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POPULAR LECTURES

ON

SCIENTIFIC SUBJECTS

HERMANN VON HELMHOLTZ

TRANSLATED BY

E. ATKINSON, PH.D., F.C.S.

FORMERLY FKOKESSOB OF EXPERIMENTAL SCIENCE, 8TAJ1? COLLEGE

SECOND SERIES

WITH AN AUTOBIOGRAPHY OF THE AUTHOR

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BIBLIOGRAPHICAL NOTE.

Issued in Silver Library, March 1893; Reprinted
July 1895, May 1898, June 1900, December 1903, August
1908.

PEEFAOB.

THE FAVOUR with which the first series of Professor
Helmholtz's Lectures has been received would justify,
if a justifi cation were needed, the publication of the
present volume.

I have to express my acknowledgments to Pro-
fessor G. Croome Eobertson, the editor, and to Messrs.
Macmillan, the publishers of ' Mind,' for permission to
use a translation of the paper on the * Axioms of
Modern Geometry ' which appeared in that journal.

The article on ' Academic Freedom in German
Universities ' contains some statements respecting the
Universities of Oxford and Cambridge to which ex-
ception has been taken. These statements were a fair
representation of the impression produced on the mind
of a foreigner by a state of things which no longer
exists in those Universities, at least to the same
extent. The reform in the University system, which

47^42

vi PREFACE.

may be said to date from the year 1854, has brought
about so many alterations both in the form and in the
spirit of the regulations, that older members of the
University have been known to speak of the place
as so changed that they could scarcely recognise it.
Hence, in respect of this article, I have availed myself
of the liberty granted by Professor Helmholtz, and
have altogether omitted some passages, and have-
slightly modified others, which would convey an erro-
neous impression of the present state of things. I
have also on these points consulted members of the
University on whose judgment I think I can rely.

In other articles, where the matter is of prime
importance, I have been anxious faithfully to repro-
duce the original ; nor have I in any such cases al-
lowed a regard for form to interfere with the plain
duty of exactly rendering the author's meaning.

E. ATKINSON.

PeRTE«BERY HlLL, CA.MBEELEY :

Dec. 1830.

CONTENTS.

tBCTTTBE PAQS

I. GUSTAV MAGNVS. IN MEMORIAM . , . i

II. ON THE ORIGIN AND SIGNIFICANCES OF GEOMETRICAL

AXIOMS 27

IIT. ON THE RELATION OF OPTICS TO PAINTING . . 73

i. Form „ 7?,

H. Shade 94

in. Colour . . . . . . . . .110

IV. Harmony of Colour 124

IV. ON THE ORIGIN OF THE PLANETARY SYSTEM . .139

V. ON THOUGHT IN MEDICINE 199

VL ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES . 237

VII. HERMANN VON HELMHOLTZ. Au AUTOBIOGRAPHICAL

SKBICH . , 206

GUSTAV MAGNUS.

Address delivered in the Leibnitz Meeting of the
Academy of Sciences on July 6, 1871. .

THE honourable duty has fallen on me of expressing in
the name of this Academy what it has lost in Gustav
Magnus, who belonged to it for thirty years. As a
grateful pupil, as a friend, and finally as his successor,
it was a pleasure to me as well as a duty to fulfil such
a task. But I find the best part of my work already
done by our colleague Hofmann at the request of the
German Chemical Society, of which he is the Pre-
sident. He has solved the difficulty of giving a pic-
ture of the life and work of Magnus in the most com-
plete and most charming manner. He has not only
anticipated me, but he stood in much closer and more
intimate personal relation to Magnus than I did ; and,
on the other hand, he is much better qualified than £

II. S

GUSTAV MAGNUS.

to pronounce a competent judgment on the principal
side of Magnus's activity, namely, the chemical.

Hence what remains for me to do is greatly re-
stricted. I shall scarcely venture to speak as the
biographer of Magnus, but only of what he was to us
and to science, to represent which is our allotted task.

His life was not indeed rich in external events
and changes ; it was the peaceful life of a man who,
free from the cares of outer circumstances, first as
member, then as leader of an esteemed, gifted, and
amiable family, sought and found abundant satisfaction
in scientific work, in the utilisation of scientific results
for the instruction and advantage of mankind. Hein-
rich Gustav Magnus was born in Berlin on May 2,
1802, the fourth of six brothers, who by their talents
have distinguished themselves in various directions.
The father, Johann Matthias, was chief of a wealthy
commercial house, whose first concern was to secure
to his children a free development of their individual
capacity and inclinations. Our departed friend showed
very early a greater inclination for the study of mathe-
matics and natural philosophy than for that of lan-
guages. His father arranged his instruction accor-
dingly, by removing him from the Werder Gymnasium
and sending him to the Cauer Private Institute, in
which more attention was paid to scientific subjects.

From 1822 to 1827 Magnus devoted himself en-

GUSTAV MAGNUS. 3

tirely to the study of natural science at the University
of Berlin. Before carrying out his original intention
of qualifying as a professor of technology, he spent two
years with that object in travelling ; he remained with
Berzelius a long time in Stockholm, then with Du-
long, Thenard and Gay-Lussac in Paris. Unusually
well prepared by these means, he qualified in the
University of this place in technology, and afterwards
also in physics ; he was appointed extraordinary pro-
fessor in 1834, and ordinary professor in 1845, and so
distinguished himself by his scientific labours at this
time, that nine years after his habilitation, on January
27, 1840, he was elected a member of this Academy.
From 1832 until 1840 he taught physics in the
Artillery and Engineering School ; and from 1850 until
1856 chemical technology in the Gewerbe Institut.
For a long time he gave the lectures in his own house,
using his own instruments, which gradually developed
into the most splendid physical collection in existence
at that time, and which the State afterwards purchased
for the University. His lectures were afterwards given
in the University, and he only retained the laboratory
in his own house for his own private work and for that
of his pupils.

His life was passed thus in quiet but unremitting
activity; travels, sometimes for scientific or technical
studies, sometimes also in the service of the State, and

» 2

4 GUSTAV MAGNUS.

occasionally for recreation, interrupted his work here
from time to time. His experience and knowledge of
business were often in demand by the State on various
commissions; among these may be especially men-
tioned the part he took in the chemical deliberations
of the Agricultural Board (Landes-Economie Colle-
gium), to which he devoted much of his time ; above
all to the great practical questions of agricultural
chemistry.

After sixty-seven years of almost undisturbed
health he was overtaken by a painful illness towards
the end of the year 1869.1 He still continued his
lectures on physics until February 25, 1870, but dur-
ing March he was scarcely able to leave his bed, and he
died on April 4.

Magnus's was a richly endowed nature, which under
happy external circumstances could develop in its own
peculiar manner, and was free to choose its activity
according to its own mind. But this mind was so
governed by reason, and so filled, I might almost say,
with artistic harmony, which shunned the*immoderate
and impure, that he knew how to choose the object of
his work wisely, and on this account almost always to
attain it. Thus the direction and manner of Magnus's
activity accorded so perfectly with his intellectual
nature as is the case only with the happy few among

1 Carcinoma recti.

GUSTAV MAGNUS. 5

mortals. The harmonious tendency and cultivation of
his mind could be recognised in the natural grace of
his behaviour, in the cheerfulness and firmness of his
disposition, in the warm amiability of his intercourse
with others. There was in all this, much more than
the mere acquisition of the outer forms of politeness
can ever reach, where they are not illuminated by a
warm sympathy and by a fine feeling for the beautiful.
Accustomed from an early age to the regulated and
prudent activity of the commercial house in which he
grew up, he retained that skill in business which he
had so frequently to exercise in the administration of the
affairs of this Academy, of the philosophical faculty, and
of the various Government commissions. He retained
from thence the love of order, the tendency towards
the actual, and towards what is practically attainable,
even although the chief aim of his activity was an ideal
one. He understood that the pleasant enjoyment of
an existence free from care, and intercourse with the
most amiable circle of relatives and friends, do not bring
a lasting satisfaction ; but work only, and unselfish work
for an ideal aim. Thus he laboured, not for the in-
crease of riches, but for science ; not as a dilettante and
capriciously, but according to a fixed aim and in
defatigably ; not in vanity, catching at striking dis-
coveries, which might at once have made his name
celebrated. He was, on the contrary, a master of faith-

6 GUSTAV MAGNTS.

fui, patient, and modest work, who tests that work
again and again, and never ceases until he knows there
is nothing left to be improved. But it is also such
work, which by the classical perfection of its methods,
by the accuracy and certainty of its results, merits and
gains the best and most enduring fame. There are
among the labours of Magnus masterpieces of finished
perfection, especially those on the expansion of gases
by heat, and on the tension of vapours. Another
master in this field, and one of the most experienced
and distinguished, namely, Eegnault of Paris, worked
at these subjects at the same time with Magnus, but
without knowing of his researches. The results of
both investigators were made public almost simul-
taneously, and showed by their extraordinarily close
agreement with what fidelity and with what skill both
had laboured. But where differences showed themselves,
they were eventually decided in favour of Magnus.

The unselfishness with which Magnus held to the
ideal aim of his efforts is shown in quite a character-
istic manner, in the way in which he attracted younger
men to scientific work, and as soon as he believed he
had discovered in them zeal and talent for such work
by placing at their disposal his apparatus, and the appli-
ances of his private laboratory. This was the way in
which I was brought in close relation to him, when I
found myself in Berlin for the purpose of passing the
Government medical examination.

GUSTAV MAGNUS. 7

He invited me at that time (I myself would not
have ventured to propose it) to extend my experiments
on fermentation and putrefaction in new directions,
and to apply other methods, which required greater
means than a young army surgeon living on his pay
could provide himself with. At that time I worked
with him almost daily for about three months, and thus
gained a deep and lasting impression of his goodness,
his unselfishness, and his perfect freedom from scientific
jealousy.

By such a course he not only surrendered the ex-
ternal advantages which the possession of one of the
richest collections of instruments would have secured
an ambitious man against competitors, but he also bore
with perfect composure the little troubles and vexations
involved in the want of skill and the hastiness with
which 3roung experimentalists are apt to handle costly
instruments. Still less could it be said that, after the
manner of the learned in other countries, he utilised
the work of the pupils for his own objects, and for the
glorification of his own name. At that time chemical
laboratories were being established according to Liebig's
precedent : of physical laboratories — which, it may be
observed, are much more difficult to organise — not one
existed at that time to my knowledge. In fact, their
institution is due to Magnus.

In such circumstances we see an essential part of
thft inner tendency of the man, which must not by

8 GJSTAV MAGNUS.

neglected in estimating his value : he was not only an
investigator, he was also a teacher of science in the
highest and widest sense of the word. He did not wish
science to be confined to the study and lecture-room,
he desired that it should find its way into all conditions
of life. 'In his active interest for technology, in his
zealous participation in the work of the Agricultural
Board, this phase of his efforts was plainly reflected, as
well as in the great trouble he took in the preparation
of experiments, and in the ingenious contrivance of the
apparatus required for them.

His collection of instruments, which subsequently
passed into the possession of the University, and is
at present used by me as his successor, is the most
eloquent testimony of this Everything is in the most
perfect order : if a silk-thread, a glass tube, or a cork,
are required for an experiment, one may safely depend
on finding them near the instrument. All the appa-
ratus which he contrived is made with the best means
at his disposal, without sparing either material, or the
labour of the workman, so as to ensure the success of
the experiment, and by making it on a sufficiently large
scale to render it visible as far off as possible. I recol-
lect very well with what wonder and admiration we
students saw him experiment, not merely because
all the experiments were successful and brilliant, but
because they scarce' y seemed to occupy or to disturb his

GUSTAV MAGNUS. 9

thoughts. The easy and clear flow of his discourse
went on without interruption; each experiment came
in its right place, was performed quickly, without haste
or hesitation, and was then put aside.

I have already mentioned that the valuable collec-
tion of apparatus came into the possession of the
University during his lifetime. He specially wished
that what he had collected and constructed as appli-
ances in his scientific work should not be scattered and
estranged from the original purpose to which he had
devoted his life. With this feeling he bequeathed to
the University the rest of the apparatus of his labora-
tory, as well as his very rich and valuable library, and
he thus laid the foundation for the further development
of a Public Physical Institute.

It is sufficient in these few touches to have recalled
the mental individuality of our departed friend, so far
as the sources of the direction of his activity are to be
found.

Personal recollections will furnish a livelier image
to all those of you who have worked with him for the
last thirty years.

If we now proceed to discuss the results of his
researches it will not be sufficient to read through
and to estimate his academical writings. I have
already shown that a prominent part of his activity was
directed to his fellow-creatures. To this must be added.

10 GUSTAV MAGNUS.

that he- lived in an age when natural science passed
through a process of development, with a rapidity such
as never occurred before in the history of science.
But the men who belonged to such a time, and co-
operated in this development are apt to appear in
wrong perspective to their successors, since the best
part of their work seems to the latter self-evident, and
scarcely worthy of mention.

It is difficult for us to realise the condition of natural
science as it existed in Germany, at least in the first
twenty years of this century. Maguus was barn in
1 802 ; I myself nineteen years later ; but when I go
back to my earliest recollections, when I began to study
physics out of the school-books in my father's posses-
sion, who was himself taught in the Cauer Institute, I
still see before me the dark image of a series of
ideas which seems now like the alchemy of the middle
ages. Of Lavoisier's and of Humphry Davy's revo-
lutionising discoveries, not much had got into the
school-books. Although oxygen was already known,
yet phlogiston, the fire element, played also its part.
Chlorine was still oxygenated hydrochloric acid; potash
and lime were still elements. Invertebrate animals
were divided into insects and reptiles ; and in botany
we still counted stamens.

It is strange to see how late and with what hesita-
tion Germans applied themselves to the study of natural

GUSTAV MAGNUS. 11

ricience in this century, whilst they had taken so promi-
nent a part in its earlier development. I need only
name Copernicus, Kepler, Leibnitz, and Stahl.

For we may indeed boast of our eager, fearless
and unselfish love of truth, which flinches before no
authority, and is stopped by no pretence ; shuns no
sacrifice and no labour, and is very modest in its claims
on worldly success. But even on this account she ever
impels us first of all to pursue the questions of prin-
ciple to their ultimate sources, and to trouble ourselves
but little about what has no connection with funda-
mental principles, and especially about practical con-
sequences and about useful applications. To this must
be added another reason, namely, that the independent
mental development of the last three hundred years,
began under political conditions which caused the
chief weight to fall on theological studies. Germany
has liberated Europe from the tyranny of the ancient
church ; but she has also paid a much dearer price for
this freedom than other nations. After the religious
wars, she remained devastated, impoverished, politi-
cally shattered, her boundaries spoiled, and arrogantly
handed over defenceless to her neighbours. To deduce
consequences from the new moral views, to prove them
scientifically, to work them out in all regions of intellec-
tual life, for all this there was no time during the storm
of war ; each man had to hold to his own party, every

12 GUSTAV MAGNUS.

incipient change of opinion was looked upon as treach-
ery, and excited bitter wrath. Owing to the Eeformation,
intellectual life had lost its old stability and cohesion ;
everything appeared in a new light, and new questions
arose. The German mind could not be quieted with
outward uniformity; when it was not convinced and
satisfied, it did not allow its doubt to remain silent.
Thus it was theology, and next to it classical philo-
logy and philosophy, which, partly as scientific aids of
theology, partly for what they could do for the solution
of the new moral, sesthetical, and metaphysical prob-
lems, laid claim almost exclusively to the interest of
scientific culture. Hence it is clear why the Protes-
tant nations, as well as that part of the Catholics
which, wavering in its old faith, only remained out-
wardly in connection with its church, threw itself with
ST tch zeal on philosophy. Ethical and metaphysical prob-
lems were chiefly to be solved ; the sources of knowledge
had to be critically examined, and this was done with
deeper earnestness than formerly. I need not enume-
rate the actual results which the last century gained
by this work. It excited soaring hopes, and it cannot
be denied that metaphysics has a dangerous attraction
for the German mind ; it could not again abandon it
until all its hiding-places had been searched through
and it had satisfied itself that for the present nothing
more is to be found there.

GUSTAV MAGNUS. 13

Then, in the second half of the last century, the
rejuvenescent intellectual life of the nation began to
cultivate its artistic flowers ; the clumsy language trans-
formed itself into one of the most expressive instru-
ments of the human mind ; out of what was still the
hard, poor, and wearisome condition of civil and political
life, the results of the religious war, in which the figure
of the Prussian hero-king only now cast the first hope
of a better future, to be again followed by the misery
of the Napoleonic war, — out of this joyless existence,
all sensitive minds gladly fled into the flowery land
opened out by German poetry, rivalling as it did the
best poetry of all times and of all peoples ; or in the
sublime aspects of philosophy they endeavoured to
sink reality in oblivion.

And the natural sciences were on the side of this
real world, so willingly overlooked. Astronomy alone
could at that time offer great and sublime prospects ;
in all other branches long and patient labour was still
necessary before great principles could be attained ;
before these subjects could have a voice in the great
questions of human life; or before they became the
powerful means of the authority of man over the
forces of nature which they have since become. The
labour of the natural philosopher seems narrow, low,
and insignificant compared with the great conceptions
of the philosopher and of the poet ; it was only those

14 GUSTAV MAGNUS.

natural philosophers who, like Ok en, rejoiced in
poetical philosophical conceptions, who found willing
auditors.

Far be it from me as a one-sided advocate of scien-
tific interests to blame this period of enthusiastic ex-
citement ; we have, in fact, to thank it for the moral
force which broke the Napoleonic yoke; we have to
thank it for the grand poetry which is the noblest
treasure of our nation ; but the real world retains its
right against every semblance, even against the most
beautiful ; and individuals, as well as nations, who wish
to rise to the ripeness of manhood must learn to look
reality in the face, in order to bend it to the purpose of
the mind. To flee into an ideal world is a false re-
source of transient success ; it only facilitates the play
of the adversary ; and when knowledge only reflects
itself, it becomes unsubstantial and empty, or resolves
itself into illusions and phrases.

Against the errors of a mental tendency, which cor-
responded at first to the natural soaring of a fresh youth-
ful ambition, but which afterwards, in the age of the
Epigones of the Romantic school and of the philosophy
of Identity, fell into sentimental straining after sub-
limity and inspiration, a reaction took place, and was
carried out not merely in the regions of science, but
also in history, in art, and in philology. In the last
departments, too, where we deal directly with products

GUSTAV MAGNUS. 15

of activity of the human mind, and where, therefore,
a construction a priori from the psychological laws
seems much more possible than in nature, it has come
to be understood that we must first know the facts, be-
fore we can establish their laws.

Gustav Magnus's development happened during the
period of this struggle ; it lay in the whole tendency
of his mind, that he whose gentle spirit usually en-
deavoured to reconcile antagonisms, took a decided part
in favour of pure experience as against speculation.
If he forbore to wound people, it must be confessed
that he did not relax one iota of the principle which,
with sure instinct, he had recognised as the true one ;
and in the most influential quarters he fought in a
twofold sense ; on the one hand, because in physics it
was a question as to the foundations of the whole of
natural sciences ; and on the other hand, because the
University of Berlin, with its numerous students, had
long been the stronghold of speculation. He con-
tinually preached to his pupils that no reasoning,
however plausible it might seem, avails against actual
fact, and that observation and experiment must de-
cide ; and he was always anxious that every practicable
experiment should be made which could give practical
confirmation or refutation of an assumed law. He did
not limit in any way the applicability of scientific
methods in the investigation of inanimate nature, but

16 GUSTAV MAONUS.

in his research on the gases of the blood (1837) he
dealt a blow at the heart of vitalistic theories. He led
physics to the centre of organic change, by laying a
scientific foundation for a correct theory of respira-
tion; a foundation upon which a great number of
more recent investigators have built, and which has
developed into one of the most important chapters
of physiology.

He cannot be reproached with having had too little
confidence in carrying out his principle ; but I must
confess that I myself and many of my companions
formerly thought that Magnus carried his distrust of
speculation too far, especially in relation to mathe-
matical physics. He had probably never dipped very
deep in the latter subject, and that strengthened our
doubts. Yet when we look around us from the stand-
point which science has now attained, it must be con-
fessed that his distrust of the mathematical physics of
that date was not unfounded. At that time no separa-
tion had been distinctly made as to what was empirical
matter of fact, what mere verbal definition, and what
only hypothesis. The vague mixture of these ele-
ments which formed the basis of calculation was put
forth as axioms of metaphysical necessity, and pos-
tulated a similar kind of necessity for the results. I
need only recall to you the great part which hypo-
theses as to the atomic structure of bodies played

GUSTAV MAGNUS. J7

In mathematical physics during the first half of this
century, whilst as good as nothing was known of
atoms ; and, for instance, hardly anything was known
of the extraordinary influence which heat has on mole-
cular forces. We now know that the expansive force of
gases depends on motion due to heat ; at that period
most physicists regarded heat as imponderable matter.
In reference to atoms in molecular physics, Sir W.
Thomson says, with much weight, that their assump-
tion can explain no property of the body which has
not previously been attributed to the atoms. Whilst
assenting to this opinion, I would in no way express
myself against the existence of atoms, but only against
the endeavour to deduce the principles of theoretical
physics from purely hypothetical assumptions as to
the atomic structure of bodies. We now know that
many of these hypotheses, which found favour in their
day, far overshot the mark. Mathematical physics
has acquired an entirely different character under the
hands of Grauss, of F. E. Neumann and their pupils,
among the Germans; as well as from those mathe
maticians who in England followed Faraday's lead,
Stokes, W. Thomson, and Clerk-Maxwell. It is now
understood that mathematical physics is a purely ex-
perimental science ; that it has no other principles to
follow than those of experimental physics. In our
immediate experience we find bodies variously formed
n. 0

18 GUSTAV MAGNUS.

and constituted ; only with such can we make our
observations and experiments. Their actions are made
up of the actions which each of their parts contributes
to the sum of the whole ; and hence, if we wish to
know the simplest and most general law of the action
of masses and substances found in nature upon one
another, and if we wish to divest these laws of the
accidents of form, magnitude, and position of the
bodies concerned, we must go back to the laws oi
action of the smallest particles, or, as mathematicians
designate it, the elementary volume. But these are
not, like the atoms, disparate and heterogeneous, but
continuous and homogeneous.

The characteristic properties of the elementary
volumes of different bodies are to be found experi-
mentally, either directly, where the knowledge ol
the sum is sufficient to discover the constituents,
or hypothetically, where the calculated sum of effects
in the greatest possible number of different cases
must be compared with actual fact by observation
and by experiment. It is thus admitted that mathe*
matical physics only investigates the laws of action
of the elements of a body independently of the acci-
dents of form, in a purely empirical manner, and is there>
fore just as much under the control of experience as
what are called experimental physics. In principle
they are not at all different, and the former only con-

GUSTAV MAGKUS. 19

tinues the function of the latter, in order to arrive at
still simpler and still more general laws of phenomena.

It cannot be doubted that this analytical tendency
of physical inquiry has assumed another character;
that it has just cast off that which was the means of
placing Magnus towards it in some degree of antago-
nism. He tried to maintain, at least in former years,
that the business of the mathematical and that of
the experimental physicist are quite distinct from one
another; that a young man who wishes to pursue
physics would have to decide between the two. It
appears to me, on the contrary, that the conviction is
constantly gaining ground, that in the present more
advanced state of science those only can experi-
mentalise profitably who have a clear-sighted know-
ledge of theory, and know how to propound and pursue
the right questions ; and, on the other hand, only
those can theorise with advantage who have great
practice in experiments. The discovery of spectrum
analysis is the most brilliant example within our
recollection of such an interpenetration of theoretical
knowledge and experimental skill.

I am not aware whether Magnus subsequently ex-
pressed other views as to the relation of experimental
and mathematical physics. In any case, those who
regard his former desertion of mathematical physics
as a reaction against the misuse of speculation carried

c2

20 GUSTAV MAGNUS.

too far, must also admit that in the older mathema-
tical physics there are many reasons for this dislike,
and that, on the other hand, he received with the
greatest pleasure the results which Kirchhoff, Sir
W. Thomson, and others had developed out of new
facts from theoretical starting-points. I may here be
permitted to adduce my own experience. My re-
searches were mostly developed in a manner against
which Magnus tried to guard; yet I never found in
him any but the most willing and friendly recognition.
It is, however, natural that every one, relying upon
his own experience, should recommend to others, as
most beneficial, the way which best suits his own
nature, and by which he has made the quickest pro-
gress. And if we are all of the same opinion that the
task of science is to find the Laws of Facts, then
each one may be left free either to plunge into facts,
and to search where he might come upon traces of
laws still unknown, or from laws already known to
search out the points where new facts are to be dis-
covered. But just as we all, like Magnus, are op-
posed to the theorist who holds it unnecessary to
prove experimentally the hypothetical results which
seem axioms to him, so would Magnus — as his works
decidedly show — pronounce with us against that kind
of excessive empiricism which sets out to discover
facts which fit to no rule, and which also try carefully

GUSTAV MAGNUS. 21

bo avoid a law, or a possible connection between newly
discovered facts.

It must here be mentioned that Faraday, another
great physicist, worked in England exactly in the
same direction, and with the same object; to whom,
on that account, Magnus was bound by the heartiest
sympathy. With Faraday, the antagonism to the phy-
sical theories hitherto held, which treated of atoms
and forces acting at a distance, was even more pro-
nounced than with Magnus.

We must, moreover, admit that Magnus mostly
worked with success on problems which seemed
specially adapted to mathematical treatment; as, for
instance, his research on the deviation of rotating shot
fired from rifled guns ; also his paper on the form of
jets of water and their resolution into drops. In the
first, he proved, by a very cleverly arranged experi-
ment, how the resistance of the air, acting on the ball
from below, must deflect it sideways as a rotating
body, in a direction depending on that of the rotation ;
and. how, in consequence of this, the trajectory is de-
flected in the same direction. In the second treatise,
he investigated the different forms of jets of water,
how they are partly changed by the form of the aper-
ture through which they flow, partly in consequence
of the manner in which they flow to it ; and how their
resolution into drops is caused by external agitation.

22 GUSTAV MAGNUS.

He applied the principle of the stroboscopic disk in
observing the phenomena, by looking at the jet
through small slits in a rotating disk. He grouped
the various phenomena with peculiar tact, so that
those among them which are alike were easily seen,
and one elucidated the other. And if a final mechanical
explanation is not always attained, yet the reason for
a great number of characteristic features of the indi-
vidual phenomena is plain. In this respect many of
his researches — I might specially commend those on
the efflux of jets of water — are excellent models of
what Goethe theoretically advanced, and in his phy-
sical labours endeavoured to accomplish, though with
only partial success.

But even where Magnus from his standpoint, and
armed with the knowledge of his time, exerted himself
in vain to seize the kernel of the solution of a difficult
question, a host of new and valuable facts is always
brought to light. Thus in his research on the thermo-
electric battery, where he correctly saw that a critical
question was to be solved, and at the conclusion de-
clared: 'When I commenced the experiment just
described, I confidently hoped to find that thermo-
electrical currents are due to a motion of heat.' In
this sense he investigated the cases in which the
thermo-electrical circuit consisted of a single metal in
which there were alternately hard portions, and such as

GUSTAV MAGNUS. 23

had been softened by heat; or those in which the
parts in contact had very different temperatures. He
was convinced that the thermo-electrical current was
due neither to the radiating power, nor to the conduc-
tivity for heat, using this expression in its ordinary
meaning, and he had to content himself with the ob-
viously imperfect explanation that two pieces of the
same metal at different temperatures acted like dis-
similar conductors, which like liquids do not fall in with
the potential series. The solution was first furnished
by the two general laws of the mechanical theory of heat.
Magnus's hope was not unfulfilled. Sir W. Thomson
discovered that alterations in the conductivity for heat,
though such as were produced by the electrical current
itself, were indeed the sources of the current.

It is the nature of the scientific direction which
Magnus pursued in his researches, that they build
many a stone into the great fabric of science, which
give it an ever broader support, and an ever growing
height, without its appearing to a fresh observer as a
special and distinctive work due to the sole exertion
of any one scientific man. If we wish to explain the
importance of each stone for its special place, how
difficult to procure it, and how skilfully it was worked,
we must presuppose either that the hearer knows the
entire history of the building, or we must explain it to
him, by which more time is lost than I can now claim.

24 GUSTAV MAGNUS.

Thus it is with Magnus's researches. Wherever he
has attacked, he has brought out a host of new and
often remarkable facts ; he has carefully and accurately
observed them, and he has brought them in connection
with the great fabric of science. He has, moreover,
bequeathed to science a great number of ingenious and
carefully devised new methods, as instruments with
which future generations will continue to discover
hidden veins of the noble metal of everlasting laws in
the apparently waste and wild chaos of accident.
Magnus's name will always be mentioned in the first
line of those on whose labours the proud edifice of the
science of Nature reposes ; of the science which has so
thoroughly remodelled the life of modern humanity by
its intellectual influence, as well as by its having subju-
gated the forces of nature to the dominion of the mind.

I have only spoken of Magnus's physical labours,
and have only mentioned those which seemed to me
characteristic for his individuality. But the number
of his researches is very great, and they extend over
wider regions than could now be grasped by any single
inquirer. He began as a chemist, but even then he
inclined to those cases which showed remarkable phy-
sical conditions ; he was afterwards exclusively a
physicist. But parallel with this he cultivated a very
extended study of technology, which of itself would
alone have occupied a man's life.

GUSTAV MAGNUS. 25

He has departed, after a rich life and a fruitful
activity. The old law that no man's life is free from
pain must have been applied to him also ; and yet his
life seems to have been especially happy. He had
what men generally desire most ; but he knew how to
ennoble external fortune by putting it at the service of
unselfish objects. To him was granted, what is dearest
to the mind of a noble spirit, to dwell in the centre of
an affectionate family, and in a circle of faithful and
distinguished friends. But I count his rarest happi-
ness to be that he could work in pure enthusiasm for
an ideal principle ; and that he saw the cause which
he served go on victoriously, and develop to unheard
of wealth and ever wider activity.

And in conclusion we must add, in so far as
thoughtfulness, purity of intention, moral and intellec-
tual tact, modesty, and true humanity can rule over the
caprices of fortune and of man, in so far was Magnui
the artificer of his own fortune ; one of the most satis-
factory and contented natures, who secure the love
and favour of men, who with sure inspiration know
how to find the right place for their activity ; and of
whom we may say, envious fact does not embitter their
successes, for, working for pure objects and with pure
wishes, they would find contentment even without
external successes.

ON THE

OKIGIN AND SIGNIFICANCE

OP

GEOMETBICAL AXIOMS.

Lecture delivered in the Docenten Verein in Heidelberg,
in the year 1870.

THE fact that a science can exist and can be de-
veloped as has been the case with geometry, has
always attracted the closest attention among those
who are interested in questions relating to the bases of
the theory of cognition. Of all branches of human
knowledge, there is none which, like it, has sprung as
a completely armed Minerva from the head of Jupiter ;
none before whose death-dealing Aegis doubt and in-
consistency have so little dared to raise their eyes. It
escapes the tedious and troublesome task of collect-
ing experimental facts, which is the province of the
natural sciences in the strict sense of the word; the

28 ORIGIN AND SIGNIFICANCE OF

sole form of its scientific method is deduction. Con-
clusion is deduced from conclusion, and yet no one
of common sense doubts but that these geometrical
principles must find their practical application in the
real world about us. Land surveying, as well as ar-
chitecture, the construction of machinery no less than
mathematical physics, are continually calculating re-
lations of space of the most varied kind by geometrical
principles ; they expect that the success of their con-
structions and experiments shall agree with these
calculations ; and no case is known in which this ex-
pectation has been falsified, provided the calculations
were made correctly and with sufficient data.

Indeed, the fact that geometry exists, and is cap-
able of all this, has always been used as a prominent
example in the discussion on that question, which
forms, as it were, the centre of all antitheses of philo-
sophical systems, that there can be a cognition of
principles destitute of any bases drawn from ex-
perience. In the answer to Kant's celebrated ques-
tion, 'How are synthetical principles a priori
possible?' geometrical axioms are certainly those
examples which appear to show most decisively that
synthetical principles are a priori possible at all.
The circumstance that such principles exist, and force
themselves on our conviction, is regarded as a proof
that space is an a priori mode of all external perception,

GEOMETRICAL AXIOMS. 29

ft appears thereby to postulate, for this a priori
form, not only the character of a purely formal scheme
of itself quite unsubstantial, in which any given result
experience would fit ; but also to include certain pe-
culiarities of the scheme, which bring it about that
only a certain content, ' and one which, as it were, is
strictly defined, could occupy it and be apprehended
by us.1

It is precisely this relation of geometry to the theory
of cognition which emboldens me to speak to you on
geometrical subjects in an assembly of those who for
the most part have limited their mathematical studies
to the ordinary instruction in schools. Fortunately,
the amount of geometry taught in our gymnasia will
enable you to follow, at any rate the tendency, of the
principles I am about to discuss.

I intend to give you an account of a series of
recent and closely connected mathematical researches
which are concerned with the geometrical axioms, their

1 In his book, On tlie Limits of Philosophy, Mr. W. Tobias main-
tains that axioms of a kind which I formerly enunciated are a
misunderstanding of Kant's opinion. But Kant specially adduces
the axioms, that the straight line is the shortest (Kritik der reinen
Vernunft, Introduction, v. 2nd ed. p. 16) ; that space has three di-
mensions (Ibid, part i. sect. i. § 3, p. 41) ; that only one straight line
is possible between two points (Ibid, part ii. sect. i. ' On the Axioms
of Intuition '), as axioms which express a priori the conditions of
intuition by the senses. It is not here the question, whether these
axioms were originally given as intuition of space, or whether they
are only the starting-points from which the understanding can
develop such axioms a priori on which my critic insists.

30 ORIGIN AND SIGNIFICANCE OF

relations to experience, with the question whether it
is logically possible to replace them by others.

Seeing that the researches in question are more
immediately designed to furnish proofs for experts in
a region which, more than almost any other, requires
a higher power of abstraction, and that they are vir-
tually inaccessible to the non-mathematician, I will
endeavour to explain to such a one the question at
issue. I need scarcely remark that my explanation
will give no proof of the correctness of the new views.
He who seeks this proof must take the trouble to
study the original researches.

Anyone who has entered the gates of the first ele-
mentary axioms of geometry, that is, the mathematical
doctrine of space, finds on his path that unbroken
chain of conclusions of which I just spoke, by which
the ever more varied and more complicated figures
are brought within the domain of law. But even in
their first elements certain principles are laid down,
with respect to which geometry confesses that she
cannot prove them, and can only assume that anyone
who understands the essence of these principles will
at once admit their correctness. These are the so-
called axioms.

For example, the proposition that if the shortest
line drawn between two points is called a straight line,
there can be only one such straight line. Again, it is

GEOMETEICAL AXIOMS. 21

an axiom that through any three points in space, not
lying in a straight line, a plane may be drawn, i.e. a
surface which will wholly include every straight line
joining any two of its points. Another axiom, about
which there has been much discussion, affirms that
through a point lying without a straight line only one
straight line can be drawn parallel to the first ; two
straight lines that lie in the same plane and never
meet, however far they may be produced, being called
parallel. There are also axioms that determine the
number of dimensions of space and its surfaces, lines
and points, showing how they are continuous ; as in
the propositions, that a solid is bounded by a surface,
a surface by a line and a line by a point, that the
point is indivisible, that by the movement of a point
a line is described, by that of a line a line or a surface,
by that of a surface a surface or a solid, but by the
movement of a solid a solid and nothing else is
described.

Now what is the origin of such propositions, un-
questionably true yet incapable of proof in a science
where everything else is reasoned conclusion? Are
they inherited from the divine source of our reason
as the idealistic philosophers think, or is it only that
the ingenuity of mathematicians has hitherto not been
penetrating enough to find the proof? Every new
votary, coming with fresh zeal to geometry, naturally

32 OEIGIN AND SIGNIFICANCE OF

strives to succeed where all before him have failed.
And it is quite right that each should make the trial
afresh ; for, as the question has hitherto stood, it is
only by the fruitlessness of one's own efforts that one
can be convinced of the impossibility of finding a
proof. Meanwhile solitary inquirers are always from
time to time appearing who become so deeply en-
tangled in complicated trains of reasoning that they
can no longer discover their mistakes and believe they
have solved the problem. The axiom of parallels
especially has called forth a great number of seeming
demonstrations.

The main difficulty in these inquiries is, and always
has been, the readiness with which results of everyday
experience become mixed up as apparent necessities of
thought with the logical processes, so long as Euclid's
method of constructive intuition is exclusively followed
in geometry. It is in particular extremely difficult, on
this method, to be quite sure that in the steps pre-
scribed for the demonstration we have not involun-
tarily and unconsciously drawn in some most general
results of experience, which the power of executing
certain parts of the operation has already taught us
practically. In drawing any subsidiary line for the
sake of his demonstration, the well-trained geometer
always asks if it is possible to draw such a line. It is
well known that problems of construction play an essen-

GEOMETRICAL AXIOMS. 33

tial part in the system of geometry. At first sight,
these appear to be practical operations, introduced for
the training of learners; but in reality they estab-
lish the existence of definite figures. They show that
points, straight lines, or circles such as the problem re-
quires to be constructed are possible under all con-
ditions, or they determine any exceptions that there
may be. The point on which the investigations turn,
that we are about to consider, is essentially of this
nature. The foundation of all proof by Euclid's
method consists in establishing the congruence of
lines, angles, plane figures, solids, &c. To make the
congruence evident, the geometrical figures are sup-
posed to be applied to one another, of course without
changing their form and dimensions. That this is
in fact possible we have all experienced from our
earliest youth. But, if we proceed to build necessities
of thought upon this assumption of the free trans-
lation of fixed figures, with unchanged form, to every
part of space, we must see whether the assumption
does not involve some presupposition of which no
logical proof is given. We shall see later on that it
does indeed contain one of the most serious import.
But if so, every proof by congruence rests upon a fact
which is obtained from experience only. %

I offer these remarks, at first only to show what
difficulties attend the complete analysis of the pre-
ii. p

34 ORIGIN AND SIGNIFICANCE OF

suppositions we make, in employing the common con-
structive method. We evade them when we apply, to
the investigation of principles, the analytical method
of modern algebraical geometry. The whole process
of algebraical calculation is a purely logical operation ;
it can yield no relation between the quantities sub-
mitted to it that is not already contained in the equa-
tions which give occasion for its being applied. The
recent investigations in question have accordingly been
conducted almost exclusively by means of the purely
abstract methods of analytical geometry.

However, after discovering by the abstract method
what are the points in question, we shall best get a
distinct view of them by taking a region of narrower
limits than our own world of space. Let us, as we
logically may, suppose reasoning beings of only two
dimensions to live and move on the surface of some
solid body. We will assume that they have not the
power of perceiving anything outside this surface, but
that upon it they have perceptions similar to ours. If
such beings worked out a geometry, they would of
course assign only two dimensions to their space.
They would ascertain that a point in moving describes
a, line, and that a line in moving describes a surface.
But they could as little represent to themselves what
further spatial construction would be generated by a
surface moving out of itself, as we can represent what

GEOMETRICAL AXIOMS. 35

would be generated by a solid moving out of the space
we know. By the much-abused expression ' to re-
present 'or ' to be able to think how something
happens ' I understand — and I do not see how any-
thing else can be understood by it without loss of all
meaning — the power of imagining the whole series of
sensible impressions that would be had in such a case.
Now as no sensible impression is known relating to
such an unheard-of event, as the movement to a fourth
dimension would be to us, or as a movement to our
third dimension would be to the inhabitants of a
surface, such a ' representation ' is as impossible as
the ' representation ' of colours would be to one born
blind, if a description of them in general terms could
be given to him.

Our surface-beings would also be able to draw
shortest lines in their superficial space. These would
not necessarily be straight lines in our sense, but what
are technically called geodetic lines of the surface on
which they live ; lines such as are described by a tense
thread laid along the surface, and which can slide upon
it freely. I will henceforth speak of such lines as the
straightest lines of any particular surface or given
space, so as to bring out their analogy with the
straight line in a plane. I hope by this expression to
make the conception more easy for the apprehension

D 2

36 OKIGIN AND SIGNIFICANCE OF

of my non-mathematical hearers without giving rise
to misconception.

