work as the ground of this belief, and to which the author, from whose
essay I have quoted, undoubtedly refers. It is well known that, in the
time of Pythagoras, the treasures of science were veiled in mystery to
all but the properly initiated, and the results of its various branches
only given to the world in the works of those who had acquired this
knowledge. So strictly was this secresy maintained amongst the disciples
and pupils of Pythagoras, that any one divulging the sacred doctrines
to the profane, was expelled the community, and none of his former
associates allowed to hold further intercourse with him; it is even
said, that one of his pupils incurred the displeasure of the philosopher
for having published the solution of a problem in geometry.[28] The
difficulty, therefore, which is expressed by writers, shortly after the
period in which Pythagoras lived, regarding a precise knowledge of his
theories, is not to be wondered at, more especially when it is considered
that he never committed them to writing. It would appear, however, that
he proceeded upon the principle, that the order and beauty so apparent
throughout the whole universe, must compel men to believe in, and refer
them to, an intelligible cause. Pythagoras and his disciples sought for
properties in the science of numbers, by the knowledge of which they
might attain to that of nature; and they conceived those properties to
be indicated in the phenomena of sonorous bodies. Observing that Nature
herself had thus irrevocably fixed the numerical value of the intervals
of musical tones, they justly concluded that, as she is always uniform
in her works, the same laws must regulate the general system of the
universe.[29] Pythagoras, therefore, considered numerical proportion as
the great principle inherent in all things, and traced the various forms
and phenomena of the world to numbers as their basis and essence.
How the principles of numbers were applied in the arts is not recorded,
farther than what transpires in the works of Plato, whose doctrines were
from the school of Pythagoras. In explaining the principle of beauty,
as developed in the elements of the material world, he commences in the
following words:—“But when the Artificer began to adorn the universe,
he first of all figured with forms and numbers, fire and earth, water
and air—which possessed, indeed, certain traces of the true elements,
but were in every respect so constituted as it becomes anything to be
from which Deity is absent. But we should always persevere in asserting
that Divinity rendered them, as much as possible, the most beautiful
and the best, when they were in a state of existence opposite to such
a condition.” Plato goes on further to say, that these elementary
bodies must have forms; and as it is necessary that every depth should
comprehend the nature of a plane, and as of plane figures the triangle
is the most elementary, he adopts two triangles as the originals or
representatives of the isosceles and the scalene kinds. The first
triangle of Plato is that which forms the half of the square, and is
regulated by the number, 2; and the second, that which forms the half
of the equilateral triangle, which is regulated by the number, 3;
from various combinations of these, he formed the bodies of which he
considered the elements to be composed. To these elementary figures I
have already referred.
Vitruvius, who studied architecture ages after the arts of Greece had
been buried in the oblivion which succeeded her conquest, gives the
measurements of various details of monuments of Greek art then existing.
But he seems to have had but a vague traditionary knowledge of the
principle of harmony and proportion from which these measurements
resulted. He says—“The several parts which constitute a temple ought
to be subject to the laws of symmetry; the principles of which should
be familiar to all who profess the science of architecture. Symmetry
results from proportion, which, in the Greek language, is termed
analogy. Proportion is the commensuration of the various constituent
parts with the whole; in the existence of which symmetry is found to
consist. For no building can possess the attributes of composition
in which symmetry and proportion are disregarded; nor unless there
exist that perfect conformation of parts which may be observed in a
well-formed human being.” After going at some length into details, he
adds—“Since, therefore, the human figure appears to have been formed
with such propriety, that the several members are commensurate with
the whole, the artists of antiquity (meaning those of Greece at the
period of her highest refinement) must be allowed to have followed the
dictates of a judgment the most rational, when, transferring to works of
art principles derived from nature, every part was so regulated as to
bear a just proportion to the whole. Now, although the principles were
universally acted upon, yet they were more particularly attended to in
the construction of temples and sacred edifices, the beauties or defects
of which were destined to remain as a perpetual testimony of their skill
or of their inability.”
