoge- 
 neous quality. 
 
 The secondary figures — the oblong, the lozenge, and the ellipse — 
 may also be divided in this manner, and the ratios in the quantity of 
 their superficies, perimeter, or circumference, under every degree of 
 modification accurately ascertained, as exemplified on Plate XIII., 
 Figures 4, 5, 6. Figure 7 is the hexagon, with its peripheral octave, 
 which latter figure will be found in Figure 3 (D) harmoniously in- 
 scribed in the equilateral triangle, and bearing to it in quantity of 
 area the relative proportion of 1 to 2. 
 
 But those secondary figures have also, in the parts of which they 
 are composed, that first requisite in proportion — variety. The har- 
 monic ratios are therefore applicable to those figures individually, 
 and their beauty will depend upon the proportioning of their length 
 to their breadth. 
 
 The oblong I have adopted as the leading figure of the secondary 
 
Fig. 7. 
 
 V.R Hcu, del T 
 
 (rt'A ' Aiknuui. sc. 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. (57 
 
 class, has peculiarities in its proportions which are worthy of remark. 
 They are as follow : — Its vertical diagonal is 60°, and its horizontal 
 diagonal 30°, consequently its proportions are, in this respect, in the 
 most consonant harmonic ratio of 1 to 2. It may, in common with 
 all other figures of this class, be divided, by two lines across its area, 
 into four of its own proportions. But it may also, by two lines drawn 
 across its shortest diameter, be divided into three figures of its own 
 proportions. If divided into two diagonally, it produces two scalene 
 triangles, which, on being joined by the longest side of the right 
 angle, form an equilateral triangle. Its vertical diagonal is to the 
 right angle in the harmonic ratio of 2 to 3, and its horizontal diagonal 
 bears the same harmonic ratio to the diagonal of the primary square. 
 Those proportions render this figure more intrinsically symmetrical, 
 and consequently more pleasing to the eye, than any other of its 
 own class. 
 
 The ellipse and rhombus, adopted as the leading figures of the 
 classes to which they respectively belong, are proportioned to this 
 figure, and consequently partake of its beauty. 
 
68 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 It has already been stated, that, by a reciprocity existing between 
 a rectangle and the curvilinear figure which it inscribes, the lines by 
 which the area of the former is divided into sixteen equal homogene- 
 ous parts, will, at the points of intersection, harmoniously divide the 
 circumference of the latter into twelve, and that, by uniting those 
 points by diagonal lines drawn across the area of the inscribed circle 
 or ellipse, the angles in harmony with the inscribing rectangle will 
 be produced. The three leading rectangles, with their curvilinear 
 and angular figures, are given in Plate XIV., on which the coinci- 
 dences of the intersections are very remarkable. 
 
 In Plate XV., I give the various triangles belonging to each of the 
 three leading rectangles, where it will be seen that the circle has on 
 its lowest chord the equilateral, [figure 1 {] on its centre chord the 
 right-angled isosceles, [figure 2 ;] and on the upper the obtuse- angled 
 isosceles, [figure 3.] The harmonic ratios they bear to the right 
 angle are those of a fifth, an octave, and a twelfth, their degrees of 
 obliquity being 60°, 45°, 30°. The first ellipse has, upon its lowest 
 chord, the right-angled isosceles, [figure 4] — upon its centre chord, 
 
PLATE XIV 
 
/) II Bay 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 the obtuse-angled isosceles, [figure 5] — and, upon its upper chord, 
 the second obtuse-angled isosceles, [figure 6.] The harmonic ratios 
 of those angles are to the right angle those of an octave, a twelfth, and 
 a seventeenth, being 45°, 30°, 18°. The second ellipse has, upon its 
 lowest chord, the obtuse-angled isosceles, [figure 7] — upon its centre 
 chord the same figure of the second class, [figure 8] — and, upon its 
 upper chord, the same figure of the third class, [figure 9.] Their 
 harmonic ratios are to the right angle those of a twelfth, a seven- 
 teenth, and a twenty-second, or treble octave, being 30°, 18°, 11° 15'. 
 The vertical triangles of the two ellipses are given in Figures 10, 11, 
 12, 13, 14, and 15. 
 
 Thus have I endeavoured to analyse the Geometric principle of 
 Beauty — proportion — by showing that it is regulated by the harmo- 
 nic ratios of numbers. And by the application of those ratios to a 
 quadrant of the circle, I have shown that an almost infinite series 
 of rectangles may be produced, bearing to one another certain 
 harmonious relations ; and that within each of those a series of six 
 other distinctive characters of figures may be systematically and 
 
TO 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 harmoniously generated. In short, that the beauty arising from 
 the harmony of form may be, on all occasions, with certainty 
 produced. 
 
