Chapter IV: Introduction (2)

Thus every musical sound is composed of a certain number of parts called pulsations, and these parts must in every scale relate harmonically to some fundamental number. When these parts are multiples of the fundamental number by 2, 4, 8, &c., like the pulsations of the sounds indicated by _c_, _c′_, _c′′_, _c′′′_, they are called tonic notes, being the most consonant; when the pulsations are similar multiples by 3, 6, 12, &c., like those of the sounds indicated by _g_, _g′_, _g′′_, they are called dominant notes, being the next most consonant; and multiples by 5, 10, &c., like those of the sounds indicated by _e_, _e′_, _e′′_, they are called mediant notes, from a similar cause. In harmonic combinations of musical sounds, the æsthetic feeling produced by their agreement depends upon the relations they bear to each other with reference to the number of pulsations produced in a given time by the fundamental note of the scale to which they belong; and it will be observed, that the more simple the numerical ratios are amongst the pulsations of any number of notes simultaneously produced, the more perfect their agreement. Hence the origin of the common chord or fundamental concord in the united sounds of the tonic, the dominant, and the mediant notes, the ratios and coincidences of whose pulsations 2:1, 3:2, 5:4, may thus be exemplified:—

In musical composition, the law of number also governs its division into parts, in order to produce upon the ear, along with the beauty of harmony, that of rhythm. Thus a piece of music is divided into parts each of which contains a certain number of other parts called bars, which may be divided and subdivided into any number of notes, and the performance of each bar is understood to occupy the same portion of time, however numerous the notes it contains may be; so that the music of art is regularly symmetrical in its structure; while that of nature is in general as irregular and indefinite in its rhythm as it is in its harmony.

Thus I have endeavoured briefly to explain the manner in which the law of numerical ratio operates in that species of beauty perceived through the ear.

The definite principles of the art of music founded upon this law have been for ages so systematised that those who are instructed in them advance steadily in proportion to their natural endowments, while those who refuse this instruction rarely attain to any excellence. In the sister arts of form and colour, however, a system of tuition, founded upon this law, is still a desideratum, and a knowledge of the scientific principles by which these arts are governed is confined to a very few, and scarcely acknowledged amongst those whose professions most require their practical application.

THE SCIENCE OF BEAUTY AS APPLIED TO FORMS.

It is justly remarked, in the “Illustrated Record of the New York Exhibition of 1853,” that “it is a question worthy of consideration how far the mediocrity of the present day is attributable to an overweening reliance on natural powers and a neglect of the lights of science;” and there is expressed a thorough conviction of the fact that, besides the evils of the copying system, “much genius is now wasted in the acquirement of rudimentary knowledge in the slow school of practical experiment, and that the excellence of the ancient Greek school of design arose from a thoroughly digested canon of form, and the use of geometrical formulas, which make the works even of the second and third-rate genius of that period the wonder and admiration of the present day.”

That such a canon of form, and that the use of such geometrical formula, entered into the education, and thereby facilitated the practice of ancient Greek art, I have in a former work expressed my firm belief, which is founded on the remarkable fact, that for a period of nearly three centuries, and throughout a whole country politically divided into states often at war with each other, works of sculpture, architecture, and ornamental design were executed, which surpass in symmetrical beauty any works of the kind produced during the two thousand years that have since elapsed. So decided is this superiority, that the artistic remains of the extraordinary period I alluded to are, in all civilised nations, still held to be the most perfect specimens of formative art in the world; and even when so fragmentary as to be denuded of everything that can convey an idea of expression, they still excite admiration and wonder by the purity of their geometric beauty. And so universal was this excellence, that it seems to have characterised every production of formative art, however humble the use to which it was applied.

The common supposition, that this excellence was the result of an extraordinary amount of genius existing among the Greek people during that particular period, is not consistent with what we know of the progress of mankind in any other direction, and is, in the present state of art, calculated to retard its progress, inasmuch as such an idea would suggest that, instead of making any exertion to arrive at a like general excellence, the world must wait for it until a similar supposed psychological phenomenon shall occur.

But history tends to prove that the long period of universal artistic excellence throughout Greece could only be the result of an early inculcation of some well-digested system of correct elementary principles, by which the ordinary amount of genius allotted to mankind in every age was properly nurtured and cultivated; and by which, also, a correct knowledge and appreciation of art were disseminated amongst the people generally. Indeed, Müller, in his “Ancient Art and its Remains,” shews clearly that some certain fixed principles, constituting a science of proportions, were known in Greece, and that they formed the basis of all artists’ education and practice during the period referred to; also, that art began to decline, and its brightest period to close, as this science fell into disuse, and the Greek artists, instead of working for an enlightened community, who understood the nature of the principles which guided them, were called upon to gratify the impatient whims of pampered and tyrannical rulers.

