THE SCIENCE OF BEAUTY AS APPLIED TO FORMS.
It is justly remarked, in the “Illustrated Record of the New York Exhibition of 1853,” that “it is a question worthy of consideration how far the mediocrity of the present day is attributable to an overweening reliance on natural powers and a neglect of the lights of science;” and there is expressed a thorough conviction of the fact that, besides the evils of the copying system, “much genius is now wasted in the acquirement of rudimentary knowledge in the slow school of practical experiment, and that the excellence of the ancient Greek school of design arose from a thoroughly digested canon of form, and the use of geometrical formulas, which make the works even of the second and third-rate genius of that period the wonder and admiration of the present day.”
That such a canon of form, and that the use of such geometrical formula, entered into the education, and thereby facilitated the practice of ancient Greek art, I have in a former work expressed my firm belief, which is founded on the remarkable fact, that for a period of nearly three centuries, and throughout a whole country politically divided into states often at war with each other, works of sculpture, architecture, and ornamental design were executed, which surpass in symmetrical beauty any works of the kind produced during the two thousand years that have since elapsed. So decided is this superiority, that the artistic remains of the extraordinary period I alluded to are, in all civilised nations, still held to be the most perfect specimens of formative art in the world; and even when so fragmentary as to be denuded of everything that can convey an idea of expression, they still excite admiration and wonder by the purity of their geometric beauty. And so universal was this excellence, that it seems to have characterised every production of formative art, however humble the use to which it was applied.
The common supposition, that this excellence was the result of an extraordinary amount of genius existing among the Greek people during that particular period, is not consistent with what we know of the progress of mankind in any other direction, and is, in the present state of art, calculated to retard its progress, inasmuch as such an idea would suggest that, instead of making any exertion to arrive at a like general excellence, the world must wait for it until a similar supposed psychological phenomenon shall occur.
But history tends to prove that the long period of universal artistic excellence throughout Greece could only be the result of an early inculcation of some well-digested system of correct elementary principles, by which the ordinary amount of genius allotted to mankind in every age was properly nurtured and cultivated; and by which, also, a correct knowledge and appreciation of art were disseminated amongst the people generally. Indeed, Müller, in his “Ancient Art and its Remains,” shews clearly that some certain fixed principles, constituting a science of proportions, were known in Greece, and that they formed the basis of all artists’ education and practice during the period referred to; also, that art began to decline, and its brightest period to close, as this science fell into disuse, and the Greek artists, instead of working for an enlightened community, who understood the nature of the principles which guided them, were called upon to gratify the impatient whims of pampered and tyrannical rulers.
By being instructed in this science of proportion, the Greek artists were enabled to impart to their representations of the human figure a mathematically correct species of symmetrical beauty; whether accompanying the slender and delicately undulated form of the Venus,—its opposite, the massive and powerful mould of the Hercules,—or the characteristic representation of any other deity in the heathen mythology. And this seems to have been done with equal ease in the minute figure cut on a precious gem, and in the most colossal statue. The same instruction likewise enabled the architects of Greece to institute those varieties of proportions in structure called the Classical Orders of Architecture; which are so perfect that, since the science which gave them birth has been buried in oblivion, classical architecture has been little more than an imitative art; for all who have since written upon the subject, from Vitruvius downwards, have arrived at nothing, in so far as the great elementary principles in question are concerned, beyond the most vague and unsatisfactory conjectures. For a more clear understanding of the nature of this application of the Pythagorean law of number to the harmony of form, it will be requisite to repeat the fact, that modern science has shewn that the cause of the impression, produced by external nature upon the sensorium, called light, may be traced to a molecular or ethereal action. This action is excited naturally by the sun, artificially by the combustion of various substances, and sometimes physically within the eye. Like the atmospheric pulsations which produce sound, the action which produces light is capable, within a limited sphere, of being reflected from some bodies and transmitted through others; and by this reflection and transmission the visible nature of forms and figures is communicated to the sensorium. The eye is the medium of this communication; and its structural beauty, and perfect adaptation to the purpose of conveying this action, must, like those of the ear, be left to the anatomist fully to describe. It is here only necessary to remark, that the optic nerve, like the auditory nerve, ends in a carefully protected fluid, which is the last of the media interposed between this peculiarly subtle action and the nerve upon which it impresses the presence of the object from which it is reflected or through which it is transmitted, and the nature of such object made perceptible to the mind. The eye and the ear are thus, in one essential point, similar in their physiology, relatively to the means provided for receiving impressions from external nature; it is, therefore, but reasonable to believe that the eye is capable of appreciating the exact subdivision of spaces, just as the ear is capable of appreciating the exact subdivision of intervals of time; so that the division of space into exact numbers of equal parts will æsthetically affect the mind through the medium of the eye.
