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.
[Illustration: 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.
[Illustration: Fig. 2.]
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.