By being instructed in this science of proportion, the Greek artists were enabled to impart to their representations of the human figure a mathematically correct species of symmetrical beauty; whether accompanying the slender and delicately undulated form of the Venus,—its opposite, the massive and powerful mould of the Hercules,—or the characteristic representation of any other deity in the heathen mythology. And this seems to have been done with equal ease in the minute figure cut on a precious gem, and in the most colossal statue. The same instruction likewise enabled the architects of Greece to institute those varieties of proportions in structure called the Classical Orders of Architecture; which are so perfect that, since the science which gave them birth has been buried in oblivion, classical architecture has been little more than an imitative art; for all who have since written upon the subject, from Vitruvius downwards, have arrived at nothing, in so far as the great elementary principles in question are concerned, beyond the most vague and unsatisfactory conjectures. For a more clear understanding of the nature of this application of the Pythagorean law of number to the harmony of form, it will be requisite to repeat the fact, that modern science has shewn that the cause of the impression, produced by external nature upon the sensorium, called light, may be traced to a molecular or ethereal action. This action is excited naturally by the sun, artificially by the combustion of various substances, and sometimes physically within the eye. Like the atmospheric pulsations which produce sound, the action which produces light is capable, within a limited sphere, of being reflected from some bodies and transmitted through others; and by this reflection and transmission the visible nature of forms and figures is communicated to the sensorium. The eye is the medium of this communication; and its structural beauty, and perfect adaptation to the purpose of conveying this action, must, like those of the ear, be left to the anatomist fully to describe. It is here only necessary to remark, that the optic nerve, like the auditory nerve, ends in a carefully protected fluid, which is the last of the media interposed between this peculiarly subtle action and the nerve upon which it impresses the presence of the object from which it is reflected or through which it is transmitted, and the nature of such object made perceptible to the mind. The eye and the ear are thus, in one essential point, similar in their physiology, relatively to the means provided for receiving impressions from external nature; it is, therefore, but reasonable to believe that the eye is capable of appreciating the exact subdivision of spaces, just as the ear is capable of appreciating the exact subdivision of intervals of time; so that the division of space into exact numbers of equal parts will æsthetically affect the mind through the medium of the eye.

We assume, therefore, that the standard of symmetry, so estimated, is deduced from the simplest law that could have been conceived—the law that the angles of direction must all bear to some fixed angle the same simple relations which the different notes in a chord of music bear to the fundamental note; that is, relations expressed arithmetically by the smallest natural numbers. Thus the eye, being guided in its estimate by direction rather than by distance, just as the ear is guided by number of vibrations rather than by magnitude, both it and the ear convey simplicity and harmony to the mind without effort, and the mind with equal facility receives and appreciates them.

_On the Rectilinear Forms and Proportions of Architecture._

As we are accustomed in all cases to refer direction to the horizontal and vertical lines, and as the meeting of these lines makes the right angle, it naturally constitutes the fundamental angle, by the harmonic division of which a system of proportion may be established, and the theory of symmetrical beauty, like that of music, rendered susceptible of exact reasoning.

Let therefore the right angle be the fundamental angle, and let it be divided upon the quadrant of a circle into the harmonic parts already explained, thus:—

Super- Sub- Sub- Sub- Semi-sub-
Right tonic Mediant dominant Dominant mediant tonic tonic Tonic
Angle. Angles. Angles. Angles. Angles. Angles. Angles. Angles. Angles.
I. (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)
II. (¹⁄₂) (⁴⁄₉) (²⁄₅) (³⁄₈) (¹⁄₃) (³⁄₁₀) (²⁄₇) (⁴⁄₁₅) (¹⁄₄)
III. (¹⁄₄) (²⁄₉) (¹⁄₅) (³⁄₁₆) (¹⁄₆) (³⁄₂₀) (¹⁄₇) (²⁄₁₅) (¹⁄₈)
IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) (³⁄₃₂) (¹⁄₁₂) (³⁄₄₀) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)

In order that the analogy may be kept in view, I have given to the parts of each of these four scales the appropriate nomenclature of the notes which form the diatonic scale in music.

When a right angled triangle is constructed so that its two smallest angles are equal, I term it simply the triangle of (¹⁄₂), because the smaller angles are each one-half of the right angle. But when the two angles are unequal, the triangle may be named after the smallest. For instance, when the smaller angle, which we shall here suppose to be one-third of the right angle, is made with the vertical line, the triangle may be called the vertical scalene triangle of (¹⁄₃); and when made with the horizontal line, the horizontal scalene triangle of (¹⁄₃). As every rectangle is made up of two of these right angled triangles, the same terminology may also be applied to these figures. Thus, the equilateral rectangle or perfect square is simply the rectangle of (¹⁄₂), being composed of two similar right angled triangles of (¹⁄₂); and when two vertical scalene triangles of (¹⁄₃), and of similar dimensions, are united by their hypothenuses, they form the vertical rectangle of (¹⁄₃), and in like manner the horizontal triangles of (¹⁄₃) similarly united would form the horizontal rectangle of (¹⁄₃). As the isosceles triangle is in like manner composed of two right angled scalene triangles joined by one of their sides, the same terminology may be applied to every variety of that figure. All the angles of the first of the above scales, except that of (¹⁄₂), give rectangles whose longest sides are in the horizontal line, while the other three give rectangles whose longest sides are in the vertical line. I have illustrated in Plate I. the manner in which this harmonic law acts upon these elementary rectilinear figures by constructing a series agreeably to the angles of scales II., III., IV. Throughout this series _a b c_ is the primary scalene triangle, of which the rectangle _a b c e_ is composed; _d c e_ the vertical isosceles triangle; and when the plate is turned, _d e a_ the horizontal isosceles triangle, both of which are composed of the same primary scalene triangle.