We assume, therefore, that the standard of symmetry, so estimated, is deduced from the simplest law that could have been conceived—the law that the angles of direction must all bear to some fixed angle the same simple relations which the different notes in a chord of music bear to the fundamental note; that is, relations expressed arithmetically by the smallest natural numbers. Thus the eye, being guided in its estimate by direction rather than by distance, just as the ear is guided by number of vibrations rather than by magnitude, both it and the ear convey simplicity and harmony to the mind without effort, and the mind with equal facility receives and appreciates them.
On the Rectilinear Forms and Proportions of Architecture.
As we are accustomed in all cases to refer direction to the horizontal and vertical lines, and as the meeting of these lines makes the right angle, it naturally constitutes the fundamental angle, by the harmonic division of which a system of proportion may be established, and the theory of symmetrical beauty, like that of music, rendered susceptible of exact reasoning.
Let therefore the right angle be the fundamental angle, and let it be divided upon the quadrant of a circle into the harmonic parts already explained, thus:—
| Right Angle. | Supertonic Angles. | Mediant Angles. | Subdominant Angles. | Dominant Angles. | Submediant Angles. | Subtonic Angles. | Semi-subtonic Angles. | Tonic Angles. | |
|---|---|---|---|---|---|---|---|---|---|
| I. | (1) | (⁸⁄₉) | (⁴⁄₅) | (³⁄₄) | (²⁄₃) | (³⁄₅) | (⁴⁄₇) | (⁸⁄₁₅) | (¹⁄₂) |
| II. | (¹⁄₂) | (⁴⁄₉) | (²⁄₅) | (³⁄₈) | (¹⁄₃) | (³⁄₁₀) | (²⁄₇) | (⁴⁄₁₅) | (¹⁄₄) |
| III. | (¹⁄₄) | (²⁄₉) | (¹⁄₅) | (³⁄₁₆) | (¹⁄₆) | (³⁄₂₀) | (¹⁄₇) | (²⁄₁₅) | (¹⁄₈) |
| IV. | (¹⁄₈) | (¹⁄₉) | (¹⁄₁₀) | (³⁄₃₂) | (¹⁄₁₂) | (³⁄₄₀) | (¹⁄₁₄) | (¹⁄₁₅) | (¹⁄₁₆) |
In order that the analogy may be kept in view, I have given to the parts of each of these four scales the appropriate nomenclature of the notes which form the diatonic scale in music.
When a right angled triangle is constructed so that its two smallest angles are equal, I term it simply the triangle of (¹⁄₂), because the smaller angles are each one-half of the right angle. But when the two angles are unequal, the triangle may be named after the smallest. For instance, when the smaller angle, which we shall here suppose to be one-third of the right angle, is made with the vertical line, the triangle may be called the vertical scalene triangle of (¹⁄₃); and when made with the horizontal line, the horizontal scalene triangle of (¹⁄₃). As every rectangle is made up of two of these right angled triangles, the same terminology may also be applied to these figures. Thus, the equilateral rectangle or perfect square is simply the rectangle of (¹⁄₂), being composed of two similar right angled triangles of (¹⁄₂); and when two vertical scalene triangles of (¹⁄₃), and of similar dimensions, are united by their hypothenuses, they form the vertical rectangle of (¹⁄₃), and in like manner the horizontal triangles of (¹⁄₃) similarly united would form the horizontal rectangle of (¹⁄₃). As the isosceles triangle is in like manner composed of two right angled scalene triangles joined by one of their sides, the same terminology may be applied to every variety of that figure. All the angles of the first of the above scales, except that of (¹⁄₂), give rectangles whose longest sides are in the horizontal line, while the other three give rectangles whose longest sides are in the vertical line. I have illustrated in [Plate I.] the manner in which this harmonic law acts upon these elementary rectilinear figures by constructing a series agreeably to the angles of scales II., III., IV. Throughout this series a b c is the primary scalene triangle, of which the rectangle a b c e is composed; d c e the vertical isosceles triangle; and when the plate is turned, d e a the horizontal isosceles triangle, both of which are composed of the same primary scalene triangle.
