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.