[Sidenote: Plate I.]

Thus the most simple elements of symmetry in rectilinear forms are the three following figures:—

The equilateral rectangle or perfect square,
The oblong rectangle, and
The isosceles triangle.

It has been shewn that in harmonic combinations of musical sounds, the æsthetic feeling produced by their agreement depends upon the relation they bear to each other with reference to the number of pulsations produced in a given time by the fundamental note of the scale to which they belong; and that the more simply they relate to each other in this way the more perfect the harmony, as in the common chord of the first scale, the relations of whose parts are in the simple ratios of 2:1, 3:2, and 5:4. It is equally consistent with this law, that when applied to form in the composition of an assortment of figures of any kind, their respective proportions should bear a very simple ratio to each other in order that a definite and pleasing harmony may be produced amongst the various parts. Now, this is as effectually done by forming them upon the harmonic divisions of the right angle as musical harmony is produced by sounds resulting from harmonic divisions of a vibratory body.

Having in previous works[7] given the requisite illustrations of this fact in full detail, I shall here confine myself to the most simple kind, taking for my first example one of the finest specimens of classical architecture in the world—the front portico of the Parthenon of Athens.

The angles which govern the proportions of this beautiful elevation are the following harmonic parts of the right angle—

Tonic Dominant Mediant Subtonic Supertonic
Angles. Angles. Angles. Angle. Angles.
(¹⁄₂) (¹⁄₃) (¹⁄₅) (¹⁄₇) (¹⁄₉)
(¹⁄₄) (¹⁄₆) (¹⁄₁₀) (¹⁄₁₈)
(¹⁄₈)
(¹⁄₁₆)

[Sidenote: Plate II.]

In Plate II. I give a diagram of its rectilinear orthography, which is simply constructed by lines drawn, either horizontally, vertically, or obliquely, which latter make with either of the former lines one or other of the harmonic angles in the above series. For example, the horizontal line AB represents the length of the base or surface of the upper step of the substructure of the building. The line AE, which makes an angle of (¹⁄₅) with the horizontal, determines the height of the colonnade. The line AD, which makes an angle of (¹⁄₄) with the horizontal, determines the height of the portico, exclusive of the pediment. The line AC, which makes an angle of (¹⁄₃) with the horizontal, determines the height of the portico, including the pediment. The line GD, which makes an angle of (¹⁄₇) with the horizontal, determines the form of the pediment. The lines EZ and LY, which respectively make angles of (¹⁄₁₆) and (¹⁄₁₈) with the horizontal, determine the breadth of the architrave, frieze, and cornice. The line _v n u_, which makes an angle of (¹⁄₃) with the vertical, determines the breadth of the triglyphs. The line _t d_, which makes an angle of (¹⁄₂), determines the breadth of the metops. The lines _c b r f_, and _a i_, which make each an angle of (¹⁄₆) with the vertical, determine the width of the five centre intercolumniations. The line _z k_, which makes an angle of (¹⁄₈) with the vertical, determines the width of the two remaining intercolumniations. The lines _c s_, _q x_, and _y h_, each of which makes an angle of (¹⁄₁₀) with the vertical, determine the diameters of the three columns on each side of the centre. The line _w l_, which makes an angle of (¹⁄₉) with the vertical, determines the diameter of the two remaining or corner columns.

In all this, the length and breadth of the parts are determined by horizontal and vertical lines, which are necessarily at right angles with each other, and the position of which are determined by one or other of the lines making the harmonic angles above enumerated.

Now, the lengths and breadths thus so simply determined by these few angles, have been proved to be correct by their agreement with the most careful measurements which could possibly be made of this exquisite specimen of formative art. These measurements were obtained by the “Society of Dilettanti,” London, who, expressly for that purpose, sent Mr F. C. Penrose, a highly educated architect, to Athens, where he remained for about five months, engaged in the execution of this interesting commission, the results of which are now published in a magnificent volume by the Society.[8] The agreement was so striking, that Mr Penrose has been publicly thanked by an eminent man of science for bearing testimony to the truth of my theory, who in doing so observes, “The dimensions which he (Mr Penrose) gives are to me the surest verification of the theory I could have desired. The minute discrepancies form that very element of practical incertitude, both as to execution and direct measurement, which always prevails in materialising a mathematical calculation made under such conditions.”[9]

Although the measurements taken by Mr Penrose are undeniably correct, as all who examine the great work just referred to must acknowledge, and although they have afforded me the best possible means of testing the accuracy of my theory as applied to the Parthenon, yet the ideas of Mr Penrose as to the principles they evolve are founded upon the fallacious doctrine which has so long prevailed, and still prevails, in the æsthetics of architecture, viz., that harmony may be imparted by ratios between the lengths and breadths of parts.