Vitruvius, however, gives no explanation of this ancient principle
of proportion, as derived from the human form; but plainly shews his
uncertainty upon the subject, by concluding this part of his essay in the
following words: “If it be true, therefore, that the decenary notation
was suggested by the members of man, and that the laws of proportion
arose from the relative measures existing between certain parts of each
member and the whole body, it will follow, that those are entitled to our
commendation who, in building temples to their deities, proportioned the
edifices, so that the several parts of them might be commensurate with
the whole.” It thus appears certain that the Grecians, at the period of
their highest excellence, had arrived at a knowledge of some definite
mathematical law of proportion, which formed a standard of perfectly
symmetrical beauty, not only in the representation of the human figure
in sculpture and painting, but in architectural design, and indeed in
all works where beauty of form and harmony of proportion constituted
excellence. That this law was not deduced from the proportions of
the human figure, as supposed by Vitruvius, but had its origin in
mathematical science, seems equally certain; for in no other way can we
satisfactorily account for the proportions of the beau ideal forms of the
ancient Greek deities, or of those of their architectural structures,
such as the Parthenon, the temple of Theseus, &c., or for the beauty that
pervades all the other formative art of the period.
This system of geometrical harmony, founded, as I have shewn it to be,
upon numerical relations, must consequently have formed part of the Greek
philosophy of the period, by means of which the arts began to progress
towards that great excellence which they soon after attained. A little
further investigation will shew, that immediately after this period a
theory connected with art was acknowledged and taught, and also that
there existed a Science of Proportion.
Pamphilus, the celebrated painter, who flourished about four hundred
years before the Christian era, from whom Apelles received the
rudiments of his art, and whose school was distinguished for scientific
cultivation, artistic knowledge, and the greatest accuracy in drawing,
would admit no pupil unacquainted with geometry.[30] The terms upon which
he engaged with his students were, that each should pay him one talent
(£225 sterling) previous to receiving his instructions; for this he
engaged “to give them, _for ten years_, lessons founded on an excellent
theory.”[31]
It was by the advice of Pamphilus that the magistrates of Sicyon ordained
that the study of drawing should constitute part of the education of the
citizens—“a law,” says the Abbé Barthélémie, “which rescued the fine arts
from servile hands.”
It is stated of Parrhasius, the rival of Zeuxis, who flourished about
the same period as Pamphilus, that he accelerated the progress of art by
purity and correctness of design; “for he was acquainted with the science
of Proportions. Those he gave his gods and heroes were so happy, that
artists did not hesitate to adopt them.” Parrhasius, it is also stated,
was so admired by his contemporaries, that they decreed him the name of
Legislator.[32] The whole history of the arts in Egypt and Greece concurs
to prove that they were based on geometric precision, and were perfected
by a continued application of the same science; while in all other
countries we find them originating in rude and misshapen imitations of
nature.
In the earliest stages of Greek art, the gods—then the only statues—were
represented in a tranquil and fixed posture, with the features exhibiting
a stiff inflexible earnestness, their only claim to excellence being
symmetrical proportion; and this attention to geometric precision
continued as art advanced towards its culminating point, and was
thereafter still exhibited in the neatly and regularly folded drapery,
and in the curiously braided and symmetrically arranged hair.[33]
These researches, imperfect as they are, cannot fail to exhibit the
great contrast that exists between the system of elementary education
in art practised in ancient Greece, and that adopted in this country at
the present period. But it would be of very little service to point out
this contrast, were it not accompanied by some attempt to develop the
principles which seem to have formed the basis of this excellence amongst
the Greeks.
But in making such an attempt, I cannot accuse myself of assuming
anything incompatible with the free exercise of that spontaneity of
genius which the learned essayist says is the parent and nurse of
idealism. For it is in no way more incompatible with the free exercise of
artistic genius, that those who are so gifted should have the advantage
of an elementary education in the science of æsthetics, than it is
incompatible with the free exercise of literary or poetic genius, that
those who possess it should have the advantage of such an elementary
education in the science of philology as our literary schools and
colleges so amply afford.
No. II.
The letter from which I have made a quotation at page 42, arose out of
the following circumstance:—In order that my theory, as applied to the
orthographic beauty of the Parthenon, might be brought before the highest
tribunal which this country afforded, I sent a paper upon the subject,
accompanied by ample illustrations, to the Royal Institute of British
Architects, and it was read at a meeting of that learned body on the 7th
of February 1853; at the conclusion of which, Mr Penrose kindly undertook
to examine my theoretical views, in connexion with the measurements he
had taken of that beautiful structure by order of the Dilettanti Society,
and report upon the subject to the Royal Institute. This report was
published by Mr Penrose, vol. xi., No. 539 of _The Builder_, and the
letter from which I have quoted appeared in No. 542 of the same journal.
It was as follows:—
“GEOMETRICAL RELATIONS IN ARCHITECTURE.
“Will you allow me, through the medium of your columns, to
thank Mr Penrose for his testimony to the truth of Mr Hay’s
revival of Pythagoras? The dimensions which he gives are to
me the surest verification of the theory that 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 under such conditions.