Thus the most simple elements of symmetry in rectilinear forms are the three following figures:—
- The equilateral rectangle or perfect square,
- The oblong rectangle, and
- The isosceles triangle.
It has been shewn that in harmonic combinations of musical sounds, the æsthetic feeling produced by their agreement depends upon the relation they bear to each other with reference to the number of pulsations produced in a given time by the fundamental note of the scale to which they belong; and that the more simply they relate to each other in this way the more perfect the harmony, as in the common chord of the first scale, the relations of whose parts are in the simple ratios of 2:1, 3:2, and 5:4. It is equally consistent with this law, that when applied to form in the composition of an assortment of figures of any kind, their respective proportions should bear a very simple ratio to each other in order that a definite and pleasing harmony may be produced amongst the various parts. Now, this is as effectually done by forming them upon the harmonic divisions of the right angle as musical harmony is produced by sounds resulting from harmonic divisions of a vibratory body.
Having in previous works[7] given the requisite illustrations of this fact in full detail, I shall here confine myself to the most simple kind, taking for my first example one of the finest specimens of classical architecture in the world—the front portico of the Parthenon of Athens.
The angles which govern the proportions of this beautiful elevation are the following harmonic parts of the right angle—
| Tonic Angles. | Dominant Angles. | Mediant Angles. | Subtonic Angle. | Supertonic Angles. |
|---|---|---|---|---|
| (¹⁄₂) | (¹⁄₃) | (¹⁄₅) | (¹⁄₇) | (¹⁄₉) |
| (¹⁄₄) | (¹⁄₆) | (¹⁄₁₀) | (¹⁄₁₈) | |
| (¹⁄₈) | ||||
| (¹⁄₁₆) |
In [Plate II.] I give a diagram of its rectilinear orthography, which is simply constructed by lines drawn, either horizontally, vertically, or obliquely, which latter make with either of the former lines one or other of the harmonic angles in the above series. For example, the horizontal line AB represents the length of the base or surface of the upper step of the substructure of the building. The line AE, which makes an angle of (¹⁄₅) with the horizontal, determines the height of the colonnade. The line AD, which makes an angle of (¹⁄₄) with the horizontal, determines the height of the portico, exclusive of the pediment. The line AC, which makes an angle of (¹⁄₃) with the horizontal, determines the height of the portico, including the pediment. The line GD, which makes an angle of (¹⁄₇) with the horizontal, determines the form of the pediment. The lines EZ and LY, which respectively make angles of (¹⁄₁₆) and (¹⁄₁₈) with the horizontal, determine the breadth of the architrave, frieze, and cornice. The line v n u, which makes an angle of (¹⁄₃) with the vertical, determines the breadth of the triglyphs. The line t d, which makes an angle of (¹⁄₂), determines the breadth of the metops. The lines c b r f, and a i, which make each an angle of (¹⁄₆) with the vertical, determine the width of the five centre intercolumniations. The line z k, which makes an angle of (¹⁄₈) with the vertical, determines the width of the two remaining intercolumniations. The lines c s, q x, and y h, each of which makes an angle of (¹⁄₁₀) with the vertical, determine the diameters of the three columns on each side of the centre. The line w l, which makes an angle of (¹⁄₉) with the vertical, determines the diameter of the two remaining or corner columns.
In all this, the length and breadth of the parts are determined by horizontal and vertical lines, which are necessarily at right angles with each other, and the position of which are determined by one or other of the lines making the harmonic angles above enumerated.
