that we could take long walks in a forest of moss, and that our eyes
were correspondingly near each other. The moss-fibres would appear like
trees. On them we should see strange, unshapely monsters creeping about.
Branches of the oak-tree, at whose base our moss-forest lay, would seem
to us dark, immovable, myriad-branched clouds, painted high on the vault
of heaven; just as the inhabitants of Saturn, forsooth, might see their
enormous ring. On the tree-trunks of our mossy woodland we should find
colossal globes several feet in diameter, brilliantly transparent,
swayed by the winds with slow, peculiar motions. We should approach
inquisitively and should find that these globes, in which here and there
animals were gaily sporting, were liquid globes, in fact that they were
water. A short, incautious step, the slightest contact, and woe betide
us, our arm is irresistibly drawn by an invisible power into the
interior of the sphere and held there unrelentingly fast! A drop of dew
has engulfed in its capillary maw a manikin, in revenge for the
thousands of drops that its big human counterparts have quaffed at
breakfast. Thou shouldst have known, thou pygmy natural scientist, that
with thy present puny bulk thou shouldst not joke with capillarity!
My terror at the accident brings me back to my senses. I see I have
turned idyllic. You must pardon me. A patch of greensward, a moss or
heather forest with its tiny inhabitants have incomparably more charms
for me than many a bit of literature with its apotheosis of human
character. If I had the gift of writing novels I should certainly not
make John and Mary my characters. Nor should I transfer my loving pair
to the Nile, nor to the age of the old Egyptian Pharaohs, although
perhaps I should choose that time in preference to the present. For I
must candidly confess that I hate the rubbish of history, interesting
though it may be as a mere phenomenon, because we cannot simply observe
it but must also _feel_ it, because it comes to us mostly with
supercilious arrogance, mostly unvanquished. The hero of my novel would
be a cockchafer, venturing forth in his fifth year for the first time
with his newly grown wings into the light, free air. Truly it could do
no harm if man would thus throw off his inherited and acquired
narrowness of mind by making himself acquainted with the world-view of
allied creatures. He could not help gaining incomparably more in this
way than the inhabitant of a small town would in circumnavigating the
globe and getting acquainted with the views of strange peoples.
* * * * *
I have now conducted you, by many paths and by-ways, rapidly over hedge
and ditch, to show you what wide vistas we may reach in every field by
the rigorous pursuit of a single scientific fact. A close examination of
the two eyes of man has conducted us not only into the dim recesses of
humanity's childhood, but has also carried us far beyond the bourne of
human life.
It has surely often struck you as strange that the sciences are divided
into two great groups; that the so-called humanistic sciences, belonging
to the so-called "higher education," are placed in almost a hostile
attitude to the natural sciences.
I must confess I do not overmuch believe in this partition of the
sciences. I believe that this view will appear as childlike and
ingenuous to a matured age as the want of perspective in the old
paintings of Egypt does to us. Can it really be that "higher culture" is
to be gotten only from a few old pots and palimpsests, which are at best
mere scraps of nature, or that more is to be learned from them alone
than from all the rest of nature? I believe that both these sciences are
simply parts of the same science, which have begun at different ends. If
these two ends still act towards each other as the Montagues and
Capulets, if their retainers still indulge in lively tilts, I believe
that after all they are not in earnest. On the one side there is surely
a Romeo, and on the other a Juliet, who, some day, it is hoped, will
unite the two houses with a less tragic sequel than that of the play.
Philology began with the unqualified reverence and apotheosis of the
Greeks. Now it has begun to draw other languages, other peoples and
their histories, into its sphere; it has, through the mediation of
comparative linguistics, already struck up, though as yet somewhat
cautiously, a friendship with physiology.
Physical science began in the witch's kitchen. It now embraces the
organic and inorganic worlds, and with the physiology of articulation
and the theory of the senses, has even pushed its researches, at times
impertinently, into the province of mental phenomena.