 But the application of this system to the various arts in which it 
 will be useful, must form the subject of another treatise, as it would 
 be premature to apply rules until their accuracy were acknow- 
 ledged. I shall, however, in the mean time, add a few general rules 
 which obviously arise out of this theory : — 
 
 1st, Rectangles, when arranged in succession, either horizontally 
 or vertically, should only differ from one another in one of their dimen- 
 sions ; so that, when vertically arranged either as solids or vacuities,* 
 their vertical sides must be in the same line ; and when horizontally 
 arranged, their horizontal lines must also be in the same line, their 
 
 * Professor Hosking, in his excellent Treatise on Architecture, amongst other valuable obser- 
 vations upon the harmony of form as applicable to that art, says, that " a degree of harmony must 
 exist between the solids and vacuities of an edifice." This I consider to form, along with the out- 
 line, the melody or leading feature in all such compositions ; for if it be inharmonious, no profusion 
 of ornament can conceal the deformity. 
 
GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 71 
 
 harmony being regulated by their diagonals alone. See Plates II. 
 and III. ; as also Plates XVI. and XVII. 
 
 2d, Triangles must on all occasions correspond to the rectangles 
 with which they are associated — acutely when the rectangle is verti- 
 cal, and obtusely when it is horizontal. As the harmonious propor- 
 tion of every rectangle, when in a vertical or horizontal position, is 
 determined by an oblique line called its diagonal, so is the proportion 
 of every regular isosceles triangle determined by a vertical line. This 
 being in a positive position, it can have no change but in its lineal 
 dimensions, which will be seen in Plate XV. to be always as 1 to 2, or 
 2 to 3. When, therefore, triangles are employed in succession, their 
 proportions must be in other respects the same. If they be not, they 
 generate between them a discordant figure. 
 
 Curvilinear figures^ in like manner, must always correspond to the 
 rectangles with which they are associated, and in succession their har- 
 mony will depend upon the ratio of their radii. Therefore, they can 
 
 L 
 
72 
 
 GEOMETRIC PRINCIPLE OF BEAUTY ANALYSED. 
 
 only differ in size, and not in degree of curvature. This difference 
 can onlv be in the ratio of 1 to 2^ or 2 to 3. The curve can never 
 be greater than what may be inscribed by the rectangle with which it 
 is associated, and can never harmoniously leave the rectangle unless 
 at the tangential point, or at right angles with it. Plate XV. 
 
 As a circle may be described within any rectangle tangential to its 
 longest sides, this peculiar curve may terminate any vertical rect- 
 angle. 
 
 Plates XVI. and XVII. are examples of harmonic combinations 
 of rectangles, divided into triangles agreeably to the Platonic sys- 
 tem.* The strong lines show how they may be formed into solids 
 and vacuities in architectural composition. These very simple ex- 
 amples are added, merely to show the mode in which the harmonic 
 angles may be applied. 
 
 * See Appendix. 
 
APPENDIX. 
 
 The careful and learned notice of my former treatise on this subject, to which I have 
 already alluded, and which appeared when the greater portion of the present Essay 
 was printed, has not only convinced me of the accuracy of what I have stated as likely to 
 have conduced to the perfection of the arts in Greece, but has given me a key to the elu- 
 cidation of my subject. The author of the notice alluded to, after having pointed out with 
 critical distinctness the errors I had committed in that treatise in regard to analogy, and 
 to point out the true cause of harmony of form, states — 
 
 " We now proceed to show that the principles we have previously developed as the Greek doc- 
 trines of beauty and proportion, give us precisely Mr Hay's elements, along with many others 
 which his system excludes. They are given in the Timseus of Plato."* — Athcnceum, No. 817, 
 p. 08G. 
 
 From this I was induced to examine the Platonic Theory, and shall here quote the 
 whole of what is in any way applicable either to my last treatise or to the present attempt. 
 This, I trust, will enable those who, like myself, are not conversant with the works of the 
 ancient philosophers, to determine in how far I have now succeeded in reducing to system 
 the elements alluded to. 
 
 " But when the Artificer began to adorn the universe, lie first of all figured with forms and num- 
 
 * In another learned review of my former treatise, the writer says : — " It is, after all, the deve- 
 loped theory of Plato, who eloquently commented on the music of beautiful forms. * * It is a 
 startling proof of the depth of the Egypto-Platonic, or geometrical philosophy, transmitted to the 
 Templar Freemasons." — Court Gazette, No. 268, p. 195. 
 