Now, the lengths and breadths thus so simply determined by these few angles, have been proved to be correct by their agreement with the most careful measurements which could possibly be made of this exquisite specimen of formative art. These measurements were obtained by the “Society of Dilettanti,” London, who, expressly for that purpose, sent Mr F. C. Penrose, a highly educated architect, to Athens, where he remained for about five months, engaged in the execution of this interesting commission, the results of which are now published in a magnificent volume by the Society.[8] The agreement was so striking, that Mr Penrose has been publicly thanked by an eminent man of science for bearing testimony to the truth of my theory, who in doing so observes, “The dimensions which he (Mr Penrose) gives are to me the surest verification of the theory I could have desired. The minute discrepancies form that very element of practical incertitude, both as to execution and direct measurement, which always prevails in materialising a mathematical calculation made under such conditions.”[9]
Although the measurements taken by Mr Penrose are undeniably correct, as all who examine the great work just referred to must acknowledge, and although they have afforded me the best possible means of testing the accuracy of my theory as applied to the Parthenon, yet the ideas of Mr Penrose as to the principles they evolve are founded upon the fallacious doctrine which has so long prevailed, and still prevails, in the æsthetics of architecture, viz., that harmony may be imparted by ratios between the lengths and breadths of parts.
I have taken for my second example an elevation which, although of smaller dimensions, is no less celebrated for the beauty of its proportions than the Parthenon itself, viz., the front portico of the temple of Theseus, which has also been measured by Mr Penrose.
The angles which govern the proportions of this elevation are the following harmonic parts of the right angle:—
| Tonic Angles. | Dominant Angles. | Mediant Angles. |
|---|---|---|
| (¹⁄₂) | (¹⁄₃) | (²⁄₅) |
| (¹⁄₄) | (¹⁄₆) | (¹⁄₅) |
| (¹⁄₁₂) |
A diagram of the rectilinear orthography of this portico is given in [Plate III.] Its construction is similar to that of the Parthenon in respect to the harmonic parts of the right angle, and I have therefore only to observe, that the line A E makes an angle of (¹⁄₄); the line A D an angle of (¹⁄₃); the line A C an angle of (²⁄₅); the line G D an angle of (¹⁄₆); and the lines E Z and L Y angles of (¹⁄₁₂) with the horizontal.
As to the colonnade or vertical part, the line a b, which determines the three middle intercolumniations, makes an angle of (¹⁄₅); the line c d, which determines the two outer intercolumniations, makes an angle of (¹⁄₆); and the line e f, which determines the lesser diameter of the columns, makes an angle of (¹⁄₁₂) with the vertical. I need give no further details here, as my intention is to shew the simplicity of the method by which this theory may be reduced to practice, and because I have given in my other works ample details, in full illustration of the orthography of these two structures, especially the first.[10]
The foregoing examples being both horizontal rectangular compositions, the proportions of their principal parts have necessarily been determined by lines drawn from the extremities of the base, making angles with the horizontal line, and forming thereby the diagonals of the various rectangles into which, in their leading features, they are necessarily resolved. But the example I am now about to give is of another character, being a vertical pyramidal composition, and consequently the proportions of its principal parts are determined by the angles which the oblique lines make with the vertical line representing the height of the elevation, and forming a series of isosceles triangles; for the isosceles triangle is the type of all pyramidal composition.
This third example is the east end of Lincoln Cathedral, a Gothic structure, which is acknowledged to be one of the finest specimens of that style of architecture existing in this country.
The angles which govern the proportions of this elevation are the following harmonic parts of the right angle:—
| Tonic. | Dominant. | Mediant. | Subtonic. | Supertonic. |
|---|---|---|---|---|
| (¹⁄₂) | (¹⁄₃) | (¹⁄₅) | (¹⁄₇) | (²⁄₉) |
| (¹⁄₄) | (¹⁄₆) | (¹⁄₁₀) | (¹⁄₉) | |
| (¹⁄₁₂) |
In [Plate IV.] I give a diagram of the vertical, horizontal, and oblique lines, which compose the orthography of this beautiful elevation.
