following parts of the right angle:—
Tonic. Dominant. Mediant. Submediant.
(¹⁄₂) (¹⁄₃) (¹⁄₅) (³⁄₁₀)
(¹⁄₄) (¹⁄₆)
When we reflect upon the variety of harmonic ellipses that may
be described, and the innumerable positions in which they may be
harmonically placed with respect to the horizontal and vertical lines,
as well as upon the various modes in which their circumferences may be
combined, the variety which may be introduced amongst such forms as the
foregoing appears almost endless. My second example is that of—
_An Ancient Grecian Marble Vase of a Vertical Composition._
I shall now proceed to another class of the ancient Greek vase, the form
of which is of a more complex character. The specimen I have chosen for
the first example of this class is one of those so correctly measured and
beautifully delineated by Tatham in his unequalled work.[25] This vase is
a work of ancient Grecian art in Parian marble, which he met with in the
collection at the Villa Albani, near Rome. Its height is 4 ft. 4¹⁄₂ in.
[Sidenote: Plate XIII.]
The following is the formula by which I endeavour to develop its harmonic
elements:—
Let A B (Plate XIII.) represent the full height of this vase. Through B
draw B D, making an angle of (¹⁄₅) with the vertical. Through D draw D O
vertical, through A draw A C, making an angle of (²⁄₅); through B draw
B L, making an angle of (¹⁄₂), and B S, making an angle of (³⁄₁₀), each
with the vertical. Through A draw A D, through B draw B O, through L draw
L N, through C draw C F, and through S draw S P, all horizontal. Through
A draw A H, making an angle of (¹⁄₁₀) with the vertical, and through
H draw H M vertical. Draw similar lines on the other side of A B, and
the rectilinear portion of the diagram is complete, and its inscribing
rectangle that of (³⁄₈).
The curvilinear portion may thus be added—
Take a cut-out ellipse of (¹⁄₃), whose greater axis is about the length
of the body of the intended vase, place it with its lesser axis upon the
line S P, and its greater axis upon the line D O, and trace the part _a
b_ of its circumference upon the diagram. Place the same ellipse with
one of its foci upon C, and its greater axis upon C F, and trace its
circumference upon the diagram. Take a cut-out ellipse of (¹⁄₅), whose
greater axis is nearly equal to that of the ellipse already used; place
it with its greater axis upon M H, and its lesser axis upon L N, and
trace its circumference upon the diagram. Make similar tracings upon the
other side of A B, and the diagram is complete. In this, as in the other
diagrams, the strong portions of the lines give the contour of the vase.
The harmonic elements of this classical form, therefore, appear to be the
right angle and its following parts:—
Tonic. Dominant. Mediant. Submediant.
(¹⁄₂) (¹⁄₃) (²⁄₅) (³⁄₁₀)
(¹⁄₅)
(¹⁄₁₀)
My third example is that of—
_An Ancient Grecian Vase of a Horizontal Composition._
This example belongs to the same class as the last, but it is of a
horizontal composition. It was carefully drawn from the original in the
museum of the Vatican by Tatham, in whose etchings it will be found with
its ornamental decorations. The diagram of its harmonic elements may be
constructed as follows:—
[Sidenote: Plate XIV.]
Let A B (Plate XIV.) represent the full height of the vase. Through B
draw B D, making an angle of (²⁄₅) with the vertical. Through A draw A H,
A L, and A C, making respectively the following angles, (¹⁄₅) with the
vertical, (⁴⁄₉) with the vertical, and (³⁄₁₀) with the horizontal. These
angles determine the horizontal lines H B, L N, and C F, which divide
the vase into its parts, and the inscribing rectangle D G K O is (³⁄₈).
This completes the rectilinear portion of the diagram. The ellipse by
which the curvilinear portion is added is one of (¹⁄₅), the greater axis
of which, at _a b_, as also at _c d_, makes an angle of (¹⁄₁₂) with
the vertical, and the same axis at _e f_ an angle of (¹⁄₁₂) with the
horizontal.
The harmonic elements of this vase, therefore, appear to be:—
Tonic. Dominant. Mediant. Submediant. Supertonic.