74 
 
 APPENDIX. 
 
 bers fire and earth, water and air, which possessed, indeed, certain traces of the true elements, but 
 were in every respect so constituted, as it becomes any thing to be from which Deity is absent. 
 But we should always persevere in asserting that Divinity rendered them as much as possible the 
 most beautiful and the best, when they were in a state of existence opposite to such a condition. 
 I shall now, therefore, endeavour to unfold to you the distribution and generation of these by a 
 discourse, unusual indeed, but to you who have trod in all the paths of erudition through which 
 demonstration is necessarily obtained, perspicuous and plain. In the first place, then, that fire and 
 earth, water and air, are bodies, is perspicuous to every one. But every species of body possesses 
 profundity ; and it is necessary that every depth should comprehend the nature of a plane. Again, 
 the rectitude of the base of a plane is composed from triangles. But all triangles originate from 
 t wo species ; one of which possesses one right angle, and the other two acute angles : and one of 
 these contains one right angle distributed with equal sides ; but, in the other, unequal angles are 
 distributed with unequal sides. Hence, proceeding according to assimilative reasons, conjoined 
 with necessity, we shall establish a principle of this kind as the origin of fire and all other bodies. 
 The supernal principles of these, indeed, are known to Divinity, and to the man who is in friend- 
 ship with Divinity. 
 
 " But it is necessary to relate by what means four most beautiful bodies were produced ; dis- 
 similar indeed to each other, but which are able, from certain dissolutions into each other, to 
 become the sources of each other's generation. For if we are able to accomplish this, we shall 
 obtain the truth concerning the generation of earth and fire, and of those elements which are 
 situated according to analogy between these. And then we shall not assent to any one who should 
 assert that there are visible bodies more beautiful than these, each of which subsists according to 
 one kind. We must endeavour, therefore, to harmonize the four sorts of bodies excelling in beauty, 
 and to evince by this means that we sufficiently comprehend the nature of these. Of the two tri- 
 angles, indeed, the isosceles is allotted one nature, but the oblong or scalene is characterized by 
 infinity. We ought, therefore, to choose the most beautiful among infinities, if we wish to com- 
 mence our investigation in a becoming manner. And if any one shall assert that he has chosen 
 something more beautiful for the composition of these, we shall suffer his opinion to prevail ; con- 
 
APPENDIX. 
 
 sidering him not as an enemy, but as a friend. Of many triangles, therefore, we shall establish one 
 as most beautiful, (neglecting the rest ;) I mean the equilateral, which is composed from three parts 
 of a scalene triangle. To assign the reason of this would, indeed, require a prolix dissertation ; 
 but a pleasant reward will remain for him who, by a diligent investigation, finds this to be the 
 case. We have, therefore, selected two triangles out of many, from which the body of fire and of 
 the other elements is fabricated ; one of which is isosceles, but the other is that which always has 
 its longer side triply greater in power than the shorter. 
 
 " But that which we formerly asserted without sufficient security, it is now necessary more accu- 
 rately to define. For it appeared to us, though improperly, that all these four natures were mutu- 
 ally generated from each other ; but they are in reality generated from the triangles which we have 
 just described — three of them, indeed, from one triangle containing unequal sides ; but the fourth 
 alone is aptly composed from the isosceles triangle. All of them, therefore, are not able, by a dis- 
 solution into each other, to produce from many small things a mighty few, or the contrary. This, 
 indeed, can be effected by three of them ; for, as all the three are naturally generated from one 
 triangle, when the greater parts are dissolved, many small parts are composed from them, receiving 
 figures accommodated to their natures. And, again, when the many small parts, being scattered 
 according to triangles, produce a number of one bulk, they complete one mighty species of a dif- 
 ferent kind. And thus much may suffice concerning their mutual generation." 
 
 " It now remains that we should speak concerning the quality of each of their kinds, and relate 
 from what concurring numbers they were collected together. The first species, indeed, is that 
 which was composed from the fewest triangles, and is the element of that which has its longer side 
 twice the length of the shorter side, which it subtends. But two of these being mutually placed 
 according to the diameter, and this happening thrice, the diameters and the shorter sides passing 
 into the same as into a centre, hence one equilateral triangle is produced from six triangles. But 
 four equilateral triangles being composed according to three plane angles, form one solid angle ; 
 and this the most obtuse of all the plane angles from which it is composed. Hence, from four tri- 
 angles of this kind receiving their completion, the first solid species was constituted, distributive of 
 the whole circumference into equal and similar parts. But the second was formed from the same 
 
76 
 
 APPENDIX. 
 
