OLIN : NA 281 .H41 1853a All books are subject to recall after two weeks Olin/Kroch Library DATE DUE JSA«-! T59?*" 1 1 c.Lmsfjf9^^r^^ ws "^fei ir^^Ssew^TPB! mmm>s, v m 01 m. d(y ( GAYLORD PRINTED IN U.S.A 3 1924 080 788 387 In compliance with current copyright law, Cornell University Library produced this replacement volume on paper that meets the ANSI Standard Z39.48-1992 to replace the inreparably deteriorated original. 1997 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924080788387 V*. "? L ■r:or (!}0titell Hntttersita Hifaraty 3tlfata, Nem ^ork FROM THE BENNO LOEWY LIBRARY COLLECTED BY BENNO LOEWY 1854-1919 BEQUEATHED TO CORNELL UNIVERSITY ORTHOGEAPHIC BEAUTY PARTHENON. I AM INCLINED TO BELIEVE SOME GENEEAX LAWS OF THE CREATOE PEE- VAILED WITH RESPECT TO THE AGEEBABLE, OE trNPLEASDlG AFFECTIONS OF ALL OUR senses; AT LEAST THE SUPPOSITION DOES NOT DEROGATE FROM THE WISDOM OR POWER OF God, AND SEEMS HIGHLY CONSONANT TO THE SmPLICITT OF THE macrocosm in general. — Sir Isaac Newton. THE OETHOGEAPHIC BEAUTY OF THE PARTHENON REPBRRED TO A LAW OF NATTJEE. TO WHICH ARE PREFIXED A FEW OBSERVATIONS ON THE IMPOETANCE OF AESTHETIC SCIENCE AS AN ELEMENT IN ARCHITECTUEAL EDUCATION. D. E. HAY, r.E.S.E; WILLIAM BLACKWOOD AND SONS, EDINBUEGH AND LONDON. M.DCCC.LIII. /^^■^ '"'i^-m^^r^'^ OLiA) HI TO THE EEVEEEND PHILIP KELLAM), M.A., F.RSS. L. & E. PKOFESSOR OP MATHEMATICS, ETC., IN THE UNIVEKSITT OP EDINBUKGH, AS A VEEY INADEQUATE EXPEESSION OF GEATITUDE JOE MUCH VALUABLE ASSISTANCE EENDEEED ON THE PEESENT AND ON SEVEEAL FOEMEE OCCASIONS, THIS TEEATISE IS DEDICATED WITH THE EESPECTFUL ESTEEM AND SINCEEE EEGAED OF THE AUTHOR. PREFACE. Being anxious that my theory, as applied simply to the orthography of architecture, should be brought before the highest tribunal which this country affords, in order that its truth might be thoroughly tested before it was offered to the pubhc, I took the liberty of sending the substance of the following pages, accompanied by ample illustrations, as a communica- tion to the Royal Institute of British Architects, and it was read at a meeting of that learned body by one of the members, on the 7th February 1853. The residt of this appeal is given at the end of an abstract of my communication printed by the Council for the use of the members, and is in the following words : — " Several members present offered observations on the paper which had been communicated, and certain VUl PREFACE. objections were made to portions of the diagrams ex- hibited ia illustration, and to the system of dividing proportions by particular angles, however convenient in some instances, on which the principle set forth by Mr Hay was based. Testimony was at the same time borne to the value of his labours in pointing out a system round which members might form their own conclusions. It was held to be very difficult to arrive at any complete or general system for the Parthenon — so much so as to induce some investigators to believe that no fixed formula in its strictest sense existed. StiU any person who produced a system with apparently fair grounds of support must be con- sidered as conferring a benefit on the study of Grecian architecture, inasmuch as it would lead to the more complete sifting of the evidence, and might thus serve to establish such points as admitted of proof. " The paper communicated by Mr Hay embraced so many points connected with geometry, mathema- tics, and aesthetics, that it was felt to be impossible to grasp the whole of so elaborate a subject upon merely hearing it read — or to _ arrive at any proper con- clusion upon its contents without mature dehberation. Subjects hke the present were of great value, by bringing the mind to investigate the laws of nature, and to discover, if possible, whether there was really PBEFACE. IX any mathematical law coinciding with the harmony always found in nature, which could be applied in the construction of buildings." Being unavoidably absent from the meeting at which my paper was read, I was prevented from hearing the observations made upon it by the mem- bers, or the objections taken to parts of some of the diagrams alluded to in the above report; and, not- withstanding many attempts to become acquainted with their speciiic nature, I have as yet received no such information as would enable me to enter into details on that part of the subject. As to the general objection made " to the system of dividing proportions by particular angles," I have only to express a hope that every unprejudiced and careful reader of this short treatise will pronounce that objection groundless. That it has been " very difficult to arrive at any complete or general system for the Parthenon," the failure of the innumerable attempts that have been made to that elFect sufficiently proves. I trust, how- ever, that the following pages, with their illustrations, will shew that the difficulty in question has not proved insurmountable, and that those investigators who were induced " to believe that no fixed formula in the strictest sense existed," simply because they X PREFACE. could not discover it, will be induced, on an acquaint- ance with this treatise, to change their opinion upon that point. As to the many points connected with geometry, mathematics, and aesthetics alluded to, I o^vn they are sufficient to render it difficult for any one to grasp the whole subject upon merely hearing it read ; but, at the same time, I feel assured that whoever takes the trouble to examine the following pages and their illustrations with , ordinary care and attention (how- ever sHght his acquaintance Avith mathematics and geometry may be), wiU find the subject of easy com- prehension, and much more simple in its nature than it at ffi'st sight appears. As I am neither a member of the Institute, nor even an architect, I cannot conclude this preface without gratefully acknowledging the kind conde- scension of the Council, in not only allowing my paper to be read, but in circulating the considerate and favourable observations which accompanied the abstract of its contents, quoted above. D. R. HAY. JoKDAif Bauk, April 1853. PART I. lMrORTA.NCE OF ESTHETIC SCIENCE AS AN ELEMENT Of AECHITECTUEAL EDUCATION. It is well known that architecture owes its excellence to power which it possesses of uniting in its productions qualities of fitness, strength, and beauty. As a useful arl demands of its professors, in the first place, a knowledge of various domestic requirements of man, both as a private ir vidual and as a member of civilised society, in order that fitr may characterise their designs — ^from that of the cottage the palace — ^from that of the cotton-mill to the cathedral ; a in the second place, it requires that they should possess a kn( ledge of the various branches of mechanical science, in or that they may be enabled to impart to their works the greai strength and durability of which the material at their disp( is susceptible. But, as a fine art, it likewise requires that its profess should be acquainted with the science of aesthetics, in or that the beauty of their works may be, like the qualities of ness and strength, of a demonstrable and intelligible charac This science of artistic beauty is understood to hold an termediate position between those sciences that are o 12 ORTHOGRAPHIC BEAUTY physical, and those that are of a metaphysical character, par- taking to some extent of both. It exclusively relates to impressions made by external nature upon the human mmd through the organs of the art-senses, as Dr Oken appro- priately denominates those of hearing and seeing; and by its means harmonic proportion is as capable of being taught, m respect to abstract forms, as it is in the art of music with respect to abstract sounds. When we reflect that, in aU civilised nations, the works of the architect, are necessarily placed before the pub- He eye in far greater number and magnitude than those of any other formative artist, and that they thus exercise a powerful influence either to improve or to pervert the perception of the beautiful — either to refine or to degrade the national taste — ^the importance of this science as an element