Now if beings of this kind lived on an infinite
plane, their geometry would be exactly the same as
our planimetry. They would affirm that only one
straight line is possible between two points ; that
through a third point lying without this line only one
line can be drawn parallel to it ; that the ends of a
straight line never meet though it is produced to
infinity, and so on. Their space might be infinitely ex-
tended, but even if there were limits to their move-
ment and perception, they would be able to represent
to themselves a continuation beyond these limits ; and
thus their space would appear to them infinitely ex-
tended, just as ours does to us, although our bodies
cannot leave the earth, and our sight only reaches as
far as the visible fixed stars.

But intelligent beings of the kind supposed might
also live on the surface of a sphere. Their shortest or
straightest line between two points would then be an
arc of the great circle passing through them. Every
great circle, passing through two points, is by these
divided into two parts ; and if they are unequal, the
shorter is certainly the shortest line on the sphere be-
tween the two points, but also the other or larger arc
of the same great circle is a geodetic or straightest
line, i.e. every small or part of it is the shortest line

GEOMETRICAL AXIOMS. 37

between its ends. Thus the notion of the geodetic or
straightest line is not quite identical with that of the
shortest line. If the two given points are the ends of
a diameter of the sphere, every plane passing through
this diameter cuts semicircles, on the surface of the
sphere, all of which are shortest lines between the
ends ; in which case there is an equal number of
equal shortest lines between the given points. Ac-
cordingly, the axiom of there being only one shortest
line between two points would not hold without a
certain exception for the dwellers on a sphere.

Of parallel lines the sphere -dwellers would know
nothing. They would maintain that any two straightest
lines, sufficiently produced, must finally cut not in one
only but in two points. The sum of the angles of a
triangle would be always greater than two right angles,
increasing as the surface of the triangle grew greater.
They could thus have no conception of geometrical
similarity between greater and smaller figures of the
same kind, for with them a greater triangle must have
different angles from a smaller one. Their space
would be unlimited, but would be found to be finite or
at least represented as such.

It is clear, then, that such beings must set up a
very different system of geometrical axioms from that
of the inhabitants of a plane, or from ours with our
space of three dimensions, though the logical power*

38 ORIGIN AND SIGNIFICANCE OF

of all were the same; nor are more examples neces-
sary to show that geometrical axioms must vary ac-
cording to the kind of space inhabited by beings
whose powers of reason are quite in conformity with
ours. But let us proceed still farther.

Let us think of reasoning beings existing on the
surface of an egg-shaped body. Shortest lines could
be drawn between three points of such a surface and
a triangle constructed. But if the attempt were made
to construct congruent triangles at different parts of
the surface, it would be found that two triangles, with
three pairs of equal sides, would not have their angles
equal. The sum of the angles of a triangle drawn at
the sharper pole of the body would depart farther from
two right angles than if the triangle were drawn at the
blunter pole or at the equator. Hence it appears that
not even such a simple figure as a triangle can be
moved on such a surface without change of form. It
would also be found that if circles of equal radii were
constructed at different parts of such a surface (the
length of the radii being always measured by shortest
lines along the surface) the periphery would be greater
at the blunter than at the sharper end.

We see accordingly that, if a surface admits of the
figures lying on it being freely moved without change
of any of their lines and angles as measured along it,
the property is a special one and does not belong to

GEOMETRICAL AXIOMS. 39

every kind of surface. The condition under which a
surface possesses this important property was pointed
out by Gauss in his celebrated treatise on the cur-
vature of surfaces.1 The * measure of curvature,' as he
called it, i.e. the reciprocal of the product of the
greatest and least radii of curvature, must be every-
where equal over the whole extent of the surface.

Gauss showed at the same time that this measure
of curvature is not changed if the surface is bent with-
out distension or contraction of any part of it. Thus
we can roll up a flat sheet of paper into the form of
a cylinder, or of a cone, without any change in the
dimensions of the figures taken along the surface of
the sheet. Or the hemispherical fundus of a bladder
may be rolled into a spindle-shape without altering the
dimensions on the surface. Geometry on a plane will
therefore be the same as on a cylindrical surface ; only
in the latter case we must imagine that any number of
layers of this surface, like the layers of a rolled sheet
of paper, lie one upon another, and that after each
entire revolution round the cylinder a new layer is
reached different from the previous ones.

These observations are necessary to give the reader a
notion of a kind of surface the geometry of which is on
the whole similar to that of the plane, but in which

1 Gauss, Werke, Bd. IV. p. 215, first published in Commcntationes
Sec, Ileg. Scientt. Gottengensis recetit'ivrcs, vol. vi., 1828.

40 OftlGIN AND SIGNIFICANCE OF

the axiom of parallels does not hold good. This is a
kind of curved surface which is, as it were, geometri-
cally the counterpart of a sphere, and which has there-
fore been called the pseudospherical surface by the
distinguished Italian mathematician E. Beltrami, who
has investigated its properties.1 It is a saddle-shaped
surface of which only limited pieces or strips can be
connectedly represented in our space, but which may
yet be thought of as infinitely continued in all direc-
tions, since each piece lying at the limit of the part
constructed can be conceived as drawn back to the
middle of it and then continued. The piece displaced
must in the process change its flexure but not its
dimensions, just as happens with a sheet of paper
moved about a cone formed out of a plane rolled up.
Such a sheet fits the conical surface in every part, but
must be more bent near the vertex and cannot be so
moved over the vertex as to be at the same time
adapted to the existing cone and to its imaginary
continuation beyond.

Like the plane and the sphere, pseudospherical sur-
faces have their measure of curvature constant, so that
every piece of them can be exactly applied to every

1 Saggio di Interpretazione della Geometria Non-Euclidea, N apoli,
1868. — Teoria Jondamentale degli Spazii di Curvatura costante, An*
nali di Matematica, Ser. II. Tom. II. pp. 232-55. Both have
been translated into French by J. Houel, AnnaLs Scienti/iqua de
Xorniale, Tom V., 1860.

GECttlETEICAL AXIOMS. 41

other piece, and therefore all figures constructed at
one place on the surface can be transferred to any
other place with perfect congruity of form, and perfect
equality of all dimensions lying in the surface itself.
The measure of curvature as laid down by Gauss,
which is positive for the sphere and zero for the plane,
would have a constant negative value for pseudo-
spherical surfaces, because the two principal curvatures
of a saddle-shaped surface have their concavity turned
opposite ways.

A strip of a pseudospherical surface may, for exam-
ple, be represented by the inner surface (turned towards
the axis) of a solid anchor-ring. If the plane figure
aabb (Fig. 1) is made to revolve on its axis of symme-
try AB, the two arcs ab will describe a pseudospherical
concave-convex surface like that of the ring. Above
and below, towards aa and 66, the surface will turn
outwards with ever-increasing flexure, till it becomes
perpendicular to the axis, and ends at the edge with one
curvature infinite. Or, again, half of a pseudospheri-
cal surface may be rolled up into the shape of a cham-
pagne-glass (Fig. 2), with tapering stem infinitely
prolonged. But the surface is always necessarily
bounded by a sharp edge beyond which it cannot be
directly continued. Only by supposing each single
piece of the edge cut loose and drawn along the surface
of the ring or glass, can it be brought to places of

42

ORIGIN AND SIGNIFICANCE OF

different flexure, at which farther continuation of the
piece is possible.

In this way too the straight est lines of the pseudo-
spherical surface may be infinitely produced. They do
not, like those on a sphere, return upon themselves,
but, as on a plane, only one shortest line is possible
between the two given points. The axiom of parallels
does not, however, hold good. If a straightest line is

Fia. 1.

given on the surface and a point without it, a whole
pencil of straightest lines may pass through the point,
no one of which, though infinitely produced, cuts the
first line; the pencil itself being limited by two
straightest lines, one of which intersects one of the
ends of the given line at an infinite distance, the other
the other end.

Such a system of geometry, which excluded the
axiom of parallels, was devised on Euclid's synthetic
method, as far back as the year 1829, by N. J. Lo-

GEOMETEICAL AXIOMS. 43

batchewsky, professor of mathematics at Kasan,1 and
it was proved that this system could be carried out as
consistently as Euclid's. It agrees exactly with the
geometry of the pseudospherical surfaces worked out
recently by Beltrami.

Thus we see that in the geometry of two dimen-
sions a surface is marked out as a plane, or a sphere, or
a pseudospherical surface, by the assumption that any
figure may be moved about in all directions without
change of dimensions. The axiom, that there is only
one shortest line between any two points, distinguishes
the plane and the pseudospherical surface from the
sphere, and the axiom of parallels marks off the plane
from the pseudosphere. These three axioms are in
fact necessary and sufficient, to define as a plane the
surface to which Euclid's planimetry has reference, as
distinguished from all other modes of space in two
dimensions.

The difference between plane and spherical geome-
try has been long evident, but the meaning of the
axiom of parallels could not be understood till Gauss
had developed the notion of surfaces flexible without
dilatation, and consequently that of the possibly in-
finite continuation of pseudospherical surfaces. In-
habiting, as we do, a space of three dimensions and
endowed with organs of sense for their perception, we
1 Principien der Geometric, Kasan, 1829-30.

44 ORIGIN AND SIGNIFICANCE OF

can represent to ourselves the various cases in which
beings on a surface might have to develop their per-
ception of space ; for we have only to limit our own
perceptions to a narrower field. It is easy to think
away perceptions that we have ; but it is very difficult
to imagine perceptions to which there is nothing ana-
logous in our experience. When, therefore, we pass to
space of three dimensions, we are stopped in our power
of representation, by the structure of our organs and
the experiences got through them which correspond
only to the space in which we live.

There is however another way of treating geometry
scientifically. All known space-relations are measur-
able, that is, they may be brought to determination of
magnitudes (lines, angles, surfaces, volumes). Problems
in geometry can therefore be solved, by finding methods
of calculation for arriving at unknown magnitudes from
known ones. This is done in analytical geometry, where
all forms of space are treated only as quantities and
determined by means of other quantities. Even the
axioms themselves make reference to magnitudes. The
straight line is defined as the shortest between two
points, which is a determination of quantity. The
axiom of parallels declares that if two straight lines in
a plane do not intersect (are parallel), the alternate
angles, or the corresponding angles, made by a third
line intersecting them, are equal; or it may be laid

GEOMETRICAL AXTOm 45

down instead that the sum of the angles of any
triangle is equal to two right angles. These, also,
are determinations of quantity.

Now we may start with this view of space, accord-
ing to which the position of a point may be deter-
mined by measurements in relation to any given
figure (system of co-ordinates), taken as fixed, and
then inquire what are the special characteristics of our
space as manifested in the measurements that have
to be made, and how it differs from other extended
quantities of like variety. This path was first entered
by one too early lost to science, B. Eiemann of Grott-
ingen.1 It has the peculiar advantage that all its
operations consist in pure calculation of quantities,
which quite obviates the danger of habitual percep-
tions being taken for necessities of thought.

The number of measurements necessary to give the
position of a point, is equal to the number of dimensions
of the space in question. In a line the distance from one
fixed point is sufficient, that is to say, one quantity ;
in a surface the distances from two fixed points must
be given ; in space, the distances from three ; or we
require, as on the earth, longitude, latitude, and height
above the sea, or, as is usual in analytical geometry,
the distances from three co-ordinate planes. Eiemann

1 Ueber die Hypothesen welche der Geometrie zu Grunde liegen,
Habilitationsschrift vom 10 Juni 1854. (AbJuindl. der koidgl,
Gesellsch. zu Gottinqen, Bd. XIII.)

4f» OEIGIX AND SIGNIFICANCE 051

calls a system of differences in which one thing can be
determined by n measurements an 'Tifold extended
aggregate ' or an * aggregate of n dimensions/ Thus
the space in which we live is a threefold, a surface is
a twofold, and a line is a simple extended aggregate of
points. Time also is an aggregate of one dimension.
The system of colours is an aggregate of three dimen-
sions, inasmuch as each colour, according to the inves-
tigations of Thomas Young and of Clerk Maxwell,1
may be represented as a mixture of three primary
colours, taken in definite quantities. The particular
mixtures can be actually made with the colour-top.

In the same way we may consider the system of
simple tones2 as an aggregate of two dimensions, if we
distinguish only pitch and intensity, and leave out of
account differences of timbre. This generalisation of
the idea is well suited to bring out the distinction be-
tween space of three dimensions and other aggregates.
We can, as we know from daily experience, compare
the vertical distance of two points with the horizontal
distance of two others, because we can apply a measure
first to the one pair and then to the other. But we
cannot compare the difference between two tones of equal
pitch and different intensity, with that between two tones
of equal intensity and different pitch. Eiemann showed,
by considerations of this kind, that the essential foun-

1 Jlelmlioltz's Popular Lectures, Series I. p. 243. « Ibid, p. 86.

GEOMETRICAL AXIOMS. 47

dation of any system of geometry, is the expression
that it gives for the distance between two points lying
in any direction towards one another, beginning with
the infinitesimal interval. He took from analytical
geometry the most general form for this expression,
that, namely, which leaves altogether open the kind of
measurements by which the position of any point is
given.1 Then he showed that the kind of free mobi-
lity without change of form which belongs to bodies
in our space can only exist when certain quantities
yielded by the calculation 2 — quantities that coincide
with Gauss's measure of surface-curvature when they
are expressed for surfaces — have everywhere an equal
value. For this reason Eiemann calls these quantities,
when they have the same value in all directions for a
particular spot, the measure of curvature of the space
at this spot. To prevent misunderstanding,3 I will
once more observe that this so-called measure of
space-curvature is a quantity obtained by purely ana-
lytical calculation, and that its introduction involves no
suggestion of relations that would have a meaning
only for sense-perception. The name is merely taken,

1 For the square of the distance of two infinitely near points the
expression is a homogeneous quadric function of the differentials of
their co-ordinates.

2 They are algebraical expressions compounded from the co-
efficients of the various terms in the expression for the square of the
distance of two contiguous points and from their differential quotients.

* As occurs, for instance, in the above-mentioned work of Tobias,
pp. 70, etc.

48 ORIGIN AND SIGNIFICANCE OP

as a short expression for a complex relation, from the
one case in which the quantity designated admits, of
sensible representation.

Now whenever the value of this measure of curva-
ture in any space is everywhere zero, that space every-
where conforms to the axioms of Euclid ; and it may be
called a flat (homaloid) space in contradistinction to
other spaces, analytically constructible, that may be
c ailed curved, because their measure of curvature has a
value other than zero. Analytical geometry may be as
completely and consistently worked out for such spaces
as ordinary geometry can for our actually existing
homaloid space.

If the measure of curvature is positive we have
spherical space, in which straightest lines return upon
themselves and there are no parallels. Such a space
would, like the surface of a sphere, be unlimited but
not infinitely great. A constant negative measure of
curvature on the other hand gives pseudo-spherical
space, in which straightest lines run out to infinity, and
a pencil of straightest lines may be drawn, in any
flattest surface, through any point which does not inter-
sect another given straightest line in that surface.

Beltrami * has rendered these last relations imagin-
able by showing that the points, lines, and surfaces of
a pseudospherical space of three dimensions, can be so
1 Teoria fondamentale, $

GEOMETRICAL AXIOMS. 49

portrayed in the interior of a sphere in Euclid's homa-
loid space, that every straightest line or flattest surface
of the pseudospherical space is represented by a
straight line or a plane, respectively, in the sphere.
The surface itself of the sphere corresponds to the
infinitely distant points of the pseudospherical space ;
and the different parts of this space, as represented in
the sphere, become smaller, the nearer they lie to the
spherical surface, diminishing more rapidly in the direc-
tion of the radii than in that perpendicular to them.
Straight lines in the sphere, which only intersect
beyond its surface, correspond to straightest lines of
the pseudospherical space which never intersect.

Thus it appeared that space, considered as a region
of measurable quantities, does not at all correspond
with the most general conception of an aggregate of
three dimensions, but involves also special conditions,
depending on the perfectly free mobility of solid
bodies without change of form to all parts of it and
with all possible changes of direction ; and, further, on
the special value of the measure of curvature which
for our actual space equals, or at least is not distin-
guishable from, zero. This latter definition is given
in the axioms of straight lines and parallels.

Whilst Eiemann entered upon this new field from
the side of the most general and fundamental questions
of analytical geometry, I myself arrived at similar

II. I

50 ORIGIN AND SIGNIFICANCE OS

conclusions,1 partly from seeking to represent in space
the system of colours, involving the comparison of one
threefold extended aggregate with another, and partly
from inquiries on the origin of our ocular measure for
distances in the field of vision. Kiemann starts by
assuming the above-mentioned algebraical expression
which represents in the most general form the distance
between two infinitely near points, and deduces there-
from, the conditions of mobility of rigid figures. I, on
the other hand, starting from the observed fact that
the movement of rigid figures is possible in our space,
with the degree of freedom that we know, deduce the
necessity of the algebraic expression taken by Biemann
as an axiom. The assumptions that I had to make as
the basis of the calculation were the following.

First, to make algebraical treatment at all possible,
it must be assumed that the position of any point A
can be determined, in relation to certain given figures
taken as fixed bases, by measurement of some kind of
magnitudes, as lines, angles between lines, angles
between surfaces, and so forth. The measurements
necessary for determining the position of A are known
as its co-ordinates. In general, the number of co-
ordinates necessary for the complete determination of
the position of a point, marks the number of the dimen-

1 Ueber die Thatsachen die der Geometric ziirn Grande liegen
{Nachrickten, yonder k'dnigl. Ges. d. Wiss.zu Gottingen, Juni 3, 1868).

GEOMETRICAL AXIOMS. 51

sions of the space in question. It is further assumed
that with the movement of the point A, the magnitudes
used as co-ordinates vary continuously.

Secondly, the definition of a solid body, or rigid
system of points, must be made in such a way as to
admit of magnitudes being compared by congruence.
As we must not, at this stage, assume any special
methods for the measurement of magnitudes, our defi-
nition can, in the first instance, run only as follows »
Between the co-ordinates of any two points belonging
to a solid body, there must be an equation which, how-
ever the body is moved, expresses a constant spatial
relation (proving at last to be the distance) between
the two points, and which is the same for congruent
pairs of points, that is to say, such pairs as can be
made successively to coincide in space with the same
fixed pair of points.

However indeterminate in appearance, this defini-
tion involves most important consequences, because
with increase in the number of points, the number of
equations increases much more quickly than the number
of co-ordinates which they determine. Five points,
A, B, C, D, E, give ten different pairs of points
AB, AC, AD, AE,
EC, BD, BE,
CD, CE,
DE,

52 ORIGIN AND SIGNIFICANCE 0£

and therefore ten equations, involving in space of three
dimensions fifteen variable co-ordinates. But of these
fifteen, six must remain arbitrary, if the system of five
points is to admit of free movement and rotation, and
thus the ten equations can determine only nine co-ordi-
nates as functions of the six variables. With six points
we obtain fifteen equations for twelve quantities, with
seven points twenty-one equations for fifteen, and so
on. Now from n independent equations we can
determine n contained quantities, and if we have
more than n equations, the superfluous ones must be
deducible from the first n. Hence it follows that the
equations which subsist between the co-ordinates of
each pair of points of a solid body must have a special
character, seeing that, when in space of three dimen-
sions they are satisfied for nine pairs of points as
formed out of any five points, the equation for the tenth
pair follows by logical consequence. Thus our assump-
tion for the definition of solidity, becomes quite suffi-
cient to determine the kind of equations holding be-
tween the co-ordinates of two points rigidly connected.
Thirdly, the calculation must further be based on
the fact of a peculiar circumstance in the movement of
solid bodies, a fact so familiar to us that but for this
inquiry it might never have been thought of as some-
thing that need not be. When in our space of three
dimensions two points of a solid body are kept fixed,

GEOMETRICAL AXIOMS. 53

its movements are limited to rotations round the
straight line connecting them. If we turn it com-
pletely round once, it again occupies exactly the po-
sition it had at first. This fact, that rotation in one
direction always brings a solid body back into its ori-
ginal position, needs special mention. A system of
geometry is possible without it. This is most easily
seen in the geometry of a plane. Suppose that with
every rotation of a plane figure its linear dimensions in-
creased in proportion to the angle of rotation, the figure
after one whole rotation through 360 degrees would no
longer coincide with itself as it was originally. But
any second figure that was congruent with the first in
its original position might be made to coincide with it
in its second position by being also turned through
360 degrees. A consistent system of geometry would
be possible upon this supposition, which does not come
under Eiemann's formula.

On the other hand I have shown that the three
assumptions taken together form a sufficient basis for
the starting-point of Eiemann's investigation, and
thence for all his further results relating to the dis-
tinction of different spaces according to their measure
of curvature.

It still remained to be seen whether the laws of
motion, as dependent on moving forces, could also be
consistently transferred to spherical or pseudospherical

54 ORIGIN AND SIGNIFICANCE OF

space. This investigation has been carried out by
Professor Lipschitz of Bonn.1 It is found that the
comprehensive expression for all the laws of dynamics,
Hamilton's principle, may be directly transferred to
spaces of which the measure of curvature is other than
zero. Accordingly, in this respect also, the disparate
systems of geometry lead to no contradiction.

We have now to seek an explanation of the special
characteristics of our own flat space, since it appears
that they are not implied in the general notion of an
extended quantity of three dimensions and of the free
mobility of bounded figures therein. Necessities of
thought, such as are involved in the conception of such
a variety, and its measurability, or from the most
general of all ideas of a solid figure contained in it,
and of its free mobility, they undoubtedly are not.
Let us then examine the opposite assumption as to
their origin being empirical, and see if they can be
inferred from facts of experience and so established, or
if, when tested by experience, they are perhaps to be
rejected. If they are of empirical origin, we must be
able to represent to ourselves connected series of facts,
indicating a different value for the measure of curva-
ture from that of Euclid's flat space. But if we can

• 'Untersuclmngen iiber die ganzen homogenen Functionen von n
Differentialen' (Borchardt's Journal filr MatJiematik, Bd. Ixx. 3, 71 ;
Ixxiii. 3, 1) ; « Untersuchung eines Problems der Variationsrechnung'
( Ibid. Bd. Ixxiv.).

GEOMETEICAL AXIOMS. 55

imagine such spaces of other sorts, it cannot be main-
tained that the axioms of geometry are necessary con-
sequences of an a priori transcendental form of intui-
tion, as Kant thought.

The distinction between spherical, pseudospherical,
and Euclid's geometry depends, as was above observed,
on the value of a certain constant called, by Eiemann,
the measure of curvature of the space in question.
The value must be zero for Euclid's axioms to hold
good. If it were not zero, the sum of the angles of
a large triangle would differ from that of the angles of
a small one, being larger in spherical, smaller in pseu-
dospherical, space. Again, the geometrical similarity
of large and small solids or figures is possible only in
Euclid's space. All systems of practical mensuration
that have been used for the angles of large rectilinear
triangles, and especially all systems of astronomical
measurement which make the parallax of the im-
measurably distant fixed stars equal to zero (in pseudo-
spherical space the parallax even of infinitely distant
points would be positive), confirm empirically the
axiom of parallels, and show the measure of curvature
of our space thus far to be indistinguishable from zero
It remains, however, a question, as Biemann observed,
whether the result might not be different if we could
use other than our limited base-lines, the greatest oi
which is the major axis of the earth's orbit.

56 ORIGIN AND SIGNIFICANCE OF

Meanwhile, we must not forget that all geometrical
measurements rest ultimately upon the principle of
congruence. We measure the distance between points
by applying to them the compass, rule, or chain. We
measure angles by bringing the divided circle or theo-
dolite to the vertex of the angle. We also determine
straight lines by the path of rays of light which in
our experience is rectilinear ; but that light travels in
shortest lines as long as it continues in a medium of
constant refraction would be equally true in space of a
different measure of curvature. Thus all our geo-
metrical measurements depend on our instruments
being really, as we consider them, invariable in form,
or at least on their undergoing no other than the small
changes we know of, as arising from variation of tem-
perature, or from gravity acting differently at different
places.

In measuring, we only employ the best and surest
means we know of to determine, what we otherwise are
in the habit of making out by sight and touch or by
pacing. Here our own body with its organs is the
instrument we carry about in space. Now it is the
hand, now the leg, that serves for a compass, or the eye
turning in all directions is our theodolite for measur-
ing arcs and angles in the visual field.

Every comparative estimate of magnitudes or mea-
surement of their spatial relations proceeds therefore

GEOMETRICAL AXIOMS. 57

upon a supposition as to the behaviour of certain phy-
sical things, either the human body or other instru-
ments employed. The supposition may be in the
highest degree probable and in closest harmony with
all other physical relations known to us, but yet it
passes beyond the scope of pure space-intuition.

It is in fact possible to imagine conditions for
bodies apparently solid such that the measurements in
Euclid's space become what they would be in spherical
or pseudospherical space. Let me first remind the
reader that if all the linear dimensions of other bodies,
and our own, at the same time were diminished or in-
creased in like proportion, as for instance to half or
double their size, we should with our means of space-
perception be utterly unaware of the change. This
would also be the case if the distension or contraction
were different in different directions, provided that
our own body changed in the same manner, and further
that a body in rotating assumed at every moment,
without suffering or exerting mechanical resistance,
the amount of dilatation in its different dimensions
corresponding to its position at the time. Think of
the image of the world in a convex mirror. The
common silvered globes set up in gardens give the
essential features, only distorted by some optical ir-
regularities. A well-made convex mirror of moderate
aperture represents the objects in front of it as ap-

58 OKiaiN AND SIGNIFICANCE 0?

parently solid and in fixed positions behind its surface.
But the images of the distant horizon and of the sun
in the sky lie behind the mirror at a limited distance,
equal to its focal length. Between these and the sur-
face of the mirror are found the images of all the other
objects before it, but the images are diminished and
flattened in proportion to the distance of their objects
from the mirror. The flattening, or decrease in the
third dimension, is relatively greater than the decrease
of the surface-dimensions. Yet every straight line or
every plane in the outer world is represented by a
straight. line or a plane in the image. The image of a
man measuring with a rule a straight line from the
mirror would contract more and more the farther he
went, but with his shrunken rule the man in the
image would count out exactly the same number of
centimetres as the real man. And, in general, all
geometrical measurements of lines or angles made
with regularly varying images of real instruments
would yield exactly the same results as in the outer
world, all congruent bodies would coincide on being
applied to one another in the mirror as in the outer
world, all lines of sight in the outer world would be
represented by straight lines of sight in the mirror.
In short I do not see how men in the mirror are
to discover that their bodies are not rigid solids and
their experiences good examples of the correctness of
Euclid's axioms. But if they could look out upon our

GEOMETKICAL AXIOMS. 59

world as we can look into theirs, without overstepping
the boundary, they must declare it to be a picture in a
spherical mirror, and would speak of us just as we
speak of them ; and if two inhabitants of the different
worlds could communicate with one another, neither,
so far as I can see, would be able to convince the other
that he had the true, the other the distorted, relations.
Indeed I cannot see that such a question would have
any meaning at all, so long as mechanical considerations
are not mixed up with it.

Now Beltrami's representation of pseudospherical
space in a sphere of Euclid's space, is quite similar, ex-
cept that the background is not a plane as in the
convex mirror, but the surface of a sphere, and that
the proportion in which the images as they approach
the spherical surface contract, has a different mathe-
matical expression.1 If we imagine then, conversely,
that in the sphere, for the interior of which Euclid's
axioms hold good, moving bodies contract as they
depart from the centre like the images in a convex
mirror, and in such a way that their representatives
in pseudospherical space retain their dimensions
unchanged, — observers whose bodies were regularly
subjected to the same change would obtain the
same results from the geometrical measurements
they could make as if they lived in pseudospherical
space.

1 Compare the Appendix at the end of this Lecture-.

60 ORIGIN AND SIGNIFICANCE OF

We can even go a step further, and infer how the
objects in a pseudospherical world, were it possible to
enter one, would appear to an observer, whose eye-
measure and experiences of space had been gained like
ours in Euclid's space. Such an observer would con-
tinue to look upon rays of light or the lines of vision
as straight lines, such as are met with in flat space,
and as they really are in the spherical representation
of pseudospherical space. The visual image of the
objects in pseudospherical space would thus make the
same impression upon him as if he were at the centre
of Beltrami's sphere. He would think he saw the
most remote objects round about him at a finite
distance,1 let us suppose a hundred feet off. But as
he approached these distant objects, they would dilate
before him, though more in the third dimension than
superficially, while behind him they would contract.
He would know that his eye judged wrongly. If he
saw two straight lines which in his estimate ran
parallel for the hundred feet to his world's end, he
would find on following them that the farther he
advanced the more they diverged, because of the
dilatation of all the objects to which he approached.
On the other hand, behind him, their distance would
seem to diminish, so that as he advanced they would

1 The reciprocal of the square of this distance, expressed in
negative quantity, would be the measure of curvature of the pseudo-
spherical space.

GEOMETRICAL AXIOMS. 61

appear always to diverge more and more. But two
straight lines which from his first position seemed to
converge to one and the same point of the background
a hundred feet distant, would continue to do this
however far he went, and he would never reach their
point of intersection.

Now we can obtain exactly similar images of our
real world, if we look through a large convex lens of
corresponding negative focal length, or even through a
pair of convex spectacles if ground somewhat prisma-
tically to resemble pieces of one continuous larger lens.
With these, like the convex mirror, we see remote ob-
jects as if near to us, the most remote appearing no
farther distant than the focus of the lens. . In going
about with this lens before the eyes, we find that the
objects we approach dilate exactly in the manner I
have described for pseudospherical space. Now any one
using a lens, were it even so strong as to have a focal
length of only sixty inches, to say nothing of a hun-
dred feet, would perhaps observe for the first moment
that he saw objects brought nearer. But after going
about a little the illusion would vanish, and in spite
of the false images he would judge of the distances
rightly. We have every reason to suppose that what
happens in a few hours to any one beginning to wear
spectacles would soon enough be experienced in pseu-
dospherical space. In short, pseudospherical space

62 OEIGIN AND SIGNIFICANCE OF

would not seem to us very strange, comparatively
speaking; we should only at first be subject to illu-
sions in measuring by eye the size and distance of the
more remote objects.

There would be illusions of an opposite description,
if, with eyes practised to measure in Euclid's space, we
entered a spherical space of three dimensions. We
should suppose the more distant objects to be more
remote and larger than they are, and should find on
approaching them that we reached them more quickly
than we expected from their appearance. But we
should also see before us objects that we can fixate
only with diverging lines of sight, namely, all those
at a greater distance from us than the quadrant of a
great circle. Such an aspect of things would hardly
strike us as very extraordinary, for we can have it even
as things are if we place before the eye a slightly pris-
matic glass with the thicker side towards the nose : the
eyes must then become divergent to take in distant
objects. This excites a certain feeling of unwonted
strain in the eyes, but does not perceptibly change the
appearance of the objects thus seen. The strangest
sight, however, in the spherical world would be the
back of our own head, in which all visual lines not
stopped by other objects would meet again, and which
must fill the extreme background of the whole per-
spective picture.

GEOMETRICAL AXIOMS. f>3

At the same time it must be noted that as a small
elastic flat disk, say of india-rubber, can only be fitted
to a slightly curved spherical surface with relative con-
traction of its border and distension of its centre, so
our bodies, developed in Euclid's flat space, could not
pass into curved space without undergoing similar
distensions and contractions of their parts, their co-
herence being of course maintained only in as far as
their elasticity permitted their bending without break-
ing. The kind of distension must be the same as in
passing from a small body imagined at the centre of
Beltrami's sphere to its pseudospherical or spherical
representation. For such passage to appear possible,
it will always have to be assumed that the body is
sufficiently elastic and small in comparison with the
real or imaginary radius of curvature of the curved
space into which it is to pass.

These remarks will suffice to show the way in
which we can infer from the known laws of our sen-
sible perceptions the series of sensible impressions
which a spherical or pseudospherical world would give
us, if it existed. In doing so, we nowhere meet with
inconsistency or impossibility any more than in the
calculation of its metrical proportions. "We can re-
present to ourselves the look of a pseudospherical
world in all directions just as we can develop the con*
ception of it. Therefore it cannot be allowed that the

64 OBiaiN AND SIGNIFICANCE 0?

axioms of our geometry depend on the native form of
our perceptive faculty, or are in any way connected
with it.

It is different with the three dimensions of space.
As all our means of sense-perception extend only to
space of three dimensions, and a fourth is not merely
a modification of what we have, but something per-
fectly new, we find ourselves by reason of our bodily
organisation quite unable to represent a fourth di-
mension.

In conclusion, I would again urge that the axioms
of geometry are not propositions pertaining only to
the pure doctrine of space. As I said before, they are
concerned with quantity. We can speak of quantities
only when we know of some way by which we can com-
pare, divide, and measure them. All space-measure-
ments, and therefore in general all ideas of quantities
applied to space, assume the possibility of figures mov-
ing without change of form or size. It is true we are
accustomed in geometry to call such figures purely
geometrical solids, surfaces, angles, and lines, because
we abstract from all the other distinctions, physical
and chemical, of natural bodies ; but yet one physical
quality, rigidity, is retained. Now we have no other
mark of rigidity of bodies or figures but congruence,
whenever they are applied to one another at any time
or place, and after any revolution. We cannot, hovv-

GEOMETKICAL AXIOMS. 65

ever, decide by pure geometry, and without mechanical
considerations, whether the coinciding bodies may not
both have varied in the same sense.

If it were useful for any purpose, we might with
perfect consistency look upon the space in which we
live as the apparent space behind a convex mirror with
its shortened and contracted background ; or we might
consider a bounded sphere of our space, beyond the
limits 'of which we perceive nothing further, as infinite
pseudospherical space. Only then we should have to
ascribe to the bodies which appear to us to be solid, and
to our own body at the same time, corresponding disten-
sions and contractions, and we should have to change
our system of mechanical principles entirely ; for even
the proposition that every point in motion, if acted upon
by no force, continues to move with unchanged velo-
city in a straight line, is not adapted to the image of
the world in the convex-mirror. The path would in-
deed be straight, but the velocity would depend upon
the place.

Thus the axioms of geometry are not concerned
with space-relations only but also at the same time
with the mechanical deportment of solidest bodies in
motion. The notion of rigid geometrical figure might
indeed be conceived as transcendental in Kant's sense,
namely, as formed independently of actual experience,
which need not exactly correspond therewith, any more

II. F

60 ORIGIN AND SIGNIFICANCE OP

than natural bodies do ever in fact correspond exactly
to the abstract notion we have obtained of them by in-
duction. Taking the notion of rigidity thus as a mere
ideal, a strict Kantian might certainly look upon the
geometrical axioms as propositions given, a priori, by
transcendental intuition, which no experience could
either confirm or refute, because it must first be decided
by them whether any natural bodies can be considered
as rigid. But then we should have to maintain that the
axioms of geometry are not synthetic propositions, as
Kant held them ; they would merely define what quali-
ties and deportment a body must have to be recognised
as rigid.

But if to the geometrical axioms we add proposi-
tions relating to the mechanical properties of natural
bodies, were it only the axiom of inertia, or the single
proposition, that the mechanical and physical proper-
ties of bodies and their mutual reactions are, other
circumstances remaining the same, independent of
place, such a system of propositions has a real import
which can be confirmed or refuted by experience, but
just for the same reason can also be gained by expe-
rience. The mechanical axiom, just cited, is in fact of
the utmost importance for the whole system of our
mechanical and physical conceptions. That rigid solids,
as we call them, which are really nothing else than elas-
tic solids of great resistance, retain the same form in

GEOMETRJCAL AXIOMS. 67

every part of space if no external force affects them,
is a single case falling under the general principle.

In conclusion, I do not, of course, maintain that man-
kind first arrived at space-intuitions, in agreement with
the axioms of Euclid, by any carefully executed systems
of exact measurement. It was rather a succession of
everyday experiences, especially the perception of the
geometrical similarity of great and small bodies, only
possible in flat space, that led to the rejection, as im-
possible, of every geometrical representation at variance
with this fact. For this no knowledge of the neces-
sary logical connection between the observed fact of
geometrical similarity and the axioms was needed ; but
only an intuitive apprehension of the typical relations
between lines, planes, angles, &c., obtained by nume-
rous and attentive observations — an intuition of the
kind the artist possesses of the objects he is to repre-
sent, and by means of which he decides with certainty
and accuracy whether a new combination, which he tries,
will correspond or not with their nature. It is true
that we have no word but intuition to mark this ; but
it is knowledge empirically gained by the aggregation
and reinforcement of similar recurrent impressions in
memory, and not a transcendental form given before
experience. That other such empirical intuitions of
fixed typical relations, when not clearly comprehended,
have frequently enough been taken by metaphysicians

72

68 ORIGIN AND SIGNIFICANCE OF

for a priori principles, is a point on which I need
not insist.

To sum up, the final outcome of the whole inquiry
may be thus expressed : —

(1.) The axioms of geometry, taken by themselves
out of all connection with mechanical propositions, re-
present no relations of real things. When thus iso-
lated, if we regard them with Kant as forms of
intuition transcendentally given, they constitute a
form into which any empirical content whatever will
fit, and which therefore does not in any way limit or
determine beforehand the nature of the content. This
is true, however, not only of Euclid's axioms, but also
of the axioms of spherical and pseudospherical geo-
metry.

(2.) As soon as certain principles of mechanics are
conjoined with the axioms of geometry, we obtain a
system of propositions which has real import, and
which can be verified or overturned by empirical obser-
vations, just as it can be inferred from experience. If
such a system were to be taken as a transcendental
form of intuition and thought, there must be assumed
fc pre-established harmony between form and reality.

GEOMETRICAL AXIOMS. 60

APPENDIX.

THE elements of the geometry of spherical space are most
easily obtained by putting for space of four dimensions the
equation for the sphere

and for the distance ds between the points (x, y, z, t) and
[(x+dx) (y+dy) (z+dz) (t + dt)] the value

...... (2.)

It is easily found by means of the methods used for three
dimensions that the shortest lines are given by equations of
the form

,3 .

in which a, b, c,f, as well as a, f3, y, <j>, are constants.

The length of the shortest arc, s, between the points
(a;, y, z, t), and (£, 17, £, r) follows, as in the sphere, from the
equation

cos J*±£!*l± ......... (4.)

One of the co-ordinates may be eliminated from the values
given in 2 to 4, by means of equation 1, and the expressions
then apply to space of three dimensions.

If we take the distances from the points

£=,,=£=0

70 ORIGIN AND SIGNIFICANCE OF GEOMETRICAL AXIOMS
from which equation 1 gives r=7?, then,

sin
in which <r=

or, *0=7? ..arc sin ( fij=R • arc tang frl. • • (5.)

In this, SQ is the distance of the point x, y, z, measured
from the centre of the co-ordinates.

If now we suppose the point x, y, z, of spherical space,
to be projected in a point of plane space whose co-ordinates
are respectively

• -T «-?«-? -

then in the plane space the equations 3, which belong to
the straightest lines of spherical space, are equations of the
straight line. Hence the shortest Hues of spherical space
are represented in the system of x, g, 3, by straight lines,
For very small values of x, y, z, t=R, and

*=*» J=y, |=*

Immediately about the centre of the co-ordinates, the
measurements of both spaces coincide. On the other hand,
we have for the distances from the centre

. arc tang f± Jj • • • (6.)

In this, r may be infinite ; but every point of plane space
must be the projection of two points of the sphere, one for
which sQ < -| T?TT, and one for which SQ > J RTT. Tha
extension in the direction of r is then

APPENDIX. 71

In order to obtain corresponding expressions for pseudo-
Bpherical space, let R and t be imaginary; that is, ^=Ii,
and t=tl Equation 6 gives then

from which, eliminating the imaginary form, we get
,0=l f log. nat. |±5

Here SQ has real values only as long as r=R; for r=1J, the
distance SQ in pseudospherical space is infinite. The image
in plane space is, on the contrary, contained in the sphere of
radius JR, and every point of this sphere forms only one
point of the infinite pseudospherical space. The extension
in the direction of r is

dsQ= W
dc $2-i»

For linear elements, on the contrary, whose direction is at
right angles to r, and for which t is unchanged, we have in
both cases

ON

THE EELATION OP OPTICS

TO

PAINTING.