“It is time that the scattered computations by which
architecture has been analysed—more than enough—be synthetised
into those formulæ which, as Mrs Somerville tells us, ‘are
emblematic of omniscience.’ The young architects of our day
feel trembling beneath their feet the ground whence men are
about to evoke the great and slumbering corpse of art. Sir, it
is food of this kind a reviving poetry demands.
——‘Give us truths,
For we are weary of the surfaces,
And die of inanition.’
“I, for one, as I listen to such demonstrations, whose scope
extends with every research into them, feel as if listening to
those words of Pythagoras, which sowed in the mind of Greece
the poetry whose manifestation in beauty has enchained the
world in worship ever since its birth. And I am sure that in
such a quarter, and in such thoughts, _we_ must look for the
first shining of that lamp of art, which even now is prepared
to burn.
“I know that this all sounds rhapsodical; but I know also that
until the architect becomes a poet, and not a tradesman, we
may look in vain for architecture: and I know that valuable
as isolated and detailed investigations are in their proper
bearings, yet that such purposes and bearings are to be found
in the enunciation of principles sublime as the generalities of
‘mathematical beauty.’
“AUTOCTHON.”
No. III.
Of the work alluded to at page 58 I was favoured with two opinions—the
one referring to the theory it propounds, and the other to its anatomical
accuracy—both of which I have been kindly permitted to publish.
The first is from Sir WILLIAM HAMILTON, Bart., professor of logic and
metaphysics in the University of Edinburgh, and is as follows:—
“Your very elegant volume is to me extremely interesting, as
affording an able contribution to what is the ancient, and,
I conceive, the true theory of the Beautiful. But though
your doctrine coincides with the one prevalent through all
antiquity, it appears to me quite independent and original
in you; and I esteem it the more, that it stands opposed to
the hundred one-sided and exclusive views prevalent in modern
times.—_16 Great King Street, March 5, 1849._”
The second is from JOHN GOODSIR, Esq., professor of anatomy in the
University of Edinburgh, and is as follows:—
“I have examined the plates in your work on the proportions
of the human head and countenance, and find the head you have
given as typical of human beauty to be anatomically correct
in its structure, only differing from ordinary nature in its
proportions being more mathematically precise, and consequently
more symmetrically beautiful.—_College, Edinburgh, 17th April
1849._”
No. IV.
I shall here shew, as I have done in a former work, how the curvilinear
outline of the figure is traced upon the rectilinear diagrams by portions
of the ellipse of (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆).
[Sidenote: Plate XIX.]
The outline of the head and face, from points (1) to (3) (fig. 1, Plate
XIX.), takes the direction of the two first curves of the diagram. From
point (3), the outline of the sterno-mastoid muscle continues to (4),
where, joining the outline of the trapezius muscle, at first concave, it
becomes convex after passing through (5), reaches the point (6), where
the convex outline of the deltoid muscle commences, and, passing through
(7), takes the outline of the arm as far as (8). The outline of the
muscles on the side, the latissimus dorsi and serratus magnus, commences
under the arm at the point (9), and joins the outline of the oblique
muscle of the abdomen by a concave curve at (10), which, rising into
convexity as it passes through the points (11) and (12), ends at (13),
where it joins the outline of the gluteus medius muscle. The outline of
this latter muscle passes convexly through the point (14), and ends at
(15), where the outline of the tensor vaginæ femoris and vastus externus
muscle of the thigh commences. This convex outline joins the concave
outline of the biceps of the thigh at (16), which ends in that of the
slight convexity of the condyles of the thigh-bone at (17). From this
point, the outline of the outer surface of the leg, which includes the
biceps, peroneus longus, and soleus muscles, after passing through the
point (18), continues convexly to (19), where the concave outline of the
tendons of the peroneus longus continues to (20), whence the outline of
the outer ankle and foot commences.
The outline of the mamma and fold of the arm-pit commences at (21),
and passes through the points (22) and (23). The circle at (24) is the
outline of the areola, in the centre of which the nipple is placed.
The outline of the pubes commences at (25), and ends at the point (26),
from which the outline of the inner surface of the thigh proceeds over
the gracilis, the sartorius, and vastus internus muscles, until it meets
the internal face of the knee-joint at (27), the outline of which ends
at (28). The outline of the inside of the leg commences by proceeding
over the gastrocnemius muscle as far as (29), where it meets that of the
soleus muscle, and continues over the tendons of the heel until it meets
the outline of the inner ankle and foot at (30).
