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


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 



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. 



Dec. 1830. 







i. Form 7?, 

H. Shade 94 

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

IV. Harmony of Colour 124 





SKBICH . , 206 


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 


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- 


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 



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 

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. 


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- 


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. 


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 

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 3 r oung 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 


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 


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 

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. 


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 


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 


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 
S T 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. 


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 


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

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 


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 


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 


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 


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- 


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 



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 


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. 


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 


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. 


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. 


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. 





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 


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, 


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. 


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 


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 

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 


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 

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- 


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 


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 


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 


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 


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* 


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 


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. 


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. 


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 



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- 


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 

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. 


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 


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.) 


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 tones 2 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. 


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. 


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, $ 


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 


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). 


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, 


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, 


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 


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.). 


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. 


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 

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 


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- 


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 


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 

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


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. 


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 


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. 


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 


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- 

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- 


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 


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 


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 



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- 

(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. 



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 

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 


from which equation 1 gives r=7?, then, 

in which <r= 

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

In this, S Q 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 s Q < -| T?TT, and one for which S Q > J RTT. Tha 
extension in the direction of r is then 


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 
, =l f log. nat. |5 

Here S Q has real values only as long as r=R; for r=1J, the 
distance S Q 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 

ds Q= 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 





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 


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 


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 


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 


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 




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 


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 


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,, 


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 


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- 


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 


sion, which impresses itself without fatigue on the 

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- 


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- 

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 


the 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 



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 


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 


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, 


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 


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. 


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- 


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 

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. 



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, 


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- 

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- 


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 


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 


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

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 


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 


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- 


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), 


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 


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 


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

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 


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 


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 



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 


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 


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. 



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, 


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

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 

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 


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 

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 


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 


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. 


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 


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. 


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 


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- 


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 


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 


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. 


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 


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. 



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 


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 


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. 


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, 


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 


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 


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 


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 


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. 


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. 


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 

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- 


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 


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 


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- 


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. 





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- 


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. 


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 


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 


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 


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. 


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 

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 


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. 


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 


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, 


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 


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 


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. 


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 


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 


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. 


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- 


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 


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- 


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. 


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 


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 


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 


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 


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 


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

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 


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 


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 


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 


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 


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 


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\ r e 
perhaps been formed by the gradual aggregation of 


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, 


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 


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 


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. 


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 


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

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 


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 


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 
r oiild unite to form water. The mass of the sun is 


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. 



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 


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 

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 


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 


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 

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, 


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 


that all stages of that process can still be found to 

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 


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. 


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. 


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 


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 


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- 


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 


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 


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- 

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. 



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 

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 


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 ? 




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- 


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.' 



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 


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 


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 


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 


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 


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- 

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 


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 


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 


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, 


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 


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 


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 


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- 


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 


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- 


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 


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 

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 


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- 


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, 


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 


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 


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 


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- 

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 


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 


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 


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 

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 


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 

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 


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 


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 (roy {jLaivo^svois oyno/ws 1 ). 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. 



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- 

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- 


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 


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 


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 


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 


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. 


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 

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. 


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 


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- 

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. 






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) 


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- 


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 


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- 


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 


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 


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 ? 


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 


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 


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 


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 


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 



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 


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 


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 


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 


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 


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 


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- 


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 



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 


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. 


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 


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 


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 


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 


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 


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 


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. 




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 


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- 


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 


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, 


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 


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 


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 

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, 


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 

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 


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 


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 



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- 


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 


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 

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- 


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, 


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 

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


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 

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 

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 


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 

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. 


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, 


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 

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.' 


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 


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. 


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 


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 


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 


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 



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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IP ?nnr 
uv ^, * dUUj 

JUM 3 1997 


AUG 1 8 2005