Mathematics may be great fun. Even simple arithmetic is not without its
comic side, as when it enables you to find, with a little management,
the Number of the Beast in the name of any one you dislike. Then there
is “the low cunning of algebra.” It became low cunning indeed when Euler
drove (so the anecdotist relates) Diderot out of Russia with a sham
algebraical formula. “Monsieur,” said Euler gravely, “_(a + bⁿ)/n = x,
donc Dieu existe; répondez_.” Diderot, no algebraist, could not answer,
and left.
But geometry furnishes the best sport. Here is a learned American
archæologist, Mr. Jay Hambidge, lecturing to that august body the
Hellenic Society and revealing to them his discovery that the secret
of classic Greek art (of the best period) is a matter of two magic
rectangles. I understand that the learned gentleman himself did not make
this extreme claim about the “secret” of “Art,” but it was at any rate
so described in the report on which my remarks are based. Mr. Hambidge
appears to have devoted years of labour and ingenuity to his researches.
The result is in any case of curious interest. But how that result can
be said to be “the secret of Greek art revealed” I wholly fail to see.
Let us look first at his rectangles. His first is 2 × √5. It is said that
these figures represent the ratio of a man’s height to the full span of
his outstretched fingers. But what man? Of what race and age? Well, let
us say an average Greek of the best period, and pass on. Mr. Hambidge has
found this rectangle over and over again in the design of the Parthenon.
“Closely akin” to it, says the report, is another fundamental rectangle,
of which the two dimensions are in the ratio of Leonardo’s famous “golden
section.” That ratio is obtained by dividing a straight line so that its
greater is to its lesser part as the whole is to the greater. Let us give
a mathematical meaning to the “closely akin.” Calling the lesser part 1
and the greater _x_, then—
_x_/1 = (_x_ + 1)/_x_ or _x_² - _x_ - 1 = 0
which gives you
_x_ = (√5 + 1)/2.
The square roots will not trouble you when you come to constructing your
rectangles, for the diagonal of the first is √(5 + 4), or 3. If AB is
your side 2, draw a perpendicular to it through B, and with A as centre
describe the arc of a circle of radius 3; the point of intersection will
give C, the other end of the diagonal. The second rectangle maintains AB,
and simply prolongs BC by half of AB or 1. Just as the dimensions of
the first rectangle are related to those of (selected) man, and to the
plan of the Parthenon, so those of the second are related, it seems, to
the arrangement of seeds in the sunflower and to the plan of some of the
Pyramids. Sir Theodore Cook writes to _The Times_ to say that both the
sunflower and the Pyramid discoveries are by no means new.
The fact is the theory of “beautiful” rectangles is not new. The classic
exponent of it is Fechner, who essayed to base it on actual experiment.
He placed a number of rectangular cards of various dimensions before his
friends, and asked them to select the one they thought most beautiful.
Apparently the “golden section” rectangle got most votes. But “most of
the persons began by saying that it all depended on the application to
be made of the figure, and on being told to disregard this, showed much
hesitation in choosing.” (Bosanquet: “History of Æsthetic,” p. 382.) If
they had been Greeks of the best period, they would have all gone with
one accord for the “golden section” rectangle.
Nor have the geometers of beauty restricted their favours to the
rectangle. Some have favoured the circle, some the square, others the
ellipse. And what about Hogarth’s “line of beauty”? I last saw it
affectionately alluded to in the advertisement of a corset manufacturer.
So, evidently, Hogarth’s idea has not been wasted.
One sympathizes with Fechner’s friends who said it all depended upon the
application to be made of the figure. The “art” in a picture is generally
to be looked for inside the frame. The Parthenon may have been planned
on the √5/2 rectangle, but you cannot evolve the Parthenon itself out of
that vulgar fraction. Fechner proceeded on the assumption that art is a
physical fact and that its “secret” could be wrung out of it, as in any
other physical inquiry, by observation and experiment, by induction from
a sufficient number of facts. But when he came to have a theory of it he
found, like anybody else, that introspection was the only way.
And whatever rectangles Mr. Hambidge may discover in Greek works of
art, he will not thereby have revealed the secret of Greek art. For
rectangles are physical facts (when they are not mere abstractions),
and art is not a physical fact, but a spiritual activity. It is in the
mind of the artist, it is his vision, the expression of his intuition,
and beauty is only another name for perfect expression. That, at any
rate, is the famous “intuition-expression” theory of Benedetto Croce,
which at present holds the field. It is a theory which, of course,
presents many difficulties to the popular mind—what æsthetic theory does
not?—but it covers the ground, as none other does, and comprehends all
arts, painting, poetry, music, sculpture, and the rest, in one. Its main
difficulty is its distinction between the æsthetic fact, the artist’s
expression, and the physical fact, the externalization of the artist’s
expression, the so-called “work” of art. Dr. Bosanquet has objected
that this seems to leave out of account the influence on the artist’s
expression of his material, his medium, but Croce, I think, has not
overlooked that objection (“Estetica,” Ch. XIII., end), though many of
us would be glad if he could devote some future paper in the _Critica_
to meeting it fairly and squarely. Anyhow, æsthetics is not a branch
of physics, and the “secret” of art is not to be “revealed” by a whole
Euclidful of rectangles.
But it is, of course, an interesting fact that certain Greeks, and before
them certain Egyptians, took certain rectangles as the basis of their
designs—rectangles which are also related to the average proportions of
the human body and to certain botanical types. If Mr. Hambidge—or his
predecessors, of whom Sir Theodore Cook speaks—have established this they
have certainly put their fingers on an engaging convention. Who would
have thought that the “golden section” that very ugly-looking (√5 + 1)/2
could have had so much in it? The builder of the Great Pyramid of Ghizeh
knew all about it in 4700 B.C. and the Greeks of the age of Pericles, and
then Leonardo da Vinci toyed with it—“_que de choses dans un menuet!_” It
is really rather cavalier of Croce to dismiss this golden section along
with Michael Angelo’s serpentine lines of beauty as the astrology of
Æsthetic.