of architectural design must at once be acknowledged, itfot but that men of great artistic genius sometimes appear who are enabled by this rare gift to compose the abstract forms of an architectiiral structure, agreeably to the successive harmony of its outUne, and the com- bined harmony of the relative proportions of its various parts. But the occurrence of such phenomena does very little towards that general improvement of which the architecture of the present day is susceptible; for such men of genius compose the forms which make up their designs by the eye alone, just as the musical amateur, unacquainted with the^ laws of har- mony, would compose an overture by the ear. It is therefore impossible for them to instruct the numerous less gifted men, who devote themselves to the study of this profession, in those simple laws by which they might at least avoid producing deformity, if they had not the genius to create beauty. Could genius be analysed, I beheve it would generally be found to exhibit a mind in which one or more of the sciences develop themselves in a more than ordinary degree. This OP THE PAETHENON. intuitive possession of science called genius, does not, howe include a knowledge of the principles which, constitute science thus developed, and, consequently, does not en those who possess it to instruct mankind generally in its ture, so as to improve their perception, by enabling them to tinguish, in works of Uterature and art, the true from the fall Poetic genius, for instance, exhibits a development of m physical, sesthetical, and physical science; and thus the po( enabled to convey to the minds of other men, in aesthetic constructed, language, the creations of his imagination respect to the workings, of the human feelings and passi and to describe, often very vividly and correctly, var objects in the animal and vegetable creation, as well as general appearance of external nature in those various asp by which the mind of man is often so much affected. Musical genius, on the other hand, exhibits more a deve] ment of eesthetical science than of any other ; because, in absence of song, a musical composition addresses the ear means of the artistic combination of abstract sounds, and 1 ing no definite language, affects the mind by its successive combined harmony alone. It is with the eye as with the ear — the genius which cor tutes the true sculptor and paintei*, like that which constiti the true poet, exhibits a development of metaphysical, sesthi and physical science ; for each in his own way conveys to minds of other men, in aesthetically constructed works, con( tions which treat of the affections and passions of the hui mind, while, at the same time, he confines himself to the ti of nature. The genius which constitutes the true architect, and by wl he is enabled to create beauty, and thereby convey the pi ing feeling of harmony to the mind, exhibits, like that of 14 OETHOGIRAPHIO BEAUTY musical composer, more an intuitive development of sesthetical science than of any other ; because an architectural work ad- dresses the eye, by means of the artistic combination of abstract forms, and, as such a combination cannot express through that organ any idea of vital action or mental emotion, in definite language, it must depend for the effect it is intended to produce on the mind, on the successive and combined harmony of its various parts. . All these fine arts are to some extent imitative. The poet imitates external nature by aesthetic description. The musical composer occasionally imitates natural sounds — from the melodious song of the bird to the discordant noise of the storm ; — ^but his imitations are not addressed to the mind so definitely as those of the poet, because they are introduced into a composition of abstract sounds, necessarily arranged accord- ing to a precise law, and to this law these imitations of natural sounds must also be subjected. Sculpture and painting are more directly imitative in addressing the mind through the eye than either poetry or music are in addressing it through the ear. But stiU their imitations, if too palpable, that is, if not aesthetically treated, cease to be works of fine art. In architecture, imitations of natural objects seem only to be admissible as embellishments to the general structure, and when so used, must be treated as Callimachus treated the acanthus in designing the Corinthian capital, that is, they must be geometrically harmonised. I believe I have now to some extent made it appear that the nature of a genius adapted to infuse beauty into a musical com- position, and a genius adapted to infuse the same quality into an architectural structure, are identical, and that the science which each of those kinds of genius develops is fundamen- tally that of aesthetics. OF THE PARTHENON. In the art of musical composition, the laws of this s have heen long understood and practically applied, the quence of which is, that the public ear has been trained to distinguish harmony from discord. In this art, theref one can become professional without a systematic kno' of those laws ; hence, although devoid of that genius by he might create beauty, he still has that knowledge by he may avoid error in the practice of his art. But in i to the combination of abstract forms in architectural stru there is not as yet any recognised method of practically ap those aesthetic laws, the existence of which all enlightent now acknowledge. The consequence of this necessarily i that numerous class of the profession who are devoid of may, and do, indulge themselves with impunity in the prod of original designs which exhibit all sorts of puerile and cious fancies ; and when they adopt a classical style, o: some ancient structure, it is found that " the soul is w; though some semblance of the external elements, like an mask, may be substituted." This being only apparent of genius, or such as have deeply studied the subje( public eye is perverted by being constantly subjected to fluence of the great exhibition of the works of architects grades, which every modern city affords. On the per of the eye in respect to proportion. Professor Cockerel the following excellent observation — " Custom, conventic often incapacity of discernment, reconcile us to those p tions we are most uSed to, and we are blind to those < which a fresh and accomplished eye is at once shocked s the sense of vision so studied by the Greeks, is to be ed like the moral sense, and every other by the diligen ture of science ;" and farther observes that — " The in! artist recognises the claim which this great element of ; 16 ORTHOGKAPHIC BEAUTY above all others on his studious attention ;" that « he seizes with extreme deUght any rule that will conduct his works to the excellence so apparent and so universally admitted in the Greek proportions— he rejoices in any the slightest elements of their grammar and syntax, by which he can attain to then: eloquent language — and he confesses that, without them, aU is caprice, hazard, and fashion." * It is, therefore, evident that architectural design, generally speaking, must depend for its improvement upon the universal inculcation of a knowledge of such a law of nature as, when appUed to abstract form, would meet a response in that inhe- rent quality of the mind which enables every man, more or less, to appreciate the beautiful. The following contains a brief outline of my system, which will be appUed- to the Parthenon in the Second Part : — The elementarv forms of architecture are, like the elemen- tary sounds in music, few and extremely simple in their nature. We are accustomed in all cases to refer direction to the hori- zontal and vertical lines. When these Hnes meet they make the right angle, which may with safety be assumed as the fundamental angle, from the divisions of which, by the simple numbers 2, 3, o, 7, or multiples of these primes, the beauty of every architectural design is to arise. AVTien two lines thus making a right angle are joined by an oblique line, the three form the right-angled triangle, which is the primary elementary figure of every other figure employed in architec- ture. "When the