| Here
under follows the transcription of the chapter Leonardo of
Houston
Stewart Chamberlain's Immanuel Kant, published by John Lane,
The
Bodley Head, 1914.
LEONARDO DA
VINCI
Painted by
himself
Drawn and
engraved
by Charles Townley
|
99
LEONARDO
CONCEPTION AND
PERCEPTION
WITH AN EXCURSUS UPON PHYSICAL OPTICS
AND THE DOCTRINE OF COLOUR
Our soul is composed of harmony,
and harmony is never bred save
in moments when the proportions
of objects are seen or heard.
Leonardo.
100
(Blank page)
101
LEONARDO
THAT
Seeing is a
passive as well as an active function is a maxim as old as Aristotle.
But
the difference between passivity and activity in different men repeats
itself in the degree and the more delicate qualities of both. It must
be
our business to bring into relief those personal peculiarities of
Kant's
way of Seeing which differentiate him from other Seers. It is with this
object in view that we resort to comparison. We desire ourselves to
observe
and See the most important of the Seers, in the
profound
conviction that this will carry us further and to greater advantage
than
if we were to go into abstract theories about them, and hedge in their
doctrines with finely pointed fences of definition.
From our first
comparison,
— that with Goethe, — we have won a significant and lasting advantage.
The intellectual individuality of Kant revealed itself in striking
contrast
to that of Goethe. In Kant we saw a peculiar quality of intuition
developed
to an absolutely astounding degree. We saw the power of appreciating
mentally
that which has been described: and that which is described is something
which has been brought to our minds parcel-wise, or, to use the
technical
expression, analytically: for it is only by degrees that words can
present
a Whole, whereas the Eye first gives us a Whole, and only by degrees
separates
it into Parts. Moreover, in Kant we found an important development of
that
method of perception which is projected from within to without,
geometrically
in accordance with formula, humanly
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LEONARDO
speaking
creative, — namely the
mathematical
method. In Goethe, on the contrary, we found as a characteristic the
insatiable
hunger of the Eye, and in connection therewith the impulse to treat
even
matters of theory as something actually seen. In spite of this,
inasmuch
as the Doctrine of Metamorphosis had furnished us with a clear
demonstration
of the relationship between passivity and activity in Goethe's manner
of
Seeing, we recognised its harmony with Kant's mental vision and his
analytical
distinction between Experience and Idea.
To-day I wish to
carry this comparison with Goethe further, for it still contains a
whole
store of instruction. I hope to convince you that without the help of
Kant
we could hardly succeed in correctly grasping Goethe's view of nature,
while at the same time no other man leads us so directly and patently
to
Kant, as does Goethe. This consideration, then, will give you a twofold
advantage. Still, I wish to associate with these two men a third, —
another
great artist. I hit upon my choice without any reference to chronology;
— simply with the intention of avoiding the ever-present danger of
allowing
our lazy thoughts to crystallise, and of contenting ourselves with some
idle phrase about the antagonism between art and philosophy.
Unfortunately
no lesser man than Schopenhauer has given encouragement to so stark a
fallacy:
he is the most read of all philosophers, and in so far justly as he is
by a long way the most readable; pity, that among his many perversities
of thought, (I can find no other word for them,) there should be the
asseveration
that “Genius and a head for mathematics“ should be contradictions. 1
It is not possible here to enter upon a refutation of this detestable
asseveration,
to which, in the very first place, the whole phenomenon of Hellenism
would
have to fall a sacrifice: it would be easy and entertaining to carry
out
such a refutation with no other help than that of Schopenhauer's own
writings:
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LEONARDO
but
one feels almost
ashamed
to enter the lists against this much too clever man when one hears him
in an important passage cite Alfieri as having been unable to master
the
fourth proposition of Euclid, and again bringing forward some unnamed
French
mathematician who, on reading through Racine's Iphigénie,
shrugged his shoulders, and asked, “Qu'est-ce que cela prouve?“ 2
If those are arguments then one might, with equal force, come to the
conclusion
that because a certain nineteenth-century poet at the age of forty, and
in spite of living in the country, did not yet know that tadpoles turn
into frogs, therefore no Poet is gifted with the power of observing
nature.
The mischief of such phrases, when they are presented with the
seductive
eloquence of a Schopenhauer, is that they are scattered far and wide,
and
establish themselves as dogmas, and so it comes to pass that to-day we
find many men who because, like Alfieri, they are incapable of
something,
pose as men of genius, and who, not content with the fact that “the
pride
of the human intellect,“ as Kant calls mathematics, is none of theirs,
plume themselves upon their impotence. Nay more, these mental waifs who
cannot even grasp the simple problem of the equilateral triangle, look
down from the height of their superiority on the most important men if
only they show any aptitude for mathematics, and catalogue them as
second-rate
goods. But we need not dwell upon this, though it is difficult to
prevent
our wrath from blazing up over the impertinence of so fundamentally
perverted
a dogma. It is time for us to enter at once upon the heart of our
subject.
3
Schopenhauer's thesis
affects genius in general. Sometimes, however, he propounds it in a
narrower
and therefore more plausible form: in this sense he writes, “Experience
has proved that men of great genius in art have no aptitude for
mathematics.“
That is an important limitation, for even in his eyes it is not the
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LEONARDO
artist
alone who can lay claim to
genius:
indeed, he is fond of quoting himself as an example, and certainly he
had
no artistic sense. Nevertheless this contention, which is to be found
in
the thirty-sixth paragraph of the first volume of his principal work,
is
so fundamentally false, that one asks oneself how Schopenhauer could
have
been blind enough to let it stand unaltered from the year 1818 to the
day
of his death. If we only think of German artists, the very first that
comes
to our thoughts is the man who was so specially admired even by Goethe,
the great, the only, I had almost said the holy, Albrecht Dürer.
He
is one of those great “men of genius in art“ of whom one can say that
they
were the beginning and the end and the culminating point, all in one.
Of
course historically they spring from what has gone before, and they
lead
to what is to follow after, but that association hangs about their
noble
forms like a mantle. Like the goddess from the sea-foam, the individual
rises out of the mass, something new, something incomparable, that
never
was before, and never can be again. At the sight of such men we are
struck
by Schopenhauer's fine saying, “Art is everywhere triumphant.“
Perfection
it is that blazes out upon us out of all the feverish struggles of
these
artists, — Peace that smiles upon us, full of trust, and resting from
the
hurry of the eternal strife for something higher. And where labour and
thought and prayer have wrought together as tireless journeymen, there
at last reigns Harmony, divinely restful, incapable of failure. Among
these
giants is Dürer — and mark this. Not only had he an aptitude for
mathematics,
but that aptitude was something quite out of the common. Dürer is
the author of the first textbook of applied geometry in the German
language!
Besides that, he devoted a whole work to the hopelessly dry, and only
mathematically
interesting, subject of Fortification; and his lectures on the
proportion
of the human figure are a little miracle
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LEONARDO
of
intricate
geometrical
descriptions. In his partiality for mathematics, in his sure eye for
the
study, and the weight that he lays upon it for the educational
equipment
of the artist — as he writes, “mensuration is the right foundation of
all
painting“ 4
—
we have the special characteristic of this great artist. This one
example
is enough to prove that Schopenhauer's contention that the great artist
has no aptitude for mathematics is an untenable generalisation from
single
instances. Further, you will easily understand that I must have it at
heart
to refute the insinuation that Kant belonged to the class of inferior
minds
lacking genius, because he had a talent for mathematics: far rather do
we see that the possession of that talent gives us no right to infer a
lack of artistic feeling.
Now, at last, I call
up the man whose radiant name I should be loath to cloud with polemics
— Leonardo da Vinci. No greater painter ever lived; and this great
painter
was like Dürer, and even more than Dürer, a pre-eminent
mathematician
and mechanician. At the same time — as we see every day more clearly —
a man of an all-embracing intellect, a Seer who penetrated all that his
eye saw, a Discoverer so inexhaustible that the world has perhaps never
seen his like, a deep, bold Thinker. Let us compare his method of
Seeing
with the methods of Goethe and Kant: that, I hope, will save us from
all
future danger of the crystallisations of the phrase-mongers.
Like Goethe, this
man is all Eye. He calls the Eye the Window of the soul, finestra
dell'
anima, 5
whose
precious qualities he is never weary of praising; the Eye is signore
de' sensi, “Lord of the senses,“ 6
the Eye is the Source of all Knowledge. Those who rely solely on the
study
of learned writings, instead of becoming acquainted with the works of
Nature
by means of their own Eyes, are only grandchildren not Sons of Nature,
that one
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LEONARDO
teacher
of all
teachers.
All Arts, all Sciences, all Thought, are according to Leonardo
“daughters
of the Eye,“ and so it is that the painter is nipote à Dio,
“the grandson of God.“ The Eye of this remarkable man is nevertheless,
like Goethe's, far from being an exclusively artistic organ, — it has
also
the power of penetrating the universe. A brilliant light radiates from
his Eye — for it is the special characteristic of the Eye of such men
that
it not only takes up light as others do, but also sends out rays of
light
illuminating the darkness, glowing through the impenetrable until it
becomes
transparent. A ray of light radiates, I say, from Leonardo's Eye so
brilliantly,
that even the most prosaic historians must admit that in him the
intuitive
power of divination of this organ verges upon the fabulous. Leonardo
anticipated
our whole modern natural science, — that is to say so far as this was
possible,
relying upon the Eye alone and without the help of the higher
mathematics,
— which were not then known, — of the new instruments, and of the mass
of observations which had to be mastered by whole generations. For
example,
this man who died in 1519, who had been brought up in the strict belief
of the Church in a flat earth laid between Heaven and Hell, knew the
principles
of the Cosmic system as Copernicus developed them thirty years later.
How
he gained this knowledge, and in what connection it came to him, we
know
not. For his observations, up to the present time far from being all
deciphered
and published, are for the most part aphoristic, often forming an
unsolvable
tangle of the most various thoughts, jotted down in the midst of, or
under,
or across his sketches, or on the backs of his sheets of drawings —
thoughts
often occurring to him in the midst of his painting, which he evidently
seizes in a hurry, in order to use them elsewhere. Sometimes he writes
expressly, “This is how I must deal with the matter in my work,“ or
something
of the same sort: or they are
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LEONARDO
clear
and neat
preparations
for books which he seems never to have written, and it is only from the
outline that we can make a guess at the direction of his thought. So in
Leonardo's writings we find no astronomical system. Yet on one sheet,
under
a number of mathematical calculations, we find written in unusually
large
letters, il sole non si muove, “the sun does not move.“ Not
another
word. Here we have clearly a sudden inspiration. But Leonardo is no
visionary:
his was throughout a positive intellect, never weary of seeking the certezza
delle scientie by the strictly empirical and mathematical road. Sperientia
è commune madre di tutte le scientie e arti. “Experience is
the common mother of all the arts and sciences,“ — and nissuna
humana
investigatione si po dimandare vera scientia, s'essa non passa per le
matematiche
dimostrationi, “no human investigation can lay claim to being true
science, unless it can stand the test of mathematical demonstration.“
Experiment,
therefore, and calculation must be brought into court as tests of the
correctness
of any assumed fact. In the same way on other sheets we find a
succession
of investigations and deductions all circling round this central idea
of
a stationary sun and an earth which is in motion. Take, for instance,
the
important recognition come la terra non è nel mezzo del
cerchio
del sole, ne nel mezzo del mondo, 7
“that the earth is not in the centre of the sun's orbit, nor in the
centre
of the universe.“ In this connection we over and over again find the
remark
that the sun is greater than the earth, together with the assertion
that
there are many stars that are many times bigger than the star which is
the earth. Molte stelle vi sono che son moltissime volte maggiore
che
la stella che è la terra. 8
The recognition of the fact that the dark earth reflects light leads
him
to the further assumption that the light of the planets is also
reflected
light, and that our earth seen from the moon would have
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LEONARDO
exactly
the same
appearance
as that which the moon gives us. 9
From this recognition there was but one step to the affirmation that
the
earth must be nearly spherical in shape and revolve round its axis.
Certainly,
so far as I know, we possess no written proof that Leonardo in any of
his
abrupt sentences ever gave expression to the further fundamental
thought
of the heliocentric system; but a great portion of his work is as yet
unpublished,
and this idea of motion follows so necessarily from the tenets which I
have cited, that we are compelled to accept the belief that it was
known
to him. If now we turn our attention from the movements of the
constellations
to the hidden inner movements of the body, we find that Leonardo with
the
help of a like magical power of vision suspected, and even had a clear
idea of, the circulation of the blood. This has been denied on the
ground
that in one passage Leonardo compared the movements of the blood with
the
ebb and flow of the tide. But the objection breaks down, because the
notes
which we possess of Leonardo's thoughts date from the most various
periods
of his life, and nothing can blot out the words which we have in his
own
hand, in black and white, concerning il continuo corso che fa il
sangue
per le sue vene, “the continuous course of the blood racing through
its veins,“ and over and above this that the blood which flows back to
the heart, il sangue che torna indirieto, differs from that
which,
when the blood is driven out, closes the valves of the heart, che
riserra
le porte del core. 10
These words suffice to prove a deep insight into the mechanism of the
circulation,
which at that time was unsuspected and not discovered until a century
later:
for Leonardo knows that the blood “runs an uninterrupted course through
the veins,“ he knows that it proceeds from the heart, and finds its way
back to the heart, and he makes a distinction between the venous blood
and the arterial blood. And here we must bear in mind that the most
important
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LEONARDO
works
of Leonardo in
this
connection, as in others, are up to the present unpublished. They lie
idle
in the dust of libraries.
I have chosen these
two examples, the astronomical and the physiological, out of the great
mass of material. Leonardo seems to have interested himself in every
branch
of science, and everywhere, through the mere penetrating power of his
sight,
coupled with the sagacity of his judgment, he appears to have
forestalled
science — often by centuries. Think only of his right appreciation of
the
significance of petrifactions and of the geological strata at a time
when
people used to explain the one as the playful products of a vis
plastica,
while for the other at best the Deluge was made responsible! But to my
regret I can give no more time to this captivating subject. If you want
more particulars I must refer you to the books upon Leonardo. 11
I must be content if, by quoting typical instances, I have made you
familiar
with the wonderful quality and astounding penetration of this power of
perception. Words are insufficient, what we need is facts — and these
facts
patent to every man, even to the unlearned, point to an intellect whose
kinship with that of Goethe is at once striking: the same ever-open
Eye,
never satiated, the Eye of the warder Lynceus (as he called it in my
first
lecture), surveying the whole world, and uninterruptedly entertaining
the
monarch imprisoned in the Tower with new pictures: at the same time it
is an Eye which creates. Yet we are struck by two important
differences.
Leonardo sees more exactly than Goethe, his Eye is sharper, and he can
do what Goethe never could: he can reproduce what he has seen so that
it
becomes something seen by others: he is a painter, and for that reason
still further removed from Kant than Goethe. But just as the outer
sense
is more refined in Leonardo than in Goethe, so it is too in the case of
that inner schematic power of perception, which Goethe hardly
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LEONARDO
possessed,
but which in
Kant was conspicuously developed. In this respect the relationship is
reversed:
Leonardo is nearer to Kant than Goethe was; in mechanics, indeed, he is
as richly gifted with genius as he is in Art. Take up the six beautiful
volumes into which Ravaisson-Mollien has divided all Leonardo's
manuscripts
in the Bibliothèque de l'Institut, in facsimile and deciphered,
and you will see that nine-tenths of these notes refer to mathematics
and
mechanics. Leonardo never ceased to calculate. His mind was busy with
the
squaring of the circle, and with groping attempts at infinitesimal
calculation;
from the flight of birds to the observation of a waterfall, in every
direction
the interest in mathematics and mechanics forces itself upon him side
by
side with that of the painter. In one place he speaks of mechanics as
“a
Paradise,“ and says of it, “the science of machinery or mechanics is
the
noblest of all the sciences“ — La scientia strumentale over
machinale
è nobilissima. On the sheet which contains perhaps the very
first sketch for the Last Supper, we find immediately under the subject
a geometrical problem drawn and solved in ciphers, and another sheet
which
contains studies for the Apostles and a pathetic sketch for the Christ,
shows under these figures a plan for a piece of machinery with
explanatory
notes. So if Leonardo and Goethe are two men in whom, in
contradistinction
to Kant, the Eye is the organ of life, still two very different
intellects
must be looking out from this finestra dell' anima, two very
different
modes of activity, to quote Aristotle, and therefore at the same time
two
very different systems of philosophy. Starting from the outward and
visible
signs we shall reach the very core of the question if we pay attention
to one thing, — that Goethe wished to paint and could not, whilst
Leonardo
presents such a culminating point of pictorial genius, that few can
reach
his level, none surpass him.
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LEONARDO
Goethe's
lack of capability in the plastic arts would be less striking if we did
not see him from childhood so passionately striving to attain mastery
in
this very direction. We know that as a student in Leipzig he painted
more
than he read. It was Oeser's studio, not the lecture-rooms of the
juristic
Faculty, that he haunted. And with what touching industry did he carry
on this struggle for the impossible!
- Doch unvermögend Streben,
Nachgelalle,
- Bracht' oft den Stift, den Pinsel
bracht's
zu Falle;
- Auf neues Wagnis endlich blieb
doch nur
- Vom besten Wollen halb und halbe
Spur.
In
the end Goethe himself was bound to
confess,
“I was lacking in the true plastic power,“ and he adds the precious
words
of irony against himself, “my attempts at representing nature were more
like distant suspicions of given forms, and my figures resembled the
light
vaporous beings in Dante's Purgatorio, shadowless themselves, and
terror-stricken
at the shadows of real bodies.“ What this defect meant Goethe
accurately
realised; for in a conversation with Eckermann he quotes with praise
words
of our Leonardo, — “If your son lacks the sense to make what he draws
stand
out in relief by powerful shadowing, so that one might grasp it in
one's
hand, then he has no talent.“ And do you know why Goethe had no talent
for drawing? Why his copies were mere “distant suspicions of given
forms“?
Because he was deficient in the sense of Geometry. Because we men are
so
built that we are incapable of accurately grasping any form which
nature
presents to us, unless, consciously or unconsciously, we have held
before
it by way of comparison the complex network of possible forms which is
innate in us, and have in this way assimilated that which is outside
all
rule, incalculable, and which has never existed, by contrasting it with
that which is regular, calculated and for ever unalterable. This happens
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LEONARDO
without
our thinking of
it during every second of our life: we are incessantly schematising.