The line A B represents the full height of this structure. The line A C, which makes an angle of (²⁄₉) with the vertical, determines the width of the design, the tops of the aisle windows, and the bases of the pediments on the inner buttresses; A G, (¹⁄₅) with the vertical, that of the outer buttress; A F, (¹⁄₉) with the vertical, that of the space between the outer and inner buttresses and the width of the great centre window; and A E, (¹⁄₁₂) with vertical, that of both the inner buttresses and the space between these. A H, which makes (¹⁄₄) with the vertical, determines the form of the pediment of the centre, and the full height of the base and surbase. A I, which makes (¹⁄₃) with the vertical, determines the form of the pediment of the smaller gables, the base of the pediment on the outer buttress, the base of the ornamental recess between the outer and inner buttresses, the spring of the arch of the centre window, the tops of the pediments on the inner buttresses, and the spring of the arch of the upper window. A K, which makes (¹⁄₂), determines the height of the outer buttress; and A Z, which makes (¹⁄₆) with the horizontal, determines that of the inner buttresses. For the reasons already given, I need not here go into further detail.[11] It is, however, worthy of remark in this place, that notwithstanding the great difference which exists between the style of composition in this Gothic design, and in that of the east end of the Parthenon, the harmonic elements upon which the orthographic beauty of the one depends, are almost identical with those of the other.
On the Curvilinear Forms and Proportions of Architecture.
Each regular rectilinear figure has a curvilinear figure that exclusively belongs to it, and to which may be applied a corresponding terminology. For instance, the circle belongs to the equilateral rectangle; that is, the rectangle of (¹⁄₂), an ellipse to every other rectangle, and a composite ellipse to every isosceles triangle. Thus the most simple elements of beauty in the curvilinear forms of architectural design are the following three figures:—
- The circle,
- The ellipse, and
- The composite ellipse.
I find it necessary in this place to go into some details regarding the specific character of the two latter figures, because the proper mode of describing these beautiful curves, and their high value in the practice of the architectural draughtsman and ornamental designer, seem as yet unknown. In proof of this assertion, I must again refer to Mr Penrose’s great work published by the “Society of Dilettanti.” At page 52 of that work it is observed, that “by whatever means an ellipse is to be constructed mechanically, it is a work of time (if not of absolute difficulty) so to arrange the foci, &c., as to produce an ellipse of any exact length and breadth which may be desired.” Now, this is far from being the case, for the method of arranging the foci of an ellipse of any given length and breadth is extremely simple, being as follows:—
Let A B C (figure 1) be the length, and D B E the breadth of the desired ellipse.
Fig. 1.
Take A B upon the compasses, and place the point of one leg upon E and the point of the other upon the line A B, it will meet it at F, which is one focus: keeping the point of the one leg upon E, remove the point of the other to the line B C, and it will meet it at G, which is the other focus. But, when the proportions of an ellipse are to be imparted by means of one of the harmonic angles, suppose the angle of (¹⁄₃), then the following is the process:—
Let A B C (figure 2) represent the length of the intended ellipse. Through B draw B e indefinitely, at right angles with A B C; through C draw the line C f indefinitely, making, with B C, an angle of (¹⁄₃).
Take B C upon the compasses, and place the point of one leg upon D where C f intersects B e, and the point of the other upon the line A B, it will meet it at F, which is one focus. Keeping the point of one leg still upon D, remove the point of the other to the line B C, and it will meet it at G, which is the other focus.
Fig. 2.
The foci being in either case thus simply ascertained, the method of describing the curve on a small scale is equally simple.