two angles made by the oblique Une with the vertical and horizontal lines are equal, this triangle may be termed simply the triangle of (i), because the smaller angles are each one-half the right angle. But when the two angles are unequal, the triangle may be named after the smaller of * On the Arcliitectural Works of William of WykehMii. Loudon, 1846. OF THE PARTHENON. these. For instance, when the smaller angle, which we si here suppose to be one- third of the right angle, is made w the vertical line, the triangle may be called the vertical seal triangle of (^), and, when made with the horizontal line, horizontal scalene triangle of (J). As every rectangle is m up of two of these right-angled triangles, the same terminol may also be applied to these figures. Thus the perfect squ is the rectangle of (^), being composed of two similar rig angled triangles of (^) ; and when two vertical scalene trianj of (J) and of similar dimensions are joined by their hypot nuses, they form the vertical rectangle of (g), as in like man two horizontal triangles of (^) united under similar circu stances would form the horizontal rectangle of (J). As isosceles triangle is, in like manner, composed of two rig angled scalene triangles joined by one of their sides, the sa terminology applies equally to every variety of that figure. Each of these rectilinear figures has a curviUnear figure t exclusively belongs to it, and to which may be applied a sim terminology. For instance : — 1st, The circle belongs to the p feet square, and may be termed simply the curvilinear figuri (^) ; because, if two axes be drawn through the centre at ri angles to each other, and the extremities be joined^ they will fc right-angled triangles of (^). 2c?, The ellipse belongs to horizontal or vertical rectangle, and may be termed simply curvilinear figure of (^), (^), &c., for a like reason ; so that vertical or horizontal ellipse of (5), (5), (^), or any other p portional part of the right angle, would be quite intelligib and, 3c?, The composite ellipse which belongs to every isosce triangle, and to which the same terminology may be applie Thus the elementary figures which belong to all the for employed in classical architecture are all ruled by a horizor or vertical right-angled triangle, and are comprised in — 18 ORTHOGRAPHIC BEAUTY The perfect square or equilateral rectangle. The oblong rectangle. The isosceles triangle. The circle. The ellipse, and The composite elhpse.* The law by which these six elementary figures may be hai moniously combined, is a simple and well understood law < nature, and the mode of its application ought to be equal natural and simple. I therefore lay down two positions, tb first of which is, that the eye is influenced in its estimation < spaces by a simplicity of proportion similar to that whic guides the ear in its appreciation of sounds ; for it is reasonabl to conclude, that an element which is so necessary to the satis faction of the one sense, should be also requisite to the com plete gratification of the other. The second is, that the eye i * The latter figure, being new and very little understood, and as I have of la' greatly improved it, I may here state that the composite ellipse is simply figure composed of arcs of various ellipses, harmonically flowing into each othe whose foci are placed on the sides of an inscribed isosceles triangle, and it thi closely resembles the parabohc and hyperbohc curves. But it has what the! curves have not, viz., the essential quality of inscribing harmonically one of tl rectilinear elements of architecture, while the parabola and hyperbok, are mere! curves of motion, which never can harmonically inscribe or resolve themselvi into a figure of any kind. The simplest method of describing it is as follows : Construct an isosceles triangle ABC, Plate I. Fig. 1 ; bisect AB in D ; throng D draw DE at right angles with AB, and through B draw BE, making with A the angle of (i). Take AD and DE as the semi-axes of an ellipse, the foci . which will be at G and F. Pix corresponding foci HI and KL in the hnes A and BC. Fix pins in each of the foci F, G, H, I, L, and K, and, in the poii E where BE and DE intersect one another, fix a seventh pin. Tie a strm flexible but free from elasticity, around the seven pins with a knot that will n( slip ; withdraw the pin from the point E, and the stiing will then necessari lie loosely about the other six points. Introduce a pencil, or any other tracii point, within the circumference of the string, by which, bring it up to ii original tension, retaining it so while you move the point around the six foe The figure thus described will be fomid to be composed of twelve arcs of ellipse thus distributed : The arcs ab, ef, and ik, ai'e those of three ellipses, whose foi OP THE PARTHENON. guided in its estimate by direction rather than by distai just as the ear is guided by number of vibrations rather tl by magnitude ; and I reckon it equally reasonable to conck that the failure of all attempts to discover the operatior harmonic ratio amongst the various parts of the architecti structures of antiquity, has occurred from length, and not dii tion, being made the standard of comparison — ^from the attei to apply simphcity of linear, not of angular, proportion. The basis, then, of my theory is this, that a figure is pie ing to the eye in the same degree as its fundamental anj bear to each other the same proportions that the vibrati bear to one another in a chord of music. IS^ow as the wl science of musical harmony depends upon the simple divig into vrhich a monochord, when in a state of vibratory moti resolves itself by nodes into (^), (^), (^), and (-f), with tl sub-multiples, (^), (^), &c. ; so, in like manner, the wl science of proportion or harmony of form arises from a sim are PI, GL, and HK. The arcs be, de,fg, hi,kl, and ma, are those of six elh whose foci are FH, GK, PL, KI, HL, and GI. The arcs cd,gh, Im, are thos three ellipses whose foci are FG, KL, and HL And upon examination it wi found that the angles of these three kinds of ellipses are to each other ic simple ratio of 3 : 4 : 5, and that the major axis of that whose arc fills the s ef is vertical ; that the major axis of that which fills the space mZ is horizo and that the major axes of the others are in the various degrees of harm obUquity of ( Jg), (/j), (/^, and (/j). I have illustrated the composite ellipsf this example, as it is the most perfectly symmetrical of all others, just as equilateral triangle which it inscribes is the most perfect of isosceles triangle! In further illustration, I shall in this place add one other example of this fi^ Let the triangle ABC, Plate I. Pig. 2, be that of (i) ; make the angle DBE and describe the composite ellipse by the process already explained. The angles which govern this figure, it will be observed, differ from each other, i much as the angle of the triangle is (3) and that which determines the axi the composite elMpse to be inscribed around it (J), so that the first bears to second the harmonic ratio of 4 : 3. By this means the ares ah, ef, and ik rendered almost imperceptible, and the inscribing curve almost touches the an] which gives a more decided character to this figure than is to be found in fijst. In all composite ellipses in which the governing angles dift'er, it is requi to amplify their terminology by naming both angles, and I therefore term the vertical composite ellipse of (i) and (J). To this figure I shall again hav refer in the sequel. 20 OETHOGRAPHIC BEAUTY division of the quadrant of a circle. The highest standard of symmetry, so estimated, is thus deduced from the law that the angles of direction must all bear to some fixed angle relations expressed arithmetically by the smallest natural numbers. 