Being the substance of a aeries of Lectures delivered in
Cologne, Berlin, and Bonn.

I FEAR that the announcement of my intention to ad-
dress you on the subject of plastic art may have created
no little surprise among some of my hearers. For I
cannot doubt that many of you have had more fre-
quent opportunities of viewing works of art, and have
more thoroughly studied its historical aspects, than I can
lay claim to have done ; or indeed have had personal
experience in the actual practice of art, in which I am
entirely wanting. I have arrived at my artistic studies
by a path which is but little trod, that is, by the phy-
siology of the senses ; and in reference to those who
have a long acquaintance with, and who are quite at
home in the beautiful fields of art, I may compare

74 ON THE KELATION OF OPTICS TO PAINTING.

myself to a traveller who has entered upon them by
a steep and stony mountain path, but who, in doing
so, has passed many a stage from which a good point
of view is obtained. If therefore I relate to you what
I consider I have observed, it is with the understand-
ing that I wish to regard myself as open to instruction
by those more experienced than myself.

The physiological study of the manner in which
the perceptions of our senses originate, how impressions
from without pass into our nerves, and how the condi-
tion of the latter is thereby altered, presents many
points of contact with the theory of the fine arts. On
a former occasion T endeavoured to establish such a
relation between the physiology of the sense of hearing,
and the theory of music. Those relations in that case
are particularly clear and distinct, because the elemen-
tary forms of music depend more closely on the nature
and on the peculiarities of our perceptions than is the
case in other arts, in which the nature of the material
to be used and of the objects to be represented has
a far greater influence. Yet even in those other
branches of art, the especial mode of perception of
that organ of sense by which the impression is taken
up is not without importance ; and a theoretical in-
sight into its action, and into the principle of its
methods, cannot be complete if this physiological ele-
ment is not taken into account. Next to music this

ON THE RELATION OF OPTICS TO PAINTING. 75

seems to predominate more particularly in painting,
and this is the reason why I have chosen painting as
the subject of my present lecture.

The more immediate object of the painter is to
produce in us by his palette a lively visual impression
of the objects which he has endeavoured to represent.
The aim, in a certain sense, is to produce a kind of
optical illusion ; not indeed that, like the birds who
pecked at the painted grapes of Apelles, we are to sup-
pose we have present the real objects themselves, and
not a picture ; but in so far that the artistic represen-
tation produces in us a conception of their objects as
vivid and as powerful as if we had them actually before
us. The study of what are called illusions of the senses
is however a very prominent and important part of
the physiology of the senses ; for just those cases in
which external impressions evoke conceptions which
are not in accordance with reality are particularly in-
structive, for discovering the laws of those means and
processes by which normal perceptions originate. We
must look upon artists as persons whose observation
of sensuous impressions is particularly vivid and accu-
rate, and whose memory for these images is particu-
larly true. That which long tradition has handed
down to the men most gifted in this respect, and
that which they have found by innumerable experi-
ments in the most varied directions, as regards means

76 ON THE RELATION OF OPTICS TO PAINTING.

and methods of representation, forms a series of import-
ant and significant facts, which the physiologist, who
has here to learn from the artist, cannot afford to ne-
glect. The study of works of art will throw great light
on the question as to which elements and relations of
our visual impressions are most predominant in deter-
mining our conception of what is seen, and what others
are of less importance. As far as lies within his power,
the artist will seek to foster the former at the cost of
the latter.

In this sense then a careful observation of the
works of the great masters will be serviceable, not only
to physiological optics, but also because the investigation
of the laws of the perceptions and of the observations
of the senses will promote the theory of art, that is,
the comprehension of its mode of action.

We have not here to do with a discussion of the
ultimate objects and aims of art, but only with an ex-
amination of the action of the elementary means with
which it works. The knowledge of the latter must,
however, form an indispensable basis for the solution
of the deeper questions, if we are to understand the
problems which the artist has to solve, and the mode
in which he attempts to attain his object.

I need scarcely lay stress on the fact, following as
it does from what I have already said, that it is not
my intention to furnish instructions according to which

ON THE EELATION OF OPTICS TO PAINTING. 77

the artist is to work. I consider it a mistake to sup-
pose that any kind of aesthetic lectures such as these
can ever do so ; but it is a mistake which those very
frequently make who have only practical objects in

view.

78 OS THE RELATION OF OPTICS TO PAINTIN&.

I. FORM.

The painter seeks to produce in his picture an image
of external objects. The first aim of our investigation
must be to ascertain what degree and what kind of
similarity he can expect to attain, and what limits are
assigned to him by the nature of his method. The
uneducated observer usually requires nothing more
than an illusive resemblance to nature : the more this
is obtained, the more does he delight in the picture.
An observer, on the contrary, whose taste in works of
art has been more finely educated, will, consciously or
unconsciously, require something more, and something
different. A faithful copy of crude Nature he will at
most regard as an artistic feat. To satisfy him, he
will need artistic selection, grouping, and even idealisa-
tion of the objects represented. The human figures
in a work of art must not be the everyday figures,
such as we see in photographs ; they must have ex-
pression, and a characteristic development, and if
possible beautiful forms, which have perhaps be-
longed to no living individuals or indeed any indi-
viduals which ever have existed, but only to such a
one as might exist, and as must exist, to produce a

ON THE RELATION OF OPTICS TO PAINTING. 79

vivid perception of any particular aspect of human
existence in its complete and unhindered development.

If however the artist is to produce an artistic
arrangement of only idealised types, whether of man
or of natural objects, must not the picture be an
actual, complete, and directly true delineation of that
which would appear if it anywhere came into being ?

Since the picture is on a plane surface, this faith-
ful representation can of course only give a faithful
perspective view of the objects. Yet our eye, which
in its optical properties 'is equivalent to a camera
obscura, the well-known apparatus of the photo-
grapher, gives on the retina, which is its sensitive
plate, only perspective views of the external world ;
these are stationary, like the drawing on a picture,
as long as the standpoint of the eye is not altered.
And, in fact, if we restrict ourselves in the first place
to the form of the object viewed, and disregard for
the present any consideration of colour, by a correct
perspective drawing we can present to the eye of an
observer, who views it from a correctly chosen point
of view, the same forms of the visual image as the
inspection of the objects themselves would present to
the same eye, when viewed from the corresponding
point of view.

But apart from the fact that any movement of the
observer, whereby his eye changes its position, will

80 ON THE EELATION OF OPTICS TO PAINTING.

produce displacements of the visual image, different
when he stands before objects from those when he
stands before the image, I could speak of only one
eye for which equality of impression is to be estab-
lished. We however see the world with two eyes,
which occupy somewhat different positions in space,
and which therefore show two different perspective
views of objects before us. This difference of the
images of the two eyes forms one of the most im-
portant means of estimating the distance of objects
from our eye, and of estimating depth, and this is
what is wanting to the painter, or even turns against
him; since in binocular vision the picture distinctly
forces itself on our perception as a plane surface.

You must all have observed the wonderful vividness
which the solid form of objects acquires when good
stereoscopic images are viewed in the stereoscope, a
kind of vividness in which either of the pictures is
wanting when viewed without the stereoscope. The
illusion is most striking and instructive with figures in
simple line ; models of crystals and the like, in
which there is no other element of illusion. The
reason of this deception is, that looking with two eyes
we view the world simultaneously from somewhat
different points of view, and thereby acquire two dif-
ferent perspective images. With the right eye we see
somewhat more of the right side of objects before us,,

ON THE KELATION OF OPTICS TO PAINTING. 81

and also somewhat more of those behind it, than we
do with the left eye ; and conversely we see with the
left, more of the left side of an object, and of the back-
ground behind its left edges, and partially concealed
by the edge. But a flat picture shows to the right eye
absolutely the same picture, and all objects represented
upon it, as to the left eye. If then we make for each
eye such a picture as that eye would perceive if itself
looked at the object, and if both pictures are combined
in the stereoscope, so that each eye sees its correspond-
ing picture, then as far as form is concerned the
same impression is produced in the two eyes as the
object itself produces. But if we look at a drawing or
a picture with both eyes, we just as easily recognise
that it is a representation on a plane surface, which is
different from that which the actual object would show
simultaneously to both eyes. Hence is due the well-
known increase in the vividness of a picture if it is
looked at with only one eye, and while quite stationary,
through a dark tube ; we thus exclude any comparison
of its distance with that of adjacent objects in the
room. For it must be observed that as we use differ-
ent pictures seen with the two eyes for the perception
of depth, in like manner as the body moves from one
place to another, the pictures seen by the same eye
serve for the same purpose. In moving, whether on
foot or riding, the nearer objects are apparently dis-
ii. O

82 ON THE RELATION OF OPTICS TO PAINTING.

placed in comparison with the more distant ones ; the
former appear to recede, the latter appear to move with
us. Hence arises a far stricter distinction between what
is near and what is distant, than seeing with one eye
from one and the same spot would ever afford us. If
we move towards the picture, the sensuous impression
that it is a flat picture hanging against the wall forces
itself more strongly upon us than if we look at it while
we are stationary. Compared with a large picture at a
greater distance, all those elements which depend on bin-
ocular vision and on the movement of the body are less
operative, because in very distant objects the differ-
ences between the images of the two eyes, or be-
tween the aspect from adjacent points of view, seem
less. Hence large pictures furnish a less distorted
aspect of their object than small ones, while the
impression on a stationary eye, of a small picture close
at hand, might be just the same as that of a large
distant one. In a painting close at hand, the fact that
it is a flat picture continually forces itself more power-
fully and more distinctly on our perception.

The fact that perspective drawings, which are taken
from too near a point of view, may easily produce a
distorted impression, is, I think, connected with this.
For here the want of the second representation for the
other eye, which would be very different, is too marked.
On the other hand, what are called geometrical pro-

ON THE RELATION OF OPTICS TO PAINTING. 83

jections, that is, perspective drawings which represent
a view taken from an infinite distance, give in many
cases a particularly favourable view of the object,
although they correspond to a point of sight which
does not in reality occur. Here the pictures of both
eyes for such an object are the same.

You will notice that in these respects there is a
primary incongruity, and one which cannot be got
over, between the aspect of a picture and the aspect
of reality. This incongruity may be lessened, but
never entirely overcome. Owing to the imperfect
action of binocular vision, the most important natural
means is lost of enabling the observer to estimate
the depth of objects represented in the picture. The
painter possesses a series of subordinate means, partly
of limited applicability, and partly of slight effect,
of expressing various distances by depth. It is not
unimportant to become acquainted with these elements,
as arising out of theoretical considerations ; for in the
practice of the art of painting' they have manifestly
exercised great influence on the arrangement, selec-
tion, and mode of illumination of the objects repre-
sented. The distinctness of what is represented is
indeed of subordinate importance when considered in
reference to the ideal aims of art ; it must not however
be depreciated, for it is the first condition by which
the observer attains an intelligibility of expres-

e 2

84. ON THE RELATION OF OPTICS TO PAINTING.

sion, which impresses itself without fatigue on the
observer.

This direct intelligibility is again the preliminary
condition for an undisturbed, and vivid action of the
picture on the feelings and mood of the observer.

The subordinate methods of expressing depth which
have been referred to, depend in the first place on per-
spective. Nearer objects partially conceal more distant
ones, but can never themselves be concealed by the
latter. If therefore the painter skilfully groups his ob-
jects, so that the feature in question comes into play,
this gives at once a very certain gradation of far and
near. This mutual concealment may even preponderate
over the binocular perception of depth, if stereoscopic
pictures are intentionally produced in which each coun-
teracts the other. Moreover, in bodies of regular or of
known form, the forms of perspective projection are for
the most part characteristic for the depth of the object.
If we look at houses, or other results of man's artistic
activity, we know at the outset that the forms are for the
most part plane surfaces at right angles to each other,
with occasional circular or even spheroidal surfaces. And
in fact, when we know so much, a correct perspective
drawing is sufficient to produce the whole shape of the
body. This is also the case with the figures of men and
animals which are familiar to us, and whose forms
moreover show two symmetrical halves. The best per-

ON THE RELATION OF OPTICS TO PAINTING. 85

spective drawing is however of but little avail in the
case of irregular shapes, rough blocks of rock and ice,
masses of foliage, and the like ; that this is so, is best
seen in photographs, where the perspective and shading
may be absolutely correct, and yet the total impression
is indistinct and confused.

When human habitations are seen in a picture, they
represent to the observer the direction of the hori-
zontal surfaces at the place at which they stand ; and
in comparison therewith the inclination of the ground,
which without them would often be difficult to repre-
sent.

The apparent magnitude which objects, whose
actual magnitude is known, present in different parts
of the picture must also be taken into account. Men
and animals, as well as familiar trees, are useful to the
painter in this respect. In the more distant centre of
the landscape they appear smaller than in the fore-
ground, and thus their apparent magnitude furnishes
a measure of the distance at which they are placed.

Shadows, and more especially double ones, are of
great importance. You all know how much more
distinct is the impression which a well-shaded drawing
gives as distinguished from an outline ; the shading is
hence one of the most difficult, but at the same time
most effective, elements in the productions of the
draughtsman and painter. It is his task to imitate

86 ON THE RELATION OP OPTICS TO PAIXTINQ.

the fi.ae gradation and transitions of light and shade
on rounded surfaces, which are his chief means of ex-
pressing their modelling, with all their fine changes of
curvature ; he must take into account the extension or
restriction of the sources of light, and the mutual
reflection of the surfaces on each other. While the
modifications of the lighting on the surface of bodies
themselves is often dubious — for instance, an intaglio
of a medal may, with a particular illumination, pro-
duce the impression of reliefs which are only illumi-
nated from the other side — double shadows, on the
contrary, are undoubted indications that the body which
throws the shadow is nearer the source of light than
that which receives the shadow. This rule is so com-
pletely without exception, that even in stereoscopic
views a falsely placed double shadow may destroy or
confuse the entire illusion.

The various kinds of illumination are not all equally
favourable for obtaining the full effect of shadows.
When the observer looks at the objects in the same
direction as that in which light falls upon them, he
sees only their illuminated sides and nothing of the
shadow; the whole relief which the shadows could give
then disappears. If the object is between the source
of light and the observer he only sees the shadows.
Hence we need lateral illumination for a picturesque
shading ; and over surfaces which like those of plane

ON THE RELATION OF OPTICS TO PAINTING. 87

+

or hilly land only present slightly moving figures, we
require light which is almost in the direction of the
surface itself, for only such a one gives shadows. This
is one of the reasons which makes illumination by the
rising or the setting sun so effective. The forms of
the landscape become more distinct. To this must
also be added the influence of colour, and of aerial
light, which we shall subsequently discuss.

Direct illumination from the sun, or from a flame,
makes the shadows sharply defined, and hard. Illu-
mination from a very wide luminous surface, such as
a cloudy sky, makes them confused, or destroys them
altogether. Between these two extremes there are
transitions; illumination by a portion of the sky,
defined by a window, or by trees, &c., allows the
shadows to be more or less prominent according to
the nature of the object. You must have seen of
what importance this is to photographers, who have to
modify their light by all manner of screens and
curtains in order to obtain well-modelled portraits.

Of more importance for the representation of
depth than the elements hitherto enumerated, and
which are more or less of local and accidental signific-
ance, is what is called aerial perspective. By this we
understand the optical action of the light, which the
illuminated masses of air, between the observer and
distant objects, give. This arises from a fine opacity

88 ON THE RELATION OF OPTICS TO PAINTING.

in the atmosphere, which never entirely disappears*
If, in a transparent medium, there are fine transparent
particles of varying density and varying refrangibility,
in so far as they are struck by it, they deflect the
light passing through such a medium, partly by reflec-
tion and partly by refraction ; to use an optical expres-
sion, they scatter it in all directions. If the opaque
particles are sparsely distributed, so that a great part
of the light can pass through them without being
deflected, distant objects are seen in sharp, well-defined
outlines through such a medium, while at the same
time a portion of the light which is deflected is dis-
tributed in the transparent medium as an opaque halo.
Water rendered turbid by a few drops of milk shows
this dispersion of the light and cloudiness very distinctly.
The light in this case is deflected by the microscopic
globules of butter which are suspended in the milk.

In the ordinary air of our rooms, this turbidity is
very apparent when the room is closed, and a ray of
sunlight is admitted through a narrow aperture. We
see then some of these solar particles, large enough to
be distinguished by the naked eye, while others form
a fine homogeneous turbidity. But even the latter
must consist mainly of suspended particles of organic
substances, for, according to an observation of Tyndall,
they can be burnt. If the flame of a spirit lamp is
placed directly below the path of these rays, the air

ON THE .RELATION OF OPTICS TO PAINTING. 89

rising from the flame stands out quite dark in the
surrounding bright turbidity; that is to say, the air
rising from the flame has been quite freed from dust.
In the open air, besides dust and occasional smoke, we
must often also take into account the turbidity arising
from incipient aqueous deposits, where the tempera-
ture of moist air sinks so far that the water retained
in it can no longer exist as invisible vapour. Part of
the water settles then in the form of fine drops, as a
kind of the very finest aqueous dust, and forms a finer
or denser fog; that is to say, cloud. The turbidity
which forms in hot sunshine and dry air may arise,
partly from dust which the ascending currents of
warm air whirl about; and partly from the irregular
mixture of cold and warm layers of air of different
density, as is seen in the tremulous motion of the
lower layers of air over surfaces irradiated by the sun.
But science can as yet give no explanation of the
turbidity in the higher regions of the atmosphere
which produces the blue of the sky ; we do not know
whether it arises from suspended particles of foreign
substances, or whether the molecules of air themselves
may not act as turbid particles in the luminous ether.

The colour of the light reflected by the opaque
particles mainly depends on their magnitude. When
a block of wood floats on water, and by a succession of
falling drops we produce small wave-rings near it,

90 ON THE RELATION OF OPTICS TO FAIN'TING

these are repelled by the floating wood as if it. were a
solid wall. But in the long waves of the sea, a block
of wood would be rocked about without the waves
being thereby materially disturbed in their progress.
Now light is well known to be an undulatory motion
of the ether which fills all space. The red and yellow
rays have the longest waves, the blue and violet the
shortest. Very fine particles, therefore, which disturb
the uniformity of the ether, will accordingly reflect
the latter rays more markedly than the red and yellow
rays. The light of turbid media is bluer, the finer
are the opaque particles ; while the larger particles of
uniform light reflect all colours, and therefore give a
whitish turbidity. Of this kind is the celestial blue,
that is, the colour of the turbid atmosphere as seen
against dark cosmical space. The purer and the more
transparent the air, the bluer is the sky. In like man
ner it is bluer and darker when we ascend high moun-
tains, partly because the air at great heights is freer
from turbidity, and partly because there is less air above
us. But the same blue, which is seen against the dark
celestial space, also occurs against dark terrestrial
objects ; for instance, when a thick layer of illuminated
air is between us and masses of deeply shaded or
wooded hills. The same aerial light makes the sky
blue, as well as the mountains ; excepting that in the
former case it is pure, while in the latter it is mixed

ON THE RELATION OF OPTICS TO PAINTING. 91

with the light from objects behind; and moreover
it belongs to the coarser turbidity of the lower regions
of the atmosphere, so that it is whiter. In hot coun-
tries, and with dry air, the aerial turbidity is also finer
m the lower regions of the air, and therefore the blue
in front of distant terrestrial objects is more like
that of the sky. The clearness and the pure colours
of Italian landscapes depend mainly on this fact. On
high mountains, particularly in the morning, the
aerial turbidity is often so slight that the colours of
the most distant objects can scarcely be distinguished
from those of the nearest. The sky may then appear
almost bluish-black.

Conversely, the denser turbidity consists mainly of
coarser particles, and is therefore whitish. As a rule,
this is the case in the lower layers of air, and in states
of weather in which the aqueous vapour in the air is
near its point of condensation.

On the other hand, the light which reaches the
eye of the observer after having passed through a long
layer of air, has been robbed of part of its violet and
blue by scattered reflections ; it therefore appears yel-
lowish to reddish-yellow or red, the former when the
turbidity is fine, the latter when it is coarse. Thus
the sun and the moon at their rising and setting, and
also distant brightly illuminated mountain-tops, espe-
cially siiow-mountains, appear coloured.

92 ON THE RELATION OF OPTICS TO PAINTING.

These colourations are moreover not peculiar to
the air, but occur in all cases in which a transparent
substance is made turbid by the admixture of another
transparent substance. We see it, as we have ob-
served, in diluted milk, and in water to which a few
drops of eau de Cologne have been added, whereby the
ethereal oils and resins dissolved by the latter, sepa-
rate out and produce the turbidity. Excessively fine
blue clouds, bluer even than the air, may be produced,
as Tyndall has observed, when the sun's light is
allowed to exert its decomposing action on the vapours
of certain carbon compounds. Groethe called attention
to the universality of this phenomenon, and ende*-
voured to base upon it his theory of colour.

By aerial perspective we understand the artistic
representation of aerial turbidity; for the greater or
less predominance of the aerial colour above the colour
of the objects, shows their varying distance very
definitely ; and landscapes more especially acquire the
appearance of depth. According to the weather, the
turbidity of the air may be greater or less, more white
or more blue. Very clear air, as sometimes met with
after continued rain, makes the distant mountains
appear small and near ; whereas, when the air contains
more vapour, they appear large and distant.

This latter is decidedly better for the landscape
painter, and the high transparent landscapes of moun-

ON THE RELATION OF OPTICS TO PAINTING. 93

tainous regions, which so often lead the Alpine climber
to under-estimate the distance and the magnitude of
the mountain-tops before him, are also difficult to turn
to account in a picturesque manner. Views from the
valleys, and from seas and plains in which the aerial
light is faintly but markedly developed, are far better ;
not only do they allow the various distances and mag-
nitudes of what is seen to stand out, but they are on
the other hand favourable to the artistic unity of
colouration.

Although aerial colour is most distinct in the
greater depths of landscape, it is not entirely wanting
in front of the near objects of a room. What is seen
to be isolated and well denned, when sunlight passes
into a dark room through a hole in the shutter, is also
not quite wanting when the whole room is lighted.
Here, also, the aerial lighting must stand out against
the background, and must somewhat deaden the
colours in comparison with those of nearer objects; and
these differences, also, although far more delicate than
against the background of a landscape, are important
for the historical, genre, or portrait painter ; and when
they are carefully observed and imitated, they greatly
heighten the distinctness of his representation.

94 ON THE RELATION OF OPTICS TO PAINTING.

II. SHADE.

The circumstances which we have hitherto dis-
cussed indicate a profound difference, and one which is
exceedingly important for the perception of solid form,
between the visual image which our eyes give, when we
stand before objects, and that which the picture gives.
The choice of the objects to be represented in pictures
is thereby at once much restricted. Artists are well
aware that there is much which cannot be represented
by the means at their disposal. Part of their artistic
skill consists in the fact that by a suitable grouping,
position, and turn of the objects, by a suitable choice
of the point of view, and by the mode of lighting,
they learn to overcome the unfavourable conditions
which are imposed on them in this respect.

It might at first sight appear that of the requisite
truth to nature of a picture, so much would remain
that, seen from the proper point of view, it would at
least produce the same distribution of light, colour,
and shadow in its field of view, and would produce in
the interior of the eye exactly the same image on the
retina as the object represented would do if we had it
actually before us, and looked at it from a definite,

ON THE EELATION OF OPTICS TO PAINTING. 95

fixed point of- view. It might seem to be an object
of pictorial skill to aim at producing, under the given
limitations, the same effect as is produced by the
object itself. /

If we proceed to examine whether, and how far,
painting can satisfy such a condition, we come upon
difficulties before which we should perhaps shrink, if
we did not know that they had been already over-
come.

Let us begin with the simplest case ; with the quan-
titative relations between luminous intensities. If the
artist is to imitate exactly the impression which the
object produces on our eye, he ought to be able to
dispose of brightness and darkness equal to that which
nature offers. But of this there can be no idea. Let
me give a case in point. Let there be, in a pic-
ture-gallery, a desert-scene, in which a procession of
Bedouins, shrouded in white, and of dark negroes,
marches under the burning sunshine; close to it a
bluish moonlight scene, where the moon is reflected in
the water, and groups of trees, and human forms, are
seen to be faintly indicated in the darkness. You
know from experience that both pictures, if they
are well done, can produce with surprising vividness
the representation of their objects; and yet, in both
pictures, the brightest parts are produced with the
same white-lead, which is but slightly altered by ad-

96 ON THE KELATION OF OPTICS TO PAINTING.

mixtures ; while the darkest parts are produced with
the same black. Both, being hung on the same wall,
share the same light, and the brightest as well as the
darkest parts of the two scarcely differ as concerns
the degree of their brightness.

How is it, however, with the actual degrees of
brightness represented? The relation between the
brightness of the sun's light, and that of the moon,
was measured by Wollaston, who compared their in-
tensities with that of the light of candles of the same
material. He thus found that the luminosity of the
sun is 800,000 times that of the brightest light of a
full moon.

An opaque body, which is lighted from any source
whatever, can, even in the most favourable case, only
emit as much light as falls upon it. Yet, from Lam-
bert's observations, even the whitest bodies only reflect
about two fifths of the incident light. The sun's rays,
which proceed parallel from the sun, whose diameter
is 85,000 miles, when they reach us, are distributed
uniformly over a sphere 195 millions of miles in dia-
meter. Its density and illuminating power is here
only the one forty-thousandth of that with which it
left the sun's surface ; and Lambert's number leads to
the conclusion that even the brightest white surface
on which the sun's rays fall vertically, has only the
one hundred- thousandth part of the brightness of the

ON THE KELAT10N OF OPTICS TO PAINTING. 97

sun's disk. The moon however is a gray body, whose
mean brightness is only about one fifth of that of the
purest white.

And when the moon irradiates a body of the purest
white on the earth, its brightness is only the hundred-
thousandth part of the brightness of the moon itself ;
hence the sun's disk is 80,000 million times brighter
than a white which is irradiated by the full moon.

Now pictures which hang in a room are not lighted
by the direct light of the sun, but by that which is re-
flected from the sky and clouds. I do not know of ^iny
direct measurements of the ordinary brightness of the
light in a picture gallery, but estimates may be made
from known data. With strong upper light and bright
light from the clouds, the brightest white on a picture
has probably l-20th of the brightness of white directly
lighted by the sun ; it will generally be only l-40th, or
even less.

Hence the painter of the desert, even if he gives
up the representation of the sun's disk, which is always
very imperfect, will have to represent the glaringly
lighted garments of his Bedouins with a white which,
in the most favourable case, shows only the 1-2 Oth part
of the brightness which corresponds to actual fact. If
he could bring it, with its lighting unchanged, into the
desert near the white there, it would seem like a dark
grey. 1 found in fact, by an experiment, tha.t lamp-

H. ii

98 ON THE RELATION OF OPTICS TO PAIXTINIK

black, lighted by the sun, is not less than balf as
bright, as shaded white in the brighter part of a
room.

On the picture of the moon, the same white which
has been used for depicting the Bedouins' garments
must be used for representing the moon's disk, and its
reflection in the water ; although the real moon ha a
only one fifth of this brightness, and its reflection in
water still less. Hence white garments in moonlight,
or marble surfaces, even when the artist gives them a
grey shade, will always be ten to twenty times as bright
in his picture as they are in reality.

On the other hand, the darkest black which the
artist could apply would be scarcely sufficient to repre-
sent the real illumination of a white object on which
the moon shone. For even the deadest black coatings
of lamp-black, black velvet, when powerfully lighted
appear grey, as we often enough know to our cost, when
we wish to shut off superfluous light. I investigated
a coating of lamp-black, and found its brightness to
be about ^fa that of white paper. The brightest
colours of a painter are only about one hundred times
as bright as his darkest shades.

The statements I have made may perhaps appear
exaggerated. But they depend upon measurements,
and you can control them by well-known observations.
According to Wollaston, the light of the full moon is

ON THE EELATION OF OPTICS TO PAINTING. 99

equal to that % of a candle burning at a distance of 12
feet. You know that we cannot read by the light of the
full moon, though we can read at a distance of three or
four feet from a candle. Now assume that you suddenly
passed from a room in daylight to a vault perfectly
dark, with the exception of the light of a single candle.
You would at first think you were in absolute darkness,
and at most you would only recognise the candle itself.
In any case, you would not recognise the slightest trace
of any objects at a distance of 12 feet from the candle.
These however are the objects whose illumination is
the same as that which the moonlight gives. You
would only become accustomed to the darkness after
some time, and you would then find your way about
without difficulty.

If, now, you return to the daylight, which before
was perfectly comfortable, it will appear so dazzling that
you will perhaps have to close the eyes, and only be
able to gaze round with a painful glare. You see
thus that we are concerned here not with minute, but
with colossal, differences. How now is it possible that,
under such circumstances, we can imagine there is any
similarity between the picture and reality ?

Our discussion of what we did not see at first, but
could afterwards see in the vault, points to the most
important element in the solution ; it is the varying
extent to which our senses are deadened by light ; a

E 2

100 ON THE RELATION OF OPTICS TO PAINTING.

process to which we can attach the same name, fatigue,
as that for the corresponding one in the muscle. Any
activity of our nervous system diminishes its power for
the time being. The muscle is tired by work, the
brain is tired by thinking, and by mental operations ;
the eye is tired by light, and the more so the more
powerful the light. Fatigue makes it dull and in-
sensitive to new impressions, so that it appreciates
strong ones only moderately, and weak ones not at all.
But now you see how different is the aim of the
artist when these circumstances are taken into account.
The eye of the traveller in the desert, who is looking
at the caravan, has been dulled to the last degree by the
dazzling sunshine ; while that of the wanderer by moon-
light has been raised to the extreme of sensitiveness.
The condition of one who is looking at a picture
differs from both the above cases by possessing a cer-
tain mean degree of sensitiveness. Accordingly, the
painter must endeavour to produce by his colours, on
the moderately sensitive eye of the spectator, the same
impression as that which the desert, on the one hand,
produces on the deadened, and the moonlight, on the
other hand, creates on the untired eye of its observer.
Hence, along with the actual luminous phenomena of
the outer world, the different physiological conditions
of the eye play a most important part in the work of
the artist. What he has to give is not a mere tran-

ON THE RELATION OF OPTICS TO PAINTING. 101

script of the object, but a translation of his impression
into another scale of sensitiveness, which belongs to a
different degree of impressibility of the observing eye,
in which the organ speaks a very different dialect in
responding to the impressions of the outer world.

In order to understand to what conclusions this
leads, I must first of all explain the law which Fechner
discovered for the scale of sensitiveness of the eye,
which is a particular case of the more general psycho-
physical law of the relations of the various sensuous
impressions to the irritations which produce them. This
law may be expressed as follows: Within very wide
limits of brightness, differences in the strength of light
are equally distinct or appear equal in sensation, if
they form an equal fraction of the total quantity of
light compared. Thus, for instance, differences in in-
tensity of one hundredth of the total amount can be
recognised without great trouble with very different
strengths of light, without exhibiting material dif-
ferences in the certainty and facility of the estimate,
whether the brightest daylight or the light of a good
candle be used.

The easiest method of producing accurately mea-
surable differences in the brightness of two white
surfaces, depends on the use of rapidly rotating disks.
If a disk, like the adjacent one in Fig. 3, is made to
rotate very rapidly (that is, 20 to 30 times in a second),

102 ON THE RELATION OF OPTICS TO PAINTING.

it appears to the eye to be covered with three grey
rings as in Fig. 4. The reader must, however, figure
to himself the grey of these rings, as it appears on

FIG. 3. FIG. 4.

the rotating disk of Fig. 3, as a scarcely perceptible
shade of the ground. When the rotation is rapid
each ring of the disk appears illuminated, as if all the
light which fell upon it had been uniformly distributed
over its entire surface. Those rings, in which are the
black bands, have somewhat less light than the quite
white ones, and if the breadth of the marks is com-
pared with the length of half the circumference of the
corresponding ring, we get the fraction by which the
intensity of the light in the white ground of the disk is
diminished in the ring in question. If the bands are all
equally broad, as in Fig. 3, the inner rings appear darker
than the outer ones, for in this latter case the same
loss of light is distributed over a larger area than in
the former. In this way extremely delicate shades of

ON THE KELATION 0*F OPTICS TO PAINTING. 103

brightness may be obtained, and by this method, when
the strength of the illumination varies, the brightness
always diminishes by the same proportion of its total
value. Now it is found, in accordance with Fechners
law, that the distinctness of the rings is nearly con-
stant for very different strengths of light. We ex-
clude, of course, the cases of too dazzling or of too dim
a light. In both cases the finer distinctions can no
longer be perceived by the eye.

The case is quite different when for different
strengths of illumination we produce differences which
always correspond to the same quantity of light. If,
for instance, we close the shutter of a room at daytime,
so that it is quite dark, and now light it by a candle,
we can discriminate without difficulty the shadows, such
as that of the hand, thrown by the candle on a sheet
of white paper. If, however, the shutters are again
opened, so that daylight enters the room, for the same
position of the hand we can no longer recognise the sha-
dow, although there falls on that part of the white sheet,
which is not struck by this shadow, . the same excess
of candle-light as upon the parts shaded by the hand.
But this small quantity of light disappears in compari-
son with the newly added daylight, provided that this
strikes all parts of the white sheet uniformly. You
see then that, while the difference between candle-light
and darkness can be easily perceived, the equally great

J04 ON THE EELATiON OF OPTICS TO PAINTING.

difference between daylight, on the one hand, and day
light plus candle-light on the other, can be no longei
recognised.

This law is of great importance in discriminating
between various degrees of brightness of natural objects.
A white body appears white because it reflects a large
fraction, and a grey body appears grey because it re-
flects a small fraction, of incident light. For different
intensities of illumination, the difference of brightness
between the two will always correspond to the same frac-
tion of their total brightness, and hence will be equally
perceptible to our eyes, provided we do not approach too
near to the upper or the lower limit of the brightness,
for which Fechner's law no longer holds. Hence, on
the whole, the painter can produce what appears an equal
difference for the spectator of his picture, notwithstand-
ing the varying strength of light in the gallery, provided
he gives to his colours the same ratio of brightness as
that which actually exists.

For, in fact, in looking at natural objects, the abso-
lute brightness in which they appear to the eye varies
within very wide limits, according to the intensity of
the light, and the sensitiveness of the eye. That which
is constant is only the ratio of the brightness in which
surfaces of various depth of colour appear to us when
lighted to the same amount. But this ratio of bright-
ness is for us the perception, from which we form our

ON THE RELATION OF OPTICS TO TAINTING. 100

judgment as Ho the lighter or darker colour of the
bodies we see. Now this ratio can be imitated by the
painter without restraint, and in conformity with na-
ture, to evoke in us the same conception as to the
nature of the bodies seen. A truthful imitation in this
respect would be attained within the limits in which
Fechner's law holds, if the artist reproduced the fully
lighted parts of the objects which he has to represent
with pigments, which, with the same light, were equal
to the colours to be represented. This is approximately
the case. On the whole, the painter chooses coloured
pigments which almost exactly reproduce the colours of
the bodies represented, especially for objects of no great
depth, such as portraits, and which are only darker in
the shaded parts. Children begin to paint on this
principle; they imitate one colour by another; and,
in like manner also, nations in which painting has
remained in a childish stage. Perfect artistic painting
is only reached when we have succeeded in imitating
the action of light upon the eye, and not merely the
pigments; and only when we look at the object of
pictorial representation from this point of view, will it
be possible to understand the variations from nature
which artists have to make in the choice of their scale
of colour and of shade.

These are, in the first case, due to the circumstance
that Fechner's law only holds for mean degrees of

106 ON THE RELATION OF OPTICS TO PAINTING.

brightness ; while, for a brightness which is too high
or too low, appreciable divergences are met with.

At both extremes of luminous intensity the eye is
less sensitive for differences in light than is required by
that law. With a very strong light it is dazzled ; that
is, its internal activity cannot keep pace with the ex-
ternal excitations ; the nerves are too soon tired. Very
bright objects appear almost always to be equally
bright, even when there are, in fact, material differ-
ences in their luminous intensity. The light at the
edge of the sun is only about half as bright as that at
the centre, yet none of you will have noticed that, if
you have not looked through coloured glasses, which
reduce the brightness to a convenient extent. With
a weak light the eye is also less sensitive, but from the
opposite reason. If a body is so feebly illuminated
that we scarcely perceive it, we shall not be able to
perceive that its brightness is lessened by a shadow
by the one hundredth or even by a tenth.

It follows from this, that, with moderate illumina-
tion, darker objects become more like the darkest
objects, while with greater illumination brighter ob-
jects become more like the brightest than should be
the case in accordance with Fechner's law, which
holds for mean degrees of illumination. From this
results, what, for painting, is an extremely characteristic

ON THE RELATION OF OPTICS TO PAINTING. 107

V

difference between the impression of very powerful
and very feeble illumination.

When painters wish to represent glowing sunshine,
they make all objects almost equally bright, and thus
produce with their moderately bright colours the im-
pression which the sun's glow makes upon the dazzled
eye of the observer. If, on the contrary, they wish
to represent moonshine, they only indicate the very
brightest objects, particularly the reflection of moon-
light on shining surfaces, and keep everything so dark
as to be almost unrecognisable ; that is to say, they
make all dark objects more like the deepest dark
which they can produce with their colours, than should
be the case in accordance with the true ratio of the
luminosities. In both cases they express, by their
gradation of the lights, the insensitiveness of the eye
for differences of tco bright or too feeble lights. If
they could employ the colour of the dazzling bright-
ness of full sunshine, or of the actual dimness of
moonlight, they would not need to represent the
gradation of light in their picture other than it is in
nature; the picture would then make the same im-
pression on the eye as is produced by equal degrees of
brightness of actual objects. The alteration in the
scale of shade which has been described is necessary
because the colours of the picture are seen in the
mean brightness of a moderately lighted room, for

108 ON THE RELATION OF OPTICS TO PAINTING.

which Fechners law holds ; and therewith objects are
to be represented whose brightness is beyond the
limits of this law.

We find that the older masters, and pre-eminently
Rembrandt, employ the same deviation, which corre-
sponds to that actually seen in moonlight landscapes ;
and this in cases in which it is by no means wished to
produce the impression of moonshine, or of a similar
feeble light. The brightest parts of the objects are
given in these pictures in bright, luminous yellowish
colours ; but the shades towards the black are made
very marked, so that the darker objects are almost lost
in an impermeable darkness. But this darkness is
covered with the yellowish haze of powerfully lighted
aerial masses, so that, notwithstanding their darkness,
these pictures give the impression of sunlight, and the
very marked gradation of the shadows, the contours of
the faces and figures, are made extremely prominent.
The deviation from strict truth to nature is very re-
markable in this shading, and yet these pictures give
particularly bright and vivid aspects of the objects.
Hence they are of particular interest for understand-
ing the principles of pictorial illumination.

In order to explain these actions we must, I think,
consider that while Fechner's law is approximately cor-
rect for those mean lights which are agreeable to the eye,
the deviations which are so marked, for too high or toe

ON THE EELATIOX OF OPTICS TO PAINTING. 109

low lights, are not without some influence in the region
of the middle lights. We have to observe more closely
in order to perceive this influence. It is found, in fact,
that when the very finest differences of shade are pro-
duced on a rotating disk, they are only visible by a
light which about corresponds to the illumination of a
white paper on a bright day, which is lighted by the
light of the sky, but is not directly struck by the
sun. With such a light, shades of y^- or y^- of
the total intensity can be recognised. The light in
which pictures are looked at is, on the contrary, much
feebler ; and if we are to retain the same distinctness
of the finest shadows and of the modelling of the
contours which it produces, the gradations of shade
in the picture must be somewhat stronger than cor-
responds to the exact luminous intensities. The
darkest objects of the picture thereby become un-
naturally dark, which is however not detrimental to
the object of the artist if the attention of the observer
is to be directed to the brighter parts. The great
artistic effectiveness of this manner shows us that the
chief emphasis is to be laid on imitating difference of
brightness and not on absolute brightness ; and that the
greatest differences in this latter respect can be borne
without perceptible incongruity, if only their grada-
tions are imitated with expression.