In short, we come to the understanding of much within us solely by
directing our glance without, and _vice versa_. Every object belongs to
both sciences. You, ladies, are very interesting and difficult problems
for the psychologist, but you are also extremely pretty phenomena of
nature. Church and State are objects of the historian's research, but
not less phenomena of nature, and in part, indeed, very curious
phenomena. If the historical sciences have inaugurated wide extensions
of view by presenting to us the thoughts of new and strange peoples, the
physical sciences in a certain sense do this in a still greater degree.
In making man disappear in the All, in annihilating him, so to speak,
they force him to take an unprejudiced position without himself, and to
form his judgments by a different standard from that of the petty human.
But if you should ask me now why man has two eyes, I should answer:
That he may look at nature justly and accurately; that he may come to
understand that he himself, with all his views, correct and incorrect,
with all his _haute politique_, is simply an evanescent shred of nature;
that, to speak with Mephistopheles, he is a part of the part, and that
it is absolutely unjustified,
"For man, the microcosmic fool, to see
Himself a whole so frequently."
FOOTNOTES:
[Footnote 17: This effect is particularly noticeable in the size of
workmen on high chimneys and church-steeples--"steeple Jacks." When
the cables were slung from the towers of the Brooklyn bridge (277
feet high), the men sent out in baskets to paint them, appeared,
against the broad background of heaven and water, like
flies.--_Trans._]
[Footnote 18: See Joh. Müller, _Vergleichende Physiologie des
Gesichtssinnes_, Leipsic, 1826.]
ON SYMMETRY.[19]
An ancient philosopher once remarked that people who cudgelled their
brains about the nature of the moon reminded him of men who discussed
the laws and institutions of a distant city of which they had heard no
more than the name. The true philosopher, he said, should turn his
glance within, should study himself and his notions of right and wrong;
only thence could he derive real profit.
This ancient formula for happiness might be restated in the familiar
words of the Psalm:
"Dwell in the land, and verily thou shalt be fed."
To-day, if he could rise from the dead and walk about among us, this
philosopher would marvel much at the different turn which matters have
taken.
The motions of the moon and the other heavenly bodies are accurately
known. Our knowledge of the motions of our own body is by far not so
complete. The mountains and natural divisions of the moon have been
accurately outlined on maps, but physiologists are just beginning to
find their way in the geography of the brain. The chemical constitution
of many fixed stars has already been investigated. The chemical
processes of the animal body are questions of much greater difficulty
and complexity. We have our _Mécanique céleste_. But a _Mécanique
sociale_ or a _Mécanique morale_ of equal trustworthiness remains to be
written.
Our philosopher would indeed admit that we have made great progress. But
we have not followed his advice. The patient has recovered, but he took
for his recovery exactly the opposite of what the doctor prescribed.
Humanity is now returned, much wiser, from its journey in celestial
space, against which it was so solemnly warned. Men, after having become
acquainted with the great and simple facts of the world without, are now
beginning to examine critically the world within. It sounds absurd, but
it is true, that only after we have thought about the moon are we able
to take up ourselves. It was necessary that we should acquire simple and
clear ideas in a less complicated domain, before we entered the more
intricate one of psychology, and with these ideas astronomy principally
furnished us.
To attempt any description of that stupendous movement, which,
originally springing out of the physical sciences, went beyond the
domain of physics and is now occupied with the problems of psychology,
would be presumptuous in this place. I shall only attempt here, to
illustrate to you by a few simple examples the methods by which the
province of psychology can be reached from the facts of the physical
world--especially the adjacent province of sense-perception. And I wish
it to be remembered that my brief attempt is not to be taken as a
measure of the present state of such scientific questions.
* * * * *
It is a well-known fact that some objects please us, while others do
not. Generally speaking, anything that is constructed according to fixed
and logically followed rules, is a product of tolerable beauty. We see
thus nature herself, who always acts according to fixed rules,
constantly producing such pretty things. Every day the physicist is
confronted in his workshop with the most beautiful vibration-figures,
tone-figures, phenomena of polarisation, and forms of diffraction.
A rule always presupposes a repetition. Repetitions, therefore, will
probably be found to play some important part in the production of
agreeable effects. Of course, the nature of agreeable effects is not
exhausted by this. Furthermore, the repetition of a physical event
becomes the source of agreeable effects only when it is connected with
a repetition of sensations.