The outline of the outer surface of the arm, as viewed in front, proceeds
from (8) over the remainder of the deltoid, in which there is here a
slight concavity, and, next, from (31) over the biceps muscle till (32),
where it takes the line of the long supinator, and passing concavely, and
almost imperceptibly, into the long and short radial extensor muscles,
reaches the wrist at (33). The outline of the inner surface of the arm
from opposite (9) commences by passing over the triceps extensor, which
ends at (34), then over the gentle convexity of the condyles of the bone
of the arm at (35), and, lastly, over the flexor sublimis which ends at
the wrist-joint (36).
The outline of the front of the figure commences at the point (1), (fig.
2, Plate II.), and, passing almost vertically over the platzsma-myoidis
muscles, reaches the point (2), where the neck ends by a concave curve.
From (2) the outline rises convexly over the ends of the clavicles,
and continues so over the pectoral muscle till it reaches (3), where
the mamma swells out convexly to (4), and returns convexly towards
(5), where the curve becomes concave. From (5) the outline follows the
undulations of the rectus muscle of the abdomen, concave at the points
(6) and (7), and having its greatest convexity at (8). This series
of curves ends with a slight concavity at the point (9), where the
horizontal branch of the pubes is situated, over which the outline is
convex and ends at (10).
The outline of the thigh commences at the point (11) with a slight
concave curve, and then swells out convexly over the extensors of the
leg, and, reaching (12), becomes gently concave, and, passing over the
patella at (13), becomes again convex until it reaches the ligament of
that bone, where it becomes gently concave towards the point (14), whence
it follows the slightly convex curve of the shin-bone, and then, becoming
as slightly concave, ends with the muscles in front of the leg at (15).
The outline of the back commences at the point (16), and, following with
a concave curve the muscles of the neck as far as (17), swells into a
convex curve over the trapezius muscle towards the point (18); passing
through which point, it continues to swell outward until it reaches
(19), half way between (18) and (20); whence the convexity, becoming
less and less, falls into the concave curve of the muscles of the loins
at (21), and passing through the point (22), it rises into convexity. It
then passes through the point (23), follows the outline of the gluteus
maximus, the convex curve of which rises to the point (24), and then
returns inwards to that of (25), where it ends in the fold of the hip.
From this point the outline follows the curve of the hamstring muscles
by a slight concavity as far as (26), and then, becoming gently convex,
it reaches (27); whence it becomes again gently concave, with a slight
indication of the condyle of the thigh-bone at (28), and, reaching (29),
follows the convex curve of the gastrocnemius muscle through the point
(30), and falling into the convex curve of the tendo Achilles at (31),
ends in the concavity over the heel at (32).
The outline of the front of the arm commences at the point (33), by a
gentle concavity at the arm-pit, and then swells out in a convex curve
over the biceps, reaching (34), where it becomes concave, and passing
through (35), again becomes convex in passing over the long supinator,
and, becoming gently concave as it passes the radial extensors, rises
slightly at (36), and ends at (37), where the outline of the wrist
commences. The outline of the back of the arm commences with a concave
curve at (38), which becomes convex as it passes from the deltoid to the
long extensor and ends at the elbow (39), from below which the outline
follows the convex curve of the extensor ulnaris, reaching the wrist at
the point (40).
It will be seen that the various undulations of the outline are regulated
by points which are determined generally by the intersections and
sometimes by directions and extensions of the lines of the diagram, in
the same manner in which I shewed proportion to be imparted, in a late
work, to the osseous structure. The mode in which the curves of (¹⁄₂),
(¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆) are thus so harmoniously blended in the
outline of the female figure, only remains to be explained.
The curves which compose the outline of the female form are therefore
simply those of (¹⁄₂), (¹⁄₃), (¹⁄₄), (¹⁄₅), and (¹⁄₆).
Manner in which these curves are disposed in the lateral outline (figure
1, Plate XIX.):—
Points. Curves.
Head from 1 to 2 (¹⁄₂)
Face ” 2 ” 3 (¹⁄₃)
Neck ” 3 ” 4 (¹⁄₅)
Shoulder ” 4 ” 6 (¹⁄₆)
” ” 6 ” 8 (¹⁄₄)
Trunk ” 9 ” 15 (¹⁄₄)
” ” 21 ” 24 (¹⁄₂)
Outer surface of thigh and leg ” 15 ” 20 (¹⁄₆)
Inner surface of thigh and leg ” 25 ” 30 (¹⁄₆)
Outer surface of the arm ” 8 ” 33 (¹⁄₆)
Inner surface of the arm ” 9 ” 36 (¹⁄₆)
Manner in which they are disposed in the outline (figure 2, Plate XIX.):—
Points. Curves.