Later
on you will learn from Kant to what degree our whole intellectual being
is under the domination of Scheme. “This schematising of our
understanding,
when we are face to face with phenomena,“ he writes, “is an art hidden
in the depths of the human soul.“ For the images which we receive from
without, that is to say the complex of the impressions of our senses,
cannot
be grasped directly, but our intellect — the “activity“ of Aristotle —
must first, as Kant says in a happy figure of speech, have impressed
its
monogram upon them. “It is only by means of the Scheme that images and
conception can be brought into union.“ You see that from without to
within
there is an intermediary action similar to that which takes place from
within to without. Our ideas, as you will remember from the
metamorphosis,
were only able to reach the Eye by borrowing a symbol, e.g. the leaf,
from
the world of the senses: but this world of the senses — so runs the new
creed — can penetrate the thinking consciousness in no other way than
by
the intermediary of schemes of the understanding; and these schemes
coincide
with the perceptions with no more exactitude than the symbols coincided
with the ideas. It is not my intention at this moment to weary you with
metaphysical disquisitions; on the contrary, I wanted only to call your
attention to the fact that the plastic artist shows us this secret
domination
of the “hidden art“ of schematisation in bright daylight, and so
smooths
the way for the understanding of one of the most difficult passages of
Kant's Critique of the Doctrine of Perception. For the great
painter,
consciously and in the sight of all men, puts into concrete form that
which
in others exists in the unconscious “depths of the soul.“
That is why
Dürer
wrote those words which perhaps may have struck you as strange a little
while ago — “The
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LEONARDO
art
of mensuration is the true
foundation
of all painting,“ and the same order of thought gives rise to what he
writes
on the next page: “The outer (practical) work must be the indication of
the inner understanding.“ And in order that you may realise how
powerfully
the geometrical and schematic principle is developed in so pre-eminent
a modeller, and how busily it is at work, I would beg you to take up
another
work of Dürer's, “the four Books of human proportion,“ — not in a
modern abbreviated edition, but in the original small folio of 1528,
with
all the charts and tables, as they left the master's hands. You will be
astounded at the world of numbers and geometrical figures in which
Dürer
lived; they are enough to make you giddy. Indeed, every complication
can
be solved by figures, yielding of its own accord without any involving
of the imagination: but one can hardly grasp how any man should have
been
able to carry in his head, as something visible, such complicated
geometrical
figures, as Dürer was obviously able to do. In the two first books
the many charts of figures and the painful precision of the
measurements
will strike you as imposing. But now look at the third book! Here
Dürer
teaches us how we may at will change the fixed proportions; for
instance,
he takes a woman of average proportions that he has already shown us,
and
makes her first long and thin, then short and monstrously fat; or else
he changes one part of the body, leaving the other parts as they were,
etc., and all this he does without ever departing from the established
foundations of geometrical schemes, and with the help of instruments
which
he calls “the perverter,“ “the falsifier,“ etc. The fourth book is
almost
more interesting: it shows “how you may distort the previously
described
images,“ and yet it is no simple doctrine of perspective in our sense
of
the word, but rather what mathematicians call the geometry of position,
bound up with that of projection. You need
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LEONARDO
only
look at the
figures
on page Y4, Z, and those that follow, in order to arrive at the
understanding
of what Dürer expects of the art-disciple.
Leonardo's brain
was organised in like fashion. To be sure he was not so self-tormenting
— for instance, look at his doctrine of perspective, how bright it
seems
by the side of Dürer's, — yet always and everywhere he paid
respect
to the mathematical relations, always bringing calculation into play,
always
displaying the geometrical Scheme between the Eye and the Object. One
hundred
and
fifty years ago, Charles Bonnet, the
Genevan botanist, introduced the so-called doctrine of Phyllotaxy, that
is to say, the exact observation of the relative position of the leaves
on the stalk.
To the most widely
spread form of this relative position he gave the name of Quincunx:
in this the sixth leaf after the stalk has been twice encircled
invariably
stands immediately above the first; accordingly every cycle of leaves
consists
of five leaves. This discovery was the result of years of study by an
experienced
professed botanist. But two hundred and fifty years earlier Leonardo's
artist-eye had observed the Quincunx, and
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LEONARDO
had
drawn it with the most painful care, and that too in his Book on
Painting.
12
You see with what mathematical precision the painter observed! and not
only with precision but also with schematisation, for as a matter of
fact
this 2/5 position is only approximatively existent. But in order that
you
may also see the geometrician at work, I have copied here a little
sketch
out of Ravaisson-Mollien — (M.S. M. f. 78 overleaf). To this Leonardo
has
added the note “all twigs possess lines which work towards the central
point of the tree.“ In order to understand him you must naturally only
take into consideration the youngest twigs, and you must realise how
this
so-called central point year by year moves upwards, quickly at first,
then
slowly. Even so there is great boldness in such a schematisation. On
other
pages you will see how Leonardo was at pains to apply to the human head
a similar law of relationship to the line of circumference. His
comparative
studies of various human heads, including monstrous deformities, are so
well known that I need do no more than allude to them. *
This cursory outline
may suffice to show you what special qualities must be at work in a man
who is capable of reproducing that which he has seen. Where these
qualities
are lacking there can be no painter, because there is no organ for the
correct assimilation of form, and every attempt yields nothing but
“distant
suspicions.“ Of such men who are willing but incapable, Leonardo says,
Multi
sono gli uomini chi anno desiderio e amore al disegno ma non
disposizione
— “many are the men who
*
Leonardo
drew a circle, by the help of which he fixed every point of the human
head
in a Scheme — nose, chin, ears, eyes, etc.
116 LEONARDO
have
the wish to draw
and
who take delight in it, but who have not the capability“ — the disposizione
lies in the aptitude for schematising. Naturally the geometrical
quality
by itself is not enough: yet it must not be wanting. The man who keeps
the scheme steadily before his Eye, notes every deviation in the form,
whereas on the other hand a Goethe, as we have seen, was rather
inclined
to underestimate points of difference, and in everything to see the
points
of union. “I was born in the school of identity,“ is his confession,
but
that is no school of painting. On the other hand, it is certainly
interesting
to discover that the thinker with closed eyes, whose stupendous
schematic
power of representation thought out the theory of the Heavens, showed
in
this respect a true intellectual relationship with a Dürer and a
Leonardo.
However much the science of mathematics may on one side root itself in
logic, and signify in many of its adepts a purely abstract logical
exercise
of the intellect, — still the living water that gives nourishment to
the
tree is the power of the Eye, and so it may happen that a Kant may in
certain
respects stand in closer relationship to Leonardo than Goethe. In order
to keep up the association of ideas in what I have to say, I shall
return
to this subject later, but I shall beg you once for all not to forget
that
the power of schematisation is a true formative power.
In the meantime we
must still linger awhile over the comparison between Leonardo and
Goethe.
I wish to show you how far-reaching is the difference that we have here
observed in the Eye at work. For this purpose Leonardo's judgments on
the
essence of Art will be of service to us. According to him the senses
are
the true agents in real art, and the man who, like the poet, excites
the
conceptions of the senses by descriptions alone, makes use of a
subordinate
and indirect species of Art. Leonardo exclaims proudly, se'l
pittore
vol vedere bellezze
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LEONARDO
che
lo innamorino
egli
n'è signore di generarle. That the poet is equally master of
producing beauty that shall be capable of exciting his love, that
Leonardo
denies. For il senso piu nobile is the Eye, and next to this
noblest
sense follows the Ear, la musica si deve chiamare sorella minore
della
pittura, whereas the artist in words is only indirectly and not
really
an artist, because he can only produce forms by roundabout ways, and by
steering clear of the impressions of the senses: e per questo il
poeta
resta, inquanto alla figurazione delle cose corporee, molto indietro al
pittore, e delle cose invisibili rimane indietro al musico, “the
Poet
in the representation of bodily things remains far behind the painter,
and in that of invisible things, behind the musician.“ But the
strongest
objection that Leonardo has to make against the poet, is che non ha
potestà in un medesimo tempo di
dire diverse cose, “that he has not
the power of saying several things at one and the same time“; but it
must
be the aim of Art to waken that Harmony of many tones which lies
slumbering
in the human soul, and that must take place with lightning rapidity,
like
an inspiration of the Deity: for armonia non
s'ingenera se non in istanti, nei quali le proportionalità degli
obietti si fan vedere, o'udire.
“Harmony cannot be bred otherwise than in instants in which the
relative
proportions of things are seen or heard.“ Here, obviously, it is the
plastic
artist who is master, for he alone reveals his whole work in one single
moment, and that is why Leonardo speaks of his art as a Divinity, una
Deità. But the musician too gives in every moment a
multiform
perfect Harmony, while on the contrary the word-poet is forced to build
up bit by bit, l'una parte nasce dall' altra successivamente, e non
nasce la succedente, se l'antecedente non muore, “one part is born
from the other in succession, and the following part is not born unless
the previous part is dead.“ It is not my intention here to discuss the
aesthetic doctrines of Leonardo: I
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LEONARDO
have
only felt
compelled
to show you with what a passionate bias this clear-sighted man paid
respect
to the Eye, and, beside the Eye, to the direct impressions of the
senses
in general as opposed to all mere reflection. The point of contact with
Richard Wagner is clear, and in any other connection would give
occasion
to useful considerations.
Here, however, we
have one immediate and special interest — Leonardo, the man whose Eye
at
once reminded us of Goethe's Eye, is not only the antipodes of Goethe
in
respect to the scientific observation of Nature, but he comes very near
to refusing altogether to recognise as true art that very art in which
Goethe rendered immortal services. From the point of view which we are
for the moment adopting, Goethe and Leonardo stand so far apart that we
should hardly bring them into relationship, were it not for Kant who
holds
out the hand to both. For as a matter of fact, Kant, whom a while ago
we
found to be of so near kin to Leonardo that the two viewed from the
distant
Goethe appeared like brothers, now, seen from point of view of
Leonardo's
aesthetics, seems to move close up to Goethe. In this method of
constructing
“parcel-wise“ — una parte nasce dall' altra — we were able to
discover
a characteristic of Kant's method of perception; now it is Leonardo who
shows us, that it is equally characteristic of every professor of the
art
of thinking, even of the Poet — and instinctively these words, una
parte
nasce dall' altra, call to our recollection Goethe's Doctrine of
Metamorphosis.
Of course Goethe examines nature with an Eye differing from that of
Kant,
yet he too is forced to construct, and in order to put the phenomena of
nature perspicuously into form and to embody them in his memory, he
cannot
help allowing one part to arise from another. That is the exact purport
of Metamorphosis. Practically there is, then, in Goethe's intellectual
personality, exactly as in
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LEONARDO
Kant's,
a preponderant
quality that we might well indicate with Kant as “understanding“ in
contradistinction
to “Sense,“ or perhaps still better as Reason (in the Kantian sense of
“the whole higher power of recognition,“) in contradistinction to the
power
of observation. In both these men, Goethe and Kant, — however various
may
have been the sources at which they drew their impressions, — the
insistence
upon the idea of dallying with theory forms a common feature. However
different
from the path trodden by Kant may be that by which Goethe reached his
Ideas,
— he is only quite at borne, only quite the master, quite the creator,
in that domain which Kant calls the Higher Power as opposed to a Lower
Power; while Leonardo looks upon this so-called Lower Power as the
Higher
Power, and takes no account of any knowledge that has not, “born of the
experience of the senses, made its way through mathematical exposition,
and found its final conclusion in experiment.“ That is why he exhorts
us
to put no faith in authors who have wished by the force of imagination
alone to make themselves interpreters between nature and man: non
vi
fidate degli autori che anno solo colla imaginatione voluto farsi
interprete
fra la natura all' uomo, and warns us not to give ourselves up to
those
things of which the human mind is incapable and which cannot be
demonstrated
by any natural example: Quelle cose di che la mente umana non
è
capace
e non si possono dimostrare per nessuno esemplo naturale. As you
see,
Leonardo will only accept in relation to Nature the most strict
empiricism
knitting together effect and cause, whereas formation by Ideas as
practised
by Goethe, and defended by Kant, seemed to him to be idle imagination,
or as he also called it bugiarda scientia, a science of lies.
Here then we discover
how far-reaching is the difference between Goethe and Leonardo; for it
is not merely con-
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LEONARDO
cerned
with art alone,
but extends to the whole method of contemplating nature. In the
previous
lecture we saw that Goethe was working with ideas when he believed
himself
to be in possession of experiences: that at once gives you an example
of
the dominant power of Reason — of the Higher Power of Recognition in
contradistinction
to empirical contemplation. For, as we saw in our investigation of the
doctrine of metamorphosis, Ideas are certainly something seen, but not
empirically seen; in other words, they are not given to us by mere
experience.
It is true that they are rooted in Impressions of the senses, though
that
is only the field which gives them nourishment: the air which surrounds
them is that of Reason, and the daylight in which we see them, radiates
from within out of a focus imaginarius.
There is a saying
of Kant's which will render us good service at this moment: for it
describes
exactly what it is that divides Goethe and Leonardo, and at the same
time
affords us a deep insight into Kant's own method of seeing; by abstract
study we might perhaps have failed altogether in understanding his
view;
but in the light and shade of Leonardo's and Goethe's methods, his view
stands out in plastic form. Kant is speaking of the essential nature of
the Poet. After having, in diametrical opposition to Leonardo, assigned
the highest of all artistic rank to the art of Poetry, he gives the
poet
the credit of encouraging “a free, personal and independent Power,
untrammelled
and unhampered, of observing and judging Nature as phenomenon,
according
to views which She herself affords neither to the senses nor to the
understanding,
and therefore to make use of her in the interests, and for the
Schematisation
of that which is transcendental“ (i.e. beyond the senses). The poet,
then,
teaches us to look upon Nature from points of view which direct
experience
does not offer us, and opens up in us a power to make use of what is
clear
to our Senses for the benefit
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LEONARDO
of
the schematisation
of
that which transcends them. This definition of the poet gives us an
exact
idea of Goethe's position in regard to Nature. In his method of
observation
there is a continual exchange between that with which the senses
furnish
us and that in which the experience of the senses only acts as a
spring-board.
Goethe is a good, trusty and, where necessary, a sober observer of
Nature;
in spite of which it is in the noblest sense of the word a poetical
longing
— I must add a yearning and a formative power — which impels him to
observation:
he wishes to put in practice that “free, personal and independent
power,“
and unconsciously he flies far beyond the boundaries of empirical
experience.
His Orphische Urworte with
its last line:
Ein
Flügelschlag! Und hinter
uns
Aeonen!
“One stroke of the wings! And behind
us aeons!“ appeared first in the Morphologie
of which the masterly Athroismos
belongs to the osteology, and here in the midst of illustrations of
bones
and comparative tables he cries out:
Nimm
vom Munde der Muse,
Dass du
schauest, nicht schwärmst,
die liebliche volle Gewissheit.
“Take from the mouth of the muse the
sweet full certainty that thou art seeing and under no delusion.“ So it
is the Muse that is to be our guardian goddess in the domain of the
investigation
of nature. Goethe, indeed, in certain moments is fully conscious of his
own method of procedure; for in his legacy of notes upon natural
science
we find the following most noteworthy passage: “Phantasy is far nearer
to nature than the senses: the latter are in nature, the former hovers
over her. Phantasy can hold its own with nature, the senses are
mastered
by her.“ There you see at work the free, personal power, of which Kant
spoke; at the same time you see the exact opposite of Leonardo's
convictions
and principles. For
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LEONARDO
according
to Leonardo
all forms of knowledge
are vain and full of errors, — vane e piene di errori, — unless
they be created from the experience of the senses and tested by
scientific
experiment. Leonardo is such a strict empiric, that he goes so far as
to
warn the artist that he must know no other aim than to gareggiare colla
natura, — literally “to compete with nature.“ How differently
the Eyes
of Goethe and Leonardo work we see not only in the Doctrines to which
their
method of Seeing gives occasion, but also in the success of their
activity.
Not only can Leonardo say of himself, “in painting I can stand
comparison
with any other man, be he who he may,“ 13
while Goethe, after toiling for
years, is obliged to confess the contrary, but Leonardo's contributions
to science are throughout of a different nature from those of Goethe. I
am far from underrating Goethe's doctrine of metamorphosis, — his
doctrine
of colour, — his other scientific thoughts; rather am I deeply
convinced
that his whole method of observing Nature possesses for the culture of
the human intellect a significance of which we are only just beginning
to be aware. In many respects Goethe is even now hardly born. But this
significance is one of culture and not of true science in the strict
sense
of the word. Goethe will teach us “to cast a free Eye upon the wide
field
of nature“ — a free Eye, that is to say the Eye of the conscious human
creator, who no longer stands in dull obedience at the command of idle
Matter, but who is able “to hold his own with Nature“: and that means
at
the same time the eye of the man who is no longer dazzled by his own
compelling
hallucinations, but who, thanks to Kant's efforts, has won together
with
his own freedom, the freedom of Nature. All this, — to which I propose
to return to-day, so soon as our observations shall have ripened
sufficiently,
— we can perceive, and yet must admit that it was Goethe's part to
excite
and spur on exact natural science rather than really to
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LEONARDO
further
it himself:
while Leonardo, on
the contrary, who saw as a schematiser, and thought as a mechanician,
was
such a master of the art of gaining knowledge, that in his guesses he
anticipated
the triumphant course of our natural history. As Kant proclaimed to us,
“experience alone is the fountain of truth in the observation of
Nature.“
Leonardo knew that full well: gareggiare
colla natura, “to compete with
nature“ — that was his maxim not only in art, but also in science; it
was
his delight and the cause of his success. That the earth revolves is no
symbolical idea, like Goethe's doctrine of metamorphosis, but a
concrete
theory; that the blood is chased from the heart through the veins, is
not,
like the discovery of the intermaxillary bone, an inference from an a priori
admission, but a fact discovered by painstaking autopsy and
observation.
In respect of pure natural science, I think we may say that Leonardo
surpassed
Goethe almost as much as he did in painting. He knows the only true
method:
one sees that in him at once: and that says everything. Observation,
experiment,
mathematical calculation, — these are the three principles which he
again
and again impresses as the foundation of all knowledge. If beyond this
we remember that he devoted a passionate interest to the technics of
instruments
(he built himself a sort of telescope for the observation of the moon a
century before Galilei), we must admit that he possessed all the
qualities
which go to make the born investigator of Nature.
By working up with more
and more
sharpness the contrast between Leonardo and Goethe, we have now reached
the critical point, that is to say the point where we shall be rewarded
if we sink a deep shaft, confident of coming upon a vein of the
precious
metal of discernment. Whoso thoroughly understands the difference
between
the value for science set upon mathematics by Leonardo and that set by
Goethe, has gained much, not only in the
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LEONARDO
estimate
of the two
great intellects,
but for his own thought-life in general. At the same time this point is
one of those which are of primary importance for the understanding of
the
intellect of Kant. For if a little while ago we saw Kant near — very
near
to Goethe — we see him quickly move back to Leonardo as soon as stress
is
laid not upon Art and Idea, but upon Science and Mathematics. Here it
is
not only the analogy of qualities between Leonardo and Kant which is
dominant,
— as it was just now in the case of the observation of schematising, —
but a true close kinship in the whole manner of looking upon the
universe.