I have taken for my second example an elevation which, although of smaller dimensions, is no less celebrated for the beauty of its proportions than the Parthenon itself, viz., the front portico of the temple of Theseus, which has also been measured by Mr Penrose.

The angles which govern the proportions of this elevation are the following harmonic parts of the right angle:—

Tonic Dominant Mediant
Angles. Angles. Angles.
(¹⁄₂) (¹⁄₃) (²⁄₅)
(¹⁄₄) (¹⁄₆) (¹⁄₅)
(¹⁄₁₂)

[Sidenote: Plate III.]

A diagram of the rectilinear orthography of this portico is given in Plate III. Its construction is similar to that of the Parthenon in respect to the harmonic parts of the right angle, and I have therefore only to observe, that the line A E makes an angle of (¹⁄₄); the line A D an angle of (¹⁄₃); the line A C an angle of (²⁄₅); the line G D an angle of (¹⁄₆); and the lines E Z and L Y angles of (¹⁄₁₂) with the horizontal.

As to the colonnade or vertical part, the line _a b_, which determines the three middle intercolumniations, makes an angle of (¹⁄₅); the line _c d_, which determines the two outer intercolumniations, makes an angle of (¹⁄₆); and the line _e f_, which determines the lesser diameter of the columns, makes an angle of (¹⁄₁₂) with the vertical. I need give no further details here, as my intention is to shew the simplicity of the method by which this theory may be reduced to practice, and because I have given in my other works ample details, in full illustration of the orthography of these two structures, especially the first.[10]

The foregoing examples being both horizontal rectangular compositions, the proportions of their principal parts have necessarily been determined by lines drawn from the extremities of the base, making angles with the horizontal line, and forming thereby the diagonals of the various rectangles into which, in their leading features, they are necessarily resolved. But the example I am now about to give is of another character, being a vertical pyramidal composition, and consequently the proportions of its principal parts are determined by the angles which the oblique lines make with the vertical line representing the height of the elevation, and forming a series of isosceles triangles; for the isosceles triangle is the type of all pyramidal composition.

This third example is the east end of Lincoln Cathedral, a Gothic structure, which is acknowledged to be one of the finest specimens of that style of architecture existing in this country.

The angles which govern the proportions of this elevation are the following harmonic parts of the right angle:—

Tonic. Dominant. Mediant. Subtonic. Supertonic.
(¹⁄₂) (¹⁄₃) (¹⁄₅) (¹⁄₇) (²⁄₉)
(¹⁄₄) (¹⁄₆) (¹⁄₁₀) (¹⁄₉)
(¹⁄₁₂)

[Sidenote: Plate IV.]

In Plate IV. I give a diagram of the vertical, horizontal, and oblique lines, which compose the orthography of this beautiful elevation.

The line A B represents the full height of this structure. The line A C, which makes an angle of (²⁄₉) with the vertical, determines the width of the design, the tops of the aisle windows, and the bases of the pediments on the inner buttresses; A G, (¹⁄₅) with the vertical, that of the outer buttress; A F, (¹⁄₉) with the vertical, that of the space between the outer and inner buttresses and the width of the great centre window; and A E, (¹⁄₁₂) with vertical, that of both the inner buttresses and the space between these. A H, which makes (¹⁄₄) with the vertical, determines the form of the pediment of the centre, and the full height of the base and surbase. A I, which makes (¹⁄₃) with the vertical, determines the form of the pediment of the smaller gables, the base of the pediment on the outer buttress, the base of the ornamental recess between the outer and inner buttresses, the spring of the arch of the centre window, the tops of the pediments on the inner buttresses, and the spring of the arch of the upper window. A K, which makes (¹⁄₂), determines the height of the outer buttress; and A Z, which makes (¹⁄₆) with the horizontal, determines that of the inner buttresses. For the reasons already given, I need not here go into further detail.[11] It is, however, worthy of remark in this place, that notwithstanding the great difference which exists between the style of composition in this Gothic design, and in that of the east end of the Parthenon, the harmonic elements upon which the orthographic beauty of the one depends, are almost identical with those of the other.

_On the Curvilinear Forms and Proportions of Architecture._

Each regular rectilinear figure has a curvilinear figure that exclusively belongs to it, and to which may be applied a corresponding terminology. For instance, the circle belongs to the equilateral rectangle; that is, the rectangle of (¹⁄₂), an ellipse to every other rectangle, and a composite ellipse to every isosceles triangle. Thus the most simple elements of beauty in the curvilinear forms of architectural design are the following three figures:—

The circle,
The ellipse, and
The composite ellipse.

I find it necessary in this place to go into some details regarding the specific character of the two latter figures, because the proper mode of describing these beautiful curves, and their high value in the practice of the architectural draughtsman and ornamental designer, seem as yet unknown. In proof of this assertion, I must again refer to Mr Penrose’s great work published by the “Society of Dilettanti.” At page 52 of that work it is observed, that “by whatever means an ellipse is to be constructed mechanically, it is a work of time (if not of absolute difficulty) so to arrange the foci, &c., as to produce an ellipse of any exact length and breadth which may be desired.” Now, this is far from being the case, for the method of arranging the foci of an ellipse of any given length and breadth is extremely simple, being as follows:—

Let A B C (figure 1) be the length, and D B E the breadth of the desired ellipse.