An excellent example that repetition of sensations is a source of
agreeable effects is furnished by the copy-book of every schoolboy,
which is usually a treasure-house of such things, and only in need of an
Abbé Domenech to become celebrated. Any figure, no matter how crude or
poor, if several times repeated, with the repetitions placed in line,
will produce a tolerable frieze.
[Illustration: Fig. 25.]
Also the pleasant effect of symmetry is due to the repetition of
sensations. Let us abandon ourselves a moment to this thought, yet not
imagine when we have developed it, that we have fully exhausted the
nature of the agreeable, much less of the beautiful.
First, let us get a clear conception of what symmetry is. And in
preference to a definition let us take a living picture. You know that
the reflexion of an object in a mirror has a great likeness to the
object itself. All its proportions and outlines are the same. Yet there
is a difference between the object and its reflexion in the mirror,
which you will readily observe.
Hold your right hand before a mirror, and you will see in the mirror a
left hand. Your right glove will produce its mate in the glass. For you
could never use the reflexion of your right glove, if it were present to
you as a real thing, for covering your right hand, but only for covering
your left. Similarly, your right ear will give as its reflexion a left
ear; and you will at once perceive that the left half of your body could
very easily be substituted for the reflexion of your right half. Now
just as in the place of a missing right ear a left ear cannot be put,
unless the lobule of the ear be turned upwards, or the opening into the
concha backwards, so, despite all similarity of form, the reflexion of
an object can never take the place of the object itself.[20]
The reason of this difference between the object and its reflexion is
simple. The reflexion appears as far behind the mirror as the object is
in front of it. The parts of the object, accordingly, which are nearest
the mirror will also be nearest the mirror in the reflexion.
Consequently, the succession of the parts in the reflexion will be
reversed, as may best be seen in the reflexion of the face of a watch or
of a manuscript.
It will also be readily seen, that if a point of the object be joined
with its reflexion in the image, the line of junction will cut the
mirror at right angles and be bisected by it. This holds true of all
corresponding points of object and image.
If, now, we can divide an object by a plane into two halves so that each
half, as seen in the reflecting plane of division, is a reproduction of
the other half, such an object is termed symmetrical, and the plane of
division is called the plane of symmetry.
If the plane of symmetry is vertical, we can say that the body is
vertically symmetrical. An example of vertical symmetry is a Gothic
cathedral.
If the plane of symmetry is horizontal, we can say that the object is
horizontally symmetrical. A landscape on the shores of a lake with its
reflexion in the water, is a system of horizontal symmetry.
Exactly here is a noticeable difference. The vertical symmetry of a
Gothic cathedral strikes us at once, whereas we can travel up and down
the whole length of the Rhine or the Hudson without becoming aware of
the symmetry between objects and their reflexions in the water. Vertical
symmetry pleases us, whilst horizontal symmetry is indifferent, and is
noticed only by the experienced eye.
Whence arises this difference? I say from the fact that vertical
symmetry produces a repetition of the same sensation, while horizontal
symmetry does not. I shall now show that this is so.
Let us look at the following letters:
d b
q p
It is a fact known to all mothers and teachers, that children in their
first attempts to read and write, constantly confound d and b, and q and
p, but never d and q, or b and p. Now d and b and q and p are the two
halves of a _vertically_ symmetrical figure, while d and q, and b and p
are two halves of a _horizontally_ symmetrical figure. The first two are
confounded; but confusion is only possible of things that excite in us
the same or similar sensations.
Figures of two flower-girls are frequently seen on the decorations of
gardens and of drawing-rooms, one of whom carries a flower-basket in her
right hand and the other a flower-basket in her left. All know how apt
we are, unless we are very careful, to confound these figures with one
another.
While turning a thing round from right to left is scarcely noticed, the
eye is not at all indifferent to the turning of a thing upside down. A
human face which has been turned upside down is scarcely recognisable as
a face, and makes an impression which is altogether strange. The reason
of this is not to be sought in the unwontedness of the sight, for it is
just as difficult to recognise an arabesque that has been inverted,
where there can be no question of a habit. This curious fact is the
foundation of the familiar jokes played with the portraits of unpopular
personages, which are so drawn that in the upright position of the page
an exact picture of the person is presented, but on being inverted some
popular animal is shown.