A pin is fixed into each of the two foci, and another into the point D. Around these three pins a waxed thread, flexible but not elastic, is tied, care being taken that the knot be of a kind that will not slip. The pin at D is now removed, and a hard black lead pencil introduced within the thread band. The pencil is then moved around the pins fixed in the foci, keeping the thread band at a full and equal tension; thus simply the ellipse is described. When, however, the governing angle is acute, say less than (¹⁄₆), it is requisite to adopt a more accurate method of description,[12] as the architectural examples which follow will shew. But architectural draughtsmen and ornamental designers would do well to supply themselves, for ordinary practice, with half a dozen series of ellipses, varying in the proportions of their axes from (⁴⁄₉) to (¹⁄₆) of the scale, and the length of their major axes from 1 to 6 inches. These should be described by the above simple process, upon very strong drawing paper, and carefully cut out, the edge of the paper being kept smooth, and each ellipse having its greater and lesser axes, its foci, and the hypothenuse of its scalene triangle drawn upon it. To exemplify this, I give [Plate V.], which exhibits the ellipses of (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆), inscribed in their rectangles, on which a b and c d are respectively the greater and lesser axes, o o the foci, and d b the angle of each. Such a series of these beautiful figures would be found particularly useful in drawing the mouldings of Grecian architecture; for, to describe the curvilinear contour of such mouldings from single points, as has been done with those which embellish even our most pretending attempts at the restoration of that classical style of architecture, is to give the resemblance of an external form without the harmony which constitutes its real beauty.
Mr Penrose, owing to the supposed difficulty regarding the description of ellipses just alluded to, endeavours to shew that the curves of all the mouldings throughout the Parthenon were either parabolic or hyperbolic; but I believe such curves can have no connexion with the elementary forms of architecture, for they are curves which represent motion, and do not, by continued production, form closed figures.
But I have shewn, in a former work,[13] that the contours of these mouldings are composed of curves of the composite ellipse,—a figure which I so name because it is composed simply of arcs of various ellipses harmonically flowing into each other. The composite ellipse, when drawn systematically upon the isosceles triangle, resembles closely parabolic and hyperbolic curves—only differing from these inasmuch as it possesses the essential quality of circumscribing harmonically one of the elementary rectilinear figures employed in architecture, while those of the parabola and hyperbola, as I have just observed, are merely curves of motion, and, consequently, never can harmonically circumscribe or be resolved into any regular figure.
The composite ellipse may be thus described.
Let A B C ([Plate VI.]) be a vertical isosceles triangle of (¹⁄₆), bisect A B in D, and through D draw indefinitely D f perpendicular to A B, and through B draw indefinitely B g, making the angle D B g (¹⁄₈), D f and B g intersecting each other in M. Take B D and D M as semi-axes of an ellipse, the foci of which will be at p and q, in each of these, and in each of the foci h t and k r in the lines A C and B C, fix a pin, and one also in the point M, tie a thread around these pins, withdraw the pin from M, and trace the composite ellipse in the manner already described with respect to the simple ellipse.
In some of my earlier works I described this figure by taking the angles of the isosceles triangle as foci; but the above method is much more correct. As the elementary angle of the triangle is (¹⁄₆), and that of the elliptic curve described around it (¹⁄₈), I call it the composite ellipse of (¹⁄₆) and (¹⁄₈), their harmonic ratio being 4:3; and so on of all others, according to the difference that may thus exist between the elementary angles.
The visible curves which soften and beautify the melody of the outline of the front of the Parthenon, as given in Mr Penrose’s great work, I have carefully analysed, and have found them in as perfect agreement with this system, as its rectilinear harmony has been shewn to be. This I demonstrated in the work just referred to[14] by a series of twelve plates, shewing that the entasis of the columns (a subject upon which there has been much speculation) is simply an arc of an ellipse of (¹⁄₄₈), whose greater axis makes with the vertical an angle of (¹⁄₆₄); or simply, the form of one of these columns is the frustrum of an elliptic-sided or prolate-spheroidal cone, whose section is a composite ellipse of (¹⁄₄₈) and (¹⁄₆₄), the harmonic ratio of these two angles being 4:3, the same as that of the angles of the composite ellipse just exemplified.