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.

The foci being in either case thus simply ascertained, the method of describing the curve on a small scale is equally simple.

[Sidenote: Plate V.]

A pin is fixed into each of the two foci, and another into the point D. Around these three pins a waxed thread, flexible but not elastic, is tied, care being taken that the knot be of a kind that will not slip. The pin at D is now removed, and a hard black lead pencil introduced within the thread band. The pencil is then moved around the pins fixed in the foci, keeping the thread band at a full and equal tension; thus simply the ellipse is described. When, however, the governing angle is acute, say less than (¹⁄₆), it is requisite to adopt a more accurate method of description,[12] as the architectural examples which follow will shew. But architectural draughtsmen and ornamental designers would do well to supply themselves, for ordinary practice, with half a dozen series of ellipses, varying in the proportions of their axes from (⁴⁄₉) to (¹⁄₆) of the scale, and the length of their major axes from 1 to 6 inches. These should be described by the above simple process, upon very strong drawing paper, and carefully cut out, the edge of the paper being kept smooth, and each ellipse having its greater and lesser axes, its foci, and the hypothenuse of its scalene triangle drawn upon it. To exemplify this, I give Plate V., which exhibits the ellipses of (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆), inscribed in their rectangles, on which _a b_ and _c d_ are respectively the greater and lesser axes, _o o_ the foci, and _d b_ the angle of each. Such a series of these beautiful figures would be found particularly useful in drawing the mouldings of Grecian architecture; for, to describe the curvilinear contour of such mouldings from single points, as has been done with those which embellish even our most pretending attempts at the restoration of that classical style of architecture, is to give the resemblance of an external form without the harmony which constitutes its real beauty.

Mr Penrose, owing to the supposed difficulty regarding the description of ellipses just alluded to, endeavours to shew that the curves of all the mouldings throughout the Parthenon were either parabolic or hyperbolic; but I believe such curves can have no connexion with the elementary forms of architecture, for they are curves which represent motion, and do not, by continued production, form closed figures.

But I have shewn, in a former work,[13] that the contours of these mouldings are composed of curves of the composite ellipse,—a figure which I so name because it is composed simply of arcs of various ellipses harmonically flowing into each other. The composite ellipse, when drawn systematically upon the isosceles triangle, resembles closely parabolic and hyperbolic curves—only differing from these inasmuch as it possesses the essential quality of circumscribing harmonically one of the elementary rectilinear figures employed in architecture, while those of the parabola and hyperbola, as I have just observed, are merely curves of motion, and, consequently, never can harmonically circumscribe or be resolved into any regular figure.

The composite ellipse may be thus described.

[Sidenote: Plate VI.]

Let A B C (Plate VI.) be a vertical isosceles triangle of (¹⁄₆), bisect A B in D, and through D draw indefinitely D _f_ perpendicular to A B, and through B draw indefinitely B _g_, making the angle D B _g_ (¹⁄₈), D _f_ and B _g_ intersecting each other in M. Take B D and D M as semi-axes of an ellipse, the foci of which will be at _p_ and _q_, in each of these, and in each of the foci _h t_ and _k r_ in the lines A C and B C, fix a pin, and one also in the point M, tie a thread around these pins, withdraw the pin from M, and trace the composite ellipse in the manner already described with respect to the simple ellipse.

In some of my earlier works I described this figure by taking the angles of the isosceles triangle as foci; but the above method is much more correct. As the elementary angle of the triangle is (¹⁄₆), and that of the elliptic curve described around it (¹⁄₈), I call it the composite ellipse of (¹⁄₆) and (¹⁄₈), their harmonic ratio being 4:3; and so on of all others, according to the difference that may thus exist between the elementary angles.

The visible curves which soften and beautify the melody of the outline of the front of the Parthenon, as given in Mr Penrose’s great work, I have carefully analysed, and have found them in as perfect agreement with this system, as its rectilinear harmony has been shewn to be. This I demonstrated in the work just referred to[14] by a series of twelve plates, shewing that the entasis of the columns (a subject upon which there has been much speculation) is simply an arc of an ellipse of (¹⁄₄₈), whose greater axis makes with the vertical an angle of (¹⁄₆₄); or simply, the form of one of these columns is the frustrum of an elliptic-sided or prolate-spheroidal cone, whose section is a composite ellipse of (¹⁄₄₈) and (¹⁄₆₄), the harmonic ratio of these two angles being 4:3, the same as that of the angles of the composite ellipse just exemplified.

[Sidenote: Plate VII.]

[Sidenote: Plate VIII.]