There, at a great distance, Goethe stands aloof.
I have already spoken of
Leonardo's
love for mathematics; but I must still claim your patience for a few
moments.
Non mi legga chi non è
matematico, “let no man read me who is not
a mathematician“! Such forcible language as this should be enough! but
we have still got to learn that in Leonardo this is no question of a
mere
predilection, nor even of an instrument indispensable to the practical
artist, but the insight of a philosopher into the essence of the human
intellect. “The man who undervalues mathematics nourishes himself upon
confusion,“ says Leonardo, chi
biasima la somma certezza della matematica,
si pasce di confusione e mai porrà silentio alle contraditioni
delle
soffistiche scientie, colle quali s'inpara uno eterno gridore.
“For truth
and the power of knowledge are contained in the mathematical sciences.“
That is a very important saying, “the power of knowledge.“ Goethe would
not have subscribed to it: Kant would have done so with both hands. And
because practical knowledge is joined to the mathematical way of
thinking,
therefore Leonardo lays down the dogma that “no human investigation can
lay claim to be considered as true science unless it will stand the
test
of mathematical demonstration.“ For the criterion of
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LEONARDO
true
science — vera scientia — is
incontrovertible
certainty, and knowledge in the sense of certainty is only afforded by
mathematics. The consequence of this is that nessuna certezza è, dove non
si puo applicare una delle scientie matematiche over che non sono unite
con esse matematiche, — therefore no investigation can lay the
foundation
of true science, unless it can and does follow the path of mathematical
exposition, that is Leonardo's impregnable conviction. It is with the
clear
recognition of the relationship between mathematics and knowledge that
this miracle of a man forestalled Kant, in the same way that in his
discoveries
he anticipated Copernicus and Harvey. In one of his ripest works, Die Anfangsgründe
der Naturwissenschaft, Kant writes in the same way, “I maintain
that in
every special nature-doctrine there will be found only so much exact
science
as it contains of mathematics.“ Certainly Kant, the thinker, analysed
more
exactly than Leonardo. The whole tenour of Kant's general philosophy
teaches
us to distinguish between “exact“ and “inexact“ science; he has shown
us
that a science which rests upon empirical observation alone, is only
worthy
of the name and dignity of a “science,“ so far as it does not deviate
from
experience, ordering its discovered facts systematically, and
dissecting
them in accordance with the relationship between cause and effect; but
that such science should preferably be called systematic art (giving as
an example the chemistry of his time), because the apodictic certainty
of any true knowledge needs something more than empirical experience.
This
Something, which Kant calls the “Pure Part,“ is exactly that inner,
human
code of laws, which, in so far as it touches intuitive vision, is
called
mathematics. Nothing, with the single exception of mathematics, gives
apodictic
certainty, and apodictic certainty alone can be called knowledge in the
strict sense of the word. Therefore, the more
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LEONARDO
we
have of mathematics,
the more we have
of exact Science.
You see what a true and
deep-reaching
kinship exists between the methods of observation of these two men, who
at first sight seemed so diametrically opposed to one another. Kant,
absolutely
devoid of all artistic gifts, has yet the power of recognising the
fundamental
significance of form and measurement in the building up of human
knowledge;
and in many of his works, and more especially in Die Metaphysischen Anfangsgründe
der Naturwissenschaft, proves himself to be a genius of the
first quality
in
the despotic domain of this schematic manner of Seeing; Leonardo, the
artist,
the painter of the Last Supper and the Mona Lisa, is none the less
devoted
to mathematics and mechanics; he compares the influence upon the
intellect
of their incontrovertible certainty, with that of light upon the Eye,
and
with the exaggeration of the hot-blooded artistic temperament, he
utters
the opinion that here alone lies the certainty of knowledge.
We have to deal here with
a true
harmony between the dispositions of the two men. And as a matter of
fact
this harmony reveals itself exactly where Goethe misses fire — for
we may legitimately here speak
of a miss-fire as well in art as in philosophy. So far as art is
concerned
we may well overlook the position, inasmuch as Goethe himself bitterly
felt his own failure. But in the matter of philosophy he was not so
clearly
conscious, and that is what has led us and him to a condition in which
the pascersi di confusione
has gained great force. That Goethe despised
mathematics is of course the foolish twaddle of the titmice that chirp
on every twig of life; a single sentence of his suffices to refute it:
“no one can set a higher value on mathematics than I do, for
mathematics
afford precisely that which it has been denied to me to accomplish.“ 14
So he too felt that here something had
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LEONARDO
been
“denied to him,“
and how highly
he often valued this “something denied“ is shown by a sentence in the Farbenlehre,
the Doctrine of Colour, a passage where any irritation against the
mathematicians
might have been excused, and where in spite of that Goethe declares
mathematics
to be “one of the noblest organs of mankind.“ Still we must admit that
Goethe was not only deficient in the power of practising mathematics,
but
was even unable fully to appreciate the essence of the science in its
inevitable
influence upon the human intellect. “It is a mistake to imagine,“ he
exclaims
pettishly, “that when I have discovered the mathematical equation for a
phenomenon I know all about it that is worth knowing, and can consider
the whole matter as sufficiently dealt with and to be laid on the
shelf.“
15 What does he
mean? The function of mathematics is to apprehend, to prove
according to the laws of motion, to reduce clearly to a science — just
as Albrecht Dürer did for the outer form of the human body, and as
Leonardo tried to do for the mechanism of the circulation of the blood
in its inner parts. “The book of Nature is written in the language of
mathematics,“
says Galilei. Goethe, on the other hand, finds a contradiction between
the phenomenon observed and the mathematical scheme. For this sentiment
he has to thank the pure power of his sight; but instead of allowing
himself
to be taught by Kant that if Image and Scheme do not exactly tally, it
is due to the essential quality of the human intellect; 16 instead of
recognising
with Leonardo the fact that mathematical representation is the
necessary
organ of everything which can be called Science in the sense of exact
knowledge,
and that what he, Goethe, is striving after is not Science but
something
different, that is to say a glorified Contemplation, — that World of
the
Eye of which we spoke in the previous lecture, — and that this World
demands
ideal exposition; instead of all this, Goethe obstinately
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LEONARDO
works
himself up into
the unfortunate
idea that there can be an unmathematical science, — that the employment
of mathematics must be kept within bounds, and that they must be
relegated
to a narrow domain in the Study of Nature, etc. Science and art, — so
he
maintains, — have “fallen into pitiable error through the wrongful
employment
of mathematics.“ 17
When we remember that Goethe's unmathematical dicta,
of which we could cite many, are chiefly in the department of optics,
and
when we consider what a famous advance mathematical optics have made
since
Goethe's time, and what a wide outlook upon comprehensive knowledge has
been opened up in this very direction in our days by the work of
Maxwell
and Hertz; when we realise the present importance of spectral analysis
to astronomy, chemistry, and physics; and then when we see Goethe
ridiculing
the spectrum as little more than a mere puerility of Newton's, we must
feel that however much the great observer of nature and Poet may have
the
right to view Nature in his own fashion, he is yet lacking in the
understanding
of the mathematical method of exact science. And this is the more
striking
when we find in Leonardo, two hundred years before Newton, a few but
astonishingly
correct remarks about the colours of the spectrum, and when we think of
Kant's high estimate of the undulation theory of Huyghens, we have then
the experimental proof that if we follow Goethe in the path of science,
we advance no further in the exact sciences, whereas by following the
mathematical
path, which he detested and which Kant looked upon as the only right
way,
we have advanced from one theoretical and practical attainment to
another.
What, then, is the essence
of
the mathematical method? That is a question which it is impossible for
us here to shirk, otherwise we should neither understand correctly
Leonardo's
extreme way of viewing Nature, nor Goethe's,
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LEONARDO
nor
should we
understand why Kant's philosophical
critique enables him to do justice to both these antagonistic views. I
shall try to answer the question at once in as few and as simple words
as possible, leaning indeed upon Kant, but without making him
responsible
for my free and illustrative exposition; we shall deal more precisely
with
the matter in observations to be added hereafter.
So soon as we
thoughtfully, —
I use the word “thoughtful“ in contradistinction to passive
contemplation,
— so soon as we thoughtfully approach Nature and construct that “unity
of objects“ without which she would no longer be Nature but Chaos,
every
single conjunction, arrange it as we may, means Motion. Think only of
the
commonest perceptions of any Bodies that you please, which you,
innocent
of any attempt at philosophising, simply join together, thinking in
contemplative
consciousness, something in the same way as the herdsman watches his
grazing
herd. Either the objects are at rest, and then our mind must move in
order
to perceive them, whereby we arrive at Form, or our mind is at rest and
the objects move before it and then we arrive at Number: in most cases
the two sorts of conjunction will take place simultaneously; and as you
see, whether we direct our observation to the proximity in space, or to
the sequence in Time, Motion is always at the bottom of it. Motion,
says
Kant, is that which unites space and time, and motion conceived, that
is
to say grasped by Reason, is Mathematics. If we look at the still
geometrical
figures in our school-books, we sometimes think that here is the very
emblem
of rest; but in the next lecture we shall see how the great Descartes
laid
the foundation of the higher mathematics, when he taught us to set free
into Motion every resting Form, whereby we attain a second gift,
namely,
the power to convert every species of Motion into a visible, permanent
Form.
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LEONARDO
But
just in the same
way as these higher mathematics proceed from the union of Geometry and
Arithmetic, so it is only by further, and, as closer observation shows,
powerful conjunctions of space and time, that a really intelligible and
logical Nature comes into existence for us, and it is from these
conjunctions
that we realise the ideas of the inter-relationship of various
perceptions,
— of the interchangeability of phenomena, — of causative cohesion.
Thus,
for example, the relationship between cause and effect signifies a
twofold
Motion in space and time. You will find that set forth with
unsurpassable
lucidity in the fourth paragraph of Schopenhauer's principal work, to
which
I refer you. 18
And with further investigation and thought you will understand
how Kant arrives at the definition, “Matter is that which is movable,“
and at the assertion that space can only be filled by motion. And that
you may not think that I am leading you on to the pin-points of the
most
abstract philosophy, but that, on the contrary, you may understand that
I am dealing here with the concrete and necessary apprehension of
Nature
by human intelligence, I will call your attention to the fact that our
modern physics, however antimetaphysical may be their attitude in their
empirical delusion, learn to recognise Kant's standpoint as the only
justifiable
one, and that the little globules of atoms are only preserved as a
deduction
and a help for coarser intellects, whereas Lord Kelvin and other
leading
spirits among the mathematical physicists speak of “centres of energy,“
and by atoms understand gyrating motion. Lord Armstrong, * in his book
Electric Movements in Air and Water,
asserts that there is no ground for
looking upon Matter as anything else but Motion. Even the hypothetical
aether he rejects as super-
* I
purposely
cite
English investigators because no others, not even Italians and
Frenchmen,
are so far removed from the influence of German metaphysics.
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LEONARDO
fluous,
and is of
opinion that “empty
space would do just as well, if we only chose to conceive a continuity
of interacting motions.“ 19
I think that this
sketch, slight as it is, will suffice to make you understand and accept
Kant's apodictic assertion, “natural science is throughout a doctrine
of
Motion, either pure or applied.“
But here comes in
a second important consideration, not, like the first one, composed of
physical elements, but purely philosophical. The highest code of this
science
of Motion is not perceived as a fact in Nature, but is rooted in the
essence
of Reason. It is we ourselves, we men, who have no other possibility of
comprehending Matter, that is to say, when we aim at a comprehension of
Nature which shall be logical, thoughtful, and capable of founding an
apodictic
certainty of knowledge; — it is we ourselves, I say, who are unable to
comprehend Matter otherwise than as Motion, and for whom in consequence
of this every vera scientia, every absolute certainty is bound
to
result in a doctrine of motion either pure or applied. The human
understanding
works out the analysis of Motion by its special gift of schematic
experience
which we call mathematics. It is by mathematics that the human
intellect
assimilates and digests that which is foreign to it and outside of its
ken. Much is rejected, but what remains from that time forth becomes
possessed
of a humanly comprehensible form. That is what Kant means when he says,
“the highest law of Nature must lie in ourselves, that is to say in our
understanding.“ To put it rather roughly, but in a way suited to the
present
standpoint of our study, Nature gives the facts, the human
understanding
gives the laws. To formulate this let me once more bring forward words
of Kant's, “the human understanding does not create its laws out of
Nature,
but imposes them upon her.“ At the first blush this remark will perhaps
strike you as
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strongly
paradoxical,
but it will
suffice
for the present if you to a certain extent clearly grasp these two
things:
all exact Science, in the
true
and
strict
meaning of the word, resolves itself into a Doctrine of Motion. All
Doctrine
of Motion is mathematics, and so far human. To try to escape from a law
of our true Being is nothing less than an attempt to creep out of our
own
skin. We may well therefore praise the acuteness of the great Leonardo,
who had so rightly and energetically grasped the fundamental law of all
exact investigation — in opposition to whom when a man comes forward,
even
should he be a Goethe, and exclaims, Friends! I will teach you a
Science
that shall be unmathematical, — then we recognise and acknowledge the
fact
that the great man is entangled in deep error. Indeed, the error is
twofold,
first inasmuch as his definition of Science cannot be called adequate,
and secondly because he does not rightly grasp the essence of
mathematics,
and their law-giving function in reference to all that constitutes
causal
conjunction, and that means Nature as it exists in our thoughts.
Quite another
question
is whether that which Goethe strove after, that is to say an
unmathematical,
and to that extent un-logical and therefore unscientific comprehension
of Nature, is not, say what you will, entitled to a profound measure of
justification. Here, too, is a question that we must not leave
unanswered,
for it is of weighty importance in the understanding of Kant. But in
order
to answer that question we must do as we did in our former lecture; we
must undertake an excursus which will furnish us with the indispensable
and self-evident material. If you were minded without any further
preparation
to plunge headlong into Kant's abstract-analytical method of thought, I
suspect that it would be very difficult for you to bring a vivid
understanding
to bear upon his exegesis of an unmathematical
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conception
of Nature,
what he calls
“Nature
as exposition“ — whereas starting from Goethe you are at once in a
position
to understand Kant, and so will be able to delight in the unexampled
profundity
of the most powerful of all thinkers. We must then take heart and
undertake
an examination of the relationship between exact Mathematical Science,
to which alone Leonardo assigned any value, and Goethe's comprehension
of Nature. In the main this excursus will result in a comparison of
physical
optics and Goethe's doctrine of colour; there are, however, some
general
remarks with which we have to set out, and which will weave themselves
into our exposition as it progresses.
*
* * * * *
The difficulty which
at the outset attaches to our task is the fact that Goethe himself was
devoid of any theoretical consciousness of his own procedure, one might
even say of his own aim. His own saying, “a man has never gained so
much
ground as when he does not know whither the way leads,“ is true of
himself;
for while he believed that he was doing no more than lending a hand in
contemporary investigations of nature, he was in reality founding a new
method. That is the naked truth, the unrecognised truth which seems to
have foundered without leaving a trace, yet not for ever, in the noise
and dust of the vulgar riot of our successful mechanical science. There
are moments in the activities of great intellects where they render
superlative
services: that is when they do not quite understand themselves, when
they
enter the lists to do battle passionately for some impossible
assertion,
in spite of being gifted with a keener sight than their fellows, and
with
more consequential thought than their censors: for it is just here
where
they entangle themselves in a mass of contradictions, that they work
like
an unconscious natural force, paving
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LEONARDO
the
way for future
knowledge: here the
intellect collects itself into an avalanche ready to sweep clean all
the
tidy paths of human frivolity, or like a volcano bursts the too heavily
weighted crust in which the idleness of tens of centuries has
imprisoned
the bright fiery element of the soul of man. Only consider Goethe, that
noble man! Is it thinkable that he with his brilliant eyes should have
looked in the light during a whole lifetime, and have seen nothing
true?
Yet, as I know that here I shall at once be tilting against unbelief
and
contradiction, I will quote the words of a pioneer in exact natural
science,
the admittedly greatest physiologist of the nineteenth century,
Johannes
Müller. He was what Louis Agassiz, Clerk Maxwell, and Heinrich
Hertz
(but with their exceptions a dwindling number of our famous natural
investigators)
were, a really lofty intellect of permanent importance. Here is what
Müller
says with reference to Goethe's essay on the skeletons of rodents, —
“It
is impossible to point to anything similar which comes up to this
projection
sketched from the centre of the organisation. Unless I am mistaken
there
lies in this outline the foreshadowing of a distant ideal of natural
history.“
Remember these words “the foreshadowing of a distant ideal“! And
Müller,
the exact investigator of nature, prizes the awakening of this
foreshadowing
so highly, that on the next page he pronounces the judgment, that
Goethe
has “reached the greatest“ not only as artist, but also as
investigator.
20 Here, too, is a
judgment which should never be forgotten. For we
moderns
have grown up under the nourishing showers of pseudo-scientific
platitudes;
Rudolf Virchow alone dared forty years ago to take Goethe as
investigator
publicly under his protection, a weighty witness indeed, upon whose
exactitude
and unimaginativeness no man will cast a doubt, but who unfortunately
was
not competent altogether to lift the veil of misunderstanding: for to
that
end would
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LEONARDO
have
been required that philosophical
training which Virchow abominated, so that his fine words raised a
great
storm of dust at the time, but soon died away leaving no influence
behind
them. 21
In these days every
tiny two-legged wheel in the great machine of science thinks himself
justified
in shrugging his shoulders over Goethe as an investigator of Nature. I
happen to possess an autograph letter from one of these celebrities,
who
rates his professorial dignity at a height which entitles him to allow
himself the following judgment of Goethe: “his conception of Nature is
just what an easy-going aesthete and collector of curiosities might
make
up out of his walks abroad.“ This is the audacity of a man of middling
capacity whom the schoolmaster's rod and the sting of hunger have
raised
by luck to the degree of Doctor of Philosophy, and finally up the three
steps of the Professorial Chair! A man whose fame may perhaps live
through
two or three editions of our encyclopaedias, dares to speak in this way
of the princely intellect of a Goethe, of that god-like Eye which for
more
than half a century never ceased in the thoughtful contemplation of
Nature,
of a man of whom a Johannes Müller could pronounce the opinion
that
as an investigator he “reached the greatest.“
But enough of this.