1 am strengthened, by high authority, in the con- viction that no other law can be devised half so simple as this ; that it accords with the other laws of nature, being but another form of that great laio of least effort which pervades and regu- lates the system of the universe ; that the labour it imposes on the eye is simply to measure round a point, and that the mesr^ sures it demands are the easiest and readiest which could pos- sibly be required, viz., to estimate the halves, the thirds, and the other simple proportions of the assigned unit. Also, that no other law could give such repose to the eye, or present its results to the mind, in a form so plain and unmistakable, for it is exhibited under a form, which we know the mind to be capable of judging from, being the identical form under which the ear presents to it an harmonious combination of sounds ; and it certainly is in perfect consonance with the wisdom of the Creator, to find sensations so widely diflferent as these brought back to the same ultimate type. Agreeably to this law of harmonic ratio, every figure in architecture, whether rectilinear or curvilinear, has an angle which regulates its indi- vidual proportions, and at the same timedetermines its propor- tional relation to such figures as enter into combination with it whether that figure represents a projecting or retiring surface. Consequently, the system I adopt is easily apphed in every formative art, and its leading elements are included in the following scale of ratios, corresponding to what would be termed in the art of music a diatonic scale of seven octaves including the flat seventh. To these ratios, therefore, the scien- tific terms employed in musical composition are applied : OP THE PAKTHENON, SCALE OF HARMONIC RATIOS. Tonics. Sui>er- touics. Mediants. Sub- doxaiuants. Dominants. Sub- mediants. Subtonic. s semi r. 1 : 1 8 : 9 4 : 5 3: 4 2 3 3: 5 4 7 8 : 11. 1 : 2 4 : 9 2 : 5 3: 8 3 3 : 10 2 7 4 : 111. 1 : 4 2 : 9 1 : 5 3 : 16 6 3 : 20 7 2 : IV. I : 8 1 : 9 1 : 10 3 : 32 12 3 : 40 14 1 : V. 1:16 1 : 18 1 :20 3: 64 24 3 : 80 28 1 : VI. 1 : 32 1 :36 1 :40 3 : 128 48 3 : 160 56 1 : VII. 1 :64 1 :72 1 :80 3 :256 96 3 :320 112 1 : These harmonic ratios may be applied in the formative a simply by dividing the circumference of the quadrant of a cii into the following parts, and using this divided quadrant a protractor : — (1) (f) ii) (f ) (f) (f ) (f ) (i% II. III. ly. y. yi. (i) (i) (i) {^) ii) (f) (i) (A) a) (f) (i) (^) (f ) (i) (t^) (f ) (n^ (A) (A) (^) (A) (t^ (^) (^) (A) (^) (^ ) (A) (tItt) (^) (^ (xf^. VII. (^) (tV) (^) (^) (^) (^) (t^) (xi Besides these, there are other ratios by which the series n be interpolated, but, being less simple, they are of a less h monic character. But the above will be found quite suffici for general practice, and those who wish to go more dee into the subject will find sufficient details in the Appendix one of my former works.* I^or let it be supposed, either that so numerous an arraj ratios will involve us in anything complicated, or that, w so many before us, the whole is simply matter of interpolati * " The Geometric Beavjty of the Human Figure defined, to which is preHxi System of Esthetic Proportion applicable to Architecture^' &c. 22 ORTHOaRAPHIC BEAIITY whereby, should the first and third fail, we have always a second at hand lying between these, and sufficiently near to both to effect every purpose. In reply to both these charges, I could fearlessly appeal to. my trea,tise on the Human Figure* where it wiU be seen that lines drawn from the two extremities of the figure formed^ with the vertical line, the ^, ^, ^, \, ^, \, and ^th of a right angle, and that these, and these only, deter- mine every point of importance in the outline. And with equal confidence could I appeal to Plate II. of the present treatise, and the description pp. 26, 27, where the angles BAG, BAD, B AE, CGD, or ^, \, I, and ^th of a right angle, determine so vast an amount of the outline, and regulate so harmoniously the proportions of the Parthenon. The theory in both cases is precisely the same. The most beautiful form in nature, and the most beautiful form created by art, rmist be governed by some one law. To one law I have attempted to reduce them. The human figure exhibits, more than any other object in nature, an equally balanced combination of those three qualities which ought to be found in every archi- tectural structure — ^fitness, strength, and beauty. If what I have advanced be correct, the mystery which has hitherto hung over the last of these quaUties has been removed, for it has been proved that the natural principles of the sesthetic deve- lopment of the human figure were but another expression of that great and universal law of nature which, while it o-o- ' to verns the motions and relative distances of those bodies which constitute the universe, also regulates the harmony by which, through the art-senses, the mind is charmed, gratified, and improved. I also shewed that the mode in which this great law operated in regulating the proportions of the human figure was characterised by perfect uniformity * " The Naiural Principles of BeaiUy as developed in the llmnan Figure:' OP THE PARTHENON. and extreme simplicity, inasmuch as, Is^, That on a given ' the human figure is developed as to its principal points entii by lines drawn eithei^ from the extremities of this line, or fi some obvious and determined localities ; 2c?, That the an; which those lines make with the given line are simple subn tiples of some given fundamental angle ; Zd, That the cont may be resolved into a series of ellipses of the same sim angles ; and, 4iA, That in the front geometrical view of figure, these eUipses, hke the lines, are incUned to the f given hne, by angles which are simple submultiples of given fundamental angle. Thus I found in the noblest work of creation, which is u versally acknowledged to exhibit the highest example of perj mechanical science, sustained in vital energy by the vl wonderful chemical processes, a development of perfect g metric harmony, not only in the relative proportions of parts, but likewise in the curvature of its outline. In or( however, to verify the truth of this discovery, I commenc under the direction, and with the valuable assistance, of E fessors Kelland and Goodsir, of the Edinburgh University careful investigation into the exact proportions of those be tiful remains of ancient Greek sculpture, the Yenus of Med and Venus of Melos, and had the satisfaction to find that th statues both agreed perfectly with the law in question. H ing thus successfully appealed for its truth to the noblest w of creation, and to the finest specimens of art in which t work has been imitated, I now turn to what mankind has, si the revival of the arts, unanimously pronounced to be noblest combination of abstract forms which human geni aided by science, has been able to achieve, and which s remains, without exception, the most magnificent ruin in world, namely, the Parthenon of Athens. 24 ORTHOGRAPHIC BEAUTY PART II. THE ORTHOGRAPHIC BEAUTY OP THE PARTHENON. The Parthenon, or temple of the virgin goddess Athene, or Minerva, the protectress of Athens, is situated on the Acro- poUs of that city. This temple was erected in the time of Pericles (ahout b.c. 448). Ictinus and CalUcrates were the architects. " The Parthenon," says the editor of the last edition of Stuart's Athens, " is one of the most highly finished edifices of Greece, and perhaps the largest octa-style temple of antiquity ever completed. It still rears its majestic columns trium- phantly over twenty-two centuries of duration. We may here figure to ourselves the muse of ideal perfection in art yet in scorn looking down on us from this her favoured seat. Other monuments of antiquity, by the magnitude of their dimensions, and the colossal character of their marbles, may exceed the Athenian Temple, which, though clad with mutilations, the sad records of every species of degradation, supernatural vio- lence, and spoUation, yet the majestic relics still rivet and even oppress the mind, by the sole influence of proportion and harmony of execution. Unlike more extensive monuments which the graver can elevate in the imagination beyond the OF THE PAETHENON. 