In Plate VII. is represented the section of such a cone, of which A B C is the isosceles triangle of (¹⁄₄₈), and B D and D M the semi-axes of an ellipse of (¹⁄₆₄). M N and O P are the entases of the column, and _d e f_ the normal construction of the capital. All these are fully illustrated in the work above referred to,[15] in which I have also shewn that the curve of the neck of the column is that of an ellipse of (¹⁄₆); the curve of the capital or echinus, that of an ellipse of (¹⁄₁₄); the curve of the moulding under the cymatium of the pediment, that of an ellipse of (¹⁄₃); and the curve of the bed-moulding of the cornice of the pediment, that of an ellipse of (¹⁄₃). The curve of the cavetto of the soffit of the corona is composed of ellipses of (¹⁄₆) and (¹⁄₁₄); the curve of the cymatium which surmounts the corona, is that of an ellipse of (¹⁄₃); the curve of the moulding of the capital of the antæ of the posticum, that of an ellipse of (¹⁄₃); the curves of the lower moulding of the same capital are composed of those of an ellipse of (¹⁄₃) and of the circle (¹⁄₂); the curve of the moulding which is placed between the two latter is that of an ellipse of (¹⁄₃); the curve of the upper moulding of the band under the beams of the ceiling of the peristyle, that of an ellipse of (¹⁄₃); the curve of the lower moulding of the same band, that of an ellipse of (¹⁄₄); and the curves of the moulding at the bottom of the small step or podium between the columns, are those of the circle (¹⁄₂) and of an ellipse of (¹⁄₃). I have also shewn the curve of the fluting of the columns to be that of (¹⁄₁₄). The greater axis of each of these ellipses, when not in the vertical or horizontal lines, makes an harmonic angle with one or other of them. In Plate VIII., sections of the two last-named mouldings are represented full size, which will give the reader an idea of the simple manner in which the ellipses are employed in the production of those harmonic curves.

Thus we find that the system here adopted for applying this law of nature to the production of beauty in the abstract forms employed in architectural composition, so far from involving us in anything complicated, is characterised by extreme simplicity.

In concluding this part of my treatise, I may here repeat what I have advanced in a late work,[16] viz., my conviction of the probability that a system of applying this law of nature in architectural construction was the only great practical secret of the Freemasons, all their other secrets being connected, not with their art, but with the social constitution of their society. This valuable secret, however, seems to have been lost, as its practical application fell into disuse; but, as that ancient society consisted of speculative as well as practical masons, the secrets connected with their social union have still been preserved, along with the excellent laws by which the brotherhood is governed. It can scarcely be doubted that there was some such practically useful secret amongst the Freemasons or early Gothic architects; for we find in all the venerable remains of their art which exist in this country, symmetrical elegance of form pervading the general design, harmonious proportion amongst all the parts, beautiful geometrical arrangements throughout all the tracery, as well as in the elegantly symmetrised foliated decorations which belong to that style of architecture. But it is at the same time worthy of remark, that whenever they diverged from architecture to sculpture and painting, and attempted to represent the human figure, or even any of the lower animals, their productions are such as to convince us that in this country these arts were in a very degraded state of barbarism—the figures are often much disproportioned in their parts and distorted in their attitudes, while their representations of animals and chimeras are whimsically absurd. It would, therefore, appear that architecture, as a fine art, must have been preserved by some peculiar influence from partaking of the barbarism so apparent in the sister arts of that period. Although its practical secrets have been long lost, the Freemasons of the present day trace the original possession of them to Moses, who, they say, “modelled masonry into a perfect system, and circumscribed its mysteries by _land-marks_ significant and unalterable.” Now, as Moses received his education in Egypt, where Pythagoras is said to have acquired his first knowledge of the harmonic law of numbers, it is highly probable that this perfect system of the great Jewish legislator was based upon the same law of nature which constituted the foundation of the Pythagorean philosophy, and ultimately led to that excellence in art which is still the admiration of the world.

Pythagoras, it would appear, formed a system much more perfect and comprehensive than that practised by the Freemasons in the middle ages of Christianity; for it was as applicable to sculpture, painting, and music, as it was to architecture. This perfection in architecture is strikingly exemplified in the Parthenon, as compared with the Gothic structures of the middle ages; for it will be found that the whole six elementary figures I have enumerated as belonging to architecture, are required in completing the orthographic beauty of that noble structure. And amongst these, none conduce more to that beauty than the simple and composite ellipses. Now, in the architecture of the best periods of Gothic, or, indeed, in that of any after period (Roman architecture included), these beautiful curves seem to have been ignored, and that of the circle alone employed.

Be those matters as they may, however, the great law of numerical harmonic ratio remains unalterable, and a proper application of it in the science of art will never fail to be as productive of effect, as its operation in nature is universal, certain, and continual.

THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE HUMAN HEAD AND COUNTENANCE.

The most remarkable characteristics of the human head and countenance are the globular form of the cranium, united as it is with the prolate spheroidal form produced by the parts which constitute the face, and the approximation of the profile to the vertical; for in none of the lower animals does the skull present so near a resemblance to a combination of these geometric forms, nor the plane of the face to this direction. We also find that although these peculiar characteristics are variously modified among the numerous races of mankind, yet one law appears to govern the beauty of the whole. The highest and most cultivated of these races, however, present only an approximation to the perfect development of those distinguishing marks of humanity; and therefore the beauty of form and proportion which in nature characterises the human head and countenance, exhibits only a partial development of the harmonic law of visible beauty. On the other hand, we find that, in their sculpture, the ancient Greeks surpassed ordinary nature, and produced in their beau ideal a species of beauty free from the imperfections and peculiarities that constitute the individuality by which the countenances of men are distinguished from each other. It may be requisite here to remark, that this species of beauty is independent of the more intellectual quality of expression. For as Sir Charles Bell has said, “Beauty of countenance may be defined in words, as well as demonstrated in art. A face may be beautiful in sleep, and a statue without expression may be highly beautiful. But it will be said there is expression in the sleeping figure or in the statue. Is it not rather that we see in these the capacity for expression?—that our minds are active in imagining what may be the motions of these features when awake or animated? Thus, we speak of an expressive face before we have seen a movement grave or cheerful, or any indication in the features of what prevails in the heart.”