110 ON THE KELATION OF OPTICS TO PAINTING.

III. COLOUR.

With these divergences in brightness are connected
certain divergences in colour, which, physiologically,
are caused by the fact that the scale of sensitiveness
is different for different colours. The strength of the
sensation produced by light of a particular colour, and
for a given intensity of light, depends altogether on
the special reaction of that complex of nerves which
are set in operation by the action of the light in
question. Now all our sensations of colour are ad-
mixtures of three simple sensations ; namely, of red,
green, and violet,1 which, by a not improbable suppo-
sition of Thomas Young, can be apprehended quite
independently of each other by three different systems
of nerve-fibres. To this independence of the different
sensations of colour corresponds their independence in
the gradation of intensity. Recent measurements 2
have shown that the sensitiveness of our eye for feeble
shadows is greatest in the blue and least in the
red. A difference of -^-^g- to -^-Q of the intensity
can be observed in the blue, and with an untired eye

1 Helmholtz's Popular Scientific Lectures, pp. 232-52.
* Dobrowolsky in Gracfe's Arckiv fur Oj)htlialmologie, vol. xviii.
part i. pp. 24-92,

ON THE RELATION OF OPTICS TO PAINTING. Ill

of ^ in the red ; or when the colour is dimmed
by being looked at for a long time, a difference of
sVto^V-

Ked therefore acts as a colour towards whose shades
the eye i? relatively less sensitive than towards that of
blue. In agreement with this, the impression of glare,
as the intensity increases, is feebler in red than in
blue. According to an observation of Dove, if a blue
and a red paper be chosen which appear of equal
brightness under a mean degree of white light, as the
light is made much dimmer the blue appears brighter,
and as the light is much strengthened, the red. I
myself have found that the same differences are seen,
and even in a more striking manner, in the red and
violet spectral colours, and, when their intensity is
increased only moderately, by the same fraction for
both.

Now the impression of white is made up of the
impressions which the individual spectral colours make
on our eye. If we increase the brightness of white,
the strength of the sensation for the red and yellow
rays will relatively be more increased than that for
the blue and violet. In bright white, therefore, the
former will produce a relatively stronger impression
than the latter; in dull white the blue and bluish
colours will have this effect. Very bright white appears
therefore yellowish, and dull white appears bluish. In

112 ON THE RELATION OF OPTICS TO PAINTING.

our ordinary way of looking at the objects about us,
we are not so readily conscious of this ; for the direct
comparison of colours of very different shade is diffi-
cult, and we are accustomed to see in this alteration in
the white the result of different illumination of one and
the same white object, so that in judging pigment-
colours we have learnt to eliminate the influence of
brightness.

If however to the painter is put the problem of imi-
tating, with faint colours, white irradiated by the sun,
he can attain a high degree of resemblance ; for by an
admixture of yellow in his white he makes this colour
preponderate just as it would preponderate in actual
bright light, owing to the impression on the nerves.
It is the same impression as that produced if we look
at a clouded landscape through a yellow glass, and
thereby give it the appearance of a sunny light. The
artist will, on the contrary, give a bluish tint to moon-
light, that is, a faint white ; for the colours on the
picture must, as we have seen, be far brighter than
the colour to be represented. In moonshine scarcely
any other colour can be recognised than blue ; the
blue starry sky or blue colours may still appear
distinctly coloured, while yellow and red can only be
seen as obscurations of the general bluish white or

I wiil again remind you that these changes of

ON TIIE RELATION OF OPTICS TO PAINTING. 113

colour TV o aid not be necessary if the artist 'had at his
disposal colours of the same brightness, or the same
faintness, as are actually shown by the bodies irradiated
by the sun or by the moon.

The change of colour, like the scale of shade, pre-
viously discussed, is a subjective action which the
artist must represent objectively on his canvas, since
moderately bright colours cannot produce them.

We observe something quite similar in regard to
the phenomena of Contrast. By this term we under-
stand cases in which the colour or brightness of a
surface appears changed by the proximity of a mass of
another colour or shade, and, in such a manner, that
the original colour appears darker by the proximity of
a brighter shade, and brighter by that of a darker
shade ; while by a colour of a different kind it tends
towards the complementary tint.

The phenomena of contrast are very various, and
depend on different causes. One class, ChevreuUs simul-
taneous Contrast, is independent of the motions of the
eyes, and occurs with surfaces where there are very
slight differences in colour and shade. This contrast
appears both on the picture and in actual objects, and
is well known to painters. Their mixtures of colours
on the palette often appear quite different to what
they are on the picture. The changes of colour which
are here met with are often very striking ; I will not,
II. 1

114 ON THE EELATION OF OPTICS TO PAINTING.

however, enter upon them, for they produce no diver-
gence between the picture and reality.

The second class of phenomena of contrast, and one
which, for us, is more important, is met with in
\x changes of direction of the glance, and more especially
between surfaces in which there are great differences
of shade and of colour. As the eye glides over bright
and dark, or coloured objects and surfaces, the impres-
sion of each colour changes, for it is depicted on por-
tions of the retina which directly before were struck
by other colours and lights, and were therefore changed
in their sensitiveness to an impression. This kind of
contrast is therefore essentially dependent on move-
ments of the eye, and has been called by Chevreul,
'successive Contrast.9

We have already seen that the retina is more sen-
sitive in the dark to feeble light than it was before.
By strong light, on the contrary, it is dulled, and is
less sensitive to feeble lights which it had before per-
ceived. This latter process is designated as < Fatigue '
of the retina ; an exhaustion of the capability of the
retina by its own activity, just as the muscles by their
activity become tired.

I must here remark that the fatigue of the
retina by light does not necessarily extend to the
whole surface ; but when only a small portion of this
membrane is struck by a minute, defined picture it

also be locally developed in this part only.

ON THE RELATION OF OPTICS TO PAINTING. 115

You must all have observed the dark spots which
move about in the field of vision, when we have been
looking for only a short time towards the setting sun,
and which physiologists call negative after-images of
the sun. They are due to the fact that only those parts
of the retina which are actually struck by the image of
the sun in the eye, have become insensitive to a new
impression of light. If, with an eye which is thus
locally tired, we look towards a uniformly bright sur-
face, such as the sky, the tired parts of the retina are
more feebly and more darkly affected than the other
portions, so that the observer thinks he sees dark spots
in the sky, which move about with his sight. We
have then in juxtaposition, in the bright parts of
the sky, the impression which these make upon the
untired parts of the retina, and in the dark spots
their action on the tired portions. Objects, bright
like the sun, produce negative after-images In the
most striking manner; but with a little attention they
may be seen even after much more moderate impres-
sions of light. A longer time is required in order to de-
velop such an impression, so that it may be distinctly
recognised, and a definite point of the bright object
must be fixed, without moving the eye, so that its image
may be distinctly formed on the retina, and only a
limited portion of the retina be excited and tired,
just as in producing sharp photographic portraits, the

i 2

116 ON TEE EELATION OF OPTICS TO PAINTING.

object must be stationary during the time of exposure
in order that its image may not be displaced on the
sensitive plate. The after-image in the eye is, as it
were, a photograph on the retina, which becomes
visible owing to the altered sensitiveness towards
fresh light, but only remains stationary for a short
time ; it is longer, the more powerful and durable was
the action of light.

If the object viewed was coloured, for instance
red paper, the after-image is of the complementary
colour on a grey ground ; in this case of a bluish green.1
Kose-red paper, on the contrary, gives a pure green
after-image, green a rose-red, blue a yellow, and
yellow a blue. These phenomena show that in the
retina partial fatigue is possible for the several
colours. According to Thomas Young's hypothesis of
the existence of three systems of fibres in the visual
nerves,2 of which one set perceives red whatever the kind
of irritation, the second green, and the third violet,
with green light, only those fibres of the retina which
are sensitive to green are powerfully excited and tired.

1 In order to see this kind of image as distinctly as possible, it
is desirable to avoid all movements of the eye. On a large sheet of
dark grey paper a small black cross is drawn, the centre of which is
steadily viewed, and a quadrangular sheet of paper of that colour
whose after-image is to be observed is slid from the side, so that one
of its corners touches the cross. The sheet is allowed to remain for
a mmute or two, the cross being steadily viewed, and it is then
drawn snddenly away, without relaxing the view. In place of the
sheet removed the after-image appears then on the dark ground.

2 See Helmholtz's Popular Lectures, first series, p. 250.

ON THE RELATION OF OPTICS TO PAINTING. 117

If this same part of the, retina is afterwards illuminated
with white light, the sensation of green is enfeebled,
while that of red and violet is vivid and predominant ;
their sum gives the sensation of purple, which mixed
with the unchanged white ground forms rose-red.

In the ordinary way of looking at light and coloured
objects, we are not accustomed to fix continuously one
and the same point ; for following with the gaze the
play of our attentiveness, we are always turning it to
new parts of the object as they happen to interest us.
This way of looking, in which the eye is continually
moving, and therefore the retinal image is also shift-
ing about on the retina, has moreover the advantage
of avoiding disturbances of sight, which powerful and
continuous after-images would bring with them. Yet
here also, after-images are not wanting ; only they are
shadowy in their contours, and of very short duration.

If a red surface be laid upon a grey ground, and if
we look from the red over the edge towards the grey,
the edges of the grey will seem as if struck by such an
after-image of red, and will seem to be of a faint
bluish green. But as the after-image rapidly disappears,
it is mostly only those parts of the grey, which are nearest
the red, which show the change in a marked degree.

This also is a phenomenon which is produced more
strongly by bright light and brilliant saturated colours
than by fainter light and duller colours. The artist

118 ON THE RELATION OF OPTICS TO PAINTING.

However, works for the most part with the latter. He
produces most of his tints by mixture; each mixed
pigment is, however, greyer and duller than the pure
colour of which it is mixed, and even the few pig-
ments of a highly saturated shade, which oil-painting
can employ, are comparatively dark. The pigments
employed in water-colours and coloured chalks are
again comparatively white. Hence such bright con-
trasts, as are observed in strongly coloured and strongly
lighted objects in nature, cannot be expected from
their representation in the picture. If, therefore,
with the pigments at his command, the artist wishes
to reproduce the impression which objects give, as
strikingly as possible, he must paint the contrasts
which they produce. If the colours on the picture
are as brilliant and luminous as in the actual objects,
the contrasts in the former case would produce them-
selves as spontaneously as in the latter. Here, also,
subjective phenomena of the eye must be objectively
introduced into the picture, because the scale of colour
and of brightness is different upon the latter.

With a little attention you will see that painters
and draughtsmen generally make a plain uniformly
lighted surface brighter, where it is close to a dark
object, and darker, where it is near a light object.
You will find that uniform grey surfaces are given
a yellowish tint at the edge where there is a back-

ON THE KELAIION OF OPTICS TO PAINTING. 119

ground of blue, and a rose-red tint where they im-
piiige on green, provided that none of the light
collected from the blue or green can fall upon the
grey. Where the sun's rays passing through the green
leafy shade of trees strike against the ground, they
appear to the eye, tired with looking at the predomi-
nant green, of a rose-red tint; the whole daylight,
entering through a slit, appears blue, compared with
reddish yellow candle-light. In this way they are re-
presented by the painter, since the colours of his pic-
tures are not bright enough to produce the contrast
without such help.

To the series of subjective phenomena, which
artists are compelled to represent objectively in their
pictures, must be associated certain phenomena of
irradiation. By this is understood cases in which
any brig at object in the field spreads its light or
colour over the neighbourhood. The phenomena are
the more marked the brighter is the radiating object,
and the halo is brightest in the immediate neighbour-
hood of the bright object, but diminishes at a greater
distance. These phenomena of irradiation are most
striking around a very bright light on a dark ground.
If the view of the flame itself is closed by a narrow
dark object such as the finger, a bright misty halo dis-
appears, which covers the whole neighbourhood, and, at
the same time, any objects there may be in the dark

J20 ON THE KELATION OF OPTICS TO PAINTING.

part of the field of view are seen more distinctly . If
the flame is partly screened by a ruler, this appears
jagged where the flame projects beyond it. The lu->
minosity in the neighbourhood of the flame is so in-
tense, that its brightness can scarcely be distinguished
from that of the flame itself; as is the case with all
bright objects, the flame appears magnified, and as if
spreading over towards the adjacent dark objects.

The cause of this phenomenon is quite similar to
that of aerial perspective. It is due to a diffusion of
light which arises from the passage of light through
dull media, excepting that for the phenomena of aerial
perspective the turbidity is to be sought in the air in
front of the eye, while for true phenomena of irradiation
it is to be sought in the transparent media of the eye.
\Vhen even the healthiest human eye is examined by
powerful light, the best being a pencil of sunlight
concentrated on the side by a condensing lens, it is
seen that the sclerotica and crystalline lens ar« not per-
fectly clear. If strongly illuminated, they both appear
whitish and as if rendered turbid by a fine mist. Both
are, in fact, tissues of fibrous structure, and are not
therefore so homogeneous as a pure liquid or a pure crys-
tal. Every inequality, however small, in the structure
of a transparent body can, however, reflect some of the
incident light— that is, can diffuse it in all directions.'
1 I disregard here the view that irradiation in the eye depends on

ON THE RELATION OF OPTICS TO TAINTING. 121

The phenomena of irradiation also occur with
moderate degrees of brightness. A dark aperture
in a sheet of paper illuminated by the sun, or .a small
dark object on a coloured glass plate which is held
against the clear sky, appear as if the colour of the
adjacent surface were diffused over them.

Hence the phenomena of irradiation are very similar
to those which produce the opacity of the air. The
only essential difference lies in this, that the opacity
by luminous air is stronger before distant objects which
have a greater mass of air in front of them than before
near ones ; while irradiation in the eye sheds its halo
uniformly over near and over distant objects.

Irradiation also belongs to the subjective pheno-
mena, of the eye which the artist represents objectively,
because painted lights and painted sunlight are not
bright enough to produce a distinct irradiation in the
eye of the observer.

The representation which the painter has to give
of the lights and colours of his object I have described
as a translation, and I have urged that, as a general
rule, it cannot give a copy true in all its details. The
altered scale of brightness which the artist must
apply in many cases is opposed to this. It is not the
colours of the objects, but the impression which they

a, diffusion of the excitation in the substance of the nerves, as this
appears to me too hj-pothetical. Moreover, we are here concerned
jvith the phenomena and not with their cause.

122 ON THE RELATION OF OPTICS TO PAIXTINO.

have given, or would give, which is to be imitated, so
as to produce as distinct and vivid a conception as pos-
sible of those objects. As the painter must change
the scale of light and colour in which he executes his
picture, he only alters something which is subject to
manifold change according to the lighting, and the
degree of fatigue of the eye. He retains the more
essential, that is, the gradations of brightness and tint.
Here present themselves a series of phenomena which
are occasioned by the manner in which the eye replies
to an external irritation ; and since they depend upon
the intensity of this irritation they are not directly
produced by the varied luminous intensity and colours
of the picture. These objective phenomena, which
occur on looking at the object, would be wanting if the
painter did not represent them objectively on his can-
vas. The fact that they are represented is particu-
larly significant for the kind of problem which is to be
solved by a pictorial representation.

Now, in all translations, the individuality of the
translator plays a part. In artistic productions many
important points are left to the choice of the artist,
which he can decide according to his individual taste,
or according to the requirements of his subject.
Within certain limits he can freely select the absolute
brightness of his colours, as well as the strength of the
shadows. Like Kembrandt, he may exaggerate them

ON THE RELATION OF OPTICS TO PAINTING. 123

in order to obtain strong relief ; or he may dimmish
them, with Fra Angelico and his modern imitators, in
order to soften earthly shadows in the representation
of sacred objects. Like the Dutch school, he may
represent the varying light of the atmosphere, now
bright and sunny, and now pale, or warm and cold,
and thereby evoke in the observer moods which
depend on the illumination and on the state of the
weather; or by means of undisturbed air he may
cause his figures to stand out objectively clear as it
were, and uninfluenced by subjective impressions. By
this means, great variety is attained in what artists call
8 style ' or ' treatment,' and indeed in their purely pic-
torial elements.

1 24 ON THE RELATION OF OPTICS TO PAINTING.

IV. HARMONY OF COLOUR.

We here naturally raise the question : If, owing to the
small quantity of light and saturation of his colours,
the artist seeks, in all kinds of indirect ways, by imi-
tating subjective impressions to attain resemblance to
nature, as close as possible, but still imperfect, would
it not be more convenient to seek for means of obvi-
ating these evils ? Such there are indeed. Frescoes
are sometimes viewed in direct sunlight ; transparen-
cies and paintings on glass cau utilise far higher
degrees of brightness, and far more saturated colours ;
in dioramas and in theatrical decorations we may
employ powerful artificial light, and, if need be, the
electric light. But when I enumerate these branches
of art, it will at once strike you that those works
which we admire as the greatest masterpieces of
painting, do not belong to this class ; but by far the
larger number of the great works of art are executed
with the comparatively dull water or oil-colours, or at
any rate for rooms with softened light. If higher
ariistic effects could be attained with colours
lighted by the sun, we should undoubtedly have pic-
tures which took advantage of this. Fresco painting

ON THE KELATION OF OPTICS TO PAINTING. 125

frould have led to this ; or the experiments of Munich's
celebrated optician Steinheil, which he made as a
matter of science, that is, to produce oil paintings
which should be looked at in bright sunshine, would
not be isolated.

Experiment seems therefore to teach, that modera-
tion of light and of colours in pictures is ever advan-
tageous, and we need only look at frescoes in direct
sunlight, such as those of the new Pinakothek in
Miinich, to learn in what this advantage consists.
Their brightness is so great that we cannot look at
them steadily for any length of time. And what in
this case is so painful and so tiring to the eye, would
also operate in a smaller degree if, in a picture, bril-
liant colours were used, even locally and to a moderate
extent, which were intended to represent bright sun-
light, and a mass of light shed over the picture.
It is much easier to produce an accurate imitation
of the feeble light of moonshine with artificial light
in dioramas and theatre decorations.

We may therefore designate truth to Nature of a
beautiful picture as an ennobled fidelity to Nature.
Such a picture reproduces all that is essential in the
impression, and attains full vividness of conception,
but without injury or tiring the eye by the nude lights
of reality. The differences between Art and Nature
are chiefly confined, as we have already seen, to those

126 ON THE RELATION OF OPTICS TO PAINTING.

matters which we can in reality only estimate in an un-
certain manner, such as the absolute intensities of light.

That which is pleasant to the senses, the beneficial
bat not exhausting fatigue of our nerves, the feeling
of comfort, corresponds in this case, as in others, to
those conditions which are most favourable for per-
ceiving the outer world, and which admit of the finest
discrimination and observation.

It has been mentioned above that the discrimina-
tion of the finest shadows, and of the modelling which
they express, is the most delicate under a certain
mean brightness. I should like to direct your atten-
tion to another point which has great importance in
painting: I refer to our natural delight in colours,
which has undoubtedly a great influence upon our
pleasure in the works of the painter. In its simplest
expression, as pleasure in gaudy flowers, feathers,
stones, in fireworks, and Bengal lights, this inclination
has but little to do with man's sense of art ; it only ap-
pears as the natural pleasure of the perceptive organism
in the varying and multifarious excitation of its various
nerves, which is necessary for its healthy continuance
and productivity. But the thorough fitness in the con-
struction of living organisms, whatever their origin,
excludes the possibility that in the majority of healthy
individuals an instinct should be developed or main-
tain itself which did not serve some definite purpose.

ON THE RELATION OF OPTICS TO PAINTING. 127

We have not far to ""seek for the delight in light
and in colours, and for the dread of darkness; this
coincides with the endeavour to see and to recognise
surrounding objects. Darkness owes the greater part
of the terror which it inspires to the fright of what
is unknown and cannot be recognised. A coloured
picture gives a far more accurate, richer, and easier
conception than a similarly executed drawing, which
only retains the contrasts of light and shade. A
picture retains the latter, but has in addition the
material for discrimination which colours afford; by
which surfaces which appear equally bright in the
drawing, owing to their different colour, are now
assigned to various objects, or again as alike in colour
are seen to be parts of the same, or of similar objects.
In utilising the relations thus naturally given, the
artist, by means of prominent colours, can direct and
enchain the attention of the observer upon the chief
objects of the picture; and by the variety of the
garments he can discriminate the figures from each
other, but complete each individual one in itself.
Even the natural pleasure in pure, strongly saturated
colours, finds its justification in this direction. The
case is analogous to that in music, with the full, pure,
well-sounding tones of a beautiful voice. Such a one
is more expressive ; that is, even the smallest change
of its pitch, or its quality — any slight interruption,

158 ON THE KELATION OF OPTICS TO PAINTING.

any tremulousness, any rising or falling in it — is at
once more distinctly recognised by the hearer than
could be the case with a less regular sound ; and it
seems also that the powerful excitation which it pro-
duces in the ear of the listener, arouses trains of ideas
and passions more strongly than does a feebler excita-
tion of the same kind. A pure, fundamental colour
bears to small admixtures the same relation as a dark
ground on which the slightest shade of light is visible.
Any of the ladies present will have known how sensi-
tive clothes of uniform saturated shades are to dirt,
in comparison with grey or greyish-brown materials.
This also corresponds to the conclusions from Young's
theory of colours. According to this theory, the per-
ception of each of the three fundamental colours
arises from the excitation of only one kind of sensitive
fibres, while the two others are at rest ; or at any rate
are but feebly excited. A brilliant, pure colour pro-
duces a powerful stimulus, and yet, at the same time,
a great degree of sensitiveness to the admixture of
other colours, in those systems of nerve-fibres which
are at rest. The modelling of a coloured surface
mainly depends upon the reflection of light of other
colours which falls upon them from without. It is
more particularly when the material glistens that the
reflections of the bright places are preferably of the
colour of the incident light. In the depth of the

ON THE RELATION OF OPTICS TO PAINTING. 129

f3lds, on the contrary, 4he coloured surface reflects
against itself, and thereby makes its own colour more
saturated. A white surface, on the contrary, of great
brightness, produces a dazzling effect, and is thereby
insensitive to slight degrees of shade. Strong colours
thus, by the powerful irritation which tt ?y produce,
can enchain the eye of the observer, and yet be ex-
pressive for the slightest change of modelling or of
illumination ; that is, they are expressive in the
artistic sense.

If, on the other hand, we coat too large surfaces,
they produce fatigue for the prominent colour, and a
diminution in sensitiveness towards it. This colour
then becomes more grey, and on all surfaces of a
different colour the complementary tint appears, espe-
cially on grey or black surfaces. Hence therefore
clothes, and more particularly curtains, which are of
too bright a single colour, produce an unsatisfactory
and fatiguing effect ; the clothes have moreover the
disadvantage for the wearer that they cover face and
hands with the complementary colour. Blue produces
yellow, violet gives greenish yellow, bright purple
gives green, scarlet gives blue, and, conversely, yellow
gives -blue, etc. There is another circumstance which
the artist has to consider, that colour is for him an
important means of attracting the attention of the
observer. To be able to do this he must be sparing in

U. K

130 ON THE RELATION OF OPTICS TO PAINTING.

the use of the pure colours, otherwise they distract
the attention, and the picture becomes glaring. It
is necessary, on the other hand, to avoid a onesided
fatigue of the eye by too prominent a colour. This is
effected either by introducing the prominent colour
to a moderate extent upon a dull, slightly coloured
ground, or by the juxtaposition of variously saturated
colours, which produce a certain equilibrium of irrita-
tion in the eye, and, by the contrast in their after-
images, strengthen and increase each other. A green
surface on which the green after-image of a purple one
falls, appears to be a far purer green than without
such an after-image. By fatigue towards purple, that
is towards red and violet, any admixture of these two
colours in the green is enfeebled, while this itself pro-
duces its full effect. In this way the sensation of A
green is purified from any foreign admixture. Even
the purest and most saturated green, which Nature
shows in the prismatic spectrum, may thus acquire a
higher degree of saturation. We find thus that the
other pairs of complementary colours, which we have
mentioned, make each other more brilliant by their
contrast, while colours which are very similar aro
detrimental to each other, and acquire a grey tint.

These relations of the colours to each other have
manifestly a great influence on the degree of pleasure
which different combinations of colours afford. Two

ON THE RELATION OF OPTICS TO PAINTING. 131

colours may, without injury, be juxtaposed, which
indeed are so similar as to look like varieties of the
same colour, produced by varying degrees of light and
shade. Thus, upon scarlet the more shaded parts ap-
pear of a carmine, or on a straw-colour they appear
of a golden yellow.

If we pass beyond these limits, we arrive at un-
pleasant combinations, such as carmine and orange, or
orange and straw-yellow. The distance of the colours
must then be increased, so as to create pleasing com-
binations once more. The complementary colours are
those which are most distant from each other. When
these are combined, such, for instance, as straw-colour
and ultramarine, or verdigris and purple, they have
something insipid but crude ; perhaps because we are
prepared to expect the second colour to appear as an
after-image of the first, and it does not sufficiently
appear to be a new and independent element in the
compound. Hence, on the whole, combinations of
those pairs are most pleasing in which the second
colour of the complementary tint is near the first,
though with a distinct difference. Thus, scarlet and
greenish blue are complementary. The combination
produced when the greenish blue is allowed to glide
either into ultramarine, or yellowish green (sap green),
is still more pleasing. In the latter case, the com-
bination tends towards yellow, and in the former,

K 2

132 ON THE RELATION OF OPTICS TO PAINTING.

towards rose-red. Still more satisfactory combinations
are those of three tints which bring about equilibrium
in the impression of colour, and, notwithstanding the
great body of colour, avoid a onesided fatigue of the
eye, without falling into the baldness of complemen-
tary tints. To this belongs the combination which
the Venetian masters used so much — red, green, and
violet; as well as Paul Veronese's purple, greenish
blue, and yellow. The former triad corresponds ap-
proximately to the three fundamental colours, in so
far as these can be produced by pigments ; the latter
gives the mixtures of each pair of fundamental colours.
It is however to be observed, that it has not yet been
possible to establish rules for the harmony of colours
with the same precision and certainty as for the con-
sonance of tones. On the contrary, a consideration of
the facts shows that a number of accessory influences
come into play,1 when once the coloured surface is
also to produce, either wholly or in part, a representa-
tion of natural objects or of solid forms, or even if it
only offers a resemblance with the representation of
a relief, of shaded and of non-shaded surfaces. It
is moreover often difficult to establish, as a matter of
fact, what are the colours which produce the harmonic
impression. This is pre-eminently the case with

1 Conf. E. Briicke, Die Physiologic der Farben fur die Zn-ecke
dcr Kunstg 'ewerbe. Leipzig, 1866. W. v. Bezold, Die Farbenlehre,
tin Hinblick auf Kunst und Kuiistgerverbe. Braunschweig, 1874.

ON THE RELATION OF OPTICS TO PAINTING. 133

pictures in which the ^aerial colour, the coloured re-
flection and shade, so variously alter the tint of each
single coloured surface when it is not perfectly smooth,
that it is hardly possible to give an indisputable de-
termination of its tint. In such cases, moreover, the
direct action of the colour upon the eye is only a
subordinate means; for, on the other hand, the
prominent colours and lights must also serve for
directing the attention to the more important points
of the representation. Compared with these more
poetical and psychological elements of the representa-
tion, considerations as to the pleasing effect of the
colours are thrown into the background. Only in the
pure ornamentation on carpets, draperies, ribbons, or
architectonic surfaces is there free scope for pure
pleasure in the colours, and only there can it develop
itself according to its own laws.

In pictures, too, there is not, as a general rule,
perfect equilibrium between the various colours, but
one of them preponderates to an extent which corre-
sponds to the dominant light. This is occasioned, in
the first case, by the truthful imitation of physical
circumstances. If the illumination is rich in yellow
light, yellow colours will appear brighter and more
brilliant than blue ones ; for yellow bodies are those
which preferably reflect yellow light ; while that of
blue is only feebly reflected, and is mainly absorbed.

134 ON THE RELATION OF OPTICS TO PAINTING.

Before the shaded parts of blue bodies, the yellow
aerial light produces its effect, and imparts to the
blue more or less of a grey tint. The same thing
happens in front of red and green, though to a less
extent, so that, in their shadows, these colours merge
into yellow. This also is closely in accordance with
the aesthetic requirements of artistic unity of compo-
sition in colour. This is caused by the fact that the
divergent colours show a relation to the predominant
colour, and point to it most distinctly in their shades.
Where this is wanting, the various colours are hard
and crude ; and, since each one calls attention to itself,
they make a motley and disturbing impression ; and,
on the other hand, a cold one, for the appearance
of a flood of light thrown over the objects is
wanting.

We have a natural type of the harmony which a
well-executed illumination of masses of air can produce
in a picture, in the light of the setting sun, which
throws over the poorest regions a flood of light and
colour, and harmoniously brightens them. The
natural reason for this increase of aerial illumination
lies in the fact, that the lower and more opaque
layers of air are in the direction of the sun, and
therefore reflect more powerfully; while at the same
time the yellowish red colour of the light which
has passed through the atmosphere becomes more dis-

ON THE RELATION OF OPTICS TO PAINTING. 135

tinct as the length of path increases which it has to
traverse, and that further, this coloration is more
pronounced as the background falls into shadow.

In summing up once more these considerations, we
have first seen what limitations are imposed on truth
to Nature in artistic representation ; how the painter
links the principal means which nature furnishes of
recognising depths in the field of view, namely binocu-
lar vision, which indeed is even turned against him,
as it shows unmistakably the flatness of the picture ;
how therefore the painter must carefully select, partly
the perspective arrangement of his subject, its posi-
tion and its aspect, and partly the lighting and
shading, in order to give us a directly intelligible
image of its magnitude, its shape, and distance, and
how a truthful representation of aerial light is one of
the most important means of attaining the object.

We then saw that even the scale of luminous
intensity, as met with in the objects, must be trans-
formed in the picture to one differing sometimes by a
hundredfold; how here, the colour of the object
cannot be simply represented by the pigment; that
indeed it is necessary to introduce important changes
in the distribution of light and dark, of yellowish and
of bluish tints.

The artist cannot transcribe Nature; he must

136 ON THE KELATION OF OPTICS TO PAIXTING.

translate her; yet this translation may give us an
impression in the highest degree distinct and forcible,
not merely of the objects themselves, but even of the
greatly altered intensities of light under which we
view them. The altered scale is indeed in many cases
advantageous, as it gets rid of everything which, in
the actual objects, is too dazzling, and too fatiguing
for the eye. Thus the imitation of Nature in the
picture is at the same time an ennobling of the im-
pression on the senses. In this respect we can often
give ourselves up more calmly and continuously, to the
consideration of a work of art, than to that of a real
object. The work of art can produce those gradations
of light, and those tints in which the modelling of the
forms is most distinct and therefore most expressive.
It can bring forward a fulness of vivid fervent colours,
and by skilful contrast can retain the sensitiveness of
the eye in advantageous equilibrium. It can fearlessly
apply the entire energy of powerful sensuous impres-
sions, and the feeling of delight associated therewith,
to direct and enchain the attention ; it can use their
variety to heighten the direct understanding of what
is represented, and yet keep the eye in a condition of
excitation most favourable and agreeable for delicate
sensuous impressions.

If, in these considerations, my having continually
laid much weight on the lightest, finest, and most

ON TEE RLLATION OF OPTICS TO PAINTING. 137

accurate sensuous intelligibility of artistic representa-
tion, may seem to many of you as a very subordinate
point — a point which, if mentioned at all by writers on
aesthetics, is treated as quite accessory — I think this
is unjustly so. The sensuous distinctness is by no
means a low or subordinate element in the action of
works of art ; its importance has forced itself the more
strongly upon me the more I have sought to discover
the physiological elements in their action.

What effect is to be produced by a work of art,
using this word in its highest sense ? It should
excite and enchain our attention, arouse in us, in easy
play, a host of slumbering conceptions and their cor-
responding feelings, and direct them towards a common
object, so as to give a vivid perception of all the fea-
tures of an ideal type, whose separate fragments lie
scattered in our imagination and overgrown by the
wild chaos of accident. It seems as if we can only
refer the frequent preponderance, in the mind, of art
over reality, to the fact that the latter mixes some-
thing foreign, disturbing, and even injurious ; while art
can collect all the elements for the desired impression,
and allow them to act without restraint. The power of
this impression will no doubt be greater the deeper,
the finer, and the truer to nature is the sensuous
impression which is to arouse the series of images
and the effects connected therewith. It must act cer-

138 ON THE KELATION OF OPTICS TO PAINTING.

tainly, rapidly, unequivocably, and with accuracy if it
is to produce a vivid and powerful impression. These
essentially are the points which I have sought to com-
prehend under the name of intelligibility of the work
of art.

Then the peculiarities of the painters' technique
(Technik), to which physiological optical investigation
have led us, are often closely connected with the highest
problems of art. We may perhaps think that even the
last secret of artistic beauty — that is, the wondrous
pleasure which we feel in its presence — is essentially
based on the feeling of an easy, harmonic, vivid stream
of our conceptions, which, in spite of manifold changes,
flow towards a common object, bring to light laws
hitherto concealed, and allow us to gaze in the deepest
depths of sensation of our own nrindg.

139

ON THE OEIGIN

OF TIIB

PLANETABY SYSTEM.

Lecture delivered in Heidelberg and in Cologne, in 1871.

IT is my intention to bring a subject before you to-day
which has been much discussed — that is, the hypothesis
of Kant and Laplace as to the formation of the celestial
bodies, and more especially of our planetary system.
The choice of the subject needs no apology. In popular
lectures, like the present, the hearers may reasonably
expect from the lecturer, that he shall bring before
them well-ascertained facts, and the complete results
of investigation, and not unripe suppositions, hypothe-
ses, or dreams.

Of all the subjects to which the thought and im-
agination of man could turn, the question as to the
origin of the world has, since remote antiquity, been
the favourite arena of the wildest speculation. Bene-

140 <JN THE ORIGIN OF TJIE PLANETARY SYSTEM.

ficent and malignant deities, giants, Kronos who
devours his children, Niflheim, with the ice-giant
Ymir, who is killed by the celestial Asas,1 that out of
him the world may be constructed — these are all figures
which fill the cosmogonic systems of the more culti-
vated of the peoples. But the universality of the fact,
that each people develops its own cosmogonies, and
sometimes in great detail, is an expression of the
interest, felt by all, in knowing what is our own origin,
what is the ultimate beginning of the things about
us. And with the question of the beginning is
closely connected that of the end of all things; for
that which may be formed, may also pass away. The
question about the end of things is perhaps of greater
practical interest than that of the beginning.

Now, I must premise that the theory which 1
intend to discuss to-day was first put forth by a man
who is known as the most abstract of philosophical
thinkers; the originator of transcendental idealism
and of the Categorical Imperative, Immanuel Kant.
The work in which he developed this, the General
Natural Philosophy and Theory of the Heavens, is one
of his first publications, having appeared in his thirty-
first year. Looking at the writings of this first period
of his scientific activity, which lasted to about his
foitieth year, we find that they belong mostly to
1 Cox's Aryan MytJwlflgy, vol. i. 372. Longmans.

ON THE OEIGIN OF THE PLANETAKY SYSTEM. 141

Natural Philosophy, and are far in advance of their
times with a number of *the happiest ideas. His
philosophical writings at this period are but few, and
partly like his introductory lecture, directly originat-
ing in some adventitious circumstance ; at the same
time the matter they contain is comparatively without
originality, and they are only important from a des-
tructive and partially sarcastic criticism. It cannot be
denied that the Kant of early life was a natural
philosopher by instinct and by inclination ; and that
probably only the power of external circumstances, the
want of the means necessary for independent scientific
research, and the tone of thought prevalent at the
time, kept him to philosophy, in which it was only
much later that he produced anything original and
important ; for the Kritik der reinen Vernunft
appeared in his fifty-seventh year. Even in the later
periods of his life, between his great philosophical
works, he wrote occasional memoirs on natural philo-
sophy, and regularly delivered a course of lectures on
physical geography. He was restricted in this to the
scanty measure of knowledge and of appliances of his
time, and of the out-of-the-way place where he lived ;
but with a large and intelligent mind he strove after
such more general points of view as Alexander von
Humboldt afterwards worked out. It is exactly an
in\ersion of the historical connection, when Kant's

142 ON THE ORIGIN OF THE PLANETARY SYSTEM.

name is occasionally misused, to recommend that
natural philosophy shall Iftave the inductive method,
by which it has become great, to revert to the windy
speculations of a so-called 'deductive method.' No
one would have attacked such a misuse, more ener-
getically and more incisively, than Kant himself
if he were still among us.

The same hypothesis as to the origin of our
planetary system was advanced a second time, but
apparently quite independently of Kant, by the most
celebrated of French astronomers, Simon, Marquis de
Laplace, It formed, as it were, the final conclusion of
his work on the mechanism of our system, executed
with such gigantic industry and great mathematical
acuteness. You see from the names of these two men,
whom we meet as experienced and tried leaders in our
course, that in a view in which they both agree, we
have not to deal with a mere random guess, but with a
careful and well-considered attempt to deduce conclu-
sions as to the unknown past from known conditions of
the present time.

It is in the nature of the case, that a hypothesis as
to the origin of the world which we inhabit, and
which deals with things in the most distant past,
cannot be verified by direct observation. It may, how-
ever, receive direct confirmation, if, in the progress of
scientific knowledge, new facts accrue to those already

ON THE ORIGIN OF THE PLANETARY SYSTEM. 143

known, and like them are explained on the hypothesis ;
and particularly if survivals of the processes, assumed
to have taken place in the formation of the heavenly
bodies, can be proved to exist in the present.

Such direct confirmations of various kinds have, in
fact, been formed for the view we are about to discuss,
and have materially increased its probability.

Partly this fact, and partly the fact that the
hypothesis in question has recently been mentioned in
popular and scientific books, in connection with philo-
sophical, ethical, and theological questions, have em-
boldened me to speak of it here. I intend not so
much to tell you anything substantially new in refer-
ence to it, as to endeavour to give, as connectedly as
possible, the reasons which have led to, and have
confirmed it.

These apologies which I must premise, only apply
to the fact that I treat a theme of this kind as a popular
lecture. Science is not only entitled, but is indeed
beholden, to make such an investigation. For her it is a
definite and important question — the question, namely,
as to the existence of limits to the validity of the laws
of nature, which rule all that now surrounds us ; the
question whether they have always held in the past,
and whether they will always hold in the future ; or
whether, on the supposition of an everlasting unifor-
mity of natural laws, our conclusions from present

144 ON THE OEIGIN OF THE PLANETAEY SYSTEM.

circumstances as to the past, and as to the future,
imperatively lead to an impossible state of things;
that is, to the necessity of an infraction of natural
laws, of a beginning which could not have been due
to processes known to us. Hence, to begin such an
investigation as to the possible or probable primeval
history of our present world, is, considered as a ques-
tion of science, no idle speculation, but a question as
to the limits of its methods, and as to the extent to
which existing laws are valid.

It may perhaps appear rash that we, restricted as
we are, in the circle of our observations in space, by our
position on this little earth, which is but as a grain of
dust in our milky way ; and limited in time by the
short duration of the human race; that we should
attempt to apply the laws which we have deduced
from the confined circle of facts open to us, to the
whole range of infinite space, and of time from
everlasting to everlasting. But all our thought and
our action, in the greatest as well as in the least,
is based on our confidence in the unchangeable order
of nature, and this confidence has hitherto been the
more justified, the deeper we have penetrated into the
interconnections of natural phenomena. And that the
general laws, which we have found, also hold for the
most distant vistas of space, has acquired strong actual
confirmation during the past half-century.