25 ^ower of reality, these surpassing ruins no pen has faithfully described, or pencil adequately depicted. Of all the monu- ments of ancient and modern magnificence which have been ■within our view, the grandeur of this alone surpassed antici- pation, leaving an impression on the mind, similar to, but more profound, than the charms of an harmonious fugue, or of a rapturous effusion of poesy." Such is the great work, by an analysis of the orthography of which I am about to test the truth of my theory of the natural laws of the harmony of form. The harmonic angles by which I was enabled to determine the true proportions of the human figure, as represented in ancient Greek Art — and approached to, because deviated from on both sides, in ordinary nature — I found to be twelve in num- ber. The right angle, being the fundamental tonic (1), was thus divided into — Tonic angles. Dominant angles. Mediant angles. Subtonic angles. Supertonic angle. (1) (i) (t) ( + ) a) (i) (i) i^) (^) (i) i^) (i) Ifow it appears that the same number of angles of an equally harmonic character determines the proportions of the eleva- tion of the principal front, or eastern portico of the Parthenon. The right angle, being the fundamental tonic (1), is divided into — Tonic angles. Dominant angles. Mediant angles. Suttonic angle. Supertonic angles. (1) (i) (i) a) (i) (i) ii) i^) (A) (i) (i) 26 ORTHOGEAPHIC BEAUTY These angles may be used in the formation of a diagram of the elevation in question, agreeably to the following simple formula : — Plate IL Let the line AB, Plate II., represent the length of the base or upper step. From the point A draw three lines making the angles CAB (^), DAB (i), and EAB (i). Through B draw BC, and through A draw AH, making the angles ABC and HAB (1), i.e., the right angle. Bisect AB in F, and draw FG parallel to CB. Through C draw CG-H par- allel to AB ; through G draw GD, making the angle CGD (^) ; through the point D, where the Hnes AD and GD in- tersect one another, draw DK parallel to AB; and through the point E, where the hnes AE and BC intersect one another, draw EL parallel to DK and EZ, making the angle ZEL Through the point m, where the Unes AE and FG intersect one another, draw ^ mp, making the angle P mp (^), draw pq parallel to FG. Make the angles qsr {^), rut (^), txv (^), W {i)> y^z (tV)» 2;A?» {\), and wl^\\). Bisect the lines qr, tv, yz, and lop in M, and draw the lines WjS perpendicular to AB. Draw similar lines on the other side ofFG. These lines represent the axes of the columns. The process by which the columns themselves may be proportioned, is as follows : — Let aJSTb be the given diameter of the column at its base. Through b draw be parallel to WB, and through a draw ac, making the angle acb (^), through c draw cf parallel to EL, and through the point e where ac and MN intersect one another draw ef, making the angle feM {-^), and produce fe until it meets AB in g, and draw go, ad, and f h parallel to ME. Through i draw ifk, making the angle ikM (^). Draw a similar line lok on the other side of MM. Through i draw OF THE PARTHENON. 27 im, making tlje angle miM (^). Draw in and Im perpen- dicular to LE; through m draw XmT parallel to LE; and through 0, where AE and the line MN drawn between w and p intersect one another, draw UoV parallel to LB. Through X draw XT, making the angle TXT {■^), and through R where XT intersects PG-, draw JW parallel to XT, join TZ, and through the point I, where EZ intersects FGr, draw gf parallel to JW. Through d, where AD and ZT^ intersect one another, draw da parallel to BC ; through b, where da intersects DK, draw bO parallel to GrD; through T draw Tic, making the angle Ybd (^). Bisect Yb in e; through e draw eP parallel to D G . Bisect B. S in w; through n draw \n\x, making the angle Enu (J), and through u draw ut parallel to FGr; also through u draw ux, making the angle tux (^), The diagram thus constructed agrees so closely with the relative measurements of the lengths and breadths of the various parts of the portico as to be sufHcient to convince any unprejudiced mind that the harmony of its proportions arises from the angles thus employed. It is true that the inclination of the oblique line of the pediment as constructed is fully half a degree greater than (^), — that is, it is nearly 13° 30', which is (^), instead of being nearly 12° 5V 2Q", which is (^). But this is very easily accounted for by the invisible curve of the base and entablature causing an angle of a less harmonic nature to be adopted, instead of that given in the normal pro- portions; for " it is easy to see that the main proportions would naturally be established in rectilinear geometrical drawing, and that the modifications due to the curvature, &c., would be an after-process."* The whole elevation is thus divided into three distinct por- tions : — 1st, A vertical portion, being a series of vertical curvi- * Penrose, Chap. II. p. 13. 28 ORTHOGRAPHIC BEAUTY Hnear forms arranged between AB and LE ; 2d, A horizontal portion, being an arrangement of rectilinear figures lying between LE and KD ; and, 3c?, An oblique portion, forming an isosceles triangle between KD and the top. It has aheady been explained how the horizontal, vertical, and oblique Unes unite in the formation of the most elementary figure in architectural composition — the right-angled scalene triangle, — ^and it wiU be seen that the lines I have drawn in constructing the diagram have necessarily formed themselves into a series of these triangles. I shall therefore now shew the harmonic ratios which the angles of these triangles bear to the horizontal and to the vertical lines. Elementary- Angles. EaSos. Ratios Fundamental Angle. I. EAB (1) rl:5 with the" 4 : 5 withthe] (1) 11. DAB (i) 1:4 horizon- ■ -3:4 vertical - (1) TTT. CAB (i) [1:3 tal line j 2:3 line ) (1) These are the harmonic ratios which govern the three first grand divisions, and the following are those which govern the details by which the divisions are filled up : — Elementary Angles. Fmp ii) qsr {^) zhw (i) wlp (i) ikM (i) 1 1 1 1 Li I. i09. 6 10 with the 8 vertical 9 line Eatios. 5: 6 9 : 10 with the Fundamental Angle. 7 : 8 horlzon- - (1) 8: 9 tal line 2: 3 - (1) OP THE PARTHENON. 29 II. ZEL (A) 1:16 with the l ri5 : 16 with the ] (1) YXT fe)H 1 : 18- horizon- - il7: 18 vertical >- (1) Mil m 1:10 talline J [ 9: 10 line J (1) vut ( ^ ) ( 1 : 3 with the ) j 2 : (1^^) (1: 2 vert-Une j" | l': 3 with the ' V 2 hor. line i (1) tux (1) III. '- CGD {^) p : 7 with the ^ f 6 : 7 with the ■) 2 vert. Une ) (1) Yhd (i) (1: 2 hor. line ) {l: (1) When we compare the ratios thus evolved from the anlaysis of the orthography of the Parthenon with those evolved from a similar analysis of the proportions of the female figure, as represented in ancient Grreek art, we can understand the meaning of Yitruvius when he says, " The several parts which constitute a temple ought to be subject to the laws of sym- metry ; the principles of which ought to be familiar to all who profess the science of architecture. For no building can possess the attributes of composition in which symmetry and proportion are disregarded, nor unless there exists that perfect conformation of parts which may be observed in a well-formed human being." And in another part, the same ancient author says, "The artists of antiquity must be allowed to have fol- lowed the dictates of a judgment the most rational when, transferring to the works of art principles derived from nature, every part was so regulated as to bear a just proportion to the whole. ISo^, although these principles were universally acted upon, yet they were more particularly attended to in the con- struction of temples and sacred edifices ; the beauties or defects of which were destined to remain as a perpetual testimony of their skill or of their inability. " 30 ORTHOGRAPHIC BEAUTY The following is the result of this comparison: — Human Figure. 1: 2, Parthenon. : 2. 1: 3. 2,: 3 1 : 3. 2: 3 1: 4. 3: 4 1 : 4. 3: 4 1: 5. 4: 5 1 : 6. 4: 5 1: 6. 5: 6 1 : 6. 5: 6 1: 7. 6: 7 1 : 7. 6: 7 1: 8. 7: 8 1 : 8. 7: 8 1: 9. 8: 9 1 : 9. 8: 9 1 : 10. 9: 10 1 : 10. 9: 10 1 : 12. 11 : 12 1 : 16. 15: 16 1: 14. 13: 14 1 : 18. 