The Right (¹⁄₁₂) (²⁄₅) (³⁄₁₀) (⁴⁄₉)
Angle. (¹⁄₅)
My remaining examples are those of—
_Etruscan Vases._
Of these vases I give four examples, by which the simplicity of the
method employed in applying the harmonic law will be apparent.
[Sidenote: Plate XV.]
The inscribing rectangle D G E K of fig. 1, Plate XV., is one of (³⁄₈),
within which are arranged tracings from an ellipse of (³⁄₁₀), whose
greater axis at _a b_ makes an angle of (¹⁄₁₂), at _c d_ an angle of
(³⁄₁₀), and at _e f_ an angle of (³⁄₄), with the vertical. The harmonic
elements of the contour of this vase, therefore, appear to be:—
Tonic. Dominant. Subdominants. Submediant.
The Right (¹⁄₁₂) (³⁄₄) (³⁄₁₀)
Angle. (³⁄₈)
The inscribing rectangle L M N O of fig. 2 is that of (¹⁄₂), within which
are arranged tracings from an ellipse of (¹⁄₃), whose greater axis, at
_a b_ and _c d_ respectively, makes angles of (¹⁄₂) and (⁴⁄₉) with the
horizontal, while that at _e f_ is in the horizontal line. The harmonic
elements of the contour of this vase, therefore, appear to be:—
Tonic. Dominant. Subtonic.
(¹⁄₂) (¹⁄₃) (⁴⁄₉)
[Sidenote: Plate XVI.]
The inscribing rectangle P Q R S of fig. 1, Plate XVI., is one of (⁴⁄₉),
within which are arranged tracings from an ellipse of (³⁄₈), whose
greater axis, at _a b_, _c d_, and _e f_, makes respectively angles
of (¹⁄₆) with the horizontal, (³⁄₅) and (⁴⁄₅) with the vertical. Its
harmonic elements, therefore, appear to be:—
Tonic. Dominant. Mediant. Supertonic. Subdominant. Submediant.
The Right (¹⁄₆) (⁴⁄₅) (⁴⁄₉) (³⁄₈) (³⁄₅)
Angle.
The inscribing rectangle T U V X of fig. 2 is one of (⁴⁄₉), within which
are arranged tracings from an ellipse of (³⁄₈) whose greater axis at _a
b_ is in the vertical line, and at _c d_ makes an angle of (¹⁄₂). The
harmonic elements of the contour of this vase, therefore, appear to be:—
Tonic. Submediant. Supertonic.
(¹⁄₂) (³⁄₈) (⁴⁄₉)
These four Etruscan vases, the contours of which are thus reduced to the
harmonic law of nature, are in the British Museum, and engravings of
them are to be found in the well-known work of Mr Henry Moses, Plates
4, 6, 14, and 7, respectively, where they are represented with their
appropriate decorations and colours.
To these, I add two examples of—
_Ancient Grecian Ornament._
I have elsewhere shewn[26] that the elliptic curve pervades the Parthenon
from the entases of the column to the smallest moulding, and we need not,
therefore, be surprised to find it employed in the construction of the
only two ornaments belonging to that great work.
[Sidenote: Plate XVII.]
In the diagram (Plate XVII.), I endeavour to exhibit the geometric
construction of the upper part of one of the ornamental apices, termed
antefixæ, which surmounted the cornice of the Parthenon.
The first ellipse employed is that of (¹⁄₃), whose greater axis _a b_ is
in the vertical line; the second is also that of (¹⁄₃), whose greater
axis _c d_ makes, with the vertical, an angle of (¹⁄₁₂); the third
ellipse is the same with its major axis _e f_ in the vertical line.
Through one of the foci of this ellipse at A the line A C is drawn, and
upon the part of the circumference C _e_, the number of parts, 1, 2, 3,
4, 5, 6, 7, of which the surmounting part of this ornament is to consist,
are set off. That part of the circumference of the ellipse whose larger
axis is _c d_ is divided from _g_ to _c_ into a like number of parts. The
third ellipse employed is one of (¹⁄₄).
Take a cut-out ellipse of this kind, whose larger axis is equal in length
to the inscribing rectangle. Place it with its vertex upon the same
ellipse at _g_, so that its circumference will pass through C, and trace
it; remove its apix first to _p_, then to _q_, and proceed in the same
way to _q_, _r_, _s_, _t_, _u_, and _v_, so that its circumference will
pass through the seven divisions on _c g_ and _e_ C: _v o_, _u n_, _t m_,
_s i_, _r k_, _q j_, _p l_, and _g x_, are parts of the larger axes of
the ellipses from which the curves are traced. The small ellipse of which
the ends of the parts are formed is that of (¹⁄₃).