ON THE OEIGIN OF THE PLANETARY SYSTEM. 145

In the front rank of all, then, is the law of gravita-
tion. The celestial bodies, as you all know, float and
move in infinite space. Compared with the enormous
distances between them, each of us is but as a grain of
dust. The nearest fixed stars, viewed even under the
most powerful magnification, have no visible diameter;
and we may be sure that even our sun, looked at from
the nearest fixed stars, would only appear as a single
luminous point ; seeing that the masses of those stars,
in so far as they have been determined, have not been
found to be materially different from that of the sun.
But, notwithstanding these enormous distances, there
is an invisible tie between them which connects them
together, and brings them in mutual interdependence.
This is the force of gravitation, with which all heavy
masses attract each other. We know this force as
gravity, when it is operative between an earthly body
and the mass of our earth. The force which causes
a body to fall to the ground is none other than that
which continually compels the moon to accompany the
earth in its path round the sun, and which keeps the
earth itself from fleeing off into space, away from the
sun.

You may realise, by means of a simple mechanical
model, the course of planetary motion. Fasten to the
branch of a tree, at a sufficient height, or to a rigid
bar, fixed horizontally in the wall, a silk cord, and at

II. L

146 ON THE ORIGIN OF THE PLANETARY SYSTEM.

its end a small heavy body— for instance, a lead ball.
If you allow this to hang at rest, it stretches the
thread. This is the position of equilibrium of the
ball. To indicate this, and keep it visible, put in
the place of the ball any other solid body — for in-
stance, a large terrestrial globe on a stand. For this
purpose the ball must be pushed aside, but it presses
against the globe, and, if taken away, it still tends to
come back to it, because gravity impels it towards its
position of equilibrium, which is in the centre of the
sphere. And upon whatever side it is drawn, the same
thing always happens. This force, which drives the
ball towards the globe, represents in our model the
attraction which the earth exerts on the moon, or the
sun on the planets. After you have convinced your-
selves of the accuracy of these facts, try to give the
ball, when it is a little away from the globe, a slight
throw in a lateral direction. If you have accurately
hit the strength of the throw, the small ball will
move round the large one in a circular path, and may
retain this motion for some time ; just as the moon
persists in its course round the earth, or the planets
about the sun. Now, in our model, the circles
described by the lead ball will be continually narrower,
because the opposing forces, the resistance of the air,
the rigidity of the thread, friction, cannot be elimi-
nated, in this case, as they are excluded in the plane-
tary system.

ON THE ORIGIN OF TEE PLANETARY SYSTEM. 147

If the path about the attracting centre is exactly
circular, the attracting force always acts on the
planets, or on the lead sphere, with equal strength.
In this case, it is immaterial according to what law
the force would increase or dimmish at other dis-
tances from the centre in which the moving body
does not come. If the original impulse has not been
of the right strength in both cases, the paths will not

Fio. 5.

be circular but elliptical, of the form of the curved
line in Fig. 5. But these ellipses lie in both cases
differently as regards the attracting centre. In our
model, the attracting force is stronger, the further the
lead sphere is removed from its position of equilibrium.
Under these circumstances, the ellipse of the path has
such a position in reference to the attracting centre,
tha,t this is in the centre, o, of the ellipse. For planets,
on the contrary, the attracting force is feebler the

L 2

148 ON THE ORIGIN OF THE PLANETARY SYSTEM.

further it is removed from the attracting body, and
this is the reason that an ellipse is described, one of
whose foci lies in the centre of attraction. The two
foci, a and 6, are two points which lie symmetrically
towards the ends of the ellipse, and are characterised
by the property that the sum of their distances,
am+6m, is the same from any given points.

Kepler had found that the paths of the planets are
ellipses of this kind ; and since, as the above example
shows, the form and position of the orbit depend on
the law according to which the magnitude of the
attracting force alters, Newton could deduce from the
form of the planetary orbits the well-known law of the
force of gravitation, which attracts the planets to the
sun, according to which this force decreases with
increase of distance as the square of that distance.
Terrestrial gravity must obey this law, and Newton
had the wonderful self-denial to refrain from publish-
ing his important discovery until it had acquired a
direct confirmation; this followed from the observa-
tions, that the force which attracts the moon towards
the earth, bears towards the gravity of a terrestrial
body the ratio required by the above law.

In the course of the eighteenth century the power
of mathematical analysis, and the methods of astrono-
mical observation, increased so far that all the compli-
cated actions, which take place between all the planets,

ON THE ORIGIN OF THE PLANETARY SYSTEM. 149

and all their satellites, in consequence of the mutual
action of each upon each, and which astronomers caH
disturbances — disturbance, that is to say, of the simpler
elliptical motions about the sun, which each one would
produce if the others were absent — that all these
could be theoretically predicted from Newton's law,
and be accurately compared with what actually takes
place in the heavens. The development of this theory
of planetary motion in detail was, as has been said,
the merit of Laplace. The agreement between this
theory, which was developed from the simple law of
gravitation, and the extremely complicated and mani-
fold phenomena which follow therefrom, was so com-
plete and so accurate, as had never previously been
attained in any other branch of human knowledge.
Emboldened by this agreement, the next step was to
conclude that where slight defects were still constantly
found, unknown causes must be at work. Thus, from
BessePs calculation of the discrepancy between the
actual and the calculated motion of Uranus, it was
inferred that there must be another planet. The
position of this planet was calculated by Leverrier and
Adams, and thus Neptune, the most distant of all
known at that time, was discovered.

But it was not merely in the region of the attrac-
tion of our sun that the law of gravitation was found
to hold. With regard to the fixed stars, it was found

150 ON THE ORIGIN OF THE PLANETARY SYSTEM.

that double stars moved about each other in elliptical
paths, and that therefore the same law of gravitation
must hold for them as for our planetary system. The
distance of some of them could be calculated. The
nearest of them, a, in the constellation of the Centaur,
is 270,000 times further from the sun than the
earth. Light, which has a velocity of 186,000 miles
a second, which traverses the distance from the sun to
the earth in eight minutes, would take four years to
travel from a Centauri to us. The more delicate
methods of modern astronomy have made it possible
to determine distances which light would take thirty-
five years to traverre ; as, for instance, the Pole Star ;
but the law of gravitation is seen to hold, ruling the
motion of the double stars, at distances in the heavens,
which all the means we possess have hitherto utterly
failed to measure.

The knowledge of the law of gravitation has here
also led to the discovery of new bodies, as in the
case of Neptune. Peters of Altona found, confirming
therein a conjecture of Bessel, that Sirius, the most
brilliant of the fixed stars, moves in an elliptical path
about an invisible centre. This must have been due
to an unseen companion, and when the excellent and
powerful telescope of the University of Cambridge, in
the United States, had been set up, this was discovered.
It is not quite dark, but its light is so feeble that it

ON THE ORIGIN OF THE PLANETARY SYSTEM. 151

can only be seen by the most perfect instruments.
The mass of Sirius is found to be 13-76, and that of
its satellite 6'71, times the mass of the sun; their
mutual distance is equal to thirty-seven times the
radius of the earth's orbit, and is therefore somewhat
larger than the distance of Neptune from the sun.

Another fixed star, Procyon, is in the same case as
Sirius, but its satellite has not yet been discovered.

You thus see that in gravitation we have dis-
covered a property common to all matter, which is not
confined to bodies in our system, but extends, as far in
the celestial space, as our means of observation have
hitherto been able to penetrate.

But not merely is this universal property of all
mass shared by the most distant celestial bodies, as
well as by terrestrial ones ; but spectrum analysis has
taught us that a number of well-known terrestrial
elements are met with in the atmospheres of the
fixed stars, and even of the nebulae.

You all know that a fine bright line of light, seen
through a glass prism, appears as a coloured band, red
and yellow at one edge, blue and violet at the other,
and green in the middle. Such a coloured image is
called a spectrum — the rainbow is such a one, produced
by the refraction of light, though not exactly by a
prism ; and it exhibits therefore the series of colours
into which white sunlight can thus be decomposed.

152 ON THE OEIGIN OF THE PLANETAKY SYSTEM.

The formation of the prismatic spectrum depends on
the fact that the sun's light, and that of most ignited
bodies, is made up of various kinds of light, which
appear of different colours to our eyes, and the rays
of which are separated from each other when refracted
by a prism.

Now if a solid or a liquid is heated to such an
extent that it becomes incandescent, the spectrum
which its light gives is, like the rainbow, a broad
coloured band without any breaks, with the well-known
series of colours, red, yellow, green, blue, and violet,
and in no wise characteristic of the nature of the body
which emits the light.

The case is different if the light is emitted by an
ignited gas, or by an ignited vapour — that is, a sub-
stance vaporised by heat. The spectrum of such a
body consists, then, of one or more, and sometimes
even a great number, of entirely distinct bright lines,
whose position and arrangement in the spectrum is
characteristic for the substances of which the gas or
vapour consists, so that it can be ascertained, by means
of spectrum analysis, what is the chemical constitution
of the ignited gaseous body. Gaseous spectra of this
kind are shown in the heavenly space by many
nebulae; for the most part they are spectra which
show the bright line of ignited hydrogen and oxygen,
and along with it a line which, as yet, has never been

ON THE ORIGIN OF THE PLANETARY SYSTEM. 153

in found in the spectrum of any terrestrial element.
Apart from the proof of two well-known terrestrial
elements, this discovery was of the utmost importance,
since it furnished the first unmistakable proof that the
cosmical nebulae are not, for the most part, small heaps
of fine stars, but that the greater part of the light
which they emit is really due to gaseous bodies.

The gaseous spectra present a different appearance
when the gas is in front of an ignited solid whose
temperature is far higher than that of the gas. The
observer sees then a continuous spectrum of a solid,
but traversed by fine dark lines, which are just visible
in the places in which the gas alone, seen in front of
a dark background, would show bright lines. The
solar spectrum is of this kind, and also that of a great
number of fixed stars. The dark lines of the solar
spectrum, originally discovered by Wollaston, were
first investigated and measured by Fraunhofer, and are
hence known as Fraunhofer's lines.

Far more powerful apparatus was afterwards used
by Kirchhoff, and then by Angstrom, to push the de-
composition of light as far as possible. Fig. 6 re-
presents an apparatus with four prisms, constructed
by Steinheil for KirchhofF. At the further end of the
telescope B is a screen with a fine slit, represent-
ing a fine slice of light, which can be narrowed or
widened by the small screw, and by which the light

154 ON THE OEIGIN OF THE PLANETARY SYSTEM.

under investigation can be allowed to enter. It then
passes through the telescope B, afterwards through the

Fio. 6.

four prisms, and finally through the telescope A, from
which it reaches the eye of the observer. Figs. 7, 8,
Fie. ?. FIG. 8.

ON THE ORIGIN OF THE PLANETARY SYSTEM. 155

and 9 represent small portions of the solar spectrum
as mapped by Kirchoff, taken from the green, yellow,
and golden-yellow, in which the chemical symbols
below — Fe (iron), Ca (calcium), Na (sodium), Pb (lead)
• — and the affixed lines, indicate the positions in which
the vapours of these metals, when made incandescent,
either in the flames or in the electrical spark, would
FIG. 9.

show bright lines. The numbers above them show
how far these fractions of KirchhofFs map of the whole
system are apart from each other. Here, also, we see
a predominance of iron lines. In the whole spectrum
Kirchhoff found not less than 450.

It follows from this, that the solar atmosphere con-
tains an abundance of the vapours of iron, which, by
the way, justifies us in concluding what an enormously
high temperature must prevail there. It shows, more-

156 ON THE ORIGIN OF THE PLANETARY SYSTEM.

over, how our figs. 7, 8, and 9 indicate iron, calcium,
and sodium, and also the presence of hydrogen, of zinc,
of copper, and of the metals of magnesia, alumina,
baryta, and other terrestrial elements. Lead, on the
other hand, is wanting, as well as gold, silver, mercury,
antimony, arsenic, and some others.

The spectra of several fixed stars are similarly con-
stituted ; they show systems of fine lines which can be
identified with those of terrestrial elements. In the
atmosphere of Aldebaran in Taurus there is, again,
hydrogen, iron, magnesium, calcium, sodium, and also
mercury, antimony, and bismuth ; and, according to
II. C. Vogel, there is in a Orionis the rare metal
thallium ; and so on.

We cannot, indeed, say that we have explained all
spectra; many fixed stars exhibit peculiarly banded
spectra, probably belonging to gases whose molecules
have not been completely resolved into their atoms by
the high temperature. In the spectrum of the sun,
also, are many lines which we cannot identify with
those of terrestrial elements. It is possible that they
may be due ty) substances unknown to us, it is also
possible that they are produced by the excessively high
temperature of the sun, far transcending anything we
can produce. But this is certain, that the known
terrestrial substances are widely diffused in space, and
especially nitrogen, which constitutes the greater part

ON T1JE ORIGIN OF TIIE PLANETARY SYSTEM. 157

of our atmosphere, and hydrogen, an element in water,
which indeed is formed by its combustion. Both have
been found in the irresolvable nebulae, and, from the
inalterability of their shape, these must be masses
of enormous dimensions and at an enormous distance.
For this reason Sir W. Herschel considered that they
did not belong to the system of our fixed stars, but
were representatives of the manner in which other
systems manifested themselves.

Spectrum analysis has further taught us more
about the sun, by which he is brought nearer to us, as it
were, than could formerly have seemed possible. You
know that the sun is an enormous sphere, whose
diameter is 112 times as great as that of the earth.
We may consider what we see on its surface as a layer
of incandescent vapour, which, to judge from the
appearances of the sun-spots, has a depth of about
500 miles. This layer of vapour, which is continually
radiating heat on the outside, and is certainly cooler
than the inner masses of the sun, is, however, hotter
than all our terrestrial flames — hotter even than the
incandescent carbon points of the electrical arc, which
represent the highest temperature attainable by terres-
trial means. This can be deduced with certainty from
Kirchhoff's law of the radiation of opaque bodies, from
the greater luminous intensity of the sun. The older
assumption, that the sun is a dark cool body, sur-

158 ON THE OKIGIN OF THE PLANETARY SYSTEM.

rounded by a photosphere which only radiates heat
and light externally, contains a physical impossibility.

Outside the opaque photosphere, the sun appears
surrounded by a layer of transparent gases, which are
hot enough to show in the spectrum bright coloured
lines, and are hence called the Chromosphere. They
show the bright lines of hydrogen, of sodium, of magne-
sium, and iron. In these layers of gas and of vapour
about the sun enormous storms occur, which are as
much greater than those of our earth in extent and in
velocity as the sun is greater than the earth. Currents
of ignited hydrogen burst out several thousands of miles
high, like gigantic jets or tongues of flame, with clouds
of srnoke above them.1 These structures could for-
merly only be viewed at the time of a total eclipse of
the sun, forming what were called the rose-red pro-
tuberances. \Ve now possess a method, devised by
MM. Jansen and Lockyer, by which they may at any
time be seen by the aid of the spectroscope.

On the other hand, there are individual darker
parts on the sun's surface, what are called sun-spots,
which were seen as long ago as by Galileo. They are
funnel-shaped, the sides of the funnel are not so dark
as the deepest part, the core. Fig. 10 represents such

1 According to H. C. Vogel's observations in Bothkamp to a height
of 70,000 miles. The spectroscopic displacement of the lines showed
velocities of 18 to 23 miles in a second; and, according to Lockyer, of
even 37 to 42 miles.

ON THE ORIGIN OF THE PLANETARY SYSTEM. 159

a spot according to Padre Secchi, as seen under power-
ful magnification. Their diameter is often more than
many tens of thousands of miles, so that two or three
earths could lie in one of them. These spots may
stand for weeks or months, slowly changing, before

FIG. 10.

they are again resolved, and meanwhile several rota-
tions of the sun may take place. Sometimes, however,
there are very rapid changes in them. That the core
is deeper than the edge of the surrounding penumbra
follows from their respective displacements as they
come neat the edge, and are therefore seen in a very

160 ON THE OKIGIN OF THE PLANETARY SYSTEM.

oblique direction. Fig. 11 represents in A to E the
different aspects of such a spot as it comes near the
edge of the sun.

Just on the edge of these spots there are spectro-
scopic indications of the most violent motion, and
in their vicinity there are often large protuberances ;
they show comparatively often a rotatory motion.
They may be considered to be places where the
FIG. 11.

* •

cooler gases from the outer layers of the sun's atmos-
phere sink down, and perhaps produce local superficial
coolings of the sun's mass. To understand the origin
of these phenomena, it must be remembered that the
gases, as they rise from the hot body of the sun, are
charged with vapours of difficultly volatile metals,
which expand as they ascend, and partly by their ex-
pansion, and partly by radiation into space, must be-
come cooled. At the same time, they deposit their

ON THE ORIGIN OF THE PLANET AR1 SYSTEM. 161

more difficultly volatile constituents as fog or cloud.
This cooling can only, of course, be regarded as com-
parative ; their temperature is probably, even then,
higher than any temperature attainable on the earth.
If now the upper layers, freed from the heavier
vapours, sink down, there will be a space over the
sun's body which is free from cloud. They appear
then as depressions, because about them are layers of
ignited vapours as much as 500 miles in height.

Violent storms cannot fail to occur in the sun's
atmosphere, because it is cooled on the outside, an'd
the coolest and comparatively densest and heaviest
parts come to lie over the hotter and lighter ones.
This is the reason why we have frequent, and at times
sudden and violent, movements in the earth's atmos-
phere, because this is heated from the ground made
hot by the sun and is cooled above. With the far
more colossal magnitude and temperature of the sun,

its meteorological processes are on a far larger scale,

•
and are far more violent.

We will now pass to the question of the perman-
ence of the present condition of our system. For a
long time the view was pretty generally held that, in
its chief features at any rate, it was unchangeable.
This opinion was based mainly on the conclusions at
which Laplace had arrived as the final results of his
long and laborious investigations, of the influence of
n. M

162 ON THE ORIGIN OF THE PLANETARY SYSTEM.

planetary disturbances. By disturbances of the plane-
tary motion astronomers understand, as I have already
mentioned, those deviations from the purely elliptical
motion which are due to the attraction of various planets
and satellites upon each other. The attraction of the
sun, as by far the largest body of our system, is indeed
the chief and preponderating force which produces the
motion of the planets. If it alone were operative,
each of the planets would move continuously in a con-
stant ellipse whose axes would retain the same direc-
tion and the same magnitude, making the revolutions
always in the same length of time. But, in point of
fact, in addition to the attraction of the sun there are
the attractions of all other planets, which, though
small, yet, in long periods of time, do effect slow
changes in the plane, the direction, and the magnitude
of the axes of its elliptical orbit. It has been asked
whether these attractions in the orbit of the planet
could go so far as to cause two adjacent planets to
encounter each other, so that individual ones fall into
the sun. Laplace was able to reply that this could not
be the case ; that all alterations in the planetary orbits
produced by this kind of disturbance must periodically
increase and decrease, and again revert to a mean
condition. But it must not be forgotten that this
result of Laplace's investigations only applies to dis-
turbances due to the reciprocal attraction of planets

ON THE ORIGIN OF THE PLANETARY SYSTEM. 163

upon each other, and on the assumption that no forces
of other kinds have any influence on their motions.

On our earth we cannot produce such an everlast-
ing motion as that of the planets seems to be ; for
resisting forces are continually being opposed to all
movements of terrestrial bodies. The best known of
these are what we call friction, resistance of the air,
and inelastic impact.

Hence the fundamental law of mechanics, accord-
ing to which every motion of a body on which no force
acts goes on in a straight line for ever with unchanged
velocity, never holds fully.

Even if we eliminate the influence of gravity in
a ball, for example, which rolls on a plane surface, we
see it go on for a while, and the further the smoother
is the path ; but at the same time we hear the rolling
ball make a clattering sound — that is, it produces waves
of sound in the surrounding bodies; there is friction
even on the smoothest surface ; this sets the surround-
ing air in vibration, and imparts to it some of its own
motion. Thus it happens that its velocity is con-
tinually less and less until it finally ceases. In like
manner, even the most carefully constructed wheel
which plays upon fine points, once made to turn, goes
on for a quarter of an hour, or even more, but then
stops. For there is always some friction on the axles,
and in addition there is the resistance of the air, which

K 2

164 ON THE OKIGlfc OF THE PLANETARY SYSTEM.

resistance is mainly due to that of the particles of air
against each other, due to their friction against the
wheel.

If we could once set a body in rotation, and keep
it from falling, without its being supported by another
body, and if we could transfer the whole arrangement to
an absolute vacuum, it would continue to move for ever
with undiminished velocity. This case, which cannot
be realised on terrestrial bodies, is apparently met
with in the planets with their satellites. They appear
to move in the perfectly vacuous cosmical space, with-
out contact with any body which could produce
friction, and hence their motion seems to be one which
never diminishes.

You see, however, that the justification of this
conclusion depends on the question whether cosmical
space is really quite vacuous. Is there nowhere any
Triction in the motion of the planets ?

From the progress which the knowledge of nature
has made since the time of Laplace, we must now
answer both questions in the negative.

Celestial space is not absolutely vacuous. In the
first place, it is filled by that continuous medium the
agitation of which constitutes light and radiant heat,
and which physicists know as the luminiferous ether.
In the second place, large and small fragments of
heavy matter, from the size of huge stones to that of

ON THE ORIGIN OF THE PLANETARY SYSTEM. 165

dust, are still everywhere scattered; at any rate, in
those parts of space which our earth traverses.

The existence of the luminiferous ether cannot be
considered doubtful. That light and radiant heat are
due to a motion which spreads in all directions has been
sufficiently proved. For the transference of such a
motion through space there must be something which
can be moved. Indeed, from the magnitude of the
action of this motion, or from that which the science
of mechanics calls its vis viva, we may indeed assign
certain limits for the density of this medium. Such
a calculation has been made by Sir W. Thomson, the
celebrated Glasgow physicist. He has found that the
density may possibly be far less than that of the air
in the most perfect exhaustion obtainable by a good
air-pump ; but that the mass of the ether cannot be
absolutely equal to zero. A volume equal to that of
the earth cannot contain less than 2,775 pounds of
luminous ether.1

The phenomena in celestial space are in conformity
with this. Just as a heavy stone flung through the
air shows scarcely any influence of the resistance of
the air, while a light feather is appreciably hindered ;
in like manner the medium which fills space is far too
attenuated for any diminution to have been perceived

1 This calculation would, however, lose its bases if Maxwell's hypo-
thesis were confirmed, according to which light depends on electrical
and magnetical oscillations

166 ON TEE ORIGIN OF THE PLANETARY SYSTEM.

in the motion of the planets since the time in which
we possess astronomical observations of their path. It
is different with the smaller bodies of our system.
Encke in particular has shown, with reference to the
well-known small comet which bears his name, that it
circulates round the sun in ever-diminishing orbits and
in ever shorter periods of revolution. Its motion is
similar to that of the circular pendulum which we have
mentioned, and which, having its velocity gradually
delayed by the resistance of the air, describes circles
about its centre of attraction, which continually become
smaller and smaller. The reason for this phenomenon
is the following: The force which offers a resistance
to the attraction of the sun on all comets and planets,
and which prevents them from getting continually
nearer to the sun, is what is called the centrifugal
force — that is, the tendency to continue their motion
in a straight line in the direction of their path. As
the force of their motion diminishes, they yield by a
corresponding amount to the attraction of the sun, and
get nearer to it. If the resistance continues, they will
continue to get nearer the sun until they fall into it.
Encke's comet is no doubt in this condition. But the
resistance whose presence in space is hereby indicated,
must act, and has long continued to act, in the same
manner on the far larger masses of the planets.

The presence of partly fine and partly coarse

ON THE ORIGIN OF THE PLANETARY SYSTEM. 167

heavy masses diffused in cosmical space is more dis-
tinctly revealed by the phenomena of asteroids and of
meteorites. We know now that these are bodies
which ranged about in cosmical space, before they came
within the region of our terrestrial atmosphere. In
the more strongly resisting medium which this atmos-
phere offers they are delayed in their motion, and at
the same time are heated by the corresponding friction.
Many of them may still find an escape from the terres-
trial atmosphere, and continue their path through
space with an altered and retarded motion. Others
fall to the earth ; the larger ones as meteorites, while
the smaller ones are probably resolved into dust by the
heat, and as such fall without being seen. According
to Alexander Herschel's estimate, we may figure shoot-
ing-stars as being on an average of the same size as
paving-stones. Their incandescence mostly occurs in
the higher and most attenuated regions of the atmos-
phere, eighteen miles and more above the surface of
the earth. As they move in space under the influence
of the same laws as the planets and comets, they
possess a planetary velocity of from eighteen to forty
miles in a second. By this, also, we observe that they
are in fact stelle cadente, falling stars, as they have
lonqr been called by poets.

This enormous velocity with which they enter our
atmosphere is undoubtedly the cause of their becoirv

168 ON THE ORIGIN OF THE PLANETARY SYSTEM.

ing heated. You all know that friction heats the
bodies rubbed. Every match that we ignite, every
badly greased coach-wheel, every auger which we work
in hard wood, teaches this. The air, like solid bodies,
not only becomes heated by friction, but also by the
work consumed in its compression. One of the most
important results of modern physics, the actual proof
of which is mainly due to the Englishman Joule, is
that, in such a case, the heat developed is exactly pro-
portional to the work expended. If, like the mechani-
cians, we measure the work done by the weight which
would be necessary to produce it, multiplied by the
height from which it must fall, Joule has shown that
the work, produced by a given weight of water falling
through a height of 425 metres, would be just suffi-
cient to raise the same weight of water through one
degree Centigrade. The equivalent in work of a
velocity of eighteen to twenty-four miles in a second
may be easily calculated from known mechanical laws ;
and this, transformed into heat, would be sufficient to
raise the temperature of a piece of meteoric iron to
900,000 to 2,500,000 degrees Centigrade, provided that
all the heat were retained by the iron, and did not, as
it undoubtedly does, mainly pass into the air. This
calculation shows, at any rate, that the velocity of the
shooting-stars is perfectly adequate to raise them to
the most violent incandescence. The temperatures

ON THE ORIGIN OF THE PLANETARY SYSTEM. 169

attainable by terrestrial means scarcely exceed 2,000
degrees. In fact, the outer crusts of meteoric stones
generally show traces of incipient fusion ; and in cases
in which observers examined with sufficient prompti-
tude the stones which had fallen they found them hot
on the surface, while the interior of detached pieces
seemed to show the intense cold of cosmical space.

To the individual observer who casually looks
towards the starry sky the meteorites appear as a rare
and exceptional phenomenon. If, however, they are
continuously observed, they are seen with tolerable
regularity, especially towards morning, when they
usually fall. But a single observer only views but a
small part of the atmosphere; and if they are calcu-
lated for the entire surface of the earth it results that
about seven and a half millions fall every day. In our
regions of space, they are somewhat sparse and distant
from each other. According to Alexander Herschel's
estimates, each stone is, on an average, at a distance of
450 miles from its neighbours. Eut the earth moves
through 18 miles every second, and has a diameter of
7,820 miles, and therefore sweeps through 876 millions
of cubic miles of space every second, and carries with
it whatever stones are contained therein.

Many groups are irregularly distributed in space,
being probably those which have already undergone
.disturbances by planets. There are also denser swarms

170 ON THE ORIGIN OF THE PLANETARY SYSTEM.

which move in regular elliptical orbits, cutting the
earth's orbit in definite places, and therefore always
occur on particular days of the year. Thus the 10th
of August of each year is remarkable, and every thirty-
three years the splendid fireworks of the 12th to the
14th of November repeats itself for a few years. It is
remarkable that certain comets accompany the paths
of these swarms, and give rise to the supposition that
the comets gradually split up into meteoric swarms.

This is an important process. What the earth
does is done by the other planets, and in a far higher
degree by the sun, towards which all the smaller
bodies of our system must fall ; those, therefore, that
are more subject to the influence of the resisting
medium, and which must fall the more rapidly, the
smaller they are. The earth and the planets have for
millions of years been sweeping together the loose
masses in space, and they hold fast what they have once
attracted. But it follows from this that the earth and
the planets were once smaller than they are now, and
that more mass was diffused in space; and if we
follow out this consideration it takes us back to a
state of things in which, perhaps, all the mass now
accumulated in the sun and in the planets, wandered
loosely diffused in space. If we consider, further, that
the small masses of meteorites as they now fall, ha\re
perhaps been formed by the gradual aggregation of

ON THE ORIGIN OF THE PLANETARY SYSTEM. 171

fine dust, we see ourselves led to a primitive condition
of fine nebulous masses.

From this point of view, that the fall of shooting-
stars and of meteorites is perhaps only a small survival
of a process which once built up worlds, it assumes far
greater significance.

This would be a supposition of which we might
admit the possibility, but which could not perhaps
claim any great degree of probability, if we did not
find that our predecessors, starting from quite different
considerations, had arrived at the same hypothesis.

You know that a considerable number of planets
rotate around the sun besides the eight larger ones,
Mercury, Venus, the Earth, Mars, Jupiter, Saturn,
Uranus, and Neptune; in the interval between Mars
and Jupiter there circulate, as far as we know, 156
small planets or planetoids. Moons also rotate about
the larger planets — that is, about the Earth and the
four most distant ones, Jupiter, Saturn, Uranus, and
Neptune; and lastly the Sun, and at any rate the
larger planets, rotate about their own axes. Now, in
the first place, it is remarkable that all the planes of
rotation of the planets and of their satellites, as well
as the equatorial planes of these planets, do not vary
much from each other, and that in these planes all the
rotation is in the same direction. The only consider-
able, exceptions known are the moons of Uranus,

172 ON THE ORIGIN OF THE PLANETAEI SYSTEM.

whose plane ib almost at right angleb to the planes of
the larger planets. It must at the same time be
remarked that the coincidence, in the direction of these
planes, is on the whole greater, the longer are the
bodies and the larger the paths in question ; while in
the smaller bodies, and for the smaller paths, espe-
cially for the rotations of the planets about their own
axes, considerable divergences occur. Thus the planes
of all the planets, with the exception of Mercury and
-of the small ones between Mars and Jupiter, differ at
most by three degrees from the path of the Earth.
The equatorial plane of the Sun deviates by only seven
and a half degrees, that of Jupiter only half as much.
The equatorial plane of the Earth deviates, it is true,
to the extent of twenty-three and a half degrees, and
that of Mars by twenty-eight and a half degrees, and
the separate paths of the small planet's satellites differ
still more. But in these paths they all move direct,
all in the same direction about the sun, and, as far as
can be ascertained, also about their own axes, like
the earth — that is, from west to east. If they had
originated independently of each other, and had
come together, any direction of the planes for each
individual one would have been equally probable; a
reverse direction of the orbit would have been just as
probable as a direct one ; decidedly elliptical paths
would have been as probable as the almost circulai

ON THE ORIGIN OF THE PLANETAEY SYSTEM. 173

ones which we meet with in all the bodies we have
named. There is, in fact, a complete irregularity in
the comets and meteoric swarms, which we .- have
much reason for considering to be formations which
have only accidentally come within the sphere of the
sun's attraction.

The number of coincidences in the orbits of the
planets and their satellites is too great to be ascribed
to accident. We must inquire for the reason of this
coincidence, and this can only be sought in a primi-
tive connection of the entire mass. Now, we are
acquainted with forces and processes which condense
an originally diffused mass, but none which could drive
into space such large masses, as the planets, in the
condition we now find them. Moreover, if they had
become detached from the common mass, at a place
much nearer the sun, they ought to have a markedly
elliptical orbit. We must assume, accordingly, that
this mass in its primitive condition extended at least
to the orbit of the outermost planets.

These were the essential features of the considera-
tions which led Kant and Laplace to their hypothesis.
In their view our system was originally a chaotic ball
of nebulous matter, of which originally, when it ex-
tended to the path of the most distant planet, many
billions of cubic miles could contain scarcely a gramme
of mass. This ball, when it had become detached from

174 OX TEE ORIGIN OF THE PLANETAHY SYSTEM.

ths nebulous balls of the adjacent fixed stars, possessed
a slow movement of rotation." It became condensed
under the influence of the reciprocal attraction of its
parts ; and, in the degree in which it condensed, the
rotatory motion increased, and formed it into a flat
disk. From time to time masses at the circumference
of this disk became detached under the influence of
the increasing centrifugal force; that which became
detached formed again into a rotating nebulous mass,
which either simply condensed and formed a planet, or
during this condensation again repelled masses from
the periphery, which became satellites, or in one case,
that of Saturn, remained as a coherent ring. In an-
other case, the mass which separated from the outside
of the chief ball, divided into many parts, detached
from each other, and furnished the swarms of small
planets between Mars and Jupiter.

Our more recent experience as to the nature of
star showers teaches us that this process of the conden-
sation of loosely diffused masses to form larger bodies
is by no means complete, but still goes on, though the
traces are slight. The form in which it now appears
is altered by the fact that meanwhile the gaseous
or dust-like mass diffused in space had united under
the influence of the force of attraction, and of the
force of crystallisation of their constituents, to larger
pieces than originally existed.

ON THE ORIGIN OF THE PLANETARY SYSTEM. 175

The showers of stars, as examples now taking place
of the process which formed the heavenly bodies, are
important from another point of view. They develop
light and heat; and that directs us to a third series
of considerations, which leads again to the same goal.

All life and all motion on our earth is, with few
exceptions, kept up by a single force, that of the sun's
rays, which bring to us light and heat. They warm
the air of the hot zones, this becomes lighter and
ascends, while the colder air flows towards the poles.
Thus is formed the great circulation of the passage-
winds. Local differences of temperature over land and
sea, plains and mountains, disturb the uniformity of
this great motion, and produce for us the capricious
change of winds. Warm aqueous vapours ascend with
the warm air, become condensed into clouds, and fall
in the cooler zones, and upon the snowy tops of the
mountains, as rain and as snow. The water collects in
brooks, in rivers, moistens the plains, and makes life
possible ; crumbles the stones, carries their fragments
along, and thus works at the geological transformation
of the earth's surface. It is only under the influence
of the sun's rays that the variegated covering of plants
of the earth grows ; and while they grow, they accumu-
late in their structure organic matter, which partly
nerveb £he whole animal kingdom as food, and serves
man more particularly as fuel. Coals and lignites, the

176 ON THE ORIGIN OF THE PLANETARY SYSTEM.

sources of power of our steam engines, are remains of
primitive plants — the ancient production of the sun's
rays.

Need we wonder if, to our forefathers of the Aryan
race in India and Persia, the sun appeared as the fittest
symbol of the Deity ? They were right in regarding it
as the giver of all life — as the ultimate source of almost
all that has happened on earth.

But whence does the sun acquire this force? It
radiates forth -a more intense light than can be attained
with any terrestrial means. It yields as much heat as
if 1,500 pounds of coal were burned every hour upon
each square foot of its surface. Of the heat which
thus issues from it, the small fraction which enters our
atmosphere furnishes a great mechanical force. Every
steam-engine teaches us that heat can produce such
force. The sun, in fact, drives on earth a kind of
steam-engine whose performances are far greater than
those of artificially constructed machines. The circu-
lation of water in the atmosphere raises, as has been
said, the water evaporated from the warm tropical
seas to the mountain heights ; it is, as it were, a water-
raising engine of the most magnificent kind, with
whose power no artificial machine can be even dis-
tantly compared. I have previously explained the
mechanical equivalent of heat. Calculated by that
standard, the work which the sun produces by ita

ON THE ORIGIN OF THE PLANETARY SYSTEM. 177

radiation is equal to the constant exertion of 7,000
horse-power for each square foot of the sun's surface.

For a long time experience had impressed on our
mechanicians that a working force cannot be produced
from nothing ; that it can only be taken from the
stores which nature possesses ; which are strictly limited
and which cannot be increased at pleasure — whether it
be taken from the rushing water or from the wind ;
whether from the layers of coal, or from men and from
animals, which cannot work without the consumption
of food. Modern physics has attempted to prove the
universality of this experience, to show that it applies
to the great whole of all natural processes, and is inde-
pendent of the special interests of man. These have
been generalised and comprehended in the all-ruling
natural law of the Conservation of Force. No natural
process, and no series of natural processes, can be
found, however manifold may be the changes which
take place among them, by which a motive force can
be continuously produced without a corresponding con-
sumption. Just as the human race finds on earth but
a limited supply of motive forces, capable of producing
work, which it can utilise but not increase, so also
must this be the case in the great whole of nature.
The universe has its definite store of force, which
works in it under ever varying forms ; is indestructible,
not to be increased, everlasting and unchangeable lifca

II. N

178 ON THE ORIGIN OF THE PLANETARY SYSTEM.

matter itself. It seems as if Groethe had an idea of
this when he makes the earth-spirit speak of himself
as the representative of natural force.

In the currents of life, in the tempests of motion,
In the fervour of art, in the fire, in the storm,

Hither and thither,

Over and under,

Wend I and wander.

Birth and the grave,

Limitless ocean,

Where the restless wave

Undulates ever

Under and over,

Their seething strife

Heaving and weaving

The changes of life.
At the whirling loom of time unawed,
I work the living mantle of God.

Let us return to the special question which con-
cerns us here : Whence does the sun derive this enor-
mous store of force which it sends out ?

On earth the processes of combustion are the
most abundant source of heat. Does the sun's heat
originate in a process of this kind? To this question
we can reply with a complete and decided negative,
for we now know that the sun contains the terrestrial
elements with which we are acquainted. I^et us select
from among them the two, which, for the smallest mass,
produce the greatest amount of heat when they com-
bine ; let us assume that the sun consists of hydrogen
and oxygen, mixed in the proportion in which they
«roiild unite to form water. The mass of the sun is

ON THE ORIGIN OF THE PLANETARY SYSTEM. 179

known, and also the quantity of heat produced by the
union of known weights of oxygen and hydrogen.
Calculation shows that under the above supposition,
the heat resulting from their combustion would be
sufficient to keep up the radiation of heat from the
sun for 3,021 years. That, it is true, is a long time,
but even profane history teaches that the sun has
lighted and warmed us for 3,000 years, and geology
puts it beyond doubt that this period must be ex-
tended to millions of years.

Known chemical forces are thus so completely in-
adequate, even on the most favourable assumption, to
explain the production of heat which takes place in
the sun, that we must quite drop this hypothesis.

We must seek for forces of far greater magnitude,
and these we can only find in cosmical attraction. We
have already seen that the comparatively small masses
of shooting-stars and meteorites can produce extra-
ordinarily large amounts of heat when their cosmical
velocities are arrested by our atmosphere. Now the
force which has produced these great velocities is
gravitation. We know of this force as one acting on
the surface of our planet when it appears as terrestrial
gravity. We know that a weight raised from the
earth can drive our clocks, and that in like manner
the gravity of the water rushing down from the moun-
tains works our mills.

x2

180 ON THE ORIGIN OF THE PLANETARY SYSTEM.

If a weight falls from a height and strikes the
ground its mass loses, indeed, the visible motion which
it had as a whole — in fact, however, this motion is not
lost; it is transferred to the smallest elementary
particles of the mass, and this invisible vibration of
the molecules is the motion of heat. Visible motion
is transformed by impact, into the motion of heat.

That which holds in this respect for gravity, holds
also for gravitation. A heavy mass, of whatever kind,
which is suspended in space separated from another
heavy mass, represents a force capable of work. For
both masses attract each other, and, if unrestrained by
centrifugal force, they move towards each other under
the influence of this attraction ; this takes place with
ever-increasing velocity ; and if this velocity is finally
destroyed, whether this be suddenly, by collision, or
gradually, by the friction of movable parts, it develops
the corresponding quantity of the motion of heat, the
amount of which can be calculated from the equiva-
lence, previously established, between heat and me-
chanical work.

Now we may assume with great probability that
very many more meteors fall upon the sun than upon
the earth, and with greater velocity, too, and therefore
give more heat. Yet the hypothesis, that the entire
amount of the sun's heat which is continually lost by
radiation, is made up by the fall of meteors, a hypothesis

ON THE ORIGIN OF THE PLANETARY SYSTEM. 181

which was propounded by Mayer, and has been favour-
ably adopted by several other physicists, is open, ac-
cording to Sir W. Thomson's investigations, to ob-
jection; for, assuming it to hold, the mass of the
sun should increase so rapidly that the consequences
would have shown themselves in the accelerated
motion of the planets. The entire loss of heat from
the sun cannot at all events be produced in this way ;
at the most a portion, which, however, may not be
inconsiderable.