In [Plate VII.] is represented the section of such a cone, of which A B C is the isosceles triangle of (¹⁄₄₈), and B D and D M the semi-axes of an ellipse of (¹⁄₆₄). M N and O P are the entases of the column, and d e f the normal construction of the capital. All these are fully illustrated in the work above referred to,[15] in which I have also shewn that the curve of the neck of the column is that of an ellipse of (¹⁄₆); the curve of the capital or echinus, that of an ellipse of (¹⁄₁₄); the curve of the moulding under the cymatium of the pediment, that of an ellipse of (¹⁄₃); and the curve of the bed-moulding of the cornice of the pediment, that of an ellipse of (¹⁄₃). The curve of the cavetto of the soffit of the corona is composed of ellipses of (¹⁄₆) and (¹⁄₁₄); the curve of the cymatium which surmounts the corona, is that of an ellipse of (¹⁄₃); the curve of the moulding of the capital of the antæ of the posticum, that of an ellipse of (¹⁄₃); the curves of the lower moulding of the same capital are composed of those of an ellipse of (¹⁄₃) and of the circle (¹⁄₂); the curve of the moulding which is placed between the two latter is that of an ellipse of (¹⁄₃); the curve of the upper moulding of the band under the beams of the ceiling of the peristyle, that of an ellipse of (¹⁄₃); the curve of the lower moulding of the same band, that of an ellipse of (¹⁄₄); and the curves of the moulding at the bottom of the small step or podium between the columns, are those of the circle (¹⁄₂) and of an ellipse of (¹⁄₃). I have also shewn the curve of the fluting of the columns to be that of (¹⁄₁₄). The greater axis of each of these ellipses, when not in the vertical or horizontal lines, makes an harmonic angle with one or other of them. In [Plate VIII.], sections of the two last-named mouldings are represented full size, which will give the reader an idea of the simple manner in which the ellipses are employed in the production of those harmonic curves.
Thus we find that the system here adopted for applying this law of nature to the production of beauty in the abstract forms employed in architectural composition, so far from involving us in anything complicated, is characterised by extreme simplicity.
In concluding this part of my treatise, I may here repeat what I have advanced in a late work,[16] viz., my conviction of the probability that a system of applying this law of nature in architectural construction was the only great practical secret of the Freemasons, all their other secrets being connected, not with their art, but with the social constitution of their society. This valuable secret, however, seems to have been lost, as its practical application fell into disuse; but, as that ancient society consisted of speculative as well as practical masons, the secrets connected with their social union have still been preserved, along with the excellent laws by which the brotherhood is governed. It can scarcely be doubted that there was some such practically useful secret amongst the Freemasons or early Gothic architects; for we find in all the venerable remains of their art which exist in this country, symmetrical elegance of form pervading the general design, harmonious proportion amongst all the parts, beautiful geometrical arrangements throughout all the tracery, as well as in the elegantly symmetrised foliated decorations which belong to that style of architecture. But it is at the same time worthy of remark, that whenever they diverged from architecture to sculpture and painting, and attempted to represent the human figure, or even any of the lower animals, their productions are such as to convince us that in this country these arts were in a very degraded state of barbarism—the figures are often much disproportioned in their parts and distorted in their attitudes, while their representations of animals and chimeras are whimsically absurd. It would, therefore, appear that architecture, as a fine art, must have been preserved by some peculiar influence from partaking of the barbarism so apparent in the sister arts of that period. Although its practical secrets have been long lost, the Freemasons of the present day trace the original possession of them to Moses, who, they say, “modelled masonry into a perfect system, and circumscribed its mysteries by land-marks significant and unalterable.” Now, as Moses received his education in Egypt, where Pythagoras is said to have acquired his first knowledge of the harmonic law of numbers, it is highly probable that this perfect system of the great Jewish legislator was based upon the same law of nature which constituted the foundation of the Pythagorean philosophy, and ultimately led to that excellence in art which is still the admiration of the world.
Pythagoras, it would appear, formed a system much more perfect and comprehensive than that practised by the Freemasons in the middle ages of Christianity; for it was as applicable to sculpture, painting, and music, as it was to architecture. This perfection in architecture is strikingly exemplified in the Parthenon, as compared with the Gothic structures of the middle ages; for it will be found that the whole six elementary figures I have enumerated as belonging to architecture, are required in completing the orthographic beauty of that noble structure. And amongst these, none conduce more to that beauty than the simple and composite ellipses. Now, in the architecture of the best periods of Gothic, or, indeed, in that of any after period (Roman architecture included), these beautiful curves seem to have been ignored, and that of the circle alone employed.
Be those matters as they may, however, the great law of numerical harmonic ratio remains unalterable, and a proper application of it in the science of art will never fail to be as productive of effect, as its operation in nature is universal, certain, and continual.