If I were to talk myself into a state of indignation over the
intellectual
decay resulting from the narrow empiricism of a tyrannical science
which
has fallen a prey to the overlearned Philistines, I should not readily
come to an end. The reaction has already begun; there are good men and
true of a younger generation at work on behalf of Goethe the
investigator,
and what is more important than the influence of these individuals is
the
fact that a universal necessity, a cultural need that cannot be put
aside,
is forcing us to enter upon the road which Goethe has pointed out to us
as the “foreshadowing of a distant ideal,“ unless we
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LEONARDO
wish
to fall into crass
barbarism. A
leading spirit among the living antimetaphysical empirics, Ernst Mach,
has disclosed what is the next thing to be annihilated, and if his
object
was to serve the ends of a purely mechanical barbarism he has not been
far out: our languages! In the interest of “science“ they are to be
abolished
in order to make room for an abstract international language! 22 The
ideal
which floats before the learned professor is the Chinese system of
writing,
because, being entirely ideographic, it throws overboard all ballast of
expression of the finer emotions. 23 After that grammar and
history are
to be “laid aside.“ Add to this the simplification of the Alphabet, and
supplement it with algebraical formulae and chemical symbols, and you
will
have collected together all that Professor Mach deems essential in a
language.
He is not far out. A science which only concerns itself with abstract
Ghosts,
is at no single point in contact with life. Goethe's desire, by means
of
his doctrine of colour, parenthetically “to enrich language and so
facilitate
the communication of the higher conceptions among the friends of
Nature,“
from this point of view must signify the ne plus ultra of folly. And
when
Mach, in conclusion, expresses the hopeful opinion that the English
language
is in a fair way to reach that ideal, we will not ignore the tiny grain
of truth which has crept into this Hellish dream, worthy of one of
Breughel's
Witches'-sabbaths, and join the standard of those who hold no
inheritance
more sacred than that of their mother-tongue. The richer, the more
illogical,
the more incomprehensible a language, the better does it hold up the
mirror
to Nature. The men who have attempted to rob us of our language, have,
so far as in them lay, robbed us of Nature; has not Lord Armstrong
taught
us that science needs no more than the assumption of empty space (vide
supra)? In contradistinction to which the man whose genius was
rooted
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LEONARDO
in
the sovereign and
creative mastery
of language, — in his much-despised teaching of nature followed the one
object, to give us side by side with his immortal poems that which was
their one eternal Source, visible, inexhaustible Nature with all the
wealth
of its many forms.
Goethe, as I said
before, did not possess a critically analytical consciousness of his
new
method, and hence it is that his judgment as to the relationship
between
his way of investigation and that of true science, is hazy and easily
misleading.
Sometimes his insight
is clear enough, for example when he cites the attraction which in his
youthful days Spinoza exercised over him, and adds, “the mathematical
method
was the very opposite of my poetical method of thought and exposition.“
This, of course, is a general statement; the mathematical method, dear
to the Jewish Thinker, seems to Goethe to be in opposition to his own
poetical
method of thought. And yet when we come to deal with the special
investigation
of Nature there are passages of decisive import which may be brought
into
court. I select one from the year 1826, which possesses the importance
of a composition with mathematics. Goethe writes, “It was not long
before
I was compelled, in deference to my own capabilities and relations, to
claim the right to view, to investigate, and to comprehend Nature in
her
simplest, most secret beginnings as in her highest and most striking
creations
without the co-operation of mathematics. That has been my contention
through
life. Any service that I may have rendered in that way is open to all:
how it may appeal to others remains to be seen.“ 24 Is not this
perfectly
clear? “In deference to my own capabilities“ — that points to the
capabilities
which are “in opposition to“ Mathematics. And Goethe claims the right
to
view, to investigate, and to comprehend Nature in accordance with these
capabilities. To view, to investigate, to com-
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LEONARDO
prehend,
that is a
perfect programme
for a personal system of Natural Philosophy. Further on in the same
disquisition
Goethe says in so many words that “a new point of view justifies new
opinions.“
This recognition explains the many passages in which Goethe declares,
with
no trace of bitterness, that his method of contemplating Nature “is
incomprehensible
to the Professors, for the simple reason that they think otherwise“; it
is in these passages that he confesses in regard to the first great
congresses
of German works on natural history, that they furnished nothing which
could
in the slightest degree touch, or move, or excite him, no new
encouragement,
no new gift, — and this was the man who “for fifty years had been
passionately
devoted to the observation of nature“; for among the German natural
scientists
there was “not one that showed so much as the slightest approach to his
own way of thinking.“ 25
And there are other passages which come under
this category, in which Goethe in his last years, — as, for example, in
the essay on the rodents quoted above — instead of as was his wont
portraying
his efforts in the domain of morphology in the bright colours of the
successful
investigators, all at once “feels most vividly that his honest
endeavours
in the observation of nature, were only presentiments and not
pioneering.“
All this leaves nothing to be wished for in the way of clearness and
true
insight. In such moments Goethe is so fully conscious that he cannot
see
eye to eye with the men of true science, that he claims it as a right
to
dare to investigate in his own way, and admits that this way is
something
which continues to be incomprehensible to them, indeed that he is
dealing
with a “new standpoint,“ — with something in the future, — of which the
significance remains half veiled even to himself.
Physics simply do
not recognise the fundamental ideas of Goethe's Doctrine of Colour. So
from the stand-
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LEONARDO
point
of physics it is
impossible to
judge of this theory. Goethe starts precisely at the point where
Physics
leave off. It is evidence of a quite superficial appreciation of the
matter
when people go on talking of Goethe's relation to Newton and modern
Physics,
and at the same time take no thought of the fact, that these two are
entirely
different. 26
Unfortunately it
is Goethe himself who with the utmost impressiveness and vehemence has
spoken about his so-called relations to Newton, and not to Newton
alone,
but to exact natural science in general. Did you remark in the
above-quoted
solemn declaration the five simple words, “without the co-operation of
mathematics“? That is where the evil fountain of misunderstanding
still
continues to flow. It is not without the co-operation of mathematics,
but
in opposition to mathematics that Goethe observes, investigates, and
comprehends
Nature. The mathematical method and Goethe's method may run parallel to
one another, but can never coalesce: no compromise between them is
possible:
they cannot at one time work together and at another time without one
another.
One instance will
serve better than a hundred to show you how deeply this
misunderstanding
penetrated in Goethe's case. One month after that fundamental
declaration
in which the practised eye alone can detect the blemish of the “without
co-operation,“ he says to Eckermann: “surely it is not the
mathematicians
who invented the metamorphosis of plants? I worked it out without
mathematics,
and the mathematicians have been forced to admit it.“ If these words
are
correctly reported, they are valid proof that we must trust to our own
powers in order to see clearly in this matter; Goethe, the herald and
founder,
leaves us in the lurch as to the true understanding of his work.
Mathematics
and metamorphosis! This would have been the place to show that we are
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LEONARDO
dealing
with two
dissimilar and
irreconcilable
subjects which nowhere come into contact with one another. The first
lecture
has shown you what was Goethe's idea of metamorphosis; we must admit
that,
like every human acceptation, it implies the conception of Motion; but
instead of trusting itself like the sailor to the stream, it hovers
like
an eagle in the empyrean from which the living hurrying flood is at
once
Motion and Rest: Motion, so far as its law of existence is concerned,
Rest
as regards form. Mathematics (and in a wider sense all true science,
inasmuch
as it everywhere obeys the one impulse to be converted into
mathematics)
have no other power and function than the analysis of the condition of
Becoming; * even that which is at rest they must set free into motion,
otherwise
they have no hold upon it. Goethe's efforts, on the contrary, do not
tend
towards analytical knowledge, but towards the most intensive
contemplation,
— “the world of the eye,“ the law of which is not Becoming, but Being.
That accounts for the peculiar permeation of that which is simultaneous
and that which is successive which has sometimes puzzled us, as indeed
it puzzled Goethe himself. For while science, whose whole essence
depends
upon the understanding of cause and effect, recognises Being as an
almost
imaginary point between something which has been growing and something
which is yet to be, the Eye, on the other hand, although not blind to
successive
alterations, can manifestly never perceive the condition of progression
or process of “Becoming,“ otherwise than as locked up in a condition of
Being. This will suffice for the moment to show the absurdity of
Goethe's
outcry against the mathematicians. How was any mathematician, as such,
to discover Meta-
*
There is, so far
as I can see, no single English word in common use which accurately
conveys
the meaning of the German Werden
as opposed to Sein. Werden
is
the
process of coming into Being — i.e. a transition state; Sein is Being —
i.e. an accomplished fact. I shall translate Werden throughout
the book as “Becoming,“ or “coming into being.“ — R.
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LEONARDO
morphosis?
He would
have been a poor
mathematician. Nor do the words, “I discovered this without
mathematics,“
hit the nail on the head any better, though Goethe rarely fails us in
that
respect. And as regards the closing remark, “they have been forced to
admit
it,“ that is simply based upon error. Goethe's doctrine of
metamorphosis
has been as much repudiated by Science as were his anti-mathematical
optics.
It may be admitted that the repudiation was not so unanimous and
immediate,
but only because in the domain of Biology the complication is far
greater,
so that room is afforded for endless misunderstandings. But open any
reliable
contemporary book on botany, for instance, Julius Sachs' History of
Botany
(chapter 4), and you will find Goethe's doctrine of metamorphosis
unconditionally
refuted. Sachs shows how Goethe was continually wavering between fact
and
idea, and he reveals the mischief of which hangers-on laid the
foundation
during long years, while they, instead of turning to account the
thought
of metamorphosis “in the deeper sense of idealistic Philosophy,“
introduced
it into exact science, which was impossible without “combining the
highest
abstractions with the most careless and rawest empiricism, and in a
measure
with quite false observations.“ The doctrine of metamorphosis has been
quite as much a hindrance as a help to the science of the nineteenth
century.
That is the judgment of a scientist whose right to be heard cannot be
called
in question. You see how peculiarly connected it is with the words
“they
have been forced to admit it.“ But the great confusion which to this
day
has existed between Goethe's doctrine of metamorphosis and exact
science,
is due, as I said before, to the nature of the subject. All the
sciences
are striving towards mathematics; yet Biology, in contradistinction to
Physics, is still far from having reached the mark. And here we must
depend
upon a schooling of the sense of sight, otherwise the subject will not
be seen at
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LEONARDO
all.
We may observe,
but we shall not
take notice. That was why before Goethe's time comparative anatomy
dragged
out a miserable existence; men like Kaspar Friedrich Wolff died
unknown.
Goethe, with several others around him like Camper and Oken, was the
first
powerfully to excite the imagination, and so compelled it to take
notice
of what was seen. Here alone lies the significance for science of a
doctrine
of metamorphosis. The whole of Goethe's natural science might be called
“an introduction to the art of seeing.“ Nor is that a small thing to
say;
for phantastic imaginings do not teach the art of seeing, but on the
contrary
lead to those false observations which Sachs blamed; on the other hand,
there is a Something which Goethe by a happy inspiration called the
“exact
phantasy of the senses.“ 27
This phantasy is — as the word describes it
— something felt by the senses, not abstract; it rests upon very
accurate
Seeing, and the unsurpassable exactness of many of Goethe's
observations
is attested by Müller and Helmholtz,
by Virchow and Gegenbaur, by
Sachs and Ferdinand Cohn; here, however, exact Phantasy must be allied
to exact Seeing. Scientific hypotheses all pass away, but Goethe's
Doctrine
of Metamorphosis and Doctrine of Colour will never pass away: they
stand
as firmly as the facts which they mirror in Reason. Hence the
importance
of Goethe's ideas in Zoology and Botany. Science has used his thoughts
as she uses the ophthalmoscope, in order to see into the depths, to
discover
facts; but, only as tools, not as an organ. It is true that the Doctors
of the theory of evolution delight in tracing their pedigree to Goethe,
but that is the innocence of the elderly child that would do better to
search for an ancestry in Moses, Sanchoniathon, Thales, and Empedocles;
venerable men who historically would render it better service, and are
also more in sympathy with its intellectual culture.
But in order exhaustively
to
lay
bare the relationship
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LEONARDO
between
Goethe's
conception of Nature
by means of the “Phantasy of the senses“ and the mathematically exact
Science
of Nature, we must avail ourselves of a concrete example. For this
Optics
and the Doctrine of Colour must serve our purpose. Naturally it is not
possible for me here to dive into the mysteries of optics, nor indeed
would
any man be able to give an exposition of Goethe's doctrine of colour
with
greater brevity than he did himself. That immortal work is all
Perception
and pure Perception; the least learned of men can study it paragraph by
paragraph, and see step by step what Goethe saw. Here sight and
understanding
are identical. One would imagine that every man would lay hold upon
it. And since Helmholtz,
whom
I may
well quote here as the universally honoured representative of the
mathematical
anti-Goethe science, has expressly affirmed, “the experiments which
Goethe
cites in his Doctrine of Colour
are accurately observed, and vividly
described;
as to their correctness there is no dispute,“ 28 no one who might wish
to
read this forbidden book need tremble for his scientific salvation.
Exact,
correct observations, which have moreover earned the praise of a
genuine
university professor as being vividly described, can certainly harm no
man. But I know that it is all of no use; no human being can be induced
to read the Doctrine of Colour.
This glorious child of a demi-god is
like
a sleeping Brünnhilde waiting for the dawn of the new day which
shall
bring the hero to awaken her. We can therefore only deal with a few
leading
principles: that however will suffice to enable you to grasp the
difference
between “Nature as Mathematics“ and “Nature as Exposition,“ and never
again
to fall into doubt as to the right of the latter to assert its right to
a place side by side with the former. In this connection Goethe,
Leonardo,
and Kant, each in his own special individuality, will arise before your
eyes.
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The first paragraph
in Helmholtz's Optics
consists of a definition of light, set forth as
follows:
“Light is looked upon by the majority of physicists as a peculiar form
of motion of a hypothetical medium, the Light-aether; and we will
accept
this view of the undulation-theory which very fully accounts for all
Phenomena.“
Here you have in a nutshell the whole method of mathematical-mechanical
science. The storied words of the dying poet, “more light!“ are often
cited
in and out of season. What would our sensitive souls have said if
Goethe
had called out “more special motive-form of the hypothetical medium!“
Do
not, however, believe that with this poor joke I wish to cast ridicule
upon the optical definition; no less a man than René Descartes
is
responsible for the conception that light must be looked upon as
Motion,
and indeed as the motion of an invisible medium pervading the universe.
It is the acceptation of this theory that has made optics the most
perfect
of all sciences: I only wished to call your attention to the fact that
the first law of all science — if science is to be exact, — is the
abolition
of that which is visible, or if not in so many words of that which is
visible,
in any case the abolition of that which is practically seen, in favour
of an abstract, mathematically available, altogether unfelt and
schematic
representation. In the same work (p. 268) Helmholtz
actually reproaches
Goethe with the fact that “in his studies in natural history he starts
upon the principle of never abandoning the domain of that which is
visible
to the senses.“ From the point of view of science, then, it is a
mistake
to dwell upon that which is visible to the senses; for, as Helmholtz
goes
on to say, “Every physical exegesis must rise to the level of all those
Forces which can naturally never be objects perceptible to the senses,
but are only objects of the apprehensive understanding.“ These
so-called
Forces are purely creatures of the imagination: even the most sober
anti-philosophical
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LEONARDO
Science
now admits that
they only exist
in our heads: and for that very reason they are unable to make use of
the
perception of the senses as such, but have to replace them by a
sensibility
of the imagination. 29
What is then seen is, as one of the greatest men
of genius among the physicists, Heinrich Hertz, admits, a delusion from
which it is impossible to exact “any conformity with things.“ 30 The
very
first step then leads from concrete perception to the abstract, and
until
that has been attained no Science is exact. To the unlearned, to the
men
of simple thoughts, Light would seem to be the most concrete of all
perceivable
things. And yet here you have a definition of Light which thrusts aside
all perception; for looked at exactly it contains only two
propositions,
of which the one says “Light is Light-aether,“ an idea which reminds
one
of the “I am that I am“ of Jehovah, and must be dealt with by the
logicians;
while the second adds the assertion that “Light is motion.“ Now since
the
most cursory reflection has already sufficed to teach us that the
conception
of Nature as Motion is an irrefutable requisite of the human intellect,
— that is to say that it is not a law of Nature but of our brain, — and
inasmuch as Leonardo as a keen thinker even in his day recognised the
justice
of this principle, — it follows that the above definition is neither
more
nor less than an universal metaphysical postulate. Since then, as you
know,
certain leading Physicists have also declared against the hypothetical
medium, the aether, as superfluous. In their minds Light is nothing but
abstract motion in empty space.
But now let us,
please,
read a little further in Helmholtz. We come to the Scheme. As Kant
acutely
remarked, “The subjective principle of Nature precedes the objective,
the
combination precedes that which is combined.“ That
“which is complex
must
be supplied by ourselves,“ for we can investigate “only that which we
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LEONARDO
have
ourselves
supplied.“ Now in optics
we supply a great deal. 31
Following the advice
of Helmholtz, let us attempt to represent the “peculiar form of motion
of the hypothetical medium“ with the help of a wet thread which we
shall
hold in our fingers and allow to hang down freely, moving to and fro.
Here you see the
diagram; the line from A to B is the thread when not in motion; so the
tiny particles of Light-aether follow one another in a straight line so
long as they are not in motion: and here the dotted line shows you how
the so-called waves behave as soon as a particle of aether begins to
rock
backwards and forwards at the top point A. And what is the reason of
the
movements of this same particle? Here again we have something peculiar.
Helmholtz writes, “exactly as the movement of the single particles of
the
thread takes place, so would be the movement of a succession of
aether-particles
along which a ray of light should transmit itself.“ If then we have
above
received a definition of Light containing nothing more than the
abstract
admission that Light is motion, the Light-ray is at once introduced as
an hypostasis transmitting itself along the particles of Light-aether,
and
so causing their motion. And whilst light is, according to the
definition,
a movement of waves, the Light-ray is, according to the mathematical
definition,
“a perpendicular line.“ Here we seem to have got into something like a
witches' caldron! Let us just imagine that the sun is at noon: the
world
is steeped in light: how many rays are we to reckon there? Certainly
more
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LEONARDO
than
you would see in
the halo of a
Byzantine
Madonna's head. Does this raise a smile? And yet here is no joke, but
your
faith is bespoken for the following proposition: “we can in such cases
consider
the movement of the particles of aether inside a ray approximately as
an
isolated mechanical Whole, progressing independently of the movements
of
the neighbouring rays.“ You perceive that the rays are an altogether
material
conception. The possibility that innumerable rays running side by side
do not take up the lateral movements of the aether-particles, so that
for
the very superabundance of light-rays there is no light, is a question
which
you must settle for yourselves, for to that Physics afford no answer.
If
we had the wave-movement alone, or the rectilinear movement alone
(which
latter movement laid the foundation of the whole optics of Newton who
knew
nothing of any wave-movement), then in that case our power of
conception
would at least possess a possible illusion; but to our mathematical
Physics
both conceptions are indispensable: Light is a rectilinear movement
which
does not spread itself like sound in all directions, and in spite of
that
Light is at the same time a wave, or undulatory movement. It is only by
the conjunction of these two contradictory hypotheses that all
phenomena
can be exhaustively and mathematically reduced to a Scheme. The great
mathematician
d'Alembert calls attention to the fact that the so-called
“cloudlessness“
of Mathematics really only holds good where they deal with the wholly
Abstract,
but that the richer the evidence of the senses to which you apply them,
the darker become the conceptions upon which you base their operations.