If, now, there is no present manifestation of force
sufficient to cover the expenditure of the sun's heat,
the sun must originally have had a store of heat which
it gradually gives out. But whence this store ? We
know that the cosmical forces alone could have pro-
duced it. And here the hypothesis, previously dis-
cussed as to the origin of the sun, comes to our aid. If
the mass of the sun had been once diffused in cosmical
space, and had then been condensed — that is, had fallen
together under the influence of celestial gravity — if
then the resultant motion had been destroyed by
friction and impact, with the production of heat, the
new world produced by such condensation must have
acquired a store of heat not only of considerable, but
even of colossal, magnitude.

Calculation shows that, assuming the thermal capa-
city of the sun to be the same as that of water, the

182 ON THE ORIGIN OF THE PLANETARY SYSTEM.

temperature might be raised to 28,000,000 of degrees,
if this quantity of heat could ever have been present
in the sun at one time. This cannot be assumed, for
such an increase of temperature would offer the
greatest hindrance to condensation. It is probable
rather that a great part of this heat, which was pro-
duced by condensation, began to radiate into space
before this condensation was complete. But the heat
tfhich the sun could have previously developed by its
condensation, would have been sufficient to cover its
present expenditure for not less than 22,000,000 of
years of the past.

And the sun is by no means so dense as it may
become. Spectrum analysis demonstrates the presence
of large masses of iron and of other known constituents
of the rocks. The pressure which endeavours to con-
dense the interior is about 800 times as great as that
in the centre of the earth ; and yet the density of the
sun, owing probably to its enormous temperature, is
less than a quarter of the mean density of the earth.

We may therefore assume with great probability
that the sun will still continue in its condensation, even
if it only attained the density of the earth — though it
will probably become far denser in the interior owing
to the enormous pressure — this would develop fresh
quantities of heat, which would be sufficient to main-
tain for an additional 17,000,000 of years the same

ON THE ORIGIN OF THE PLANETARY SYSTEM. 183

intensity of sunshine as that which is now the source
of all terrestrial life.

The smaller bodies of our system might become
less hot than the sun, because the attraction of the
fresh masses would be feebler. A body like the earth
might, if even we put its thermal capacity as high as
that of water, become heated to even 9,000 degrees,
to more than our flames can produce. The smaller
bodies must cool more rapidly as long as they are still
liquid. The increase in temperature, with the depth,
is shown in bore-holes and in mines. The existence of
hot wells and of volcanic eruptions shows that in the
interior of the earth there is a very high temperature,
which can scarcely be anything than a remnant of the
high temperature which prevailed at the time of its
production. At any rate, the attempts to discover for
the internal heat of the earth a more recent origin in
chemical processes, have hitherto rested on very arbi-
trary assumptions ; and, compared with the general uni-
form distribution of the internal heat, are somewhat
insufficient.

On the other hand, considering the huge masses of
Jupiter, of Saturn, of Uranus, and of Neptune, their
small density, as well as that of the sun, is surprising,
while the smaller planets and the moon approximate to
the density of the earth. We are here reminded of
the higher initial temperature, and the slower cooling,

184 ON THE ORIGIN OF TIIE PLANETARY SYSTEM.

which characterises larger masses.1 The moon, on the
contrary, exhibits formations on its surface which are
strikingly suggestive of volcanic craters, and point to
a former state of ignition of our satellite. The mode
of its rotation, moreover, that it always turns the
same side towards the earth, is a peculiarity which
might have been produced by the friction of a fluid.
At present no trace of such a one can be perceived.
You see, thus, by what various paths we are con-
FIG ]2 stantly led to the same

primitive conditions.
The hypothesis of Kant
and Laplace is seen to
be one of the happiest
ideas in science, which
at first astounds us, and
then connects us in all
directions with other dis-
coveries, by which the
conclusions are confirmed until we have confidence in
them. In this case another circumstance has con-
tributed— that is, the observation that this process of
transformation, which the theory in question presup-
poses, goes on still, though on a smaller scale, seeing

1 Mr. Zoellner concludes from photometric measurements, which,
however, need confirmation, that Jupiter still possesses a light of iis
own.

ON THE ORIGIN OF THE PLANETARY SYSTEM. 185

that all stages of that process can still be found to
exist.

For as we have already seen, the larger bodies
which are already formed go on increasing with the

Fio. 13.

development of heat, by the attraction of the meteorio
masses already diffused in space. Even now the
smaller bodies are slowly drawn towards the sun by
the resistance in space. We still find in the firma-
ment of fixed stars, according to Sir J. Herschel's
newest catalogue, over 5,000 nebulous spots, of which

!86 ON THE ORIGIN OF THE PLANETARY SYSTEM.

those whose light is sufficiently strong give for the
most part a coloured spectrum of fine bright lines, as
they appear in the spectra of the ignited gases. The
nebulae are partly rounded structures, which are called

Fio. 14.

planetary nebula; (fig. 12); sometimes wholly irregular
in form, as the large nebula in Orion, represented in
fig. 13; they are partly annular, as in the figures in
fig. 14, from the Canes Venatici. They are for the
most part feebly luminous over their whole surface,
while the fixed stars only appear as luminous points.

ON THE ORIGIN OF THE PLANETARY SYSTEM 187

In many nebulae small stars can be seen, as in figs.
\5 and 16, from Sagittarius and Aurigo. More stars
are continually being discovered in them, the better
are the telescopes used in their analysis. Thus, before
the discovery of spectrum analysis, Sir W. Herschel's
former view might be regarded as the most probable,
that that which we see to be nebulae are pnly heaps of
Fio. 15. FIG. 16.

very fine stars, of other Milky Ways. Now, however,
spectrum analysis has shown a gas spectrum in many
nebulae which contains stars, while actual heaps of stars
show the continuous spectrum of ignited solid bodies.
Nebulae have in general three distinctly recognisable
lines, one of which, in the blue, belongs to hydrogen,
a second in bluish-green to nitrogen,1 while the third,
between the two, is of unknown origin. Fig. 1 7 shows

1 Or perhaps also to oxygen. The line occurs in the spectrum of
atmospheric air, and according to H. C. Vogel's observations was want-
ing in the spectrum of pure oxygen.

188 ON THE ORIGIN OF THE PLANETARY SYSTEM.

such a spectrum of a small but bright nebula in the
Dragon. Traces of other bright lines are seen along
<vith them, and sometimes also, as in fig. 17, traces
of a continuous spectrum ; all of which, however, are
too feeble to admit of accurate investigation. It must
be observed here that the light of very feeble objects
which give a continuous spectrum are distributed by
the spectroscope over a large surface, and are there-
fore greatly enfeebled or even extinguished, while the

FIG. 17.

undecomposable light of bright gas lines remains unde-
composed, and hence can still be seen. In any case,
the decomposition of the light of the nebulae shows
that by far the greater part of their luminous surface
is due to ignited gases, of which hydrogen forms a
prominent constituent. In the planetary masses, the
spherical or discoidal, it might be supposed that the
gaseous mass had attained a condition of equilibrium ;
but most other nebulas exhibit highly irregular forms,
which by no means correspond to such a condition.
As, however, their shape has either not at all altered,
or not appreciably, since they have been known and
observed, they must either have very little mass, or

ON THE OKIGIN OF THE PLANETARY SYSTEM. 189

they must be of colossal size and distance. The former
does not appear very probable, because small masses
very soon give out their heat, and hence we are left
to the second alternative, that they are of huge di-
mensions and distances. The same conclusion had
been originally drawn by Sir \V. Herschel, on the
assumption that the nebulae were heaps of stars.

With those nebulae which, besides the lines of gases,
also show the continuous spectrum of ignited denser
bodies, are connected spots which are partly irresolv-
able and partly resolvable into heaps of stars, which
only show the light of the latter kind.

The countless luminous stars of the heavenly firma-
ment, whose number increases with each newer and
more perfect telescope, associate themselves with this
primitive condition of the worlds as they are formed.
They are like our sun in magnitude, in luminosity, and
on the whole also in the chemical condition of their
surface, although there may be differences in the quan-
tity of individual elements.

But we find also in space a third stadium, that of
extinct suns; and for this also there are actual evi-
dences. In the first place, there are, in the course of
history, pretty frequent examples of the appearance of
new stars. In 1572 Tycho Brahe observed such a one,
which, though gradually burning paler, was visible
for two years, stood still like a fixed star, and finally

190 ON THE ORIGIN OF THE PLANETARY SYSTEM.

reverted to the darkness from which it had FO suddenly
emerged. The largest of them all seems to have been
that observed by Kepler in the year 1604, which was
brighter than a star of the first magnitude, and was
observed from September 27, 1604, until March 1606.
The reason of its luminosity was probably the collision
with a smaller world. In a more recent case, in which
on May 12, 1866, a small star of the tenth magnitude
in the Corona suddenly burst out to one of the second
magnitude, spectrum analysis showed that it was an
outburst of ignited hydrogen which produced the light.
This was only luminous for twelve days.

In other cases obscure heavenly bodies have dis-
covered themselves by their attraction on adjacent
bright stars, and the motions of the latter thereby pro-
duced. Such an influence is observed in Sirius and
Procyon. By means of a new refracting telescope
Messrs. Alvan Clarke and Pond, of Cambridge, U.S.,
have discovered in the case of Sirius a scarcely visible
star, which has but little luminosity, but is almost
seven times as heavy as the sun, has about half the
mass of Sirius, and whose distance from Sirius is about
equal to that of Neptune from the sun. The satellite
of Procyon has not yet been seen; it appears to be
quite dark.

Thus there are extinct suns. The fact that there
are such lends new weight to the reasons which per-

ON THE ORIGIN OF THE PLANETARY SYSTEM. 191

mit us to conclude that our sun also is a body which
slowly gives out its store of heat, and thus will some
time become extinct.

The term of 17,000,000 years which I have given
may perhaps become considerably prolonged by the
gradual abatement of radiation, by the new accretion of
falling meteors, and by still greater condensation than
that which I have assumed in that calculation. But
we know of no natural process which could spare our
sun the fate which has manifestly fallen upon other
suns. This is a thought which we only reluctantly
admit; it seems to us an insult to the beneficent
Creative Power which we otherwise find at work in
organisms and especially in living ones. But we must
reconcile ourselves to the thought that, however we
may consider ourselves to be the centre and final
object of Creation, we are but as dust on the earth ;
which again is but a speck of dust in the immensity
of space ; and the previous duration of our race, even
if we follow it far beyond our written history, into the
era of the lake dwellings or of the mammoth, is but
an instant compared with the primeval times of our
planet; when living beings existed upon it, whose
strange and unearthly remains still gaze at us from
their ancient tombs ; and far more does the duration
of our race sink into insignificance compared with the
enormous periods during which worlds have been iu

192 ON THE ORIGIN OF THE PLANETARY SYSTEM.

process of formation, and will still continue to form
when our sun is extinguished, and our earth is either
solidified in cold or is united with the- ignited central
body of our system.

But who knows whether the first living inhabitants
of the warm sea on the young world, whom we ought
perhaps to honour as our ancestors, would not have
regarded our present cooler condition with as much
horror as we look on a world without a sun ? Consider-
ing the wonderful adaptability to the conditions of life
which all organisms possess, who knows to what degree
of perfection our posterity will have been developed in
17,000,000 of years, and whether our fossilised bones
will not perhaps seem to them as monstrous as those of
the Ichthyosaurus now do ; and whether they, adjusted
for a more sensitive state of equilibrum, will not con-
sider the extremes of temperature, within which we now
exist, to be just as violent and destructive as those of the
older geological times appear to us ? Yea, even if sun and
earth should solidify and become motionless, who could
say what new worlds would no- oe ready to develop
life ? Meteoric stones sometimes contain hydrocarbons ;
the light of the heads of comets exhibits a spectrum
which is most like that of the electrical light in gases
containing hydrogen and carbon. But carbon is the
element, which is characteristic of organic compounds,
from which living bodies are built up. Who knows

ON THE OKIGIN OF THE PLANET AKY SYSTEM. 193

whether these bodies, which everywhere swarm through
space, do not scatter germs of life wherever there is a
new world, which has become capable of giving a dwell-
ing-place to organic bodies ? And this life we might
perhaps consider as allied to ours in its primitive
germ, however different might be the form which it
would assume in adapting itself to its new dwelling-
place.

However this may be, that which most arouses our
moral feelings at the thought of a future, though pos-
sibly very remote, cessation of all living creation on the
earth, is more particularly the question whether all this
life is not an aimless sport, which will ultimately fall a
prey to destruction by brute force ? Under the light of
Darwin's great thought we begin to see that not only
pleasure and joy, but also pain, struggle, and death, are
the powerful means by which nature has built up her
finer and more perfect forms of life. And we men
know more particularly that in our intelligence, our
civic order, and our morality we are living on the in-
heritance which our . efathers have gained for us, and
that which we acquire in the same way, will in like
manner ennoble the life of our posterity. Thus the
individual, who works for the ideal objects of humanity,
even if in a modest position, and in a limited sphere
of activity, may bear without fear the thought that the
thiead of his own consciousness will one day break.

II. 0

ON THE ORIGIN OF THE PLANETARY SYSTEM.

But even men of such free and large order of minds
as Lessing and David Strauss could not reconcile them-
selves to the thought of a final destruction of the
living race, and with it of all the fruits of all past
generations.

As yet we know of no fact, which can be established
by scientific observation, which would show that the
finer and complex forms of vital motion could exist
otherwise than in the dense material of organic life ;
that it can propagate itself as the sound-movement
of a string can leave its originally narrow and fixed
home and diffuse itself in the air, keeping all the time
its pitch, and the most delicate shade of its colour-tint ;
and that, when it meets another string attuned to it,
starts this again or excites a flame ready to sing to the
same tone. The flame even, which, of all processes in
inanimate nature, is the closest type of life, may
become extinct, but the heat which it produces con-
tinues to exist — indestructible, imperishable, as an in-
visible motion, now agitating the molecules of ponder-
able matter, and then radiating into boundless space as
the vibration of an ether. Even there it retains the
characteristic peculiarities of its origin, and it reveals its
history to the inquirer who questions it by the spectro-
scope. United afresh, these rays may ignite a new
flame, and thus, an it were, acquire a new bodily
existence

ON THE ORIGIN OF THE PLANETAEY SYSTEM. 195

Just as the flame remains the same in appearance,
and continues to exist with the same form and struc-
ture, although it draws every minute fresh combustible
vapour, and fresh oxygen from the air, into the vortex
of its ascending current; and just as the wave goes
on in unaltered form, and is yet being reconstructed
every moment from fresh particles of water, so also in
the living being, it is not the definite mass of substance,
which now constitutes the body, to which the con-
tinuance of the individual is attached. For the material
of the body, like that of the flame, is subject to con-
tinuous and comparatively rapid change — a change the
more rapid, the livelier the activity of the organs in
question. Some constituents are renewed from day to
day, some from month to month, and others only after
years. That which continues to exist as a particular
individual is like the flame and the wave — only the
form of motion which continually attracts fresh matter
into its vortex and expels the old. The observer with
a deaf ear only recognises the vibration of sound
as long as it is visible and can be felt, bound up with
heavy matter. Are our senses, in reference to life, like
the deaf ear in this respect ?

0 23

196 ON THE ORIGIN OF THE PLANETARY SYSTEM.

ADDENDUM.

THE sentences on page 193 gave rise to a controversial
attack by Mr. J. C. F. Zoellner, in his book 'On the
Nature of the Comets/ on Sir W. Thomson, on which I
took occasion to express myself briefly in the preface to the
second part of the German translation of the ' Handbook of
Theoretical Physics/ by Thomson and Tait. I give here the
passage in question : —

* I will mention here a farther objection. It refers to the
question as to the possibility that organic germs may occur
in meteoric stones, and be conveyed to the celestial bodies
which have been cooled. In his opening Address at the
Meeting of the British Association in Edinburgh, in August
1871, Sir "W. Thomson had described this as " not unscien-
tific." Here also, if there is an error, I must confess that I
also am a culprit. I had mentioned the same view as a
possible mode of explaining the transmission of organisms
through space, even a little before Sir W. Thomson, in a
lecture delivered in the spring of the same year at Heidel-
berg and Cologne, but not published. I cannot object if any-
one considers this hypothesis to be in a high, or even in the
highest, degree improbable. But to me it seems a perfectly
correct scientific procedure, that when all our attempts fail
in producing organisms from inanimate matter, we may
inquire whether life has ever originated at all or not, and
whether its germs have not been transported from one
world to another, and have developed themselves wherever
they found a favourable soil.

' Mr. Zoellner's so-called physical objections are but of
very small weight. He recalls the history of meteoric stone,
and adds (p. xxvi.): " If, therefore, that meteoric stones covered
with organisms had escaped with a whole skin in the smash-

ON THE ORIGIN OF THE PLANETARY SYSTEM. 197

up of its mother-body, and had not shared the general rise
of temperature, it must necessarily have first passed through
the atmosphere of the earth, before it could deliver itself of
its organisms for the purpose of peopling the earth."

* Now, in the first place, we know from repeated observa-
tions that the larger meteoric stones only become heated in
their outside layer during their fall through the atmosphere,
while the interior is cold, or even very cold. Hence all
germs which there might be in the crevices would be safe
from combustion in the earth's atmosphere. But even those
germs which were collected on the surface when they reached
the highest and most attenuated layer of the atmosphere would
long before have been blown away by the powerful draught
of air, before the stone reached the denser parts of the gaseous
mass, where the compression would be sufficient to produce
an appreciable heat. And, on the other hand, as far as the
impact of two bodies is concerned, as Thomson assumes,
the first consequences would be powerful mechanical motions,
and only in the degree in which this would be destroyed by
friction would heat be produced. We do not know whether
that would last for hours, for days, or for week^. The frag-
ments, which at the first moment were scattered with planet-
ary velocity, might escape without any disengagement of
heat. I consider it even not improbable, that a stone, or
shower of stones, flying through the higher regions of the
atmosphere of a celestial body, carries with it a mass of air
which contains unburned germs.

1 As I have already remarked I am not inclined to suggest
that all these possibilities are probabilities. They are ques-
tions the existence and signification of which we must re-
member, in order that if the case arise they may be solved
by actual observations or by conclusions therefrom.'

OH

THOUGHT IN MEDICINE.

An Address delivered August 2, 1877, on the Anniversary of

the Foundation of the Institute for the Education of

Army Surgeons.

ir is now thirty-five years since, on the 2nd August, I
stood on the rostrum in the Hall of this Institute, before
another such audience as this, and read a paper on
the operation of Venal Tumours. I was then a pupil of
this Institution, and was just at the end of my studies.
I had never seen a tumour cut, and the subject-matter
of my lecture was merely compiled from books ; but
book knowledge played at that time a far wider and
a far more influential part in medicine than we are at
present disposed to assign to it. It was a period of
fermentation, of the fight between learned tradition and
the new spirit of natural science, which would have
no more of tradition, but wished to depend upon
individual experience. The authorities at that time

200 ON THOUGHT IN MEDICINE.

judged more favourably of my Essay than I did myself,
and I still possess the books which were awarded to me
as the prize.

The recollections which crowd in upon me on this
occasion have brought vividly before my mind a picture
of the then condition of our science, of our endeavours
and of our hopes, and have led me to compare the
past state of things with that into which it has de-
veloped. Much indeed has been accomplished.

Although all that we hoped for has not been ful-
filled, and many things have turned out differently from
what we wished, yet we have gained much for which we
could not have dared to hope. Just as the history
of the world has made one of its few giant steps
before our eyes, so also has our science ; hence an old
student, like myself, scarcely recognises the somewhat
matronly aspect of Dame Medicine, when he accident-
ally comes again in relation to her, so vigorous and
so capable of growth has she become in the fountain of
youth of the Natural Sciences.

I may, perhaps, retain the impression of this an-
tagonism, more freshly than those of my contemporaries
whom I have the honour to see assembled before me ;
and who, having remained permanently connected with
science and practice, have been less struck and less
surprised by great changes, taking place as they do by
blow steps. This must be my excuse for speaking to

ON THOUGHT IN MEDICINE. 201

you about the metamorphosis which has taken place in
medicine during this period, and with the results of
whose development you are better acquainted than I
am. I should like the impression of this development
and of its causes not to be quite lost on the younger of
my hearers. They have no special incentive for con-
sulting the literature of that period ; they would meet
with principles which appear as if written in a lost
tongue, so that it is by no means easy for us to transfer
ourselves into the mode of thought of a period which
is so far behind us. The course of development of
medicine is an instructive lesson on the true principles
of scientific inquiry, and the positive part of this
lesson has, perhaps, in no previous time been so im-
pressively taught as in the last generation.

The task falls to me, of teaching that branch of the
natural sciences which has to make the widest gene-
ralisations, and has to discuss the meaning of funda-
mental ideas ; and which has, on that account, been
not unfitly termed Natural Philosophy by the English-
speaking peoples. Hence it does not fall too far out of
the range of my official duties and of my own studies, if I
attempt to discourse here of the principles of scientific
method, in reference to the sciences of experience.

As regards my acquaintance with the tone of
thought of the older medicine, independently of the
general obligation, incumbent on every educated

202 ON THOUGHT IN MEDICINE.

physician, of understanding the literature of his science
and the direction as well as the conditions of its
progress, there was in my case a special incentive. In
my first professorship at Konigsberg, from the year
1849 to 1856, I had to lecture each winter on general
pathology — that is, on that part of the subject which
contains the general theoretical conceptions of the
nature of disease, and of the principles of its treatment.

General pathology was regarded by our elders as
the fairest blossom of medical science. But in fact,
that which formed its essence possesses only historical
interest for the disciples of modern natural science.

Many of my predecessors have broken a lance for
the scientific defence of this essence, and more especially
Jlenle and Lotz. The latter, whose starting-point was
also medicine, had, in his general pathology and thera-
peutics, arranged it very thoroughly and methodically
and with great critical acumen.

My own original inclination was towards physics ;
external circumstances compelled me to commence the
study of medicine, which was made possible to me by
the liberal arrangements of this Institution. It had,
however, been the custom of a former time to combine
the study of medicine with that of the Natural Sciences,
and whatever in this was compulsory I must consider
fortunate ; not merely that I entered medicine at a
time in which any one who was even moderately at

ON THOUGHT IN MEDICINE. 203

tome in physical considerations found a fruitful vir-
gin soil for cultivation; but I consider the study of
medicine to have been that training which preached
more impressively and more convincingly than any
other could have done, the everlasting principles of all
scientific work; principles which are so simple and yet
are ever forgotten again ; so clear and yet always hidden
by a deceptive veil.

Perhaps only he can appreciate the immense im-
portance a nd the fearful practical scope of the problems
of medical theory, who has watched the fading eye of
approaching death, and witnessed the distracted grief
of affection, and who has asked himself the solemn
questions, Has all been done which could be done to
ward off the dread event ? Have all the resources and
all the means which Science has accumulated become
exhausted ?

Provided that he remains undisturbed in his study,
the purely theoretical inquirer may smile with calm
contempt when, for a time, vanity and conceit seek
to swell themselves in science and stir up a commo-
tion. Or he may consider ancient prejudices to be
interesting and pardonable, as remains of poetic ro-
mance, or of youthful enthusiasm. To one who has to
contend with the hostile forces of fact, indifference
and romance disappear ; that which he knows and can
do, is exposed to severe tests ; he can only use the

204 OX THOUGHT IN MEDICINE.

hard and clear light of facts, and must give up the notion
of lulling himself in agreeable illusions.

I rejoice, therefore, that I can once more address
an assembly consisting almost exclusively of medical
men who have gone through the same school. Medicine
was once the intellectual home in which I grew up, and
even the emigrant best understands and is best under-
fit ood by his native land.

If I am called upon to designate in one word the
fundamental error of that former time, I should be in-
clined to say that it pursued a false ideal of science in
a one-sided and erroneous reverence for the deductive
method. Medicine, it is true, was not the only science
which was involved in this error, but in no other
science have the consequences been so glaring, or have
so hindered progress, as in medicine. The history of
this science claims, therefore, a special interest in the
history of the development of the human mind. None
other is, perhaps, more fitted to show that a true
criticism of the sources of cognition is also prac-
tically an exceedingly important object of true philo-
sophy.

The proud word of Hippokrates,

irjTpoz <f>t\6(TO(j)o^ IffodeoQ,

4 Godlike is the physician who is a philosopher,' served,
as it were, as a banner of the old deductive medicine.
We may admit this if only we once agree what wo

. ON THOUGHT IN MEDICINE. 205

are to understand as a philosopher. For the ancients,
philosophy embraced all theoretical knowledge ; their
philosophers pursued Mathematics, Physics, Astronomy,
Natural History, in close connection with true philo-
sophical or metaphysical considerations. If, therefore,
we are to understand the medical philosopher of Hip-
pokrates to be a man who has a perfected insight into
the causal connection of natural processes, we shall in
fact be able to say with Hippokrates, Such a one can
give help like a god.

Understood in this sense, the aphorism describes
in three words the ideal which our science has to strive
after. But who can allege that it will ever attain
this ideal ?

But those disciples of medicine who thought them-
selves divine even in their own lifetime, and who
wished to impose themselves upon others as such, were
not inclined to postpone their hopes for so long a
period. The requirements for the (f>L\6a-o^>os were
considerably moderated. Every adherent of any given
cosmological system, in which, for well or ill, facts
must be made to correspond with reality, felt himself to
be a philosopher. The philosophers of that time knew
little more of the laws of Nature than the unlearned
layman; but the stress of their endeavours was laid upon
thinking, upon the logical consequence and complete-
ness of the system. It is not difficult to understand

206 ON THOUGHT IN MEDICINE. •

how in periods of youthful development, such a one-
sided over-estimate of thought could be arrived at,
The superiority of man over animals, of the scholar
over the barbarian, depends upon thinking ; sensation,
feeling, perception, on the contrary, he shares with his
lower fellow-creatures, and in acuteness of the senses
many of these are even superior to him. That man
strives to develop his thinking faculty to the utmost
is a problem on the solution of which the feeling of
his own dignity, as well as of his own practical power,
depends ; and it is a natural error to have considered
unimportant the dowry of mental capacities which
Nature had given to animals, and to have believed that
thought could be liberated from its natural basis,
observation and perception, to begin its Icarian flight
of metaphysical speculation.

It is, in fact, no easy problem to ascertain com-
pletely the origins of our knowledge. An enormous
amount is transmitted by speech and writing. This
power which man possesses of gathering together the
stores of knowledge of generations, is the chief reason
of his superiority over the animal, who is restricted
to an inherited blind instinct and to its individual
experience. But all transmitted knowledge is handed
on already formed; whence the reporter has derived
it, or how much criticism he has bestowed upon it,
can seldom be made out, especially if the tradition haw

OjST THOUGHT IN MEDICINE. 207

been handed down through several generations. We
must admit it all upon good faith; we cannot arrive
at the source ; and when many generations have con-
tented themselves with such knowledge, have brought
no criticism to bear upon it ; have, indeed, gradually
added all kinds of small alterations, which ultimately
grew up to large ones — after all this, strange things are
often reported and believed under the authority of
primeval wisdom. A curious case of this kind is the
history of the circulation of the blood, of which we
shall still have to speak.

But another kind of tradition by speech, which
long remained undetected, is even still more confusing
for one who reflects upon the origin of knowledge.
Speech cannot readily develop names for classes of
objects or for classes of processes, if we have not been ac-
customed very often to mention together the correspond-
ing individuals, things, and separate cases, and to assert
what there is in common about them. They must,
therefore, possess many points in common. Or if we,
reflecting scientifically upon this, select some of these
characteristics, and collate them to form a definition,
the common possession of these selected characteristics
must necessitate that in the given cases a great num-
ber of other characteristics are to be regularly met
with ; there must be a natural connection between the
(kst and the last-named characteristics. If, for instance,

208 ON THOUGHT IN MEDICINE.

we assign the name of mammals to those animals which >
when young, are suckled by their mothers, we can
assert further, in reference to them, that they are all
warm-blooded animals, born alive, that they have a
spinal column but no quadrate bone, breathe through
lungs, have separate divisions of the heart, &c. Hence
the fact, that in the speech of an intelligent observing
people a certain class of things are included in one
name, indicates that these things or cases fall under a
common natural relationship ; by this alone a host of
experiences are transmitted from preceding generations
without this appearing to be the case.

The adult, moreover, when he begins to reflect upon
the origin of his knowledge, is in possession of a huge
mass of every-day experiences, which in great part
reach back to the obscurity of his first childhood.
Everything individual has long been forgotten, but
the similar traces which the daily repetition of similar
cases has left in his memory have deeply engraved
themselves. And since only that which is in con-
formity with law is always repeated with regularity,
these deeply impressed remains of all previous con-
ceptions are just the conceptions of what is conform-
able to law in the things and processes.

Thus man, when he begins to reflect, finds that he
possesses a wide range of acquirements of which he
knows not whence they came, which he has possessed

ON THOUGHT IN MEDICINE. 209

as long as lie can remember. We need not refer even
to the possibility of inheritance by procreation.

The conceptions which he has formed, which his
mother tongue has transmitted, assert themselves as
regulative powers, even in the objective world of fact,
and as he does not know that he or his forefathers have
developed these conceptions from the things them-
selves, the world of facts seems to him, like his con-
ceptions, to be governed by intellectual forces. We
recognise this psychological anthropomorphism, from
the Ideas of Plato, to the immanent dialectic of the
cosmical process of Hegel, and to the unconscious will
of Schopenhauer.

Natural science, which in former times was virtually
identical with medicine, followed the path of philoso-
phy ; the deductive method seemed to be capable of
doing everything. Socrates, it is true, had developed
the inductive conception in the most instructive
manner. But the best which he accomplished remained
virtually misunderstood.

I will not lead you through the motley confusion of
pathological theories which, according to the varying
inclination of their authors, sprouted up in consequence
of this or the other increase of natural knowledge, and
were mostly put forth by physicians, who obtained
fame and renown as great observers and empirics, inde-
pendently of their theories. Then came the less gifted

210 ON THOUGHT IN MEDICINE.

pupils, who copied their master, exaggerated his theory ,
made it more one-sided and more logical, without
regard to any discordance with Nature. The more
rigid the system, the fewer and the more thorough
were the methods to which the healing art was re-
stricted. The more the schools were driven into a
corner by the increase in actual knowledge, the more
did they depend upon the ancient authorities, and the
more intolerant were they against innovation. The
great reformer of anatomy, Vesalius, was cited before
the Theological faculty of Salamanca; Servetus was
burned at Geneva along with his book, in which he
described the circulation of the lungs ; and the Paris
faculty prohibited the teaching of Harvey's doctrine of
the circulation of the blood in its lecture rooms.

At the same time the bases of the systems from
which these schools started were mostly views on
natural science which it would have been quite right
to utilise within a narrow circle. What was not
right was the delusion that it was more scientific to
refer all diseases to one kind of explanation, than to
several. What was called the solidar pathology wanted to
deduce everything from the altered mechanism of the
solid parts, especially from their altered tension ; from
the strictum and laxum, from tone and want of tone,
and afterwards from strained or relaxed nerves and from
obstructions in the vessels. Humoral pathology was

ON THOUGHT IN MEMCINE. 211

only acquainted with alterations in mixture. The four
cardinal fluids, representatives of the classical four
elements, blood, phlegm, black and yellow gall ; with
others, the acrimonies or dyscrasies, which had to be
expelled by sweating and purging ; in the beginning of
our modern epoch, the acids and alkalies or the alchy-
mistic spirits, and the occult qualities of the substances
assimilated — all these were the elements of this chem-
istry. Along with these were found all kinds of phy-
siological conceptions, some of which contained remark-
able foreshadowings, such as the sjufrvrov 0e/?/*oz/, *the
inherent vital force of Hippokrates, which is kept up
by nutritive substances, this again boils in the stomach
and is the source of all motion ; here the thread is
begun to be spun which subsequently led a physician
to the law of the conservation of force. On the other
hand, the irvsv/Aa, which is half spirit and half air,
which can be driven from the lungs into the arteries
and fills them, has produced much confusion. The
fact that air is generally found in the arteries of
dead bodies, which indeed only penetrates in the
moment in which the vessels are cut, led the ancients
to the belief that air is also present in the arteries
during life. The veins only remained then in which
blood could circulate. It was believed to be formed
in the liver, to move from there to the heart, and
through the veins to the organs. Any careful ob-

212 ON THOUGHT IN MEDICINE.

serration of the operation of blood-letting must have
taught that, in the veins, it comes from the periphery,
and flows towards the heart. But this false theory
had become so mixed up with the explanation of fever
and of inflammation, that it acquired the authority of
a dogma, which it was dangerous to attack.

Yet the essential and fundamental error of this
system was, and still continued to be, the false kind of
logical conclusion to which it was supposed to lead ;
the conception that it must be possible to build a
complete system which would embrace all forms of dis-
ease, and their cure, upon any one such simple explana-
tion. Complete knowledge of the causal connection of
one class of phenomena gives finally a logical coherent
system. There is no prouder edifice of the most exact
thought than modern astronomy, deduced even to the
minutest of its small disturbances, from Newton's law of
gravitation. But Newton had been preceded by Kepler,
who had by induction collated all the facts; and the
astronomers have never believed that Newton's force
excluded the simultaneous action of other forces. They
have been continually on the watch to see whether
friction, resisting media, and swarms of meteors have
not also some influence. The older philosophers and
physicians believed they could deduce, before they had
settled their general principles by induction. They
forgot that a deduction can have no more certainty than

ON THOUGHT IN MEDICINE. 213

the principle from which it is deduced ; and that each
new induction must in the first place be a new test, by
experience, of its own bases. That a conclusion is de-
duced by the strictest logical method from an uncertain
premiss does not give it a hair's breadth of certainty
or of value.

One characteristic of the schools which built up
their system on such hypotheses, which they assumed
as dogmas, is the intolerance of expression which I have
already partially mentioned. One who works upon a well-
ascertained foundation may readily admit an error ; he
loses, by so doing, nothing more than that in which he
erred. If, however, the starting-point has been placed
upon a hypothesis, which either appears guaranteed by
authority, or is only chosen because it agrees with that
which it is wished to believe true, any crack may then
hopelessly destroy the whole fabric of conviction. The
convinced disciples must therefore claim for each
individual part of such a fabric the same degree of
infallibility; for the anatomy of Hippokrates just as
much as for fever crises; every opponent must only
appear then as stupid or depraved, and the dispute will
thus, according to old precedent, be so much the more
passionate and personal, the more uncertain is the basis
which is defended. We have frequent opportunities
of confirming these general rules in the schools of
dogmatic deductive medicine. They turned their in-

214 ON THOUGHT IN MEDICINE.

tolerance partly against each other, and partly against
the eclectics who found various explanations for various
forms of disease. This method, which in its essence is
completely justified, had, in the eyes of systcmatists,
the defect of being illogical. And yet the greatest
physicians and observers, Hippokrates at the head,
Aretseus, Galen, Sydenham, and Boerhaave, had become
eclectics, or at any rate very lax systematists.

About the time when we seniors commenced ihe
study of medicine, it was still under the influence of
the important discoveries which Albrecht von Haller had
made on the excitability of nerves ; and which he had
placed in connection with the vitalistic theory of the
nature of life. Haller had observed the excitability in
the nerves and muscles of amputated members. The
most surprising thing to him was, that the most varied
external actions, mechanical, chemical, thermal, to
which electrical ones were subsequently added, had
always the same result; namely, that they produced
muscular contraction. They were only quantitatively
distinguished as regards their action on the organism,
that is, only by the strength of the excitation; he
designated them by the common name of stimulus;
he called the altered condition of the nerve the exci-
tation, and its capacity of responding to a stimulus
the excitability, which was lost at death. This entire
condition of things, which physically speaking asserts

ON THOUGHT IN MEDICINE. 215

no mere than that the nerves, as concerns the changes
which take place in them after excitation, are in an
exceedingly unstable state of equilibrium ; this was
looked upon as the fundamental property of animal
life, and was unhesitatingly transferred to the other
organs and tissues of the body, for which there was no
similar justification. It was believed that none of
them were active of themselves, but must receive an
impulse by a stimulus from without; air and nourish-
ment were considered to be the normal stimuli. The
kind of activity seemed, on the contrary, to be con-
ditioned by the specific energy of the organ, under the
influence of the vital force. Increase or diminution
of the excitability was the category under which the
whole of the acute diseases were referred, and from
which indications were taken as to whether the treat-
ment should be lowering or stimulating. The rigid
one-sidedness and the unrelenting logic with which
Robert Brown had once worked out this system was
broken, but it always furnished the leading points of
view.

The vital force had formerly lodged as ethereal
spirit, as a Pneuma in the arteries ; it had then with
Paracelsus acquired the form of an Archeus, a kind
of useful Kobold, or indwelling alchymist, and had
acquired its clearest scientific position as ' soul of
life, oni^rta inscia, in Georg Ernst Stahl, who, in

216 ON THOUGHT IN MEDICINE.

the first half of the last century, was professor of
chemistry and pathology in Halle. Stahl had a clear
and acute mind, which is informing and stimulating,
from the way in which he states the proper question,
even in those cases in which he decides against our pre-
sent views. He it is who established the first compre-
hensive system of chemistry, that of phlogiston. If we
translate his phlogiston into latent heat, the theoretical
bases of his system passed essentially into the system
of Lavoisier ; Stahl did not then know oxygen, which
occasioned some false hypotheses ; for instance, on the
negative gravity of phlogiston. Stahl's * soul of life '
is, on the whole, constructed on the pattern on which
the pietistic communities of that period represented to
themselves the sinful human soul; it is subject to
errors and passions, to sloth, fear, impatience, sorrow^
indiscretion, despair. The physician must first appease
it, or then incite it, or punish it, and compel it to
repent. And the way in which, at the same time, he
established the necessity of the physical and vital
actions was well thought out. The soul of life governs
the body, and only acts by means of the physico-
chemical forces of the substances assimilated. But it
.has the power to bind and to loose these forces, to allow
them full play or to restrain them. After death the
restrained forces become free, and evoke putrefaction
and decomposition. For the refutation of this hypo-

ON THOUGHT IN MEDICINE. 217

thesis of binding and loosing, it was necessary to dis
cover the law of the conservation of force.

The second half of the previous century was too
much possessed by the principles of rationalism to recog-
nise openly Stahl's 'soul of life.' It was presented
more scientifically as vital force, Vis vitalis, while in
the main it retained its functions, and under the name
of ' Nature's healing power ' it played a prominent part
in the treatment of diseases.

The doctrine of vital force entered into the patho
logical system of changes in irritability. The attempt
was made to separate the direct actions of the virus
which produce disease, in so far as they depended on
the play of blind natural forces, the symptomata morbi,
from those which brought on the reaction of vital force,
the symptomata reactionis. The latter were princi-
pally seen in inflammation and in fever. It was the
function of the physician to observe the strength of
this reaction, and to stimulate or moderate it accord-
ing to circumstances.

The treatment of fever seemed at that time to be
the chief point ; to be that part of medicine which had a
real scientific foundation, and in which the local treat-
ment fell comparatively into the background. The the-
rapeutics of febrile diseases had thereby become very
monotonous, although the means indicated by theory
were still abundantly used, and especially blood-letting,

213 ON THOUGHT IN MEDICINE.

which since that time has almost been entirely
doned. Therapeutics became still more impoverished as
the younger and more critical generation grew up, and
tested the assumptions of that which was considered to
be scientific. Among the younger generation were
many who, in despair as to their science, had almost
entirely given up therapeutics, or on principle had
grasped at an empiricism such as Rademacher then
taught, which regarded any expectation of a scientific
explanation as a vain hope.

What we learned at that time were only the ruins
of the older dogmatism, but their doubtful features
soon manifested themselves.

The vitalistic physician considered that the essen-
tial part of the vital processes did not depend upon
natural forces, which, doing their work with blind
necessity and according to a fixed law, determined the
result. What these forces could do appeared quite
subordinate, and scarcely worthy of a minute study.
He thought that he had to deal with a soul-like being,
to which a thinker, a philosopher, and an intelligent
man must be opposed. May I elucidate this by a few
outlines ?