This capacity for expression certainly enhances our admiration of the human countenance; but it is more a concomitant of the primary cause of its beauty than the cause itself. This cause rests on that simple and secure basis—the harmonic law of nature; for the nearer the countenance approximates to an harmonious combination of the most perfect figures in geometry, or rather the more its general form and the relation of its individual parts are arranged in obedience to that law, the higher its degree of beauty, and the greater its capacity for the expression of the passions.

Various attempts have been made to define geometrically the difference between the ordinary and the ideal beauty of the human head and countenance, the most prominent of which is that of Camper. He traced, upon a profile of the skull, a line in a horizontal direction, passing through the foramen of the ear and the exterior margin of the sockets of the front teeth of the upper jaw, upon which he raised an oblique line, tangential to the margin of these sockets, and to the most prominent part of the forehead. Agreeably to the obliquity of this line, he determined the relative proportion of the areas occupied by the brain and by the face, and hence inferred the degree of intellect. When he applied this measurement to the heads of the antique statues, he found the angle much greater than in ordinary nature; but that this simple fact afforded no rule for the reproduction of the ideal beauty of ancient Greek art, is very evident from the heads and countenances by which his treatise is illustrated. Sir Charles Bell justly remarks, that although, by Camper’s method, the forehead may be thrown forward, yet, while the features of common nature are preserved, we refuse to acknowledge a similarity to the beautiful forms of the antique marbles. “It is true,” he says, “that, by advancing the forehead, it is raised, the face is shortened, and the eye brought to the centre of the head. But with all this, there is much wanting—that which measurement, or a mere line, will not shew us.”—“The truth is, that we are more moved by the features than by the form of the whole head. Unless there be a conformity in every feature to the general shape of the head, throwing the forehead forward on the face produces deformity; and the question returns with full force—How is it that we are led to concede that the antique head of the Apollo, or of the Jupiter, is beautiful when the facial line makes a hundred degrees with the horizontal line? In other words—How do we admit that to be beautiful which is not natural? Simply for the same reason that, if we discover a broken portion of an antique, a nose, or a chin of marble, we can say, without deliberation—This must have belonged to a work of antiquity; which proves that the character is distinguishable in every part—in each feature, as well as in the whole head.”

Dr Oken says upon this subject:[17]—“The face is beautiful whose nose is parallel to the spine. No human face has grown into this estate; but every nose makes an acute angle with the spine. The facial angle is, as is well known, 80°. What, as yet, no man has remarked, and what is not to be remarked, either, without our view of the cranial signification, the old masters have felt through inspiration. They have not only made the facial angle a right angle, but have even stepped beyond this—the Romans going up to 96°, the Greeks even to 100°. Whence comes it that this unnatural face of the Grecian works of art is still more beautiful than that of the Roman, when the latter comes nearer to nature? The reason thereof resides in the fact of the Grecian artistic face representing nature’s design more than that of the Roman; for, in the former, the nose is placed quite perpendicular, or parallel to the spinal cord, and thus returns whither it has been derived.”

Other various and conflicting opinions upon this subject have been given to the world; but we find that the principle from which arose the ideal beauty of the head and countenance, as represented in works of ancient Greek art, is still a matter of dispute. When, however, we examine carefully a fine specimen, we find its beauty and grandeur to depend more upon the degree of harmony amongst its parts, as to their relative proportions and mode of arrangement, than upon their excellence taken individually. It is, therefore, clear that those (and they are many) who attribute the beauty of ancient Greek sculpture merely to a selection of parts from various models, must be in error. No assemblage of parts from ordinary nature could have produced its principal characteristic, the excess in the angle of the facial line, much less could it have led to that exquisite harmony of parts by which it is so eminently distinguished; neither can we reasonably agree with Dr Oken and others, who assert that it was produced by an exclusive degree of the inspiration of genius amongst the Greek people during a certain period.

That the inspiration of genius, combined with a careful study of nature, were essential elements in the production of the great works which have been handed down to us, no one will deny; but these elements have existed in all ages, whilst the ideal head belongs exclusively to the Greeks during the period in which the schools of Pythagoras and Plato were open. Is it not, therefore, reasonable to suppose, that, besides genius and the study of nature, another element was employed in the production of this excellence, and that this element arose from the precise mathematical doctrines taught in the schools of these philosophers?