Front of neck from 1 to 2 (¹⁄₆)
” trunk ” 2 ” 10 (¹⁄₄)
Back of neck ” 16 ” 18 (¹⁄₆)
” trunk ” 18 ” 23 (¹⁄₄)
” ” ” 23 ” 25 (¹⁄₃)
Front of thigh and leg ” 11 ” 13 (¹⁄₄)
” ” ” ” 13 ” 15 (¹⁄₆)
Back of thigh and leg ” 25 ” 32 (¹⁄₆)
Front of the arm ” 33 ” 37 (¹⁄₆)
Back of the arm ” 38 ” 40 (¹⁄₆)
Foot ” 0 ” 0 (¹⁄₆)
[Sidenote: Plate XX.]
In order to exemplify more clearly the manner in which these various
curves appear in the outline of the figure, I give in Plate XX. the whole
curvilinear figures, complete, to which these portions belong that form
the outline of the sides of the head, neck, and trunk, and of the outer
surface of the thighs and legs.
The various angles which the axes of these ellipses form with the
vertical, will be found amongst other details in the works I have just
referred to.
No. V.
At page 85 I have remarked upon the variety that may be introduced into
any particular form of vase; and, in order to give the reader an idea of
the ease with which this may be done without violating the harmonic law,
I shall here give three examples:—
[Sidenote: Plate XXI.]
The first of these (Plate XXI.) differs from the Portland vase, in the
concave curve of the neck flowing more gradually into the convex curve of
the body.
[Sidenote: Plate XXII.]
The second (Plate XXII.) differs from the same vase in the same change
of contour, as also in being of a smaller diameter at the top and at the
bottom.
[Sidenote: Plate XXIII.]
The third (Plate XXIII.) is the most simple arrangement of the elliptic
curve by which this kind of form may be produced; and it differs from the
Portland vase in the relative proportions of height and diameter, and in
having a fuller curve of contour.
The following comparison of the angles employed in these examples, with
the angles employed in the original, will shew that the changes of
contour in these forms, arise more from the mode in which the angles are
arranged than in a change of the angles themselves:—
Line Line Line Line
Plate VIII. _A C_ (¹⁄₂) _B C_ (¹⁄₃) _o p_ (H) _v u_ (³⁄₁₀)
Plate XXI. (¹⁄₂) (¹⁄₃) (²⁄₉) (¹⁄₄)
Plate XXII. (¹⁄₂) (¹⁄₃) (¹⁄₈) (⁴⁄₉)
Plate XXIII. (¹⁄₂) (¹⁄₄) (H) (-)
Line Line
Plate VIII. _m n_ (¹⁄₃) _i k_ (¹⁄₅) ellipse (¹⁄₄) rectangle (²⁄₅)
Plate XXI. (²⁄₉) (¹⁄₅) (¹⁄₄) (²⁄₅)
Plate XXII. (¹⁄₃) (¹⁄₅) (¹⁄₄) (²⁄₅)
Plate XXIII. (¹⁄₅) (¹⁄₅) ellipses {(¹⁄₃)} (¹⁄₃)
{(¹⁄₄)}
The harmonic elements of each are therefore simply the following parts of
the right angle:—
Tonic. Dominant. Mediant. Submediant.
Plate VIII. (¹⁄₂) (¹⁄₃) (¹⁄₅) (³⁄₁₀)
(¹⁄₄)
Tonic. Dominant. Mediant. Supertonic.
Plate XXI. (¹⁄₂) (¹⁄₃) (¹⁄₅) (²⁄₉)
(¹⁄₄)
Tonic. Dominant. Mediant. Supertonic.
Plate XXII. (¹⁄₂) (¹⁄₃) (¹⁄₅) (⁴⁄₉)
(¹⁄₄)
(¹⁄₈)
Tonic. Dominant. Mediant.
Plate XXIII. (¹⁄₂) (¹⁄₃) (¹⁄₅)
(¹⁄₄)
No. VI.
So far as I know, there has been only one attempt in modern times,
besides my own, to establish a universal system of proportion, based on a
law of nature, and applicable to art. This attempt consists of a work of
457 pages, with 166 engraved illustrations, by Dr Zeising, a professor in
Leipzic, where it was published in 1854.
One of the most learned and talented professors in our Edinburgh
University has reviewed that work as follows:—