At this time auscultation and percussion of the
organs of the chest were being regularly practised in
the clinical wards. But I have often heard it main-
tained that they were a coarse mechanical means of

ON THOUGHT IN MEDICINE. 219

investigation which a physician with a clear mental
vision did not need ; and it indeed lowered and debased
the patient, who was anyhow a human being, by treat-
ing him as a machine. To feel the pulse seemed the
most direct method of learning the mode of action of
the vital force, and it was practised, therefore, as by far
the most important means of investigation. To count
with a repeater was quite usual, but seemed to the
old gentlemen as a method not quite in good taste.
There was, as yet, no idea of measuring temperature in
cases of disease. In reference to the ophthalmoscope, a
celebrated surgical colleague said to me that he would
never use the instrument, it was too dangerous to admit
crude light into diseased eyes ; another said the mirror
might be useful for physicians with bad eyes, his, how-
ever, were good, and he did not need it.

A professor of physiology of that time, celebrated
for his literary activity, and noted as an orator and
intelligent man, had a dispute on the images in the eye
with his colleague the physicist. The latter challenged
the physiologist to visit him and witness the experi-
ment. The physiologist, however, refused his request
with indignation ; alleging that a physiologist had
nothing to do with experiments ; they were of no good
but for the physicist. Another aged and learned pro-
fessor of therapeutics, who occupied himself much with
the reorganisation of the Universities, was urgent willi

220 ON THOUGHT IN MEDICINE.

me to divide physiology, in order to restore the good
old time; that I myself should lecture on the really
intellectual part, and should hand over the lower
experimental part to a colleague whom he regarded as
good enough for the purpose. He quite gave me up
ivhen I said that I myself considered experiments to
be the true basis of science.

I mention these points, which I myself have ex-
perienced, to elucidate the feeling of the older schools,
and indeed of the most illustrious representatives of
medical science, in reference to the progressive set
of ideas of the natural sciences; in literature these
ideas naturally found feebler expression, for the old
gentlemen were cautious and worldly wise.

You will understand how great a hindrance to
progress such a feeling on the part of influential and
respected men must have been. The medical education
of that time was based mainly on the study of books ;
there were still lectures, which were restricted to mere
dictation ; for experiments and demonstrations in the
laboratory the provision made was sometimes good and
sometimes the reverse; there were no physiological
and physical laboratories in which the student himself
might go to work. Liebig's great deed, the founda-
tion of the chemical laboratory, was complete, as far
as chemistry was concerned, but his example had
net been imitated elsewhere. Yet medicine possessed

ON THOUGHT IN MEDICINE. 221

in anatomical dissections a great means of education
for independent observation, which is wanting in the
other faculties, and to which I am disposed to attach
great weight. Microscopic demonstrations were iso-
lated and infrequent in the lectures. Microscopic
instruments were costly and scarce. I came into pos-
session of one by having spent my autumn vacation
in 1841 in the Charite, prostrated by typhoid fever; as
pupil, I was nursed without expense, and on my re-
covery I found myself in possession of the savings of
my small resources. The instrument was not beautiful,
yet I was able to recognise by its means the prolonga-
tions of the ganglionic cells in the invertebrata, which
I described in my dissertation, and to investigate the
vibrions in my research on putrefaction and fermenta-
tion.

Any of my fellow-students who wished to make
experiments had to do so at the cost of his pocket-
money. One thing we learned thereby, which the
younger generation does not, perhaps, learn so well in
the laboratories — that is, to consider in all directions
the ways and means of attaining the end, and to ex-
haust all possibilities, in the consideration, until a prac-
ticable path was found. We had, it is true, an almost
uncultivated field before us, in which almost every
stroke of the spade might produce remunerative results.

It was one man more especially who aroused our

222 ON THOUGHT IN MEDICINE.

enthusiasm for work in the right direction — that is,
Johannes Miiller, the physiologist. In his theoretical
views he favoured the vitalistic hypothesis, but in the
most essential points he was a natural philosopher, firm
and immovable; for him, all theories were but hy-
potheses, which had to be tested by facts, and about
which facts could alone decide. Even the views upon
those points which most easily crystallise into dogmas,
on the mode of activity of the vital force and the activity
of the conscious soul, he tried continually to define more
precisely, to prove or to refute by means of facts.

And, although the art of anatomical investigation was
most familiar to him, and he therefore recurred most
willingly to this, yet he worked himself into the chemical
and physical methods which were more foreign to him.
He furnished the proof that fibrine is dissolved in blood ;
he experimented on the propagation of sound in such
mechanisms as are found in the drum of the ear;
he treated the action of the eye as an optician. His
most important performance for the physiology of the
nervous system, as well as for the theory of cognition,
was the actual definite establishment of the doctrine of
the specific energies of the nerves. In reference to the
separation of the nerves of motor and sensible energy,
he showed how to make the experimental proof of
Bell's law of the roots of the spinal cord so as to
be free from errors; and in regard to the sensible

ON THOUGHT IN MKDICIN'K 223

energies he not only established the general law, but
carried out a great number of separate investigations, to
eliminate objections, and to refute false indications and
evasions. That which hitherto had been imagined from
the data of every-day experience, and which had been
sought to be expressed in a vague manner, in which the
true was mixed up with the false ; or which had just
been established for individual branches, such as by Dr.
Young for the theory of colours, or by Sir Charles Bell
for the motor nerves, that emerged from Miiller's hands
in a state of classical perfection — a scientific achieve-
ment whose value I am inclined to consider as equal to
that of the discovery of the law of gravitation.

His scientific tendency, and more especially his ex-
ample, were continued in his pupils. We had been
preceded by Schwann, Henle, Eeichert, Peters, Eemak ;
I met as fellow-students E. Du Bois-Keymond, Virchow,
Briicke, Ludwig, Traube, J. Meyer, Lieberkiihn, Hall-
mann ; we were succeeded by A. von Graefe, W. Busch,
Max Schultze, A. Schneider.

Microscopic and pathological anatomy, the study of
organic types, physiology, experimental pathology and
therapeutics, ophthalmology, developed themselves in
Germany under the influence of this powerful impulse
far beyond the standard of rival adjacent countries.
This was helped by the labours of those of similar
ten iencies among Miiller's contemporaries, among whom

224 ON THOUGHT IN MEDICINE,

the three brothers Weber of Leipzig must first of all be
mentioned, who have built solid foundations in the
mechanism of the circulation, of the muscles, of the
joints, and of the ear.

The attack was made wherever a way could be
perceived of understanding one of the vital processes ;
it was assumed that they could be understood, and
success justified this assumption. A delicate and
copious technical apparatus has been developed in the
methods of microscopy, of physiological chemistry, and
of vivisection ; the latter greatly facilitated more par-
ticularly by the use of anaesthetic ether and of the para-
lysing curara, by which a number of deep problems
became open to attack, which to our generation seemed
hopeless. The thermometer, the ophthalmoscope, the
auricular speculum, the laryngoscope, nervous irritation
on the living body, opened out to the physician possibi-
lities of delicate and yet certain diagnosis where there
seemed to be absolute darkness. The continually in-
creasing number of proved parasitical organisms substi-
tute tangible objects for mystical entities, and teach
the surgeon to forestall the fearfully subtle diseases of
decomposition.

But do not think, gentlemen, that the struggle is at
an end. As long as there are people of such astound-
ing conceit as to imagine that they can effect, by a few
clever strokes, that which man can otherwise only hope

ON THOUGHT IN MEDICINE. 225

to achieve by toilsome labour, hypotheses will be started
which, propounded as dogmas, at once promise to
solve all riddles. And as long as there are people who
believe implicitly in that which they wish to be true,
so long will the hypotheses of the former find credence.
Both classes will certainly not die out, and to the latter
the majority will always belong.

There are two characteristics more particularly
which metaphysical systems have always possessed.
In the first place man is always desirous of feeling
himself to be a being of a higher order, far beyond the
standard of the rest of nature ; this wish is satisfied by
the spiritualists. On the other hand, he would like
to believe that by his thought he was unrestrained
lord of the world', and of course by his thinking with
those conceptions, to the development of which he
has attained ; this is attempted to be satisfied by the
materialists.

But one who, like the physician, has actively to face
natural forces which bring about weal or woe, is also
under the obligation of seeking for a knowledge of
the truth, and of the truth only ; without considering
whether, what he finds, is pleasant in one way or the
other. His aim is one which is firmly settled ; for him
the success of facts is alone finally decisive. He must
endeavour to ascertain beforehand, what will be the
result of his attack if he pursues this or that course.

ii. VI

226 ON THOUGHT IN MEDICINE.

In order to acquire this foreknowledge of what is
coming, but of what has not been settled by obser-
vations, no other method is possible than that of
endeavouring to arrive at the laws of facts by observa-
tions; and we can only learn them by induction, by the
careful selection, collation, and observation of those cases
which fall under the law. When we fancy that we have
arrived at a law, the business of deduction commences.
It is then our duty to develop the consequences of our
law as completely as may be, but in the first place only
to apply to them the test of experience, so far as they
can be tested, and then to decide by this test whether
the law holds, and to what extent. This is a test
which really never ceases. The true natural philo-
sopher reflects at each new phenomenon, whether the
best established laws of the best known forces may
not experience a change; it can of course only be a
question of a change which does not contradict the whole
store of our previously collected experiences. It never
thus attains unconditional truth, but such a high degree
of probability that it is practically equal to certainty.
The metaphysicians may amuse themselves at this ; we
will take their mocking to heart when they are in a
position to do better, or even as well. The old words
of Socrates, the prime master of inductive definitions,
in reference to them are just as fresh as they were
2,000 years ago : ( They imagined they knew what they

ON THOUGHT IN MEDICINE. 227

did not know, and he at any rate had the advantage
of not pretending to know what he did not know.'
And again, he was surprised at its not being clear to
them that it is not possible for men to discover such
things ; since even those who most prided themselves
on the speeches made on the matter, did not agree
among themselves, but behaved to each other like
madmen (ro£y {jLaivo^svois oyno/ws1).1 Socrates calls
them rovs {JU^ICTTOV fypovovvras. Schopenhauer 2 calls
himself a Mont Blanc, by the side of a mole-heap,
when he compares himself with a natural philosopher.
The pupils admire these big words and try to imitate
the master.

In speaking against the empty manufacture of hy-
potheses, do not by any means suppose that I wish to
diminish the real value of original thoughts. The first
discovery of a new law, is the discovery of a similarity
which has hitherto been concealed in the course of
natural processes. It is a manifestation of that which
our forefathers in a serious sense described as 'wit';
it is of the same quality as the highest performances
of artistic perception in the discovery of new types of
expression. It is something which cannot be forced,
and which cannot be acquired by any known method.

1 Xenophon, Nemorabil. I. i. 11.

2 Arthur Schopenhauer, Von ihm. uber ihn von Fraiicnstadt und
Lindner* Berlin, 1863, p. 653.

02

228 ON THOUGHT IN MEDICINE.

Hence all those aspire after it who wish to pass as
the favoured children of genius. It seems, too, so
easy, so free from trouble, to get by sudden mental
flashes an unattainable advantage over our contem-
poraries. The true artist and the true inquirer knows
that great works can only be produced by hard work.
The proof that the ideas formed do not merely scrape
together superficial resemblances, but are produced by
a quick glance into the connection of the whole, can
only be acquired when these ideas are completely de-
veloped— that is, for a newly discovered natural law,
only by its agreement with facts. This estimate must
by no means be regarded as depending on external
success, but the success is here closely connected with
the depth and completeness of the preliminary per-
ceptions.

To find superficial resemblances is easy; it is
amusing in society, and witty thoughts soon procure for
their author the name of a clever man. Among the
great number of such ideas, there must be some which
are ultimately found to be partially or wholly correct ;
it would be a stroke of skill always to guess falsely.
In such a happy chance a man can loudly claim his
priority for the discovery; if otherwise, a lucky
oblivion conceals the false conclusions. The adherents
of such a process are glad to certify the value of a first
thought. Conscientious workers who are shy at bring-

ON THOUGHT IN MEDICINE. 229

ing their thoughts before the public before they have
tested them in all directions, solved all doubts, and
have firmly established the proof, these are at a decided
disadvantage. To settle the present kind of questions
of priority, only by the date of their first publication,
and without considering the ripeness of the research,
has seriously favoured this mischief.

In the * type case ' of the printer all the wisdom of
the world is contained which has been or can be dis-
covered ; it is only requisite to know how the letters
are to be arranged. So also, in the hundreds of books
and pamphlets which are every year published about
ether, the structure of atoms, the theory of perception,
as well as on the nature of the asthenic fever and
carcinoma, all the most refined shades of possible hy-
potheses are exhausted, and among these there must
necessarily be many fragments of the correct theory.
But who knows how to find them ?

I insist upon this in order to make clear to you that
all this literature, of untried and unconfirmed hypo-
theses, has no value in the progress of science. On the
contrary, the few sound ideas which they may contain
are concealed by the rubbish of the rest ; and one who
wants to publish something really new — facts — sees
himself open to the danger of countless claims of
priority, unless he is prepared to waste time and power
in reading beforehand a quantity of absolutely useless

230 ON THOUGHT IN MEDICINE.

books, and to destroy his* readers' patience by a multitude
of useless quotations.

Our generation has had to suffer under the tyranny
of spiritualistic metaphysics ; the newer generation will
probably have to guard against that of the materialistic
hypotheses. Kant's rejection of the claims of pure
thought has gradually made some impression, but Kant
allowed one way of escape. It was as clear to him as
to Socrates that all metaphysical systems which up to
that time had been propounded were tissues of false
conclusions. His Kritik der reinen Vemunft is a
continual sermon against the use of the category of
thought beyond the limits of possible experience. But
geometry seemed to him to do something which meta-
physics was striving after; and hence geometrical
axioms, which he looked upon as a priori principles
antecedent to all experience, he held to be given by
transcendental intuition, or as the inherent form of
all external intuition. Since that time, pure a priori
intuition has been the anchoring-ground of metaphy-
sicians. It is even more convenient than pure thought,
because everything can be heaped on it without going
into chains of reasoning, which might be capable of
proof or of refutation. The nativistic theory of per-
ception of the senses is the expression of this theory
in physiology. All mathematicians united to fight
against any attempt to resolve the intuitions into their

ON THOUGHT IN MEDICINE. 231

natural elements ; whether the so-called pure or the
empirical, the axioms of geometry, the principles of
mechanics, or the perceptions of vision. For this
reason, therefore, the mathematical investigations of
Lobatschewsky, Gauss, and Riemann on the altera-
tions which are logically possible in the axioms of
geometry; and the proof that the axioms are principles
which are to be confirmed or perhaps even refuted by
experience, and can accordingly be acquired from ex-
perience— these I consider to be very important steps.
That all metaphysical sects get into a rage about this
must not lead you astray, for these investigations lay
the axe at the bases of apparently the firmest supports
which their claims still possess. Against those investi-
gators who endeavour to eliminate from among the per-
ceptions of the senses, whatever there may be of the
actions of memory, and of the repetition of similar im-
pressions, which occur in memory; whatever, in short,
is a matter of experience, against them it is attempted
to raise a party cry that they are spiritualists. As if
memory, experience, and custom were not also facts,
whose laws are to be sought, and which are not to be
explained away because they cannot be glibly referred
to reflex actions, and to the complex of the prolonga-
tion of ganglionic cells, and of the connection of nerve-
fibres in the brain.

Indeed, however self-evident, and however important

232 ON THOUGHT IN MEDICINE.

the principle may appear to be, that natural science has
to seek for the laws of facts, this principle is neverthe-
less often forgotten. In recognising the law found, as a
force which rules the processes in nature, we conceive it
objectively as a force, and such a reference of individual
cases to a force which under given conditions produces
a definite result, that we designate as a causal explana-
tion of phenomena. We cannot always refer to the
forces of atoms ; we speak of a refractive force, of electro-
motive and of electrodynamic force. But do not forget
the given conditions and the given result. If these
cannot be given, the explanation attempted is merely
a modest confession of ignorance, and then it is decidedly
better to confess this openly.

If any process in vegetation is referred to forces in
the cells, without a closer definition of the conditions
among which, and of the direction in which, they work,
this can at most assert that the more remote parts of
the organism are without influence; but it would be
difficult to confirm this with certainty in more than a
few cases. In like manner, the originally definite sense
which Johannes Miiller gave to the idea of reflex action,
i? gradually evaporated into this, that when an impres-
sion has been made on any part of the nervous system,
and an action occurs in any other part, this is supposed
to have been explained by saying that it is a reflex
action. Much mny be imposed upon the irresolvable

ON THOUGHT IN MEDICINE. 233

complexity of the nerve-fibres of the brain. But the
resemblance to the qualitates occultce of ancient
medicine is very suspicious.

From the entire chain of my argument it fol-
lows that what I have said against metaphysics is
not intended against philosophy. But metaphysicians
have always tried to plume themselves on being philo-
sophers, and philosophical amateurs have mostly taken
an interest in the high-flying speculations of the meta-
physicians, by which they hope in a short time, and
at no great trouble, to learn the whole of what is worth
knowing. On another occasion * I compared the rela-
tionship of metaphysics to philosophy with that of
astrology to astronomy. The former had the most
exciting interest for the public at large, and especially
for the fashionable world, and turned its alleged con-
noisseurs into influential persons. Astronomy, on the
contrary, although it had become the ideal of scientific
research, had to be content with a small number of
quietly working disciples.

In like manner, philosophy, if it gives up meta-
physics, still possesses a wide and important field, the
knowledge of mental and spiritual processes and their
laws. Just as the anatomist, when he has reached the
limits of microscopic vision, must try to gain an in-

1 Preface to the German translation of Tyndall's Scientific
> p. xxii.

234 ON THOUGHT IN MEDICINE.

sight into the action of his optical instrument, in like
manner every scientific enquirer must study minutely
the chief instrument of his research as to its capabili-
ties. The groping of the medical schools for the last
two thousand years is, among other things, an illus-
tration of the harm of erroneous views in this respect.
And the physician, the statesman, the jurist, the
clergyman, and the teacher, ought to be able to build
upon a knowledge of physical processes if they wish
to acquire a true scientific basis for their practical ac-
tivity. But the true science of philosophy has had,
perhaps, to suffer more from the evil mental habits and
the false ideals of metaphysics than even medicine
itself.

One word of warning. I should not like you to
think that my statements are influenced by personal
irritation. I need not explain that one who has such
opinions as I have laid before you, who impresses on
his pupils, whenever he can, the principle that ' a
metaphysical conclusion is either a false conclusion or
a concealed experimental conclusion,' that he is not
exactly beloved by the votaries of metaphysics or of
intuitive conceptions. Metaphysicians, like all those
who cannot give any decisive reasons to their oppo-
nents, are usually not very polite in their controversy ;
one's own success may approximately be estimated
from the increasing want of politeness in the replies.

ON THOUGHT IN MEDICINE. 235

My own researches have led me more than other
disciples of the school of natural science into contro-
versial regions; and the expressions of metaphysical
discontent have perhaps concerned me even more than
my friends, as many of you are doubtless aware.

In order, therefore, to leave my own personal opinions
quite on one side, I have allowed two unsuspected war-
rantors to speak for me — Socrates and Kant — both of
whom were certain that all metaphysical systems estab-
lished up to their time were full of empty false con-
clusions, and who guarded themselves against adding
any new ones. In order to show that the matter has
not changed, either in the last 2,000 years or in the
last 100 years, let me conclude with a sentence of one
who was unfortunately too soon taken away from us,
Frederick Albert Lange, the author of the ' History of
Materialism.' In his posthumous 'Logical Studies,'
which he wrote in anticipation of his approaching end,
he gives the following picture, which struck me because
it would hold just as well in reference to solidar or
humoral pathologists, or any other of the old dogmatic
schools of medicine.

Lange says : The Hegelian ascribes to the Herbartian
a less perfect knowledge than to himself, and conversely ;
but neither hesitates to consider the knowledge of the
other to be higher compared with that of the empiricist,
and to recognise in it at any rate an approximation to

236 ON THOUGHT IN MEDICINE.

the only true knowledge. It is seen, also, that here no
regard is paid to the validity of the proof, and that a
mere statement in the form of a deduction from the
entirety of a system is recognised as * apodictic know-
ledge.'

Let us, then, throw no stones at our old medical
predecessors, who in dark ages, and with but slight
preliminary knowledge, fell into precisely the same
errors as the great intelligences of what wishes to be
thought the illuminated nineteenth century. They did
no worse than their predecessors except that the non-
cense of their method was more prominent in the
matter of natural science. Let us work on. In this work
of true intelligence physicians are called upon to play
a prominent part. Among those who are continually
called upon actively to preserve and apply their know-
ledge of nature, you are those who begin with the best
mental preparation, and are acquainted with the most
varied regions of natural phenomena.

In order, finally, to conclude our consultation on the
condition of Dame Medicine correctly with the epikri-
sis, I think we have every reason to be content with the
success of the treatment which the school of natural
science has applied, and we can only recommend the
younger generation to continue the same therapeutics.

237

ON

ACADEMIC FBEEDOM

IX

GEBMAN UNIVEESITIES.

Inaugural Address as Rector of the Frederick William
University of Berlin. Delivered October 15, 1877.

IN entering upon the honourable office to which the
confidence of my colleagues has called me, my first
duty is once more openly to express my thanks to
those who have thus honoured me by their confidence.
I have the more reason to appreciate it highly, as it
was conferred upon me, notwithstanding that I have
been but few years among you, and notwithstanding
that I belong to a branch of natural science which
has come within the circle of University instruction
in some sense as a foreign element ; which has necessi-
tated many changes in the old order of University
teaching, and which will, perhaps, necessitate other
changes. It is indeed just in that branch (Physics)

238 ON ACADEMIC FREEDOM IN QEBMAN UNIVERSITIES,

which I represent, and which forms the theoretical
basis of all other branches of Natural Science, that
the particular characteristics of their methods are
most definitely pronounced. I have already been seve-
ral times in the position of having to propose altera-
tions in the previous regulations of the University,
and I have always had the pleasure of meeting with
the ready assistance of my colleagues in the faculty,
and of the Senate. That you have made me the
Director of the business of this University for this
year, is a proof that you regard me as no thought-
less innovator. For, in fact, however the objects, the
methods, the more immediate aims of investigations
in the natural sciences may differ externally from
those of the mental sciences, and however foreign
their results and however remote their interest may
often appear, to those who are accustomed only to the
direct manifestations and products of mental activity,
there is in reality, as I have endeavoured to show in my
discourse as Kector at Heidelberg, the closest connec-
tion in the essentials of scientific methods, as well as
in the ultimate aims of both classes of the sciences,
Even if most of the objects of investigation of the
natural sciences are not directly connected with the
interests of the mind, it cannot, on the other hand, be
forgotten that the power of true scientific method
stands ant in the natural sciences far more promi-

ON ACADEMIC FREEDOM IN GEKMAN UNIVERSITIES. 239

nently — that the real is far more sharply separated
from the unreal, by the incorruptible criticism of
facts, than is the case with the more complex problems
of mental science.

And not merely the development of this new side
of scientific activity, which " was almost unknown to
antiquity, but also the influence of many political,
social, and even international relationships make
themselves felt, and require to be taken into account.
The circle of our students has had to be increased ;
a changed national life makes other demands upon
those who are leaving; the sciences become more and
more specialised and divided ; exclusive of the libraries,
larger and more varied appliances for study are re-
quired. We can scarcely foresee what fresh demands
and what new problems we may have to meet in the
more immediate future.

On the other hand, the German Universities have
conquered a position of honour not confined to their
fatherland; the eyes of the civilised world are upon them.
Scholars speaking the most different languages crowd
towards them, even from the farthest parts of the
earth. Such a position would be easily lost by a false
step, but would be difficult to regain.

Under these circumstances it is our duty to get a
clear understanding of the reason for the previous pro-
sperity of our Universities ; we must try to find what is

240 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES.

the feature in their arrangements which we must seek
to retain as a precious jewel, and where, on the contrary,
we may give way when changes are required. I consider
myself by no means entitled to give a final opinion on
this matter. The point of view of any single indi-
vidual is restricted; representatives of other sciences
will he able to contribute something. But I think
that a final result can only be arrived at when each one
becomes clear as to the state of things as seen from his
point of view.

The European Universities of the Middle Age had
their origin as free private unions of their students,
who came together under the influence of celebrated
teachers, and themselves arranged their own affairs.
In recognition of the public advantage of these unions
they soon obtained from the State, privileges and
honourable rights, especially that of an independent
jurisdiction, and the right of granting academic de-
grees. The students of that time were mostly men
of mature years, who frequented the University more
immediately for their own instruction and without any
direct practical object; but younger men soon began to
be sent, who, for the most part, were placed under the
superintendence of the older members. The separate
Universities split again into closer economic unions,
under the name of 'Nations,' 'Bursaries,' 'Colleges,'
v.'hose older members, the seniors, governed the com-

ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 241

mon affairs of each such union, and also met together
for regulating the common affairs of the University.
In the courtyard of the University of Bologna are still
to be seen the coats-of-arms, and lists of members and
seniors, of many such Nations in ancient times. The
older graduated members were regarded as permanent
life members of such Unions, and they retained the
right of voting, as is still the case in the College of
Doctors in the University of Vienna, and in the Col-
leges of Oxford and of Cambridge, or was until recently.
Such a free confederation of independent men, in
which teachers as well as taught were brought together
by no other interest than that of love of science ;
some by the desire of discovering the treasure of
mental culture which antiquity had bequeathed, others
endeavouring to kindle in a new generation the ideal
enthusiasm which had animated their lives. Such was
the origin of Universities, based, in the conception,
and in the plan of their organisation, upon the most
perfect freedom. But we must not think here of
freedom of teaching in the modern sense. The majority
was usually very intolerant of divergent opinions. Not
^infrequently the adherents of the minority were com-
pelled to quit the University in a body. This was not
restricted to those cases in which the Church inter-
meddled, and where political or metaphysical proposi-
tions were in question. Even the medical faculties —
n K

242 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES.

that of Paris, the most celebrated of all at the bead
— allowed no divergence from that which they re-
garded as the teaching of Hippocrates. Anyone who
used the medicines of the Arabians or who believed
in the circulation of the blood was expelled.

The change, in the Universities, to their present
constitution, was caused mainly by the fact that the
State granted to them material help, but required, on
the other hand, the right of co-operating in their
management. The course of this development was
different in different European countries, partly owing
to divergent political conditions and partly to that of
national character.

Until lately, it might have been said that the
least change has taken place in the old English Uni-
versities, Oxford and Cambridge. Their great endow-
ments, the political feeling of the English for the reten-
tion of existing rights, had excluded almost all change,
even in directions in which such change was urgently
required. Until of late both Universities had in great
measure retained their character as schools for the
clergy, formerly of the Eoman and now of the Anglican
Church, whose instruction laymen might also share in
so far as it could serve the general education of the
mind ; they were subjected to such a control and mode
of life, as was formerly considered to be good for young
priests. They lived, as they still live, in colleges, under

ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 243

the superintendence of a number of older graduate mem-
bers (Fellows) of the College ; in other respects in the
style and habits of the well-to-do classes in England.

The range and the method of the instruction is a
more highly developed gymnasial instruction ; though
in its limitation to what is afterwards required in the
examination, and in the minute study of the contents
of prescribed text-books, it is more like the Kepeti-
toria which are here and there held in our Univer-
sities. The acquirements of the students are controlled
by searching examinations for academical degrees, in
which very special knowledge is required, though only
for limited regions. By such examinations the aca-
demical degrees are acquired.

While the English Universities give but little for
the endowment of the positions of approved scientific
teachers, and do not logically apply even that little for
this object, they have another arrangement which is
apparently of great promise for scientific study, but
which has hitherto not effected much; that is the
institution of Fellowships. Those who have passed
the best examinations are elected as Fellows of their
college, where they have a home, and along with this,
a respectable income, so that they can devote the whole
of their leisure to scientific pursuits. Both Oxford and
Cambridge have each more than 500 such fellowships.
The Fellows may, but need not act as tutors for the

H ?

2-14 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES.

students. They need not even live in the University
Town, but may spend their stipends where they like,
and in many cases may retain the fellowships for an
indefinite period. With some exceptions, they only lose
it in case they marry, or are elected to certain offices.
They are the real successors of the old corporation
of students, by and for which the University was
founded and endowed. But however beautiful this
plan may seem, and notwithstanding the enormous
sums devoted to it, in the opinion of all unprejudiced
Englishmen it does but little for science ; manifestly
because most of these young men, although they are
the pick of the students, and in the most favourable
conditions possible for scientific work, have in their
student-career not come sufficiently in contact with
the living spirit of inquiry, to work on afterwards on
their own account, and with their own enthusiasm.

In certain respects the English Universities do
a great deal. They bring up their students as cul-
tivated men, who are expected not to break through
the restrictions of their political and ecclesiastical
party, and, in fact, do not thus break through. Tn
two respects we might well endeavour to imitate
them. In the first place, together with a lively feeling
for the beauty and youthful freshness of antiquity,
they develop in a high degree a sense for delicacy
and precision in writing which shows itself in the

ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 245

way in which they handle their mother-tongue. I
fear that one of the weakest sides in the instruction
of German youth is in this direction. In the second
place the English Universities, like their schools, take
greater care of the bodily health of their students.
They live and work in airy, spacious buildings, sur-
rounded by lawns and groves of trees ; they find much
of their pleasure in games which excite a passionate
rivalry in the development of bodily energy and skill,
and which in this respect are far more efficacious
than our gymnastic and fencing exercises. It must
not be forgotten that the more young men are cut off
from fresh air and from the opportunity of vigorous
exercise, the more induced will they be to seek an
apparent refreshment in the misuse of tobacco and of
intoxicating drinks. It must also be admitted that
the English Universities accustom their students to
energetic and accurate work, and keep them up to
the habits of educated society. The moral effect of the
more rigorous control is said to be rather illusory.

The Scotch Universities and some smaller English
foundations of more recent origin — University College
and King's College in London, and Owens College in
Manchester — are constituted more on the German and
Dutch model.

The development of French Universities has been
quite different, and indeed almost in the opposite

246 ON ACADEMIC FEEEDOM IN GERMAN UNIVERSITIES.

direction. In accordance with the tendency of the
French to throw overboard everything of historic de-
velopment to suit some rationalistic theory, their
faculties have logically become purely institutes for
instruction — special schools, with definite regulations
for the course of instruction, developed and quite dis-
tinct from those institutions \vhich are to further the
progress of science, such as the College de France, the
Jardin des Plantes, and the Ecole des Etudes Su~
perieures. The faculties are entirely separated from
one another, even when they are in the same town.
The course of study is definitely prescribed, and is
controlled by frequent examinations. French teaching
is confined to that which is clearly established, and
transmits this in a well-arranged, well worked-out
manner, which is easily intelligible, and does not ex-
cite doubt nor the necessity for deeper inquiry. The
teachers need only possess good receptive talents.
Thus in France it is looked upon as a false step when
a young man of promising talent takes a professorship
in a faculty in the provinces. The method of instruc-
tion in France is well adapted to give pupils, of even
moderate capacity, sufficient knowledge for the routine
of their calling. They have no choice between different
teachers, and they swear in verba magistri; this gives
a happy self-satisfaction and freedom from doubts. If
the teacher has been well chosen, this is sufficient in

ON ACADEMIC FKEEDOM IN GERilAN UNIVERSITIES. 247

ordinary cases, in which the pupil does what he has
seen his teacher do. It is only unusual cases that test
how much actual insight and judgment the pupil has
acquired. The French people are moreover gifted,
vivacious, and ambitious, and this corrects many de-
fects in their system of teaching.

A special feature in the organisation of French
Universities consists in the fact that the position of
the teacher is quite independent of the favour of his
hearers ; the pupils who belong to his faculty are
generally compelled to attend his lectures, and the far
from inconsiderable fees which they pay flow into the
chest of the Minister of Education ; the regular salaries
of the University professors are defrayed from this
source ; the State gives but an insignificant contri-
bution towards the maintenance of the University.
When, therefore, the teacher has no real pleasure in
teaching, or is not ambitious of having a number of
pupils, he very soon becomes indifferent to the success
of his teaching, and is inclined to take things easily.

Outside the lecture-rooms, the French students
live without control, and associate with young men of
other callings, without any special esprit de corps or
common feeling.

The development of the German Universities differs
characteristically from these two extremes. Too poor
in their own possessions not to be compelled, with

248 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES.

increasing demands for the means of instruction, eagerly
to accept the help of the State, and too weak to re-
sist encroachments upon their ancient rights in times
in which modern States attempt to consolidate them-
selves, the German Universities have had to submit
themselves to the controlling influence of the State.
Owing to this latter circumstance the decision in all
important University matters has in principle been
transferred to the State, and in times of religious or
political excitement this supreme power has occasionally
been unscrupulously exerted. But in most cases the
States which were working out their own independence
were favourably disposed towards the Universities ;
they required intelligent officials, and the fame of their
country's University conferred a certain lustre upon the
Government. The ruling officials were, moreover, for
the most part students of the University; they re
mained attached to it. It is very remarkable how
among wars and political changes in the States fight-
ing with the decaying Empire for the consolidation of
their young sovereignties, while almost all other privi-
leged orders were destroyed, the Universities of Germany
saved a far greater nucleus of their internal freedom
and of the most valuable side of this freedom, than in
conscientious Conservative England, and than in France
with its wild chase after freedom.

We have retained the old conception of students, as

.

ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 24D

that of young men responsible to themselves, striving
after science of their own free will, and to whom it is
left to arrange their own plan of studies as they think
best. If attendance on particular lectures was enjoined
for certain callings — what are called ' compulsory lec-
tures ' — these regulations were not made by the Univer-
sity, but by the State, which was afterwards to admit
candidates to these callings. At the same time the
students had, and still have, perfect freedom to migrate
from one German University to another, from Dorpat
to Zurich, from Vienna to Gratz; and in each University
they had free choice among the teachers of the same
subject, without reference to their position as ordinary
or extraordinary professors or as private docents. The
students are, in fact, free to acquire any part of their
instruction from books ; it is highly desirable that the
works of great men of past times should form an essen-
tial part of study.

Outside the University there is no control over the
proceedings of the students, so long as they do not
come in collision with the guardians of public order.
Beyond these cases the only control to which they are
subject is that of their colleagues, which prevents
them from doing anything which is repugnant to the
feeling of honour of their own body. The Universities
of the Middle Ages formed definite close corporations,
with their own jurisdiction, which extended to the

250 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES.

right over life and death of their own members. As they
lived for the- most part on foreign soil, it was necessary
to have their own jurisdiction, partly to protect the
members from the caprices of foreign judges, partly to
keep up that degree of respect and order, within the
society, which was necessary to secure the continuation
of the rights of hospitality on a foreign soil; and
partly, again, to settle disputes among the members.
In modern times the remains of this academic juris-
diction have by degrees been completely transferred
to the ordinary courts, or will be so transferred ; but it
is still necessary to maintain certain restrictions on a
union of strong and spirited young men, which guar-
antee the peace of their fellow-students and that of
the citizens. In cases of collision this is the object of
the disciplinary power of the University authorities.
This object, however, must be mainly attained by the
sense of honour of the students ; and it must be con-
sidered fortunate that German students have retained
a vivid sense of corporate union, and of what is inti-
mately connected therewith, a requirement of honour-
able behaviour in the individual. I am by no means
prepared to defend every individual regulation in the
Codex of Students' Honour; there are many Middle
Age remains among them which were better swept
away; but that can only be done by the students
themselves.

ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 251

For most foreigners the uncontrolled freedom of
German students is a subject of astonishment; the
more so as it is usually some obvious excrescences
of this freedom which first meet their eyes ; they are
unable to understand how young men can be so left
to themselves without the greatest detriment. The
German looks back to his student life as to his golden
age ; our literature and our poetry are full of expres-
sions of this feeling. Nothing of this kind is but
even faintly suggested in the literature of other Euro-
pean peoples. The German student alone has this
perfect joy in the time, in which, in the first delight in
youthful responsibility, and freed more immediately
from having to work for extraneous interests, he can
devote himself to the task of striving after the best and
noblest which the human race has hitherto been able to
attain in knowledge and in speculation, closely joined
in friendly rivalry with a large body of associates of
similar aspirations, and in daily mental intercourse
with teachers from whom he learns something of the
workings of the thoughts of independent minds.

When I think of my own student life, and of the
impression which a man like Johannes Miiller. the
physiologist, made upon us, I must place a very high
value upon this latter point. Anyone who has once
come in contact with one or more men of the first rank
must have had his whole mental standard altered for

2.52 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES.

the rest of his life. Such intercourse is, moreover, the
most interesting that life can offer.

You, my younger friends, have received in this
freedom of the German students a costly and valuable
inheritance of preceding generations. Keep it — and
hand it on to coming races, purified and ennobled
if possible. You have to maintain it, by each, in his
place, taking care that the body of German students is
worthy of the confidence which has hitherto accorded
such a measure of freedom. But freedom necessarily
implies responsibility. It is as injurious a present for
weak, as it is valuable for strong characters. Do not
wonder if parents and statesmen sometimes urge that a
more rigid system of supervision and control, like that of
the English, shall be introduced even among us. There
is no doubt that, by such a system, many a one would
be saved who is ruined by freedom. But the State and
the Nation is best served by those who can bear free-
dom, and have shown that they know how to work and
to struggle, from their own force and insight and from
their own interest in science.

My having previously dwelt on the influence of
mental intercourse with distinguished men, leads me
to discuss another point in which German Universities
are distinguished from the English and French ones.
It is that we start with the object of having instruc-
tion given, if possible, only by teachers who have proved

ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 253

their own power of advancing science. This also is a
point in respect to which the English and French often
express their surprise. They lay more weight than the
Germans on what is called the ' talent for teaching '--
that is, the power of explaining the subjects of instruc-
tion in a well-arranged and clear manner, and, if pos-
sible, with eloquence, and so as to entertain and to
fix the attention. Lectures of eloquent orators at the
College de France, Jardin des Plantes, as well as in
Oxford and Cambridge, are often the centres of the
elegant and the educated world. In Germany we are
not only indifferent to, but even distrustful of, oratorical
ornament, and often enough are more negligent than
we should be of the outer forms of the lecture. There
can be no doubt that a good lecture can be followed
with far less exertion than a bad one ; that the matter
of the first can be more certainly and completely ap-
prehended ; that a well-arranged explanation, which
develops the salient points and the divisions of the sub-
ject, and which brings it, as it were, almost intuitively
before us, can impart more information in the same
time than one which has the opposite qualities. I am
by no means prepared to defend what is, frequently, our
too great contempt for form in speech and in writing.
It cannot also be doubted that many original men,
who have done considerable scientific work, have often
an uncouth, heavy, and hesitating delivery. Yet I have

254 ON ACADEMIC FKEEDOM IN GEftMAN UNIVERSITIES.

not infrequently seen that such teachers had crowded
lecture-rooms, while empty-headed orators excited
astonishment in the first lecture, fatigue in the
second, and were deserted in the third. Anyone
who desires to give his hearers a perfect conviction
of the truth of his principles must, first of all, know
from his own experience how conviction is acquired and
how not. He must have known how to acquire con-
viction where no predecessor had been before him —
that is, he must have worked at the confines of human
knowledge, and have conquered for it new regions. A
teacher who retails convictions which are foreign to
him, is sufficient for those pupils who depend upon
authority as the source of their knowledge, but not for
such as require bases for their conviction which extend
to the very bottom.

You will see that this is an honourable confidence
which the nation reposes in you. Definite courses
and specified teachers are not prescribed to you. You
are regarded as men whose unfettered conviction is
to be gained ; who know how to distinguish what
is essential from what is only apparent; who can no
longer be appeased by an appeal to any authority, and
who no longer let themselves be so appeased. Care is
also always taken that you yourselves should penetrate
to the sources of knowledge in so far as these consist

ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 255

in books and monuments, or in experiments, and in the
observation of natural objects and processes.