It is a fact, then, that the two halves of a vertically symmetrical
figure are easily confounded and that they therefore probably produce
very nearly the same sensations. The question, accordingly, arises,
_why_ do the two halves of a vertically symmetrical figure produce the
same or similar sensations? The answer is: Because our apparatus of
vision, which consists of our eyes and of the accompanying muscular
apparatus is itself vertically symmetrical.[21]
Whatever external resemblances one eye may have with another they are
still not alike. The right eye of a man cannot take the place of a left
eye any more than a left ear or left hand can take the place of a right
one. By artificial means, we can change the part which each of our eyes
plays. (Wheatstone's pseudoscope.) But we then find ourselves in an
entirely new and strange world. What is convex appears concave; what is
concave, convex. What is distant appears near, and what is near appears
far.
The left eye is the reflexion of the right. And the light-feeling retina
of the left eye is a reflexion of the light-feeling retina of the right,
in all its functions.
The lense of the eye, like a magic lantern, casts images of objects on
the retina. And you may picture to yourself the light-feeling retina of
the eye, with its countless nerves, as a hand with innumerable fingers,
adapted to feeling light. The ends of the visual nerves, like our
fingers, are endowed with varying degrees of sensitiveness. The two
retinæ act like a right and a left hand; the sensation of touch and the
sensation of light in the two instances are similar.
Examine the right-hand portion of this letter T: namely, T. Instead of
the two retinæ on which this image falls, imagine feeling the object, my
two hands. The T, grasped with the right hand, gives a different
sensation from that which it gives when grasped with the left. But if we
turn our character about from right to left, thus: T, it will give the
same sensation in the left hand that it gave before in the right. The
sensation is repeated.
If we take a whole T, the right half will produce in the right hand the
same sensation that the left half produces in the left, and _vice
versa_.
The symmetrical figure gives the same sensation twice.
If we turn the T over thus: T, or invert the half T thus: L, so long as
we do not change the position of our hands we can make no use of the
foregoing reasoning.
The retinæ, in fact, are exactly like our two hands. They, too, have
their thumbs and index fingers, though they are thousands in number; and
we may say the thumbs are on the side of the eye near the nose, and the
remaining fingers on the side away from the nose.
With this I hope to have made perfectly clear that the pleasing effect
of symmetry is chiefly due to the repetition of sensations, and that
the effect in question takes place in symmetrical figures, only where
there is a repetition of sensation. The pleasing effect of regular
figures, the preference which straight lines, especially vertical and
horizontal straight lines, enjoy, is founded on a similar reason. A
straight line, both in a horizontal and in a vertical position, can cast
on the two retinæ the same image, which falls moreover on symmetrically
corresponding spots. This also, it would appear, is the reason of our
psychological preference of straight to curved lines, and not their
property of being the shortest distance between two points. The straight
line is felt, to put the matter briefly, as symmetrical to itself, which
is the case also with the plane. Curved lines are felt as deviations
from straight lines, that is, as deviations from symmetry.[22] The
presence of a sense for symmetry in people possessing only one eye from
birth, is indeed a riddle. Of course, the sense of symmetry, although
primarily acquired by means of the eyes, cannot be wholly limited to the
visual organs. It must also be deeply rooted in other parts of the
organism by ages of practice and can thus not be eliminated forthwith by
the loss of one eye. Also, when an eye is lost, the symmetrical muscular
apparatus is left, as is also the symmetrical apparatus of innervation.
It appears, however, unquestionable that the phenomena mentioned have,
in the main, their origin in the peculiar structure of our eyes. It will
therefore be seen at once that our notions of what is beautiful and ugly
would undergo a change if our eyes were different. Also, if this view is
correct, the theory of the so-called eternally beautiful is somewhat
mistaken. It can scarcely be doubted that our culture, or form of
civilisation, which stamps upon the human body its unmistakable traces,
should not also modify our conceptions of the beautiful. Was not
formerly the development of all musical beauty restricted to the narrow
limits of a five-toned scale?