An application of the great harmonic law seems to prove that there is no object in nature in which the science of beauty is more clearly developed than in the human head and countenance, nor to the representations of which the same science is more easily applied; and it is to the mode in which this is done that the varieties of sex and character may be imparted to works of art. Having gone into full detail, and given ample illustrations in a former work,[18] it is unnecessary for me to enter upon that part of the subject in this _résumé_; but only to shew the typical structure of beauty by which this noble work of creation is distinguished.

The angles which govern the form and proportions of the human head and countenance are, with the right angle, a series of seven, which, from the simplicity of their ratios to each other, are calculated to produce the most perfect concord. It consists of the right angle and its following parts—

Tonic. Dominant. Mediant. Subtonic.
(¹⁄₂) (¹⁄₃) (¹⁄₅) (¹⁄₇)
(¹⁄₄) (¹⁄₆)

These angles, and the figures which belong to them, are thus arranged:—

[Sidenote: Plate IX.]

The vertical line A B (Plate IX. fig. 2) represents the full length of the head and face. Taking this line as the greater axis of an ellipse of (¹⁄₃), such an ellipse is described around it. Through A the lines A G, A K, A L, A M, and A N, are drawn on each side of the line A B, making, with the vertical, respectively the angles of (¹⁄₃), (¹⁄₄), (¹⁄₅), (¹⁄₆), and (¹⁄₇). Through the points G, K, L, M, and N, where these straight lines meet the curved line of the ellipse, horizontal lines are drawn by which the following isosceles triangles are formed, A G G, A K K, A L L, A M M, and A N N. From the centre X of the equilateral triangle A G G the curvilinear figure of (¹⁄₂), viz., the circle, is described circumscribing that triangle.

The curvilinear plane figures of (¹⁄₂) and (¹⁄₃), respectively, represent the solid bodies of which they are sections, viz., a sphere and a prolate spheroid. These bodies, from the manner in which they are here placed, are partially amalgamated, as shewn in figures 1 and 3 of the same plate, thus representing the form of the human head and countenance, both in their external appearance and osseous structure, more correctly than they could be represented by any other geometrical figures. Thus, the angles of (¹⁄₂) and (¹⁄₃) determine the typical form.

From each of the points _u_ and _n_, where A M cuts G G on both sides of A B, a circle is described through the points _p_ and _q_, where A K cuts G G on both sides of A B, and with the same radius a circle is described from the point _a_, where K K cuts A B.

The circles _u_ and _n_ determine the position and size of the eyeballs, and the circle _a_ the width of the nose, as also the horizontal width of the mouth.

The lines G G and K K also determine the length of the joinings of the ear to the head. The lines L L and M M determine the vertical width of the mouth and lips when at perfect repose, and the line N N the superior edge of the chin. Thus simply are the features arranged and proportioned on the facial surface.

It must, however, be borne in mind, that in treating simply of the æsthetic beauty of the human head and countenance, we have only to do with the external appearance. In this research, therefore, the system of Dr Camper, Dr Owen, and others, whose investigations were more of a physiological than an æsthetic character, can be of little service; because, according to that system, the facial angle is determined by drawing a line tangential to the exterior margin of the sockets of the front teeth of the upper jaw, and the most prominent part of the forehead. Now, as these sockets are, when the skull is naturally clothed, and the features in repose, entirely concealed by the upper lip, we must take the prominent part of it, instead of the sockets under it, in order to determine properly this distinguishing mark of humanity. And I believe it will be found, that when the head is properly poised, the nearer the angle which this line makes with the horizontal approaches 90°, the more symmetrically beautiful will be the general arrangement of the parts (see line _y z_, figure 3, Plate IX.).

THE SCIENCE OF BEAUTY, AS DEVELOPED IN THE FORM OF THE HUMAN FIGURE.

The manner in which this science is developed in the symmetrical proportions of the entire human figure, is as remarkable for its simplicity as it has been shewn to be in those of the head and countenance. Having gone into very full details, and given ample illustration in two former works[19] upon this subject, I may here confine myself to the illustration of one description of figure, and to a reiteration of some facts stated in these works. These facts are, _1st_, That on a given line the human figure is developed, as to its principal points, entirely by lines drawn either from the extremities of this line, or from some obvious or determined localities. _2d_, That the angles which these lines make with the given line, are all simple sub-multiples of some given fundamental angle, or bear to it a proportion expressible under the most simple relations, such as those which constitute the scale of music. _3d_, That the contour is resolved into a series of ellipses of the same simple angles. And, _4th_, That these ellipses, like the lines, are inclined to the first given line by angles which are simple sub-multiples of the given fundamental angle. From which four facts, and agreeably to the hypothesis I have adopted, it results as a natural consequence that the only effort which the mind exercises through the eye, in order to put itself in possession of the data for forming its judgment, is this, that it compares the angles about a point, and thereby appreciates the simplicity of their relations. In selecting the prominent features of a figure, the eye is not seeking to compare their relative distances—it is occupied solely with their relative positions. In tracing the contour, in like manner, it is not left in vague uncertainty as to what is the curve which is presented to it; unconsciously it feels the complete ellipse developed before it; and if that ellipse and its position are both formed by angles of the same simple relative value as those which aided its determination of the positions of the prominent features, it is satisfied, and finds the symmetry perfect.