17: 18 In proceeding to analyse the visible curves of the Parthe- non, I find that they are in perfect agreement with the ele- mentary rectilinear figures which have arisen from the combin- ation of the harmonic angles illustrated in Plate 11., for they are simply the following : — (i) (i) a) a) i^) In addition to the angles in the first scale, the following are now reqtiired to be added in order to complete the har- mony of this portion of the design : — Tonic. Dominant. Mediant. M (A) (*) Sabmediant. Subtonic Semi-subtonic Supertonica. (t^) (tV) (A) a) (i) I shall commence with the curve called the entasis of the column, upon the nature of which so much has already been written, and so many conflicting opinions advanced. OF THE PAETHENON. 31 It appears clear to rae that this ieurve is simply an arc of a composite ellipse of (^) and (^), which is the fifth of a series commencing with that of (^) and (|), each having the harmo- nic rJatio of 3:4 between the angle of the curve and the angle of the inclination of its niajor axis to the vertical line. The particular arc of this figure is not selected empirically, but is determined by certain definite points. In order to exempUfy this on a sufficiently large scale, I shall take the second of the series, a composite ellipse of (^) and {^). Therefore let ABO, Plate III., be a vertical isosceles tri-Plai angle of (^), bisect AB in D, and through D draw D/ perpendi- cular to AB, and through B draw Bf making the angle DBf (^). Taking BD and Df as semi-axes, describe the composite ellipse as already explained whose foci are Gr, F, K, L, H, and I. Through the foci Gr and L draw Ga parallel to B and L b parallel to AB. Join ab, and through f draw fo parallel to ab. The relative proportions of the lower and upper diame- ters of the shaft of the column, and the nature of the curve called its entasis, will now be found between ab and fo. The shafts of the columns of this portico are therefore simply firustra of an elliptic sided, or prolate spheroidal, cone, whose section is a composite ellipse of (-^) and (^), and whose curve is to its inclination in the harmonic ratio of 3 : 4. This Plate exhibits the composite ellipse of (^) and (^), Plai as connected with the rectilinear diagram of the column given in Plate II. This Plate exhibits a section of the shaft of the column, Plai drawn agreeably to the same formula, but on a larger scale. This Plate represents a section of the face of the capital ^ Plal the full size. The curve of the neck is an arc of an ellipse of (^), whose major axis AB is in the vertical line. The curve of 32 ORTHOGRAPHIC BEAUTY the echinus is composed of 3 arcs of an ellipse of {^), whose major axis at HF is incHned {^), at GrD {^), and at XT (^) to the horizontal line. The cavettos of the annulets are like- wise arcs of the same ellipse, whose major axes,* a J, cd, ef, and gh, are inclined (^), and their fillets arranged upon the samf elUpse whose major axis IK is inclined (^) to the vertical line The lines drawn perpendicular to if represent Mr Penrose's offsets. The lines LE and U V are portions of the same lines on Plate II, Plate vn. Fig. 1 of this Plate represents a full-sized section of the moulding under the cymatium of the pediment, consisting oJ the parts termed respectively a cavetto and cyma reversa. The curves of this moulding are arcs of elhpses of (^), the major axis ah of that which forms the cavetto is in the hori- zontal hue, and the major axes cd and cy"of those which form the cyma reversa are inclined (;|) to the horizontal line. Fig. 2 is a full-sized section of the bed moulding of the cornice of the pediment, the curves of which are arcs of an ellipse of (J), whose major axes, gh, ik, and Im are inclined (f ) to the vertical line. Fig. 3 represents the cavetto of the soffit of the corona ^ the full size. Its curve is composed of arcs of two ellipses ; one of (^), whose major axis AB is in the horizontal line, and another of {^, whose major axis CD is also in the horizontal line. The hnes drawn perpendicular ito no are Mr Penrose's offsets. Plate VIII. Fig. 1 of Plate yill. represents a section of the cymatium which surmounts the corona \ the full size. Its curve is simply an arc of an elhpse of (g), whose semiraajor axis A B is in the vertical line, and whose focus C determines the length of the arc. Fig. 2 represents a full-sized section of the corona, with the moulding over it of the capital of the autaj of the posticum. OP THE PARTHENON. 33 The curves of the moulding are arcs of an elHpse of {^), whose major axis at cd is first Inclined (f), and then at ef (f) to the horizontal line. Fig. 3 represents a full-sized section of the lower moulding of the same capital. It is composed of the parts termed an ovola, a fillet, and a bead ; the curve of the first is the arc of an ellipse of (^), whose major axis mu is inclined (^) to the " vertical line. The fillet under this is incUned {^\) to the vertical line ; and the curve of the bead is a semicircle, whose axis is also inclined. {-^) to the vertical line. Plate IX. represents a full-sized section of the moulding ph which lies between the two just described, and completes the capital. It is composed of arcs of an ellipse of (^), "whose major axes gh, ik, Im, is inchned, ist, (f ) to the horizontal line ; 2d, (t^) *o the vertical line ; and, dd, (f) to the vertical line. This Plate represents a full-sized section of the upper pi£ moulding of the band under the beams of the ceiling of the peristyle. Its curves are arcs of ellipses of (^), the major axis of the first op is in the vertical line, and its minor axis qr \s equal to the projection of the whole moulding. The major axis of the other ellipse as placed at st and uv is inclined {\) to the vertical line. This Plate, Fig. 1, represents a full-sized section of the Pia lower moulding of the same band. Its curves are arcs of an ellipse of (^), whose major axis at xy and wz is inclined (^). Fig. 2 represents a full-sized section of the moulding at the bottom of the small step or podium, which extended between the columns of the prpnaos and posticum. It has a semi- circular bead, whose axis a 5 is inclined (^) to the vertical line, and the curves of the lower part are arcs of an ellipse of (-^), and whose major axes at cd and ef are in the horizontal line, and whose foci and centres are in the same vertical line. c 34 ORTHOGRAPHIC BEAUTY Thus have the orthographic proportions and curvilinear details of the principal front of this celebrated edifice been for the first time reduced to a natural principle of beauty. I shall now look into the methods adopted to discover this law respectively, by the earhest and latest investigators. The work of Yitruvius is the only one on the subject of architecture which has come down to us from antiquity. This author is supposed to have flourished about the times of Julius Caesar and Augustus. Although I have already quoted his words as bearing testimony to the fact, that a natural law of proportion was systematically applied in architecture by the ancient Greeks, yet it is quite evident that the proper method of its application was unknown to him, for he tries to account for the analogy between the proportions of the human figure and those of a temple in a manner equally fallacious and un- satisfactory. It, therefore, appears that a practical knowledge of this law did not reach down to the period of Vitruvius, which was about four hundred years posterior to the best period of Greek art. The recovery of this important knowledge has been the aim and occupation of critical inquiry among us from the revival to the present day, a period of nearly four hun- dred years ; but nothing has been elicited that can be called a natural and systematic law of harmonic proportion, a law which wiU, with certainty, produce the fundamental element of the beautiful in architecture, and awaken what Professor CockereU terms " that sense of rhythmical proportion, and that harmony which affects the mind like a mathematical truth, and hke a concord of musical sounds on the ear, is perceived and con- fessed as obvious and unalterable." The latest investigator is Mr Penrose, an architect of London who, while at Athens in 1846, observed some peculiarities in the curvature of the Parthenon, of which, he says, no adequate OF THE PARTHENON. 