32 You see here how
true are his words. Mathematical Physics are
practical,
useful, infallible, grand, bewildering — I
would gladly grant
their right to all the laudatory Predicates in the Dictionary, — with
one
exception, clear. Whoever in
agreement with d'Alembert searches
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LEONARDO
into
their foundations,
will find them
obscure. When Goethe wished to
lay before a friend a few ideas
concerning
light he began by storming against the unhappy conception of the Rays.
“There is absolutely no question of Rays, — they are an abstraction
invented
to explain the phenomenon in its greatest simplicity, an abstraction
worked
up, built upon, or rather piled up, until the whole matter at last
became
muddled into incomprehensibility.“ But let us leave the so-called Rays
and return to our Waves, only following the exposition so far until we
meet a practical conception clear to the senses, so as to justify the
belief
that we are floating down from the cloudy Olympus of hypothetical
constructions,
and are setting foot upon the solid ground of empirics.
And here I would
crave your attention: the time has come for us, as Kant taught us, “to
supply the synthesis.“ In proportion as I move the thread more or less
violently, so the serpentine line which it forms in the air will show
greater
or lesser curves; in the same way the aether-particles in the
Light-aether,
in proportion to the violence of their motion, will deviate more or
less
from their original position, — in other words, the Waves will be
higher
or lower: this variation in the height of the Waves is known as the
Amplitude
of Wave-motion. Besides the height of the Waves their length has to be
taken into consideration. The distance between a1 and a2, from
wave-crest
to wave-crest, or from b1 to b2, from wave-valley to wave-valley, may
vary
in length: that is called the Wave-length.
Thirdly, the movement of
each
aether-particle, which can be imagined as rocking to and fro, may take
place with varying speed: that is called the Duration of oscillation.
Pray
hold fast to the conception that we must admit in these Waves a varying
height, length, and duration of oscillation. And inasmuch as thought,
as
the saying goes, pays no duty, we may besides all this assume various
directions
of motion.
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LEONARDO
Once
more let us
consider the wet
thread.
I can move my hand from right to left and from left to right, and then
the single particles of the string which is curving into waves will
also
move rectilinearly to and fro; the same holds good with the
aether-particles;
in this case we say that the light is “rectilinearly polarised.“ But
just
as I move my hand rectilinearly from left to right, I might equally
move
it in a right line to and fro from front to back; I must therefore
assume
at least two directions of oscillation, and indeed might, if necessary,
assume as many more as I please; in the simplest case we speak of two
perpendicular
directions of waves mutually polarised. Again, I might move my hand in
a circle or in an ellipse. In that case the single particle of the
string
would, instead of a straight line, describe a circular line, or an
elliptic
line from one wave-crest to another, or from one wave-valley to the
next:
here too we must assume the same of the Light-aether particles; in the
one
case we speak of circular polarised light, in the other of elliptic
polarised
light. There are several other complications, but for our present
purpose
this is sufficient. We can therefore conceive waves of varying
amplitude
(that is to say height), waves of different length, waves of different
duration of oscillation, waves rectilinearly polarised,
perpendicularly
polarised in opposition, circularly polarised, and elliptically
polarised.
But now I have to make a last and highest demand upon your imagination.
Represent to yourselves all these differences with all their various
prepositions
as before, “with,“ “in,“ “upon,“ etc. — high waves and low waves, short
and long, swift and slow in their oscillations, in endless gradations,
pressing upon one another in all directions, — and in addition the
aether-particles
in various straight lines, and also working through each other in
circles
and ellipses: and then do you know what you have arrived at? Why, the
Natural
Light of the Physicists, as the
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LEONARDO
sun,
the candle, the
match brings it
into existence! This again is a serious matter. Cross-examine Helmholtz:
he will furnish you with
information.
“Natural light,“ he writes, “is a uniform compound of all sorts of
differently
polarised Light,“ moreover, “it contains wave-features of an endless
number
of continually intermingling values of oscillatory duration.“
It seems to me that
we have fairly carried out Kant's behests as to ourselves supplying the
synthesis. Yet I must here insert a remark. Nothing would be more
unjustifiable
than to ridicule this Scheme of the physicists, far rather would I with
complete confidence vote with Kant that such intellectual constructions
are “the pride of human reason.“ It is only by degrees that the
monstrous
complexity of the illusion established itself, as new phenomena, which
it was necessary to incorporate in the one great Scheme, gradually
became
known, new ones have cropped up since Helmholtz's time: as for instance
the Röntgen Rays, which compelled the conclusion that the
oscillations
do not only take place perpendicularly to the direction of
transmission,
but also parallel to it; that is to say as if we were not only to move
our wet thread to and fro rectilinearly and in a circular line, but
also
from top to bottom and conversely from bottom to top, — not only
therefore
in the direction of floor and ceiling, but also in the direction of the
room's walls. There will always be new additional matter coming to the
front, until in the end the Scheme of Undulation will become useless by
reason of its growing and alarming complication, and then some genius
will
enrich us with a new Image which will combine Light, radiating energy,
chemical agency, electricity, magnetism, in one single practical scheme
paving the way for new discoveries. The new theory is already at hand,
widely developed, only lacking as yet a presentation as Image. 33
Enthusiastic
admiration is the due of those men who like
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LEONARDO
Democritus
and
Descartes and Kant endow
the human brain with such schematic and creative illusions; and
unquestionable
recognition is the meed of those men of exact Science who, like Newton
and Helmholtz,
by their scorn of fatigue, their gift of observation,
acuteness,
power of sensitiveness, and talent, not only enrich the treasury of
knowledge,
— adding to the already existing thoughts of genius which they have
inherited,
— but also render to mankind services of imperishable value. One need
only
think of the ophthalmoscope! The depreciation of exact science, as we
meet
with it here and there in the works of various fanatics and
obscurantists,
makes one so indignant because it denies manifest demonstrable services
which every lamp-cleaner can see even if he cannot understand them;
whereas
the depreciation of philosophy and art is pardonable where it is due to
stupidity or faulty
education.
We must
have no misunderstanding upon this point. The one thing against which I
defend myself is this, that an invisible church served by a priesthood
of narrow-minded, arrogant, and intolerant professors, who under the
honourable
title of “learned“ enjoy a quite unjustified respect, — since learning
and power of judgment by no means of necessity go hand in hand 34 —
that
these enemies of nature, this tribe of fanatics should seize upon my
understanding
even in childhood, should annihilate its healthy power of observation,
should hold in a scientific vice its healthy thought, and compel my
belief
in silly dogmas with a tyranny more cruel than the tribunal of the
Inquisition.
There is no need for me to believe in God: it matters little whether I
am a morally strong, energetic, and free man: but if I refuse to
believe
in the hypothetical medium, the waves that are rays and the rays that
are waves, in the amplitudes and oscillations and polarisations and
such
abominations,
together with the descent of man from apes and of apes from jelly-fish,
then I am outside
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LEONARDO
the
pale. Heinrich
Hertz gives a
striking
example with reference to our modern Physics. A piece of iron lies upon
the table. Why does this iron not fly into the air, or pierce the table
in order to fall to the ground, or burst asunder into millions of
atoms?
Just as in the case of Light, physics here set out such innumerable
so-called
“forces,“ all of them busily at work dragging the piece of iron hither
and thither, that a mathematician would have to work for weeks before
he
could give a scientifically plausible proof that the piece of iron is
really
lying peacefully upon the table. Hertz writes, “But the truth is that
all
powers are so compensated as against one another, that the whole
arsenal
of them comes to nothing: that in spite of a thousand causes of motion
which are present, no motion takes place: that the iron just remains
quiet.
If we lay these conceptions before unprejudiced thinkers, who will
believe
us? Whom shall we convince that we are talking of something real and
not
of the hallucinations of an extravagant imagination? 35 “We may accept the
power of imagination of science even though it becomes extravagant; but
that we should sacrifice our independence, our reason, our phantasy
nourished
at the fountain of perception, and lay it upon the altar of this
Goddess
of Abstraction, — that is something that we must fight against with
might
and main before it is too late, before this scientific barbarism shall
have plunged us into the darkness of night.
But we had proposed
to ourselves to follow the course of the exegesis of Physics until we
should
at last reach something real and tangible, and not a mere imaginary
perception.
And here on the very page where there is all this talk of waves and
polarisations,
a little lower down a well-known idea beams upon me. I see the word
Colour!
and what do I read? “the most striking peculiarity by which Light of
varying
oscillatory duration distinguishes itself is Colour.“ 36 The
unprejudiced
thinker to whom
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LEONARDO
Heinrich
Hertz appealed
will at first,
as it seems to me, be staggered by this. Colour is oscillatory
duration?
Yet there is no mistake; for here is the definition: “when every
aether-particle
in the motion of light always, over and over again, follows the same
course
at the same time and at the same speed, then the Light is called
simple,
monochromatic, or homogeneous.“ The unprejudiced thinker is more and
more
puzzled. He remembers that to the Physicist Light and Visibility have
absolutely
no common signification; in every ray of Natural Light the Physicist
detects
a great quantity of “Unseen Light“ — there is the ultra-violet and the
infra-red (or ultra-red): it follows that every wave must have its
colour,
a colour which no human eye can see. What is colour outside of the
circle
of Red, Yellow, Green, Blue? What is an invisible Colour? Neither a
perception
nor a conception in any way possible. Besides this, Physics are
compelled,
as we have seen, to premise that there exists a boundless number of
continually
interchanging values of oscillatory duration: this claim, raised in a
somewhat
different form by Newton, who admitted an endless number of material
light-corpuscules,
cannot be denied by any one who is possessed of the elements of
mathematical
Physics; now the physicists reckon the number of oscillations in the
deepest
red at 400 billions in the second, and in the brightest violet at about
800 billions: according to the definition of the physicists, therefore,
there must be within the visible spectrum some 400 billions of
different
colours. In truth, however, in theory as in practice, we can well do
with
the acceptation of four primitive colours, as Leonardo has shown in
many
passages; many people, Helmholtz
among the number, have thought it
sufficient
to distinguish three colours. 37
Besides this there is no relationship
between the accepted number of oscillations and the order of
progression
of the colours: the numbers rise by
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LEONARDO
a
hundred billions and
you are still
in the Red; on the other hand, a few beggarly billions, perhaps ten or
twelve, suffice to lead you out of the loveliest green into the darkest
blue. It is a still greater tax upon you that you should believe that
Red
and Violet, two colours that so imperceptibly merge into one another
that
no art can draw a line between them, are the extreme opposites of one
another,
the one called into being by the slowest of all oscillations, the other
by the absolute fastest. Again, spectral analysis has taught us that
flames
which present exactly the same colour to the eye consist of rays which
occupy a totally different place in the Spectrum, and must therefore,
according
to the Physicists, correspond with different numbers of oscillations. 38
I might go on for
half an hour upon this subject; for so soon as mathematical physics
tread
upon the domain of colours, we wade up to our mouths in the thick slime
of impossibilities and irreconcilable contradictions. As, however, I am
not prepared to go minutely into this matter, and as it is nevertheless
my duty to convince you that it is no private opinion of my own which
is
forcing itself upon you, but that I am expounding undeniable facts, I
should
wish to recommend to you an excellent Kompendium
der Physik, of which I
made special use in my student days: it is comprehensive, clear, and
strictly
scientific. 39 In
the first lines of page 536 you will find these
words,
“our eye distinguishes different colours, which originate in the fact
that
the number of oscillations which strike our eye in the same unit of
time
is different.“ Now does not that give the impression of a perfectly
concrete,
reliable fact? The Dogma, like the credo
in the catechism, comes first.
Then the honest author brings forward a whole string of considerations,
which are certainly not meant as objections, for that Colours are
oscillations
is a Dogma — Cursed be he
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LEONARDO
who
would look upon the
sacrosanct
oscillations
as a mere scheme for calculation! — but these very considerations
compel
the author to a confession which you will find on the last line of the
same page, — “Hence the perception of colour must be looked upon as a
purely
physiological fact for which physics have no further explanation.“ Thus
in the first line the physicists' explanation of our perception of
colour
is given in concise language, and in the last line comes the confession
that physics can give no possible explanation of the fact of the sense
of colour. If we are to bring the two assertions into harmony we must
premise
that the Physicist makes a distinction between colour and the
perception
of colour. If he is speaking of colour, the word means no more to him
than
an epithetum ornans for
“duration of oscillation,“ there exists no
interdependence
between Colour and Eye, colour is an objective physical phenomenon; he
talks quite calmly of rays which are sensitive to red, — not made
sensitive
by red — as if every wave-length wore its own livery; 40 but so soon as
his Physics take the eye into consideration the whole artificial
thought-phantom collapses. Light, the whole fabric of waves, of rays,
and of
polarisations, — all is well and good until it clashes with the human
retina;
but as for colour, there your hitherto all-powerful juggler must
confess
himself beaten: here is a sensation which his physics will not help him
to explain, and now comes the physiologist whom he himself called in,
and
if the physiologist is like Johannes Müller, a true philosophical
spirit, he will tell the physicist, “with the exception of the purely
optical
mathematical definitions upon the subject of elementary motions, your
doctrines
all rest upon the most obvious contradictions: Light is energy of the
senses,
and colour is an affection of the optic nerve.“ 41
We need go no
further.
We have reached the core of the matter. If you had already studied
Kant,
a word
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LEONARDO
would
have sufficed,
and you would be
able to survey the whole lie of the land, as one looks down from a
lofty
mountain peak upon the structure of the country below. That is the
point
to which you must be led, and you must once more be guided to an
understanding
of Kant before you have had the opportunity of studying the philosopher
himself.
Stand at the windows
of this room and look out! What do you see? The green of the meadows,
the
blue of Heaven, the yellow of the corn, the white of the snow
mountains,
and the grey of the clouds. All colour! The whole of your Seeing
consists
in a Seeing of Colours; the conception Light is an abstract one, it is
a collective name for all Colours. For if you consider the sources of
light,
such as the sun, the stars, and the flames and lamps which we use for
the
production of Light, they are in reality generators of Colour. Light
without
colour would be a contradictio in
adjecto. Indeed, there is no such
thing
in existence as a white light. If a source of light appears to us as
white,
it is only a question of relative brilliancy, or else it is owing to
the
absence of any object of comparison. The old-fashioned street lamps of
oil appeared to be of a deep orange-red colour when the gas lamps were
near them: gas flames are orange-coloured, incandescent lamps red,
incandescent
gaslight that seems so dazzlingly white is blue when seen near powerful
arc lights. 42
Whatever the eye takes in is Colour: everywhere Colour.
And
even white and black, notwithstanding that the careful observer is
obliged
to consider them as something special which he cannot without further
elucidation
put into the same category with the other colours, and though optical
analysis
teaches us that no active light contains them, still will be regarded
by
every independent thinker, as something related to Colour — as
something
positive. White is just as much as black a privatione de colori;
physically
it is the result of
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LEONARDO
every
exact mixture of
two antagonistic
colours, for instance, of yellow and blue, or of red and green, because
the one visual impression cancels the other. It is impossible for us
even
to think of anything absolutely devoid of Colour; it would simply
become
at once invisible. But I would not have you believe that I am trying to
prove Light to be a mere cobweb of the brain. That would be nothing but
sophistry. But in the same way as out of the experience of various
tones
I construct for myself the idea“ Sound,“ which thenceforth assumes
consistency,
and under which all the phenomena which the sense of hearing perceives
may practically and theoretically be gathered together, so out of the
experiences
of my eye which one and all can never assume any other form than that
of
Colour, — because every affection of the nerves of the eye is Colour, —
I abstract the universal conception of “Light,“ and if the expression
“abstract“ should seem too strong, — remember I am developing no
system,
and am not weighing my words in a scale — we will substitute the word
“derive“
— the conception of Light is a thought derived from the perceptions of
Colours. Do not run away with the idea that these are hair-splittings;
rather are we dealing with a real distinction, with a distinction which
the slightest reflection makes clear, and once made clear, may of
itself
suffice to render impossible the eternal confusion between the
mathematical
optics of the Physicists and the doctrine of Colour of Goethe, which
rests
upon a close observation of the perceptions of our sight. Colours are a
rock against which not even the force of a Hercules can prevail. We can
neither add to nor detract anything from the conception of red and
blue;
and moreover they defy any attempt at definition. As Descartes says in
his simple language: En vain nous
définirions ce que c'est que
le
blanc pour le faire comprendre à celui qui ne verrait absolument
rien, tandis que pour le connaître Il ne faut
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LEONARDO
qu'ouvrir les yeux et voir du blanc.
43 The conception
“Colour“ possesses no comprehensible element; it is,
to speak physiologically, pure energy of the senses, and to speak
philosophically
it is a sensation and empirical perception. Light, on the contrary, is
a matter of comprehension: in this case the understanding meditates
upon
material afforded to it by the senses, and for that reason the circle
of
this comprehension is uncertain and unstable: it even lies in our power
to widen it or contract it. Under examination you would probably hold
Light
and Visibility to be the same, and Goethe's startling assertion that
Light
and the Eye are one and the same, 44 would at once charm you
as a sound
truth: yet, as you have seen, to Physics the conception of an invisible
Light — therefore of a day that is night — is familiar, and science for
the moment finds itself at a critical point where a new extension is
gradually
being acquired by this already widened conception, and not only
invisible
visibility, but Light which is not Light is accepted within the circle.
For we are so far on the road towards comprehending in one united idea
the phenomena of Light with those of electricity, of magnetism, and
other
molecular phenomena. This thought is not so new as the gentlemen of the
Press, upon whom the modern world depends for its culture, imagine. You
might almost say that it is seen as a germ in Plato's Timaios, and at
any
rate Descartes saw it floating before him, even though the phenomena of
electricity were too little known in his time for him to entertain more
than something like a general presentiment upon the subject. Herder, in
the 2nd chapter of the 5th book of his Ideen, makes mystical allusions
to the Identity of Light, Aether, the Warmth of Life, allusions which
have
no scientific value, but which show how near the same thought was to
him.