[Sidenote: Plate XVIII.]
In the diagram (Plate XVIII.), I endeavour to exhibit the geometric
construction of the ancient Grecian ornament, commonly called the
_Honeysuckle_, from its resemblance to the flower of that name. The first
part of the process is similar to that just explained with reference to
the antefixæ of the Parthenon, although the angles in some parts differ.
The contour is determined by the circumference of an ellipse of (¹⁄₃),
whose major axis A B makes an angle of (¹⁄₉) with the vertical, and
the leaves or petals are arranged upon a portion of the perimeter of a
similar ellipse whose larger axis E F is in the vertical line, and these
parts are again arranged upon a similar ellipse whose larger axis C D
makes an angle of (¹⁄₁₂) with the vertical. The first series of curved
lines proceeding from 1, 2, 3, 4, 5, 6, 7, and 8, are between K E and H
C, part of the circumference of an ellipse of (¹⁄₃); and those between C
H and A G are parts of the circumference of four ellipses, each of (¹⁄₃),
but varying as to the lengths of their larger axes from 5 to 3 inches.
The change from the convex to the concave, which produces the ogie forms
of which this ornament is composed, takes place upon the line C H, and
the lines _a b_, _c d_, _e f_, _g h_, _i k_, _l m_, _n o_, and _p q_, are
parts of the larger axis of the four ellipses the circumference of which
give the upper parts of the petals or leaves.
This peculiar Grecian ornament is often, like the antefixæ of the
Parthenon, combined with the curve of the spiral scroll. But the volute
is so well understood that I have not rendered my diagrams more complex
by adding that figure. Many varieties of this union are to be found in
Tatham’s etchings, already referred to. The antefixæ of the Parthenon,
and its only other ornament the honeysuckle, as represented on the soffit
of the cornice, are to be found in Stewart’s “Athens.”
APPENDIX.
No. I.
In pages 34, 35, and 58, I have reiterated an opinion advanced in several
of my former works, viz., that, besides genius, and the study of nature,
an additional cause must be assigned for the general excellence which
characterises such works of Grecian art as were executed during a period
commencing about 500 B.C., and ending about 200 B.C. And that this cause
most probably was, that the artists of that period were instructed in
a system of fixed principles, based upon the doctrines of Pythagoras
and Plato. This opinion has not been objected to by the generality of
those critics who have reviewed my works; but has, however, met with an
opponent, whose recondite researches and learned observations are worthy
of particular attention. These are given in an essay by Mr C. Knight
Watson, “On the Classical Authorities for Ancient Art,” which appeared in
the _Cambridge Journal of Classical and Sacred Philology_ in June 1854.
As this essay is not otherwise likely to meet the eyes of the generality
of my readers, and as the objections he raises to my opinion only occupy
two out of the sixteen ample paragraphs which constitute the first part
of the essay, I shall quote them fully:—
“The next name on our list is that of the famous Euphranor
(B.C. 362). For the fact that to the practice of sculpture
and of painting he added an exposition of the theory, we are
indebted to Pliny, who says (xxxv. 11, 40), ‘Volumina quoque
composuit de symmetria et coloribus.’ When we reflect on the
_critical_ position occupied by Euphranor in the history
of Greek art, as a connecting link between the idealism of
Pheidias and the naturalism of Lysippus, we can scarcely
overestimate the value of a treatise on art proceeding from
such a quarter. This is especially the case with the first
of the two works here assigned to Euphranor. The inquiries
which of late years have been instituted by Mr D. R. Hay of
Edinburgh, on the proportions of the human figure, and on the
natural principles of beauty as illustrated by works of Greek
art, constitute an epoch in the study of æsthetics and the
philosophy of form. Now, in the presence of these inquiries,
or of such less solid results as Mr Hay’s predecessors in
the same field have elicited, it naturally becomes an object
of considerable interest to ascertain how far these laws of
form and principles of beauty were consciously developed in
the mind, and by the chisel, of the sculptor: how far any
such system of curves and proportions as Mr Hay’s was used
by the Greek as a practical manual of his craft. Without in
the least wishing to impugn the accuracy of that gentleman’s
results—a piece of presumption I should do well to avoid—I must
be permitted to doubt whether the ‘Symmetria’ of Euphranor
contained anything analogous to them in kind, or indeed equal
in value. It must not be forgotten that the truth of Mr Hay’s
theory is perfectly compatible with the fact, that of such
theory the Greek may have been utterly ignorant. It is on this
fact I insist: it is here that I join issue with Mr Hay, and
with his reviewer in a recent number of _Blackwood’s Magazine_.