 triangles, but at the same time constituted according to eight equilateral triangles, which produced 
 one solid angle from four planes ; so that the second body received its completion from the compo- 
 sition of six triangles of this kind. And the third arose from the conjunction of twice sixty ele- 
 ments, and twelve solid angles, each of which having twenty equilateral bases, is contained by five plane 
 equilateral triangles. In this manner, then, the other elements generated these. But the isosceles 
 triangle, being constituted according to four triangles, and collecting the right angles at the centre, 
 and forming one equilateral quadrangle, generated the nature of the fourth element. But six such 
 as these being conjoined, produced eight solid angles, each of which is harmonized together, accord- 
 ing to three plane right angles. Hence the figure of the body thus composed is cubical, obtaining 
 six plane quadrangular equilateral bases. There is also a certain fifth composition, which Divinity 
 employed in the fabrication of the universe, and when he delineated those forms, the contemplation 
 of which may justly lead some one to doubt whether it is proper to assert that the number of worlds 
 is infinite or finite ; — though, indeed, to affirm that there arc infinite worlds, can only be the dogma 
 of one who is ignorant about things in which it is highly proper to be skilful. But it may with 
 much less absurdity be doubted whether there is in reality but one world, or whether there are 
 five. According to our opinion, indeed, which is founded on assimilative reasons, there is but one 
 world ; though some one, regarding in a certain respect other particulars, may be of a different 
 opinion. But it is proper to dismiss any further speculations of this kind." — The Works of Plato, 
 translated by Taylor, Vol. II. p. 526. 
 
 The writer in the Athenmim, already quoted,* thus explains and simplifies what is said 
 in the above quotation regarding triangles : — 
 
 " The two symmetrical triangles of Plato are — 1, that triangle which is half of the square, and, 
 
 2, that which is half of the equilateral triangle." * * * " The first exemplifies the relations 
 of the number 2, and the second Platonic triangle adds to these the properties of the number 
 
 3. " 
 
 * No. 817, p. 586. 
 
Plate W'l. 
 
 ./ com/iositiofi <'/ /{/'/■/</ //////■.*■ /br///f</ n/ion /ho fliai/oaa/s of Jo' ~ 4 .'' ffff* a tut 
 To? three first arc in /he harmonic ra/iox /o the ric/ht aiu//c of A' /o ft '/ to J ana" 
 2 /c .7 and '/? " trwa ///<■ horizontal of ft " if in the ratio of / to i.J. 
 
 The aloor /iW/(//it//t:c i/i/i/rAv/ /////> sea/cat /ciaiu/lc.r aocccalli/ ta //if /la tonic 
 Theory of the elements, and ' surmounted 61/ the horizontal of ft " which. /ias 
 {to angle of />)' o/nl is canseauentla in the harmonic ratio to the right 'an fie 0/ r i 'too. 
 
 -if' 
 
APPENDIX. 
 
 This is precisely the situation in which these two angles occur upon the division of the 
 quadrant by the numbers 2 and 3, namely, 45° and 60°, which have been shown to 
 be in the harmonic ratio to the right angle of 1 to 2 in the first instance, and of 2 to 3 in 
 the second. Those angles were, therefore, adopted by me as the tonic and dominant of 
 my present system, long before I was aware of the high authority by which this funda- 
 mental part of my theory is supported. 
 
 It seems clear, however, that the mediant angle, as well as the tonic and dominant, must 
 have been known to Plato ; for he mentions a triangle " which always has its longest side 
 triply greater in power than the shorter." Now, this triangle is neither the half of the square 
 nor of the equilateral triangle ; but if the quadrant be divided by five, a triangle of this 
 description is the result, and this completes the primaries. 
 
 Fig. 1. Fig. 2. Fig. 3. 
 
 I therefore consider this authority quite conclusive of the fact, that the perfection of the 
 arts in Greece was the result of a system of proportion similar to what I have endeavoured 
 to develop in the foregoing pages. 
 
 All the combinations of the harmonic angles that can be treated of in an essay of this 
 kind, are only such as may be depicted upon the retina — namely, plane polygons. There- 
 fore the combination of the solid triangles, producing the polyhedrons which Plato con- 
 
78 
 
 APPENDIX. 
 
 ceived to form the atomic particles of the four elements 3 can in no way apply to the object 
 of this treatise ; while, on the other hand, it includes numerous plane figures unnoticed in 
 the Platonic Theory of triangles. 
 
 THE END. 
 
 EDINBURGH: PRINTED BY ISALLANTYNE AND HUGHES 
 PAUL'S WORK, CANONGATE. 
 
ERRATA. 
 
 Page 29, last line, ./or second, read secondary. 
 Pages 38 and 48, for harmonies, read harmonics. 
 Page 00, 9th line from bottom, for figure 3, read figur 
 
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