Kant, however, in an earlier writing (1763), says, “There are strong
reasons
for presuming that the expansion of
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LEONARDO
bodies
by warmth,
Light, electric power,
thunderstorms, perhaps also the power of magnetism, may be various
manifestations
of one and the same energetic matter which is distributed in all space,
namely of the aether.“ 45
And in his last unfinished work, he had very
accurately
served as pioneer to the modern theory, by the hypothetical acceptation
of his “material of Heat or Light,“ as a universal primitive movens. 46
This idea is one which it is no longer possible to lay aside, and if we
then claim for Light that it is a “peculiar form of motion“ of the
electric
waves, or for Electricity that it is a “peculiar form of motion“ of the
Light-aether, that is bonnet blanc,
blanc bonnet, and is decided by
practical
considerations or arbitrarily. The conception Light is either so
extended
that visibility only becomes one phenomenon among many, or so
contracted
that Light itself only forms one special case inside a greater complex
of molecular phenomena of movement. The various Colours, red, green,
blue,
yellow, orange, black, and white, on the contrary, remain what they
have
been from the beginning of time: on the one hand something entirely
objective,
a perception grasped by the understanding which no thought could have
generated,
but only the practical sensation caused by the object, — and at the
same
time entirely and utterly subjective in so far as Colour lies
altogether
in my eye, and is an expression of my purely personal relation to the
object.
Since then Light possesses the elasticity of all that is thought,
Colour
is an immovable phenomenon firmly wedged in between Object and Subject,
coyly rejecting any arbitrary handling.
You now know exactly
why our exact science has had such noble successes in the investigation
of Light, whereas its dealings with Colour have led to such a jumble of
impossibilities and contradictions, that the world is puzzled and the
specialists who have any literary scholar-
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LEONARDO
ship
begin to look up
to Goethe as to
a “Paradise Lost.“ 47
The conception Light is from the first
derivative,
inferential, the child of our human brain, torn away before its time
from
its mother, — the perception of the senses, — and so we may deal with
it
as may seem good to us. Not, be it noted, in the sense that would allow
us to invent or to ignore experiences or to turn them upside down, but
in the sense that we should guide our experiences from the outset on
the
road which they should follow in order to reach the arsenal of our
Knowledge.
That is the “supplying of synthesis“ of which Kant spoke. The changes
with
which the Scheme has from time to time to put up, are but adjustments
to
facts which cannot be forced into the chosen road. Thus Newton's idea
of
Light as motion differs from that of Descartes, — Huyghens', again,
from
that of Newton, — Young's from that of Huyghens; and here again we are
brought face to face with deep-reaching changes. In this connection it
is neither the most philosophical intellect nor the powerfully seeing
Eye
that will work with the greatest success, but the man who like Newton,
gifted with the greatest aptitude for mathematical combination,
possesses
into the bargain the sure instinct for the practical adaptation of what
he sees to that which is abstract and capable of calculation. John
Locke
long ago made the remark that Newton's greatness consisted in the
discovery
of intermediate ideas. 48
As an observer of nature, Newton is not
worthy
to loosen the latchet of Descartes' shoe. Descartes' inspired thought
of
the movement of a propagating medium was too lofty for him, he could
not
conceive of Light otherwise than as matter thrown out by luminous
bodies;
and his eye was so innocently unsophisticated that he felt compelled,
two
hundred years after Leonardo, to hold fast to the biblical number seven
in relation to colours, and in his remarks about the colours of
shadows,
fell into blunders about contrasting colours and the like, which any
decent
student of Painting
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LEONARDO
in
Italy could have
pointed out to him.
And so his painting theory of emission with his doctrine of colour has
gone the way of all manifest falsehood (not to say absurdity), and at
the
same time his whole conception of Light, including what was imagination
and what was perception, tumbled to pieces. And then what remained? Why
do we all honour Newton as an immortal investigator? In the first place
there remained the calculations of one of the most marvellous masters
of
figures in the world's history, of an intellect specially incomparable
in combinations; for Newton is an unerring teacher when he remains
within
his own mathematical domain, the domain of observation peculiar to the
human understanding: all that lay to the right and to the left of these
calculations, the thoughts which gave birth to them, the perceptions to
the explanation of which they were directed, all might be false, but
the
calculations themselves were none the less correct. The next point is
that
Newton not only employed calculations which never can be upset, but
that
his intellect proved to be true in everything which might even
indirectly
be calculable, especially in the invention and ordering of experiments
having for their aim the reference of phenomena to movements capable of
analysis. For example, the phenomenon of Colours, as to which Leonardo
had made such keen observations, had no meaning for him. Never would
he,
like Goethe, have been led on the path of science by the sight of an
aquamarine
landscape with a purple sky in the snow country: the great
Colour-phenomena
of nature, — the blue of Heaven, the green of the thicket, the white of
the snow — are all emblems of rest: Newton hardly saw them. But, on the
other hand, he did remark in his dealings with optical glasses, that if
you press one glass surface against another phenomena of Colour arise.
Here Colour stood in relationship to motion, and to a measurable
manifestation
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LEONARDO
of
power: here Colour
must be pressed
into the service in order that it should give birth to mathematical
Physics!
In Newton we have to deal with a mode of sight which is practical and
also
at the same time intuitive, on account of which, even as a
contradictory
contrast, it may be placed in a parallel line with the method of seeing
of the artistic genius. That is why Newton's calculations remained a
great
generative principle for future science; that is to say, what he
created — not
its theory, but its practical ideas, and that implies the setting out
of
a number of points of contact between the mechanism of human thought
and
the mechanism of nature, the inventive achievement of an abstract
artist.
This side of the Newtonian intellect, — and this alone, — it is to
which
the word “genius“ applies, for here we see intuition and a bold
combination
of elements lying far apart. This recognition of the incomparable
importance
of Newton has been expressed in poetry by Albrecht von Haller, the
great
investigator of Nature, when he says that he — “Find't die Natur im
Werk
und scheint sie selbst zu meistern“ — finds nature at work and seems
himself
to master her. 49
To master Nature! that is not only the goal, but also
the method of exact science, which shrinks from no violence of thought!
Thus, for example, Newton's theory of gravitation rests upon two
directly
irrational assumptions, empty space and forces working at a distance: *
and it strides away over every observation of the senses, as we have
just
seen in the case of Colours: with this intent it builds for itself a
kingdom,
a kingdom of its own, in which observation of the senses has no place,
which is at once quite abstract and quite practical.
This contrast leads
us at once to a clear view of what Goethe's natural investigation
strove
for: a kingdom of that which is purely seen and unconditionally true.
The
* Forces working at a distance: e.g. the moon
acting upon the ocean.
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LEONARDO
two
methods are
diametrically opposed
to one another. They take up no challenge: there would be no need for
them
to fight, if it were not that passion is apt to take the place of
insight:
indeed, the one might consciously help on the other, which so far has
only
occurred involuntarily here and there; above all, it would be necessary
that all men of culture should as clearly recognise the relation, as
the
relation itself is clear.
But before applying
all this to the understanding of Goethe, I must say one word more. For
sufficient reasons I have in this lecture hitherto only incidentally
alluded
to Kant, yet as a matter of fact it is he who has been my guide: it is
to him that you owe any intelligence that you may have gained. To show
this more accurately would require a too minute enquiry into pure
philosophy.
Still, I should be loath to conclude this consideration of Exact
Science
without having, at least aphoristically, made two points, first to
prove
how correctly Kant grasped the essence of Science, and secondly how
undeniably
Science itself, insensibly and involuntarily, bears witness to the
truth
of his philosophy.
From what I have
already said you know how devoted Kant was to Science, subordinated to
the exact method of mathematics; how entirely he was at one with
Leonardo
in the belief that that alone was vera
scientia, and that nulla
certezza
was to be found except under its sovereign rule; you would therefore
not
suspect that he could possibly wish to degrade it. In one monumental
sentence
he gathers together all that we have been learning about it, and I
should
like you once for all to impress that sentence upon your memories,
because
in it he is laying down something that hardly any man knows, and yet
which
we all need to know. “Physics are the investigation of Nature, not by
experience,
but on behalf of experience.“ Here we have the essence and the value of
exact science enunciated and defined. I take it that
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LEONARDO
after
what has
gone before any
commentary
is superfluous. Your own knowledge now bears witness that Kant is
right.
But if the wisest heads amongst us are in doubt about it, if they go on
confusing method and matter, if they imagine that it is by experience
that
they have laid it down as a law that colours are “a varying number of
oscillations
of aether,“ whereas all these putative oscillations are, lock, stock,
and
barrel, only a method “on behalf of experience,“ that is to say, a
method
invented to widen the domain of experience, but not a method for coming
nearer to what is experienced by so much as an inch: — then there
arises
the lamentable confusion by which we are now surrounded, and by which
that
principle of our being, which may be described as the innocent, the
feminine,
the receptive, and the parturient principle, namely Perception, is
cruelly
imperilled.
I will say no more
about this at present. Even if Kant is here only speaking of Physics,
you
know that all science of necessity strives after Physics — and so this
method of investigation, not “through“ but “on behalf of“ experience,
forces
its way even into those sciences which are still at pains to tear
themselves
away from the matter of experience. For example, the essence and value
of Darwinism consists in the fact that this doctrine revealed a method
on behalf of experience. Darwin, like Newton, did not see clearly, and
still less did he think deeply; his, like Newton's, was a practical
inventive
intellect, utterly without reference to Nature, and that is why the
success
of his labours was an enormous addition to the matter of experience. 50
We shall return to this in a later lecture.
And now one more
word about the way in which the whole history of our exact sciences
bears
witness on behalf of Kant. We are indeed standing upon the highest peak
of a metaphysical mountain; I wish to
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LEONARDO
make
just a tiny rift
in the mist which
shrouds every man who has not yet grasped the thought of Kant, I wish
just
to open out a little streak of the blue sky, and by way of commentary
to
bring into play not Abstraction, but the practical history of our
sciences.
Democritus, up to
whose time the philosophers had regarded the original characteristics
of matter as qualitative, looked upon the “Qualitative“ as being really
quantitative.
That was a bold stroke, but it was a bold stroke which made Science
possible.
It was from him that Newton took the two important conceptions of the
Atoms
and Empty Space. Once admit that everything must be Quantity, then, in
order that it may have form everything must be Motion: hence these two
suppositions. This method of investigating “on behalf of experience,“
is
called the Mechanical Method. Inside the same frame another method
stands
in opposition to it, the Dynamic: this was founded by Descartes,
preferred
by Kant, introduced by Faraday into Physics in opposition to the
Newtonian
conceptions, and Heinrich Hertz was intending to establish it in
extenso
when he was snatched away by death. It is the method of the more
profound
thinkers among the investigators of the exact school, which makes havoc
of the absurdities of the mechanistic views. In this dynamic method a
space
filled without a gap is presupposed in which not hypostasised forces
acting
in empty space, but displacements, are the cause of all motion, and
since
experience is insufficient to make the calculation correspond,
invisible
masses and unseen movements have been invented in aid. 51 Outside these
two methods there is no possible mathematical interpretation of Nature.
52 The exclusive
stress laid upon Motion is common to both. But what is
it that motion presupposes? Time and Space, nothing else. Space for the
“outer sense,“ Time for the “inner Sense.“ And yet there is a third
presumption:
for “in
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LEONARDO
Space,
taken as such,
there is nothing
movable,“ and “it is not Time itself that changes, but that which Time
contains.“ In order then to be able to speak of Motion we require,
outside
of Space and Time, “the perception of a presence and of the succession,
or consecutive ordering, of its rules, consequently of Experience.“ If
then we follow up the history of our exact sciences, — whether we build
upon Mechanism or Dynamism, — we discover that it is their principle to
adopt a minimum in the matter of Experience. Time and Space, with
Motion
as third: for anything more they have no use. They remove from
Experience
everything which has no reference to Time and Space, and consequently
cannot
be brought into any relationship to Motion. The sense of the colour
red,
of the colour blue, is certainly Experience, but it is not the
Experience
of a “consecutive ordering.“ Blue is blue, blue is not red — and even
if
I construct for myself a scale of colour, it still hardly possesses a
greater
value than does the idea of the metamorphosis of the bones of the
vertebrae.
For this conception of colour and of the scale of colour has nothing to
do with Space, and contains no imaginable relation to Time, and so
affords
not the slightest point for mathematics to lay hold on. The Physicist
therefore starts not from Colour but from Light, and even that he only
seizes hold of where it suits him. The mirroring of outlines, the
refraction
of images, e.g. when seen in water; that is his starting-point, and
indeed
because there are here angles, and therefore something capable of
measurement
and calculation. The so-called Dioptrics, or science of refraction,
preceded
the mathematical theory of colours by a century: Kepler founded it in
1604,
Newton's experiments upon the “Colours of Light“ appeared in 1704. It
then
became a point to discover some relationship between refraction and
colour.
You can easily obtain a simple and correct idea of the nature of
Newton's
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LEONARDO
work
with the prism in
the interests
of this discovery. If you construct a light-tight box, and make a hole
in it with a fine needle, you will obtain on a photographic plate, if
placed
at the right distance, a beautifully sharp picture of the whole
landscape.
If in the same way you catch a so-called Ray of the Sun, then you will
obtain a picture of the sun. But if you enter the camera obscura
yourself,
and draw this Ray through a prism, projecting the broken Light upon a
screen,
you will no longer see the picture, for it is ruined past recognition,
but in its place you will see colours, and those colours will be in a
fixed
consecutive series. That this experiment does not carry us very deep
into
the essence of colour, as colour, is shown by what follows: you saw,
just
now, how lamentably the Physicist fails as soon as he reaches the point
where colour really exists, that is to say, the Eye; but that troubles
him little; for his principle is, as we have seen, to give a minimum of
importance to experience: he does not work “through“ experience, but
“on
behalf of“ it: and now he has what he wants: the Colours at which he
could
in no other way arrive are brought into relative position in space
fixed
by law, that is to say into a geometrical consecutive progression, and
that again means geometrical Motion; and so he can also measure and
calculate. 53
“The mathematician,“ says Kant, “can enter upon his construction of a
conception
from any datum that he
pleases, without thereby being under any
obligation
again to explain that datum.“
54 Not only does
the mathematician pay no
attention to anything further, but he consciously and of his own free
will
pushes it aside, together with everything which makes Colour what it
is:
all that concerns him is Space, Time, and therein Motion: colour is for
him a number of oscillations and nothing more; not indeed because he
has
in this fashion fathomed the depths of the matter, but because he lacks
the power to move one step nearer
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LEONARDO
to
the true essence of
anything, by
means
of scientific methods.
Rather than follow
Goethe in his indignation over all this, we will learn to look upon the
methods and successes of the exact investigator as a testimony to the
correctness
of Kant's fundamental conceptions of the human intellect. Kant teaches
us, as you heard in the previous lecture, that there are “two branches
of human intelligence; namely, the senses and the understanding, by the
first of which objects are given to us, while by the second they are
thought.“
What our senses give us we call Perceptions. For to-day let us leave on
one side the one branch — the understanding. Let us talk only of the
senses,
the source of our perceptions.
Within the senses
we must learn to distinguish between the two parts of which Perception
is composed, for that is the foundation-stone of the Kantian building:
in every perception of the senses one part is empirical, the other part
pure. The Greek word empeiria
means nothing more than experience, but
our more refined analysis needs the word experience for a special
meaning.
We will therefore not rebel against the expression “empirical.“ The
empirical
part of Perception is then that which we receive by sensation;
everything
that you see, smell, hear, etc., is, — in so far as you take into
account
this impression only — empirical Perception. “The impressions of the
senses
give us the first occasion to bring about experience.“ But before you
can
perceive as object an object afforded by the senses you must add
something
which is equally Perception, though not empirical Perception, — that is
to say, not an impression of the senses, not a feeling received from
outside,
but something which you yourself contribute as man, and which Kant
calls
Pure Perception in contradistinction to the other Perception. This pure
Perception is the idea of space. As Kant says, “the conception of space
is the
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LEONARDO
form
in which our
senses perceive, and
is innate in us before ever a concrete object has impressed our senses
in any one particular direction.“ 55
I do not wish to-day
to embark upon metaphysical discussions: and so I lay hold upon a
concrete
argument. You are aware that Natural Science has been developing
itself,
unhappily, out of touch with Kant, — for the most part in violent
opposition
to all philosophy: even such a man as Helmholtz,
who busied himself
much
with Kant, yet in many essential points utterly misunderstood him; 56
now
I should like you to take up the work of one of the most rabid
anti-metaphysicians
of our day, yet a pre-eminent and trustworthy investigator; Mach's
Analyse
der Empfindungen (The Analysis of Sensations). Here you will
find, p.
93
of the 2nd edition, 1900 (104 of the 4th), the assurance that the
biological
and psychological investigations of the nineteenth century have led to
the conviction that “the perception
of space is born with us.“ As we do
not propose to go deeper into this subject, this testimony, which is
above
suspicion, may suffice; it comes from a quarter in which for a whole
century
men have been labouring to prove the contrary. Mach and the men of his
intellectual school are certainly of a very different opinion from that
of Kant: there are millipedes that crawl upon the ground and eagles
that
soar in the air; both have the right to live, and it would be foolish
to
exact that they should view the world from the same point of sight; yet
the recognition acquired with painful honesty that “the perception of
space
is born with us,“ expresses the same fact as Kant's irrefutable
metaphysical
creed — “the conception of space is present as a form of our
sense-perception,“
— and that means the conditional possibility of all experience —
“before
a real object has fixed itself upon our senses by perception.“ You must
not fall into the absurd mistake of supposing that Kant meant that
Space
is not something
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LEONARDO
really
present; on the
contrary, he
calls
it on that very account pure
perception, because Space is the
fundamental
condition under which things in general “manifest themselves to us,“
and
thus at the same time determines the root of all perception. Moreover,
you must understand that this pure perception by itself would serve us
little; for, says Kant, “the Material or Real which is to be seen in
Space,
necessarily presupposes perception, and independently of that
perception,
which exhibits the reality of something in space, can by no force of
the
imagination be invented and brought into existence.“ Indeed, Kant gives
here a fine definition of sensation when he says, “it is that which
describes
a reality in Space.“ We are not then floating in the clouds, but are
working
on behalf of knowledge attainable by every thinking man, and without
which
he can rightly grasp neither Goethe's investigation of Nature nor exact
Science in its essence — and what we recognise is that in everything
that
nature in such generous measure brings to our senses, we must, within
the
limits of sensitive perception, and without reckoning all that our
understanding
afterwards adds to it, distinguish between a “pure perception“ which
constitutes
form, and an “empirical perception“ which constitutes the matter of
perception.
We can apply the saying of Aristotle which I quoted at the beginning of
the lecture: within the limits of the perception of the senses there is
passivity and activity; the conception of space is an “activity“ of the
human intellect, it is the condition upon which that which is perceived
by sensation (and that is “passivity“) can be viewed.
Now for the
application
of these considerations. Everything that is size, form, and quantity,
manifestly
belongs to the conception Space, and that means to the domain of pure
perception,
to the domain of form, to the domain of the necessary purely human
condition
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LEONARDO
of
activity. Here arises the plain
certainty
of mathematics. There are people who cannot see red, others who cannot
see blue; empirical perception, that is to say the capability of
grasping
sensations, differs in different individuals; but there is no man for
whom
the sum of the three angles of a triangle means more or less than two
right
angles. Again, I can in my mind construct a cone, that is to say
perceive
it, and out of this perception develop all its mathematical essentials,
without ever having had a cone presented to my empirical perception:
while,
on the contrary, I could never invest the cone with a colour or a
smell,
unless they had been previously known to me by the perception of the
senses.