The fact that a repetition of sensations is productive of pleasant
effects is not restricted to the realm of the visible. To-day, both the
musician and the physicist know that the harmonic or the melodic
addition of one tone to another affects us agreeably only when the added
tone reproduces a part of the sensation which the first one excited.
When I add an octave to a fundamental tone, I hear in the octave a part
of what was heard in the fundamental tone. (Helmholtz.) But it is not my
purpose to develop this idea fully here.[23] We shall only ask to-day,
whether there is anything similar to the symmetry of figures in the
province of sounds.
Look at the reflexion of your piano in the mirror.
You will at once remark that you have never seen such a piano in the
actual world, for it has its high keys to the left and its low ones to
the right. Such pianos are not manufactured.
If you could sit down at such a piano and play in your usual manner,
plainly every step which you imagined you were performing in the upward
scale would be executed as a corresponding step in the downward scale.
The effect would be not a little surprising.
For the practised musician who is always accustomed to hearing certain
sounds produced when certain keys are struck, it is quite an anomalous
spectacle to watch a player in the glass and to observe that he always
does the opposite of what we hear.
But still more remarkable would be the effect of attempting to strike a
harmony on such a piano. For a melody it is not indifferent whether we
execute a step in an upward or a downward scale. But for a harmony, so
great a difference is not produced by reversal. I always retain the same
consonance whether I add to a fundamental note an upper or a lower
third. Only the order of the intervals of the harmony is reversed. In
point of fact, when we execute a movement in a major key on our
reflected piano, we hear a sound in a minor key, and _vice versa_.
It now remains to execute the experiments indicated. Instead of playing
upon the piano in the mirror, which is impossible, or of having a piano
of this kind built, which would be somewhat expensive, we may perform
our experiments in a simpler manner, as follows:
1) We play on our own piano in our usual manner, look into the mirror,
and then repeat on our real piano what we see in the mirror. In this way
we transform all steps upwards into corresponding steps downwards. We
play a movement, and then another movement, which, with respect to the
key-board, is symmetrical to the first.
2) We place a mirror beneath the music in which the notes are reflected
as in a body of water, and play according to the notes in the mirror. In
this way also, all steps upwards are changed into corresponding, equal
steps downwards.
3) We turn the music upside down and read the notes from right to left
and from below upwards. In doing this, we must regard all sharps as
flats and all flats as sharps, because they correspond to half lines and
spaces. Besides, in this use of the music we can only employ the bass
clef, as only in this clef are the notes not changed by symmetrical
reversal.
You can judge of the effect of these experiments from the examples which
appear in the annexed musical cut. (Page 102.) The movement which
appears in the upper lines is symmetrically reversed in the lower.
The effect of the experiments may be briefly formulated. The melody is
rendered unrecognisable. The harmony suffers a transposition from a
major into a minor key and _vice versa_. The study of these pretty
effects, which have long been familiar to physicists and musicians, was
revived some years ago by Von Oettingen.[24]
[Music: Fig. 26.
(See pages 101 and 103.)]
Now, although in all the preceding examples I have transposed steps
upward into equal and similar steps downward, that is, as we may justly
say, have played for every movement the movement which is symmetrical to
it, yet the ear notices either little or nothing of symmetry. The
transposition from a major to a minor key is the sole indication of
symmetry remaining. The symmetry is there for the mind, but is wanting
for sensation. No symmetry exists for the ear, because a reversal of
musical sounds conditions no repetition of sensations. If we had an ear
for height and an ear for depth, just as we have an eye for the right
and an eye for the left, we should also find that symmetrical
sound-structures existed for our auditory organs. The contrast of major
and minor for the ear corresponds to inversion for the eye, which is
also only symmetry for the mind, but not for sensation.
By way of supplement to what I have said, I will add a brief remark for
my mathematical readers.
Our musical notation is essentially a graphical representation of a
piece of music in the form of curves, where the time is the abscissæ,
and the logarithms of the number of vibrations the ordinates. The
deviations of musical notation from this principle are only such as
facilitate interpretation, or are due to historical accidents.