35 notice had been taken in the works of any former investigator. He, therefore, obtained, through the Society of Dilettanti, the necessary faciUties for a careful examination of this great work of ancient art, the results of which have been since published,* and evince on the part of the author the most careful and minute research ; and I believe no one can examine the con- tents of his splendid book, without being convinced of the accuracy of the various measurements, and the care that must have been bestowed on every detail. ^Notwithstanding this, the conclusions he has arrived at with respect to the SBSthetic developments of the Parthenon appear to me to be far from satisfactory. This, evidently, arises in the first place from the method (which seems to have originated with Vitmvius) of making length and not direction the standard of comparison — from the attempt to find Unear instead of angular harmony. Mr Penrose, like all who have studied the subject with care, seems firmly to believe that a law of harmonic proportion governed the orthography of this structure ; and in regard to that of the principal front, he has anxiously searched for its development in the ratios which exist among the lengths and breadths of the various parts of which it is composed, and all the proofs of the existence of harmony which he has been able to produce by these means consist of the following facts : — 1st, That the entire height to the top of the cymatium of the pediment is to its breadth in the ratio of 7 : 12. 2d, That the height of the column is to the entire height in the same ratio. M, That, if the height of the pediment had been 12*556 ft. instead of 12-643 ft., it would have been to the length of the horizontal cornice in the ratio of 6 : 25. * An Investigation of the Principles of Athenian Architecture, &c., by Prancis Cranmer Penrose, Architect, M,A., &o. London: Longman & Co. and John Murray. 1851. 36 OETHOGRAPHIG BEAUTY ^th, That, if the length of the architrave had befell 1- 126 ft. less than the upper step, instead of l-llO ft., which it mea- sures, the length of the architrave to that of the upper step would have been in the ratio of 89 : 90. 5ih, That .the diminution of the columns is to their height in the ratio of 1 : 50. This is a most unsatisfactory result; for the ratios thus found are only four in number, and are not exactly harmonic in their nature; whereas, there are eighteen ratios evolved by the angular system, all of which are truly harmonic in their nature, and are to be found in the scale of the leading ele- ments of harmony, p. 21. These I shall place in juxtaposition with the above four :■ — Evolvei by the Angular System, and arranged in the order of their simplicity. Evolved by Lineal Measurement. 1 :2 3 : 4 5: 6 7 : 12 1 :3 1: 7 6: 7 6:25 1 :4 1 : 8 1 : 16 89:90 2:3 4: 5 8: 9 1 : 50 1 :5 1 : 9 1 : 18 1 :6 1 : 10 9: 10 The comparison of these two results shews that the univer- sally acknowledged harmony which pervades the orthography of the Parthenon is found in the angles, and not in the lengths and breadths of its parts, where they have been hitherto searched for ; and the conclusion to which this fact necessarily leads is, that the beauty of proportion could only have been imparted to that " most harmonious structure " by the system I have pointed out. Doubtless, in the course of erecting any great structure, upon whatever system of proportion the orthography may have OF THE PARTHENON. ' 37 been arranged; many accidental circumstances must occur to occasion deviations from exact symmetry, but wbicb, if sligbt, do not at all injure tbe general effect; and such appears to have been the case in the building of the Parthenon. For instance, if we take the arrangement erf the columns and of the metopes of the principal front, according to Mr Penrose's measurements, we find the following deviations : — I. II. III. Feet. ' The lineal distance between the axes of the first and second columns on the north side of the centre, being .... ll'OSi and that between the axes of the first and second columns on the south side of the centre, being 14*106 there is a deviation from perfect symmetry of ....... -022 The Uneal distance, between the axes of the second and third columns on the north side of the centre, being .... 14-115 and that between the axes of the second and third columns on the south side of the centre, being 14*078 there is a deviation from perfect symmetry of -037 The lineal distance between the axes of the third and fourth columns on the north side of the centre, being . . • ..,12*041 and that between the axes of the thud and fourth columns on the south side of the centre, being 12*147 there is a deviation from perfect symmetry of -106 38 ORTHOGRAPHIC BEAUTY In like manner the corresponding metopes on each side of the centre differ, as follows : — TThe breadths of the two first are respectively I. s and ...... I shewing a deviation from sjrmmetry of {Those of the two second are respectively and ...... shewing a deviation from symmetry of Those of the two third are respectively III. < and shewing a deviation from symmetry of ( Those of the two fourth are respectively ly. ^ and ...... I shewing a deviation from symmetry of {Those of the two fifth are respectively and ...... shewing a deviation from symmetry of ( Those of the two sixth are respectively VI. < and L shewing a deviation from symmetry of ( Those of the two seventh are respectively VII. i and I shewing a deviation from symmetry of Feet. 4-320 4-375 •055 4-295 4-169 •126 4-281 4-186 •095 4-050 4-195 -145 4-064 4-192 •128 4-066 4-120 -054 4-160 4-121 •039 It could not, therefore, have been by a fastidious attention to the minutiae of lineal measurements that the beauty of harmonic proportion was imparted to the orthography of the Parthenon, as Mr Penrose's investigations would lead us to suppose, but to the application of that universal law of nature which governs all visible beauty. And I may add, that as nature seldom, if ever, produces a human being so perfect in OF THE PARTHENON. 39 symmetrical beauty, that a careful and minute scrutiny vdU not detect in it some deviation from a perfect development of this law, but that this fact neither causes us to doubt the ex- istence of the law, nor is sufficient to destroy our satisfaction in the work ; so we are not bound down to absolute perfection in an artistic combination of abstract forms, however strongly we may feel convinced that the structure owes its harmonic beauty and grandeur to the application of a definite law in its original design. With reference to this point. Professor Donaldson, amongst his excellent Architectural Maxims and Theorems* says : — '' If a difference of size in parts or details exist in a building, and be not discoverable unless one takes a compass, a rule, a square, or a level, they are no longer defects. Buildings are not made for measurement, but dehght. When these dis- crepancies escape the eye, and minute examination can alone detect the irregularity, the beauty of the whole does not cease to afiect us. In the Pantheon at Eome, the angular columns are anti-Vitruvian, being smaller, instead of larger, than the others. At the top of the pediment there is a double modillion ; and on one side of the sloping cornice, twenty-four modillions, on the other, twenty-two. ' Bravo,' says Milizia, ' to him who has counted them ; but piit hravo to him who turns up his nose at such microscopic criticism.' " With respect to the nature of the visible curves employed in the Doric order of architecture, Vitruvius says nothing definitely ; and since his time many conflicting opinions have been advanced by those who have written upon the subject. * Architectural Maxims and Theorems, &c. By Thomas Leverton Donaldson, M.E..I.B-A.., Professor of Architecture in the University College, London, &c. Lon- don : J. Weale. I OETHOGEAPHIC BEAUTY • Penrose says ithat the entases of the columns are hyper- lic curves ; that the soffit of the corona of the pediment is !urve of the same kind ; and that the echinus of the capital jomposed of two different hyperbolic curves and one circular rve. In order to prove this, he gives offsets, carefully mea- :ed from a stretched wire, along with