If then in my investigations of Nature I confine myself as far as may
be
to the pure side of perception with the utmost possible neglect of the
empirical side, I shall be in the enjoyment of two great advantages. In
the first place, I take into consideration only that which is
absolutely
certain and universally valid, — the Formal, as you have seen, in
opposition
to the Material; secondly, as I am, so far as possible, limiting myself
to my own peculiar human domain, I am able on the basis of fewer
experiments
to hurry on to further experiments, just as I investigated the
essentials
of the cone (that is to say, its mathematical essentials) in my brain.
Empirical perception, that is, the perception of the outer senses,
brings
me at every step something new, something that I never had seen before;
whereas pure perception is despotically confined to sure and fixed
ways.
Every voyage of discovery, every net that is sunk in the depths of the
ocean, brings to light new forms of life, forms never suspected, never
anticipated: every year chemistry discovers new elements; with modern
telescopes
the number of the celestial problems has only been multiplied: on the
other
hand, Newton's calculations are to-day what they were yesterday, and
ten
thousand
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LEONARDO
years
hence they will
be just as true:
they are more firmly built than the Pyramids of Egypt; they lay down
the
tyrannical law of our own human intellect, the law from which we cannot
escape, and with which we “master Nature.“ Here, therefore, in the
domain
of pure perception, mixed up as little as possible with empirical data,
I can work on behalf of experience, and can give to the results of
experiment
a safe, incontrovertible expression. For “empirical perception is only
possible with the help of the pure contemplation of space and time;
what
Geometry says of the former, therefore, holds good also, without any
possibility
of contradiction, of the latter.“ Here, and here only, we obtain a firm
grip of the latter. That is the somma
certezza della matematica which
Leonardo so rightly saw and honoured as the ideal for all scientific
investigation.
While in other fields the attempts at exact research are subject to
change,
so that as Kant says “only fleeting steps are possible, of which time
preserves
not the slightest trace, in mathematics, on the contrary, its progress
is along a high road which the most remote posterity will be able to
tread
with confidence.“ That is why exact science, — and to be exact is the
strenuous
endeavour of all science, — confines itself to Size, Quantity, Form,
Motion:
in its ultimate perfection it postulates empty Space and Quantity —
nothing
more (see page 131): so when you see that it cannot altogether brush
away
the qualities of which empirical contemplation tells the tale, as for
instance
Colour, it bends and forcibly changes them into Motion, true to the
principle
formulated by Kant, “everything that is real in the objects of the
outer
senses must be looked upon as Force in Motion.“
In this little
exegesis
I have for simplicity's sake always made use of Kant in order to render
intelligible the essence and progress of our exact Science: but now you
need only invert the whole story, — you need only
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LEONARDO
recall
what I have said about physical
optics, and you will understand how I was justified in maintaining the
proposition that our science bears witness to the correctness of Kant's
analysis of the human intellect. It is the proof derived from
experience
that he saw aright.
I have spent so much
time over Leonardo's goddess, exact science, — that I have hardly any
time
left for Goethe's unmathematical method of perception. Yet I must hold
this to be but a small evil. For as soon as you have grasped the
essence
of exact Science, you almost automatically obtain as a result the
essence
of that observation of Nature which prefers the empirical method, the
impression
of the senses, while it as far as possible pushes on one side the
so-called
“pure method of perception“ as a mere formal principle, and only takes
it into consideration where it touches the empirical and unites itself
therewith, namely in the case of Form. “Quantity and mensuration in
their
nakedness,“ writes Goethe, “annihilate Form and banish the spirit of
living
contemplation.“ 57
Red is 400 billions of oscillations of the
hypothetical
light-aether in the second: we may well agree with Goethe in calling
that
a banishment of the spirit of living contemplation. It is in this
spirit,
in the spirit of living contemplation which has been banished by his
opponents,
that Goethe's observation of Nature is rooted. In order not to break
through
the circle, we will hold fast to his Doctrine of Colour.
You remember how
the physicist Helmholtz
tackled the subject. First came an abstract
definition
of Light, then a rich mass of possible constructions of the “illusion,“
as Hertz calls it, finally came the question of Colour. Goethe on the
contrary
starts with Colour. “All Nature,“ he says, “reveals itself by Colour to
the sense of the eye.“ 58
He declines to speak of the essence of Light:
“no mortal will ever be able to explain the nature of Light; and even
should
any man be able so to
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LEONARDO
do,
he would find no
one to understand
him or his Light.“ More comprehensive still is the passage in the
preface
to the Farbenlehre (doctrine
of colour). “For really it is a vain
undertaking
to pretend to express in words the essence of any thing. We perceive
results,
and an exhaustive history of these results might at the most embrace
the
essence of the thing. In vain we take pains to portray the character of
a man: but show us his dealings and his deeds together, and a picture
of
the character will arise. Colours are the deeds of Light, its
activities
and passivities. In this sense we can hope from them to obtain
disclosures
about Light.“
I would not add a
syllable to this: the Master has in these few words said all. My one
wish
would be that you should make friends with that beautiful, precious
work,
and learn to see through Goethe's eyes.
I should gladly have
said a little
more about the Doctrine of Colours, but it would lead us too far. Only
one thing I must say. If a brilliant process of discovery has testified
to the value of the mathematical method, equally a century of
experiments
has led to the result that Goethe, and Goethe alone, has correctly
observed
the phenomena of Colour. In the matter of the Doctrine of Colour,
Johannes
Müller is already out of date; Helmholtz, whom we have only just
lost
— out of date; Hering, who is still with us — out of date; 59 Goethe,
on
the contrary, as a younger professor has recently assured us,
“comprises
the foundations of the most modern opinions“; and that will hold good a
thousand years hence. It is no part of Goethe's endeavour to find a
theory,
that is to say a mathematical formularisation. When his brother-in-law,
Schlosser, asked him how far his Doctrine of Colour might agree with
the
hypothesis of oscillations, “I had unfortunately to confess that my
method
took no notice of the matter, but that the only object was to focus
innumerable
experiences, to set them in order, to
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LEONARDO
discover
their
inter-relationship and
their position as opposed to and as agreeing with one another, to make
them generally comprehensible.“ 60
My further lectures will show that
this
position of Goethe's corresponds exactly with Plato's and Kant's
critical
method of the understanding of phenomena, in opposition to every
childish
attempt at their explanation. Goethe's Doctrine of Colour is the almost
spotlessly clear reflection of empirical observations, and this is a
more
difficult undertaking, and needs more training than the use of
mathematical
instruments. The student in his very first term can make
spectroscopical
experiments — I know it from my own experience; but to have such a
clear
insight into Nature as Goethe had, that is a matter of genius and
self-education.
Goethe himself bore witness that he was “not gifted with keen sight“;
Leonardo's
eye, on the contrary, pierced like a dagger into the very heart of
phenomena;
but I think that our theoretical endeavours will have rewarded you by
enabling
you henceforth to distinguish between keen sight and clear sight: if we
accurately consider the schematising of the plastic artist, with which
we dealt at the beginning of this lecture, we shall find it to be under
the domination of the same despotic spirit that rules the schematising
of an exact investigation: that Goethe could not paint does not only
originate
from any deficiency, it may also be looked upon as the positive quality
of a spotlessly clear sight. And it may well be that this most rare
quality
accounts for the fact that people have not even understood how to read
Goethe. To this day you will find in every book about Goethe, whether
it
be the work of friend or foe, the assertion that Goethe taught the
existence
of three primitive colours, Red, Yellow, and Blue, and that he held
green
to be a mixed colour. Now if the book in question is under the
influence
of Helmholtz,
you will be taught that Goethe was mistaken, and that the
three
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LEONARDO
primitive
colours are
Red, Green, and
Violet: if the book is more modern, it will probably prove to you with
ease, that the idea of three colours is nothing but an absurdity, since
all phenomena of contrast show that colours run in pairs, and therefore
that, as main ideas, we must in every case accept two, or four, or six,
or some other even number of colours. Of these, as Leonardo was the
first
clearly to recognise, Red-Green on the one hand, Blue-Yellow on the
other
hand, are without a doubt to be accepted as primitive colours. To set
up
primitive colours, on the contrary, as Helmholtz did referring to
Young,
and at the same time to leave out Yellow and Blue, means a
ne-plus-ultra
of the art of combination devoid of all observation. 61 If you take
Goethe
himself in hand, you will be amazed to discover that it never occurred
to him to teach the doctrine of three primitive colours. It is true
that
he asserts that painters and colour-makers start from three colours
because
out of them they can obtain all the others; 62 but he himself fixes no
number as a general proposition, but only affirms that colour proceeds
from two extreme starting-points; nearest to the light a colour arises
which we call Yellow: another one nearest to darkness arises which we
describe
by the word Blue. 63
And so far as the culminating point of these two
extremes,
leading through Orange on the one side, through Violet on the other, is
Red (the Zenith as Goethe calls it); 64 while the depression of
these
same
extremes through Yellow-Green and Blue-Green, reaches a furthest point
called Green (which Goethe calls the Nadir): — so far we may certainly
talk of four primitive colours as Goethe sometimes does. We might
therefore
in Goethe's case speak of two or of four primitive colours, but never
of
three. But the truth is that in his view colour is a unity; that is why
he once suggests that Red includes all other colours. 65 But colour
might
equally be considered as a duality, inasmuch as “there are only two
quite
pure
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LEONARDO
colours,“
Yellow and
Blue. 66 Here, as
you will surely observe, the fundamental idea is form, not the
conception
of Numbers. And for that reason, unless I am mistaken, our most modern
physiologists with their purely mechanical conception of colours are
not
so near to Goethe as they themselves fancy. It is true that their
colour-cross
|
Red
|
|
Yellow
|
|
Blue
|
|
Green
|
|
has set us free from the silly colour
triangle,
|
Red
|
|
|
|
Red |
|
|
|
|
or
|
|
|
|
| Yellow
|
|
Blue
|
|
Green |
|
Violet |
and
every empirical truth here signifies
an approach towards Goethe, but I am afraid that Goethe must still wait
a while before he becomes quite modern. True, he has said that to
understand
his teaching “needs nothing more than clear vision and a healthy
brain.“
At the same time he has expressly declared that his teaching “is
harder to comprehend than
Newton's.“
67 Clear Vision is
as good as non-existent among us; we had to wait for
Goethe to teach us that.
And so we arrive at
the answer
to the question which rendered this excursus necessary — whether what
Goethe
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aimed
at, in express
opposition to
Leonardo,
namely an anti-mathematical, and so far illogical and unscientific,
comprehension
of Nature, was not profoundly justified? I hope that the question now
presents
itself to you in a quite different guise and stripped of all
phrase-mongering.
You have seen with your eyes the might, and at the same time the
beggarly
poverty, of the whole system of pure science. “The mathematician is
master
over Nature,“ says Kant rightly enough, but what does the master know
of
his slave? Nothing but the work with which he has entrusted him. Goethe
faces nature in a quite different spirit, and therefore with a
different
intellectual conception. His is not the ambition to master Nature, but
to possess her intimately: she is not to work for him, but he for her;
he wishes to re-create her and so make her his own. Exactly as we just
now recognised Colour as something at the same time quite subjective
and
quite objective, so he paves the way for a view of Nature that shall be
quite Human (without which it would be incomprehensible), and at the
same
time Pure Nature, or perhaps it were better to say as nearly Pure
Nature
as possible. Mathematical Physics are, as you have seen, something
painfully
abstract: for whilst, as far as may be practicable, it pushes on one
side
empirical observation, it not only denudes things of their essential
nature,
but it robs me as man of all the direct perceptions of the senses.
There
remain nothing but ghosts flitting hither and thither between object
and
subject. That Red can be understood as 400 billions of oscillations in
the second is a most important formula for science, and therefore for
practice:
for life it is absolutely without significance. La meccanica è
il
paradiso was no conviction of Goethe's; he said, “mechanical
formulae
change
the living into the dead.“ 68
His wish was to teach men to look upon
life
as something living.
Here again there must
be Method,
otherwise there
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would
be no unity and
no goal to attain.
The saying about phantasy which I quoted at page 121 — that it is very
near
to Nature and is the offspring of Nature — gives you the key. The
relationship
with Art is patent and sure, comforting and inspiring. But never forget
that other saying about the “exact phantasy of the senses,“ do not
forget
that Goethe saw with incomparably greater accuracy than Newton and
Helmholtz.
We are not dealing here with creatures of fancy, but with that which
Goethe
calls “the productive power of imagination.“ 69 Without imagination we
men
are lost:
only think of the waves
and
the rays,
and the polarisation of the hypothetical medium! But while mathematical
science works with Schemes which have only been invented in the
interest
of the human brain, Goethe is striving to come on the track of Nature,
and by the means of Symbols to discover and explain, not her mechanism,
but her Ideas. On one occasion his language is as clear as daylight, —
“my impulse is the embodiment of Ideas.“
As soon as we have
obtained a
clear notion of Goethe's goal and method, we understand what Kant means
when he demands of us that “we shall judge phenomena not only as
belonging
to Nature in her purposeless mechanism, but also to analogy with Art.“
But before speaking in this connection of Kant who in such a peculiar
fashion
goes hand in hand with Leonardo and Goethe, it will fit our purpose to
sum up briefly the result at which we have arrived in regard to the two
ways of contemplating Nature. The riskiness of such an undertaking is
well
known to you, but this is not the case of a building in which we
purpose
to abide, but only of a milestone on the road towards the attainment of
a living idea of the personal way of thinking of Immanuel Kant.
There is one mode of
Seeing,
analytical,
aiming at a mathematical dissection of Motions, — and there is another
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mode
which is intuitive
and directed
towards an imaginary reproduction of Nature. Neither has any value
unless
it is exact. The material, both objective and subjective, is in both
cases
the same; but the direction of sight implies as condition a deeply
reaching
difference; one man is unable to see the one end of the spectrum,
another
man cannot see the other end: that accounts for Goethe's inability to
understand
the essence of mathematics, — while it equally accounts for Leonardo's
one-sided preference for mathematical interpretations.
The analysis of
motions
leads to true exact Science. The principle of Science is the lordship
of
the human intellect, which imposes its law upon Nature empirically
perceived.
The organs of Science are Mathematics for that which is seen, the Logic
of cause and effect for synthesis outside that which is seen. 70 All
matter
of perception that cannot be assigned to one or other of these Schemes
is put aside and ignored. We may therefore describe Science as
systematic
Anthropomorphism. From this there result two deductions. Inasmuch as
the
Anthropos himself is a portion of nature, it is manifestly probable
that
he will be able to assimilate an important part of the phenomena of
Nature
according to the scheme which is specially his own; to this the history
of science bears witness. What he has assimilated according to this
method
is unconditional Knowledge: it is available at all times and to all
men.
This knowledge is not the same as reality: it is only a Scheme; it
hardly
touches the essence of things; yet it suffices for theory and for
practice.
That we should call the first deduction. And now for the second. On
such
a foundation it is possible to construct a flawless consistent building
as a system of Nature, without the human intellect ever falling into
contradiction
with itself, without therefore any injury to the principle of thought,
— correct logical sequence — and it is in spite of that possible that
from
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LEONARDO
beginning
to end, at
every single stage,
it may only have grasped a fragment of the truth. For the revelation of
this fallacy which gives the lie to the whole essence of the world —
(we
might call it the error ex incomperto) — science affords no
handle.
A Goethe is therefore fully justified in demanding that Nature should
be
considered as not only mechanically logical, but also according to
another
method; he is only wrong when he calls this other method of observation
“exact Science.“ 71
The essence of the
other form of seeing is more difficult to define, just because it is a
purer method of seeing. Here the characteristic feature is devotion to
Nature, the struggle to escape from the serfdom of anthropomorphism.
The
Principle is Love, the Aim “the contemplation of Nature's own
thoughts,“
the Organ the senses in partnership with Phantasy. That which is
thought
is here incapable of being known; as the Indian sage Bartrihari says,
“there are no words for this thought.“ Think only of the doctrine of
Metamorphosis:
the doctrine of colour runs on all fours with it. Yet we must always
keep
before our eyes what is the meaning of our so-called knowledge, and
within
what narrow bounds it is fixed; rightly viewed, as we have seen, the
knowledge
of our Science is rather a method of investigating and mastering Nature
than true knowledge. By science, our powers, the physical conditions of
our existence, our arsenal of the material of knowledge, are enriched.
But it is only its subjective acceptation, and the elaboration into
something
personal, that enriches our intellect. And this is the way that Goethe
adopts. Of his doctrine of colour he himself confesses that it cannot
be
taught, “it must be held as practical, not as theory.“
Let me here make
my own profession of faith. You know how greatly devoted I am to exact
science; that is above any other the sphere in which, had the fates
been
kind, I might have been in a position to render
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LEONARDO
some
service. At the
same time I am
firmly
convinced, that the greater the prosperity of the development of exact
science, the more indispensable must become a purely contemplative
conception
of Nature, — and that moreover in the interest of human culture. And if
we wish to define this method of contemplation from the standpoint of
Leonardo,
that is to say from the standpoint of a strictly and logically
synthetic
understanding, then we must say that it is a contemplation of Nature
that
is devoid of causality. Pure contemplation tells us nothing of the
past,
nothing of the future, nothing of cause and effect. “The tracing back
of
effect to cause is a mere historical proceeding,“ says Goethe. His
doctrine
of Metamorphosis is not the discovery of something that has taken
place,
but the setting forth of an Idea, — Idea of Nature, Idea of Man, which
meet at this point: and in his studies of colour he is so far from
wishing
to supplant an old theory by a new one, that he blames the inclination
of men to “set aside phenomena“ by an explanation, instead “of making
themselves
acquainted with Detail by intimate sympathy, and so building up a
Whole.“
72 But our whole
educational training makes us rather “historical“
beings
than creatures of the present, and those two conceptions, Moment and
Sight,
are nearly related. The man who in the contemplation of Nature busies
himself
to trace back a so-called Effect to a so-called Cause, follows the path
of the mathematician: for just as the one prefers the forms of pure
perception,
so the other prefers the forms of empirical perception to the prejudice
of experience afforded by the senses: he will see keenly, but not
clearly,
for his vision is troubled by thought, that is to say, by systematic
synthesis
according to human Laws. Experience is, according to Kant, “a product
of
the understanding from materials furnished by the senses“; that it is
in
every case; but it makes a great difference whether the preference be
given
to the
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LEONARDO
element
of the
understanding or to that
of the senses; exact science does the one, Goethe the other. For Goethe
the question is one of reproductive, or let us say boldly, of artistic
vision, — (which can only be the case when it is allied to Genius) — in
contradistinction
to an abstract observation of Nature which exclusively busies itself
with
dissection and peering into causes. If any one should teach us to
suppress,
not for ever, but at pleasure, this involuntarily schematising activity
of the intellect in the interest of vision, and as Goethe said, of
“construction,“
he would be endowing us with new eyes. Just as the mathematical method
has been enriched with new instruments, so would this method of pure
observation
enrich us with new thoughts and images. Richer than ever before would
the
source of phantasy flow, because science has in the meanwhile been
extending
the field of the Visible. Unless we follow Goethe's example our
civilisation
will evaporate into mere equations; Goethe has shown the road to
culture.