Or, to speak more accurately,—while I am quite prepared to find
that the Elgin marbles will best of all stand the test which Mr
Hay has hitherto applied, I believe, to works of a later age, I
am none the less convinced that it is precisely that golden age
of Hellenic art to which they belong, precisely that first and
chief of Hellenic artists by whom they were executed, to which
and to whom any such line of research on the laws of form would
have been pre-eminently alien. Pheidias, remember, by the right
of primogeniture, is the ruling spirit of idealism in art. Of
spontaneity was that idealism begotten and nurtured: by any
such system as Mr Hay’s, that spontaneity would be smothered
and paralysed. Pheidias copied an idea in his own mind—‘Ipsius
in mente insidebat species pulchritudinis eximia quædam’
(_Cic._);—later ages copied _him_. He created: they criticised.
He was the author of Iliads: they the authors of Poetics.
Doubtless, if you unsphere the _spirit_ of Mr Hay’s theories,
you will find nothing discordant with what I have here said.
That is a sound view of Beauty which makes it consist in that
due subordination of the parts to the whole, that due relation
of the parts to each other, which Mendelssohn had in his
mind when he said that the essence of beauty was ‘unity in
variety’—variety beguiling the imagination, the perception of
unity exercising the thewes and sinews of the intellect. On
such a view of beauty, Mr Hay’s theory may, _in spirit_, be
said to rest. But here, as in higher things, it is the letter
that killeth, while the spirit giveth life. And accordingly I
must enter a protest against any endeavour to foist upon the
palmy days of Hellenic art systems of geometrical proportions
incompatible, as I believe, with those higher and broader
principles by which the progress of ancient sculpture was
ordered and governed—systems which will bear nothing of that
‘felicity and chance by which’—and not by rule—‘Lord Bacon
believed that a painter may make a better face than ever was:’
systems which take no account of that fundamental distinction
between the schools of Athens and of Argos, and their
respective disciples and descendants, without which you will
make nonsense of the pages of Pliny, and—what is worse—sense
of the pages of his commentators;—systems, in short, which may
have their value as instruments for the education of the eye,
and for instructions in the arts of design, but must be cast
aside as matters of learned trifling and curious disputation,
where they profess to be royal roads to art, and to map the
mighty maze of a creative mind. And even as regards the
application of such a system of proportions to those works of
sculpture which are posterior to the Pheidian age, only partial
can have been the prevalence which it or any other _one_ system
can have obtained. The discrepancies of different artists
in the treatment of what was called, technically called,
_Symmetria_ (as in the title of Euphranor’s work) were, by the
concurrent testimony of all ancient writers, far too salient
and important to warrant the supposition of any uniform scale
of proportions, as advocated by Mr Hay. Even in Egypt, where
one might surely have expected that such uniformity would have
been observed with far greater rigour than in Greece, the
discoveries of Dr Lepsius (_Vorläufige Nachricht_, Berlin,
1849) have elicited three totally different κανόνες, one of
which is identical with the system of proportions of the human
figure detailed in Diodorus. While we thus venture to differ
from Mr Hay on the historical data he has mixed up with his
inquiries, we feel bound to pay him a large and glad tribute of
praise for having devised a system of proportions which rises
superior to the idiosyncracies of different artists, which
brings back to one common type the sensations of eye and ear,
and so makes a giant stride towards that _codification_, if
I may so speak, of the laws of the universe which it is the
business of the science to effect. I have no hesitation in
saying, that, for scientific precision of method and importance
of results, Albert Durer, Da Vinci, and Hogarth, not to mention
less noteworthy writers, must all yield the palm to Mr Hay.