the iUustratioris of )se parts ; and this mode of proof would, at first sight, seem iclusive, but it can only be so in the absence of a knowledge the composite ellipse, and of the various other modes in lich ellipses may be combined; for an acquaintance with 3se will shew that arcs of the composite, mixed, or inchned ipse resemble so closely those of the hyperbola and para- la, that the one may very easily be mistaken for the other, proof of this, I have applied to two of my elliptic istrations (Plates VI. and YII.) the offsets by which Mr mrose endeavours to prove the hyperbolic nature of their rves, and they seem to agree as well with the one as with 3 other. The fact appears to be, that the nature even of 3 regular ellipse is little known, and seems to form no part architectural education at the present period. Mr Penrose nself says, " By whatever means an ellipse is to be con- •ucted mechanically, it is a work of time (if not of absolute ficulty) so to arrange the foci, &c., as to produce an ellipse any exact length and breadth which may be desired." This wever, is far from being the case ; for any one acquainted th the nature of this beautiful curvilinear figure can in one nute arrange the foci of any one of its almost endless rieties by the following simple process: — ^Let ABC be e length of the ellipse, and DBE its breadth. Take AB ion the compasses, and place the point of one leg upon E, d the point of the other upon the line AB — it will meet at F, which is one focus ; keeping the point of the one leg OP THE PARTHENON. 41 upon E, remove the.point of the other to the line BO, and it will meet it at Gr, which is the other focus.* D -ii -X — 'C E In the absence, therefore, of this knowledge of the nature of the regular ellipse, it is not to be wondered at that those curves which are evidently derived from its harmonic com- bination should be mistaken for those of a hyperbolic or para- bolic kind, which they sometimes so closely resemble. Mr Penrose says, that the Greek architects of the Periclean period, from this assumed difficulty, were forced, in forming the flutes of columns, to adopt an approximate ellipse (made up of arcs of circles), instead of a true one ; but this is a mistake, for, having obtained a correct mould or templet from a cast of the upper part of one of the columns of the Parthe- non I found the curve of the flute to be two arcs of an ellipse of (i), of which I*fO, OP, QE, and ST, Plate XII., are semi-^^' axes and u, v, x, y foci; It is surely much more reasonable to suppose that the Greek architects of that period were thoroughly acquainted with the nature of the elUpse, the simplest method of mechanically fixing its foci and describing * By a very simple machine, which I have lately invented, an ellipse of any given proportions, and of any size from half an inch to fifty feet or upwards, can be easily and correctly described — the given length and angle of the intended ellipse being all that is required to be known. 42 ORTHOGRAPHIC BEAUTY its circumference, than to suppose that, for want of a know- ledge so easily acquired, they were compelled to imitate elliptic curves by combining arcs of circles of various diame- ters. And it is equally reasonable to believe that they were sufficiently acquainted with the best modes of harmoniously composing arcs of this simple curve, in the production of new and beautifiil forms, to prevent the necessity of their adopting the more complex and less manageable mixtures of arcs of parabolas and hyperbolas, even if these curves were known at the period, which is doubtful. But I beg here to state, that in thus shewing Mr Penrose to be unacquainted with the nature and use of the elliptic curve, and that he was thereby induced to adopt curves which have no connexion with the elementary forms of architecture, I have no other motive than the support of truth, and trust my doing so can have no tendency in the slightest degree to lower the estimation in which that gentleman's very laborious and highly valuable Investigation of the Principles of Athe- nian Architecture must be held by aU who take an interest in the subject. In fact, I quoted Mr Penrose's words as those of a highly educated architect, in order to prove the fact, that the nature of the ellipse forms no part of architectural educa- tion at the present period. Indeed, it would appear that a knowledge of the nature of this simple and beautiful figure, and of its value in architectm-al orthography, formed no ele- ment either in the education of the Eoman architects, or in that of those of the middle ages, for the curve of the ellipse is equally ignored in the comparatively crude works of the former, and in the semi-barbarous productions of the latter. In conclusion, I beg to offer a few remarks upon that quality which constitutes the orthographic beauty of all architecture. OP THE PARTHENON. 43 viz. harmony— a quality, the existence of which is more or less felt by all, and thp feeling often eloquently acknowledged by the best writers on the subject. I have already quoted the words of Professor Cockerell and of Mr Kinnard in proof of the existence of harmony in the orthographic beauty of archi- tecture ; the first of these gentlemen observing, that true proportion is, like a concord of musical sounds on the ear, conceived and confessed by the eye to be obvious and un- alterable ; and the second assuring us, that when he first viewed the Parthenon its proportions left an impression on his mind similar to the charm of an harmonious fugue. In addition to these, I may add the words of Professor T. L. Donaldson, who, in his Architectural Maxims and Theorems, observes, " Who shall say that Greek architecture is poor, because its mouldings are few? Seven musical notes have sufficed for thousands of exquisite melodies, and each nation has its own peculiar airs. . . . Beauty consists in the complete- ness and harmonious relation of all the parts and properties of any object." Mr Hosking, the author of the article " Architecture" in the EncyclopcBdia Britannica, truly says, that " simphcity and har- mony are the elements of beauty in architecture ;" and Mr Penrose terms the Parthenon a " most harmonious structure." When, however, we inquire into the true nature of harmony, we find that it is not of a vague, indefinite nature, but of a specific character, and founded on a natural law, discovered by Pythagoras, m respect to sound, and since acknowledged as a scientific fact. This discovery proved that the mind is affected by the successive or combined relations of various sounds according to the numerical ratios of the pulsations or vibratory motions which they individually occasion in the surrounding atmosphere, and that these affections are most 44 ORTHOGEAPHIC BEAUTY OF THE PARTHENON. pleasing when their numerical relations are the most simple. Harmony being thus the expression of a law of nature, we cannot impart it to thfe form of any visible object, unless we find some means of actually applying that law in the construc- tion of such an object. l>reither can we feel certain of the existence of harmony in the construction of a work of art, unless we can also trace that quality specifically to the opera- tion of the same law. For the eye does not so readily convey to the mind those harmonic impressions made upon it by external objects, as the ear does in respect to sounds. That harmony exists in respect to the efiects produced by colour has been systematically proved by various authors, from Sir Isaac Newton downwards. But with respect to its existence in form, no system has as yet been established such as would enable the architect systematically to impart it to his works. In regard to my present attempt to supply this desideratum, I have only to request, that it may be carefully examined, and to assure my readers, in the words of a great philosopher of antiquity, quoted in a former work, that " if any one can prove that he has found any method yet better and more suitable, he shall be treated, not as an enemy, but as a friend, and his ideas shall prevail." THE END. liiLlABTVKl!, ntlMTKll, KBIKEIIKGH. e f HI. w. T. YL. Pi X. jSifc.y XL.