Even the smaller German Universities have their
own libraries, collections of casts, and the like. And
in the establishment of laboratories for chemistry,
microscopy, physiology, and physics, Germany has
preceded all other European countries, who are now be-
ginning to emulate her. In our own University we may
in the next few weeks expect the opening of two new
institutions devoted to instruction in natural science.

The free conviction of the student can only be
acquired when freedom of expression is guaranteed to
the teacher's own conviction — the liberty of teaching.
This has not always been ensured, either in Germany
or in the adjacent countries. In times of political and
ecclesiastical struggle the ruling parties have often
enough allowed themselves to encroach ; this has
always been regarded by the German nation as an
attack upon their sanctuary. The advanced political
freedom of the new German Empire has brought a
cure for this. At this moment, the most extreme con-
sequences of materialistic metaphysics, the boldest
speculations upon the basis of Darwin's theory of evo-
lution, may be taught in German Universities with as
little restraint as the most extreme deification of Papal
Infallibility. As in the tribune of European Parlia-

256 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES.

ments it is forbidden to suspect motives or indulge in
abuse of the personal qualities of our opponents, so
also is any incitement to such acts as are legally for-
bidden. But there is no obstacle to the discussion of
a scientific question in a scientific spirit. In English
and French Universities there is less idea of liberty of
teaching in this sense. Even in the College de France
the lectures of a man of Kenan's scientific impor-
tance and earnestness are forbidden.

I have to speak of another aspect of our liberty of
teaching. That is, the extended sense in which Ger-
man Universities have admitted teachers. In the
original meaning of the word, a doctor is a ' teacher,' or
one whose capacity as teacher is recognised. In the
Universities of the Middle Ages any doctor who found
pupils could set up as teacher. In course of time the
practical signification of the title was changed. Most
of those who sought the title did not intend to act as
teachers, but only needed it as an official recognition
of their scientific training. Only in Germany are
there any remains of this ancient right. In accord-
ance with the altered meaning of the title of doctor,
and the minuter specialisation of the subjects of in-
struction, a special proof of more profound scientific
proficiency, in the particular branch in which they wish
to habilitate, is required from those doctors who desire
to exercise the right of teaching. In most German

.

ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 257

Universities, moreover, the legal status of these habili-
tated doctors as teachers is exactly the same as that of
the ordinary professors. In a few places they are
subject to some slight restrictions which, however,
have scarcely any practical effect. The senior teachers
of the University, especially the ordinary professors,
have this amount of favour, that, on the one hand, in
those branches in which special apparatus is needed
for instruction, they can more freely dispose of the
means belonging to the State ; while on the other it
falls to them to hold the examinations in the faculty,
and, as a matter of fact, often also the State examina-
tion. This naturally exerts a certain pressure on the
weaker minds among the students. The influence of
examinations is, however, often exaggerated. In the
frequent migrations of our students, a great number
of examinations are held in which the candidates have
never attended the lectures of the examiners.

On no feature of our University arrangements do
foreigners express their astonishment so much as about
the position of private decent s. They are surprised,
and even envious, that we have such a number of
young men who, without salary, for the most part with
insignificant incomes from fees, and with very un-
certain prospects for the future, devote themselves to
strenuous scientific work. And, judging us from the
point of view of basely practical interests, they are

II. 8

258 ON ACADEMIC FKEEDOM IN GERMAN UNIVEBSITIES.

equally surprised that the faculties so readily admit
young men who at any moment may change from
assistants to competitors ; and further, that only in
the most exceptional cases is anything ever heard of
unworthy means of competition in what is a matter of
some delicacy.

The appointment to vacant professorships, like the
admission of private docents, rests, though not uncon-
ditionally, and not in the last resort, with the faculty,
that is with the body of ordinary professors. These
form, in German Universities, that residuum of former
colleges of doctors to which the rights of the old
corporations have been transferred. They form as it
were a select committee of the graduates of a former
epoch, but established with the co-operation of the
Government. The usual form for the nomination of
new ordinary professors is that the faculty proposes
three candidates to Government for its choice ; where
the Government, however, does not consider itself
restricted to the candidates proposed. Excepting in
times of heated party conflict it is very unusual for the
proposals of the faculty to be passed over. If there is
not a very obvious reason for hesitation it is always a
serious personal responsibility for the executive officials
to elect, in opposition to the proposals of competent
judges, a teacher who has publicly to prove his
capacity before large circles.

ON ACADEMIC FREEDOM IN GEKMAN UNIVEESITIES. 259

The professors have, however, the strongest motives
for securing to the faculty the best teachers. The
most essential condition for being able to work with
pleasure at the preparation of lectures is the con-
sciousness of having not too small a i lumber of intelli-
gent listeners ; moreover, a considerable fraction of the
income of many teachers depends upon the number of
their hearers. Each one must wish that his faculty, as
a whole, shall attract as numerous and as intelligent a
body of students as possible. That, however, can only
be attained by choosing as many able teachers, whether
professors or docents, as possible. On the other hand,
a professor's attempt to stimulate his hearers to
vigorous and independent research can only be suc-
cessful when it is supported by his colleagues;
besides this, working with distinguished colleagues
makes life in University circles interesting, instructive,
and stimulating. A faculty must have greatly sunk,
it must not only have lost its sense of dignity, but also
even the most ordinary worldly prudence, if other
motives could preponderate over these ; and such a
faculty would soon ruin itself.

With regard to the spectre of rivalry among Uni-
versity teachers with which it is sometimes attempted
to frighten public opinion, there can be none such if
the students and their teachers are of the right kind.
In the first place, it is only in large Universities that

8 2

2 CO ON ACADEMIC FKEEDOM IN GEKMAN UNIVERSITIES.

there are two to teach one and the same branch ; and
even if there is no difference in the official definition
of the subject, there will be a difference in the scien-
tific tendencies of the teachers ; they will be able to
divide the work in such a manner that each has that
side which he most completely masters. Two distin-
guished teachers who are thus complementary to each
other, form then so strong a centre of attraction for
the students that both suffer no loss of hearers, though
they may have to share among themselves a number of
the less zealous ones.

The disagreeable effects of rivalry will be feared
by a teacher who does not feel quite certain in his
scientific position. This can have no considerable
influence on the official decisions of the faculty when
it is only a question of one, or of a small number, of
the voters.

The predominance of a distinct scientific school in
a faculty may become more injurious than such per-
sonal interests. When the school has scientifically out-
lived itself, students will probably migrate by degrees
to other Universities. This may extend over a long
period, and the faculty in question will suffer during
that time.

We see best how strenuously the Universities under
this system have sought to attract the scientific ability
of Germany when we consider how many pioneers have

ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 261

remained outside the Universities. The answer to such
an inquiry is given in the not infrequent jest or sneer
that all wisdom in Germany is professorial wisdom. If
we look at England, we see men like Humphry Davy,
Faraday, Mill, Grote, who have had no connection with
English Universities. If, on the other hand, we deduct
from the list of German men of science those who,
like David Strauss, have been driven away by Govern-
ment for ecclesiastical or for political reasons, and those
who, as members of learned Academies, had the right
to deliver lectures in the Universities, as Alexander and
Wilhclm von Humboldt, Leopold von Buch, and others,
the rest will only form a small fraction of the number
of the men of equal scientific standing who have been
at work in the Universities ; while the same calculation
made for England would give exactly the opposite result.
I have often wondered that the Royal Institution of
London, a private Society, which provides for its mem-
bers and others short courses of lectures on the Progress
of Natural Science, should have been able to retain
permanently the services of men of such scientific
importance as Humphry Davy and Faraday. It was
no question of great emoluments; these men were
manifestly attracted by a select public consisting of
men and women of independent mental culture. In
Germany the Universities are unmistakably the insti-
tutions which exert the most powerful attraction on

262 ON ACADEMIC FEEEDOM IN GERMAN UNIVERSITIES,

the taught. But it is clear that this attraction depends
on the teacher's hope that he will not only find in the
University a body of pupils enthusiastic and accus-
tomed to work, but such also as devote themselves to
the formation of an independent conviction. It is only
with such students that the intelligence of the teacher
bears any further fruit.

The entire organisation of our Universities is thus
permeated by this respect for a free independent con-
viction, which is more strongly impressed on the
Germans than on their Aryan kindred of the Celtic and
Romanic branches, in whom practical political motives
have greater weight. They are able, and as it would
seem with perfect conscientiousness, to restrain the
inquiring mind from the investigation of those prin-
ciples which appear to them to be beyond the range of
discussion, as forming the foundation of their political,
social, and religious organisation; they think them-
selves quite justified in not allowing their youth to
look beyond the boundary which they themselves are
not disposed to overstep.

If, therefore, any region of questions is to be con-
sidered as outside the range of discussion, however
remote and restricted it may be, and however good
may be the intention, the pupils must be kept in the
prescribed path, and teachers must be appointed who

OX ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 2G3

do not rebel against authority. We can then, however,
only speak of free conviction in a very limited sense.

You see how different was the plan of our fore-
fathers. However violently they may at times have
interfered with individual results of scientific inquiry,
they never wished to pull it up by the roots. An
opinion which was not based upon independent con-
viction appeared to them of no value. In their hearts
they never lost faith that freedom alone could cure the
errors of freedom, and a riper knowledge the errors of
what is unripe. The same spirit which overthrew the
yoke of the Church of Eome, also organised the Ger-
man Universities.

But any institution based upon freedom must also
be able to calculate on the judgment and reasonable-
ness of those to whom freedom is granted. Apart
from the points which have been previously discussed,
where the students themselves are left to decide on
the course of their studies and to select their teachers,
the above considerations show how the students react
upon their teachers. To produce a good course of
lectures is a labour which is renewed every term.
New matter is continually being added which necessi-
tates a reconsideration and a rearrangement of the
old from fresh points of view. The teacher would
poon be dispirited in his work if he could not count
upon the zeal and the interest of his hearers. The

264 ON ACADEMIC FREEDOM IN GERMAN UNIVERSITIES,

estimate which he places on his task will depend on
how far he is followed by the appreciation of a suffi-
cient number of, at any rate, his more intelligent
hearers. The influx of hearers to the lectures of a
teacher, has no slight influence upon his fame and
promotion, and, therefore, upon the composition of
the body of teachers. In all these respects, it is
assumed that the general public opinion among the
students cannot go permanently wrong. The majority
of them — who are, as it were, the representatives cf
the general opinion — must come to us with a suffi-
ciently logically trained judgment, with a sufficient
habit of mental exertion, with a tact sufficiently de-
veloped on the best models, to be able to discriminate
truth from the babbling appearance of truth. Among
the students are to be found those intelligent heads
who will be the mental leaders of the next generation,
and who, perhaps, in a few years, will direct to them-
selves the eyes of the world. Occasional errors in
youthful and excitable spirits naturally occur ; but, on
the whole, we may be pretty sure that they will soon
set themselves right.

Thus prepared, they have hitherto been sent to
us by the Gymnasiums. It would be very dangerous
for the Universities if large numbers of students fre-
quented them, who were less developed in the above
respects. The general self-respect of the studeius

OX ACADEMIC FREEDOM IN GERMAN UNIVERSITIES. 265

must not be allowed to sink. If that were the case,
the dangers of academic freedom would choke its
blessings. It must therefore not be looked upon as
pedantry, or arrogance, if the Universities are scrupu-
lous in the admission of students of a different style
of education. It would be still more dangerous if,
for any extraneous reasons, teachers were introduced
into the faculty, who have not the complete qualifica-
tions of an independent academical teacher.

Do not forget, my dear colleagues, that you are
in a responsible position. You have to preserve the
noble inheritance of which I have spoken, not only
for your own people, but also as a model to the
widest circles of humanity. You will show that
youth also is enthusiastic, and will work for inde-
pendence of conviction. I say work; for indepen-
dence of conviction is not the facile assumption of
untested hypotheses, but can only be acquired as
the fruit of conscientious inquiry and stremious
labour. You must show that a conviction which
you yourselves have worked out is a in ore fruitful
germ of fresh insight, and a better guide for action,
than the best-intentioned guidance by authority.
Germany — which in the sixteenth century first re-
volted for the right of such conviction, and gave its
witness in blood — is still in the van of this fight.
To Germany has fallen an exalted historical task, and
in it you are called upon to co-operate.

HERMANN VOX HELfflOLTZ

AN

AUTOBIOGRAPHICAL SKETCH.

An Address delivered on the occasion of his Jubilee, 1891,

IN the course of the past year, and most recently on
the occasion of the celebration of my seventieth
birthday, and the subsequent festivities, I have been
overloaded with honours, with marks of respect and of
goodwill in a way which could never have been ex-
pected. My own sovereign, his Majesty the German.
Emperor, has raised me to the highest rank in the
Civil Service ; the Kings of Sweden and of Italy, my
former sovereign, the Grand Duke of Baden, and the
President of the French Eepublic, have conferred Grand
Crosses on me ; many academies, not only of science,
but also of the fine arts, faculties, and learned societies
spread over the whole world, from Tomsk to Melbourne,
have sent me diplomas, and richly illuminated addresses,
expressing in elevated language their recognition of
my scientific endeavours, and their thanks for those
endeavours, in terms which I cannot read without a

AN AUTOBIOGRAPHICAL SKETCH. 267

feeling of shame. My native town, Potsdam, has con-
ferred its freedom on me. To all this must be added
countless individuals, scientific and personal friends,
pupils, and others personally unknown to me, who have
sent their congratulations in telegrams and in letters.

But this is not all. You desire to make my name
the banner, as it were, of a magnificent institution
which, founded by lovers of science of all nations, is
to encourage and promote scientific inqufry in all
countries. Science and art are, indeed, at the present
time the only remaining bond of peace between
civilised nations. Their ever-increasing development is
a common aim of all ; is effected by the common work
of all, and for the common good of all. A great and a
sacred work ! The founders even wish to devote their
gift to the promotion of those branches of science
which all my life I have pursued, and thus bring me,
with my shortcomings, before future generations almost
as an exemplar of scientific investigation. This is the
proudest honour which you could confer upon me, in
so much as you thereby show that I possess your un-
qualified favourable opinion. But it would border on
presumption were I to accept it without a quiet ex-
pectation on my part that the judges of future
centuries will not be influenced by considerations of
personal favour.

My personal appearance even, you have had repre-

268 HERMANN VON HELMIIOLTZ :

sented in marble by a master of the first rank, so that
I shall appear to the present and to future generations
in a more ideal form ; and another master of the
etching needle has ensured that faithful portraits of
me shall be distributed among my contemporaries.

I cannot fail to remember that all you have done is
an expression of the sincerest and warmest goodwill on
your part, and that I am most deeply indebted to you
for it.

I must, however, be excused if the first effect of
these abundant honours is rather surprising and con-
fusing to me than intelligible. My own consciousness
does not justify me in putting a measure of the value
of what I have tried to do, which would leave such a
balance in my favour as you have drawn. I know
how simply everything I have done has been brought
about ; how scientific methods worked out by my pre-
decessors have naturally led to certain results, and how
frequently a fortunate circumstance or a lucky acci-
dent has helped me. But the chief difference is this
— that which I have seen slowly growing from small
beginnings through months and years of toilsome and
tentative work, all that suddenly starts before you like
Pallas fully equipped from the head of Jupiter. A feel-
ing of surprise has entered into your estimate, but not
into mine. At times, and perhaps even frequently, my
own estimate may possibly have been unduly lowered

AN AUTOBIOGRAPHICAL SKETCH. 2G9

by the fatigue of the work, and by vexation about all
kinds of futile steps which I had taken. My colleagues,
as well as the public at large, estimate a scientific or
artistic work according to the utility, the instruction, or
the pleasure which it has afforded. An author is usually
disposed to base his estimate on the labour it has
cost him, and it is but seldom that both kinds of
judgment agree. It can, on the other hand, be seen
from incidental expressions of some of the most cele-
brated men, especially of artists, that they lay but small
weight on productions which seem to us inimitable,
compared with others which have been difficult, and
yet which appear to readers and observers as much less
successful. I need only mention Groethe, who once
stated to Eckermann that he did not estimate his
poetical works so highly as what he had done in the
theory of colours.

The same may have happened to me, though in a
more modest degree, if I may accept your assurances
and those of the authors of the addresses which have
reached me. Permit me, therefore, to give you a short
account of the manner in which I have been led to
the special direction of my work.

In my first seven years I was a delicate boy, for
long confined to my room, and often even to bed ; but,
nevertheless, I had a strong inclination towards occupa-
tion and mental activity. My parents busied them-
selves a good deal with me ; picture books and games,

270 HERMANN VON HELMIIOLTZ:

especially with wooden blocks, filled up the rest of the
time. Reading came pretty early, which, of course,
greatly increased the range of my occupations. But a
defect of my mental organisation showed itself almost
as early, in that I had a bad memory for disconnected
things. The first indication of this I consider to be
the difficulty I had in distinguishing between left and
right ; afterwards, when at school I began with
languages, I had greater difficulties than others in
learning words, irregular grammatical forms, and
peculiar terms of expression. History as then taught
to us I could scarcely master. To learn prose by heart
was martyrdom. This defect has, of course, only in-
creased, and is a vexation of my mature age.

But when I possessed small mnemotechnical me-
thods, or merely such as are afforded by the metre
and rhyme of poetry, learning by heart, and the reten-
tion of what I had learnt, went on better. I easily
remembered poems by great authors, but by no means
so easily the somewhat artificial verses of authors of
the second rank. I think that is probably due to
the natural flow of thought in good poems, and I am
inclined to think that in this connection is to be
found an essential basis of aesthetic beauty. In the
higher classes of the Gymnasium I could repeat some
books of the Odyssey, a considerable number of the
oles of Horace, and large stores of German poetry.
In other directions I was just in the position of our

AN AUTOBIOGRAPHICAL SKETCH. 271

older ancestors, who were not able to write, arid hence
expressed their laws and their history in verse, so as
to learn them by heart.

p. That which a man does easily he usually does
willingly ; hence I was first of all a great admirer and
lover of poetry. This inclination was encouraged by
my father, who, while he had a strict sense of duty,
was also of an enthusiastic disposition, impassioned
for poetry, and particularly for the classic period of
German Literature. He taught German in the upper
classes of the Gymnasium, and read Homer with us.
Under his guidance we did, alternately, themes in
German prose and metrical exercises — poems as we
called them. But even if most of us remained in-
different poets, we learned better in this way, than in
any other I know of, how to express what we had to say
in the most varied manner.

But the most perfect mnemotechnical help is a
knowledge of the laws of phenomena. This I first got
to know in geometry. From the time of my childish
playing with wooden blocks, the relations of special
proportions to each other were well known to me from
actual perception. What sort of figures were produced
when bodies of regular shape were laid against each
other I knew well without much consideration. When
I began the scientific study of geometry, all the facts
which I had to learn were perfectly well known and

272 HERMANN VON HELMHOLTZ :

familiar" to me, much to the astonishment of my
teachers. So far as I recollect, that came out in-
cidentally in the elementary school attached to the
Potsdam Training College, which I attended up to my
eighth year. Strict scientific methods, on the con-
trary, were new to me, and with their help I saw the
difficulties disappear which had hindered me in other
regions.

One thing was wanting in geometry ; it dealt ex-
clusively with abstract forms of space, and I delighted
in complete reality. As I became bigger and stronger
I went about with my father and my schoolfellows a
great deal in the neighbourhood of rny native town,
Potsdam, and I acquired a great love of Nature. This
is perhaps the reason why the first fragments of physics
which I learned in the Grymnasium engrossed me
much more closely than purely geometrical and alge-
braical studies. Here there was a copious and multi-
farious region, with the mighty fulness of Nature, to be
brought under the dominion of a mentally appre-
hended law. And, in fact, that which first fascinated
me was the intellectual mastery over Nature, which at
first confronts us as so unfamiliar, by the logical force
of law. But this, of course, soon led to the recognition
that knowledge of natural processes was the magical key
which places ascendency over Nature in the hands of its
possessor. In this order of ideas I felt myself at home,

A!N AUTOBIOGRAPHICAL SKETCH. 273

I plunged then with great zeal and pleasure into
the study of all the books on physics I found in
my father's library. They were very old-fashioned ;
phlogiston still held sway, and galvanism had not
grown beyond the voltaic pile. A young friend and
myself tried, with our small means, all sorts of experi-
ments about which we had read. The action of acids
on our mothers' stores of linen we investigated
thoroughly ; we had otherwise but little success.
Most successful was, perhaps, the construction of
optical instruments by means of spectacle glasses,
which were to be had in Potsdam, and a small
botanical lens belonging to my father. The limitation
of our means had at that time the value that I was
compelled always to vary in all possible ways my plans
for experiments, until I got them in a form in which I
could carry them out. I must confess that many a
time when the class was reading Cicero or Virgil,
both of which I found very tedious, I was calculating
under the desk the path of rays in a telescope, and I
discovered, even at that time, some optical theorems,
not ordinarily met with in text-books, but which I
afterwards found useful in the construction of the
ophthalmoscope.

Thus it happened that I entered upon that special
line of study to which I have subsequently adhered, and

which, in the conditions I have mentioned, grew into
IL T

274 HERMANN VON HELMHOLTZt

an absorbing impulse, amounting even to a passion.
This impulse to dominate the actual world by acquiring
an understanding of it, or what, I think, is only another
expression for the same thing, to discover the causal
connection of phenomena, has guided me through my
whole life, and the strength of this impulse is possibly
the reason why I found no satisfaction in apparent
solutions of problems so long as I felt there were still
obscure points in them.

And now I was to go to the university. Physics
was at that time looked upon as an art by which a
living could not be made. My parents were compelled
to be very economical, and my father explained to me
that he knew of no other way of helping me to the
study of Physics, than by taking up the study of
medicine into the bargain. I was by no means averse
from the study of living Nature, and assented to this
without much difficulty. Moreover, the only influential
person in our family had been a medical man, the late
Surgeon-General Mursinna ; and this relationship was
a recommendation in my favour among other ap-
plicants for admission to our Army Medical School, the
Friedrich Wilhelms Institut, which very materially
helped the poorer students in passing through their
medical course.

In this study I came at once under the influence
of a profound teacher — Johannes Miiller; he who at

AN AUTOBIOGRAPHICAL SKETCH. 275

the same time introduced E. Du Bois Reymond,
E. Briicke, C. Ludwig, and Virchow to the study of
anatomy and physiology. As respects the critical
questions about the nature of life, Miiller still
struggled between the older — essentially the meta-
physical— view and the naturalistic one, which was
then being developed; but the conviction that nothing
could replace the knowledge of facts forced itself upon
him with increasing certainty, and it may be that his
influence over his students was the greater because he
still so struggled.

Young people are ready at once to attack the
deepest problems, and thus I attacked the perplexing
question of the nature of the vital force. Most
physiologists had at that time adopted Gr. E. Stahl's
way out of the difficulty, that while it is the physical
and chemical forces of the organs and substances of the
living body which act on it, there is an indwelling vital
soul or vital force which could bind and Icose the
activity of these forces ; that after death the free
action of these forces produces decomposition, while
during life their action is continually being controlled
by the soul of life. I had a misgiving that there was
something against nature in this explanation; but it
took me a good deal of trouble to state my misgiving
in the form of a definite question. I found ultimately,
in the latter years of my career as a student, that

r>

276 HEKMANN VON HELMHOLTZ :

Stahl's theory ascribed to every living body the nature
of a perpetuum mobile. I was tolerably well acquainted
with the controversies on this latter subject. In my
school days I had heard it discussed by my father and
our mathematical teachers, and while still a pupil of
the Friedrich Wilhelms Institut I had helped in the
library, and in my spare moments had looked through
the works of Daniell, Bernouilli, D'Alembert, and other
mathematicians of the last century. I thus came
upon the question, { What relations must exist be-
tween the various kinds of natural forces for a per-
petual motion to be possible?' and the further one,
c Do those relations actually exist ? ' In my essay,
c On the Conservation of Force/ my aim was merely to
give a critical investigation and arrangement of the
facts for the benefit of physiologists.

I should have been quite prepared if the experts
had ultimately said, ' We know all that. What is this
young doctor thinking about, in considering himself
called upon to explain it all to us so fully ?' But, to my
astonishment, the physical authorities with whom I
came in contact took up the matter quite differently.
They were inclined to deny the correctness of the law,
and in the eager contest in which they were engaged
against Hegel's Natural Philosophy were disposed to
declare my essay to be a fantastical speculation.
Jacobi, the mathematician, who recognised the con-

AN AUTOBIOGRAPHICAL SKETCH. 277

nection of my line of thought with that of the mathe-
maticians of the last century, was the only one who
took an interest in my attempt, and protected me
from being misconceived. On the other hand, I met
with enthusiastic applause and practical help from
my younger friends, and especially from E. Du Bois
Reymond. These, then, soon brought over to my side
the members of the recently formed Physical Society
of Berlin. About Joule's researches on the same sub-
ject I knew at that time but little, and nothing at all
of those of Eobert Mayer.

Connected with this were a few smaller experi-
mental researches on putrefaction and fermentation, in
which I was able to furnish a proof, in opposition to
Liebig's contention, that both were by no means purely
chemical decompositions, spontaneously occurring, or
brought about by the aid of the atmospheric oxygen ;
that alcoholic fermentation more especially was bound
up with the presence of yeast spores which are only
formed by reproduction. There was, further, my
work on metabolism in muscular action, which after-
wards was connected with that on the development of
heat in muscular action ; these being processes which
were to be expected from the law of the conservation
of force.

These researches were sufficient to direct upon me
the attention of Johannes Miiller as well as of the

278 HEKMANN VON HELMHOLTZ:

Prussian Ministry of Instruction, and to lead to my
being called to Berlin as Briicke's successor, and
immediately thereupon to the University of Konigs-
berg. The Army medical authorities, with thank-
worthy liberality, very readily agreed to relieve me
from the obligation to further military service, and
thus made it possible for me to take up a scientific
position.

In Konigsberg I had to lecture on general
jpathology and physiology. A university professor
undergoes a very valuable training in being compelled
to lecture every year, on the whole range of his science,
in such a manner that he convinces and satisfies the
intelligent among his hearers — the leading men of
the next generation. This necessity yielded me, first
of all, two valuable results.

For in preparing my course of lectures, I hit
directly on the possibility of the ophthalmoscope, and
then on the plan of measuring the rate of propagation
of excitation in the nerves.

The ophthalmoscope is, perhaps, the most popular
of my scientific performances, but I have already
related to the oculists how luck really played a com-
paratively more important part than my own merit.
I had to explain to my hearers Briicke's theory of
ocular illumination. In this, Briicke was actually
within a hair's breadth of the invention of the ophthal-

AN AUTOBIOGRAPHICAL SKETCH. 279

moscope. He had merely neglected to put the ques-
tion, To what optical image do the rays belong, which
come from the illuminated eye ? For the purpose he
then had in view it was not necessary to propound this
question. If he had put it, he was quite the man to
answer it as quickly as I could, and the plan of the
ophthalmoscope would have been given. I turned the
problem about in various ways, to see how I could best
explain it to my hearers, and I thereby hit upon the
question I have mentioned. I knew well, from my
medical studies, the difficulties which oculists had
about the conditions then comprised under the name
of Amaurosis, and I at once set about constructing the
instrument by means of spectacle glasses and the glass
used for microscope purposes. The instrument was at
first difficult to use, and without an assured theoretical
conviction that it must work, I might, perhaps, not
have persevered. But in about a week I had the great
joy of being the first who saw clearly before him a
living human retina.

The construction of the ophthalmoscope had a very
decisive influence on my position in the eyes of the
world. From this time forward I met with the most
willing recognition and readiness to meet my wishes on
the part of the authorities and of my colleagues, so that
for the future I was able to pursue far more freely the
secret impulses of my desire for knowledge. I must,

280 HEKMANN VON HELMHOLTZ :

however, say that I ascribed my success in great measure
to the circumstance that, possessing some geometrical
capacity, and equipped with a knowledge of physics, I
had, by good fortune, been thrown among medical men,
where I found in physiology a virgin soil of great fer-
tility ; while, on the other hand, I was led by the con-
sideration of the vital processes to questions and points
of view which are usually foreign to pure mathematicians
and physicists. Up to that time I had only been able
to compare my mathematical abilities with those of my
fellow-pupils and of my medical colleagues ; that I was
for the most part superior to them in this respect did
not, perhaps, say very much. Moreover, mathematics
was always regarded in the school as a branch of
secondary rank. In Latin composition, on the con-
trary, which then decided the palm of victory, more
than half my fellow-pupils were ahead of me.

In my own consciousness, my researches were
simple logical applications of the experimental and
mathematical methods developed in science, which by
plight modifications could be easily adapted to the
particular object in view. My colleagues and friends,
who, like myself, had devoted themselves to the phy-
sical aspect of physiology, furnished results no less
surprising.

But in the course of time matters could not remain
in that stage. Problems which might be solved by

AN AUTOBIOGKAPHICAL SKETCH. 281

known methods I had gradually to hand over -to the
pupils in my laboratory, and for my own part turn to
more difficult researches, where success was uncertain,
where general methods left the investigator in the
lurch, or where the method itself had to be worked
out.

In those regions also which come nearer the boun-
daries of our knowledge I have succeeded in many
things experimental and mechanical— I do not know if
I may add philosophical. In respect of the former,
like any one who has attacked many experimental
problems, I had become a person of experience, who was
acquainted with many plans and devices, and I had
changed my youthful habit of considering things geo-
metrically into a kind of mechanical mode of view. I
felt, intuitively as it were, how strains and stresses
were distributed in any mechanical arrangement, a
faculty also met with in experienced mechanicians and
machine constructors. But I had the advantage over
them of being able to make complicated and specially
important relations perspicuous, by means of theoretical
analysis.

I have also been in a position to solve several
mathematical physical problems, and some, indeed, on
which the great mathematicians, since the time of Euler,
had in vain occupied themselves ; for example, questions
as to vortex motion and the discontinuity of motion in

232 HERMANN VON HELMHOLTZ:

liquids, the question as to the motion of sound at the
open ends of organ pipes, &c. &c. But the pride
which I might have felt about the final result in these
cases was considerably lowered by my consciousness
that I had only succeeded in solving such problems
after many devious ways, by the gradually increasing
generalisation of favourable examples, and by a series
of fortunate guesses. I had to compare myself with
an Alpine climber, who, not knowing the way, ascends
slowly and with toil, and is often compelled to retrace
his steps because his progress is stopped ; sometimes
by reasoning, and sometimes by accident, he hits upon
traces of a fresh path, which again leads him a little
further; and finally, when he has reached the goal, be
finds to his annoyance a royal road on which he might
have ridden up if he had been clever enough to find
the right starting-point at the outset. In my memoirs
I have, of course, not given the reader an account of
my wanderings, but I have described the beaten path
on which he can now reach the summit without
trouble.

There are many people of narrow views, who greatly
admire themselves, if once in a way, they have had a
happy idea, or believe they have had one. An investi-
gator, or an artist, who is continually having a great
number of happy ideas, is undoubtedly a privileged
being, and is recognised as a benefactor of humanity.

AN AUTOBIOGBAPHICAL SKETCH. 233

Bat who can count or measure such mental flashes ?
Who can follow the hidden tracts by which conceptions
are connected ?

That which man had never known,
Or had not thought out,
Through the labyrinth of mind
Wanders in the night.

I must say that those regions, in which we have not
to rely on lucky accidents and ideas, have always been
most agreeable to me, as fields of work.

But, as I have often been in the unpleasant position
of having to wait for lucky ideas, ITiave had some ex-
perience as to when and where they came to me, which
will perhaps be useful to others. They often steal into
the line of thought without their importance being at
first understood ; then afterwards some accidental cir-
cumstance shows how and under what conditions they
have originated ; they are present, otherwise, without
our knowing whence they came. In other cases they
occur suddenly, without exertion, like an inspiration.
As far as my experience goes, they never came at the
desk or to a tired brain. I have always so turned my
problem about in all directions that I could see in my
mind its turns and complications, and run through them
freely without writing them down. But to reach that
stage was not usually possible without long preliminary
work. Then, after the fatigue from this had passed away,

284 HERMANN VON HELMHOLTZ:

an hour of perfect bodily repose and quiet comfort was
necessary before the good ideas came. They often
came actually in the morning on waking, as expressed
in Goethe's words which I have quoted, and as Gauss
also has remarked.1 But, as I have stated in Heidel-
berg, they were usually apt to come when comfortably
ascending woody hills in sunny weather. The smallest
quantity of alcoholic drink seemed to frighten them
away.

Such moments of fruitful thought were indeed very
delightful, but not so the reverse, when the redeeming
ideas did not come. For weeks or months I was gnaw-
ing at such a question until in my mind I was

Like to a beast upon a barren heath
Dragged in a circle by an evil spirit,
While all around are pleasant pastures green.

And, lastly, it was often a sharp attack of headache
which released me from this strain, and set me free for
other interests.

I have entered upon still another region to which I
was led by investigation on perception and observation
of the senses, namely, the theory of cognition. Just
as a physicist has to examine the telescope and galva-
nometer with which he is working ; has to get a clear
conception of what he can attain with them, and how

1 Gauss, Wcrke, vol. v. p. 609. * The law of induction discovered
Jan. 23, 1835, at 7 A.M., before rising.'

AN AUTOBIOGRAPHICAL SKETCH. 285

they may deceive him ; so, too, it seemed to me necessary
to investigate likewise the capabilities of our power of
thought. Here, also, we were concerned only with, a
series of questions of fact about which definite answers
could and must be given. We have distinct impres-
sions of the senses, in consequence of which we know
how to act. The success of the action usually agrees
with that which was to have been anticipated, but
sometimes also not, in what are called subjective im-
pressions. These are all objective facts, the laws regu-
lating which it will be possible to find. My principal
result was that the impressions of the senses are only
signs for the constitution of the external world, the in-
terpretation of which must be learned by experience.
The interest for questions of the theory of cognition,
had been implanted in me in my youth, when I had
often heard my father, who had retained a strong impres-
sion from Fichter's idealism, dispute with his colleagues
who believed in Kant or Hegel. Hitherto I have had
but little reason to be proud about those investigations.
For each one in my favour, I have had about ten
opponents ; and I have in particular aroused all the
metaphysicians, even the materialistic ones, and all
people of hidden metaphysical tendencies. But the
addresses of the last few days have revealed a host of
friends whom as yet I did not know ; so that in this
respect also I am indebted to this festivity for pleasure

286 HERMANN VON HELMHOLTZ :

and for fresh hope. Philosophy, it is true, has been
for nearly three thousand years the battle-ground for
the most violent differences of opinion, and it is not to
be expected that these can be settled in the course of a
single life.

I have wished to explain to you how the history of
my scientific endeavours and successes, so far as they
go, appears when looked at from my own point of view,
and you will perhaps understand that I am surprised
at the universal profusion of praise which you have
poured out upon me. My successes have had primarily
this value for my own estimate of myself, that they
furnished a standard of what I might further attempt;
but they have not, I hope, led me to self-admiration.
I have often enough seen how injurious an exaggerated
sense of self-importance may be for a scholar, and
hence- 1 have always taken great care not to fall a prey
to this enemy. I well knew that a rigid self-criticism
of my own work and my own capabilities was the
protection and palladium against this fate. But it is
only needful to keep the eyes open for what others can
do, and what one cannot do oneself, to find there is
no great danger ; and, as regards my own work, I do
not think I have ever corrected the last proof of a
memoir without finding in the course of twenty-four
hours a few points which I could have done better or
more carefully.

AN AUTOBIOGRAPHICAL SKETCH. 287

As regards the thanks which you consider you owe
me, I should be unjust if I said that the good of
humanity appeared to me, from the outset, as the
conscious object of my labours. It was, in fact, the
special form of my desire for knowledge which im-
pelled me and determined me, to employ in scientific
research all the time which was not required by my
official duties and by the care for my family. These
two restrictions did not, indeed, require any essential
deviation from the aims I was striving for. My office
required me to make myself capable of delivering
lectures in the University ; my family, that I should
establish and maintain my reputation as an investi-
gator. The State, which provided my maintenance,
scientific appliances, and a great share of my free
time, had, in my opinion, acquired thereby the right
that I should communicate faithfully and completely
to my fellow-citizens, and in a suitable form, that
which I had discovered by its help.

The writing out of scientific investigations is
usually a troublesome affair ; at any rate it has been so
to me. Many parts of my memoirs I have rewritten
five or six times, and have changed the order about
until I was fairly satisfied. But the author has a great
advantage in such a careful wording of his work. It
compels him to make the severest criticism of each
sentence and each conclusion, more thoroughly even

238 AN AUTOBIOGRAPHICAL SKETCH.

than the lectures at the University which I have men-
tioned. I have never considered an investigation
finished until it was formulated in writing, completely
and without any logical deficiencies.

Those among my friends who were most conversant
with the matter represented to my mind, my conscience
as it were. I asked myself whether they would approve
of it. They hovered before me as the embodiment of
the scientific spirit of an ideal humanity, and furnished
me with a standard.

In the first half of my life, when I had still to work
for my external position, I will not say that, along with
a desire for knowledge and a feeling of duty as servant
of the State, higher ethical motives were not also at
work ; it was, however, in any case difficult to be
certain of the reality of their existence so long as
selfish motives were still existent. This is, perhaps,
the case with all investigators. But afterwards, when
an assured position has been attained, when those
who have no inner impulse towards science may quite
cease their labours, a higher conception of their relation
to humanity does influence those who continue to
work. They gradually learn from their own experience
how the thoughts which they have uttered, whether
through literature or through oral instruction, continue
to act on their fellow-men, and possess, as it were, an
independent life j how these thoughts, further worked

AN AUTOBIOGRAPHICAL SKETCH. 289

out by their pupils, acquire a deeper significance and
a more definite form, and, reacting on their originators,
furnish them with fresh instruction. The ideas of an
individual, which he himself has conceived, are of
course more closely connected with his mental field
of view than extraneous ones, and he feels more
encouragement and satisfaction when he sees the
latter more abundantly developed than the former.
A kind of parental affection for such a mental child
ultimately springs up, which leads him to care and to
struggle for the furtherance of his mental offspring as
he does for his real children.

But, at the same time, the whole intellectual world
of civilised humanity presents itself to him as a con-
tinuous and spontaneously developing whole, the dura-
tion of which seems infinite as compared with that
of a single individual. With his small contributions
to the building up of science, he sees that he is in the
service of something everlastingly sacred, with which
he is connected by close bands of affection. His work
thtLeuy appears to him more sanctified. Anyone can,
perhaps, apprehend this theoretically, but actual per-
sonal experience is doubtless necessary to develop this
idea into a strong feeling.

The world, which is not apt to believe in ideal
motives, calls this feeling love of fame. But there is a

decisive criterion by which both kinds of sentiment
Ji. u

290 HERMANN VON HELMHOLTZ :

can be discriminated. Ask the question if it is the
same thing to you whether the results of investigation
which you have obtained are recognised as belonging
to you or not when there are no considerations of
external advantage bound up with the answer to this
question. The reply to it is easiest in the case of
chiefs of laboratories. The teacher must usually
furnish the fundamental idea of the research as well
as a number of proposals for overcoming experimental
difficulties, in which more or less ingenuity comes into
play. All this passes as the work of the student, and
ultimately appears in his name when the research is
finished. Who can afterwards decide what one or the
other has done ? And how many teachers are there
not who in this respect are devoid of any jealousy?

Thus, gentlemen, I have been in the happy position
that, in freely following my own inclination, I have been
led to researches for which you praise me, as having
been useful and instructive. I am extremely fortunate
that I am praised and honoured by my contemporaries,
in so high a degree, for a course of work which is to
me the most interesting I could pursue. But mv
contemporaries have afforded me great and essential
help. Apart from the care for my own existence and
that of my family, of which they have relieved me,
and apart from the external means with which they
have provided me, I have found in them a standard

-

AN AUTOBIOGRAPHICAL SKETCH. 2,91

of the intellectual capacity of man; and by their
sympathy for my work they have evoked in me a vivid
conception of the universal mental life of humanity
which has enabled me to see the value of my own
researches in a higher light. In these circum-
stances, I can only regard as a free gift the thanks
which you desire to accord to me, given unconditionally
and without counting on any return.

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