But it is not only
Goethe who has shown it; Kant has done the same. If only you should
obtain
a living insight into the fact that Kant, — whose eye so plainly
differed
from Leonardo's and Goethe's, — yet shares the diametrically opposed
views
of both men in relation to the observation of Nature, and so leads to a
harmonious adjustment between them, then the patience which you have
shown
to-day would be richly rewarded. You know how far he agrees with
Leonardo:
there is no difficulty about that: to perceive the agreement with
Goethe
requires a finer analysis. Yet it is such a distinct feature in Kant's
being, that a few words will suffice to direct your attention to it. A
later lecture will go further into this.
Kant knew exactly
what constitutes the principles of exact science. He writes, “we can
speak
of Light-matter, Heat-matter, etc., because they are mere fictions of
forces
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LEONARDO
which
contain no more
than a relative
conception“ (Ur, III, 598). And the same man who said, “there can be
no
true recognition of Nature without presupposing mechanism as the
foundation
of natural investigation“ — you know exactly what he meant by that, and
how he meant it, — the same man writes in the same place, “this is in
no
way opposed to the maxim to seek and reflect upon a principle in
certain
forms of Nature, which is a quite different matter from explanation in
accordance with the mechanism of Nature“ (Ur, § 70). It is true
that
here Kant has in his head only a single other principle, that of Final
Causes; still in this work upon the power of judgment other principles
come into play which only indirectly affect the Final Causes, as for
instance
in the great discussions over the ideas of genus and species,
metamorphosis
and persistence of form, etc. The Final Cause is in general, as we see
in many passages, considered by Kant as of identical meaning with
architectonics.
We are dealing then with “Nature as a presentation of Ideas,“ as the
same
work says, “an effort,“ as Kant had already said in the essay on Pure
Reason,
“which deserves respect and following up.“ I would bespeak your whole
attention
for the following sentence. Kant says, “Taken literally and considered
logically, it is impossible to represent ideas. But if we widen our
power
of conception ... for the contemplation of Nature, Reason inevitably
comes
in, and brings forward the effort of the mind, vain though it may be,
to
make the representation of the senses adequate to these ideas“ (Ur,
note
to § 29). Here is a sentence which might have been coined from
Goethe's
mode of contemplating nature. To widen the empirical power of
representation
must be its keynote. It is just this very effort to widen empirical
contemplation,
— the share of the senses, in contradistinction to the preference of a
one-sided and so to speak abstract contemplation — which fundamentally
distinguishes
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LEONARDO
Goethe's
method from
that of science.
And notice how our dear dry Kant speaks: “taken literally and looked at
logically“ ideas cannot be expressed, and the attempt remains “in
vain“;
yet no one hinders us from casting away the livery of the Literal, and
the serf's chain of a Logic ultimately leading to absurdities, as Kant
himself has shown in his famous exposition of the Widerstreit der
Vernunft
(the Antagonism of Reason). The idea “species“ taken literally cannot
be
shown to the senses, yet in spite of that it forms the foundation of
all
science of beasts and plants: the idea metamorphosis cannot hold out
against
logical investigation, yet it is an idea which lies at the root of all
comparative anatomy. Science only widens extensively, while on the
contrary
a fashion of seeing after the manner of Goethe widens our “power of
representing
the perception of Nature“ intensively.
It would of course
be an absurdity to expect to find ideas working in a man like Kant, who
kept his eyes closed, in the same sense as they did in Goethe.
Descriptions
alone gave him any lively conceptions; in no other way could he see
anything.
A direct contemplation of Nature, of the widening of which he speaks
here,
so far as Nature in the concrete is concerned, could not exist for him.
Face to face with nature he could neither discover with Leonardo nor
invent
with Goethe. His ideas — ideas of genius — in regard to surrounding
Nature,
are accordingly purely schematic, purely mechanical. His theory of the
heavens is a good example of the way in which the Physicist searches
“for“
experience. But there is also an inner Empiricism, an inner nature. And
here, here where Kant is quite at home, he stands in exactly the same
position
as Goethe stands in relation to surrounding, concrete Nature. There is
indeed a form of idea which only in a case where the eye is directed
inwards
could develop itself to such a brilliant clearness,
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LEONARDO
that
it might at last,
like the Holy
Grail, blaze through the cavernous darkness, an idea whose counterpart,
— what Goethe called the God-Nature, — is the idea of Freedom. The man
who had followed the mechanism of Nature, not only in the construction
of the world, but extending into the innermost folds of the
intellectual
activity of man, recognised moral independence as the highest power of
human personality. That is an idea, an idea which may not be capable of
being expressed, but which may well be “lived,“ exactly as Goethe's
endeavour
strove towards “living“ the Nature around him. So long as I take as
principle
the mechanism of natural investigation, so long I must look upon myself
as a mere machine, or as Kant puts it “a Nature which the will
subjugates“;
I must be able to explain every most delicate impulse of my thought and
feeling just as mechanically as the essence of Light, even though I
should
be compelled to premise for the purpose many “peculiar forms of motion
of hypothetic media.“ Whoever denies that is working in the interest of
obscurantism, and shows that he has no share in the blessing of true
Teutonic
science; for him the whole development of our knowledge of Nature from
the fifteenth to the twentieth century has never existed. But does
mechanism
suffice for me? As a thinking and moral being am I not compelled to
feel
that if I go no further I am lying to myself? Does not the most
intimate
experience of every moment bear witness to freedom and responsibility?
Does it not prove the reality of “a Nature which is subjected to a
Will“?
Remember how Goethe undertook the investigation of Light, not by the
presumption
of a hypothetical being, but by the faithful exposition of his
dealings,
his activities, and passivities. In the same way Kant repudiates the
idea
that we should search for the essence and the importance of being a man
in the study of anatomy and in the comparison of the
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human
skeleton with
that of other genera
of animals — the rather “are they only to be found in his dealings
whereby
he reveals his character.“ 73
Here you have, as you see, almost
literally
Goethe's words about Light. And what does Kant, the mechanician and
analyst,
discover when he puts these dealings to the test? “Man's freedom and
independence
of the mechanism of all Nature.“ But this view determines him to hold
up
before us men an idea (here called ideal
because it leads to dealings
which
it is our business to make perfect), which is not taken up passively
from
so-called revelations, — which as a general proposition is not
pre-existent
and waiting for our coming any more than is the idea of metamorphosis,
“but which may become reality through that which we do and that which
we
leave undone.“ Instead of theoretically debating about Freedom it is
our
duty to prove it by deeds; by Freedom we must realise ideals in
defiance
of Nature.
Here you see Kant
himself enlarging upon that which in respect of man's relation to
surrounding
Nature he had only discussed theoretically, namely the representation
of
Nature in ideas, as opposed to an explanation of her as mechanism. The
connection with Goethe, which certainly does not lie upon the surface,
is here so intimate and deep, that I hardly think that it is possible
to
understand the one man apart from the other. I do not think that you
can
attain a fully concrete notion, void of phrasemongering, of what Kant
understands
by “Independence of mechanism,“ and “Freedom,“ unless you dip deeply
into
Goethe's study of nature; and, on the other hand, I am convinced that
our
conception of Goethe's method of considering Nature remains dull,
insufficient,
and false, until we have realised that with him it is a question of
something
quite as direct, quite as truly experienced, as Kant's idea of the
freedom
of the human Will. Goethe, like Kant, wishes to carry out a work of
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LEONARDO
salvation,
and in the
case of both men
the method is the same. Kant on one occasion writes, “Two things fill
the
soul with ever new and increasing admiration and awe, the oftener and
more
persistently my thought busies itself with them: the Star-studded
Heaven
above me, and the Moral Law within me. It is not permitted to me to
seek
and only guess at either of these as shrouded in a veil of darkness, or
in the Transcendental beyond the range of my vision; I see them before
me and connect them directly with the consciousness of my existence.“
These
words should remind you of Goethe's words, “Nature has neither Kernel
nor
Shell, she is everything at one and the same time.“ But for us men
there
certainly is a distinction between Kernel and Shell, and the
inalienable
tendency of our mechanical science, in other ways so admirable, is to
make
every thing into Shell, the star-studded heaven as well as the moral
law:
whereas Kant and Goethe are at one in this, — that they teach us how we
are to begin to show everything as Kernel, in which — each following
the
nature of his special gifts — the one by preference fixes his eye upon
the starry heaven, the other upon the moral law.
Goethe's saying,
“Instinctively I followed the same road as Kant,“ has grown more and
more
significant. It is my special hope that from Goethe's standpoint you
are
beginning clearly to see in its organic consistency the marvellous
personality
of Kant, so rich in what the superficial observer calls contradictions.
Now, when you read the Critique of
Pure Reason, and come across the
often
quoted and almost always misunderstood sentence, “I had to set aside
Knowledge
in order to make room for Faith,“ you will, I think, understand it, and
that too, exactly in the sense in which Kant understood it. There we
have
a touchstone; for no man ever had a more glowing respect for exact
knowledge
than
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LEONARDO
Kant,
and no man read
into the
conception
“Faith“ so little History and so much force of living actuality. To
separate
these two, Knowledge and Faith, and having done so once more directly
to
reunite them, — that was his special distinguishing gift.
For to-day my task
is accomplished, and at the same time I leave the sphere of these two
first
lectures. With René Descartes and Giordano Bruno we shall climb
new heights, and much upon which we have already touched will appear to
us in another light. Let me, however, in conclusion, say a few words
about
the great artist who has rendered us such conspicuous service. For it
is
Leonardo whom we have to thank for the incitement, indeed for the
peculiar
driving power, towards those considerations which have opened up to us
so clear and deep an insight into Kant's inmost heart. Unfortunately we
cannot in these lectures busy ourselves more minutely with that
wonderful
man: still a most cursory glance will now suffice us for the discovery
of more than one feature of his relationship to Goethe and Kant, and
this
last recapitulation will at the same time spread a bright, transparent,
and protecting varnish over the colours of the picture that we have
obtained.
Leonardo, who so
exactly agreed with Kant in his estimate of mathematical and mechanical
investigation of Nature, has left behind him proofs that he too knew
how
to distinguish between a Nature which subjugates Will, and a Nature
which
is obliged to subordinate itself to Will. He writes, La
necessità
è maestra e tutrice della Natura, “necessity is mistress
and
guardian
of Nature,“ — that is Nature as Mechanism; and yet in another place he
writes, il dono principal di Natura
è libertà, “the chief
gift of Nature is Liberty,“ that is Nature as Idea; and the
confirmation
of this Idea in man he sees with Kant in the signoria di se medesimo,
“the
Lordship over himself,“ that is to say, what the German sage
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LEONARDO
called
“Freedom from
the mechanism of
all Nature.“ This agreement is of itself interesting; it points to the
fact that whoever understands mechanism as the principle of the
explanation
of Nature, and, without leaving the slightest loophole through which
any
extravagance of idea might slip in, maintains it consistently, must
inevitably
in this way reach a healthy idealism: the conceptions Necessity and
Freedom
do not exclude one another, on the contrary they mutually depend upon
one
another. 74 In the
clearness of this recognition Leonardo is nearer to
Kant than Goethe; the latter was not sufficiently a mechanist to be a
pure
idealist.
But we reach greater
depths in this parallel if we take into consideration the relationship
of Leonardo the working artist to Leonardo the man. Leonardo the
theorist
is a keen but inordinately strict, dry intellect. Often, in the none
too
easy reading of his works, I have been compelled to think of Kant.
There
is the same hatred of exaggeration, the same distrust of all that might
believe itself to be intuition and inspiration. To his disciples he
never
speaks of anything else but measurements and calculations and technical
practices, and he never tires of impressing upon them the “copying of
Nature
as in a mirror.“ And now let us turn from his books to his works. Is
what
you find a copy, line by line, of a mechanically seen form? Is it not
far
rather a revelation of all that is invisible, indescribable,
unthinkable,
“seeking in an instant of vision to concentrate a thousand
experiences“?
75 (Walter Pater).
Never, with the possible exception of Rembrandt, has
personality been so brought home to us: the most intimate secret of the
soul rests, only half veiled, upon the peaceful features; his female
heads
are the living representation of what Goethe calls das ewig weibliche
“the
eternal feminine“; his figure of Christ has the significance of a fifth
Gospel.
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LEONARDO
The
same magic rests
upon some of the
pen-and-ink landscapes from his hand:
als ob
da drinnen ganze Weltenraüme
wären
Wald und Wiesen,
Bäche, Seen
unerforschte Tiefen
“as if in them lay whole world-spaces,
Forest and Meadows, Brooks, Lakes, ... unplumbed Depths.“
And he, the empiric
and mechanician, knew full well what secrets of the human spirit are
here
disclosed. Of works of genius, he says: questa non s'insegna come fan
le
matematiche, ... non si copia come si fa le lettere ... questa sola si
resta
nobile, questa sola onora il suo autore e resta pretiosa e unica, e non
partorisce mai figluoli eguali à se. “This cannot be
taught like
mathematics or copied like letters.... This alone remains noble, this
alone
does honour to its author and remains precious and unique, and never
gives
birth to children that shall be the equals of Itself.“ It is useless to
degrade the phenomenon of the human intellect to a matter of small
importance,
or to deny its existence, as we daily see attempted: it is a beggarly
life
which we prepare for ourselves in that way. True, the merciless realist
calls man Rè delle bestie, “the King of Beasts,“ and he
laughs
at the monks and “other liars,“ who talk of the miracles of “the soul,“
where he, as anatomist and mechanician, has only found nerves leading
to
a brain; but now he steps up to the canvas and produces upon it an
Idea — the
Saviour, the rè degli uomini, the King of men, as his
mind's
eye has beheld him, a work which cannot be learnt and cannot be copied,
and which can never give birth to a son like to itself. And we — we
draw
near full of awe, blest with happiness, enriched for all time. We do
not
doubt the secret connection between the great man of art, the man who
investigated
the geometry of space and perspective, who measured with compass and
rule
the position of every leaf on the tree,
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LEONARDO
of
every muscle in the
face, — and the
man who gave form to immortal ideas; but here again the relationship is
the same as that of Kant's example of the starry heaven above me, the
moral
law within me: in combination they complete my being; but as cause and
effect they stand in no relationship to one another.
There are just two
worlds. Two
worlds which stand in opposition and in contraposition to one another,
which at the same time imply and exclude one another. You may perhaps
remember
an important conclusion at which we arrived at the very outset of the
present
lecture: namely that these two worlds are sharply divided from one
another:
they lie separated as it were by a broad stream: on the one side a
bridge,
on the other a ferry, leads from bank to bank: there are no other means
of crossing. The World of the Senses can in no way directly obtain
access
to Thought and Reason otherwise than by means of Schemes of the
understanding,
and on the other hand the world of Ideas can only attain visibility on
condition of borrowing Symbols from the World of the Senses. That is a
fundamental fact of the human intellect, a fact long suspected, and
laid
down by Kant for all time. Every attempt to deny this twofold
existence,
which is of the very essence of our nature, and therefore of universal
nature, is offering up the one half of our being as a sacrifice to the
other. The Mystics, among whom we must reckon intellects like
Schopenhauer,
either dispute the mechanical law, or violate it at many points, and
thus
play havoc with our understanding: the scientific Monist acts even more
despotically: for while the one only repudiates the abstract
mathematical,
the other rejects the concrete evidence of the senses. The course of
exact
science refutes the first, every genius convicts the second of lying.
And
what I wish to impress upon your minds is that Goethe's creative ideas
about organic Nature and the essence of Colour, and Kant's creative
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ideas
about the construction of the
human
intellect, and about the connection between the two natures, should be
accurately examined and judged precisely in the same way, as we examine
and judge an original artistic production due to the pure creative
power
of Leonardo. We have sufficient evidence that men of true genius occupy
themselves with exact Science: yet it is foolish to expect creative
achievements
from science as such. Whatever inspirations science brings to light are
not original but practical discoveries of Nature; none the less does
science
deserve praise on that account: but it is a higher matter when Nature
in
man crosses over to the work of discovery, and in a paroxysm of the
intellect
gives birth to a new thing. But to measure this with a yard-measure is
a ridiculous undertaking. It is only the free, open eye which will
convince
us here, not the Logic which is like the crutch of the blind man
groping
his way: the two methods, combined yet ever estranged, stand over
against
one another like the oscillations and colour. Granted that Goethe and
Kant
were both technicians, yet here too there is a mysterious connection
between
passivity and activity, between experience and idea, between empiricism
and creative power, just as in what concerns the unconscious method of
life, there is a connection between Scheme and Symbol, Technics and
Phantasy.
No man is further removed from the bungler, no man is more industrious,
than the Genius. Fifteen volumes has the Weimar edition already devoted
to works of Goethe on investigations of nature; turn over those leaves
if you are minded to learn how tirelessly, with what painful accuracy,
how soberly, the great man studied nature. Kant on his side has
surrounded his solitude with such gruesome moats and fortifications of
philosophical
technicalities that much courage and persistence are needed to
penetrate
into the interior. And just as I could not avoid criticising Goethe the
194
LEONARDO
technician,
so it will seem difficult
to many people to subscribe altogether to the Kantian technics. But
exactly
as Leonardo, the mighty artist, placed ideas in living form before our
eyes, so also did Goethe, the pure Seer of Nature, the founder
according
to Johannes Müller of a new ideal of Natural History, and so also
did Kant, the august enlightener of the soul of man. Their works are
creative
achievements of Genius. These guide us to the inmost mysteries of
nature
very differently from the ways of mathematical science. For as Goethe
teaches
us, “the way of Nature is the way upon which you will of necessity meet
Roger Bacon, Homer, and Shakespeare.“ This way, gentlemen, is the one
which
we seek to tread in these Lectures, the way upon which we meet Nature
herself
as she reveals herself in her noblest creatures. If I have succeeded,
in
ever so modest a manner, in leading you upon this road to which Goethe
points, then there should arise before you the vision of three giants,
Leonardo, Goethe, Kant, each standing out as a perfectly distinct
personality:
thrice the glorious eye of genius meets your own: a threefold stream of
Light floods over your world.
End
of page.
Last update July 25th, 2004.