“I am quite aware that in the digression I have here allowed
myself, on systems of proportions prevalent among ancient
artists, and on the probable contents of such treatises as that
of Euphranor, _De Symmetria_, I have laid myself open to the
charge of treating an intricate question in a very perfunctory
way. At present the exigencies of the subject more immediately
in hand allow me only to urge in reply, that, as regards the
point at issue—I mean the ‘solidarité’ between theories such as
Mr Hay’s and the practice of Pheidias—the _onus probandi_ rests
with my adversaries.”
I am bound, in the first place, gratefully to acknowledge the kind and
complimentary notice which, notwithstanding our difference of opinion,
this author has been pleased to take of my works; and, in the second, to
assure him that if any of them profess to be “royal roads to art,” or
to “map the mighty maze of a creative mind,” they certainly profess to
do more than I ever meant they should; for I never entertained the idea
that a system of æsthetic culture, even when based upon a law of nature,
was capable of effecting any such object. But I doubt not that this too
common misapprehension of the scope and tendency of my works must arise
from a want of perspicuity in my style.
I have certainly, on one occasion,[27] gone the length of stating
that as poetic genius must yield obedience to the rules of rhythmical
measure, even in the highest flights of her inspirations; and musical
genius must, in like manner, be subject to the strictly defined laws of
harmony in the most delicate, as well as in the most powerfully grand
of her compositions; so must genius, in the formative arts, either
consciously or unconsciously have clothed her creations of ideal beauty
with proportions strictly in accordance with the laws which nature has
set up as inflexible standards. If, therefore, the laws of proportion, in
their relation to the arts of design, constitute the harmony of geometry,
as definitely as those that are applicable to poetry and music produce
the harmony of acoustics; the former ought, certainly, to hold the same
relative position in those arts which are addressed to the eye, that is
accorded to the latter in those which are addressed to the ear. Until
so much science be brought to bear upon the arts of design, the student
must continue to copy from individual and imperfect objects in nature, or
from the few existing remains of ancient Greek art, in total ignorance of
the laws by which their proportions are produced, and, what is equally
detrimental to art, the accuracy of all criticism must continue to rest
upon the indefinite and variable basis of mere opinion.
It cannot be denied that men of great artistic genius are possessed of
an intuitive feeling of appreciation for what is beautiful, by means
of which they impart to their works the most perfect proportions,
independently of any knowledge of the definite laws which govern that
species of beauty. But they often do so at the expense of much labour,
making many trials before they can satisfy themselves in imparting to
them the true proportions which their minds can conceive, and which,
along with those other qualities of expression, action, or attitude,
which belong more exclusively to the province of genius. In such
cases, an acquaintance with the rules which constitute the science of
proportion, instead of proving fetters to genius, would doubtless afford
her such a vantage ground as would promote the more free exercise of
her powers, and give confidence and precision in the embodiment of her
inspirations; qualities which, although quite compatible with genius, are
not always intuitively developed along with that gift.
It is also true that the operations of the conceptive faculty of the mind
are uncontrolled by definite laws, and that, therefore, there cannot
exist any rules by the inculcation of which an ordinary mind can be
imbued with genius sufficient to produce works of high art. Nevertheless,
such a mind may be improved in its perceptive faculty by instruction in
the science of proportion, so as to be enabled to exercise as correct and
just an appreciation of the conceptions of others, in works of plastic
art, as that manifested by the educated portion of mankind in respect
to poetry and music. In short, it appears that, in those arts which are
addressed to the ear, men of genius communicate the original conceptions
of their minds under the control of certain scientific laws, by means
of which the educated easily distinguish the true from the false, and
by which the works of the poet and musical composer may be placed above
mere imitations of nature, or of the works of others; while, in those
arts that are addressed to the eye in their own peculiar language, such
as sculpture, architecture, painting, and ornamental design, no such laws
are as yet acknowledged.
Although I am, and ever have been, far from endeavouring “to foist upon
the palmy days of Hellenic art” any system incompatible with those higher
and more intellectual qualities which genius alone can impart; yet, from
what has been handed down to us by writers on the subject, meagre as it
is, I cannot help continuing to believe that, along with the physical and
metaphysical sciences, æsthetic science was taught in the early schools
of Greece.